PATENT ABSTRACT
The above and other needs are met by a method of determining actual properties of a film stack by directing an incident beam of light towards the film stack, such that the incident beam of light is reflected from the film stack as a reflected beam of light. The actual properties of the reflected beam of light are measured, and properties of the film stack are estimated. A mathematical model of the film stack is solved with the estimated properties of the film stack, to yield theoretical properties of the reflected beam of light. The mathematical model is solved in part using a fast Z-matrix algorithm. The theoretical properties of the reflected beam of light are compared to the actual properties of the reflected beam of light, to yield a cost function. The estimated properties of the film stack are iteratively adjusted, and the mathematical model is iteratively solved, until the cost function is within a desired tolerance. The estimated properties of the film stack are reported as the actual properties of the film stack.

PATENT DESCRIPTION
FIELD 
   This invention relates to the field of film measurement. More particularly, this invention relates to a system for improving the speed and accuracy of multi layered film stack thickness measurement and other property measurement, such as in the integrated circuit fabrication industry. 
   BACKGROUND 
   Integrated circuits are formed of many layers of different materials, which layers are patterned so as to form desired structures that interact with one another according to predetermined designs. Thus, it is of vital importance that many of these layers be formed to very exacting tolerances, such as in their shape, thickness, and composition. If the various structures so formed during the integrated circuit fabrication process are not precisely formed, then the integrated circuit tends to not function in the intended manner, and may not function at all. 
   As the term is used herein, “integrated circuit” includes devices such as those formed on monolithic semiconducting substrates, such as those formed of group IV materials like silicon or germanium, or group III-V compounds like gallium arsenide, or mixtures of such materials. The term includes all types of devices formed, such as memory and logic, and all designs of such devices, such as MOS and bipolar. The term also comprehends applications such as flat panel displays, solar cells, and charge coupled devices. 
   Because the layers of which integrated circuits are formed are so thin and patterned to be so small, they cannot be inspected without the aid of instrumentation. The precision of the instrumentation used is, therefore, vital to the successful production of integrated circuits. Thus, any improvement that can be made in the accuracy of such instrumentation is a boon to the integrated circuit fabrication industry. In addition, any improvement in the speed at which such instrumentation can take its readings is also of benefit to the industry, as such speed enhancements tend to reduce the production bottlenecks at inspection steps, or alternately allow for the inspection of a greater number of integrated circuits at such inspection steps. 
   Spectral ellipsometers and dual beam spectrophotometers are typically used to measure properties such as thickness and refractive index of individual layers within a multilayered film stack. Such instruments work by directing one or more beams of light toward the surface of the film stack, and then sensing the properties of the light as it is variously reflected off of the different surfaces of the individual layers within the film stack. By adjusting the properties of the incident beam of light, and detecting the associated changes in the reflected beam of light, the properties of the film stack, such as the materials of which the various layers are formed and the thicknesses to which they are formed, can be determined. Such methods typically involve solving Maxwell&#39;s equations, which provide a model for such systems. 
   This film measurement process can be broken down into two basic steps, being 1) the measurement of the properties of the reflected light beam, and 2) the mathematical fitting of reflectance property values from Maxwell&#39;s equations, which are solved or estimated, to the measured results attained in step 1. Step 2 typically consists of the iterated steps of computing one or more theoretical values by plugging estimates of the film stack parameters, such as thickness and refractive index, into the model film stack equations, comparing the theoretical values obtained to the actual measured property values of the reflected beam of light, and if the theoretical values and the measured values do not agree to within a desired tolerance, then adjusting the estimated film stack parameters and recomputing the theoretical values. 
   This process is performed again and again, each time making some adjustment to the estimated film stack parameters that are fed into the model, until the theoretical values computed by the model agree with the actual measured values within the desired precision limits. When this agreement is attained, then there is some confidence that the estimated film stack parameters that were used to produce the theoretical values are very nearly the same as the actual film stack parameters. 
   For film stacks containing non-isotropic, inhomogeneous layers, such as layers containing metal patterns or gratings, the theoretical values are commonly generated by mathematical models based on either the rigorous coupled wave analysis or modal method. 
   The reflectance equations that are derived for film stacks that generate multiple plane waves traveling at different angles are typically formulated using at least one of the S matrix algorithm or the R matrix algorithm, as understood by those with skill in the art. However, these algorithms are computationally burdensome, and require a longer than desirable period of time—and greater than desirable computational resources—to solve. Another well-known alternate formulation for computing the reflectance is the T matrix algorithm. Although this algorithm is more computationally efficient, it is highly susceptible to computational overflow error and is therefore seldom used in practical applications. 
   What is needed is a method to improve the efficiency of the electromagnetic field computation of multilayer structures containing patterned dielectric structures. 
   SUMMARY 
   The above and other needs are met by a method of determining actual properties of a film stack by directing an incident beam of light towards the film stack, such that the incident beam of light is reflected from the film stack as a reflected beam of light. The actual properties of the reflected beam of light are measured, and properties of the film stack are estimated. A mathematical model of the film stack is solved with the estimated properties of the film stack, to yield theoretical properties of the reflected beam of light. The mathematical model is solved in part using a fast Z-matrix algorithm. The theoretical properties of the reflected beam of light are compared to the actual properties of the reflected beam of light, to yield a cost function. The estimated properties of the film stack are iteratively adjusted, and the mathematical model is iteratively solved, until the cost function is within a desired tolerance. The estimated properties of the film stack are reported as the actual properties of the film stack. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further advantages of the invention are apparent by reference to the detailed description when considered in conjunction with the FIGURE, which is not to scale so as to more clearly show the details, wherein like reference numbers indicate like elements, and wherein is depicted a multi-layered film stack as both a non-inverted and an inverted film stack. 
   

   DETAILED DESCRIPTION 
   The purpose of this invention is to improve the efficiency of the spectral ellipsometer (SE) and the dual-beam spectrophotometer (DBS and polarized DBS) that measure thickness and material parameters of multilayered film stacks containing patterned metal and non-metal lines, with or without using precomputed libraries. 
   The various embodiments of this invention make it possible for a fast Z-Matrix method to compute the total electromagnetic fields in any arbitrary layer, while incurring negligible computational overhead. The embodiments of the invention improve the efficiency of computing the fields inside a film layer relative to the conventionally used S-Matrix method by about four times. This invention can be implemented as an add-in to existing instrument software. 
   The unique features of this invention include: 1) the definition of a novel field transmittance that allows us to efficiently compute the transmittance, and 2) the judicious ordering of matrix-vector multiplications that eliminate O(N 3 ) overhead. 
   The embodiments of the present invention differ from conventional methods in that we have developed a more efficient method of computing the S-Matrix. Of note, we use a variant of the fast Z Matrix algorithm to compute the S-Matrix. By doing so, we realize two important advantages over the conventional S-Matrix computation algorithm. The first is that we can reduce the computational time by about 20-35%, when the number of layers is greater than four. The second—and possibly more significant—advantage is reduced code management and maintenance. That is, we can leverage existing software infrastructure based on the highly efficient fast Z-Matrix algorithm and, with minor modification, make the same code as computationally efficient as the S-Matrix for special film stack configurations and library generation. Details of the S-Matrix computation using a fast Z-Matrix algorithm are given below. 
   Two-Dimensional RCWA Formulation 
   Maxwell&#39;s equations (uniaxial media):
 
 ∇xh=jωε   0     ε e   (1a)
 
 ∇xe=−jωμ   0     μ h   (1b)
 
   where e=electric field, h=magnetic field, 
               ɛ   _     _     =     [         ɛ       0       0           0       ɛ       0           0       0         ɛ   z           ]           
=relative electric permittivity tensor,
 
               μ   _     _     =     [         μ       0       0           0       μ       0           0       0         μ   z           ]           
=relative magnetic permeability tensor, (ε 0 ,μ 0 )=electric permittivity and magnetic permeability of free space, respectively.
 
   Assume the dielectric and magnetic permittivity has a periodic distribution in x and y directions of a Cartesian coordinate system. Mathematically, we can write 
   
     
       
         
           
             
               
                 A 
                 = 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       
                         - 
                         N 
                       
                     
                     N 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         
                           - 
                           M 
                         
                       
                       M 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         a 
                         
                           m 
                           , 
                           n 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               m 
                             
                             
                               P 
                               x 
                             
                           
                           ⁢ 
                           x 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               n 
                             
                             
                               P 
                               y 
                             
                           
                           ⁢ 
                           y 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   where A=Fourier transform representation of the distribution v, where: 
           υ   ⁢           ∈     (     ɛ   ,   μ   ,     ɛ   z     ,     μ   z     ,     1   ɛ     ,     1   μ     ,     1     ɛ   z       ,     1     μ   z         )           
P x , P y =period in x, y directions, respectively,
 
(M,N)=number of positive and negative modes in the x and y directions, respectively.
 
