# Patent

## Title
Method and apparatus for optically measuring periodic structures using orthogonal azimuthal sample orientation

## Abstract

An optical metrology apparatus for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) incident angles is described. One embodiment provides the enhanced calculation speed for the special case of phi=90 incidence for 1-D (line and space) structures, which has the incident plane parallel to the grating lines, as opposed to the phi=0 classical mounting, which has incident plane perpendicular to the grating lines. The enhancement reduces the computation time of the phi=90 case to the same order as the corresponding phi=0 case, and in some cases the phi=90 case can be significantly faster. One advantageous configuration consists of two measurements for each sample structure, one perpendicular to the grating lines and one parallel. This provides additional information about the structure, equivalent to two simultaneous angles of incidence, without excessive increase in computation time. Alternately, in cases where the computation for phi=90 is faster than the corresponding phi=0 incidence, it may be advantageous to measure parallel to the grating lines only. In the case where two sets of incident angles are used, the incident light can be polarized to provide a total of four sets of data—Rs0, Rp0, Rs90, Rp90—for each incident polar angle, all from the same structure.

## Background
This application is a continuation of U.S. patent application Ser. No. 11/998,263 filed on Nov. 29, 2007 now abandoned which claims priority to Provisional Patent Application No. 60/872,010 filed Nov. 30, 2006; the disclosure of which is expressly incorporated herein by reference in its entirety.


TECHNICAL FIELD OF THE INVENTION
This invention relates to an optical metrology apparatus and methods and systems for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) angles, and particularly to enhanced calculation speed for a special case of phi=90 incidence for 1-D (line and space) gratings, having an incident plane parallel to grating lines. This results in additional datasets to supplement data collected in the phi=0 classical mount configuration without an untenable increase in computation cost.
BACKGROUND OF THE INVENTION
A widely referenced source on a rigorous coupled wave (RCW) algorithm is that of Moharam and Gaylord (M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, J. Opt. Soc. Am. A, Vol. 12, No. 5, p. 1068 (1995)). A schematic from their paper showing the diffraction problem is reproduced in FIG. 1. In particular, FIG. 1 defines the various incident conditions for the diffraction problem. The plane of incidence is defined by the polar angle, theta, and the azimuthal angle, phi. The azimuthal angle defines the angle the incident plane makes with the plane perpendicular to the grating lines, so that phi=0 corresponds to the classical incidence case. The angle psi defines the direction of the electric field with respect to the plane of incidence, with psi=90 corresponding to s polarized and psi=0 to p polarized incident light.
As shown in FIG. 1, a diffraction grating 100 has a grating region 104 formed in a substrate 102 (the substrate being designated as Region II). The grating region 104 has a height d as indicated in the figure. Region I is comprised of the material above the grating region, in this case in air or vacuum space. As indicated in FIG. 1, the grating region may be formed of alternating grating lines 106 and grating spaces 108. The grating lines 106 may have a width 109. The grating periodicity is characterized by the grating period 110 as indicated.
It will be recognized that a diffraction grating may be formed in other manners than that of FIG. 1 and that FIG. 1 is only one exemplary diffraction grating as known to those skilled in the art. For example, a diffraction grating need not be formed utilizing spaces. FIG. 1B shows one such alternative diffraction grating. As shown in FIG. 1A, the diffraction grating 100 may be comprised of grating lines 106A and 106B, again having a grating region 104 with height d. In this example, grating lines 106A and 106B will be formed in a manner in which the lines have different optical properties. Further, though the examples shown include gratings having two different optical properties within each period of the diffraction grating, it will be recognized that the diffraction grating may comprise three or more different materials within each period. Likewise, though each grating line is shown as a single material, it will be recognized that the grating lines may be formed of multiple layers of the same or different materials. In addition, though the grating lines are shown as being “squared off,” it will be recognized that each line may have sloped sides, curved edges, etc.
With reference again to FIG. 1, an x-y-z coordinate system is shown having a frame of reference in which the x-direction is shown as being perpendicular to the original alignment of the grating lines. The plane of incidence 112 of the incident light is defined by the polar angle 114, theta and the azimuthal angle 116, phi. The electric field 120 has a propagation vector 122 (k) of the incident wave. The unit vectors 124 (t) and 126 (n) are tangent and normal to the plane of incidence, respectively. As mentioned above, the angle 128, psi, defines the direction of the electric field with respect to the plane of incidence.
The RCW method involves the expansion of the field components inside and outside the grating region in terms of generalized Fourier series. The method consists of two major parts—an eigen-problem to determine a general solution inside the grating layer, and a boundary problem to determine the reflected and transmitted diffracted amplitudes along with the specific solution for the fields inside the grating region. The Fourier series are truncated after a finite number of terms. The truncation is usually characterized by the truncation order, N, which means that 2N+1 spatial harmonics are retained in the series (positive and negative terms to ±N, and the 0 term).
Standard methods for solving the eigen-problem, boundary problem, and the various other matrix multiplications and inversions involved are order N3 operations. This means that an increase of the truncation order by a factor of two results in an increase in overall computation time by a factor of approximately 8. The truncation order required for convergence is determined by the specifics of the diffraction problem, and generally increases for larger pitch to incident wavelength ratios and larger optical contrast between grating lines and spaces. The result is that while some diffraction problems are very tenable, others quickly become impractical to solve due to a large computation cost.
In the case of the phi=0 classical mount, the diffraction problem decouples into TE and TM components, which can be solved separately (for the phi=0 mount, TE polarization corresponds to s polarized incident light, and TM polarization corresponds to p polarized incident light). Any arbitrary polarization is decomposed into a combination of the TE and TM problems. In practice, the incident light is often purely TE or TM polarized, and only one case needs to be solved. For given truncation order N and classical mount the eigen-problem is of size 2N+1, and the boundary problem is of size 2(2N+1).
The general case where phi≠0 is known as conical diffraction. In this case, the s and p components are coupled, with a corresponding increase in the amount of computation time. The boundary problem involves 4(2N+1) sized matrices. The eigen-problem has been successfully decoupled into two smaller eigen-problems, each of size 2N+1 (see Moharam and Gaylord 1995 referenced above, or S. Peng and G. M. Morris, J. Opt. Soc. Am. A, Vol. 12, No. 5, p. 1087 (1995)). Therefore, the computation time for the general conical incidence case suffers a factor of 2 increase for the eigen-problem and a factor of 8 increase for the boundary problem compared to the corresponding classical mount case with same polar incident angle, theta.
Analysis of a diffraction grating problem is of particular use to determining the various characteristics of the diffraction grating structure. For example, critical dimensions of a device (such as in semiconductor processing in one exemplary use) may be monitored by evaluating the characteristics of a diffraction grating as is known in the art. By evaluating data from known optical metrology tools using regression and/or library methods, the diffraction analysis may lead to, for example, a determination of the grating line widths, the grating height/depth, the period of the grating, the slopes and profiles of the grating, the material composition of the grating, etc. As known in the art, such grating characteristics may be related to the characteristics of a device that is being analyzed, such as for example but not limited to widths, heights, depths, profiles, etc. of transistors, metallization lines, trenches, dielectric layers, or the like, all as is known to those skilled in the art. Since the regression and/or library methods may require many calculations of diffraction efficiencies, special consideration must be given to computation expense in such applications.
SUMMARY OF THE INVENTION
An optical metrology apparatus for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) incident angles is described. One embodiment provides the enhanced calculation speed for the special case of phi=90 incidence for 1-D (line and space) structures, which has the incident plane parallel to the grating lines, as opposed to the phi=0 classical mounting, which has incident plane perpendicular to the grating lines. The enhancement reduces the computation time of the phi=90 case to the same order as the corresponding phi=0 case, and in some cases the phi=90 case can be significantly faster. One advantageous configuration consists of two measurements for each sample structure, one perpendicular to the grating lines and one parallel. This provides additional information about the structure, equivalent to two simultaneous angles of incidence, without excessive increase in computation time. Alternately, in cases where the computation for phi=90 is faster than the corresponding phi=0 incidence, it may be advantageous to measure parallel to the grating lines only. In the case where two sets of incident angles are used, the incident light can be polarized to provide a total of four sets of data—Rs0, Rp0, Rs90, Rp90—for each incident polar angle, all from the same structure (Rs0 being a data set having incident phi=0 and incident polarization normal to the plane of incidence, Rp90 being a data set having incident phi=90 and incident polarization within the plane of incidence, etc.).
In one embodiment, the techniques described herein provide an optical metrology apparatus and methods and systems for measuring periodic structures using multiple incident azimuthal (phi) and polar (theta) angles, and particularly to enhanced calculation speed for a special case of phi=90 incidence for 1-D (line and space) structures, having an incident plane parallel to grating lines.
In one embodiment, a method of reducing an RCW calculation for the phi=90 incidence mount by exploiting the degeneracy in the resulting diffraction problem is provided. The method may include using this calculation in the regression part of an optical grating measurement. Alternately, the method may include using the enhanced speed in the generation of a database library to be used in conjunction with an optical grating measurement. The optical method could be any in existence, such as reflectometry, polarized reflectometry, ellipsometry, polarimetry, etc. and can be broadband or single wavelength.
The method may include illuminating a grating structure with polarized or unpolarized, monochromatic or broadband light. The incident light may be at one or more polar angles, theta, at the phi=0 and phi=90 azimuthal directions for each of the polar angles. The method may further include detecting the response, for a total of up to four datasets per grating sample per incident polar angle, which are then simultaneously analyzed in order to take advantage of the enhanced information content contained in the multiple datasets. The calculation time is reduced compared to conventional RCW formulations due to the reduced calculation requirements for the phi=90 cases.
One or more diffracted orders may be detected along with or instead of the 0'th order. Further when the detected response is reflected or diffracted intensity, one or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the tool unnecessary.
One or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the optical tool unnecessary, and the inverse ratio is substituted in calculations for specific wavelength regions where the denominator of the original ratio is near zero.
One or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the tool unnecessary, and the inverse ratio is substituted in calculations for specific wavelength regions where the denominator of the original ratio is near zero, and a weighting function is used to equalize the contribution to the merit function regardless of reflectance ratio magnitude.
In addition, one or more of the datasets may be used to normalize the other datasets, making an absolute calibration of the optical tool unnecessary, and the data regions where the denominator of the ratio is near zero are dropped from the analysis.
Data collected from the diffracting structure may be normalized by data from a nearby uniform film structure having the same stack layer structure as the diffracting structure.
The angle of incidence may be explicitly varied by changing the polar angle of incidence of the optical plane, or by rotating the optical plane or sample at fixed polar angle to generate phi=0 and phi=90 incident data.
The multiple angle of incidence data may be generated through use of a high numerical aperture optic (so it contains a spread of angles) and selecting specific angles using an aperture stop to allow light incident at only specific angles.
In addition, multiple angles of incidence may be allowed, and the data simultaneously analyzed.
Further, the method may include only measuring and analyzing the phi=90 incidence data.
As described below, other features and variations can be implemented, if desired, and a related method can be utilized, as well.



DESCRIPTION OF THE DRAWINGS
It is noted that the appended drawings illustrate only exemplary embodiments of the invention and are, therefore, not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments.
FIG. 1 shows a schematic diagram of diffraction problem illustrating polar (theta) and azimuthal (phi) incident angles.
FIG. 1B shows an exemplary alternative diffraction grating.
FIGS. 2A and 2B shows rotation of the stage/sample/objective by 90 degrees and back for each measurement.
FIGS. 3A-3E illustrate using a high NA objective with an aperture stop that allows light incident at specific angles. Multiple incident angles are achieved by rotating the sample, aperture, or objective. One modification might simultaneously illuminate the grating from both directions (phi=0 and phi=90) as shown in FIG. 3D and FIG. 3E.
FIG. 4 illustrates an apparatus for collecting VUV-Vis reflectance data at multiple incident angles.



DETAILED DESCRIPTION OF THE INVENTION
One way to directly attack the time required for the RCW method is to reduce the number of spatial harmonics involved in the eigen-problem, the boundary problem, or both. This was done for the general case to a large extent in the work of Moharam and Gaylord in their papers: M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068-1076 (1995) and M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077-1086 (1995). Subsequent modifications to enhance the TM convergence have included those techniques shown in: P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779-784 (1996); G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019-1023 (1996); and L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996). However, for certain incidence conditions where the incident plane wave has an x-periodicity that is the same as or is a multiple of the grating period, the diffraction problem benefits from additional degeneracy, and the coupled equations can be even further reduced.
One case where this occurs is when the plane of incidence is in the phi=90 mount so that the incident wave is constant with x, and the resulting degeneracy can be exploited to reduce the total number of unknowns. This reduces the total number of spatial harmonics required for the conical diffraction case from 4(2N+1) to 4(N+1). The result is a reduction of the computation time required by a factor of approximately 8 compared to the standard phi=90 conical diffraction case. In addition, a small but significant reduction in computation time compared to the classical mount with same theta is achieved. There are still two eigen-problems, but each of size N+1, leading to an overall reduction of the eigen-problem by a factor of approximately 4 compared to the comparable phi=0 case. The boundary problems for the two cases require approximately the same computation time (2(2N+1) vs. 4(N+1) matrix sizes). Therefore, the computation time for the phi=90 case is reduced to the same order as the corresponding phi=0 case. The phi=90 case can sometimes be significantly faster than the phi=0 case, depending on how much influence the eigen-problem has on the overall computation time. The result is that the additional azimuthal angle phi=90 can be added to the data without an excessive increase in computation time. It will be recognized that although the concepts described herein may refer to analysis at particular angles such as phi=0 and phi=90, the concepts are not limited to use of these exact angles. For example, equipment and sample tolerances may result in other angles being actually used as variability from an anticipated angle is to be expected in real world applications. In addition, variations from the angles of best choice may be purposefully allowed beyond such tolerances while still obtaining the benefits of the techniques described herein. For example, the optical system and grating may be of such a nature that variations from the most desirable angles will still provide a sufficiently accurate calculation at other angles such that the data collected and the calculations may be effectively similar to the use of phi=0 and phi=90 to the extent that sufficient accuracy for a particular application is obtained. In this manner angles that deviate from the phi=0 and phi=90 may be considered to be effectively phi=0 and phi=90 for the purpose of utilizing a metrology tool implementing the techniques described herein for a given application.
An effective way of increasing the amount of information that can be extracted from a single sample is to collect more data sets. This is often done by using multiple angles of incidence theta or, in the case of grating structures, using multiple azimuthal angles. For instance, T. Novikova, A. De Martino, S. B. Hatit, and B. Drevillon, “Application of Mueller polarimetry in conical diffraction for critical dimension measurements in microelectronics,” Appl. Opt. 45, 3688-3697 (2006) shows a technique to extract more information about grating line-shapes using multiple azimuthal angles in conjunction with Mueller polarimetry.
This is significant since the hardware implementation for multiple phi configurations is considerably simpler than cases where multiple polar angles are used. In the simplest case, a fixed angle theta can be used, while the sample or stage is rotated through the azimuths. Alternately, the optic objective could be rotated. To take advantage of the above mentioned RCW improvements, the system would simply have to rotate the stage/sample/objective by 90 degrees and back for each measurement (FIGS. 2A-2B), generating two datasets for each sample, at orthogonal incident conditions.
As shown in FIG. 2A, a substrate is provided with a diffraction grating 100. A source 206 provides an incident light wave 200 at a polar angle 114 (theta) and an azimuthal angle 116 (phi). As shown in FIG. 2A phi=0 so no angle is indicated. A detector 208 may be provided to detect the diffracted orders of reflected light 202. As used herein, light detected from a diffraction grating may be referred to as reflected light, reflected data, or the like and may include one or both of specular reflection of the zero order light and higher order diffracted light.
Though not shown in the figures provided herein, a computer system, processor, or the like may be coupled to the detector to process the collected data according to the analysis techniques described herein. The rotation of the diffraction grating with reference to the position of FIG. 2A is shown in FIG. 2B. As shown in FIG. 2B, the diffraction grating has been rotated (for example by rotating the stage, the sample, or the incident light or a combination thereof) so as to create an azimuthal angle 116 (phi) that is phi=90.
Another means for collecting data at multiple angles of incidence could utilize the large cone angle inherent in high numeric aperture (NA) normal incident objective systems in conjunction with an aperture stop to allow light incident at specific angles, as shown in FIGS. 3A-C. One advantage of this configuration is that multiple polar angles can be more easily incorporated, in addition to the orthogonal azimuthal angles. As shown in FIG. 3A an aperture 300 provides collimated light to an objective 302 which focuses the light on a diffraction grating 100 at a polar angle 114 theta. Different aperture settings are shown between FIGS. 3A and 3B to illustrate utilizing different polar angles theta. As shown in FIGS. 3A and 3B, the incident wave is not rotated from the classical phi=0 orientation. FIG. 3C illustrates a rotation such that phi does not equal zero, for example phi=90.
Another possibility might simultaneously illuminate the sample from both directions (phi=0 and phi=90). This could be implemented using two separate, orthogonal optic planes, or a modified version of the high NA system (FIG. 3D). The configuration of FIG. 3D is similar to that of FIG. 3A except a modified aperture 300A is provided. A top view of the modified aperture 300A is shown in FIG. 3E. As can be seen, the aperture 300A allows for light to pass in multiple directions, more particularly in a manner such that phi=0 and phi=90 may be simultaneously illuminated.
It will be recognized by those in the art that the embodiments of FIGS. 2A-2B and 3A-3D are merely illustrative so as to demonstrate the technique of changing the azimuthal angle. A wide range of optical metrology tool configurations may be utilized to achieve the desired incidence conditions so as to acquire the data sets described herein. These data sets may then be utilized in a manner that yields desired characteristics of a diffraction grating from the reduced computational complexity analysis that is achieved by collecting the rotated data as described in more detail below. It will also be recognized that although the present disclosure generally is described in reference to data collected at two phi angles (phi=0 and 90 angles), the techniques described herein are not limited to such techniques. In particular, the techniques described herein provide a reduced computation technique that may be utilized for measurements collected at a single phi angle. Further, additional phi angles beyond two may also be utilized to collect data while still obtaining the benefits described herein. Thus, the concepts described herein are not merely limited to collection of data at two phi angles.
Note that collecting multiple broadband datasets using different azimuthal incidence conditions is distinct from the old method of “phi scatterometry”, where the entire dataset consists of the diffraction spectrum of a single wavelength as a function of the azimuthal angle, phi.
The light can additionally be polarized parallel and perpendicular to the plane of incidence, so that four simultaneous data sets per incident theta can be obtained for a given 1-D grating structure, with corresponding enhancement in information. Alternately, each of the four configurations can be explored for a particular structure, and the most promising configuration employed in practice for measuring that particular structure.
Further, the additional datasets may enhance the information content to the extent that fewer wavelengths in a broadband system can be used in the analysis, and in this way the measurement can actually be made faster than a single angle of incidence configuration broadband measurement. In other-words, the total number of calculations with respect to incident conditions, including wavelength and angle, may be reduced over that required for a single incident condition over many more wavelengths, while still extracting the same information about the grating structure.
Another advantageous configuration is to collect and analyze the ratio of reflected (0 order, for instance) intensity from the phi=0 scan to that of the reflected (0 order) intensity of the phi=90 scan. The intensity ratio is the same as the reflectance ratio, as long as the intensities are measured in quick succession so that there is minimal system drift between the two measurements. In this way the system calibration can be skipped, and the intensity ratio can be analyzed according to the above methods, taking advantage of the reduction in computation expense for the phi=90 case. In some wavelength ranges, the denominator may be close to zero. Those regions need not be analyzed, or the inverse ratio can be analyzed instead. This implementation can be particularly advantageous when using Vacuum Ultra-Violet (VUV) incident light. One implementation of a VUV metrology apparatus is described in U.S. Pat. No. 7,126,131, Broad Band Referencing Reflectometer, by Harrison, the disclosure of which is incorporated herein by reference in its entirety. For such systems, contaminant buildup on calibration standards over time causes difficulties for traditional calibration methods.
An example of a VUV metrology apparatus configured to collect multiple broadband data sets using different azimuthal incident angles is presented in FIG. 4. The instrument is separated into two environmentally controlled chambers, the instrument chamber and the sample chamber. The instrument chamber houses most of the system optics and is not opened to the atmosphere on a regular basis. The sample chamber houses the sample, the sample focusing optic M-2, the reference focusing optic M-4 and the reference plane mirror M-5. This chamber is opened regularly to facilitate changing samples. The instrument is configured to enable collection of sample and reference data sets. The reference data set can be used to correct for system and/or environmental changes which may occur between calibration and sample measurement times. The system may be configured with multiple sources and spectrometers/detectors that are selected using flip-in mirrors FM-1, FM-2, FM-3 and FM-4.
In operation the VUV data is first obtained by switching flip-in source mirrors FM-1 and FM-3 into the “out” position so as to allow light from the VUV source to be collected, collimated and redirected towards beam splitter element BS by focusing mirror M-1. Light striking the beam splitter is divided into two components, the sample beam and the reference beam, using a balanced Michelson interferometer arrangement. The sample beam is reflected from the beam splitter BS and travels through shutter S-1, aperture A-1 and VUV-transparent window W-1. Aperture A-1 is configured to restrict illumination of the sample to some azimuthal plane(s). Shutter S-2 is closed during this time.
Light entering the sample chamber is focused by focusing optic M-2 onto the sample. Light collected from the sample is collimated and redirected by mirror M-2 back through window W-1, aperture A-1 and beam splitter BS. Light passing through the beam splitter encounters aperture A-2, which is configured to selectively pass some fraction of the collected sample response. Light passing through aperture A-2 is redirected and focused onto the entrance slit of the VUV spectrometer by focusing mirror M-3. Flip-in detector mirrors FM-2 and FM-4 are switched to the “out” position during this time.
Following collection of the sample beam, the reference beam is measured by closing shutter S-1 and opening shutter S-2. Once the reference signal has been recorded, data from other spectral regions can be collected in a similar manner using the appropriate flip-in mirrors.
In totality, the use of VUV incident radiation, large polar incident angle or angles, and multiple azimuthal angles, can greatly enhance sensitivity to grating line shape parameters. An analysis using a series of simulations can be done for any given grating structure in order to determine which combination of incident polar angles, azimuthal angles, and wavelengths yields the most information for smallest computation cost.
For faster measurements, where a smaller amount of information may be desired (e.g. line height and average width only), it may be sufficient to use only the phi=90 incidence case and take advantage of the improved calculation speed over the corresponding classical phi=0 incidence case.
Another technique disclosed herein analyzes the multiple sets using different models. For instance, the phi=90 data might be analyzed using a simpler rectangular line shape model with a course parameter search to narrow down the average parameter values, and then a more thorough analysis done using another one of the spectrum (or all of the spectra together) with a more complicated model to further refine the line shape.
Review of the RCW Method for Conical Incidence
This review follows the notation of the Moharam and Gaylord references cited above. Many publications exist on the basic RCW method, some with notation differences and some with modifications to the formulations/derivation procedures such as that shown in the P. Lalanne and G. M. Morris reference cited above. It should be noted that the phi=90 case reduction described in this disclosure is can be applied to any of these formulations, and is not limited to just that of Moharam and Gaylord.
Note that unreduced eigen-problem matrix and vector indices run from −N to N, with the (−N, −N) matrix element at the top left corner, in order to be consistent with a symmetric diffraction problem with positive and negative orders. When creating a computer algorithm, the indices are labeled from 1 to 2N+1 (or 0 to 2N), depending on the programming language used. It will be recognized, this is a notation preference and has no effect on the outcome. The indices of the reduced matrices will run from 0 to N in either case.
Following the Moharam and Gaylord references, the RCW method expands the fields in each region of FIG. 1 as a generalized Fourier series:




  
    
      
        
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                        ⁢
                        z
                      
                    
                  
                
                ]
              
              ⁢
              
                exp
                ⁡
                
                  [
                  
                    -
                    
                      j
                      ⁡
                      
                        (
                        
                          
                            
                              k
                              xi
                            
                            ⁢
                            x
                          
                          +
                          
                            
                              k
                              y
                            
                            ⁢
                            y
                          
                        
                        )
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        4
      
    
  




in the grating region, where




  
    
      
        
          
            k
            xi
          
          =
          
            
              k
              0
            
            ⁡
            
              [
              
                
                  
                    n
                    I
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                  ⁢
                  
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
                -
                
                  i
                  ⁡
                  
                    (
                    
                      
                        λ
                        0
                      
                      /
                      Λ
                    
                    )
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        5
      
    
  
  
    
      
        
          
            k
            y
          
          =
          
            
              k
              0
            
            ⁢
            
              n
              I
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            θ
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            φ
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        6
      
    
  
  
    
      
        
          k
          
            l
            ,
            zi
          
        
        =
        
          {
          
            
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                (
                                
                                  
                                    k
                                    0
                                  
                                  ⁢
                                  
                                    n
                                    l
                                  
                                
                                )
                              
                              2
                            
                            -
                            
                              k
                              xi
                              2
                            
                            -
                            
                              k
                              y
                              2
                            
                          
                          ]
                        
                        
                          1
                          /
                          2
                        
                      
                    
                    
                      
                        
                          
                            (
                            
                              
                                k
                                xi
                                2
                              
                              +
                              
                                k
                                y
                                2
                              
                            
                            )
                          
                          
                            1
                            /
                            2
                          
                        
                        <
                        
                          
                            k
                            0
                          
                          ⁢
                          
                            n
                            l
                          
                        
                      
                    
                  
                  
                    
                      
                        -
                        
                          
                            j
                            ⁡
                            
                              [
                              
                                
                                  k
                                  xi
                                  2
                                
                                +
                                
                                  k
                                  y
                                  2
                                
                                -
                                
                                  
                                    (
                                    
                                      
                                        k
                                        0
                                      
                                      ⁢
                                      
                                        n
                                        l
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ]
                            
                          
                          
                            1
                            /
                            2
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              (
                              
                                
                                  k
                                  xi
                                  2
                                
                                +
                                
                                  k
                                  y
                                  2
                                
                              
                              )
                            
                            
                              1
                              /
                              2
                            
                          
                          >
                          
                            
                              k
                              0
                            
                            ⁢
                            
                              n
                              l
                            
                          
                        
                        ,
                      
                    
                  
                
                ⁢
                
                  

                
                ⁢
                l
              
              =
              I
            
            ,
            II
            ,
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        7
      
    
  




k0=(2π/λ0), λ0 is the incident wavelength, and Λ is the grating pitch. Note that for a 1D grating, ky is constant. In eqs. 3 and 4, εf is the permittivity of free space, and μf is the magnetic permeability of free space.
In equations 1 and 2, the Ri and Ti are the Fourier coefficients of the electric field in regions I and II, and correspond to the amplitudes of the reflected and transmitted diffraction orders. The diffracted orders can be propagating or evanescent. The corresponding magnetic fields can be obtained from Maxwell's relations

∇×E=−jωμfH, 

∇×H=jωεfε(x)E,  eq. 8

where ω is the angular frequency, and μ is the magnetic permeability. Usually, one assumes μ=μf.

The complex permittivity in the grating region is also expanded as a Fourier series, which is




  
    
      
        
          
            ɛ
            ⁡
            
              (
              x
              )
            
          
          =
          
            
              ∑
              h
            
            ⁢
            
              
            
            ⁢
            
              
                ɛ
                h
              
              ⁢
              
                exp
                ⁡
                
                  (
                  
                    j
                    ⁢
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        hx
                      
                      Λ
                    
                  
                  )
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            ɛ
            0
          
          =
          
            
              
                n
                
                  r
                  ⁢
                  
                    
                  
                  ⁢
                  d
                
                2
              
              ⁢
              f
            
            +
            
              
                n
                
                  g
                  ⁢
                  
                    
                  
                  ⁢
                  r
                
                2
              
              ⁡
              
                (
                
                  1
                  -
                  f
                
                )
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            ɛ
            h
          
          =
          
            
              (
              
                
                  n
                  
                    r
                    ⁢
                    
                      
                    
                    ⁢
                    d
                  
                  2
                
                -
                
                  n
                  
                    g
                    ⁢
                    
                      
                    
                    ⁢
                    r
                  
                  2
                
              
              )
            
            ⁢
            
              
                sin
                ⁡
                
                  (
                  
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    hf
                  
                  )
                
              
              
                π
                ⁢
                
                  
                
                ⁢
                h
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        9
      
    
  




for the binary grating structure of the first Moharam and Gaylord reference cited above and shown in FIG. 1. In eq. 9, nrd and ngr are the complex indices of refraction for the lines and spaces, respectively.

Eqs. 3, 4, 8, and 9 combine to give a set of coupled equations:




  
    
      
        
          [
          
            
              
                
                  
                    ∂
                    
                      S
                      y
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      S
                      x
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      U
                      y
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      U
                      x
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
          
          ]
        
        =
        
          
            [
            
              
            
            ⁢
            
              
                
                  0
                
                
                  0
                
                
                  
                    
                      K
                      y
                    
                    ⁢
                    
                      E
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      K
                      x
                    
                  
                
                
                  
                    I
                    -
                    
                      
                        K
                        y
                      
                      ⁢
                      
                        E
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        K
                        y
                      
                    
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    
                      
                        K
                        x
                      
                      ⁢
                      
                        E
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        K
                        x
                      
                    
                    -
                    I
                  
                
                
                  
                    
                      -
                      
                        K
                        x
                      
                    
                    ⁢
                    
                      E
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      K
                      y
                    
                  
                
              
              
                
                  
                    
                      K
                      x
                    
                    ⁢
                    
                      K
                      y
                    
                  
                
                
                  
                    E
                    -
                    
                      K
                      y
                      2
                    
                  
                
                
                  0
                
                
                  0
                
              
              
                
                  
                    
                      K
                      x
                      2
                    
                    -
                    E
                  
                
                
                  
                    
                      -
                      
                        K
                        x
                      
                    
                    ⁢
                    
                      K
                      y
                    
                  
                
                
                  0
                
                
                  0
                
              
            
            ]
          
          ×
          
             
            
              [
              
                
                  
                    
                      S
                      y
                    
                  
                
                
                  
                    
                      S
                      x
                    
                  
                
                
                  
                    
                      U
                      y
                    
                  
                
                
                  
                    
                      U
                      x
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        10
      
    
  




where Kx is a diagonal matrix with elements kxi/k0, Ky is a diagonal matrix with elements ky/k0, E is the permittivity matrix (not to be confused with the electric field), with Ei,j=ε(i−j), and z′=k0z.

When a truncation order of N is used, eq. 10 is a system of 4(2N+1)×4(2N+1) coupled equations. The authors of the first Moharam and Gaylord reference cited above further reduce eq. 10 to two 2(2N+1)×2(2N+1) sets of equations:




  
    
      
        
          
        
        ⁢
        
          
            
              [
              
                
                  
                    
                      
                        
                          ∂
                          2
                        
                        ⁢
                        
                          S
                          y
                        
                      
                      
                        ∂
                        
                          
                            (
                            
                              z
                              ′
                            
                            )
                          
                          2
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          ∂
                          2
                        
                        ⁢
                        
                          S
                          x
                        
                      
                      
                        ∂
                        
                          
                            (
                            
                              z
                              ′
                            
                            )
                          
                          2
                        
                      
                    
                  
                
              
              ]
            
            =
            
              
                [
                
                  
                    
                      
                        
                          K
                          x
                          2
                        
                        +
                        DE
                      
                    
                    
                      
                        
                          K
                          y
                        
                        ⁡
                        
                          [
                          
                            
                              
                                E
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                K
                                x
                              
                              ⁢
                              E
                            
                            -
                            
                              K
                              x
                            
                          
                          ]
                        
                      
                    
                  
                  
                    
                      
                        
                          K
                          x
                        
                        ⁡
                        
                          [
                          
                            
                              
                                E
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                K
                                y
                              
                              ⁢
                              E
                            
                            -
                            
                              K
                              y
                            
                          
                          ]
                        
                      
                    
                    
                      
                        
                          K
                          y
                          2
                        
                        +
                        BE
                      
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        S
                        y
                      
                    
                  
                  
                    
                      
                        S
                        x
                      
                    
                  
                
                ]
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        11
      
    
  
  
    
      
        
          
        
        ⁢
        or
      
    
    
      
        
      
    
  
  
    
      
        
          
            [
            
              
                
                  
                    
                      
                        ∂
                        2
                      
                      ⁢
                      
                        U
                        y
                      
                    
                    
                      ∂
                      
                        
                          (
                          
                            z
                            ′
                          
                          )
                        
                        2
                      
                    
                  
                
              
              
                
                  
                    
                      
                        ∂
                        2
                      
                      ⁢
                      
                        U
                        x
                      
                    
                    
                      ∂
                      
                        
                          (
                          
                            z
                            ′
                          
                          )
                        
                        2
                      
                    
                  
                
              
            
            ]
          
          =
          
            
              [
              
                
                  
                    
                      
                        K
                        y
                        2
                      
                      +
                      EB
                    
                  
                  
                    
                      
                        [
                        
                          
                            K
                            x
                          
                          -
                          
                            
                              EK
                              x
                            
                            ⁢
                            
                              E
                              
                                -
                                1
                              
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        K
                        y
                      
                    
                  
                
                
                  
                    
                      
                        [
                        
                          
                            K
                            y
                          
                          -
                          
                            
                              EK
                              y
                            
                            ⁢
                            
                              E
                              
                                -
                                1
                              
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        K
                        x
                      
                    
                  
                  
                    
                      
                        K
                        x
                        2
                      
                      +
                      ED
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    
                      U
                      y
                    
                  
                
                
                  
                    
                      U
                      x
                    
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        12
      
    
  
  
    
      
        
          
        
        ⁢
        
          Where
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            B
            =
            
              
                
                  
                    K
                    x
                  
                  ⁢
                  
                    E
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    K
                    x
                  
                
                -
                
                  I
                  ⁢
                  
                    
                  
                  ⁢
                  and
                  ⁢
                  
                    
                  
                  ⁢
                  D
                
              
              =
              
                
                  
                    K
                    y
                  
                  ⁢
                  
                    E
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    K
                    y
                  
                
                -
                
                  I
                  .
                
              
            
          
        
      
    
    
      
        
      
    
  




These last equations are reduced still further into two (2N+1)×(2N+1) sets of equations:

[∂2Ux/∂(z′)2]=[Ky2+A][Ux]  eq. 13

and

[∂2Sx/∂(z′)2]=[Ky2+BE][Sx],  eq. 14

where A=Kx2−E.

Later, Lalanne and Morris (cited above) were able to improve the convergence of the conical case by replacing the matrix E in the third row, second column of eq. 10 with the inverse of the inverse permittivity matrix, Einv, where Einvi,j=(1/ε)i,j:




  
    
      
        
          [
          
            
              
                
                  
                    ∂
                    
                      S
                      y
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      S
                      x
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      U
                      y
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      U
                      x
                    
                  
                  
                    ∂
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                  
                
              
            
          
          ]
        
        =
        
           
          
            
              [
              
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      
                        K
                        y
                      
                      ⁢
                      
                        E
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        K
                        x
                      
                    
                  
                  
                    
                      I
                      -
                      
                        
                          K
                          y
                        
                        ⁢
                        
                          E
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          K
                          y
                        
                      
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      
                        
                          K
                          x
                        
                        ⁢
                        
                          E
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          K
                          x
                        
                      
                      -
                      I
                    
                  
                  
                    
                      
                        -
                        
                          K
                          x
                        
                      
                      ⁢
                      
                        E
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        K
                        y
                      
                    
                  
                
                
                  
                    
                      
                        K
                        x
                      
                      ⁢
                      
                        K
                        y
                      
                    
                  
                  
                    
                      
                        Einv
                        
                          -
                          1
                        
                      
                      -
                      
                        K
                        y
                        2
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    
                      
                        K
                        x
                        2
                      
                      -
                      E
                    
                  
                  
                    
                      
                        -
                        
                          K
                          x
                        
                      
                      ⁢
                      
                        K
                        y
                      
                    
                  
                  
                    0
                  
                  
                    0
                  
                
              
              ]
            
            ⁢
            
               
              
                ×
                
                  [
                  
                    
                      
                        
                          S
                          y
                        
                      
                    
                    
                      
                        
                          S
                          x
                        
                      
                    
                    
                      
                        
                          U
                          y
                        
                      
                    
                    
                      
                        
                          U
                          x
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        15
      
    
  




which lead to

[∂2Ux/∂(z′)2]=[Ky2+A][Ux],  eq. 16

and

[∂2Sx/∂(z′)2]=[Ky2+BEinv−1][Sx]  eq. 17

in place of eqs. 13 and 14.