   An incident plane wave solution to equations (1) is given by
 
 e   inc   , h   inc   ∝e   ±jk     0     k·r   (3)
 
where k=k x0 {circumflex over (x)}+k y0 ŷ+k z0 {circumflex over (z)}=wave vector, r=x{circumflex over (x)}+yŷ+z{circumflex over (z)}.
 
   Phase matching at the boundary requires that the electric and magnetic fields are expressible as a superposition of Floquet harmonics, i.e., 
   
     
       
         
           
             
               
                 
                   f 
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         
                           - 
                           N 
                         
                       
                       N 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           
                             - 
                             M 
                           
                         
                         M 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             f 
                             
                               m 
                               , 
                               n 
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               j 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       m 
                                     
                                     
                                       P 
                                       x 
                                     
                                   
                                   - 
                                   
                                     k 
                                     xo 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             x 
                           
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               j 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                     
                                     
                                       P 
                                       y 
                                     
                                   
                                   ⁢ 
                                   
                                     k 
                                     yo 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             y 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           where 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   f 
                   ⁢ 
                   
                       
                   
                   ∈ 
                   
                     { 
                     
                       
                         e 
                         x 
                       
                       , 
                       
                         e 
                         y 
                       
                       , 
                       
                         e 
                         z 
                       
                       , 
                       
                         h 
                         x 
                       
                       , 
                       
                         h 
                         y 
                       
                       , 
                       
                         h 
                         z 
                       
                     
                     } 
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   For purposes of keeping track of the cross-modal interactions, it is convenient to remove the explicit dependence on the Fourier and Floquet harmonics and rewrite equations (3) and (4) as a matrix and (5) as a column vector, i.e., 
   
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             A 
                             0 
                             ′ 
                           
                         
                         
                           
                             A 
                             
                               - 
                               1 
                             
                             ′ 
                           
                         
                         
                           … 
                         
                         
                           
                             A 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               N 
                             
                             ′ 
                           
                         
                       
                       
                         
                           
                             A 
                             1 
                             ′ 
                           
                         
                         
                           
                             A 
                             0 
                             ′ 
                           
                         
                         
                           … 
                         
                         
                           
                             A 
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     N 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             ′ 
                           
                         
                       
                       
                         
                           . 
                         
                         
                           . 
                         
                         
                           … 
                         
                         
                           . 
                         
                       
                       
                         
                           
                             A 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               N 
                             
                             ′ 
                           
                         
                         
                           
                             A 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 N 
                               
                               - 
                               1 
                             
                             ′ 
                           
                         
                         
                           … 
                         
                         
                           
                             A 
                             0 
                             ′ 
                           
                         
                       
                     
                     ] 
                   
                 
                 , 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             f 
                             
                               - 
                               N 
                             
                           
                         
                       
                       
                         
                           
                             f 
                             
                               - 
                               
                                 ( 
                                 
                                   N 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           . 
                         
                       
                       
                         
                           
                             f 
                             N 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   where the subscripts correspond to the Floquet harmonic numbers and the matrix elements of equation (5) are themselves matrices and vectors, i.e., 
   
     
       
         
           
             
               
                 
                   
                     A 
                     n 
                     ′ 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             a 
                             
                               0 
                               , 
                               n 
                             
                           
                         
                         
                           
                             a 
                             
                               
                                 - 
                                 1 
                               
                               , 
                               n 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             a 
                             
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 M 
                               
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             a 
                             
                               1 
                               , 
                               n 
                             
                           
                         
                         
                           
                             a 
                             
                               0 
                               , 
                               n 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             a 
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       M 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           . 
                         
                         
                           . 
                         
                         
                           … 
                         
                         
                           . 
                         
                       
                       
                         
                           
                             a 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 M 
                               
                               , 
                               n 
                             
                           
                         
                         
                           
                             a 
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   M 
                                 
                                 - 
                                 1 
                               
                               , 
                               n 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             a 
                             
                               0 
                               , 
                               n 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 , 
                 
                     
                 
                 ⁢ 
                 
                   
                     f 
                     n 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             f 
                             
                               
                                 - 
                                 M 
                               
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             f 
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     M 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           . 
                         
                       
                       
                         
                           
                             f 
                             
                               M 
                               , 
                               n 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   We note that the ordering of the elements of equations (5) and (6) is arbitrary provided we maintain 1) the correspondence between the vector elements and the Floquet harmonics, and 2) the correspondence between the matrix elements and the Fourier harmonics. For one dimensional periodicity (e.g., x periodic, y constant) N=0, and equations (5) and (6) reduce to A=A′ 0  and f=f 0 . 
   Expanding equation (1) yields: 
   
     
       
         
           
             
               
                 
                   e 
                   x 
                 
                 = 
                 
                   
                     ( 
                     
                       1 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ɛ 
                           0 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ∂ 
                             
                               h 
                               z 
                             
                           
                           
                             ∂ 
                             y 
                           
                         
                         - 
                         
                           
                             ∂ 
                             
                               h 
                               y 
                             
                           
                           
                             ∂ 
                             z 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   6 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 
                   e 
                   y 
                 
                 = 
                 
                   
                     ( 
                     
                       1 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ɛ 
                           0 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ∂ 
                             
                               h 
                               x 
                             
                           
                           
                             ∂ 
                             z 
                           
                         
                         - 
                         
                           
                             ∂ 
                             
                               h 
                               z 
                             
                           
                           
                             ∂ 
                             x 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   6 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
           
             
               
                 
                   e 
                   z 
                 
                 = 
                 
                   
                     ( 
                     
                       1 
                       
                         jωɛ 
                         0 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ∂ 
                             
                               h 
                               y 
                             
                           
                           
                             ∂ 
                             x 
                           
                         
                         - 
                         
                           
                             ∂ 
                             
                               h 
                               x 
                             
                           
                           
                             ∂ 
                             y 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   6 
                   ⁢ 
                   c 
                 
                 ) 
               
             
           
           
             
               
                 
                   h 
                   x 
                 
                 = 
                 
                   
                     ( 
                     
                       1 
                       
                         jωμ 
                         o 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       A 
                       
                         1 
                         / 
                         μ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           - 
                           
                             
                               ∂ 
                               
                                 e 
                                 z 
                               
                             
                             
                               ∂ 
                               y 
                             
                           
                         
                         + 
                         
                           
                             ∂ 
                             
                               e 
                               y 
                             
                           
                           
                             ∂ 
                             z 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   6 
                   ⁢ 
                   d 
                 
                 ) 
               
             
           
           
             
               
                 
                   h 
                   y 
                 
                 = 
                 
                   
                     ( 
                     
                       1 
                       
                         jωμ 
                         o 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       A 
                       
                         1 
                         / 
                         μ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ∂ 
                             
                               e 
                               x 
                             
                           
                           
                             ∂ 
                             z 
                           
                         
                         + 
                         
                           
                             ∂ 
                             
                               e 
                               z 
                             
                           
                           
                             ∂ 
                             x 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   6 
                   ⁢ 
                   e 
                 
                 ) 
               
             
           
           
             
               
                 
                   h 
                   z 
                 
                 = 
                 
                   
                     - 
                     
                       1 
                       
                         jωμ 
                         o 
                       
                     
                   
                   ⁢ 
                   
                     
                       A 
                       
                         1 
                         / 
                         
                           μ 
                           z 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ∂ 
                             
                               ⅇ 
                               y 
                             
                           
                           
                             ∂ 
                             x 
                           
                         
                         - 
                         
                           
                             ∂ 
                             
                               e 
                               x 
                             
                           
                           
                             ∂ 
                             y 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   6 
                   ⁢ 
                   f 
                 
                 ) 
               
             
           
         
       
     
   
   where the subscript of A denotes the permittivity and permeability distribution function being approximated by the Fourier transform. Substituting equation (6f) into equation (1) yields: 
   
     
       
         
           
             
               
                 
                   
                     e 
                     x 
                   
                   - 
                   
                     
                       1 
                       
                         k 
                         o 
                         2 
                       
                     
                     ⁢ 
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                     
                     ⁢ 
                     
                       ∂ 
                       
                         ∂ 
                         y 
                       
                     
                     ⁢ 
                     
                       
                         A 
                         
                           1 
                           / 
                           
                             μ 
                             z 
                           
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ∂ 
                               
                                 e 
                                 y 
                               
                             
                             
                               ∂ 
                               x 
                             
                           
                           - 
                           
                             
                               ∂ 
                               
                                 e 
                                 x 
                               
                             
                             
                               ∂ 
                               y 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       jωμ 
                       o 
                     
                     
                       k 
                       o 
                       2 
                     
                   
                   ⁢ 
                   
                     A 
                     
                       1 
                       / 
                       ɛ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         h 
                         y 
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                 
               
             
             
               
                 ( 
                 7 
                 ) 
               
             
           
         
       
     
   
   Recalling the implicit dependence of e and h on the Floquet harmonics, i.e., equation (4), substituting derivatives with the appropriate diagonal matrices yields: 
   
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         I 
                         - 
                         
                           
                             A 
                             
                               1 
                               / 
                               ɛ 
                             
                           
                           ⁢ 
                           
                             K 
                             y 
                           
                           ⁢ 
                           
                             A 
                             
                               1 
                               / 
                               
                                 μ 
                                 z 
                               
                             
                           
                           ⁢ 
                           
                             K 
                             y 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       e 
                       x 
                     
                   
                   + 
                   
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                     
                     ⁢ 
                     
                       K 
                       y 
                     
                     ⁢ 
                     
                       A 
                       
                         1 
                         / 
                         
                           μ 
                           z 
                         
                       
                     
                     ⁢ 
                     
                       K 
                       x 
                     
                     ⁢ 
                     
                       e 
                       y 
                     
                   
                 
                 = 
                 
                   
                     
                       jωμ 
                       o 
                     
                     
                       k 
                       o 
                       2 
                     
                   
                   ⁢ 
                   
                     A 
                     
                       1 
                       / 
                       ɛ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         h 
                         y 
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                 
               
             
             
               
                 ( 
                 8 
                 ) 
               
             
           
         
       
     
   
   Interchanging the x,y components and changing the sign to derive the x component yields: 
   
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         ( 
                         
                           I 
                           - 
                           
                             
                               A 
                               
                                 1 
                                 / 
                                 ɛ 
                               
                             
                             ⁢ 
                             
                               K 
                               x 
                             
                             ⁢ 
                             
                               A 
                               
                                 1 
                                 / 
                                 
                                   μ 
                                   z 
                                 
                               
                             
                             ⁢ 
                             
                               K 
                               x 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       e 
                       y 
                     
                   
                   - 
                   
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                     
                     ⁢ 
                     
                       K 
                       x 
                     
                     ⁢ 
                     
                       A 
                       
                         1 
                         / 
                         
                           μ 
                           z 
                         
                       
                     
                     ⁢ 
                     
                       K 
                       y 
                     
                     ⁢ 
                     
                       e 
                       x 
                     
                   
                 
                 = 
                 
                   
                     
                       jωμ 
                       o 
                     
                     
                       k 
                       o 
                       2 
                     
                   
                   ⁢ 
                   
                     A 
                     
                       1 
                       / 
                       ɛ 
                     
                   
                   ⁢ 
                   
                     
                       ∂ 
                       
                         h 
                         x 
                       
                     
                     
                       ∂ 
                       z 
                     
                   
                 
               
             
             
               
                 ( 
                 9 
                 ) 
               
             
           
         
       
     
   
   Defining the tangential field vectors as: 
                     e   t     =     [           e   x               e   y           ]       ,           ⁢       h   t     =     [           h   x               h   y           ]               (   10   )               
Therefore:
 
                     A   h     ′   -   1       ⁢     e   t       =         jωμ   o       k   o   2       ⁢       ∂     h   t         ∂   z                 (     11   ⁢   a     )               
where
 
                   A   h     ′   -   1       =         A     1   /   ɛ       -   1       ⁡     [             -     A     1   /   ɛ         ⁢     K   x     ⁢     A     1   /     μ   z         ⁢     K   y             (       -   I     +       A     1   /   ɛ       ⁢     K   x     ⁢     A     1   /     μ   z         ⁢     K   x         )               (     I   -       A     1   /   ɛ       ⁢     K   y     ⁢     A     1   /     μ   z         ⁢     K   y         )             A     1   /   ɛ       ⁢     K   y     ⁢     A     1   /     μ   z         ⁢     K   x             ]       .             (     11   ⁢   b     )               
From E,H duality:
 
                     A   e     ′   -   1       ⁢     h   t       =       -       jωɛ   o       k   o   2         ⁢       ∂     e   t         ∂   z                 (     12   ⁢   a     )               
where
 
   
     
       
         
           
             
               
                 
                   A 
                   e 
                   
                     ′ 
                     - 
                     1 
                   
                 
                 = 
                 
                   
                     
                       A 
                       
                         1 
                         / 
                         μ 
                       
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 - 
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     μ 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   ɛ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   - 
                                   I 
                                 
                                 + 
                                 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       μ 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     x 
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       ɛ 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     x 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 I 
                                 - 
                                 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       μ 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     y 
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       
                                         ɛ 
                                         z 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 A 
                                 
                                   1 
                                   / 
                                   μ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 
                   12 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
         
       
     
   
   Substituting equation (11) into equation (10) to form a differential equation yields: 
   
     
       
         
           
             
               
                 
                   
                     S 
                     h 
                     2 
                   
                   ⁢ 
                   
                     h 
                     t 
                   
                 
                 = 
                 
                   
                     
                       A 
                       h 
                       
                         ′ 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       A 
                       e 
                       
                         ′ 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       h 
                       t 
                     
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         
                           k 
                           o 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         ⁢ 
                         
                           h 
                           t 
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 13 
                 ) 
               
             
           
         
       
     
   
   The general solutions to (13) include:
 
 h   t   =e   jk     0     S     h     z   d   +   +e   −jk     0     S     h     z   d   —   (14)
 
   From (12a) we see that: 
   
     
       
         
           
             
               
                 
                   e 
                   t 
                 
                 = 
                 
                   
                     
                       
                         jωμ 
                         0 
                       
                       
                         k 
                         o 
                         2 
                       
                     
                     ⁢ 
                     
                       A 
                       h 
                       ′ 
                     
                     ⁢ 
                     j 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       o 
                     
                     ⁢ 
                     
                       
                         S 
                         h 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   h 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               d 
                               + 
                             
                           
                           - 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   h 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               d 
                               - 
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             μ 
                             0 
                           
                         
                         
                           k 
                           o 
                         
                       
                     
                     ⁢ 
                     
                       
                         η 
                         
                           h 
                           , 
                           n 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   h 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               d 
                               + 
                             
                           
                           - 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   h 
                                 
                                 ⁢ 
                                 z 
                               
                             
                             ⁢ 
                             
                               d 
                               - 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
   
   where η h =A′ h S h =intrinsic impedance. 
   Using duality we can obtain a parallel set of equations in terms of e t , i.e.: 
   
     
       
         
           
             
               
                 
                   
                     
                       S 
                       e 
                       2 
                     
                     ⁢ 
                     
                       e 
                       t 
                     
                   
                   = 
                   
                     
                       
                         
                           A 
                           ′ 
                         
                         e 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         A 
                         h 
                         
                           ′ 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         e 
                         t 
                       
                     
                     = 
                     
                       
                         - 
                         
                           1 
                           
                             k 
                             o 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             2 
                           
                           ⁢ 
                           
                             e 
                             t 
                           
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     e 
                     t 
                   
                   = 
                   
                     
                       
                         ⅇ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             S 
                             e 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         d 
                         + 
                       
                     
                     + 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             S 
                             e 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         d 
                         - 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
           
             
               
                 
                   h 
                   t 
                 
                 = 
                 
                   
                     
                       ωɛ 
                       0 
                     
                     
                       k 
                       o 
                     
                   
                   ⁢ 
                   
                     
                       η 
                       
                         e 
                         , 
                         n 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             ⅇ 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               
                                 S 
                                 e 
                               
                               ⁢ 
                               z 
                             
                           
                           ⁢ 
                           
                             d 
                             + 
                           
                         
                         - 
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 j 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               
                                 s 
                                 e 
                               
                               ⁢ 
                               z 
                             
                           
                           ⁢ 
                           
                             d 
                             - 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
         
       
     
   
   where η e =A′ e S e =intrinsic impedance. 
   By simplifying for non-magnetic media A 1/μ =A 1/μ     z   =I, i.e.: 
   
     
       
         
           
             
               
                 
                   S 
                   h 
                   2 
                 
                 = 
                 
                   
                     
                       A 
                       
                         1 
                         / 
                         ɛ 
                       
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 - 
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     ɛ 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     μ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   - 
                                   I 
                                 
                                 + 
                                 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       ɛ 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     x 
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       
                                         μ 
                                         z 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     x 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 I 
                                 - 
                                 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       ɛ 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     y 
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       
                                         μ 
                                         z 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 A 
                                 
                                   1 
                                   / 
                                   ɛ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     μ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               
                                 - 
                                 
                                   K 
                                   x 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   - 
                                   I 
                                 
                                 + 
                                 
                                   
                                     K 
                                     x 
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       
                                         ɛ 
                                         z 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     x 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 I 
                                 - 
                                 
                                   
                                     K 
                                     y 
                                   
                                   ⁢ 
                                   
                                     A 
                                     
                                       1 
                                       / 
                                       
                                         ɛ 
                                         z 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   and expanding equation (16) produces: 
   
     
       
         
           
             
               
                 
                   S 
                   h 
                   2 
                 
                 = 
                 
                   
                     A 
                     
                       1 
                       / 
                       ɛ 
                     
                     
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 A 
                                 
                                   1 
                                   / 
                                   ɛ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                                 2 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     - 
                                     I 
                                   