The new formulation eqs. 16 and 17 are advantageous and it may be noted that E and Einv−1 are not the same matrices when they are truncated. The details can be found in references cited above from P. Lalanne and G. M. Morris; G. Granet and B. Guizal; and L. Li.
Equations 16 and 17 are solved by finding the eigenvalues and eigenvectors of the matrices [Ky2+A] and [Ky2+BEinv−1], which leads to




  
    
      
        
          
        
        ⁢
        
          
            
              U
              xi
            
            =
            
              
                ∑
                
                  m
                  =
                  1
                
                
                  
                    2
                    ⁢
                    N
                  
                  +
                  1
                
              
              ⁢
              
                
                   
                  
                    w
                    
                      1
                      ,
                      i
                      ,
                      m
                    
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              -
                              
                                c
                                
                                  1
                                  ,
                                  m
                                
                                +
                              
                            
                            ⁢
                            
                              exp
                              ⁡
                              
                                (
                                
                                  
                                    -
                                    
                                      k
                                      0
                                    
                                  
                                  ⁢
                                  
                                    q
                                    
                                      1
                                      ,
                                      m
                                    
                                  
                                  ⁢
                                  z
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              1
                              ,
                              m
                            
                            -
                          
                          ⁢
                          
                            exp
                            ⁡
                            
                              [
                              
                                
                                  k
                                  0
                                
                                ⁢
                                
                                  
                                    q
                                    
                                      1
                                      ,
                                      m
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      d
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                        
                      
                    
                  
                  }
                
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        18
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                S
                xi
              
              ⁡
              
                (
                z
                )
              
            
            =
            
              
                ∑
                
                  m
                  =
                  1
                
                
                  
                    2
                    ⁢
                    N
                  
                  +
                  1
                
              
              ⁢
              
                
                  w
                  
                    2
                    ,
                    i
                    ,
                    m
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              c
                              
                                2
                                ,
                                m
                              
                              +
                            
                            ⁢
                            
                              exp
                              ⁡
                              
                                (
                                
                                  
                                    -
                                    
                                      k
                                      0
                                    
                                  
                                  ⁢
                                  
                                    q
                                    
                                      2
                                      ,
                                      m
                                    
                                  
                                  ⁢
                                  z
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              2
                              ,
                              m
                            
                            -
                          
                          ⁢
                          
                            exp
                            ⁡
                            
                              [
                              
                                
                                  k
                                  0
                                
                                ⁢
                                
                                  
                                    q
                                    
                                      2
                                      ,
                                      m
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      d
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                        
                      
                    
                  
                  }
                
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        19
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                
                  
                    S
                    yi
                  
                  ⁡
                  
                    (
                    z
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      
                        
                          2
                          ⁢
                          N
                        
                        +
                        1
                      
                    
                    ⁢
                    
                      
                        v
                        
                          11
                          ,
                          i
                          ,
                          m
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              
                                
                                  
                                    c
                                    
                                      1
                                      ,
                                      m
                                    
                                    +
                                  
                                  ⁢
                                  
                                    exp
                                    ⁡
                                    
                                      (
                                      
                                        
                                          -
                                          
                                            k
                                            0
                                          
                                        
                                        ⁢
                                        
                                          q
                                          
                                            1
                                            ,
                                            m
                                          
                                        
                                        ⁢
                                        z
                                      
                                      )
                                    
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  c
                                  
                                    1
                                    ,
                                    m
                                  
                                  -
                                
                                ⁢
                                
                                  exp
                                  ⁡
                                  
                                    [
                                    
                                      
                                        k
                                        
                                          0
                                          ⁢
                                          
                                            
                                          
                                        
                                      
                                      ⁢
                                      
                                        
                                          q
                                          
                                            1
                                            ,
                                            m
                                          
                                        
                                        ⁡
                                        
                                          (
                                          
                                            z
                                            -
                                            d
                                          
                                          )
                                        
                                      
                                    
                                    ]
                                  
                                
                              
                            
                          
                        
                        }
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    
                      
                        2
                        ⁢
                        N
                      
                      +
                      1
                    
                  
                  ⁢
                  
                    
                      v
                      
                        1
                        ,
                        2
                        ,
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            
                              
                                
                                  c
                                  
                                    2
                                    ,
                                    m
                                  
                                  +
                                
                                ⁢
                                
                                  exp
                                  ⁡
                                  
                                    (
                                    
                                      
                                        -
                                        
                                          k
                                          0
                                        
                                      
                                      ⁢
                                      
                                        q
                                        
                                          2
                                          ,
                                          m
                                        
                                      
                                      ⁢
                                      z
                                    
                                    )
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                c
                                
                                  2
                                  ,
                                  m
                                
                                -
                              
                              ⁢
                              
                                exp
                                ⁡
                                
                                  [
                                  
                                    
                                      k
                                      0
                                    
                                    ⁢
                                    
                                      
                                        q
                                        
                                          2
                                          ,
                                          m
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          z
                                          -
                                          d
                                        
                                        )
                                      
                                    
                                  
                                  ]
                                
                              
                            
                          
                        
                      
                      }
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        20
      
    
  
  
    
      
        
          
            
              U
              yi
            
            ⁡
            
              (
              z
              )
            
          
          =
          
            
              
                ∑
                
                  m
                  =
                  1
                
                
                  
                    2
                    ⁢
                    N
                  
                  +
                  1
                
              
              ⁢
              
                
                  v
                  
                    21
                    ,
                    i
                    ,
                    m
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              -
                              
                                c
                                
                                  1
                                  ,
                                  m
                                
                                +
                              
                            
                            ⁢
                            
                              exp
                              ⁡
                              
                                (
                                
                                  
                                    -
                                    
                                      k
                                      0
                                    
                                  
                                  ⁢
                                  
                                    q
                                    
                                      1
                                      ,
                                      m
                                    
                                  
                                  ⁢
                                  z
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              1
                              ,
                              m
                            
                            -
                          
                          ⁢
                          
                            exp
                            ⁡
                            
                              [
                              
                                
                                  k
                                  0
                                
                                ⁢
                                
                                  
                                    q
                                    
                                      1
                                      ,
                                      m
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      d
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                        
                      
                    
                  
                  }
                
              
            
            +
            
              
                ∑
                
                  m
                  =
                  1
                
                
                  
                    2
                    ⁢
                    N
                  
                  +
                  1
                
              
              ⁢
              
                
                  v
                  
                    22
                    ,
                    i
                    ,
                    m
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              -
                              
                                c
                                
                                  2
                                  ,
                                  m
                                
                                +
                              
                            
                            ⁢
                            
                              exp
                              ⁡
                              
                                (
                                
                                  
                                    -
                                    
                                      k
                                      0
                                    
                                  
                                  ⁢
                                  
                                    q
                                    
                                      2
                                      ,
                                      m
                                    
                                  
                                  ⁢
                                  z
                                
                                )
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            c
                            
                              2
                              ,
                              m
                            
                            -
                          
                          ⁢
                          
                            exp
                            ⁡
                            
                              [
                              
                                
                                  k
                                  0
                                
                                ⁢
                                
                                  
                                    q
                                    
                                      2
                                      ,
                                      m
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      z
                                      -
                                      d
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                        
                      
                    
                  
                  }
                
              
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        21
      
    
  
  
    
      
        
          
        
        ⁢
        where
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              V
              11
            
            =
            
              
                A
                
                  -
                  1
                
              
              ⁢
              
                W
                1
              
              ⁢
              
                Q
                1
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        22
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              V
              12
            
            =
            
              
                (
                
                  
                    k
                    y
                  
                  /
                  
                    k
                    0
                  
                
                )
              
              ⁢
              
                A
                
                  -
                  1
                
              
              ⁢
              
                K
                x
              
              ⁢
              
                W
                2
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        23
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              V
              21
            
            =
            
              
                (
                
                  
                    k
                    y
                  
                  /
                  
                    k
                    0
                  
                
                )
              
              ⁢
              
                B
                
                  -
                  1
                
              
              ⁢
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
              ⁢
              
                W
                1
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        24
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              V
              22
            
            =
            
              
                B
                
                  -
                  1
                
              
              ⁢
              
                W
                2
              
              ⁢
              
                Q
                2
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        25
      
    
  




Q1 and Q2 are diagonal matrices with elements q1,m and q2,m, which are the square roots of the 2N+1 eigenvalues of the matrices [Ky2+A] and [Ky2+BEinv−1], and W1 and W2 are the (2N+1)×(2N+1) matrices formed by the corresponding eigenvectors, with elements w1,i,m and w2,i,m. Eqs. 16-25 constitute the eigen-problem portion of the RWC method given in the first the Moharam and Gaylord reference cited above. It is noted that there are other equivalent formulations of the same eigen-problem that will lead to the same final results.
The constants c1,m+, c1,m−, c2,m+, c2,m− are determined by matching the tangential electric and magnetic field components at the two boundary regions of the grating. The first of the Moharam and Gaylord reference cited above uses a boundary formulation where the field components are rotated into the corresponding diffraction plane, φi, for each diffracted order:

sin ψδi0+Rs,i=cos φiSyi(0)−sin φiSxi(0),  eq. 26

j[sin ψnI cos θδi0−(kI,zi/k0)Rs,i]=−[cos φiUxi(0)+sin φiUyi(0)],  eq. 27

cos ψ cos θδi0−j[kI,zi/(k0nI2)]Rp,i=cos φiSxi(0)+sin φiSyi(0)  eq. 28

−jnI cos ψδi0+Rp,i=−[cos φiUyi(0)−sin φiUxi(0)],  eq. 29

where

φi=tan−1(ky/kxi),  eq. 30

Rs,i=cos φiRyi−sin φiRxi,  eq. 31

Rp,i=(j/k0)[cos φi(kI,ziRxi+kxiRzi)+sin φi(kyRzi+kI,ziRyi)],  eq. 32

at the z=0 boundary, and

cos φiSyi(d)−sin φiSxi(d)=Ts,i,  eq. 33

−[cos φiUxi(d)+sin φiUyi(d)]=j(kI,zi/k0)Ts,i,  eq. 34

−[cos φiUyi(d)−sin φiUxi(d)]=Tp,i,  eq. 35

cos φiSxi(d)+sin φiSyi(d)=j(kI,zi/k0nI2)Tp,i,  eq. 36

Ts,i=cos φiTyi−sin φiTxi,  eq. 37

Tp,i=(−j/k0)[cos φi(kII,ziTxi−kxiTzi)−sin φi(−kII,ziTyi+kyTzi)]  eq. 38

at the z=d boundary. Note that there is one equation for each spatial harmonic retained in the Fourier expansions. Rs,i and Rp,i are the components of the reflected electric and magnetic field amplitudes normal to the diffraction plane, and Ts,i and Tp,i are the transmitted amplitudes.

In matrix form, eqs. 26-29 are




  
    
      
        
          [
          
            
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  ψ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
              
            
            
              
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  ψ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    n
                    I
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
              
            
            
              
                
                  
                    -
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  ψ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    n
                    I
                  
                  ⁢
                  
                    δ
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
              
            
            
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  ψ
                  ⁢
                  
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
              
            
          
          ]
        
        +
        
          
            [
            
              
                
                  I
                
                
                  0
                
              
              
                
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Y
                      I
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  I
                
              
              
                
                  0
                
                
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Z
                      I
                    
                  
                
              
            
            ]
          
          ⁢
          
             
            
              
                [
                
                  
                    
                      
                        R
                        s
                      
                    
                  
                  
                    
                      
                        R
                        p
                      
                    
                  
                
                ]
              
              =
              
                
                  [
                  
                    
                      
                        
                          V
                          ss
                        
                      
                      
                        
                          V
                          sp
                        
                      
                      
                        
                          
                            V
                            ss
                          
                          ⁢
                          
                            X
                            1
                          
                        
                      
                      
                        
                          
                            V
                            sp
                          
                          ⁢
                          
                            V
                            
                              2
                              ⁢
                              
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          W
                          ss
                        
                      
                      
                        
                          W
                          sp
                        
                      
                      
                        
                          
                            -
                            
                              W
                              ss
                            
                          
                          ⁢
                          
                            X
                            
                              1
                              ⁢
                              
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              W
                              sp
                            
                          
                          ⁢
                          
                            X
                            
                              2
                              ⁢
                              
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          W
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            s
                          
                        
                      
                      
                        
                          W
                          pp
                        
                      
                      
                        
                          
                            -
                            
                              W
                              
                                p
                                ⁢
                                
                                  
                                
                                ⁢
                                s
                              
                            
                          
                          ⁢
                          
                            X
                            1
                          
                        
                      
                      
                        
                          
                            -
                            
                              W
                              pp
                            
                          
                          ⁢
                          
                            X
                            2
                          
                        
                      
                    
                    
                      
                        
                          V
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            s
                          
                        
                      
                      
                        
                          V
                          pp
                        
                      
                      
                        
                          
                            V
                            
                              p
                              ⁢
                              
                                
                              
                              ⁢
                              s
                            
                          
                          ⁢
                          
                            X
                            1
                          
                        
                      
                      
                        
                          
                            V
                            pp
                          
                          ⁢
                          
                            X
                            2
                          
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          c
                          1
                          +
                        
                      
                    
                    
                      
                        
                          c
                          2
                          +
                        
                      
                    
                    
                      
                        
                          c
                          1
                          -
                        
                      
                    
                    
                      
                        
                          c
                          2
                          -
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        39
      
    
  




for the z=0 boundary and eqs. 33-36 are





  
    
      
        
          
            [
            
              
                
                  
                    
                      V
                      ss
                    
                    ⁢
                    
                      X
                      1
                    
                  
                
                
                  
                    
                      V
                      sp
                    
                    ⁢
                    
                      X
                      
                        2
                        ⁢
                        
                          
                        
                      
                    
                  
                
                
                  
                    V
                    ss
                  
                
                
                  
                    V
                    sp
                  
                
              
              
                
                  
                    
                      W
                      ss
                    
                    ⁢
                    
                      X
                      1
                    
                  
                
                
                  
                    
                      W
                      sp
                    
                    ⁢
                    
                      X
                      2
                    
                  
                
                
                  
                    -
                    
                      W
                      
                        ss
                        ⁢
                        
                          
                        
                      
                    
                  
                
                
                  
                    -
                    
                      W
                      sp
                    
                  
                
              
              
                
                  
                    
                      W
                      
                        p
                        ⁢
                        
                          
                        
                        ⁢
                        s
                      
                    
                    ⁢
                    
                      X
                      1
                    
                  
                
                
                  
                    
                      W
                      pp
                    
                    ⁢
                    
                      X
                      2
                    
                  
                
                
                  
                    -
                    
                      W
                      
                        p
                        ⁢
                        
                          
                        
                        ⁢
                        s
                      
                    
                  
                
                
                  
                    -
                    
                      W
                      pp
                    
                  
                
              
              
                
                  
                    
                      V
                      
                        p
                        ⁢
                        
                          
                        
                        ⁢
                        s
                      
                    
                    ⁢
                    
                      X
                      1
                    
                  
                
                
                  
                    
                      V
                      pp
                    
                    ⁢
                    
                      X
                      2
                    
                  
                
                
                  
                    V
                    
                      p
                      ⁢
                      
                        
                      
                      ⁢
                      s
                    
                  
                
                
                  
                    V
                    pp
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    c
                    1
                    +
                  
                
              
              
                
                  
                    c
                    2
                    +
                  
                
              
              
                
                  
                    c
                    1
                    -
                  
                
              
              
                
                  
                    c
                    2
                    -
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  I
                
                
                  0
                
              
              
                
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Y
                      II
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  I
                
              
              
                
                  0
                
                
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Z
                      II
                    
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    T
                    
                      s
                      ⁢
                      
                        
                      
                    
                  
                
              
              
                
                  
                    T
                    p
                  
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        40
      
    
  




for the z=d boundary, where

Vss=FcV11 Wpp=FcV22 

Wss=FcW1+FsV21 Vpp=FcW2+FsV12 

Vsp=FcV12−FsW2 Wps=FcV21−FsW1 

Wsp=FsV22 Vps=FsV11  eq. 41

YI, YII, ZI, and ZII are diagonal matrices with elements (kI,zi/k0), (kII,zi/k0), (kI,zi/k0nI2), and (kII,zi/k0nII2), respectively, X1 and X2 are diagonal matrices with elements exp(−k0q1,md) and exp(−k0q2,md), respectively, and Fc and Fs are diagonal matrices with elements cos φi and sin φi, respectively.
Eqs. 39 and 40 are typically solved by eliminating Rs and Rp from eq. 39, Ts and Tp from eq. 40 and solving the resulting 4(2N+1) equations for the 4(2N+1) coefficients c1,m+, c1,m−, c2,m+, c2,m− which can be substituted back into 39 and 40 to solve for the reflected and transmitted amplitudes.
Alternately, a procedure similar to the partial solution approach given in the second Moharam and Gaylord reference cited above can be used to determine reflected amplitudes only, giving a 2(2N+1)×2(2N+1) system of equations for the c1,m+ and c2,m+ coefficients:




  
    
      
        
          
            
              
                j
                ⁡
                
                  (
                  
                    Y
                    I
                  
                  )
                
              
              
                0
                ,
                0
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            ψ
            ⁢
            
              
            
            ⁢
            
              δ
              
                i
                ⁢
                
                  
                
                ⁢
                0
              
            
          
          +
          
            j
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            ψ
            ⁢
            
              
            
            ⁢
            
              n
              I
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            θ
            ⁢
            
              
            
            ⁢
            
              δ
              
                i
                ⁢
                
                  
                
                ⁢
                0
              
            
          
        
        =
        
          
            [
            
              
                j
                ⁢
                
                  
                
                ⁢
                
                  Y
                  I
                
                ⁢
                
                  f
                  T
                
              
              +
              
                f
                B
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    c
                    1
                    +
                  
                
              
              
                
                  
                    c
                    2
                    +
                  
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        42
      
    
  
  
    
      
        
          
            
              
                (
                
                  Z
                  I
                
                )
              
              
                0
                ,
                0
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            ψ
            ⁢
            
              
            
            ⁢
            
              n
              I
            
            ⁢
            
              δ
              
                i
                ⁢
                
                  
                
                ⁢
                0
              
            
          
          +
          
            cos
            ⁢
            
              
            
            ⁢
            ψ
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            θ
            ⁢
            
              
            
            ⁢
            
              δ
              
                i
                ⁢
                
                  
                
                ⁢
                0
              
            
          
        
        =
        
          
            [
            
              
                j
                ⁢
                
                  
                
                ⁢
                
                  Z
                  I
                
                ⁢
                
                  g
                  T
                
              
              +
              
                g
                B
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    c
                    1
                    +
                  
                
              
              
                
                  
                    c
                    2
                    +
                  
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        43
      
    
  




which are related to the reflected amplitudes:





  
    
      
        
          
            R
            s
          
          =
          
            
              
                f
                T
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        
                          1
                          ⁢
                          
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        c
                        2
                        +
                      
                    
                  
                
                ]
              
            
            -
            
              sin
              ⁢
              
                
              
              ⁢
              ψ
              ⁢
              
                
              
              ⁢
              
                δ
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        44
      
    
  
  
    
      
        
          R
          p
        
        =
        
          
            
              g
              T
            
            ⁡
            
              [
              
                
                  
                    
                      c
                      1
                      +
                    
                  
                
                
                  
                    
                      c
                      2
                      +
                    
                  
                
              
              ]
            
          
          +
          
            j
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            ψ
            ⁢
            
              
            
            ⁢
            
              n
              I
            
            ⁢
            
              δ
              
                i
                ⁢
                
                  
                
                ⁢
                0
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        45
      
    
  
  
    
      where
    
    
      
        
      
    
  
  
    
      
        
          
            [
            
              
                
                  
                    f
                    T
                  
                
              
              
                
                  
                    f
                    B
                  
                
              
              
                
                  
                    g
                    
                      T
                      ⁢
                      
                        
                      
                    
                  
                
              
              
                
                  
                    g
                    B
                  
                
              
            
            ]
          
          =
          
            
              [
              
                
                  
                    
                      V
                      
                        ss
                        ⁢
                        
                          
                        
                      
                    
                  
                  
                    
                      V
                      sp
                    
                  
                
                
                  
                    
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                        p
                        ⁢
                        
                          
                        
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                      W
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                        s
                      
                    
                  
                  
                    
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                        ⁢
                        
                          
                        
                      
                    
                  
                
              
              ]
            
            +
            
              
                [
                
                  
                    
                      
                        
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                        ⁢
                        
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                          1
                        
                      
                    
                    
                      
                        
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                            sp
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                            2
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                        ⁢
                        
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                          2
                        
                      
                    
                  
                  
                    
                      
                        
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                            W
                            
                              p
                              ⁢
                              
                                
                              
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                          X
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                          -
                          