                                   + 
                                   
                                     
                                       A 
                                       
                                         1 
                                         / 
                                         ɛ 
                                       
                                     
                                     ⁢ 
                                     
                                       K 
                                       x 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   I 
                                   - 
                                   
                                     
                                       K 
                                       y 
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         1 
                                         / 
                                         
                                           ɛ 
                                           z 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       K 
                                       y 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     ɛ 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                               ⁢ 
                               
                                 
                                   K 
                                   y 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       - 
                                       I 
                                     
                                     + 
                                     
                                       
                                         K 
                                         x 
                                       
                                       ⁢ 
                                       
                                         A 
                                         
                                           1 
                                           / 
                                           
                                             ɛ 
                                             z 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         K 
                                         x 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     - 
                                     I 
                                   
                                   + 
                                   
                                     
                                       A 
                                       
                                         1 
                                         / 
                                         ɛ 
                                       
                                     
                                     ⁢ 
                                     
                                       K 
                                       x 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   ( 
                                   
                                     I 
                                     - 
                                     
                                       
                                         A 
                                         
                                           1 
                                           / 
                                           ɛ 
                                         
                                       
                                       ⁢ 
                                       
                                         K 
                                         y 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                             + 
                             
                                 
                             
                             ⁢ 
                             
                                 
                               
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     ɛ 
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   y 
                                 
                                 ⁢ 
                                 
                                   
                                     K 
                                     x 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       I 
                                       - 
                                       
                                         
                                           K 
                                           y 
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             1 
                                             / 
                                             
                                               ɛ 
                                               z 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           K 
                                           y 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                     
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   I 
                                   - 
                                   
                                     
                                       A 
                                       
                                         1 
                                         / 
                                         ɛ 
                                       
                                     
                                     ⁢ 
                                     
                                       K 
                                       y 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     - 
                                     I 
                                   
                                   + 
                                   
                                     
                                       K 
                                       x 
                                     
                                     ⁢ 
                                     
                                       A 
                                       
                                         1 
                                         / 
                                         ɛ 
                                       
                                     
                                     ⁢ 
                                     
                                       K 
                                       x 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 A 
                                 
                                   1 
                                   / 
                                   ɛ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                                 2 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 19 
                 ) 
               
             
           
           
             
               
                 
                   S 
                   h 
                   2 
                 
                 = 
                 
                   
                     A 
                     
                       1 
                       / 
                       ɛ 
                     
                     
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 I 
                               
                               + 
                               
                                 
                                   K 
                                   y 
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     
                                       ɛ 
                                       z 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   y 
                                 
                               
                               + 
                               
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     ɛ 
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   x 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         
                           
                             
                               
                                 A 
                                 
                                   1 
                                   / 
                                   ɛ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                             - 
                             
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   K 
                                   x 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   1 
                                   / 
                                   
                                     ɛ 
                                     z 
                                   
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                             
                             + 
                             
                               
                                 A 
                                 
                                   1 
                                   / 
                                   ɛ 
                                 
                               
                               ⁢ 
                               
                                 K 
                                 y 
                               
                               ⁢ 
                               
                                 K 
                                 x 
                               
                             
                           
                         
                         
                           
                             ( 
                             
                               
                                 - 
                                 I 
                               
                               + 
                               
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     ɛ 
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   y 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   K 
                                   x 
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     1 
                                     / 
                                     
                                       ɛ 
                                       z 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   x 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
         
       
     
   
   A similar simplification can be found for S e   2  using duality. 
   Fast Z-Matrix Algorithm 
   The goal is to derive an algorithm based on field impedance (admittance) to compute the reflectance, transmittance, and fields inside a multi-layered film stack. 
   Derivation 
   2.1 Recursive Impedance Computation 
   Starting from: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           h 
                           
                             t 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           
                             I 
                           
                           
                             I 
                           
                         
                         
                           
                             
                               η 
                               
                                 h 
                                 , 
                                 n 
                               
                             
                           
                           
                             
                               - 
                               
                                 η 
                                 
                                   h 
                                   , 
                                   n 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             d 
                             
                               + 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                       
                         
                           
                             d 
                             
                               - 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 1 
                 ) 
               
             
           
         
       
     
   
   where 
   
     
       
         
           
             
               
                 
                   
                     e 
                     
                       t 
                       , 
                       n 
                     
                     ′ 
                   
                   = 
                   
                     
                       
                         k 
                         o 
                       
                       
                         ωμ 
                         0 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         t 
                         , 
                         n 
                       
                     
                   
                 
                 , 
                 
                     
                 
                 ⁢ 
                 and 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             h 
                             
                               t 
                               , 
                               n 
                             
                           
                         
                       
                       
                         
                           
                             e 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 d 
                                 
                                   + 
                                   
                                     , 
                                     n 
                                   
                                 
                               
                               + 
                               
                                 d 
                                 
                                   - 
                                   
                                     , 
                                     n 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 η 
                                 
                                   h 
                                   , 
                                   n 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     d 
                                     
                                       + 
                                       
                                         , 
                                         n 
                                       
                                     
                                   
                                   - 
                                   
                                     d 
                                     
                                       - 
                                       
                                         , 
                                         n 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁢ 
                                 
                                   d 
                                   
                                     + 
                                     
                                       , 
                                       n 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁢ 
                                 
                                   d 
                                   
                                     - 
                                     
                                       , 
                                       n 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 η 
                                 
                                   h 
                                   , 
                                   n 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       ⅇ 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           S 
                                           
                                             h 
                                             , 
                                             n 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ⁢ 
                                     
                                       d 
                                       
                                         + 
                                         
                                           , 
                                           n 
                                         
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           S 
                                           
                                             h 
                                             , 
                                             n 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ⁢ 
                                     
                                       d 
                                       
                                         - 
                                         
                                           , 
                                           n 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
     
   
   find d ±,n  in terms of tangential fields at the bottom of the layer z=0, i.e., h t,n−1, , e t,n−1 . We have:
 
 d   +,n   +d   −,n   =h   t,n−1   (4)
 
 d   +,n   −d   −,n =η h,n   −1   e′   t.n−1   (5)
 
Therefore:
 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           d 
                           
                             + 
                             
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           d 
                           
                             - 
                             
                               , 
                               n 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               
                                 - 
                                 
                                   η 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               I 
                             
                             
                               
                                 η 
                                 
                                   h 
                                   , 
                                   n 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               h 
                               
                                 t 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               
                                 t 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ′ 
                             
                           
                         
                       
                       ] 
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   Relate n th  and (n−1) th  layer fields, i.e.: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           h 
                           
                             t 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 I 
                               
                               
                                 I 
                               
                             
                             
                               
                                 
                                   η 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     η 
                                     
                                       h 
                                       , 
                                       n 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     S 
                                     
                                       h 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     S 
                                     
                                       h 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             I 
                           
                           
                             
                               η 
                               
                                 h 
                                 , 
                                 n 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             I 
                           
                           
                             
                               - 
                               
                                 η 
                                 
                                   h 
                                   , 
                                   n 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             h 
                             
                               t 
                               , 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             e 
                             
                               t 
                               , 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                             ′ 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 
                   7 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
         
       
     
   
   Recall η h,n =A′ h,n S h,n , 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           h 
                           
                             t 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     1 
                     2 
                   
                   ⁢ 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 W 
                                 n 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   A 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   W 
                                   n 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ︸ 
                     
                     
                       W 
                       A 
                     
                   
                   ⁢ 
                   
                     
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               I 
                             
                           
                           
                             
                               
                                 K 
                                 
                                   z 
                                   , 
                                   n 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   K 
                                   
                                     z 
                                     , 
                                     n 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ︸ 
                     
                     P 
                   
                   ⁢ 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   Z 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ︸ 
                     
                     
                       
                         K 
                         ph 
                       
                       = 
                       phasematrix 
                     
                   
                   ⁢ 
                   
                       
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   I 
                                 
                                 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                               
                               
                                 
                                   I 
                                 
                                 
                                   
                                     - 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ︸ 
                         
                         
                           P 
                           
                             - 
                             1 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         W 
                                         n 
                                         
                                           - 
                                           1 
                                         
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         
                                           W 
                                           n 
                                           
                                             - 
                                             1 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           
                                             h 
                                             , 
                                             n 
                                           
                                           
                                             ′ 
                                             - 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               ︸ 
                             
                             
                               W 
                               A 
                               
                                 - 
                                 1 
                               
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     h 
                                     
                                       t 
                                       , 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     e 
                                     
                                       t 
                                       , 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     ′ 
                                   
                                 
                               
                             
                             ] 
                           
                         
                         . 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   7 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
         
       
     
   
   Define an impedance (admittance), i.e., 
             [           h   t               e   t   ′           ]     =       [           Y   ⁢           ⁢     e   t                 e   t   ′           ]     .           
Therefore, we can relate Y n  and Y n−1 , i.e.:
 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Y 
                             
                               n 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             e 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               W 
                               n 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 A 
                                 
                                   h 
                                   , 
                                   n 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 W 
                                 n 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                       
                     
                       
                         
                           [ 
                           
                             
                               
                                 I 
                               
                               
                                 I 
                               
                             
                             
                               
                                 
                                   K 
                                   
                                     z 
                                     , 
                                     n 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                               
                               
                                 
                                   
                                     ⅇ 
                                     
                                       j 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         k 
                                         o 
                                       
                                       ⁢ 
                                       
                                         K 
                                         
                                           z 
                                           , 
                                           n 
                                         
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           K 
                                           
                                             z 
                                             , 
                                             n 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   W 
                                   n 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   Y 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     t 
                                     , 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   W 
                                   n 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                   
                                     ′ 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     t 
                                     , 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   8 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
         
       
     
   
   Multiplying out: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Y 
                             n 
                           
                           ⁢ 
                           
                             e 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 W 
                                 n 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   A 
                                   
                                     h 
                                     , 
                                     n 
                                   
                                   ′ 
                                 
                                 ⁢ 
                                 
                                   W 
                                   n 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                               + 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     ⅇ 
                                     
                                       j 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         k 
                                         o 
                                       
                                       ⁢ 
                                       
                                         K 
                                         
                                           z 
                                           , 
                                           n 
                                         
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                   - 
                                   
                                     ⅇ 
                                     
                                       
                                         - 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         k 
                                         o 
                                       
                                       ⁢ 
                                       
                                         K 
                                         
                                           z 
                                           , 
                                           n 
                                         
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 K 
                                 
                                   z 
                                   , 
                                   n 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     ⅇ 
                                     
                                       j 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         k 
                                         o 
                                       
                                       ⁢ 
                                       
                                         K 
                                         
                                           z 
                                           , 
                                           n 
                                         
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                   - 
                                   
                                     ⅇ 
                                     
                                       
                                         - 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         k 
                                         o 
                                       
                                       ⁢ 
                                       
                                         K 
                                         
                                           z 
                                           , 
                                           n 
                                         
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 K 
                                 
                                   z 
                                   , 
                                   n 
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 + 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               
                                 W 
                                 n 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 Y 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               
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                                 n 
                                 
                                   - 
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                               ⁢ 
                               
                                 A 
                                 
                                   h 
                                   , 
                                   n 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   8 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               Y 
                               n 
                             
                             ⁢ 
                             
                               e 
                               
                                 t 
                                 , 
                                 n 
                               
                               ′ 
                             
                           
                         
                       
                       
                         
                           
                             e 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   W 
                                   n 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       ⅇ 
                                       
                                         j 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           K 
                                           
                                             z 
                                             , 
                                             n 
                                           
                                         
                                         ⁢ 
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                                     + 
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           K 
                                           
                                             z 
                                             , 
                                             n 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 
                                   
                                     W 
                                     n 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         ⅇ 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             K 
                                             
                                               z 
                                               , 
                                               n 
                                             
                                           
                                           ⁢ 
                                           z 
                                         
                                       
                                       - 
                                       
                                         ⅇ 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             K 
                                             
                                               z 
                                               , 
                                               n 
                                             
                                           
                                           ⁢ 
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                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   K 
                                   
                                     z 
                                     , 
                                     n 
                                   
                                   
                                     - 
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                                   A 
                                   
                                     h 
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                                     n 
                                   
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                                     ( 
                                     
                                       
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                                             o 
                                           
                                           ⁢ 
                                           
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                                               z 
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                                           ⁢ 
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                                       - 
                                       
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                                             - 
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                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
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                                               z 
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                                           ⁢ 
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                                     ) 
                                   
                                 
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                                     z 
                                     , 
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                                 ⁢ 
                                 
                                   
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                                     n 
                                   
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                                     ( 
                                     
                                       
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                                           j 
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                                           ⁢ 
                                           
                                             k 
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                                               z 
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                                               n 
                                             
                                           
                                           ⁢ 
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                                       + 
                                       
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                                             - 
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                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
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                                               z 
                                               , 
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                                           ⁢ 
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                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 W 
                                 n 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 Y 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 W 
                                 n 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 A 
                                 
                                   h 
                                   , 
                                   n 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
               
             
             
               
                 ( 
                 
                   8 
                   ⁢ 
                   c 
                 
                 ) 
               
             
           
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Y 
                             n 
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 t 
                                 &#39; 
                               
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             
                               t 
                               &#39; 
                             
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                     
                   
                       
                     
                         
                       
                           
                       
                       ⁢ 
                       
                         
                             
                             
                         
                         ⁢ 
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                             
                           
                               
                             
                               [ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       
                                         W 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     ⅇ 
                                                     
                                                       j 
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         k 
                                                         o 
                                                       
                                                       ⁢ 
                                                       
                                                         K 
                                                         
                                                           z 
                                                           , 
                                                           n 
                                                         
                                                       
                                                       ⁢ 
                                                       z 
                                                     
                                                   
                                                   + 
                                                   
                                                     ⅇ 
                                                     
                                                       
                                                         - 
                                                         j 
                                                       
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         k 
                                                         o 
                                                       
                                                       ⁢ 
                                                       
                                                         K 
                                                         
                                                           z 
                                                           , 
                                                           n 
                                                         
                                                       
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                                                 ) 
                                               
                                               ⁢ 
                                               
                                                 W 
                                                 
                                                   - 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 Y 
                                                 
                                                   n 
                                                   - 
                                                   1 
                                                 
                                               
                                             
                                             + 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     ⅇ 
                                                     
                                                       j 
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         k 
                                                         o 
                                                       
                                                       ⁢ 
                                                       
                                                         K 
                                                         
                                                           z 
                                                           , 
                                                           n 
                                                         
                                                       
                                                       ⁢ 
                                                       z 
                                                     
                                                   
                                                   - 
                                                   
                                                     ⅇ 
                                                     
                                                       
                                                         - 
                                                         j 
                                                       
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         k 
                                                         o 
                                                       
                                                       ⁢ 
                                                       
                                                         K 
                                                         
                                                           z 
                                                           , 
                                                           n 
                                                         
                                                       
                                                       ⁢ 
                                                       z 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               
                                                 K 
                                                 
                                                   z 
                                                   , 
                                                   n 
                                                 
                                                 
                                                   - 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 W 
                                                 
                                                   - 
                                                   1 
                                                 
                                               
                                               ⁢ 
                                               
                                                 A 
                                                 
                                                   h 
                                                   , 
                                                   n 
                                                 
                                                 
                                                   ′ 
                                                   - 
                                                   1 
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           
                                             t 
                                             &#39; 
                                           
                                           , 
                                           
                                             n 
                                             - 
                                             1 
                                           
                                         
                                         ′ 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         A 
                                         
                                           h 
                                           , 
                                           n 
                                         
                                         ′ 
                                       
                                       ⁢ 
                                       
                                         W 
                                         [ 
                                         
                                           
                                             
                                               
                                                 K 
                                                 
                                                   z 
                                                   , 
                                                   n 
                                                 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
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                                                       j 
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         k 
                                                         o 
                                                       
                                                       ⁢ 
                                                       
                                                         K 
                                                         z 
                                                       
                                                       ⁢ 
                                                       z 
                                                     
                                                   
                                                   - 
                                                   
                                                     ⅇ 
                                                     
                                                       
                                                         - 
                                                         j 
                                                       
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       
                                                         k 
                                                         o 
                                                       
                                                       ⁢ 
                                                       
                                                         K 
                                                         z 
                                                       
                                                       ⁢ 
                                                       z 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ⁢ 
                                             
                                               W 
                                               
                                                 - 
                                                 1 
                                               
                                             
                                             ⁢ 
                                             
                                               Y 
                                               
                                                 n 
                                                 - 
                                                 1 
                                               
                                             
                                           
                                           + 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             
                                                 
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       ⅇ 
                                                       
                                                         j 
                                                         ⁢ 
                                                         
                                                             
                                                         
                                                         ⁢ 
                                                         
                                                           k 
                                                           o 
                                                         
                                                         ⁢ 
                                                         
                                                           K 
                                                           
                                                             z 
                                                             , 
                                                             n 
                                                           
                                                         
                                                         ⁢ 
                                                         z 
                                                       
                                                     
                                                     + 
                                                     
                                                       ⅇ 
                                                       
                                                         
                                                           - 
                                                           j 
                                                         
                                                         ⁢ 
                                                         
                                                             
                                                         
                                                         ⁢ 
                                                         
                                                           k 
                                                           o 
                                                         
                                                         ⁢ 
                                                         
                                                           K 
                                                           
                                                             z 
                                                             , 
                                                             n 
                                                           
                                                         
                                                         ⁢ 
                                                         z 
                                                       
                                                     
                                                   
                                                   ) 
                                                 
                                                 ⁢ 
                                                 
                                                   W 
                                                   
                                                     - 
                                                     1 
                                                   
                                                 
                                                 ⁢ 
                                                 
                                                   A 
                                                   
                                                     h 
                                                     , 
                                                     n 
                                                   