                            W
                            pp
                          
                        
                        ⁢
                        
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                            2
                            ⁢
                            
                              
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          V
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
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                            1
                            ⁢
                            
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          V
                          pp
                        
                        ⁢
                        
                          X
                          2
                        
                      
                    
                  
                
                ]
              
              ·
              a
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        46
      
    
  




and the matrix a is defined as the top half of





  
    
      
        
          
            
              [
              
                
                  
                    
                      -
                      
                        V
                        ss
                      
                    
                  
                  
                    
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              ]
            
            
              -
              1
            
          
          ⁡
          
            [
            
              
                
                  
                    
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                      ss
                    
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                      sp
                    
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                        ⁢
                        
                          
                        
                        ⁢
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                    ⁢
                    
                      X
                      1
                    
                  
                
                
                  
                    
                      V
                      pp
                    
                    ⁢
                    
                      X
                      2
                    
                  
                
              
            
            ]
          
        
        ≡
        
          
            [
            
              
                
                  a
                
              
              
                
                  b
                
              
            
            ]
          
          .
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        47
      
    
  




Note that (YI)0,0 and (ZI)0,0 refer to the center elements of the matrices YI, or (kI,z0/k0), and ZI, or (kI,z0/k0nI2), respectively.
The boundary matching can be generalized to multiple layers using (for example) the enhanced transmittance matrix approach outlined in the second the Moharam and Gaylord reference cited above. Given an L layer stack, where L+1 refers to the substrate, start by setting




  
    
      
        
          [
          
            
              
                
                  
                    
                      
                        
                          
                            
                              f
                              
                                
                                  L
                                  +
                                  1
                                
                                ,
                                T
                              
                            
                          
                        
                        
                          
                            
                              f
                              
                                
                                  L
                                  +
                                  1
                                
                                ,
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                        g
                        
                          
                            L
                            +
                            1
                          
                          ,
                          T
                        
                      
                    
                  
                
              
            
            
              
                
                  g
                  
                    
                      L
                      +
                      1
                    
                    ,
                    B
                  
                
              
            
          
          ]
        
        =
        
          
            [
            
              
                
                  I
                
                
                  0
                
              
              
                
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Y
                      II
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  I
                
              
              
                
                  0
                
                
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Z
                      II
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        48
      
    
  




The matrices for aL and bL are constructed for layer L,




  
    
      
        
          
            
              
                [
                
                  
                    
                      
                        -
                        
                          V
                          
                            ss
                            ,
                            L
                          
                        
                      
                    
                    
                      
                        -
                        
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                            sp
                            ,
                            L
                          
                        
                      
                    
                    
                      
                        f
                        
                          
                            L
                            +
                            1
                          
                          ,
                          T
                        
                      
                    
                  
                  
                    
                      
                        W
                        
                          ss
                          ,
                          L
                        
                      
                    
                    
                      
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                          sp
                          ,
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                            L
                            +
                            1
                          
                          ,
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                            ⁢
                            
                              
                            
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                          ,
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                        g
                        
                          
                            L
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                          ,
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                          L
                        
                      
                    
                    
                      
                        g
                        
                          
                            L
                            +
                            1
                          
                          ,
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                ]
              
              
                -
                1
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        V
                        
                          ss
                          ,
                          L
                        
                      
                      ⁢
                      
                        X
                        
                          1
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                        V
                        
                          sp
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                      ⁢
                      
                        X
                        
                          2
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                          ss
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                        X
                        
                          1
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                        W
                        
                          sp
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                      ⁢
                      
                        X
                        
                          2
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                            p
                            ⁢
                            
                              
                            
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                          ,
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                          1
                          ,
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                        W
                        
                          pp
                          ,
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                      ⁢
                      
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                          2
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                        V
                        
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            s
                          
                          ,
                          L
                        
                      
                      ⁢
                      
                        X
                        
                          1
                          ,
                          L
                        
                      
                    
                  
                  
                    
                      
                        V
                        
                          pp
                          ,
                          L
                        
                      
                      ⁢
                      
                        X
                        
                          2
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              ]
            
          
          ≡
          
            [
            
              
                
                  
                    a
                    L
                  
                
              
              
                
                  
                    b
                    L
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        49
      
    
  




where WL and VL come from the solution to the eigen-problem for layer L, X1,L=exp(−k0q1,m,LdL), and X2,L=exp(−k0q2,m,LdL), where dL is the thickness of layer L. fL and gL are then obtained from





  
    
      
        
          [
          
            
              
                
                  f
                  
                    L
                    ,
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                  f
                  
                    L
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                  g
                  
                    L
                    ,
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                  g
                  
                    L
                    ,
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          ]
        
        =
        
          
            [
            
              
                
                  
                    V
                    
                      ss
                      ,
                      L
                    
                  
                
                
                  
                    V
                    
                      sp
                      ,
                      L
                    
                  
                
              
              
                
                  
                    W
                    
                      ss
                      ,
                      L
                    
                  
                
                
                  
                    W
                    
                      sp
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                        ⁢
                        
                          
                        
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          +
          
            
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                        V
                        
                          ss
                          ,
                          L
                        
                      
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                        X
                        
                          1
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                          sp
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                          2
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                            ss
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                            sp
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                          2
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                        -
                        
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                              p
                              ⁢
                              
                                
                              
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                            ,
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                        X
                        
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                          ,
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                            pp
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                      ⁢
                      
                        X
                        
                          2
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                        V
                        
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            s
                          
                          ,
                          L
                        
                      
                      ⁢
                      
                        X
                        
                          1
                          ,
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                          pp
                          ,
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                      ⁢
                      
                        X
                        
                          
                            2
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                          ⁢
                          
                            
                          
                        
                      
                    
                  
                
              
              ]
            
            ·
            
              
                a
                L
              
              .
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        50
      
    
  




fL and gL are fed back into eq. 49 along with the solution to the eigen-problem for layer L-1 to find aL-1 and bL-1, and so on until at the top layer f1T, f1B, g1T, and g1B are obtained. These are substituted into eqs. 42-45 in place of fT, fB, gT, and gB to solve for the coefficients c1,m+ and c2,m+ for the top layer, and finally for the reflection coefficients for the diffracted orders via eqs. 44 and 45.
Eqs. 42 and 43 reduce the boundary problem to a 2(2N+1)×2(2N+1) set of equations. For large truncation order, the boundary problem can still be dominated by the 4(2N+1)×4(2N+1) matrix inversion in eq. 49, but efficient inversion techniques can be employed since only the top half of the matrix is used.
Therefore, for a given incident polar angle theta, the computational expense incurred by using nonzero azimuthal incidence phi is two (2N+1)×(2N+1) eigen-problems versus one (2N+1)×(2N+1) eigen-problem in the corresponding classical (same theta, phi=0) case, a 2(2N+1)×2(2N+1) linear system of equations for the boundary problem (to solve for reflected amplitudes only) versus a (2N+1)×(2N+1) system of equations in the corresponding classical incidence case, and a 4(2N+1)×4(2N+1) matrix inversion in the boundary problem versus a 2(2N+1)×2(2N+1) matrix inversion in the corresponding classical mount. Since these operations are governed by order n3 operations, the conical mount requires approximately 2 times the computing time as the corresponding classical mount case for the eigen-problem, and approximately 8 times the computing time for the boundary problem.
Details of the Reduction in RCW Computation Time for the phi=90 Conical Mount
For the purposes of this description, the symmetry properties of the Fourier series are assumed a priori, and not proved. The initial assumptions can be derived through symmetry arguments, or by experimentation with the conventional formulation given above. In particular, for the phi=90 mount and s polarized incident light (psi=90),

Ex,i=Ex,−i  eq. 51

Ey,i=−Ey,−i  eq. 52

Hx,i=−Hx,−i  eq. 53

Hy,i=Hy,−i,  eq. 54

while for p polarized incident light (psi=0),

Ex,i=−Ex,−i  eq. 55

Ey,i=Ey,−i  eq. 56

Hx,i=Hx,−i  eq. 57

Hy,i=−Hy,−i.  eq. 58

In equations 51-58, the subscript i refers to the expansion term, which in the incident region corresponds to the diffraction order.
These relationships are valid in all regions of the grating problem, and all of the Fourier expansions can be reduced accordingly. In addition to these relationships, there is a 180 degree phase difference between opposite odd orders, but this can be ignored when not considering interference between multiple gratings.
Also, for phi=90, eq. 5 becomes

kxi=−ik0(λ0/Λ)  eq. 59

This gives

kxi=−kx−i  eq. 60

kl,zi=kl,z−i  eq. 61

The relations 51-61 show that for the phi=90 incidence case:

    
    
        i) The generalized Fourier expansions in eqs. 1-4 become regular Fourier expansions, and
        ii) The Fourier expansions for the fields have either even or odd symmetry, depending on the particular field component.
    
    


This means that a complex Fourier series representation is not necessary, and the field components can be expressed as cosine series for even symmetry cases or sine series for odd symmetry cases—although the Fourier coefficients themselves will still in general be complex. In either case, the entire content of the 2N+1 terms of a truncated complex Fourier series is contained in the N+1 terms of a cosine or sine series, depending on the symmetry. The usefulness of this for the grating problem arises from the fact that the fields have this symmetry in every region. When re-expressed as cosine and sine series, all of the information about the grating problem is contained in roughly half the number of terms required for the traditional formulation. This leads to a reduction in computation time by a factor of approximately 8 compared with the usual phi=90 formulation.
Each incident polarization case will be treated separately. For s polarized light, incident at polar angle theta and the phi=90 conical plane, eqs. 51-54 give

Sx,i=Sx,−i  eq. 62

Sy,i=−Sy,−i  eq. 63

Ux,i=−Ux,−i  eq. 64

Uy,i=Uy,−i,  eq. 65

in the grating region.

The reduced Eqs. 26-29 and eqs. 33-36 may be derived by substituting the symmetry relations for Ri and Ti into eqs. 1-4 (this requires determining further symmetry relations for Rz,i and Tz,i), reducing the fields everywhere to the appropriate Fourier cosine or sine series, applying the boundary conditions to the tangential components of the fields at z=0 and z=d, and rotating the boundary equations into the diffraction plane. However, the notation is unnecessarily cumbersome, and it is easier to apply eqs. 62-65 directly to eqs. 26-29 and eqs. 33-36 and use

Rs,i=Rs,−i  eq. 66

Rp,i=−Rp,−i  eq. 67

Ts,i=Ts,−i  eq. 68

Tp,i=−Tp,−i,  eq. 69

to derive the same thing. Eqs. 66-69 can again be verified using the conventional formulation with the phi=90 mount.

In addition, eq. 30 for phi=90 gives

cos φi=−cos φ−i  eq. 70

and

sin φi=sin φ−i.  eq. 71

Applying the symmetry relations, the i=0 terms in eqs. 26-29 and eqs. 33-36 remain the same, but the nonzero i terms can be combined by adding the i and −i terms of eqs. 26, 27, 33, and 34, and subtracting the −ith from the ith terms in eqs. 28, 29, 35, and 36.
For example, eq. 26 gives

sin ψ+Rs,0=−Sx,0(0)  eq. 72

for i=0, and

Rs,i+Rs,−i=cos φiSy,i(0)+cos φ−iSy,−i(0)−sin φiSx,i(0)−sin φ−iSx,−i(0),

2Rs,i=2 cos φiSy,i(0)−2 sin φiSx,i(0),

Rs,i=cos φiSy,i(0)−sin φiSx,i(0),  eq. 73

which is the same as eq. 26, except that i>0.

Similarly, eq. 28 gives

cos ψ cos θ−j[kI,z0/(k0nI2)]Rp,0=Sy,0(0)  eq. 74

for i=0, and

−j[kI,zi/(k0nI2)]Rp,i+j[kI,z(−i)/(k0nI2)]Rp,−i=cos φiSx,i(0)−cos φ−iSx,−i(0)+sin φiSy,i(0)−sin φ−iSy,−i(0)

−2j[kI,zi/(k0nI2)]Rp,i=2 cos φiSx,i(0)+2 sin φiSy,i(0),

−j[kI,zi/(k0nI2)]Rp,i=cos φiSx,i(0)+sin φiSy,i(0),  eq. 75

which is eq. 28, but with i>0.

The other boundary equations can be similarly reduced, and the form of the boundary problem is the same as the conventional one, except that only the i=0 and i>0 terms occur in the matrix equations. This reduces the matrix boundary problem (eqs. 39 and 40) to 4(N+1)×4(N+1) systems of equations, but leaves the form the same as in eqs. 26-41, as long as it is possible to also reduce the solution in the grating region to determining 4(N+1) coefficients c1,m+, c1,m−, c2,m+, c2,m− instead of 4(2N+1) coefficients. This is shown to be the case below. Therefore, except for modifying the matrices to consist of N+1 harmonic terms (and therefore N+1 diffraction orders), the boundary problem is the same as previously defined. Now, Rsi, Rpi, Tsi, and Tpi are the amplitudes of both the +i and −i diffracted orders.
To reduce the eigen-system, apply eqs. 60 and 62-65 directly to equations 15-17, reducing the total number of unknowns from 4(2N+1) to 4(N+1). The eigen-problems specified by eqs. 16 and 17 are each reduced from size (2N+1)×(2N+1) to size (N+1)×(N+1), for a total reduction of a factor of approximately 8 over the previous conical descriptions, and a factor of 4 over the corresponding classical mount eigen-problem.
Aside from reducing eqs. 16 and 17, reduced matrices for eqs. 22-25 will also need to be found. This will reduce the solution in the grating region to the determination of 4(N+1) coefficients instead of 4(2N+1).
Note that the symmetry of the lamellar grating also implies that the elements of the permittivity matrix satisfy

Ei,j=E−i,−j.  eq. 76

The i th row of equation 16 can be written as




  
    
      
        
          
            
              ∂
              2
            
            ⁢
            
              U
              
                xi
                ⁢
                
                  
                
              
            
          
          
            ∂
            
              
                (
                
                  z
                  ′
                
                )
              
              
                2
                ⁢
                
                  
                
              
            
          
        
        =
        
          
            
              
                k
                
                  y
                  ⁢
                  
                    
                  
                
                2
              
              
                k
                
                  0
                  ⁢
                  
                    
                  
                
              
            
            ⁢
            
              U
              xi
            
          
          +
          
            
              
                k
                xi
                2
              
              
                k
                0
                2
              
            
            ⁢
            
              U
              xi
            
          
          -
          
            
              ∑
              
                m
                =
                
                  -
                  ∞
                
              
              ∞
            
            ⁢
            
              
                E
                
                  i
                  ,
                  m
                
              
              ⁢
              
                
                  U
                  xm
                
                .
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        77
      
    
  




Due to eq. 64, eq. 16 obeys the following symmetry condition:




  
    
      
        
          
            
              ∂
              2
            
            ⁢
            
              U
              xi
            
          
          
            ∂
            
              
                (
                
                  z
                  ′
                
                )
              
              2
            
          
        
        =
        
          -
          
            
              
                
                  ∂
                  2
                
                ⁢
                
                  U
                  
                    x
                    -
                    i
                  
                
              
              
                ∂
                
                  
                    (
                    
                      z
                      ′
                    
                    )
                  
                  2
                
              
            
            .
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        78
      
    
  




Subtracting the −i th row from the i th row gives




  
    
      
        
          
        
        ⁢
        
          
            
              
                
                  
                    ∂
                    
                      2
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                
                  ∂
                  
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                    2
                  
                
              
              =
              
                
                  
                    
                      k
                      y
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        0
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
              
            
            ⁢
            
              

            
            ⁢
            
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                
                  ∂
                  
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                    2
                  
                
              
              =
              
                
                  
                    
                      
                        k
                        y
                        2
                      
                      
                        k
                        0
                        2
                      
                    
                    ⁢
                    
                      U
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                  -
                  
                    
                      E
                      
                        0
                        ,
                        0
                      
                    
                    ⁢
                    
                      U
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                  -
                  
                    
                      ∑
                      
                        m
                        =
                        
                          -
                          ∞
                        
                      
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        E
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        U
                        xm
                      
                    
                  
                  -
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                        E
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        U
                        xm
                      
                    
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                
                =
                
                  
                    
                      
                        k
                        y
                        2
                      
                      
                        k
                        0
                        2
                      
                    
                    ⁢
                    
                      U
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                  -
                  
                    
                      E
                      
                        0
                        ,
                        0
                      
                    
                    ⁢
                    
                      U
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                        E
                        
                          0
                          ,
                          
                            -
                            m
                          
                        
                      
                      ⁢
                      
                        U
                        xm
                      
                    
                  
                  -
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                        E
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        U
                        xm
                      
                    
                  
                
              
            
          
          ,
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        79
      
    
  
  
    
      
        
          
        
        ⁢
        so
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                
                  ∂
                  2
                
                ⁢
                
                  U
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
              
                ∂
                
                  
                    (
                    
                      z
                      ′
                    
                    )
                  
                  2
                
              
            
            =
            
              
                
                  
                    k
                    y
                    2
                  
                  
                    k
                    0
                    2
                  
                
                ⁢
                
                  U
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
              -
              
                {
                
                  
                    
                      E
                      
                        0
                        ,
                        0
                      
                    
                    ⁢
                    
                      U
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        0
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                        (
                        
                          
                            E
                            
                              0
                              ,
                              m
                            
                          
                          -
                          
                            E
                            
                              0
                              ,
                              
                                -
                                m
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        U
                        xm
                      
                    
                  
                
                }
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                for
                ⁢
                
                  
                
                ⁢
                i
              
              =
              0
            
            ,
            and
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                2
                ⁢
                
                  
                    
                      ∂
                      2
                    
                    ⁢
                    
                      U
                      xi
                    
                  
                  
                    ∂
                    
                      
                        (
                        
                          z
                          ′
                        
                        )
                      
                      2
                    
                  
                
              
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                      k
                      y
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      xi
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      im
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      y
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      xi
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                -
                
                  
                    E
                    
                      i
                      ,
                      0
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      E
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    E
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      y
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      xi
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                -
                
                  
                    E
                    
                      i
                      ⁢
                      
                        
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        i
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    E
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      y
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      xi
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                -
                
                  
                    E
                    
                      i
                      ,
                      0
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        i
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    E
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      E
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      y
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      k
                      xi
                      2
                    
                    
                      k
                      0
                      2
                    
                  
                  ⁢
                  
                    U
                    xi
                  
                
                -
                
                  
                    (
                    
                      
                        E
                        
                          i
                          ,
                          0
                        
                      
                      -
                      
                        E
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    U
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                      (
                      
                        
                          E
                          
                            i
                            ,
                            m
                          
                        
                        +
                        
                          E
                          
                            
                              -
                              i
                            
                            ,
                            
                              -
                              m
                            
                          
                        
                        -
                        
                          E
                          
                            i
                            ,
                            
                              -
                              m
                            
                          
                        
                        -
                        
                          E
                          
                            
                              -
                              i
                            
                            ,
                            m
                          
                        
                      
                      )
                    
                    ⁢
                    
                      U
                      xm
                    
                  
                
                ,
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        so
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              ∂
              2
            
            ⁢
            
              U
              xi
            
          
          
            ∂
            
              
                (
                
                  z
                  ′
                
                )
              
              2
            
          
        
        =
        
          
            
              
                k
                y
                2
              
              
                k
                0
                2
              
            
            ⁢
            
              U
              xi
            
          
          +
          
            
              
                k
                
                  xi
                  ⁢
                  
                    
                  
                
                2
              
              
                k
                0
                2
              
            
            ⁢
            
              U
              xi
            
          
          -
          
            {
            
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            E
                            
                              i
                              ,
                              0
                            
                          
                          -
                          
                            E
                            
                              
                                -
                                i
                              
                              ,
                              0
                            
                          
                        
                        )
                      
                      ⁢
                      
                        U
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          0
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                          (
                          
                            
                              
                                
                                  
                                    E
                                    
                                      i
                                      ,
                                      m
                                    
                                  
                                  +
                                  
                                    E
                                    
                                      
                                        -
                                        i
                                      
                                      ,
                                      
                                        -
                                        m
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  
                                    E
                                    
                                      
                                        -
                                        i
                                      
                                      ,
                                      
                                        -
                                        m
                                      
                                    
                                  
                                  -
                                  
                                    E
                                    
                                      i
                                      ,
                                      
                                        -
                                        m
                                      
                                    
                                  
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          U
                          xm
                        
                      
                    
                  
                
              
            
            }
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        80
      
    
  




for i>0. Note that i now runs from 0 to ∞ instead of −∞ to ∞.