                                                   
                                                     ′ 
                                                     - 
                                                     1 
                                                   
                                                 
                                               
                                               ] 
                                             
                                             ⁢ 
                                             
                                               e 
                                               
                                                 
                                                   t 
                                                   &#39; 
                                                 
                                                 , 
                                                 
                                                   n 
                                                   - 
                                                   1 
                                                 
                                               
                                               ′ 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   8 
                   ⁢ 
                   d 
                 
                 ) 
               
             
           
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Y 
                             n 
                           
                           ⁢ 
                           
                             e 
                             
                               
                                 t 
                                 &#39; 
                               
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     
                       
                         
                           e 
                           
                             
                               t 
                               &#39; 
                             
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     1 
                     2 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             
                               W 
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         + 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       D 
                                       
                                         o 
                                         , 
                                         n 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         - 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       O 
                                       n 
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   t 
                                   &#39; 
                                 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ′ 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 O 
                                 n 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               z 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         - 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               z 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       D 
                                       
                                         o 
                                         , 
                                         n 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             j 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         + 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               j 
                                             
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               k 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       O 
                                       n 
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   t 
                                   &#39; 
                                 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ′ 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 
                   8 
                   ⁢ 
                   e 
                 
                 ) 
               
             
           
         
       
     
   
   Solving for Y n :
 
 Y   n   =W   n   Y′   n   O   n   −1   (9)
 
where
 
   
     
       
         
           
             
               
                 
                   Y 
                   n 
                   ′ 
                 
                 = 
                 
                   
                       
                       
                   
                   ⁢ 
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 + 
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               D 
                               
                                 o 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 - 
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     
                                       K 
                                       
                                         z 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               O 
                               n 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ⅇ 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             K 
                                             
                                               z 
                                               , 
                                               n 
                                             
                                           
                                           ⁢ 
                                           z 
                                         
                                       
                                       + 
                                       
                                         ⅇ 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             K 
                                             
                                               z 
                                               , 
                                               n 
                                             
                                           
                                           ⁢ 
                                           z 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     D 
                                     
                                       o 
                                       , 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⅇ 
                                         
                                           j 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             K 
                                             
                                               z 
                                               , 
                                               n 
                                             
                                           
                                           ⁢ 
                                           z 
                                         
                                       
                                       + 
                                       
                                         ⅇ 
                                         
                                           
                                             - 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             K 
                                             
                                               z 
                                               , 
                                               n 
                                             
                                           
                                           ⁢ 
                                           z 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     O 
                                     n 
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                               
                               ] 
                             
                             
                               - 
                               1 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 10 
                 ) 
               
             
           
         
       
     
   
   Of practical note, equation (10) is, in general, catastrophically susceptible to round-off error related to the presence of large exponential values, i.e., e jk     0     K     z,n     z . Specifically, the problem arises whenever the ratio of a large exponential relative to the magnitudes of the Y′ n  elements exceeds the floating point accuracy of the machine (this can be verified by going through an LU decomposition/back substitution for a 2×2 matrix). This problem, for example, prevents us from simply computing the inverse and performing a matrix multiplication. We can reduce this error to the limits of floating point precision by noting that for large exponentials, the solution Y′ n  is the identity matrix, i.e.: 
   
     
       
         
           
             
               
                 
                   
                     lim 
                     
                       
                         
                           k 
                           o 
                         
                         ⁢ 
                         
                           K 
                           
                             z 
                             , 
                             n 
                           
                         
                         ⁢ 
                         z 
                       
                       → 
                       ∞ 
                     
                   
                   ⁢ 
                   
                     Y 
                     n 
                     ′ 
                   
                 
                 = 
                 
                   
                     
                       
                         
                           
                             ⅇ 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               
                                 K 
                                 
                                   z 
                                   , 
                                   n 
                                 
                               
                               ⁢ 
                               z 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 D 
                                 
                                   o 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                               
                               + 
                               
                                 O 
                                 n 
                                 
                                   - 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               D 
                               
                                 o 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                             + 
                             
                               O 
                               n 
                               
                                 - 
                                 1 
                               
                             
                           
                           ] 
                         
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           o 
                         
                         ⁢ 
                         
                           K 
                           
                             z 
                             , 
                             n 
                           
                         
                         ⁢ 
                         z 
                       
                     
                   
                   = 
                   I 
                 
               
             
             
               
                 ( 
                 11 
                 ) 
               
             
           
         
       
     
   
   Consequently, we can use equation (11) to solve for column vectors of Y′ n  corresponding to large exponentials, while solving for the remaining column vectors corresponding to the small exponentials, using the usual LU decomposition and back substitution. 
   Alternately, we can rewrite equation (9) as:
 
 Y   n   =W   n   Y″   n   (12)
 
where:
 
                   Y   n   ″     =         Y   n   ′     ⁢     O   n       =                 ⁢       [           ⁢         (       ⅇ     j   ⁢           ⁢     k   o     ⁢     K     z   ,   n       ⁢   z       +     ⅇ       -   j     ⁢           ⁢     k   o     ⁢     K     z   ,   n       ⁢   z         )     ⁢     D     o   ,     n   -   1           +           ⁢     
     ⁢                 (       ⅇ     j   ⁢           ⁢     k   o     ⁢     K     z   ,   n       ⁢   z       -     ⅇ       -   j     ⁢           ⁢     k   o     ⁢     K     z   ,   n       ⁢   z         )     ⁢     O   n     -   1         ]     [         (       ⅇ     j   ⁢           ⁢     k   o     ⁢     K   z     ⁢   z       -     ⅇ       -   j     ⁢           ⁢     k   o     ⁢     K   z     ⁢   z         )     ⁢     D       o   .   n     -   1         +           ⁢     
     ⁢           ⁢     (       ⅇ     j   ⁢           ⁢     k   o     ⁢     K     z   ,   n       ⁢   z       +     ⅇ       -   j     ⁢           ⁢     k   o     ⁢     K     z   ,   n       ⁢   z         )     ⁢     O   n     -   1           ]     ⁢           ⁢             -   1       .                   (   13   )               
2.2 Computing Reflectance From Impedance
 
   Defining reflectance as: 
   
     
       
         
           
             
               
                 
                   
                     R 
                     
                       t 
                       , 
                       n 
                     
                   
                   : 
                   
                     [ 
                     
                       
                         
                           
                             d 
                             
                               + 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                       
                         
                           
                             d 
                             
                               - 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           d 
                           
                             + 
                             
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               t 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             d 
                             
                               + 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 14 
                 ) 
               
             
           
         
       
     
   
   Relate Y n  to R t,n  using equations (5), (8) and (13): 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           d 
                           
                             + 
                             
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               t 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             d 
                             
                               + 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   
                                     jk 
                                     o 
                                   
                                   ⁢ 
                                   
                                     S 
                                     
                                       h 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     
                                       jk 
                                       o 
                                     
                                   
                                   ⁢ 
                                   
                                     S 
                                     
                                       h 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             I 
                           
                           
                             
                               η 
                               
                                 h 
                                 , 
                                 n 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         
                           
                             I 
                           
                           
                             
                               - 
                               
                                 η 
                                 
                                   h 
                                   , 
                                   n 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             
                               Y 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 t 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ′ 
                             
                           
                         
                       
                       
                         
                           
                             e 
                             
                               t 
                               , 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                             ′ 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
         
       
     
   
   Because the n−1 layer is semi-infinite, η h,n−1 =Y n−1   −1 . Therefore: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           d 
                           
                             + 
                             
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               t 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             d 
                             
                               + 
                               
                                 , 
                                 n 
                               
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       jk 
                                       o 
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Y 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     + 
                                     
                                       η 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       
                                         jk 
                                         o 
                                       
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Y 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     - 
                                     
                                       η 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
         
       
     
   
   Therefore, an equation for R t,n  is given by:
 
 R   t,n   =e   −jk     0     S     h,n     z ( Y   n−1   −η   h,n   −1 )( Y   n−1   +η   h,n   −1 ) −1   e   −jk     0     S     h,n     z   (17)
 
2.3 Computing Transmittance From Impedance
 
Defining transmittance as:
 
 T   N,n     0     : T   N,n     0     d   +,N   =d   +,n     0     (18)
 
To compute T N,n     0    we introduce a field transmittance:
 
 C   n,n−1   : e′   t,n−1   =C   n,n−1   e′   t,n .  (19)
 
   Substituting equation (18) into equation (8e) yields: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Y 
                             n 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               
                                 t 
                                 ′ 
                               
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                     
                       
                         
                           ⅇ 
                           
                             
                               t 
                               ′ 
                             
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       2 
                     
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             
                               w 
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             
                                               jk 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         + 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               
                                                 jk 
                                                 o 
                                               
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       D 
                                       
                                         o 
                                         , 
                                         n 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             
                                               jk 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         - 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               
                                                 jk 
                                                 o 
                                               
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       O 
                                       n 
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               C 
                               
                                 n 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   t 
                                   ′ 
                                 
                                 , 
                                 n 
                               
                               ′ 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 O 
                                 n 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           e 
                                           
                                             
                                               jk 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               z 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         - 
                                         
                                           e 
                                           
                                             
                                               - 
                                               
                                                 jk 
                                                 o 
                                               
                                             
                                             ⁢ 
                                             
                                               K 
                                               z 
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       D 
                                       
                                         o 
                                         , 
                                         n 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⅇ 
                                           
                                             
                                               jk 
                                               o 
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                         + 
                                         
                                           ⅇ 
                                           
                                             
                                               - 
                                               
                                                 jk 
                                                 o 
                                               
                                             
                                             ⁢ 
                                             
                                               K 
                                               
                                                 z 
                                                 , 
                                                 n 
                                               
                                             
                                             ⁢ 
                                             z 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       O 
                                       n 
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               C 
                               
                                 n 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 
                                   t 
                                   ′ 
                                 
                                 , 
                                 n 
                               
                               ′ 
                             
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   . 
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
         
       
     
   
   From equation (20) we get: 
   
     
       
         
           
             
               
                 
                   C 
                   
                     n 
                     , 
                     
                       n 
                       - 
                       1 
                     
                   
                 
                 = 
                 
                   
                     2 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 ⅇ 
                                 
                                   
                                     jk 
                                     o 
                                   
                                   ⁢ 
                                   
                                     K 
                                     z 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                               - 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     
                                       jk 
                                       o 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     z 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             D 
                             
                               o 
                               , 
                               n 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 ⅇ 
                                 
                                   
                                     jk 
                                     o 
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                               + 
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     
                                       jk 
                                       o 
                                     
                                   
                                   ⁢ 
                                   
                                     K 
                                     
                                       z 
                                       , 
                                       n 
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             O 
                             n 
                             
                               - 
                               1 
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                     
                       O 
                       n 
                       
                         - 
                         1 
                       
                     
                     . 
                   
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   Using the recursive nature of equation (19), we can compute the field transmission between layer N−1 and n 0 , i.e.:
 
 C   N−1,n     0     =C   N−2,n   0   C   N−1,N−2 .  (22)
 
   To compute the transmittance T N,n   0  from the field transmittance C n,n     0   , we need to define two conversion matrices C N,N−1  and C n     0     ,n     0     −1  at the top (n=N) and bottom (n=n 0 ) layers that convert fields into propagating waves and vice-versa, i.e.:
 
 e′   t,N−1   =C   N,N−1   d   +,N ,  (23a)
 
 d   +,n     0     =C   n     0     ,n     0     −1   e′   t,n     0   .  (23b)
 
   Deriving C N,N−1  and substituting equation (23a) into equation (16) yields: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           d 
                           
                             + 
                             
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               t 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             d 
                             
                               + 
                               
                                 , 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       jk 
                                       o 
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Y 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     + 
                                     
                                       η 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       
                                         jk 
                                         o 
                                       
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Y 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     - 
                                     
                                       η 
                                       
                                         h 
                                         , 
                                         n 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       jk 
                                       o 
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         N 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Y 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     + 
                                     
                                       η 
                                       
                                         h 
                                         , 
                                         N 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 C 
                                 
                                   N 
                                   , 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 d 
                                 
                                   + 
                                   
                                     , 
                                     N 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       
                                         jk 
                                         o 
                                       
                                     
                                     ⁢ 
                                     
                                       S 
                                       
                                         h 
                                         , 
                                         N 
                                       
                                     
                                     ⁢ 
                                     z 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       Y 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                     
                                     - 
                                     
                                       η 
                                       
                                         h 
                                         , 
                                         N 
                                       
                                       
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 C 
                                 
                                   N 
                                   , 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                               
                               ⁢ 
                               
                                 d 
                                 
                                   + 
                                   
                                     , 
                                     N 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
         
       
     
   
   Therefore,
 
2= e   jk     0     S     h,N     z ( Y   N−1   +η   h,N   −1 ) C   N,N−1 ,  (25)
 
 C   N,N−1 =2( Y   N−1   +η   h,N   −1 ) −1   e   −jk     0     S     h,N     z .  (26)
 
   Derive C n     0     ,n     0     −1 . In this case, the bottom layer is semi-infinite, Y n     0     −1 =η h,n     0     −1  or equivalently R t,n     0     −1 =0. At this point, we could also assume that z=0, however, we will retain this term to accommodate for possible reference plane shifting (virtual interface) associated with the sector S-matrix. Therefore: 
   
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               C 
                               
                                 
                                   n 
                                   o 
                                 
                                 , 
                                 
                                   
                                     n 
                                     o 
                                   
                                   - 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 t 
                                 , 
                                 
                                   n 
                                   o 
                                 
                               
                               ′ 
                             
                           
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                     ⅇ 
                                     
                                       
                                         jk 
                                         o 
                                       
                                       ⁢ 
                                       
                                         S 
                                         
                                           h 
                                           , 
                                           
                                             n 
                                             o 
                                           
                                         
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         η 
                                         
                                           h 
                                           , 
                                           
                                             n 
                                             o 
                                           
                                         
                                         
                                           - 
                                           1 
                                         
                                       
                                       + 
                                       
                                         η 
                                         
                                           h 
                                           , 
                                           
                                             n 
                                             o 
                                           
                                         
                                         
                                           - 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     t 
                                     , 
                                     
                                       n 
                                       o 
                                     
                                   
                                   ′ 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ⅇ 
                                     
                                       
                                         - 
                                         
                                           jk 
                                           o 
                                         
                                       
                                       ⁢ 
                                       
                                         S 
                                         
                                           h 
                                           , 
                                           
                                             
                                               n 
                                               o 
                                             
                                             - 
                                             1 
                                           
                                         
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         η 
                                         
                                           h 
                                           , 
                                           
                                             n 
                                             o 
                                           
                                         
                                         
                                           - 
                                           1 
                                         
                                       
                                       - 
                                       
                                         η 
                                         
                                           h 
                                           , 
                                           
                                             n 
                                             o 
                                           
                                         
                                         
                                           - 
                                           1 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     t 
                                     , 
                                     
                                       n 
                                       o 
                                     
                                   
                                   ′ 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   
                                     jk 
                                     o 
                                   
                                   ⁢ 
                                   
                                     S 
                                     
                                       h 
                                       , 
                                       
                                         n 
                                         o 
                                       
                                     
                                   
                                   ⁢ 
                                   z 
                                 
                               
                               ⁢ 
                               
                                 η 
                                 
                                   h 
                                   , 
                                   
                                     n 
                                     o 
                                   
                                 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   t 
                                   , 
                                   
                                     n 
                                     o 
                                   
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 
                   27 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 
                   
                     ⅇ 
                     
                       
                         - 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       
                         S 
                         
                           h 
                           , 
                           
                             n 
                             o 
                           
                         
                       
                       ⁢ 
                       z 
                     
                   
                   ⁢ 
                   
                     η 
                     
                       h 
                       , 
                       
                         n 
                         o 
                       
                     
                     
                       - 
                       1 
                     
                   
                 
                 = 
                 
                   
                     C 
                     
                       
                         n 
                         o 
                       
                       , 
                       
                         
                           n 
                           o 
                         
                         - 
                         1 
                       
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 
                   27 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
         
       
     
   
   Summarizing, the transmittance between the top layer (N) to the bottom layer (n 0 ) is given by the product 
   
     
       
         
           
             
               
                 
                   
                     T 
                     
                       N 
                       , 
                       
                         n 
                         o 
                       
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         n 
                         = 
                         
                           n 
                           o 
                         
                       
                       N 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       C 
                       
                         n 
                         , 
                         
                           n 
                           - 
                           1 
                         
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 28 
                 ) 
               
             
           
         
       
     
   
   where: 
                   C     n   ,     n   -   1         =     {                 ⅇ       -     jk   o       ⁢     S     h   ,   n       ⁢   z       ⁢     η     h   ,   n       -   1       ⁢           ⁢   for   ⁢           ⁢   n     =     bottom   ⁢           ⁢   layer   ⁢           ⁢   index   ⁢           ⁢     (     n   o     )                     2   ⁢       (       Y     n   -   1       +     η     h   ,   n       -   1         )       -   1       ⁢     ⅇ       -     jk   o       ⁢     S     h   ,   n       ⁢   z       ⁢           ⁢   for   ⁢           ⁢   n     =     top   ⁢           ⁢   layer   ⁢           ⁢   index   ⁢           ⁢     (   N   )                         2   ⁡     [         (       ⅇ       jk   o     ⁢     K     z   ,   n       ⁢   z       -     ⅇ       -     jk   o       ⁢     K     z   ,   n       ⁢   z         )     ⁢     D     o   ,   n         +       (       ⅇ       jk   o     ⁢     K     z   ,   n       ⁢   z       +     ⅇ       -     jk   o       ⁢     K     z   ,   n       ⁢   z         )     ⁢     O   n     -   1           ]         -   1       ⁢     O   n     -   1       ⁢           ⁢   for   ⁢           ⁢     n   o       &lt;   n   &lt;   N           .               (   29   )               
2.4 Efficient Computation Of Total Field At Any Layer
 