The first two terms in eq. 79 and 80 indicate that the matrices Ky2 and Kx2 in eq. 16 should simply be replaced by diagonal matrices consisting of the 0 and positive terms of the original matrices. In fact, this will turn out to be the case for Kx and Ky throughout, and the subscripts and superscripts on these matrices distinguishing reduced from unreduced will hereafter be omitted.
The terms




  
    
      
        
          
            E
            
              i
              ,
              0
            
          
          ⁢
          
            U
            
              x
              ⁢
              
                
              
              ⁢
              0
            
          
        
        +
        
          
            ∑
            
              m
              =
              1
            
            ∞
          
          ⁢
          
            
              (
              
                
                  E
                  
                    0
                    ,
                    m
                  
                
                -
                
                  E
                  
                    0
                    ,
                    
                      -
                      m
                    
                  
                
              
              )
            
            ⁢
            
              U
              xm
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        81
      
    
  




from eq. 79 and





  
    
      
        
          
            1
            2
          
          ⁢
          
            (
            
              
                E
                
                  i
                  ,
                  0
                
              
              -
              
                E
                
                  
                    -
                    i
                  
                  ,
                  0
                
              
            
            )
          
          ⁢
          
            U
            
              x
              ⁢
              
                
              
              ⁢
              0
            
          
        
        +
        
          
            1
            2
          
          ⁢
          
            
              ∑
              
                m
                =
                1
              
              ∞
            
            ⁢
            
              
                (
                
                  
                    
                      
                        
                          E
                          
                            i
                            ,
                            m
                          
                        
                        +
                        
                          E
                          
                            
                              -
                              i
                            
                            ,
                            
                              -
                              m
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          E
                          
                            
                              -
                              i
                            
                            ,
                            m
                          
                        
                        -
                        
                          E
                          
                            i
                            ,
                            
                              -
                              m
                            
                          
                        
                      
                    
                  
                
                )
              
              ⁢
              
                U
                xm
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        82
      
    
  




from eq. 80 are the rows of the reduced matrix that replaces the matrix E in eq. 16:





  
    
      
        
          E
          reduced
          s
        
        =
        
           
          
            
              [
              
                
              
              ⁢
              
                
                  
                    
                      E
                      
                        0
                        ,
                        0
                      
                    
                  
                  
                    
                      
                        E
                        
                          0
                          ,
                          1
                        
                      
                      -
                      
                        E
                        
                          0
                          ,
                          
                            -
                            1
                          
                        
                      
                    
                  
                  
                    
                      
                        E
                        
                          0
                          ,
                          2
                        
                      
                      -
                      
                        E
                        
                          0
                          ,
                          
                            -
                            2
                          
                        
                      
                    
                  
                  
                    ...
                  
                
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            E
                            
                              1
                              ,
                              0
                            
                          
                          -
                          
                            E
                            
                              
                                -
                                1
                              
                              ,
                              0
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            
                              
                                
                                  E
                                  
                                    1
                                    ,
                                    1
                                  
                                
                                +
                                
                                  E
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    
                                      -
                                      1
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  E
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    1
                                  
                                
                                -
                                
                                  E
                                  
                                    1
                                    ,
                                    
                                      -
                                      1
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            
                              
                                
                                  E
                                  
                                    1
                                    ,
                                    2
                                  
                                
                                +
                                
                                  E
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    
                                      -
                                      2
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  E
                                  
                                    
                                      -
                                      1
                                    
                                    ,
                                    2
                                  
                                
                                -
                                
                                  E
                                  
                                    1
                                    ,
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  
                    ⋯
                  
                
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            E
                            
                              2
                              ,
                              0
                            
                          
                          -
                          
                            E
                            
                              
                                -
                                2
                              
                              ,
                              0
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            
                              
                                
                                  E
                                  
                                    2
                                    ,
                                    1
                                  
                                
                                +
                                
                                  E
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    
                                      -
                                      1
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  E
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    1
                                  
                                
                                -
                                
                                  E
                                  
                                    2
                                    ,
                                    
                                      -
                                      1
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            
                              
                                
                                  E
                                  
                                    2
                                    ,
                                    2
                                  
                                
                                +
                                
                                  E
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    
                                      -
                                      2
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  E
                                  
                                    
                                      -
                                      2
                                    
                                    ,
                                    2
                                  
                                
                                -
                                
                                  E
                                  
                                    1
                                    ,
                                    
                                      -
                                      2
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                  
                    ...
                  
                
                
                  
                    ⋮
                  
                  
                    
                      
                    
                  
                  
                    
                      
                    
                  
                  
                    ⋱
                  
                
              
              ⁢
              
                
              
              ]
            
            ⁢
            
              

            
            ⁢
            with
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        83
      
    
  
  
    
      
        
          
            A
            reduced
            s
          
          =
          
            
              K
              x
              2
            
            -
            
              E
              reduced
              s
            
          
        
        ⁢
        
          

        
        ⁢
        and
      
    
    
      
        eq
        .
        
          
        
        ⁢
        84
      
    
  
  
    
      
        
          
            [
            
              
                
                  ∂
                  2
                
                ⁢
                
                  U
                  x
                
              
              /
              
                ∂
                
                  
                    (
                    
                      z
                      ′
                    
                    )
                  
                  2
                
              
            
            ]
          
          =
          
            
              [
              
                
                  K
                  y
                  2
                
                +
                
                  A
                  reduced
                  s
                
              
              ]
            
            ⁡
            
              [
              
                U
                x
              
              ]
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        85
      
    
  




where the subscript s refers to the incident polarization case. All of the vectors in eq. 85 are of size N+1, and the matrices are of size (N+1)×(N+1) for a given truncation order, N.

Many of the terms in eq. 83 can be reduced using eq. 76, but it is more useful to assume nothing about the elements of the matrices being reduced. This way, other matrices that may not necessarily obey eq. 76 can be reduced using the same formulas. Along these lines, more general reductions can be formulated, which can be applied to a variety of matrices or even the products of matrices that will be required to find the reduced matrices of eqs. 22-25.
Disregarding the simpler diagonal matrices Ky and Kx, the unreduced equations have the general form




  
    
      
        
          l
          ⁡
          
            [
            
              P
              i
            
            ]
          
        
        =
        
          
            ∑
            
              m
              =
              
                -
                ∞
              
            
            ∞
          
          ⁢
          
            
          
          ⁢
          
            
              ɛ
              
                i
                ,
                m
              
            
            ⁢
            
              Q
              m
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        86
      
    
  




Where l is a linear operator, such as




  
    
      
        ∂
        
          
        
      
      
        ∂
        
          (
          
            z
            ′
          
          )
        
      
    
    ⁢
    
      
    
    ⁢
    or
    ⁢
    
      
    
    ⁢
    
      
        
          ∂
          2
        
        ⁢
        
          
        
      
      
        ∂
        
          
            (
            
              z
              ′
            
            )
          
          2
        
      
    
  
  ,




and the elements of the vectors P and Q are spatial harmonic coefficients of the Fourier expansions for the corresponding fields. The goal is to find a reduced matrix for ε through application of symmetry relations to the vectors P and Q. Without making any assumptions about the elements of the matrix ε, there are in general four types of reductions:

    
    
        1) Both P and Q are even and the corresponding Fourier series can be reduced to cosine series,
        2) Both P and Q are odd and the corresponding Fourier expressions can be reduced to sine series,
        3) P is even and Q is odd,
        4) P is odd and Q is even.
    
    


Note that if P has even or odd symmetry, then




  
    
      
        ∂
        P
      
      ⁢
      
        
      
    
    
      ∂
      
        (
        
          z
          ′
        
        )
      
    
  
  ⁢
  
    
  
  ⁢
  and
  ⁢
  
    
  
  ⁢
  
    
      
        
          ∂
          2
        
        ⁢
        P
      
      ⁢
      
        
      
    
    
      ∂
      
        
          (
          
            z
            ′
          
          )
        
        2
      
    
  




are also even or odd, respectively.

The reduction leading to eqs. 81 and 82 belongs to category 2. The same argument can be applied to eq. 86 to give




  
    
      
        
          
            l
            ⁡
            
              [
              
                P
                0
              
              ]
            
          
          =
          
            
              
                ɛ
                
                  0
                  ,
                  0
                
              
              ⁢
              
                Q
                0
              
            
            +
            
              
                ∑
                
                  m
                  =
                  1
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  
                    
                      ɛ
                      
                        0
                        ,
                        m
                      
                    
                    -
                    
                      ɛ
                      
                        0
                        ,
                        
                          -
                          m
                        
                      
                    
                  
                  )
                
                ⁢
                
                  Q
                  m
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          i
          =
          0
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        eq
        .
        
          
        
        ⁢
        87
      
    
  
  
    
      
        
          
            l
            ⁡
            
              [
              
                P
                i
              
              ]
            
          
          =
          
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    ɛ
                    
                      i
                      ,
                      0
                    
                  
                  -
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                
                )
              
              ⁢
              
                Q
                0
              
            
            +
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        
                          
                            
                              ɛ
                              
                                i
                                ,
                                m
                              
                            
                            +
                            
                              ɛ
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            -
                          
                        
                      
                      
                        
                          
                            
                              ɛ
                              
                                
                                  -
                                  i
                                
                                ,
                                m
                              
                            
                            -
                            
                              ɛ
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    m
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          i
          >
          0
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        88
      
    
  




for any matrix ε and field harmonics P and Q having odd symmetry.

The other 3 cases are developed below.
For case 1, both P and Q have even symmetry. Therefore the i and −i rows can be added together:




  
    
      
        
          
            
              
                
                  
                    l
                    ⁡
                    
                      [
                      
                        P
                        0
                      
                      ]
                    
                  
                  =
                  
                  ⁢
                  
                    
                      ∑
                      
                        m
                        =
                        
                          -
                          ∞
                        
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ɛ
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                
              
            
            
              
                
                  =
                   
                  ⁢
                  
                    
                      
                        ɛ
                        
                          0
                          ,
                          0
                        
                      
                      ⁢
                      
                        Q
                        0
                      
                    
                    +
                    
                      
                        ∑
                        
                          m
                          =
                          
                            -
                            ∞
                          
                        
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ɛ
                          
                            0
                            ,
                            m
                          
                        
                        ⁢
                        
                          Q
                          m
                        
                      
                    
                    +
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ɛ
                          
                            0
                            ,
                            m
                          
                        
                        ⁢
                        
                          Q
                          m
                        
                      
                    
                  
                
              
            
            
              
                
                  =
                  
                  ⁢
                  
                    
                      
                        ɛ
                        
                          0
                          ,
                          
                            0
                            ⁢
                            m
                          
                        
                      
                      ⁢
                      
                        Q
                        0
                      
                    
                    +
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ɛ
                          
                            0
                            ,
                            
                              -
                              m
                            
                          
                        
                        ⁢
                        
                          Q
                          m
                        
                      
                    
                    +
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ɛ
                          
                            0
                            ,
                            m
                          
                        
                        ⁢
                        
                          Q
                          m
                        
                      
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            l
            ⁡
            
              [
              
                P
                0
              
              ]
            
          
          =
           
          ⁢
          
            
              
                ɛ
                
                  0
                  ,
                  0
                
              
              ⁢
              
                Q
                0
              
            
            +
            
              
                ∑
                
                  m
                  =
                  1
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  
                    
                      ɛ
                      
                        0
                        ,
                        m
                      
                    
                    +
                    
                      ɛ
                      
                        0
                        ,
                        
                          -
                          m
                        
                      
                    
                  
                  )
                
                ⁢
                
                  Q
                  m
                
              
            
          
        
        ,
        
          
        
        ⁢
        
          i
          =
          0
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        eq
        .
        
          
        
        ⁢
        89
      
    
  
  
    
      
        
          
            
              
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      i
                    
                    ]
                  
                
                +
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      
                        -
                        i
                      
                    
                    ]
                  
                
              
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ɛ
                    
                      i
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          m
                        
                      
                      +
                      
                        ɛ
                        
                          i
                          ,
                          
                            -
                            m
                          
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    m
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  2
                  ⁢
                  
                    l
                    ⁡
                    
                      [
                      
                        P
                        i
                      
                      ]
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        90
      
    
  
  
    
      
        
          so
          ⁢
          
            

          
          ⁢
          
            l
            ⁡
            
              [
              
                P
                i
              
              ]
            
          
          =
          
          ⁢
          
            
              
                1
                2
              
              ⁢
              
              ⁢
              
                (
                
                  
                    ɛ
                    
                      i
                      ,
                      0
                    
                  
                  +
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                
                )
              
              ⁢
              
                Q
                0
              
            
            +
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        
                          
                            
                              ɛ
                              
                                i
                                ,
                                m
                              
                            
                            +
                            
                              ɛ
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              ɛ
                              
                                
                                  -
                                  i
                                
                                ,
                                m
                              
                            
                            +
                            
                              ɛ
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    m
                  
                
              
            
          
        
        ,
        
          
        
        ⁢
        
          

        
        ⁢
        
          i
          >
          0.
        
      
    
    
      
        
      
    
  




Eqs. 89 and 90 define a reduced matrix




  
    
      
        
          ɛ
          reduced
        
        =
        
           
          
            [
            
              
                
                  
                    ɛ
                    
                      0
                      ,
                      0
                    
                  
                
                
                  
                    
                      ɛ
                      
                        0
                        ,
                        1
                      
                    
                    +
                    
                      ɛ
                      
                        0
                        ,
                        
                          -
                          1
                        
                      
                    
                  
                
                
                  
                    
                      ɛ
                      
                        0
                        ,
                        2
                      
                    
                    +
                    
                      ɛ
                      
                        0
                        ,
                        
                          -
                          2
                        
                      
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          ɛ
                          
                            1
                            ,
                            0
                          
                        
                        +
                        
                          ɛ
                          
                            
                              -
                              1
                            
                            ,
                            0
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                ɛ
                                
                                  1
                                  ,
                                  1
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                ɛ
                                
                                  1
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  1
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                ɛ
                                
                                  1
                                  ,
                                  2
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                ɛ
                                
                                  1
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  2
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          ɛ
                          
                            2
                            ,
                            0
                          
                        
                        +
                        
                          ɛ
                          
                            
                              -
                              2
                            
                            ,
                            0
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                ɛ
                                
                                  2
                                  ,
                                  1
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                ɛ
                                
                                  2
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  1
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                ɛ
                                
                                  2
                                  ,
                                  2
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                ɛ
                                
                                  2
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                              
                                ɛ
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  2
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  
                    
                  
                
                
                  ⋱
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        91
      
    
  




For case 3, the i and −i rows are again added:




  
    
      
        
          
            
              
                
                  
                    
                      l
                      ⁡
                      
                        [
                        
                          P
                          0
                        
                        ]
                      
                    
                    =
                    
                    ⁢
                    
                      
                        ∑
                        
                          m
                          =
                          
                            -
                            ∞
                          
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ɛ
                          
                            0
                            ,
                            m
                          
                        
                        ⁢
                        
                          Q
                          m
                        
                      
                    
                  
                
              
              
                
                  
                    =
                    
                    ⁢
                    
                      
                        
                          ɛ
                          
                            0
                            ,
                            0
                          
                        
                        ⁢
                        
                          Q
                          0
                        
                      
                      +
                      
                        
                          ∑
                          
                            m
                            =
                            
                              -
                              ∞
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            ɛ
                            
                              0
                              ,
                              m
                            
                          
                          ⁢
                          
                            Q
                            m
                          
                        
                      
                      +
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            ɛ
                            
                              0
                              ,
                              m
                            
                          
                          ⁢
                          
                            Q
                            m
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      =
                      
                      ⁢
                      
                        
                          
                            ɛ
                            
                              0
                              ,
                              0
                            
                          
                          ⁢
                          
                            Q
                            0
                          
                        
                        -
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              ɛ
                              
                                0
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              Q
                              m
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              ɛ
                              
                                0
                                ,
                                m
                              
                            
                            ⁢
                            
                              Q
                              m
                            
                          
                        
                      
                    
                    ,
                  
                
              
            
            ⁢
            
              

            
            ⁢
            
              
                l
                ⁡
                
                  [
                  
                    P
                    0
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    ɛ
                    
                      0
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      (
                      
                        
                          ɛ
                          
                            0
                            ,
                            m
                          
                        
                        -
                        
                          ɛ
                          
                            0
                            ,
                            
                              -
                              m
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
          ,
          
            
          
          ⁢
          
            i
            =
            0
          
          ,
        
        ⁢
        
          

        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        92
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      i
                    
                    ]
                  
                
                +
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      
                        -
                        i
                      
                    
                    ]
                  
                
              
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ɛ
                    
                      i
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      m
                    
                  
                  ⁢
                  
                    Q
                    m
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      m
                    
                  
                  ⁢
                  
                    Q
                    m
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          m
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      -
                      
                        ɛ
                        
                          i
                          ,
                          
                            -
                            m
                          
                        
                      
                      -
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    m
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  2
                  ⁢
                  
                    l
                    ⁡
                    
                      [
                      
                        P
                        i
                      
                      ]
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        
      
    
  
  
    
      giving
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                l
                ⁡
                
                  [
                  
                    P
                    i
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      +
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                        (
                        
                          
                        
                        ⁢
                        
                          
                            
                              
                                
                                  ɛ
                                  
                                    i
                                    ,
                                    m
                                  
                                
                                +
                                
                                  ɛ
                                  
                                    
                                      -
                                      i
                                    
                                    ,
                                    m
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  ɛ
                                  
                                    i
                                    ,
                                    
                                      -
                                      m
                                    
                                  
                                
                                -
                                
                                  ɛ
                                  
                                    
                                      -
                                      i
                                    
                                    ,
                                    
                                      -
                                      m
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                
                ,
                
                  
                
                ⁢
                
                  i
                  >
                  0.
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        93
      
    
  




For case 4, subtract the −i th row from the i th row:




  
    
      
        
          
            
              
                l
                ⁡
                
                  [
                  
                    P
                    0
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        0
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                =
                
                  
                    
                      ɛ
                      
                        0
                        ,
                        0
                      
                    
                    ⁢
                    
                      Q
                      0
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        
                          -
                          ∞
                        
                      
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ɛ
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ɛ
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      ɛ
                      
                        0
                        ,
                        0
                      
                    
                    ⁢
                    
                      Q
                      0
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ɛ
                        
                          0
                          ,
                          
                            -
                            m
                          
                        
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        ɛ
                        
                          0
                          ,
                          m
                        
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                
              
              ,
            
          
        
        
          
            
              
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      0
                    
                    ]
                  
                
                =
                
                ⁢
                
                  
                    
                      ɛ
                      
                        0
                        ,
                        0
                      
                    
                    ⁢
                    
                      Q
                      0
                    
                  
                  +
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          ɛ
                          
                            0
                            ,
                            m
                          
                        
                        +
                        
                          
                            ɛ
                            
                              0
                              ,
                              
                                -
                                m
                              
                            
                          
                          ⁢
                          
                            Q
                            m
                          
                        
                      
                      )
                    
                  
                
              
              ,
              
                
              
              ⁢
              
                i
                =
                0
              
              ,
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        94
      
    
  
  
    
      and
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      i
                    
                    ]
                  
                
                -
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      
                        -
                        i
                      
                    
                    ]
                  
                
              
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ɛ
                    
                      i
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                -
                
                  
                    ɛ
                    
                      
                        -
                        i
                      
                      ,
                      0
                    
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      -
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        
                          -
                          m
                        
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
                -
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ɛ
                      
                        
                          -
                          i
                        
                        ,
                        m
                      
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      -
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            
                              
                                ɛ
                                
                                  i
                                  ,
                                  m
                                
                              
                              -
                              
                                ɛ
                                
                                  i
                                  ,
                                  
                                    -
                                    m
                                  
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                ɛ
                                
                                  
                                    -
                                    i
                                  
                                  ,
                                  m
                                
                              
                              -
                              
                                ɛ
                                
                                  
                                    -
                                    i
                                  
                                  ,
                                  
                                    -
                                    m
                                  
                                
                              
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      Q
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                2
                ⁢
                
                  l
                  ⁡
                  
                    [
                    
                      P
                      i
                    
                    ]
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      giving
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                l
                ⁡
                
                  [
                  
                    P
                    i
                  
                  ]
                
              
              =
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        ɛ
                        
                          i
                          ,
                          0
                        
                      
                      -
                      
                        ɛ
                        
                          
                            -
                            i
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    Q
                    0
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        
                          
                            
                              
                                
                                  ɛ
                                  
                                    i
                                    ,
                                    m
                                  
                                
                                +
                                
                                  ɛ
                                  
                                    i
                                    ,
                                    
                                      -
                                      m
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  ɛ
                                  
                                    
                                      -
                                      i
                                    
                                    ,
                                    m
                                  
                                
                                -
                                
                                  ɛ
                                  
                                    
                                      -
                                      i
                                    
                                    ,
                                    
                                      -
                                      m
                                    
                                  
                                
                              
                            
                          
                        
                        )
                      
                      ⁢
                      
                        Q
                        m
                      
                    
                  
                
                ,
                
                  
                
                ⁢
                
                  i
                  >
                  0.
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        95
      
    
  




Application of case 2 with ε=E leads directly to eqs. 81 and 82 for Ereduced and leads to the reduced eigenproblem of eqs. 84 and 85. To reduce eq. 17, case 1 can be applied directly to the product BEinv−1, giving




  
    
      
        
          
            
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    S
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                
                  ∂
                  
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                    2
                  
                
              
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        k
                        y
                        2
                      
                      
                        k
                        0
                      
                    
                    )
                  
                  ⁢
                  
                    S
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                +
                
                  
                    
                      (
                      
                        BEinv
                        
                          -
                          1
                        
                      
                      )
                    
                    
                      0
                      ,
                      0
                    
                  
                  ⁢
                  
                    S
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        
                          (
                          
                            BEinv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          0
                          ,
                          m
                        
                      
                      +
                      
                        
                          (
                          
                            BEinv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          0
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    S
                    xm
                  
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        96
      
    
  
  
    
      
        
          
            
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    S
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      i
                    
                  
                
                
                  ∂
                  
                    
                      (
                      
                        z
                        ′
                      
                      )
                    
                    2
                  
                
              
              =
              
              ⁢
              
                
                  
                    (
                    
                      
                        k
                        y
                        2
                      
                      
                        k
                        0
                      
                    
                    )
                  
                  ⁢
                  
                    S
                    
                      x
                      ,
                      
                        
                      
                      ⁢
                      i
                    
                  
                
                +
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            (
                            
                              BEinv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            i
                            ,
                            0
                          
                        
                        +
                        
                          
                            (
                            
                              BEinv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            
                              -
                              i
                            
                            ,
                            0
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  
                    S
                    
                      x
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            
                              
                                (
                                
                                  BEinv
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                m
                              
                            
                            +
                            
                              
                                (
                                
                                  BEinv
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                (
                                
                                  BEinv
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                m
                              
                            
                            +
                            
                              
                                (
                                
                                  BEinv
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                    S
                    xm
                  
                  .
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        97
      
    
  




This involves (2N+1)×(2N+1) matrix multiplications to find the elements of BEinv−1. A slightly more efficient way to construct the reduced eigen-problem is to reduce the components of the product first, and multiply the reduced (N+1)×(N+1) matrices together to form Breduced(Einv−1)reduced.
To do this one can go back to eq. 15 and apply the appropriate reductions to the third column of the second row and second column of the third row for B and Einv−1, respectively.
For B, explicitly reduce the product KxE−1Kx:




  
    
  
  ⁢
  
    
      
        
          
            
              
                ∂
                
                  S
                  xi
                
              
              
                ∂
                
                  (
                  
                    z
                    ′
                  
                  )
                
              
            
            =
            
              ...
              +
              
                
                  
                    k
                    xi
                  
                  
                    k
                    0
                  
                
                ⁢
                
                  
                    ∑
                    
                      m
                      =
                      
                        -
                        ∞
                      
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        (
                        
                          E
                          
                            -
                            1
                          
                        
                        )
                      
                      
                        i
                        ,
                        m
                      
                    
                    ⁢
                    
                      
                        k
                        xm
                      
                      
                        k
                        0
                      
                    
                    ⁢
                    
                      U
                      ym
                    
                  
                
              
            
          
          ,
        
      
      
        
          
            eq
            .
            
              
            
            ⁢
            98
          
          ⁢
          
            
          
        
      
    
  




where the dots replace other terms in eq. 15 that are not relevant for the purpose of finding the reduced matrix.

Adding the i th and −i th rows:




  
    
      
        
          
            
              ∂
              
                S
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
          ⁢
          
            ...
            +
            0
          
        
        ,
        
          
        
        ⁢
        
          i
          =
          0
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        99
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                
                  since
                  ⁢
                  
                    
                  
                
                
                  
                
              
              ⁢
              
                k
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
            =
            0
          
          ,
          and
        
      
    
    
      
        
      
    
  
  
    
      
        
          2
          ⁢
          
            
              ∂
              
                S
                xi
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
        
        =
        
        ⁢
        
          ...
          +
          
            
              
                k
                xi
              
              
                k
                0
              
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  
                    -
                    ∞
                  
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    (
                    
                      E
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    i
                    ,
                    m
                  
                
                ⁢
                
                  
                    k
                    xm
                  
                  
                    k
                    0
                  
                
                ⁢
                
                  U
                  ym
                
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
                            
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁢
          
            
              ∑
              
                m
                =
                
                  -
                  ∞
                
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                
                  (
                  
                    E
                    
                      -
                      1
                    
                  
                  )
                
                
                  
                    -
                    i
                  
                  ,
                  m
                
              
              ⁢
              
                
                  k
                  xm
                
                
                  k
                  0
                
              
              ⁢
              
                U
                ym
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              
                                -
                                ∞
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                m
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
                            
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      
                        ∑
                        
                          m
                          =
                          
                            -
                            ∞
                          
                        
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            
                              -
                              i
                            
                            ,
                            m
                          
                        
                        ⁢
                        
                          
                            k
                            xm
                          
                          
                            k
                            0
                          
                        
                        ⁢
                        
                          U
                          ym
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              
                                k
                                
                                  x
                                  -
                                  m
                                
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
                            
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            
                              -
                              i
                            
                            ,
                            
                              -
                              m
                            
                          
                        
                        ⁢
                        
                          
                            k
                            
                              x
                              -
                              m
                            
                          
                          
                            k
                            0
                          
                        
                        ⁢
                        
                          U
                          ym
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          -
                          
                            
                              ∑
                              
                                m
                                =
                                1
                              
                              ∞
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  i
                                  ,
                                  
                                    -
                                    m
                                  
                                
                              
                              ⁢
                              
                                
                                  k
                                  xm
                                
                                
                                  k
                                  0
                                
                              
                              ⁢
                              
                                U
                                ym
                              
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
                            
        ⁢
        
          
            
              k
              xi
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      -
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              
                                -
                                i
                              
                              ,
                              
                                -
                                m
                              
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          -
                          
                            
                              ∑
                              
                                m
                                =
                                1
                              
                              ∞
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  i
                                  ,
                                  
                                    -
                                    m
                                  
                                
                              
                              ⁢
                              
                                
                                  k
                                  xm
                                
                                
                                  k
                                  0
                                
                              
                              ⁢
                              
                                U
                                ym
                              
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                m
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              
                                -
                                i
                              
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                m
                              
                            
                            +
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            -
                          
                        
                      
                      
                        
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            -
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                m
                              
                            
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      k
                      xm
                    
                    
                      k
                      0
                    
                  
                  ⁢
                  
                    U
                    ym
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            =
            
            ⁢
            
              ...
              +
              
                
                  
                    k
                    xi
                  
                  
                    k
                    0
                  
                
                ⁢
                
                  
                    ∑
                    
                      m
                      =
                      1
                    
                    ∞
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      [
                      
                        
                          2
                          ⁢
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              
                                -
                                m
                              
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      
                        k
                        xm
                      
                      
                        k
                        0
                      
                    
                    ⁢
                    
                      U
                      ym
                    
                  
                
              
            
          
          ,
        
      
    
    
      
        
      
    
  
  
    
      giving
    
    
      
        
      
    
  
  
    
      
        
          
            2
            ⁢
            
              
                ∂
                
                  S
                  xi
                
              
              
                ∂
                
                  (
                  
                    z
                    ′
                  
                  )
                
              
            
          
          =
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          i
                          ,
                          m
                        
                      
                      -
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          i
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      k
                      xm
                    
                    
                      k
                      0
                    
                  
                  ⁢
                  
                    U
                    ym
                  
                
              
            
          
        
        ,
        
          
        
        ⁢
        
          

        
        ⁢
        
          i
          >
          0
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        100
      
    
  
  
    
      or
    
    
      
        
      
    
  
  
    
      
        
          
            (
            
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
              ⁢
              
                K
                x
              
            
            )
          
          reduced
          S
        
        =
        
           
          
            [
            
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
              
              
                
                  0
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    1
                                    ,
                                    1
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    1
                                    ,
                                    2
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  ...
                
              
              
                
                  0
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    2
                                    ,
                                    1
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    2
                                    ,
                                    2
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  
                    
                  
                
                
                  ⋱
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        101
      
    
  




in explicit form. Then

Breduceds=(KxE−1Kx)reduceds−I  eq. 102

For Einv−1, use




  
    
      
        
          
            ∂
            
              U
              yi
            
          
          
            ∂
            
              z
              ′
            
          
        
        =
        
          ...
          +
          
            
              ∑
              
                m
                =
                
                  -
                  ∞
                
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                
                  (
                  
                    Einv
                    
                      -
                      1
                    
                  
                  )
                
                im
              
              ⁢
              
                S
                xm
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        103
      
    
  




to which case 1 may be directly applied:





  
    
      
        
          
            
              ∂
              
                U
                
                  y
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
            
              ∂
              
                z
                ′
              
            
          
          =
          
            ...
            +
            
              
                
                  (
                  
                    Einv
                    
                      -
                      1
                    
                  
                  )
                
                
                  0
                  ,
                  0
                
              
              ⁢
              
                S
                
                  x
                  ,
                  0
                
              
            
            +
            
              
                ∑
                
                  m
                  =
                  1
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                2
                ⁢
                
                  
                    (
                    
                      Einv
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    0
                    ,
                    m
                  
                
                ⁢
                
                  S
                  xm
                
              
            
          
        
        ,
        
          
        
        ⁢
        
          

        
        ⁢
        
          i
          =
          0
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        104
      
    
  
  
    
      
        
          
            
              ∂
              
                U
                
                  y
                  ⁢
                  
                    
                  
                  ⁢
                  i
                
              
            
            
              ∂
              
                z
                ′
              
            
          
          =
          
            ...
            +
            
              
                
                  (
                  
                    Einv
                    
                      -
                      1
                    
                  
                  )
                
                
                  i
                  ,
                  0
                
              
              ⁢
              
                S
                
                  x
                  ,
                  0
                
              
            
            +
            
              
                ∑
                
                  m
                  =
                  1
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              (
                              
                                Einv
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            i
                            ,
                            
                              -
                              m
                            
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  S
                  xm
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          i
          >
          
            
          
          ⁢
          0
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        105
      
    
  




which implies





  
    
      
        
          
            (
            
              Einv
              
                -
                1
              
            
            )
          
          reduced
          S
        
        =
        
          [
          
            
              
                
                  
                    (
                    
                      Einv
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    0
                    ,
                    0
                  
                
              
              
                
                  2
                  ⁢
                  
                    
                      (
                      
                        Einv
                        
                          -
                          1
                        
                      
                      )
                    
                    
                      0
                      ,
                      1
                    
                  
                
              
              
                
                  2
                  ⁢
                  
                    
                      (
                      
                        Einv
                        
                          -
                          1
                        
                      
                      )
                    
                    
                      0
                      ,
                      2
                    
                  
                
              
              
                ...
              
            
            
              
                
                  
                    (
                    
                      Einv
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    1
                    ,
                    0
                  
                
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            1
                            ,
                            1
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          1
                          ,
                          
                            -
                            1
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            1
                            ,
                            2
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          1
                          ,
                          
                            -
                            2
                          
                        
                      
                    
                  
                
              
              
                ...
              
            
            
              
                
                  
                    (
                    
                      Einv
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    2
                    ,
                    0
                  
                
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            2
                            ,
                            1
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          2
                          ,
                          
                            -
                            1
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            2
                            ,
                            2
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          2
                          ,
                          
                            -
                            2
                          
                        
                      
                    
                  
                
              
              
                ...
              
            
            
              
                ⋮
              
              
                
                  
                
              
              
                
                  
                
              
              
                ⋱
              
            
          
          ]
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        106
      
    
  




where one makes use of the fact that (Einv−1)i,m=(Einv−1)−i,−m.

Eq. 17 becomes

[∂2Sx/∂(z′)2]=[Ky2+Breduceds(Einv−1)reduceds][Sx]  eq. 107

In eqs. 85 and 107 the vectors Sx and Ux and diagonal matrices Ky and Kx are trivially reduced to consist of the zeroth and positive terms of the original vectors/matrices. When truncated with truncation order N, the size of the eigen-problems are (N+1)×(N+1) instead of (2N+1)×(2N+1), and require much less computation time to solve.
The solution to the reduced eigen-problems has the same form as eqs. 18-25, but with 4(N+1) coefficients to be determined instead of 4(2N+1). The correct reduced matrices to use in eqs. 22-25 should still be found, so that the reduced form of eq. 15 is satisfied. Here again one could have derived the entire reduced set of eqs. for eq. 15, but it is really only necessary to reduce a few specific terms in order to find A−1, B−1, A−1Kx, and B−1KxE−1 to use in eqs. 22-25.
Substituting eqs. 18-21 into the second row of eq. 15 gives

W2Q2=(KxE−1Kx−I)V22  eq. 108

and

(KxE−1Kx−I)V21=KxE−1KyW1.  eq. 109

Substituting eq. 25 into eq. 108 gives

W2Q2=BV22=BB−1W2Q2,  eq. 110

which implies that B−1 in eq. 25 should be replaced by the inverse of the reduced matrix Breduced found earlier.

Eqs. 24 and 109 give




  
    
      
        
          
            BV
            21
          
          =
          
            
              
                
                  BB
                  
                    -
                    1
                  
                
                ⁡
                
                  (
                  
                    
                      k
                      y
                    
                    
                      k
                      0
                    
                  
                  )
                
              
              ⁢
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
              ⁢
              
                W
                1
              
            
            =
            
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
              ⁢
              
                K
                y
              
              ⁢
              
                W
                1
              
            
          
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        111
      
    
  




Which again implies that B−1→(Breduced)−1 in eq. 24, and KxE−1 is found by reducing




  
    
      
        
          
            ∂
            
              S
              x
            
          
          
            ∂
            
              (
              
                z
                ′
              
              )
            
          
        
        =
        
          ...
          -
          
            
              K
              x
            
            ⁢
            
              E
              
                -
                1
              
            
            ⁢
            
              K
              y
            
            ⁢
            
              
                U
                x
              
              .
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        112
      
    
  




Since Sx is even in x and Ux is odd, the reduced matrix for KxE−1 can be found by applying case 3 with ε=KxE−1:




  
    
      
        
          
            
              ∂
              
                S
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
            ...
            -
            
              {
              
                
                  
                    
                      
                        
                          
                            (
                            
                              
                                K
                                x
                              
                              ⁢
                              
                                E
                                
                                  -
                                  1
                                
                              
                            
                            )
                          
                          
                            0
                            ,
                            0
                          
                        
                        ⁢
                        
                          U
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          [
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                0
                                ,
                                m
                              
                            
                            -
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                0
                                ,
                                
                                  -
                                  m
                                
                              
                            
                          
                          ]
                        
                        ⁢
                        
                          U
                          xm
                        
                      
                    
                  
                
              
              }
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              for
              ⁢
              
                
              
              ⁢
              i
            
            =
            0
          
          ,
          and
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        113
      
    
  
  
    
      
        
          
            
              ∂
              
                S
                xi
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
            ...
            ⁢
            
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          
                            1
                            2
                          
                          [
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                i
                                ,
                                0
                              
                            
                            +
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                0
                              
                            
                          
                          )
                        
                        ⁢
                        
                          U
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            [
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  K
                                                  x
                                                
                                                ⁢
                                                
                                                  E
                                                  
                                                    -
                                                    1
                                                  
                                                
                                              
                                              )
                                            
                                            
                                              i
                                              ,
                                              m
                                            
                                          
                                          +
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  K
                                                  x
                                                
                                                ⁢
                                                
                                                  E
                                                  
                                                    -
                                                    1
                                                  
                                                
                                              
                                              )
                                            
                                            
                                              
                                                -
                                                i
                                              
                                              ,
                                              m
                                            
                                          
                                          -
                                        
                                      
                                    
                                  
                                
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  K
                                                  x
                                                
                                                ⁢
                                                
                                                  E
                                                  
                                                    -
                                                    1
                                                  
                                                
                                              
                                              )
                                            
                                            
                                              i
                                              ,
                                              
                                                -
                                                m
                                              
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            (
                                            
                                              
                                                K
                                                x
                                              
                                              ⁢
                                              
                                                E
                                                
                                                  -
                                                  1
                                                
                                              
                                            
                                            )
                                          
                                          
                                            
                                              -
                                              i
                                            
                                            ,
                                            
                                              -
                                              m
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          ⁢
                          
                            U
                            xm
                          
                        
                      
                    
                  
                
              
              }
            
          
        
        ,
        
          i
          >
          0.
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        114
      
    
  




This gives




  
    
      
        
          
            (
            
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
            
            )
          
          reduced
          S
        
        =
        
           
          
            [
            
              
                
                  
                    
                      (
                      
                        
                          K
                          x
                        
                        ⁢
                        
                          E
                          
                            -
                            1
                          
                        
                      
                      )
                    
                    
                      0
                      ,
                      0
                    
                  
                
                
                  
                    
                      
                        (
                        
                          
                            K
                            x
                          
                          ⁢
                          
                            E
                            
                              -
                              1
                            
                          
                        
                        )
                      
                      
                        0
                        ,
                        1
                      
                    
                    -
                    
                      
                        (
                        
                          
                            K
                            x
                          
                          ⁢
                          
                            E
                            
                              -
                              1
                            
                          
                        
                        )
                      
                      
                        0
                        ,
                        
                          -
                          1
                        
                      
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  0
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                
                                  -
                                  1
                                
                                ,
                                0
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  1
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  1
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  0
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                
                                  -
                                  2
                                
                                ,
                                0
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  1
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  1
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  
                    ⋱
                    ⁢
                    
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        115
      
    
  




Substituting eqs. 18-20, 22, and 23 into the fourth row of eq. 15 gives

W1Q1=(Kx2−E)V11=AV11=AA−1W1Q1  eq. 116

and

(Kx2−E)V12=AV12=AA−1KxKyW2.  eq. 117

Since A is replaced by Areduced in eqs. 116 and 117, A−1 should be replaced by (Areduced)−1 in both eqs. 22 and 23. Kx is simply replaced by a diagonal matrix with the (Kx)00, (Kx)11, . . . , (Kx)NN components of the original Kx matrix, as always.
Therefore eqs. 22-25 are replaced by

V11=(Areduceds)−1W1Q1,  eq. 118

V12=(ky/k0)(Areduceds)−1KxW2,  eq. 119

V21=(ky/k0)(Breduceds)−1(KxE−1)reducedsW1,  eq. 120

V22=(Breduceds)−1W2Q2,  eq. 121

where Q1, W1, Q2, and W2 are the eigenvalue and eigenvector matrices for the new, reduced eigen-problems of eqs. 85 and 107.

This new, reduced eigen-system, combined with the reduced boundary problem, gives exactly the same diffracted amplitudes and diffraction efficiencies as the old formulation for phi=90 for any given truncation order, N, but with much improved computational efficiency. For a given order, N, the computation speed is reduced by a factor of approximately 8 compared to the old formulation.
In some cases, the new, reduced phi=90 algorithms can be significantly faster than even the corresponding classical mount problem with the same polar incidence angle. In the theoretical best case limit, the phi=90 case requires about 62.5% the time as the corresponding classical case. This assumes that the eigen-problem and boundary value problem require equal amounts of time to solve for a given truncation order, N. In practice, this is more or less realized for lower truncation orders. Such a speed advantage can quickly add up when considering the amount of time that may be required to generate a library of several million spectra. In such cases, it may be beneficial to use the phi=90 mount only.
In the other limiting case where a very large truncation order is required, the computation time is basically dominated by the large matrix inversion in the boundary problem (eq. 47). In this limit, the phi=90 case requires approximately 92.5% of the computation time as the phi=0 case. Steps can be taken to make the matrix inversion more efficient, since only the top half is used, which is of some help.
These estimates ignore the fact that there is a little more overhead when constructing the various matrices for the phi=90 case than with the phi=0 case. In practice, the differences in computation speed ranges from being about equal for the phi=90 and phi=0 cases to a 20-30% speed improvement for the phi=90 mount over the corresponding phi=0 case. Either way, the improvement over the old phi=90 formulation is quite significant, and the ideas outlined in the introduction section involving multiple azimuthal datasets can be employed without a disabling increase in computation cost.
To complete the description, the reduced eigen-system is derived for p polarized incident light in the phi=90 conical mount. In this case, the fields satisfy

Rs,i=−Rs,−i  eq. 122

Rp,i=Rp,−i  eq. 123

Ts,i=−Ts,−i  eq. 124

Tp,i=Tp,−i,  eq. 125

in regions I and II, and

Sx,i=−Sx,−i  eq. 126

Sy,i=Sy,−i  eq. 127

Ux,i=Ux,−i  eq. 128

Uy,i=−Uy,−i,  eq. 129

in the grating region.

Eqs. 122-129 applied to the boundary problem lead to the same conclusion as in the s-polarized incidence case, except in this case add the i and −i terms for eqs. 28, 29, 35, and 36, and subtract the −ith from the ith terms in eqs. 26, 27, 33, and 34.
Again, the boundary matching at z=0 and z=d leads to eqs. 26-41 for the boundary equations, but with N+1 sized vectors Rs, Rp, Ts, and Tp, so long as it is again possible to reduce the eigen-problem as well.
To do this, start with eq. 16 and apply the case 1 reduction:




  
    
      
        
          
            
              ∂
              2
            
            ⁢
            
              U
              x0
            
          
          
            ∂
            
              
                (
                
                  z
                  ′
                
                )
              
              2
            
          
        
        =
        
          
            
              
                k
                y
                2
              
              
                k
                0
                2
              
            
            ⁢
            
              U
              
                x
                ⁢
                
                  
                
                ⁢
                0
              
            
          
          -
          
            {
            
              
                
                  E
                  
                    0
                    ,
                    0
                  
                
                ⁢
                
                  U
                  
                    x
                    ,
                    0
                  
                
              
              +
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        
                          
                            
                              E
                              
                                0
                                ,
                                m
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            E
                            
                              0
                              ,
                              
                                -
                                m
                              
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    U
                    xm
                  
                
              
            
            }
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        130
      
    
  
  
    
      
        
          
            
              ∂
              2
            
            ⁢
            
              U
              xi
            
          
          
            ∂
            
              
                (
                
                  z
                  ′
                
                )
              
              2
            
          
        
        =
        
          
            
              
                k
                y
                2
              
              
                k
                0
                2
              
            
            ⁢
            
              U
              
                x
                ⁢
                
                  
                
                ⁢
                i
              
            
          
          +
          
            
              
                k
                xi
                2
              
              
                k
                0
                2
              
            
            ⁢
            
              U
              
                x
                ⁢
                
                  
                
                ⁢
                i
              
            
          
          -
          
            {
            
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        (
                        
                          
                            E
                            
                              i
                              ,
                              0
                            
                          
                          +
                          
                            E
                            
                              
                                -
                                i
                              
                              ,
                              0
                            
                          
                        
                        )
                      
                      ⁢
                      
                        U
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          0
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          (
                          
                            
                              
                                
                                  
                                    E
                                    
                                      i
                                      ,
                                      m
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    E
                                    
                                      
                                        -
                                        i
                                      
                                      ,
                                      
                                        -
                                        m
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  E
                                  
                                    i
                                    ,
                                    
                                      -
                                      m
                                    
                                  
                                
                              
                            
                          
                          )
                        
                        ⁢
                        
                          U
                          xm
                        
                      
                    
                  
                
              
            
            }
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        131
      
    
  




which shows that Ereduced is given by





  
    
      
        
          
            
              E
              
                i
                ,
                0
              
            
            ⁢
            
              U
              
                x
                ⁢
                
                  
                
                ⁢
                0
              
            
          
          +
          
            
              ∑
              
                m
                =
                1
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                (
                
                  E
                  
                    0
                    ,
                    
                      m
                      ⁢
                      
                        
                      
                      +
                      
                        E
                        
                          0
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                  
                
                )
              
              ⁢
              
                U
                xm
              
            
          
        
        ,
        
          i
          =
          0
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        eq
        .
        
          
        
        ⁢
        132
      
    
  
  
    
      
        
          
            
              1
              2
            
            ⁢
            
              (
              
                
                  E
                  
                    i
                    ,
                    0
                  
                
                +
                
                  E
                  
                    
                      -
                      i
                    
                    ,
                    0
                  
                
              
              )
            
            ⁢
            
              U
              
                x
                ⁢
                
                  
                
                ⁢
                0
              
            
          
          +
          
            
              1
              2
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  1
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  
                    
                      
                        
                          
                            E
                            
                              i
                              ,
                              m
                            
                          
                          +
                          
                            E
                            
                              
                                -
                                i
                              
                              ,
                              
                                -
                                m
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            E
                            
                              
                                -
                                i
                              
                              ,
                              m
                            
                          
                          +
                          
                            E
                            
                              i
                              ,
                              
                                -
                                m
                              
                            
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  U
                  xm
                
              
            
          
        
        ,
        
          i
          >
          0
        
        ,
        
          

        
        ⁢
        or
      
    
    
      
        eq
        .
        
          
        
        ⁢
        133
      
    
  
  
    
      
        
          E
          reduced
          p
        
        =
        
           
          
            [
            
              
                
                  
                    E
                    
                      0
                      ,
                      0
                    
                  
                
                
                  
                    
                      E
                      
                        0
                        ,
                        1
                      
                    
                    +
                    
                      E
                      
                        0
                        ,
                        
                          -
                          1
                        
                      
                    
                  
                
                
                  
                    
                      E
                      
                        0
                        ,
                        2
                      
                    
                    +
                    
                      E
                      
                        0
                        ,
                        
                          -
                          2
                        
                      
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                E
                                
                                  1
                                  ,
                                  0
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              E
                              
                                
                                  -
                                  1
                                
                                ,
                                0
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                E
                                
                                  1
                                  ,
                                  1
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                E
                                
                                  1
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  1
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                E
                                
                                  1
                                  ,
                                  2
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                E
                                
                                  1
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  2
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  ⋯
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                E
                                
                                  2
                                  ,
                                  0
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              E
                              
                                
                                  -
                                  2
                                
                                ,
                                0
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                E
                                
                                  2
                                  ,
                                  1
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                E
                                
                                  2
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  1
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                E
                                
                                  2
                                  ,
                                  2
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                E
                                
                                  2
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                              +
                              
                                E
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  2
                                
                              
                            
                          
                        
                      
                      )
                    
                  
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  
                    
                  
                
                
                  ⋱
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        134
      
    
  




Using

Areducedp=Kx2−Ereducedp,  eq. 135

eq. 16 becomes

[∂2Ux/∂(z′)2]=[Ky2+Areducedp][Ux]  eq. 136

where the indices on Ux run from 0 to N, and Kx and Ky are reduced as in the s polarization case.

For eq. 17, case 2 could be applied directly to the product BEinv−1, but a more efficient set of operations is to proceed as in the s polarization case. Einv−1 is reduced by applying case 2 to eq. 103:




  
    
      
        
          
        
        ⁢
        
          
            
              
                ∂
                
                  U
                  
                    y
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
              
                ∂
                
                  z
                  ′
                
              
            
            =
            
              ...
              +
              
                
                  
                    (
                    
                      Einv
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    0
                    ,
                    0
                  
                
                ⁢
                
                  S
                  
                    x
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                
              
            
          
          ,
          
            i
            =
            0
          
          ,
        
        ⁢
        
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        137
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                ∂
                
                  U
                  yi
                
              
              
                ∂
                
                  z
                  ′
                
              
            
            =
            
              ...
              +
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          i
                          ,
                          m
                        
                      
                      -
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          i
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    S
                    xm
                  
                
              
            
          
          ,
          
            i
            >
            0
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          or
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        138
      
    
  
  
    
      
        
          
            (
            
              Einv
              
                -
                1
              
            
            )
          
          reduced
          p
        
        =
        
          [
          
            
              
                
                  
                    (
                    
                      Einv
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    0
                    ,
                    0
                  
                
              
              
                0
              
              
                0
              
              
                ...
              
            
            
              
                0
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            1
                            ,
                            1
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          1
                          ,
                          
                            -
                            1
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            1
                            ,
                            2
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          1
                          ,
                          
                            -
                            2
                          
                        
                      
                    
                  
                
              
              
                ...
              
            
            
              
                0
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            2
                            ,
                            1
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          2
                          ,
                          
                            -
                            1
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          
                            (
                            
                              Einv
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            2
                            ,
                            2
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            Einv
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          2
                          ,
                          
                            -
                            2
                          
                        
                      
                    
                  
                
              
              
                ...
              
            
            
              
                ⋮
              
              
                
                  
                
              
              
                
                  
                
              
              
                ⋱
              
            
          
          ]
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        139
      
    
  




where the fact that (Einv−1)i,m=(Einv−1)−i,−m is utilized.

For B, explicitly reduce the product KxE−1Kx:




  
    
      
        
          
            ∂
            
              S
              xi
            
          
          
            ∂
            
              (
              
                z
                ′
              
              )
            
          
        
        =
        
          ...
          +
          
            
              
                k
                xi
              
              
                k
                0
              
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  
                    -
                    ∞
                  
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    (
                    
                      E
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    i
                    ,
                    m
                  
                
                ⁢
                
                  
                    k
                    xm
                  
                  
                    k
                    0
                  
                
                ⁢
                
                  
                    U
                    ym
                  
                  .
                
              
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        140
      
    
  




Subtracting the −i th row from the i th row:




  
    
      
        
          
            
              ∂
              
                S
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
            ...
            +
            0
          
        
        ,
        
          i
          =
          0
        
        ,
        
          

        
        ⁢
        
          
            
              si
              ⁢
              nce
            
            ⁢
            
              
            
            ⁢
            
              k
              x0
            
          
          =
          0
        
        ,
        
          
        
        ⁢
        and
      
    
    
      
        eq
        .
        
          
        
        ⁢
        141
      
    
  
  
    
      
        
          2
          ⁢
          
            
              ∂
              
                S
                xi
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
        
        =
        
        ⁢
        
          ...
          +
          
            
              
                k
                xi
              
              
                k
                0
              
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  
                    -
                    ∞
                  
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    (
                    
                      E
                      
                        -
                        1
                      
                    
                    )
                  
                  
                    i
                    ,
                    m
                  
                
                ⁢
                
                  
                    k
                    xm
                  
                  
                    k
                    0
                  
                
                ⁢
                
                  U
                  ym
                
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
                             
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁢
          
            
              ∑
              
                m
                =
                
                  -
                  ∞
                
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                
                  (
                  
                    E
                    
                      -
                      1
                    
                  
                  )
                
                
                  i
                  ,
                  m
                
              
              ⁢
              
                
                  k
                  xm
                
                
                  k
                  0
                
              
              ⁢
              
                U
                ym
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              
                                -
                                ∞
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            
                              -
                              ∞
                            
                          
                          ∞
                        
                        ⁢
                        
                          
                            
                          
                          ⁢
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
                          
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      
                        ∑
                        
                          m
                          =
                          
                            -
                            ∞
                          
                        
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            
                              -
                              i
                            
                            ,
                            m
                          
                        
                        ⁢
                        
                          
                            k
                            xm
                          
                          
                            k
                            0
                          
                        
                        ⁢
                        
                          U
                          ym
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          -
                          
                            
                              ∑
                              
                                m
                                =
                                1
                              
                              ∞
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  i
                                  ,
                                  
                                    -
                                    m
                                  
                                
                              
                              ⁢
                              
                                
                                  k
                                  
                                    x
                                    -
                                    m
                                  
                                
                                
                                  k
                                  0
                                
                              
                              ⁢
                              
                                U
                                ym
                              
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
                          
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      -
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              
                                -
                                i
                              
                              ,
                              
                                -
                                m
                              
                            
                          
                          ⁢
                          
                            
                              k
                              
                                x
                                -
                                m
                              
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  
                    x
                    -
                    i
                  
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              
                                k
                                
                                  x
                                  -
                                  m
                                
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
                           
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            
                              -
                              i
                            
                            ,
                            
                              -
                              m
                            
                          
                        
                        ⁢
                        
                          
                            k
                            
                              x
                              -
                              m
                            
                          
                          
                            k
                            0
                          
                        
                        ⁢
                        
                          U
                          ym
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
            ⁢
            
              
                
                  k
                  
                    x
                    -
                    i
                  
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              i
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
                          
        ⁢
        
          
            
              k
              
                x
                -
                i
              
            
            
              k
              0
            
          
          ⁡
          
            [
            
              
                
                  
                    
                      
                        ∑
                        
                          m
                          =
                          1
                        
                        ∞
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            
                              -
                              i
                            
                            ,
                            
                              -
                              m
                            
                          
                        
                        ⁢
                        
                          
                            k
                            xm
                          
                          
                            k
                            0
                          
                        
                        ⁢
                        
                          U
                          ym
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        m
                        =
                        1
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          
                            -
                            i
                          
                          ,
                          m
                        
                      
                      ⁢
                      
                        
                          k
                          xm
                        
                        
                          k
                          0
                        
                      
                      ⁢
                      
                        U
                        ym
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
            ⁢
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁡
              
                [
                
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                m
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            ⁢
                            
                              
                                k
                                xm
                              
                              
                                k
                                0
                              
                            
                            ⁢
                            
                              U
                              ym
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              (
                              
                                E
                                
                                  -
                                  1
                                
                              
                              )
                            
                            
                              
                                -
                                i
                              
                              ,
                              m
                            
                          
                          ⁢
                          
                            
                              k
                              xm
                            
                            
                              k
                              0
                            
                          
                          ⁢
                          
                            U
                            ym
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
            ⁢
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            +
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                i
                                ,
                                m
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                
                                  -
                                  m
                                
                              
                            
                            +
                            
                              
                                (
                                
                                  E
                                  
                                    -
                                    1
                                  
                                
                                )
                              
                              
                                
                                  -
                                  i
                                
                                ,
                                m
                              
                            
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      k
                      xm
                    
                    
                      k
                      0
                    
                  
                  ⁢
                  
                    u
                    ym
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
          ⁢
          
            ...
            +
            
            ⁢
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        2
                        ⁢
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            i
                            ,
                            m
                          
                        
                      
                      +
                      
                        2
                        ⁢
                        
                          
                            (
                            
                              E
                              
                                -
                                1
                              
                            
                            )
                          
                          
                            i
                            ,
                            
                              -
                              m
                            
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      k
                      xm
                    
                    
                      k
                      0
                    
                  
                  ⁢
                  
                    u
                    ym
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      giving
    
    
      
        
      
    
  
  
    
      
        
          
            
              ∂
              
                S
                xi
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
          ⁢
          
            ...
            +
            
              
                
                  k
                  xi
                
                
                  k
                  0
                
              
              ⁢
              
                
                  ∑
                  
                    m
                    =
                    1
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    [
                    
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          i
                          ,
                          m
                        
                      
                      +
                      
                        
                          (
                          
                            E
                            
                              -
                              1
                            
                          
                          )
                        
                        
                          i
                          ,
                          
                            -
                            m
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      k
                      xm
                    
                    
                      k
                      0
                    
                  
                  ⁢
                  
                    U
                    ym
                  
                
              
            
          
        
        ,
        
          i
          >
          0
        
        ,
      
    
    
      
        eq
        .
        
          
        
        ⁢
        142
      
    
  




which implies





  
    
      
        
          
            (
            
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
              ⁢
              
                K
                x
              
            
            )
          
          reduced
          p
        
        =
        
           
          
            [
            
              
                
                  0
                
                
                  0
                
                
                  0
                
                
                  ...
                
              
              
                
                  0
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    1
                                    ,
                                    1
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    1
                                    ,
                                    2
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  ...
                
              
              
                
                  0
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    2
                                    ,
                                    1
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  
                    
                      
                        
                          k
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        
                          k
                          0
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                    )
                                  
                                  
                                    2
                                    ,
                                    2
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  
                                    -
                                    2
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                        k
                        
                          x
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      
                        k
                        0
                      
                    
                  
                
                
                  ...
                
              
              
                
                  ...
                
                
                  
                    
                  
                
                
                  
                    
                  
                
                
                  ⋱
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        143
      
    
  




in explicit form. Then

Breducedp=(KxE−1Kx)reducedp−I  eq. 144

and eq. 17 becomes

[∂2Sx/∂(z′)2]=[Ky2+Breducedp(Einv−1)reducedp][Sx],  eq. 145

where again the indices run from 0 to N and Ky is reduced as in the s polarization case.

The corresponding equations to replace eqs 21-25 are found in a similar manner as before. Most of the verification steps are omitted here. A−1 and B−1 are replaced by (Areduced)−1 and (Breduced)−1 as before. To find KxE−1 in eq. 24, use eq. 111 with case 4:




  
    
      
        
          
            
              ∂
              
                S
                
                  x
                  ⁢
                  
                    
                  
                  ⁢
                  0
                
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
            
              ...
              -
              
                
                  {
                  
                    
                      
                        
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                0
                                ,
                                0
                              
                            
                            ⁢
                            
                              U
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                0
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            ∑
                            
                              m
                              =
                              1
                            
                            ∞
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              [
                              
                                
                                  
                                    
                                      
                                        
                                          (
                                          
                                            
                                              K
                                              x
                                            
                                            ⁢
                                            
                                              E
                                              
                                                -
                                                1
                                              
                                            
                                          
                                          )
                                        
                                        
                                          0
                                          ,
                                          m
                                        
                                      
                                      +
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            K
                                            x
                                          
                                          ⁢
                                          
                                            E
                                            
                                              -
                                              1
                                            
                                          
                                        
                                        )
                                      
                                      
                                        0
                                        ,
                                        
                                          -
                                          m
                                        
                                      
                                    
                                  
                                
                              
                              ]
                            
                            ⁢
                            
                              U
                              xm
                            
                          
                        
                      
                    
                  
                  }
                
                ⁢
                i
              
            
            =
            0
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        eq
        .
        
          
        
        ⁢
        146
      
    
  
  
    
      
        
          
            
              ∂
              
                S
                xi
              
            
            
              ∂
              
                (
                
                  z
                  ′
                
                )
              
            
          
          =
          
            -
            
              {
              
                
                  
                    
                      
                        
                          
                            1
                            2
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            K
                                            x
                                          
                                          ⁢
                                          
                                            E
                                            
                                              -
                                              1
                                            
                                          
                                        
                                        )
                                      
                                      
                                        i
                                        ,
                                        0
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    
                                      (
                                      
                                        
                                          K
                                          x
                                        
                                        ⁢
                                        
                                          E
                                          
                                            -
                                            1
                                          
                                        
                                      
                                      )
                                    
                                    
                                      
                                        -
                                        i
                                      
                                      ,
                                      0
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁢
                        
                          U
                          
                            x
                            ⁢
                            
                              
                            
                            ⁢
                            0
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            m
                            =
                            1
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            [
                            
                              
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            K
                                            x
                                          
                                          ⁢
                                          
                                            E
                                            
                                              -
                                              1
                                            
                                          
                                        
                                        )
                                      
                                      
                                        i
                                        ,
                                        m
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            K
                                            x
                                          
                                          ⁢
                                          
                                            E
                                            
                                              -
                                              1
                                            
                                          
                                        
                                        )
                                      
                                      
                                        i
                                        ,
                                        
                                          -
                                          m
                                        
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            K
                                            x
                                          
                                          ⁢
                                          
                                            E
                                            
                                              -
                                              1
                                            
                                          
                                        
                                        )
                                      
                                      
                                        
                                          -
                                          i
                                        
                                        ,
                                        m
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    
                                      (
                                      
                                        
                                          K
                                          x
                                        
                                        ⁢
                                        
                                          E
                                          
                                            -
                                            1
                                          
                                        
                                      
                                      )
                                    
                                    
                                      
                                        -
                                        i
                                      
                                      ,
                                      
                                        -
                                        m
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                          ⁢
                          
                            U
                            xm
                          
                        
                      
                    
                  
                
              
              }
            
          
        
        ,
        
          i
          >
          0.
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        147
      
    
  




This gives




  
    
      
        
          
            (
            
              
                K
                x
              
              ⁢
              
                E
                
                  -
                  1
                
              
            
            )
          
          reduced
          p
        
        =
        
           
          
            [
            
              
                
                  
                    
                      (
                      
                        
                          K
                          x
                        
                        ⁢
                        
                          E
                          
                            -
                            1
                          
                        
                      
                      )
                    
                    
                      0
                      ,
                      0
                    
                  
                
                
                  
                    
                      
                        (
                        
                          
                            K
                            x
                          
                          ⁢
                          
                            E
                            
                              -
                              1
                            
                          
                        
                        )
                      
                      
                        0
                        ,
                        1
                      
                    
                    +
                    
                      
                        (
                        
                          
                            K
                            x
                          
                          ⁢
                          
                            E
                            
                              -
                              1
                            
                          
                        
                        )
                      
                      
                        0
                        ,
                        
                          -
                          1
                        
                      
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  0
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                
                                  -
                                  1
                                
                                ,
                                0
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  1
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  1
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  1
                                
                              
                              -
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    1
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  ...
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  0
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                (
                                
                                  
                                    K
                                    x
                                  
                                  ⁢
                                  
                                    E
                                    
                                      -
                                      1
                                    
                                  
                                
                                )
                              
                              
                                
                                  -
                                  2
                                
                                ,
                                0
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  
                    
                      1
                      2
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  1
                                
                              
                              +
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  2
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  1
                                
                              
                              -
                              
                                
                                  (
                                  
                                    
                                      K
                                      x
                                    
                                    ⁢
                                    
                                      E
                                      
                                        -
                                        1
                                      
                                    
                                  
                                  )
                                
                                
                                  
                                    -
                                    2
                                  
                                  ,
                                  
                                    -
                                    1
                                  
                                
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  ...
                
              
              
                
                  ⋮
                
                
                  
                    
                  
                
                
                  ⋱
                
              
            
            ]
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        148
      
    
  




Putting all of this together, eqs. 22-25 for p polarized incidence are replaced by

V11=(Areducedp)−1W1Q1,  eq. 149

V12=(ky/k0)(Areducedp)−1KxW2,  eq. 150

V21=(ky/k0)(Breducedp)−1(KxE−1)reducedpW1,  eq. 151

V22=(Breducedp)−1W2Q2,  eq. 152

where Q1, W1, Q2, and W2 are the eigenvalue and eigenvector matrices for the new, reduced eigen-problems of eqs. 136 and 145. The speed improvement is very similar to the s polarization case.

After solving the reduced boundary problem for the particular s or p incidence case, the diffraction efficiencies can be obtained from




  
    
      
        
          DE
          ri
        
        =
        
          
            
              
                
                
                  R
                  
                    s
                    ,
                    i
                  
                
                
              
              2
            
            ⁢
            
              Re
              ⁡
              
                (
                
                  
                    k
                    
                      I
                      ,
                      zi
                    
                  
                  
                    
                      k
                      0
                    
                    ⁢
                    
                      n
                      I
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
            
          
          +
          
            
              
                
                
                  R
                  
                    p
                    ,
                    i
                  
                
                
              
              2
            
            ⁢
            
              
                Re
                ⁡
                
                  (
                  
                    
                      
                        k
                        
                          I
                          ,
                          zi
                        
                      
                      /
                      
                        n
                        I
                        2
                      
                    
                    
                      
                        k
                        0
                      
                      ⁢
                      
                        n
                        I
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        153
      
    
  




For i=0, eq. 153 is just the specular reflectance for the given incident condition.

It should be pointed out that the only assumption about the grating permittivity expansion coefficients was the symmetry exploited in Eq. 76. In other words, the specific form of the permittivity Fourier coefficients for a binary grating shown in Eq. 9 were not explicitly used in the above descriptions. The grating can consist of more than 2 different materials with differing optical properties, the only difference being that the permittivity Fourier coefficients are different from the coefficients given for the binary structure in Eq. 9. The grating should still satisfy Eq. 76 where required. Additionally, as with the conventional formulation, profile shapes other than rectangular can be treated using a staircase approximation consisting of multiple rectangular grating layers.
The calculated diffraction efficiencies or amplitudes can be used to compute polarized or unpolarized reflectance data, ellipsometric data, or polarimetric data. During an optical grating measurement, one or more datasets are generated by varying the incident wavelength, polar angle of incidence, theta, and rotating the azimuthal angle of incidence between 0 degrees and 90 degrees. The optical data of the one or more datasets are compared to data generated from a theoretical model of the grating using the above calculation methods. A regression analysis is used to optimize the parameters of the theoretical grating model. The result of the optical measurement is given by the optimized grating parameters. The average of the s and p incident calculations can be used to analyze unpolarized reflectance.
The regression algorithm can be the Simplex or Levenberg-Marquardt algorithms, described in W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (2nd Edition), Cambridge University Press, Cambridge, 1992, among others, or can even consist of a simple parameter grid search. The model calculation can be performed in real-time (at the time of measurement), using one or multiple CPUs. The theoretical model spectra can also be pre-calculated ahead of time, generating a library database of spectra, from which the calculation result can be rapidly extracted during the measurement. A neural network can be pre-generated, from which the best-fit model can be directly extracted using a fixed number of relatively simple calculation steps during the measurement.
In many cases, specularly reflected, transmitted, and/or diffracted intensities are detected, and the optical system can be calibrated to give reflectance (0 R diffraction efficiency), transmittance, or diffraction efficiency. However, in some cases, particularly for VUV reflectance work, it may be beneficial to normalize some of the intensities with intensities from other structures or from different incidence conditions. These ratios are independent of incident intensity, and a system calibration that involves determining incident intensity may be skipped. The analysis can be done by calculating the corresponding reflectance or diffraction efficiency ratios. For example, a first dataset may be reflected (0 order) intensity I(0) due to unpolarized light incident at phi=0, and the second dataset may be reflected (0 order) intensity I(90) due to unpolarized light incident at phi=90. The incident intensity will typically not change over short time periods, so if the datasets are collected in close succession, the intensity ratio is the same as the reflectance ratio:




  
    
      
        
          
            I
            ⁡
            
              (
              0
              )
            
          
          
            I
            ⁡
            
              (
              90
              )
            
          
        
        =
        
          
            R
            ⁡
            
              (
              0
              )
            
          
          
            R
            ⁡
            
              (
              90
              )
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        154
      
    
  




R(0) and R(90) can be calculated using the conventional phi=0 calculation and new phi=90 calculation presented above. A regression procedure might use the following merit function:





  
    
      
        
          κ
          2
        
        -
        
          
            ∑
            
              i
              =
              1
            
            N
          
          ⁢
          
            
          
          ⁢
          
            
              
                (
                
                  1
                  
                    σ
                    i
                  
                
                )
              
              2
            
            ⁢
            
              
                (
                
                  
                    
                      (
                      
                        
                          R
                          ⁡
                          
                            (
                            0
                            )
                          
                        
                        
                          R
                          ⁡
                          
                            (
                            90
                            )
                          
                        
                      
                      )
                    
                    
                      i
                      ,
                      measured
                    
                  
                  -
                  
                    
                      (
                      
                        
                          R
                          ⁡
                          
                            (
                            0
                            )
                          
                        
                        
                          R
                          ⁡
                          
                            (
                            90
                            )
                          
                        
                      
                      )
                    
                    
                      i
                      ,
                      calculated
                    
                  
                
                )
              
              2
            
          
        
      
    
    
      
        eq
        .
        
          
        
        ⁢
        155
      
    
  




where the subscript i refers to incident condition (usually wavelength), σi is the estimated uncertainty of the measured reflectance ratio, and N is the total number of data points included for the ratio. The merit function is minimized by the regression procedure, thereby optimizing the grating parameters, which affect the calculated values for both numerator and denominator of the ratio. Note that in this case, the grating parameters are the same for both numerator and denominator.

As describe above the analysis of a diffraction grating problem is of particular use to determining the various characteristics of the diffraction grating structure including, for example, the critical dimensions and the composition of a diffraction grating. The analysis techniques described herein are of particular use in reducing the complexity and increase the speed of such analysis, which is of particular importance in high volume manufacturing processes. It will be recognized that the diffraction problem analysis techniques described herein may be utilized in a wide range of applications where is desirable to analysis a diffraction grating to obtain any of a wide range of types of characteristics of the grating structure.
Further modifications and alternative embodiments of this invention will be apparent to those skilled in the art in view of this description. It will be recognized, therefore, that the present invention is not limited by these example arrangements. Accordingly, this description is to be construed as illustrative only and is for the purpose of teaching those skilled in the art the manner of carrying out the invention. It is to be understood that the forms of the invention herein shown and described are to be taken as the presently preferred embodiments. Various changes may be made in the implementations and architectures. For example, equivalent elements may be substituted for those illustrated and described herein, and certain features of the invention may be utilized independently of the use of other features, all as would be apparent to one skilled in the art after having the benefit of this description of the invention.

## Claims

1. A method of characterizing a diffraction grating structure, comprising
collecting a first set of reflected data from the grating structure by providing incident light at a first angle of azimuthal incidence with respect to the grating structure;
collecting a second set of reflected data from the grating structure by providing incident light at a second angle of azimuthal incidence with respect to the grating structure, the first and second angles being effectively orthogonal and the second angle of azimuthal incidence being different from zero;
analyzing a combination of at least the first and second set of reflected data; and
utilizing symmetrical characteristics of a diffraction analysis of the second angle of azimuthal incidence reflected data so as to reduce the computation complexity of the analysis of the second set of reflected data during the determination of at least one geometrical characteristic of the grating structure,
where one or more of the sets of data are used to normalize other set(s) of data so that optical metrology data comprises ratios of reflected data collected for different incident conditions, avoiding the need to determine incident intensity via an absolute calibration process.

2. The method of claim 1, wherein the optical metrology data comprises a first ratio of at least a portion of the first set of reflected data and at least a portion of the second set of reflected data.

3. The method of claim 2, wherein the diffraction analysis comprises a regression or library lookup procedure that minimizes the difference between a calculated reflectance or diffraction efficiency ratio and a measured intensity ratio.

4. The method of claim 3, wherein an inverse ratio is substituted in specific wavelength regions where the denominator of the first ratio is near zero.

5. The method of claim 4, wherein a weighting function is used to equalize a contribution to a merit function regardless of a reflectance ratio magnitude.

6. The method of claim 3, wherein data regions at which a denominator of the first ratio is near zero are dropped from the diffraction analysis.

7. The method of claim 1, where one or more diffracted orders of reflected data are detected along with or instead of the 0′th order.

8. The method of claim 1, wherein data is only collected at the first azimuthal angle and the second azimuthal angle.

9. The method of claim 8, wherein the diffraction analysis comprises utilizing a rigorous coupled wave analysis.

10. The method of claim 1, wherein four data sets are utilized in the diffraction analysis for each polar angle, the data sets being comprised of two different polarizations at each of the first and second azimuthal angles.

11. The method of claim 1, further comprising utilizing multiple polar angles at each of the first and second azimuthal angles.

12. The method of claim 1, wherein the diffraction analysis comprises utilizing a rigorous coupled wave analysis.

13. The method of claim 12, wherein the symmetry properties comprise symmetry properties of the Fourier expansions of the rigorous coupled wave (RCW) analysis for the second azimuthal angle, allowing RCW eigen- and boundary problems to be reduced in complexity.

14. The method of claim 13, wherein a second azimuthal angle boundary problem is reduced to a 4(N+1)×4(N+1) system of equations and an eigen-problem is reduced to two (N+1)×(N+1) eigen-systems for a given truncation order, N.

15. The method of claim 14, where each of the various matrices of the eigenproblems are reduced according to one of
1) ε0,0 for i=m=0, (ε0,m+ε0,−m) for i=0 and m>0, 1⁄2(εi,0+ε−i,0) for i>0 and m=0, and 1⁄2(εi,m+ε−i,−m+εi,−m+ε−i,m) for i,m>0,
2) ε0,0 for i=m=0, (ε0,m−ε0,−m) for i=0 and m>0, 1⁄2(εi,0−ε−i,0) for i>0 and m=0, and 1⁄2(εi,m+ε−i,−m−ε−i,m−εi,−m) for i,m>0,
3) ε0,0 for i=m=0, (ε0,m−ε0,−m) for i=0 and m>0, 1⁄2(εi,0+ε−i,0) for i>0 and m=0, and 1⁄2(εi,m+ε−i,m−εi,−m−ε−i,−m) for i,m>0, or
4) ε0,0 for i=m=0, (ε0,m+ε0,−m) for i=0 and m>0, 1⁄2(εi,0−ε−i,0) for i>0 and m=0, and 1⁄2(εi,m+εi,−m−ε−i,m−ε−i,−m) for i,m>0,

depending on the symmetry conditions obeyed by the original, unreduced coupled equations, and where εi,m, −N≦i,m≦N, are the elements of the unreduced matrices.

16. The method of claim 14, where one of the eigenproblem matrices is reduced according to ε0,0 for i=m=0, (ε0,m+ε0,−m) for i=0 and m>0, 1⁄2(εi,0+ε−i,0) for i>0 and m=0, and 1⁄2(εi,m+ε−i,−m+εi,−m+ε−i,m) for i,m>0, and the second matrix is reduced according to γ0,0 for i=m=0, (γ0,m−γ0,−m) for i=0 and m>0, 1⁄2(γi,0−γ−i,0) for i>0 and m=0, and 1⁄2(γi,m+γ−i,−m−γ−i,m−γi,−m) for i,m>0, where εi,m and γi,m, −N≦i,m≦N, are the elements of the unreduced matrices.

17. The method of claim 16, where the first matrix is BE and the second matrix is E for s incident polarization, and vice versa for p incident polarization, resulting in reduced eigenproblems Ky2+(BE)reduced and Ky2+Kx2−Ereduced, where Ky2 and Kx2 are diagonal matrices with elements consisting of the 0, . . . , N elements of the unreduced diagonal matrices.

18. The method of claim 16, where the first matrix is BEinv−1 and the second matrix is E for s incident polarization, and vice versa for p incident polarization, resulting in reduced eigenproblems Ky2+(BEinv−1)reduced and Ky2+Kx2−Ereduced, where Ky2 and Kx2 are diagonal matrices with elements consisting of the 0, . . . , N elements of the unreduced diagonal matrices.