   This computation boils down to computing d ±,m . Denote the layer in which we want to compute the field as m. First consider the case when m≠n 0 , m≠N. From equations (1) and (18) of this section, the fields at the bottom of layer m (z=0) can be expressed as: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Y 
                             m 
                           
                           ⁢ 
                           
                             T 
                             
                               N 
                               , 
                               m 
                             
                           
                           ⁢ 
                           
                             d 
                             
                               + 
                               
                                 , 
                                 N 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             η 
                             
                               h 
                               , 
                               m 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             T 
                             
                               N 
                               , 
                               m 
                             
                           
                           ⁢ 
                           
                             d 
                             
                               + 
                               
                                 , 
                                 N 
                               
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               d 
                               
                                 + 
                                 
                                   , 
                                   m 
                                 
                               
                             
                             + 
                             
                               d 
                               
                                 - 
                                 
                                   , 
                                   m 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 
                                   + 
                                   
                                     , 
                                     m 
                                   
                                 
                               
                               + 
                               
                                 d 
                                 
                                   - 
                                   
                                     , 
                                     m 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                   . 
                 
               
             
             
               
                 ( 
                 30 
                 ) 
               
             
           
         
       
     
   
   Solving for d ±,m  in terms of a known incident propagating wave d +,N  yields: 
   
     
       
         
           
             
               
                 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           Y 
                           m 
                         
                         + 
                         
                           η 
                           
                             h 
                             , 
                             m 
                           
                           
                             - 
                             1 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       T 
                       
                         N 
                         , 
                         m 
                       
                     
                     ⁢ 
                     
                       d 
                       
                         + 
                         
                           , 
                           N 
                         
                       
                     
                   
                   = 
                   
                     d 
                     
                       + 
                       
                         , 
                         m 
                       
                     
                   
                 
                 , 
               
             
             
               
                 ( 
                 
                   31 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 
                   
                     1 
                     2 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         Y 
                         m 
                       
                       - 
                       
                         η 
                         
                           h 
                           , 
                           m 
                         
                         
                           - 
                           1 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     T 
                     
                       N 
                       , 
                       m 
                     
                   
                   ⁢ 
                   
                     d 
                     
                       + 
                       
                         , 
                         N 
                       
                     
                   
                 
                 = 
                 
                   
                     d 
                     
                       - 
                       
                         , 
                         m 
                       
                     
                   
                   . 
                 
               
             
             
               
                 ( 
                 
                   31 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
         
       
     
   
   Substituting equations (31a) and (31b) into equation (1), we obtain a general expression for the fields inside the m th  layer in terms of known quantities, i.e.: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 h 
                                 
                                   t 
                                   , 
                                   m 
                                 
                               
                             
                           
                           
                             
                               
                                 e 
                                 
                                   t 
                                   , 
                                   m 
                                 
                                 ′ 
                               
                             
                           
                         
                         ] 
                       
                       = 
                     
                       
                   
                   ⁢ 
                   
                       
                   
                     
                 
                 ⁢ 
                 
                   
                     
                         
                         
                     
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             
                               
                                 1 
                                 2 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           jk 
                                           o 
                                         
                                         ⁢ 
                                         
                                           S 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           Y 
                                           m 
                                         
                                         + 
                                         
                                           η 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                           
                                             - 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           
                                             jk 
                                             o 
                                           
                                         
                                         ⁢ 
                                         
                                           S 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           Y 
                                           m 
                                         
                                         - 
                                         
                                           η 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                           
                                             - 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               T 
                               
                                 N 
                                 , 
                                 m 
                               
                             
                             ⁢ 
                             
                               d 
                               
                                 + 
                                 
                                   , 
                                   N 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               
                                 η 
                                 
                                   h 
                                   , 
                                   m 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           jk 
                                           o 
                                         
                                         ⁢ 
                                         
                                           S 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           Y 
                                           m 
                                         
                                         + 
                                         
                                           η 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                           
                                             - 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           
                                             jk 
                                             o 
                                           
                                         
                                         ⁢ 
                                         
                                           S 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           Y 
                                           m 
                                         
                                         - 
                                         
                                           η 
                                           
                                             h 
                                             , 
                                             m 
                                           
                                           
                                             - 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               T 
                               
                                 N 
                                 , 
                                 m 
                               
                             
                             ⁢ 
                             
                               d 
                               
                                 + 
                                 
                                   , 
                                   N 
                                 
                               
                             
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   . 
                 
               
             
             
               
                 ( 
                 32 
                 ) 
               
             
           
         
       
     
   
   Consider the case when m=n 0  (the bottom layer). Equation (28) becomes precisely equal to the propagating wave transmission coefficient. Consequently:
 
 T   N,m   d   +,N   =d   +,n     0   .  (33)
 
   When m=N (the top layer), the field solution is trivial since d +,N  is given by the known incident propagating wave, while d −,N  can be computed directly from the multimodal reflectance R t,N , i.e.:
 
 d   −,N   =R   t,N   d   +,N .  (34)
 
   At first glance, equations (5), (6), (7), and (2b) appear to suggest that a field computation will increase the computational overhead of the fast Z-Matrix by requiring additional full matrix operations. We can, however, eliminate this overhead by performing the matrix-vector multiplication first, i.e., begin multiplication from d +,N  and proceed from right to left. Moreover, we can store the product of equation (28), and this reduces the operation count of using equation (5) from O(N 3 ) to O(N 2 ). 
   Fast Z-Matrix Algorithm 
   The goal is to present an algorithm, based on the fast Z-matrix algorithm, to compute the S-Matrix of an arbitrary multi-layered film stack. 
     FIG. 1  depicts the decomposition of the S-Matrix computation of a multi-layered film stack into a non-inverted and an inverted film stack. Compare the computational efficiency of the new algorithm with the conventional algorithm 
   Algorithm 
   Step 1: Decompose the original film stack into two film stacks, namely, a non-inverted and inverted film stack, as depicted in  FIG. 1 . 
   Step 2: Compute the reflectance and transmittance at the top of both film stacks using the fast Z-matrix algorithm. 
   Step 3: By definition, the S-matrix S of the original film stack is given by: 
   
     
       
         
           
             S 
             = 
             
               [ 
               
                 
                   
                     
                       T 
                       
                         0 
                         , 
                         N 
                       
                       ′ 
                     
                   
                   
                     
                       R 
                       
                         t 
                         , 
                         N 
                       
                     
                   
                 
                 
                   
                     
                       R 
                       
                         t 
                         , 
                         0 
                       
                       ′ 
                     
                   
                   
                     
                       T 
                       
                         N 
                         , 
                         0 
                       
                     
                   
                 
               
               ] 
             
           
           , 
         
       
     
   
   ]where: R t,N , T N,0 =the reflectance and transmittance of the non-inverted film stack, and R′ t,0 , T′ 0,N =the reflectance and transmittance of the inverted film stack, where the prime denotes the inverted stack. 
   Operation Count and Improved Efficiency 
   Fast Z-Matrix 
   We can derive a formula for the total major operation count of the fast Z-matrix algorithm as applied to the S-matrix computation, by noting that any given N-layered film stack will have one top layer, one bottom layer, and N-2 middle layers. Therefore, the total operation count=M z =M bot +M top +(N−2)*M mid , where M bot , M top , M mid  are the major operation counts for the computation at the top, middle and bottom layers. 
   Conventional Algorithm 
   A formula for the total operation count can be derived by noting that for any given N-layered film stack we must build the full S-matrix by propagating sector S-matrices. Since each sector S-matrix characterizes a two-layer media, it is easy to show that there are (N−1) sector S-matrices and (N−2) S-matrix propagations to combine the sector S-matrices. Therefore, the total operation count M standard =M bot +(N−1)*M sec +(N−2)*M prop , where M bot , M sec , M prop  are the major operation counts for the computation at the bottom, each sector and for the propagation. 
   Conclusion 
   The Z-matrix algorithm is more computationally efficient than the conventional algorithm for computing the S-matrix of a film stack with more than two layers. The improved efficiency of the Z-matrix algorithm is proportional to the number of layers. In the worst case, the improvement saturates to about 22%. In a more typical case the improvement saturates to about 32%. The improvement saturates (approximately) when the number of layers exceeds about ten. For a two-layer films stack, i.e., a sector S-matrix film stack, the Z-matrix algorithm has the same efficiency as the conventional algorithm. 
   The foregoing description of preferred embodiments for this invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Obvious modifications or variations are possible in light of the above teachings. The embodiments are chosen and described in an effort to provide the best illustrations of the principles of the invention and its practical application, and to thereby enable one of ordinary skill in the art to utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. All such modifications and variations are within the scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled.