PATENT ABSTRACT
A method of determining the actual properties of a film stack. An incident beam of light is directed 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 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. A method based on analytical derivatives, and not numerically computed derivatives, of solutions to Maxwell&#39;s equations that are at least partially expressible as complex exponential matrices is used to iteratively adjust the estimated 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  
   As used herein, the term “integrated circuits” generally refers to monolithic semiconducting devices, such as those formed of group IV materials like silicon or germanium or mixtures thereof, or group III–V compounds such as gallium arsenide. The term “integrated circuits” includes all known configurations of such devices, such as memory and logic, and all designs of such devices, such as CMOS and bipolar. 
   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. 
   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 thicknesses to which they are formed, can be determined. 
   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 computed reflectance property values based on some theoretical model, such as Rigorous Coupled Wave Analysis, to the measured results attained in step 1. Step 2 typically consists of the iterated steps of computing one or more theoretical value 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 many film stacks, the step of iterative computing as described above consumes by far the most time during the process. For some film stacks, such as those film stacks that contain patterned lines, this iterative steps become extremely complex, because of the tremendous overhead of computing and tracking reflections at multiple diffracting angles. Currently, the theoretical reflectance values are determined using numerical derivatives for the reflectance, because exact, closed-form expressions for these derivatives has heretofore not been found. 
   Unfortunately, the current methods tend to suffer from several problems. First, the extra mathematical computations that are required by the numerical methods tends to limit the speed at which measurements can be taken, thus reducing the throughput of the instrument. Second, such numerical methods tend to have a reduced degree of accuracy resulting in decreased robustness and reliability. 
   What is needed, therefore, is a method of spectral film stack measurement that tends to reduce at least some of the problems described above. 
   SUMMARY  
   The above and other needs are met by a method of determining the actual properties of a film stack. An incident beam of light is directed 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 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. A method based on analytical derivatives, and not numerically computed derivatives, of solutions to Maxwell&#39;s equations that are at least partially expressible as complex exponential matrices is used to iteratively adjust the estimated properties of the film stack. 
   Thus, the present invention defines over the prior art in that an analytical method is used to determine the solutions to the mathematical model of the film stack, instead of relying on less accurate and more time consuming numerical methods. In particular, for models of inhomogeneous film stacks that are at least partially expressible as non-diagonal complex exponential matrix solutions to Maxwell&#39;s equation, such a solution was previously thought to be intractable or impossible. However, a method has been discovered by the present inventors to manipulate the equations for this case in a manner where they can be solved analytically. That method is described herein. 
   In various embodiments, the method is implemented in an ellipsometer. The film stack may include layers of more than one material, and more than one layer. The film stack preferably includes a patterned layer. The actual properties of the film stack preferably include at least one of layer thickness and layer refractive index. Preferably, the mathematical model is computationally solved using matrix operations. Most preferably, the mathematical model is a reflectance model that is derived at least in part from rigorous coupled wave analysis. The estimated properties of the film stack are preferably derived exactly and in closed form. The equation to be used iteratively estimating properties of the film stack based on reflectance is preferably given by: 
   
     
       
         
           
             
               
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   A generalized expression of the above is given by: 
   
     
       
         
           
             
               ∂ 
               
                 C 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         R 
                         meas 
                       
                       ⁡ 
                       
                         ( 
                         
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                 ⁢ 
                 
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                         ⁡ 
                         
                           ( 
                           
                             
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                         n 
                       
                     
                   
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   where C is the cost function, R theory  is the theoretical reflectance, R meas  is the measured reflectance, P n  is a variable of the theoretical reflectance model (i.e. film stack parameters), λ m  is the m th  wavelength, m is the integer index, M is the total number of wavelength values, and σ m  is the measurement uncertainty at λ m . 
   The preferred embodiments of the present invention differ from prior art methods in that the reflectance derivatives, with respect to the parameters of the film stack, including without limitation pitch, critical dimension, and refractive index, needed for regression are done exactly and in a closed form. An operation count of numerical derivatives based on a three-point central difference approximation and the analytic derivatives shows that this method reduces the number of full matrix multiplications by a factor of up to about four, depending on the amount of caching that is made available. Moreover, this method greatly improves the robustness, speed, and reliability of the measurement, because it eliminates the numerical errors and potential convergence problems associated with using numerical derivatives. 
   Relative to the methods that use numerical derivatives, it is believed that the preferred embodiments of the invention reduce the computation time needed for measurement by approximately 15% to 40%. The reduction can be significantly greater in those cases where the numerical derivatives do not converge. Many of the current methods of measuring films containing patterned metal are based on rigorous coupled wave analysis. Consequently, the preferred embodiments of the present invention provide an immediate low-cost advantage with respect to the reliable, accurate, and fast real-time measurement of multi-layered film stacks containing patterned lines. 
   This invention is preferably implemented as an add-in to existing instrument software. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Further advantages of the invention are apparent by reference to the detailed description when considered in conjunction with the figures, which are not to scale so as to more clearly show the details, wherein like reference numbers indicate like elements throughout the several views, and wherein: 
       FIG. 1  is a representation of a plane wave incident on a grating structure. 
       FIG. 2  is a representation of an N-layer film stack with multiple grating layers. 
   

   DETAILED DESCRIPTION 
   Ellipsometry methods generally begin by computing the cost or error function, for example:
 
C(R meas (λ),R theory (λ, P))   1
 
   Where R meas  is a function of wavelength (λ), and R theory  is a multivariable function in λ and P, and where:
 
P=(P 0 , P 1 , P 2 , . . . P N ).
 
   Minimize equation 1 with respect to P. There are two ways to do this. One method requires derivatives and the other does not. In general, methods that require derivatives converge faster than those that do not use derivatives and are, for that reason, preferable. The derivative approach requires finding values that make the first derivatives equal to zero, for example, 
   
     
       
         
           
             
               
                 
                   
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                     ⁢ 
                     
                         
                     
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   Where:
 
n=0, 1, . . . N.
 
   Equation 2 is N equations in N unknowns, because there are N derivatives corresponding to N parameters which must be set to zero. To solve equation 2 using linear algebra theory, it is generally necessary to approximate R theory  with its linear approximation R′ theory  using a Taylor series expansion, for example, 
   
     
       
         
           
             
               
                 
                   
                     
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                     ⁡ 
                     
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               3 
             
           
         
       
     
   
   Where:
 
P=(P 00 , P 10  . . . P N0 ).
 
   As alluded to above, one novel aspect of the various embodiments according to the present invention lies in the regression algorithm and, in particular, the derivatives used therein. R theory  derivatives in equations 2 and 3 can be computed numerically or analytically. Preferably, these derivatives are computed analytically so as to not only improve the accuracy of the method, but also to remove the nonconvergence problems that are associated with numerical derivatives. Moreover, analytical derivatives tend to be more amenable to computational efficiency improvements associated with the caching of intermediate variables. Thus, embodiments of the present invention tend to be both faster than numerical methods and more accurate than numerical methods. 
   In practice, analytical derivatives are not always derivable due to the mathematical complexity of R theory . An important case in point is the R theory  derivatives based on the Rigorous Coupled Wave Analysis. The Rigorous Coupled Wave Analysis model is one of the most if not the most widely used physical model used in ellipsometry and scatterometry measurements over patterned inhomogeneous media. The expressions for the derivatives of R theory , however, are not readily computed because of the numerical eigenvalue/eigenvector computations involved. One of the novelties in the various embodiments according to the present invention is that analytic expressions for R theory  derivatives are derived based on the Rigorous Coupled Wave Analysis. A generalized expression of the equation is given by: 
   
     
       
         
           
             
               
                 
                   
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                       ⁡ 
                       
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                             ⁡ 
                             
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               4 
             
           
         
       
     
   
   where C is the cost function, R theory  is the theoretical reflectance, R meas  is the measured reflectance, P n  is a variable of the theoretical reflectance model (i.e. film stack parameters), λ m  is the m th  wavelength, m is the integer index, M is the total number of wavelength values, and σ m  is the measurement uncertainty at λ m . 
   The first step in the process to produce this analytic expression is to derive governing partial differential equations for a TE (E field parallel to the grating grooves) and TM plane (E field perpendicular to the grating grooves) wave incident on the planar grating shown in  FIG. 1 . It is noted that the discussions provided herein assume that the reader is generally familiar with the art of ellipsometry, and thus most terms are not specifically defined herein. However, the basic equations, variables, and methods are known to those with skill in the art. 
   RIGOROUS COUPLED WAVE ANALYSIS FORMULATION 
   The TE case is defined by (E x ,H y ,H z )≠0, and where all other field components are zero. E and H are the electric and magnetic fields, respectively, and is preferably solved using Maxwell&#39;s equations. For the TE case the equations simplify to: 
   
     
       
         
           
             
               
                 
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   Where ε=relative dielectric permittivity, μ=relative magnetic permeability in the (x, y) direction, μ z =relative magnetic permeability in the z direction, and ε 0  and μ 0 =dielectric permittivity and magnetic permeability of free space, respectively. 
   Because the grating structure is periodic in the y axis, the electromagnetic fields and the grating profile is preferably expresses as a Fourier series, for example: 
   
     
       
         
           
             
               
                 
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   From orthogonality, each Fourier harmonic preferably independently satisfies Maxwell&#39;s equations. Therefore, Maxwell&#39;s equations are preferably expressed in the following matrix form: 
   
     
       
         
           
             
               
                 
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   Where: 
   
     
       
         
           
             
               
                 
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                       ] 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       h 
                       y 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 h 
                                 
                                   y 
                                   , 
                                   
                                     - 
                                     N 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 h 
                                 
                                   y 
                                   , 
                                   
                                     - 
                                     
                                       ( 
                                       
                                         N 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               … 
                             
                           
                           
                             
                               
                                 h 
                                 
                                   y 
                                   , 
                                   o 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 h 
                                 
                                   y 
                                   , 
                                   
                                     ( 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 h 
                                 
                                   y 
                                   , 
                                   N 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         h 
                         z 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               h 
                               
                                 z 
                                 , 
                                 
                                   - 
                                   N 
                                 
                               
                             
                           
                         
                         
                           
                             
                               h 
                               
                                 z 
                                 , 
                                 
                                   - 
                                   
                                     ( 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             … 
                           
                         
                         
                           
                             
                               h 
                               
                                 z 
                                 , 
                                 o 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               h 
                               
                                 z 
                                 , 
                                 
                                   ( 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               h 
                               
                                 z 
                                 , 
                                 N 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               12 
             
           
         
       
     
   
   and 
   
     
       
         
           
             
               
                 
                   E 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             ɛ 
                             0 
                           
                         
                         
                           
                             ɛ 
                             1 
                           
                         
                         
                           … 
                         
                         
                           
                             ɛ 
                             
                               - 
                               N 
                             
                           
                         
                       
                       
                         
                           
                             ɛ 
                             1 
                           
                         
                         
                           
                             ɛ 
                             0 
                           
                         
                         
                           … 
                         
                         
                           
                             ɛ 
                             
                               - 
                               
                                 ( 
                                 
                                   N 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋯ 
                         
                         
                           ⋯ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             ɛ 
                             N 
                           
                         
                         
                           
                             ɛ 
                             
                               N 
                               - 
                               1 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             ɛ 
                             0 
                           
                         
                       
                     
                     ] 
                   
                 
                 , 
                 
                   M 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             μ 
                             0 
                           
                         
                         
                           
                             μ 
                             1 
                           
                         
                         
                           … 
                         
                         
                           
                             μ 
                             
                               - 
                               N 
                             
                           
                         
                       
                       
                         
                           
                             μ 
                             1 
                           
                         
                         
                           
                             μ 
                             0 
                           
                         
                         
                           … 
                         
                         
                           
                             μ 
                             
                               - 
                               
                                 ( 
                                 
                                   N 
                                   - 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           … 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             μ 
                             N 
                           
                         
                         
                           
                             μ 
                             
                               N 
                               - 
                               1 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             μ 
                             0 
                           
                         
                       
                     
                     ] 
                   
                 
                 , 
                 
                   
                     
                       M 
                       z 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 0 
                               
                             
                           
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 1 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 
                                   - 
                                   N 
                                 
                               
                             
                           
                         
                         
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 1 
                               
                             
                           
                           
                             
                               μ 
                               0 
                             
                           
                           
                             … 
                           
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 
                                   - 
                                   
                                     ( 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             … 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 N 
                               
                             
                           
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               μ 
                               
                                 z 
                                 , 
                                 0 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   \13 
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   Equation 11 is preferably alternately written as: 
   
     
       
         
           
             
               
                 
                   
                     
                       - 
                       jω 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       μ 
                       o 
                     
                     ⁢ 
                     
                       h 
                       y 
                     
                   
                   = 
                   
                     
                       A 
                       u 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           e 
                           x 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                   
                 
                 ; 
                 
                   
                     jω 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       μ 
                       o 
                     
                     ⁢ 
                     
                       h 
                       z 
                     
                   
                   = 
                   
                     
                       A 
                       
                         μ 
                         z 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           e 
                           x 
                         
                       
                       
                         ∂ 
                         y 
                       
                     
                   
                 
               
             
             
               14 
             
           
         
       
     
   
   Where: 
   
     
       
         
           
             
               A 
               μ 
             
             = 
             
               [ 
               
                 
                   
                     
                       a 
                       0 
                     
                   
                   
                     
                       a 
                       1 
                     
                   
                   
                     … 
                   
                   
                     
                       a 
                       
                         - 
                         N 
                       
                     
                   
                 
                 
                   
                     
                       a 
                       1 
                     
                   
                   
                     
                       a 
                       0 
                     
                   
                   
                     … 
                   
                   
                     
                       a 
                       
                         - 
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     … 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       a 
                       N 
                     
                   
                   
                     
                       a 
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       a 
                       o 
                     
                   
                 
               
               ] 
             
           
           , 
           
             
               A 
               
                 μ 
                 z 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       a 
                       
                         z 
                         , 
                         0 
                       
                     
                   
                   
                     
                       a 
                       
                         z 
                         , 
                         1 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       a 
                       
                         z 
                         , 
                         
                           - 
                           N 
                         
                       
                     
                   
                 
                 
                   
                     
                       a 
                       
                         z 
                         , 
                         1 
                       
                     
                   
                   
                     
                       a 
                       
                         z 
                         , 
                         0 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       a 
                       
                         z 
                         , 
                         
                           - 
                           
                             ( 
                             
                               N 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     … 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       a 
                       
                         z 
                         , 
                         N 
                       
                     
                   
                   
                     
                       a 
                       
                         z 
                         , 
                         
                           N 
                           - 
                           1 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       a 
                       
                         z 
                         , 
                         o 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
   
   and 
   
     
       
         
           
             1 
             μ 
           
           = 
           
             
               
                 ∑ 
                 
                   n 
                   = 
                   
                     - 
                     N 
                   
                 
                 N 
               
               ⁢ 
               
                 
                   a 
                   n 
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       p 
                     
                     ⁢ 
                     y 
                   
                 
                 ⁢ 
                 
                   1 
                   
                     μ 
                     z 
                   
                 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   
                     - 
                     N 
                   
                 
                 N 
               
               ⁢ 
               
                 
                   a 
                   
                     z 
                     , 
                     n 
                   
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         n 
                       
                       p 
                     
                     ⁢ 
                     y 
                   
                 
               
             
           
         
       
     
   
   Equations 11 and 14 are self-consistent with respect to their usage of M or A μ . For the present purposes, the following hybrid is preferably used: 
   
     
       
         
           
             
               
                 
                   
                     h 
                     y 
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             μ 
                             o 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       A 
                       μ 
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           e 
                           x 
                         
                       
                       
                         ∂ 
                         z 
                       
                     
                   
                 
                 ; 
                 
                   
                     h 
                     z 
                   
                   = 
                   
                     
                       ( 
                       
                         1 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             μ 
                             o 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       M 
                       z 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         
                           e 
                           x 
                         
                       
                       
                         ∂ 
                         y 
                       
                     
                   
                 
               
             
             
               15 
             
           
         
       
     
   
   In general, equation 15 is not self-consistent, since M −1 ≠A μ . However, equation 15 is preferably used in the present method, because it has been determined to have more rapid convergence of the field solutions as the number of field harmonics is increased. Combining equations 5 and 15 yields: 
   
     
       
         
           
             
               
                 
                   
                     - 
                     
                       
                         A 
                         μ 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               - 
                               
                                 ∂ 
                                 
                                   ∂ 
                                   y 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   M 
                                   z 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   ∂ 
                                   
                                     ∂ 
                                     y 
                                   
                                 
                               
                               ) 
                             
                           
                           - 
                           E 
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     e 
                     x 
                   
                 
                 = 
                 
                   
                     - 
                     
                       1 
                       
                         k 
                         o 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         ∂ 
                         2 
                       
                       ⁢ 
                       
                         e 
                         x 
                       
                     
                     
                       ∂ 
                       
                         z 
                         2 
                       
                     
                   
                 
               
             
             
               16 
             
           
         
       
     
   
   Note that 
               ∂     ∂   y       →     j   ⁢           ⁢     k   ym         ,         
where the m subscript denotes the m th  harmonic.
 
   
     
       
         
           
             
               
                 
                   
                     - 
                     
                       S 
                       2 
                     
                   
                   ⁢ 
                   
                     e 
                     x 
                   
                 
                 = 
                 
                   
                     
                       
                         A 
                         μ 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               K 
                               y 
                             
                             ⁢ 
                             
                               M 
                               z 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               K 
                               y 
                             
                           
                           - 
                           E 
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       e 
                       x 
                     
                   
                   = 
                   
                     
                       1 
                       
                         k 
                         o 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         ⁢ 
                         
                           e 
                           x 
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                   
                 
               
             
             
               17 
             
           
         
       
     
   
   For isotropic magnetic permeability, M −1 =M z   −1 =A μ =I. Therefore, the TE case is preferably given by: 
   
     
       
         
           
             
               
                 
                   
                     - 
                     
                       S 
                       2 
                     
                   
                   ⁢ 
                   
                     e 
                     x 
                   
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           K 
                           y 
                           2 
                         
                         - 
                         E 
                       
                       ] 
                     
                     ⁢ 
                     
                       e 
                       x 
                     
                   
                   = 
                   
                     
                       1 
                       
                         k 
                         o 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         ⁢ 
                         
                           e 
                           x 
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                   
                 
               
             
             
               18 
             
           
         
       
     
   
   The TM case is defined by (H x ,E y ,E z )≠0, and where all other field components are zero. The TM case equations are preferably obtained from the TE case equations by duality. Equation 17 thereby yields: 
   
     
       
         
           
             
               
                 
                   
                     - 
                     
                       S 
                       2 
                     
                   
                   ⁢ 
                   
                     h 
                     x 
                   
                 
                 = 
                 
                   
                     
                       
                         A 
                         ɛ 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               K 
                               y 
                             
                             ⁢ 
                             
                               E 
                               z 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               K 
                               y 
                             
                           
                           - 
                           M 
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       h 
                       x 
                     
                   
                   = 
                   
                     
                       1 
                       
                         k 
                         o 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         ⁢ 
                         
                           h 
                           x 
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                   
                 
               
             
             
               19 
             
           
         
       
     
   
   For isotropic magnetic permeability, M=A μ =I. Therefore, the TM case is preferably given by: 
   
     
       
         
           
             
               
                 
                   
                     - 
                     
                       S 
                       2 
                     
                   
                   ⁢ 
                   
                     h 
                     x 
                   
                 
                 = 
                 
                   
                     
                       
                         A 
                         ɛ 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               K 
                               y 
                             
                             ⁢ 
                             
                               E 
                               z 
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               K 
                               y 
                             
                           
                           - 
                           I 
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       h 
                       x 
                     
                   
                   = 
                   
                     
                       1 
                       
                         k 
                         o 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ∂ 
                           2 
                         
                         ⁢ 
                         
                           h 
                           x 
                         
                       
                       
                         ∂ 
                         
                           z 
                           2 
                         
                       
                     
                   
                 
               
             
             
               20 
             
           
         
       
     
   
   Both the TE and the TM differential equations as derived above are preferably summarized as: 
   
     
       
         
           
             
               
                 
                   
                     - 
                     
                       S 
                       2 
                     
                   
                   ⁢ 
                   f 
                 
                 = 
                 
                   
                     
                       
                         1 
                         
                           k 
                           o 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             ∂ 
                             2 
                           
                           ⁢ 
                           f 
                         
                         
                           ∂ 
                           
                             z 
                             2 
                           
                         
                       
                     
                     : 
                     f 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 TE 
                                 : 
                                 
                                   e 
                                   x 
                                 
                               
                             
                           
                           
                             
                               
                                 TM 
                                 : 
                                 
                                   h 
                                   x 
                                 
                               
                             
                           
                         
                         : 
                         
                           - 
                           
                             S 
                             2 
                           
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 TE 
                                 : 
                               
                             
                             
                               
                                 
                                   A 
                                   μ 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         K 
                                         y 
                                       
                                       ⁢ 
                                       
                                         M 
                                         z 
                                         
                                           - 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         K 
                                         y 
                                       
                                     
                                     - 
                                     E 
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 TM 
                                 : 
                               
                             
                             
                               
                                 
                                   A 
                                   ɛ 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       
                                         K 
                                         y 
                                       
                                       ⁢ 
                                       
                                         E 
                                         z 
                                         
                                           - 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         K 
                                         y 
                                       
                                     
                                     - 
                                     M 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               21 
             
           
         
       
     
   
   By inspection, the complex exponential matrix functions given by e ±jk     o     Sz  are solutions to 21, therefore:
 
 f=e   jk     o     Sz   d   +   +e   −jk     o     Sz   d _  22
 
   Where d ±  are arbitrary column vector constants. 
   Using DeMoivre&#39;s identity, equation 22 can alternately be expressed as:
 
f=cos( k   o   Sz ) d   + +sin( k   o   Sz ) d   — 
 
   Equation 22 can also be alternately expressed as:
 
 f=W   −1   e   jk     o     K     z     z   Wd   +   +W   −1   e   −jk     o     K     z     z   Wd   — 
 
   where W=eigenvector matrix and K z =eigenvalue (diagonal) matrix such that S 2 =WK z   2 W −1    
   A REFLECTANCE UPDATE FORMULATION 
   The next step in the method is preferably to develop a method to compute the TE and TM reflectance of an N-layer film stack containing multiple gratings, as depicted in  FIG. 2 . 
   For the purposes of example, the detailed analysis for only the TE case is presented. Analogous results for the TM case can be derived from the TE case, using the same duality principle applied earlier in 19. The first step for the TE case is preferably to consider the electromagnetic fields, i.e., the solution to Maxwell&#39;s equations, in the form given by 22: 
   
     
       
         
           
             
               
                 
                   e 
                   x 
                 
                 = 
                 
                   
                     
                       ⅇ 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           o 
                         
                         ⁢ 
                         Sz 
                       
                     
                     ⁢ 
                     
                       d 
                       + 
                     
                   
                   + 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           o 
                         
                         ⁢ 
                         Sz 
                       
                     
                     ⁢ 
                     
                       d 
                       - 
                     
                   
                 
               
             
             
               23 
             
           
           
             
               
                 
                   h 
                   
                     y 
                     ′ 
                   
                 
                 = 
                 
                   
                     - 
                     
                       h 
                       y 
                     
                   
                   = 
                   
                     
                       A 
                       μ 
                     
                     ⁢ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 Sz 
                               
                             
                             ⁢ 
                             
                               d 
                               + 
                             
                           
                           - 
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 Sz 
                               
                             
                             ⁢ 
                             
                               d 
                               - 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               24 
             
           
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           x 
                         
                       
                     
                     
                       
                         
                           h 
                           
                             y 
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           
                             I 
                           
                           
                             I 
                           
                         
                         
                           
                             
                               
                                 A 
                                 μ 
                               
                               ⁢ 
                               S 
                             
                           
                           
                             
                               
                                 - 
                                 
                                   A 
                                   μ 
                                 
                               
                               ⁢ 
                               S 
                             
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 Sz 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 Sz 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             d 
                             + 
                           
                         
                       
                       
                         
                           
                             d 
                             - 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               25 
             
           
         
       
     
   
   The next step is preferably to find c ±  in terms of tangential fields at z=0, for example, e xo , h y′o , as next given: 
   
     
       
         
           
             
               
                 
                   
                     d 
                     + 
                   
                   + 
                   
                     d 
                     - 
                   
                 
                 = 
                 
                   e 
                   xo 
                 
               
             
             
               26 
             
           
           
             
               
                 
                   
                     d 
                     + 
                   
                   - 
                   
                     d 
                     - 
                   
                 
                 = 
                 
                   
                     S 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       A 
                       μ 
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     h 
                     
                       
                         y 
                         ′ 
                       
                       ⁢ 
                       o 
                     
                   
                 
               
             
             
               27 
             
           
           
             
               
                 
                   d 
                   ± 
                 
                 = 
                 
                   
                     1 
                     2 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         e 
                         xo 
                       
                       ± 
                       
                         
                           S 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           A 
                           μ 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           h 
                           
                             
                               y 
                               ′ 
                             
                             ⁢ 
                             o 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             
               28 
             
           
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           d 
                           + 
                         
                       
                     
                     
                       
                         
                           d 
                           - 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               
                                 
                                   S 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   μ 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                           
                             
                               I 
                             
                             
                               
                                 
                                   - 
                                   
                                     S 
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   A 
                                   μ 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               e 
                               xo 
                             
                           
                         
                         
                           
                             
                               h 
                               
                                 
                                   y 
                                   ′ 
                                 
                                 ⁢ 
                                 o 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       A 
                                       μ 
                                     
                                     ⁢ 
                                     S 
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               I 
                             
                             
                               
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         μ 
                                       
                                       ⁢ 
                                       S 
                                     
                                     ) 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               e 
                               xo 
                             
                           
                         
                         
                           
                             
                               h 
                               
                                 
                                   y 
                                   ′ 
                                 
                                 ⁢ 
                                 o 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               29 
             
           
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           x 
                         
                       
                     
                     
                       
                         
                           h 
                           
                             y 
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       2 
                     
                     ⁢ 
                     
                       G 
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 e 
                                 xo 
                               
                             
                           
                           
                             
                               
                                 h 
                                 
                                   
                                     y 
                                     ′ 
                                   
                                   ⁢ 
                                   o 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             1 
                             2 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   I 
                                 
                                 
                                   I 
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         μ 
                                       
                                       ⁢ 
                                       S 
                                     
                                     ) 
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       ( 
                                       
                                         
                                           A 
                                           μ 
                                         
                                         ⁢ 
                                         S 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     Sz 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   ⅇ 
                                   
                                     
                                       - 
                                       j 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       k 
                                       o 
                                     
                                     ⁢ 
                                     Sz 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       A 
                                       μ 
                                     
                                     ⁢ 
                                     S 
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               I 
                             
                             
                               
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         A 
                                         μ 
                                       
                                       ⁢ 
                                       S 
                                     
                                     ) 
                                   
                                   - 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               e 
                               xo 
                             
                           
                         
                         
                           
                             
                               h 
                               
                                 
                                   y 
                                   ′ 
                                 
                                 ⁢ 
                                 o 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               30 
             
           
         
       
     
   
   An impedance Z n  is preferably defined, which relates the tangential magnetic field to the tangential electric field at the top of the nth layer, for example: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Z 
                             n 
                           
                           ⁢ 
                           
                             h 
                             
                               
                                 y 
                                 ′ 
                               
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           h 
                           
                             
                               y 
                               ′ 
                             
                             , 
                             n 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           e 
                           
                             x 
                             , 
                             n 
                           
                         
                       
                     
                     
                       
                         
                           h 
                           
                             
                               y 
                               ′ 
                             
                             , 
                             n 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               31 
             
           
         
       
     
   
   Equation 30 is preferably rewritten as: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Z 
                             n 
                           
                           ⁢ 
                           
                             h 
                             
                               
                                 y 
                                 ′ 
                               
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           h 
                           
                             
                               y 
                               ′ 
                             
                             , 
                             n 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         
                           1 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 I 
                               
                               
                                 I 
                               
                             
                             
                               
                                 
                                   V 
                                   n 
                                 
                               
                               
                                 
                                   - 
                                   
                                     V 
                                     n 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                           
                       
                       [ 
                       
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   n 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               ⅇ 
                               
                                 
                                   - 
                                   j 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   k 
                                   o 
                                 
                                 ⁢ 
                                 
                                   S 
                                   n 
                                 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                         
                     
                     [ 
                     
                       
                         
                           I 
                         
                         
                           
                             V 
                             n 
                             
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           I 
                         
                         
                           
                             - 
                             
                               V 
                               n 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                   [ 
                   
                     
                       
                         
                           
                             Z 
                             
                               n 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             h 
                             
                               
                                 y 
                                 ′ 
                               
                               , 
                               
                                 n 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           h 
                           
                             
                               y 
                               ′ 
                             
                             , 
                             
                               n 
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               32 
             
           
         
       
     
   
   Where
 
 V   n   =A   μ,n   S   n .
 
   At this point there is preferably defined a value R′ b,n  as given below: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           
                             b 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               b 
                               , 
                               n 
                             
                             ′ 
                           
                           ⁢ 
                           
                             e 
                             
                               b 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               
                                 V 
                                 n 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                           
                             
                               I 
                             
                             
                               
                                 - 
                                 
                                   V 
                                   n 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       [ 
                       
                         
                           
                             
                               
                                 Z 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 h 
                                 
                                   
                                     y 
                                     ′ 
                                   
                                   , 
                                   
                                     n 
                                     - 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               h 
                               
                                 
                                   y 
                                   ′ 
                                 
                                 , 
                                 
                                   n 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⇒ 
                     
                       R 
                       
                         b 
                         , 
                         n 
                       
                       ′ 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           Z 
                           
                             n 
                             - 
                             1 
                           
                         
                         - 
                         
                           V 
                           n 
                           
                             - 
                             1 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             Z 
                             
                               n 
                               - 
                               1 
                             
                           
                           + 
                           
                             V 
                             n 
                             
                               - 
                               1 
                             
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
             
             
               33 
             
           
         
       
     
   
   Equation 23 is then preferably rewritten as:
 
 R′   b,n =( Z   n−1   −V   n   −1 )( Z   n−1   +V   n   −1 ) −1   =W   n   R   b,n   W   n   −1    34
 
   Where
 
 R   b,n =( D   0,n   −O   n   −1 )( D   0,n   +O   n   −1 ) −1    35
 
 D   0,n   =W   n′   −1   Z   n−1    36
 
 O   n   =A   μ,n   W   n   K   z,n   =V   n   W   n    37
 
   There is then preferably defined a value R′ t,n  as given below: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                           ⁢ 
                           
                             e 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     S 
                                     n 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     S 
                                     n 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       [ 
                       
                         
                           
                             
                               e 
                               
                                 b 
                                 , 
                                 n 
                               
                               ′ 
                             
                           
                         
                         
                           
                             
                               
                                 R 
                                 
                                   b 
                                   , 
                                   n 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   b 
                                   , 
                                   n 
                                 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⇒ 
                     
                       R 
                       
                         t 
                         , 
                         n 
                       
                       ′ 
                     
                   
                   = 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           o 
                         
                         ⁢ 
                         
                           S 
                           n 
                         
                         ⁢ 
                         z 
                       
                     
                     ⁢ 
                     
                       R 
                       
                         b 
                         , 
                         n 
                       
                       ′ 
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           o 
                         
                         ⁢ 
                         
                           S 
                           n 
                         
                         ⁢ 
                         z 
                       
                     
                   
                 
               
             
             
               38 
             
           
         
       
     
   
   or equivalently,
 
 R′   t,n   =e   −jk     o     S     n     z   R′   b,n   e   −jk     o     S     n     z   =W   n   e   −jk     o     K     z,n     z   R   b,n   e   −jk     o     K     z,n     z   W   n   −1   =W   n   R   t,n   W   n   −1    39
 
   Where:
 
 R   t,n   =e   −jk     o     K     z,n     z   R   b,n   e   −jk     o     K     z,n     z    40
 
   Substituting equation 38 into equation 32 yields: 
   
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             Z 
                             n 
                           
                           ⁢ 
                           
                             h 
                             
                               
                                 y 
                                 ′ 
                               
                               , 
                               n 
                             
                           
                         
                       
                     
                     
                       
                         
                           h 
                           
                             
                               y 
                               ′ 
                             
                             , 
                             n 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       2 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             I 
                           
                           
                             I 
                           
                         
                         
                           
                             
                               V 
                               n 
                             
                           
                           
                             
                               - 
                               
                                 V 
                                 n 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   ⁢ 
                   
                       
                   
                   [ 
                   
                     
                       
                         
                           e 
                           
                             t 
                             , 
                             n 
                           
                           ′ 
                         
                       
                     
                     
                       
                         
                           
                             R 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                           ⁢ 
                           
                             e 
                             
                               t 
                               , 
                               n 
                             
                             ′ 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               41 
             
           
         
       
     
   
   Equation 41 then implies:
 
 Z   n =( I+R′   t,n )[ V   n ( I−R′   t,n )] −1    42
 
   With substitution into equation 39 then yielding:
 
 Z   n   =W   n   Z′   n   O   n   −1    43
 
   Where:
 
 Z′   n =( I+R   t,n )( I−R   t,n ) −1    44
 
RIGOROUS COUPLED WAVE ANALYSIS GRATING LAYER DERIVATIVE FORMULATION
 
   The next step of the method is preferably to derive equations to analytically compute the derivatives of the reflectance coefficient, for example R b,N , in isotropic space at the top of an N-layer film stack with respect to the parameters of a grating layer, as depicted in  FIG. 2 . 
   The derivative computation is preferably assumed to consist of three steps, for example:
     1. Computation of the impedance derivative at the grating layer, i.e., ∂Z n  computation.   2. Computation of the derivative of the impedance matrix at N−1 using iteration, i.e., ∂Z N−1 =f d (f d ( . . . f d (∂Z n ) . . . )) where f d ( ) is a function which relates ∂Z n  to ∂Z n+1  for 0&lt;n&lt;N.   3. Computation of ∂R b,N  from ∂Z N−1      

   The computations associated with these three steps are given in order below. 
   IMPEDANCE DERIVATIVE AT THE GRATING LAYER: ∂Z n  COMPUTATION 
   First, there is preferably derived an expression for ∂Z n , when the parameter over which the derivative taken is in the n th  layer. This preferably starts with the impedance equations 42, 43, 39, 34, and 35 respectively, as given below:
 
 Z   n =( I+R′   t,n )[ V   n ( I−R′   t,n )] −1   =W   n   Z′   n   O   n   −1    45
 
 R′   t,n   =e   −k     o     S     n     z   R′   b,n   e   −k     o     S     n     z   =W   n   e   −k     o     K     z,n     z   W   n   −1   R′   b,n   W   n   e   −k     o     K     z,n     z   W   n   −1    46
 
 R′   b,n= ( Z   n−1   −V   n   −1 )( Z   n−1   +V   n   −1 ) −1   =W   n ( D   n   −O   n   −1 )( D   n   +O   n   −1 ) −1   W   n   −1   =W   n   R   b,n   W   n   −1    47
 
   Where:
 
 D   n   =W   n   −1   Z   n−1    48
 
 V   n   =A   μ,n   S   n   =O   n   W   n   −1    49
 
   Equation 46 is preferably rewritten as:
 
 R′   t,n   =e   −jk     o     S     n     z   R′   b,n   e   −jk     o     S     n     z   =W   n   e   −jk     o     K     z,n     z   R   b,n   e   −jk     o     K     z,n     z   W   n   −1   =W   n   R   t,n   W   n   −1    50
 
   It is noted that equation 1 is used instead of the expression Z n =W n (I+R t,n )(I−R t,n ) −1 O n   −1 , because preferably only analytical expressions are used for the derivatives of e −jk     o     S     n     z  and V n . 
   Therefore,
 
∂ Z   n   =∂R′   t,n   [V   n ( I−R′   t,n )] −1 −( I+R′   t,n )[ V   n ( I−R′   t,n )] −1   ∂[V   n ( I−R′   t,n )] V   n ( I−R′   t,n )] −1    51
 
∂ Z   n (∂ R′   t,n   −Z   n   ∂[V   n ( I−R′   t,n )])[ V   n ( I−R′   t,n )] −1    52
 
   Expanding the expression ∂└V n (I−R′ t,n )┘ yields:
 
∂ Z   n =(∂ R′   t,n   −Z   n   [∂V   n ( I−R′   t,n )− V   n   ∂R′   t,n ])[ V   n ( I−R′   t,n )] −1    53
 
 ∂Z   n =(∂ R′   t,n   +Z   n   V   n   ∂R′   t,n   −Z   n   ∂V   n ( I−R′   t,n ))[ V   n ( I−R′   t,n )] −1    54
 
   Substituting in the expression V n =Z μ,n S n =O n W n   −1  yields:
 
∂ Z   n =(∂ R′   t,n   +Z   n   O   n   W   n   −1   ∂R′   t,n   −Z   n   ∂V   n   +Z   n   ∂V   n   R′   t,n )[ O   n   W   n   −1 ( I−R′   t,n )] −1    55
 
   Substituting in equation 45 from above, Z n =W n Z′ n O n   −1  yields: 56
 
∂ Z   n =(∂ R′   t,n   +W   n   Z′   n   W   n   −1   ∂R′   t,n   −W   n   Z′   n   O   n   −1   ∂V   n ( I−R′   t,n ))[ W   n   −1 ( I−R′   t,n )] −1   O   n   −1    57
 
   Substituting in R′ t,n =W n R t,n W n   −1  yields:
 
∂ Z   n =( ∂R′   t,n   +W   n   Z′   n   W   n   −1   ∂R′   t,n   −W   n   Z′   n   O   n   −1   ∂V   n ( I−W   n   R   t,n   W   n   −1 ))[ W   n   −1 ( I−W   n   R   t,n   W   n   −1 )] −1   O   n   −1    58
 
   Then by factoring W n  to the left yields:
 
∂ Z   n   =W   n ( W   n   −1   ∂R′   t,n   +Z′   n   W   n   −1   ∂R′   t,n   −Z′   n   O   n   −1   ∂V   n   W   n ( I−R   t,n ) W   n   −1 )[ W   n   −1   W   n ( I−R   t,n ) W   n   −1 ] −1   O   n   −1    59
 
∂ Z   n   =W   n ( W   n   −1   ∂R′   t,n   +Z′   n   W   n   −1   ∂R′   t,n   −Z′   n   O   n   −1   ∂V   n   W   n ( I−R   t,n ) W   n   −1 ) W   n ( I−R   t,n ) −1   O   n   −1    60
 
   The next step is preferably to multiply through with W n  from the right side, yielding:
 
 ∂Z   n   =W   n ( W   n   −1   ∂R′   t,n   W   n   +Z′   n   W   n   −1   ∂R′   t,n   W   n   −Z′   n   O   n   −1   ∂V   n   W   n ( I−R   t,n ))( I−R   t,n ) −1   O   n   −1    61
 
   Find
 
∂ V   n   =∂A   μ,n   S   n   +A   μ,n   ∂S   n   =∂A   μ,n   W   n   K   z,n   W   n   −1   +A   μ,n   ∂S   n    62
 
   Recall
 
 S   n   =W   n   K   z,n   W   n   −1    63
 
 ∂S   n   =W   n   X   n   W   n   −1    64
 
   Where F n =Fourier series coefficient matrix (frequency independent). 
   
     
       
         
           
             X 
             n 
           
           = 
           
             
               
                 
                   
                     W 
                     n 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     ∂ 
                     
                       S 
                       n 
                       2 
                     
                   
                   ⁢ 
                   
                     W 
                     n 
                   
                 
                 
                   K 
                   
                     s 
                     , 
                     n 
                   
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 { 
                 
                   
                     
                       C 
                       ≡ 
                       
                         A 
                         B 
                       
                     
                     : 
                     
                       c 
                       mn 
                     
                   
                   = 
                   
                     
                       a 
                       mn 
                     
                     
                       b 
                       mn 
                     
                   
                 
                 } 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   K 
                   
                     s 
                     , 
                     n 
                   
                 
                 : 
                 
                   k 
                   sij 
                 
               
             
             = 
             
               ( 
               
                 
                   k 
                   zi 
                 
                 + 
                 
                   k 
                   zj 
                 
               
               ) 
             
           
         
       
     
   
   Therefore:
 
∂ V   n   =∂A   μ,n   W   n   K   z,n   W   n   −1   +A   μ,n   W   n   X   n   W   n   −1    65
 
Factoring out O n  yields ∂ V   n   =O   n ( O   n   −1   ∂A   μ,n   W   n   K   z,n   +K   z,n   −1   X   n ) W   n   −1   =O   n   Φ   n   W   n   −1    66
 
   Substituting equation 66 into equation 61 yields:
 
∂ Z   n   =W   n ( W   n   −1   ∂R′   t,n   W   n   +Z′   n   W   n   −1   ∂R′   t,n   W   n   −Z′   n   O   n   −1   ∂V   n   W   n ( I−R   t,n ))( I−R   t,n ) −1   O   n   −1    67
 
∂ Z   n   =W   n ( W   n   −1   ∂R′   t,n   W   n   +Z′   n   W   n   −1   ∂R′   t,n   W   n   −Z′   n   Φ   n ( I−R   t,n ))( I−R   t,n ) −1   O   n   −1    68
 
   Finding ∂R′ t,n , and recalling from equation 46 that:
 
 R′   t,n   =e   −jk     o     S     n     z   R′   b,n   e   −jk     o     S     n     z    69
 
   Therefore:
 
 ∂R′   t,n   =∂e   −jk     o     S     n     z   R′   b,n   e   −jk     o     S     n     z   +e   −jk     o     S     n     z   ∂R′   b,n   e   −jk     o     S     n     z   +e   −jk     o     S     n     z   R′   b,n   ∂e   −jk     o     S   n   z    70
 
   Finding ∂R′ b,n , and recalling from equation 47 that:
 
 R′   b,n =( Z   n−1   −V   n   −1 )( Z   n−1   +V   n   −1 ) −1    71
 
 ∂R′   b,n   =−∂V   n   −1 ( Z   n−1   +V   n   −1 ) −1 −( Z   n−1   −V   n   −1 )( Z   n−1   +V   n   −1 ) −1   ∂V   n   −1 ( Z   n−1   +V   n   −1 ) −1    72
 
   With substitution from equation 47, this yields:
 
∂ R′   b,n   =−∂V   n   −1 ( Z   n−1   +V   n   −1 ) −1   −R′   b,n   ∂V   n   −1 ( Z   n−1   +V   n   −1 ) −1    73
 
   Factoring yields:
 
∂ R′   b,n −−( I+R′   b,n )∂ V   n   −1 ( Z   n−1   +V   n   −1 ) −1 =−( I+R′   b,n )∂ V   n   −1 ( D   n   +O   n   −1 ) −1   W   n   −1    74
 
   Finding
 
∂ V   n   −1   =−V   n   −1   O   n   Φ   n   W   n   −1   V   n   −1 =−( W   n   O   n   −1 ) O   n   Φ   n   W   n   −1 ( W   n   O   n   −1 )=− W   n   Φ   n   O   n   −1    75
 
   And then substituting back into equation 74 yields:
 
∂ R′   b,n =( I+R′   b,n ) W   n   Φ   n   O   n   −1 ( D   n   +jO   n   −1 ) −1   W   n   −1    76
 
   Substitute from equation 47 yields:
 
 R′   b,n   =W   n   R   b,n   W   n   −1    77
 
 ∂R′   b,n   =W   n ( I+R   b,n )Φ n   O   n   −1 ( D   n   +jO   n   −1 ) −1   W   n   −1    78
 
 ∂R′   b,n   =W   n ( I+R   b,n )Φ n   O   n   −1   D   l,n   W   n   −1    79
 
   Where:
 
 D   l,n =( D   n   +O   n   −1 ) −1  
 
   It is noted that:
 
 ∂e   −jk     o     S     n     z   =W   n   Y   —,n   W   n   −1    80
 
   Where:
 
 Y   ±,n   =X   n   ×K   −,n  where { C≡A×B: c   mn   =a   mn   b   mn }  81
 
   
     
       
         
           
             
               
                 
                   
                     K 
                     
                       ± 
                       
                         , 
                         n 
                       
                     
                   
                   : 
                   
                     k 
                     
                       
                         ± 
                         p 
                       
                       , 
                       q 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           p 
                           ≠ 
                           
                             q 
                             : 
                             
                               ± 
                               
                                 
                                   ( 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           ± 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           k 
                                           zp 
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     - 
                                     
                                       ⅇ 
                                       
                                         
                                           ± 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           k 
                                           qj 
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   ( 
                                   
                                     
                                       k 
                                       zp 
                                     
                                     - 
                                     
                                       k 
                                       zq 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           p 
                           = 
                           
                             q 
                             : 
                             
                               
                                 ± 
                                 j 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   
                                     ± 
                                     j 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     k 
                                     o 
                                   
                                   ⁢ 
                                   
                                     k 
                                     zp 
                                   
                                   ⁢ 
                                   z 
                                 
                               
                               ⁢ 
                               z 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               82 
             
           
         
       
     
   
   Substituting equations 80 and 81 and R′ b,n =W n R b,n W n   −1  into equation 70 yields:
 
∂ R′   t,n   =W   n ( Y   −,n   R   b,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   W   n   −1   ∂R′   b,n   W   n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   R   b,n   Y   −,n ) W   n   −1    83
 
   Substituting equation 79 into equation 83 yields:
 
∂ R′   t,n   =W   n ( Y   −,n   R   b,n   e   −jk     o     K   z,n   z   +e   −jk     o     K     z,n     z ( I+R   b,n )Φ n   O   n   −1   D   l,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   R   b,n   Y   −,n ) W   n   −1    84
 
∂ R′   t,n   =W   n ( Y   −,n   R   b,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   Φ   n   O   n   −1   D   l,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   R   b,n   Φ   n   O   n   −1   D   l,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   R   b,n   Y   −,n ) W   n   −1    85
 
 ∂R′   t,n   =W   n ( Y   −,n   R   b,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z Φ n   O   n   −1   D   l,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z   R   b,n (Φ n   O   n   −1   D   l,n   e   −jk     o     K     z,n     z   +Y   −,n )) W   n   −1    86
 
 ∂R′   t,n   =W   n ( Y   −,n   R   b,n   e   −jk     o     K     z,n     z   +e   −jk     o     K     z,n     z Θ n   +e   −jk     o     K     z,n     z   R   b,n (Θ n   +Y   −,n )) W   n   −1    87
 
∂ R′   t,n   =W   n   A   n   W   n   −1    88
 
   Where:
 
Θ n   =Φ   n   O   n   −1   D   l,n   e   −jk     o     K     z,n     z    89
 
   Substituting equation 88 into equation 68 yields:
 
 ∂Z   n   =W   n ( W   n   −1   ∂R′   t,n   W   n   +Z′   n   W   n   −1   ∂R′   t,n   W   n   −Z′   n Φ n ( I−R   t,n ))( I−R   t,n ) −1   O   n   −1    90
 
 ∂Z   n   =W   n ( A   n   +Z′   n   A   n   −Z′   n Φ n ( I−R   t,n ))( I−R   t,n ) −1   O   n   −1    91
 
 ∂Z   n   =W   n (( I+Z′   n ) A   n ( I−R   t,n ) −1   −Z′   n Φ n ) O   n   −1    92
 
   Rearranging these equations yields:
 
 ∂Z   n   =W   n (( I+Z′   n )Ω n   −Z′   n Φ n ) O   n   −1   =W   n (Ω n   +Z′   n (Ω n   −Φ   n )) O   n   −1    93
 
   Where:
 
Ω n   =Λ   n ( I−R   t,n ) −1    94
 
PROPAGATION OF IMPEDANCE DERIVATIVE ABOVE GRATING LAYER: F D( )  
 
   This process preferably begins with the expression for the impedance from equation 45, given below again:
 
 Z   n   =W   n   Z′   n   O   n   −1    95
 
 Z′   n =( I+R   t,n )( I−R   t,n ) −1    96
 
 R   t,n   =e   −jk     o     K     z,n     z   R   b,n   e   −jk     o     K     z,n     z    97
 
 R   b,n =( D   0,n   −O   n   −1 ) D   1,n    98
 
 D   l,n =( D   0,n   +O   n   −1 ) −1    99
 
 D   0,n   =W   n   −1   Z   n−1    100
 
 O   n   −1   =K   z,n   −1   W   n   −1   A   μ,n   −1    101
 
 C   n   =└K   y,n   M   n   −1   K   y,n   −E   n ┘  102
 
   F n =Fourier series coefficient matrix (frequency independent)
 
 A   μ,n =(μ line,n   −1 −μ sp,n   −1 ) F   n +(μ sp,n   −1 ) I    103
 
 S   n   2   =A   μ,n   −1   C   n    104
 
   Taking the derivative of equation 95 yields:
 
 ∂Z   n   =W   n   ∂Z′   n   O   n   −1    105
 
   Taking the derivative of equation 96 yields:
 
∂ Z′   n   =∂R   t,n ( I−R   t,n ) −1 +( I+R   t,n )( I−R   t,n ) −1   ∂R   t,n ( I−R   t,n ) −1    106
 
∂ Z′   n =( I+Z′   n )∂ R   t,n ( I−R   t,n ) −1    107
 
   Taking the derivative of equation 47 yields:
 
∂ R   t,n   =e   −jk     o     K     z,n     z   ∂R   b,n   e   −jk     o     K     z,n     z    108
 
   Taking the derivative of equation 48 yields:
 
∂ R   b,n   =∂D   0,n   D   1,n +( D   0,n   −O   n   −1 )∂ D   1,n    109
 
   Taking the derivative of equation 49 yields:
 
 ∂D   1,n   =−D   1,n   ∂D   0,n   D   1,n    110
 
   Taking the derivative of equation 50 yields:
 
∂ D   0,n   =W   n   −1   ∂Z   n−1    111
 
   Substituting equation 110 into equation 111 finally yields:
 
∂ R   b,n   =∂D   0,n   D   1,n −( D   0,n   −O   n   −1 ) D   1,n   ∂D   0,n   D   1,n    112
 
∂ R   b,n =( ∂D   0,n   −R   b,n   ∂D   0,n ) D   1,n =( I−R   b,n )∂ D   0,n   D   1,n    113
 
VARIOUS DERIVATIVE DERIVATIONS
 
   The proofs below are offered as support for the derivations provided above. 
   First proof: Prove: 
   
     
       
         
           
             
               
                 ∂ 
                 
                   ⅇ 
                   
                     
                       ± 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       o 
                     
                     ⁢ 
                     Sz 
                   
                 
               
               
                 ∂ 
                 p 
               
             
             ⁢ 
             
               ❘ 
               
                 p 
                 ∉ 
                 
                   { 
                   
                     
                       k 
                       o 
                     
                     , 
                     z 
                   
                   } 
                 
               
             
           
           = 
           
             
               W 
               ⁡ 
               
                 [ 
                 
                   X 
                   × 
                   
                     K 
                     ± 
                   
                 
                 ] 
               
             
             ⁢ 
             
               W 
               
                 - 
                 1 
               
             
           
         
       
     
   
   Where: 
   
     
       
         
           
             X 
             = 
             
               
                 W 
                 
                   - 
                   1 
                 
               
               ⁢ 
               
                 ∂ 
                 
                   k 
                   o 
                 
               
               ⁢ 
               SW 
             
           
           , 
           
             { 
             
               
                 C 
                 ≡ 
                 
                   A 
                   × 
                   
                     B 
                     : 
                     
                       c 
                       mn 
                     
                   
                 
               
               = 
               
                 
                   a 
                   mn 
                 
                 ⁢ 
                 
                   b 
                   mn 
                 
               
             
             } 
           
           , 
           
             
 
           
           ⁢ 
           
             
               
                 K 
                 ± 
               
               : 
               
                 k 
                 
                   ± 
                   mn 
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       m 
                       ≠ 
                       
                         n 
                         : 
                         
                           ± 
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         ⅇ 
                                         
                                           
                                             ± 
                                             j 
                                           
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             k 
                                             o 
                                           
                                           ⁢ 
                                           
                                             k 
                                             zm 
                                           
                                           ⁢ 
                                           z 
                                         
                                       
                                       - 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           ± 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           k 
                                           zn 
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               ( 
                               
                                 
                                   k 
                                   zm 
                                 
                                 - 
                                 
                                   k 
                                   zn 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       m 
                       = 
                       
                         n 
                         : 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               j 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               
                                 k 
                                 zn 
                               
                               ⁢ 
                               z 
                             
                           
                           ⁢ 
                           z 
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Proof: 
   
     
       
         
           
             
               
                 
                   ∂ 
                   
                     e 
                     
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Sz 
                     
                   
                 
                 = 
                 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       j 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         z 
                       
                       ⁢ 
                       
                         
                           e 
                           
                             
                               
                                 + 
                                 _ 
                               
                               ⁢ 
                               
                                 jk 
                                 o 
                               
                             
                             ⁢ 
                             
                               S 
                               ⁡ 
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   u 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               S 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           e 
                           
                             
                               
                                 + 
                                 _ 
                               
                               ⁢ 
                               
                                 jk 
                                 o 
                               
                             
                             ⁢ 
                             Su 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           u 
                         
                       
                     
                   
                   = 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       j 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         z 
                       
                       ⁢ 
                       
                         
                           We 
                           
                             
                               
                                 + 
                                 _ 
                               
                               ⁢ 
                               
                                 jk 
                                 o 
                               
                             
                             ⁢ 
                             
                               
                                 K 
                                 z 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   u 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           W 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               S 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           We 
                           
                             
                               
                                 + 
                                 _ 
                               
                               ⁢ 
                               
                                 jk 
                                 o 
                               
                             
                             ⁢ 
                             
                               K 
                               z 
                             
                             ⁢ 
                             u 
                           
                         
                         ⁢ 
                         
                           W 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           u 
                         
                       
                     
                   
                 
               
             
             
               
                 i 
                 ) 
               
             
           
         
       
     
   
   (From Wilcox, 1967). 
   Define: 
   ii) X=W −1 ∂(k o S)W 
   Hence: 
   
     
       
         
           
             ∂ 
             
               ⅇ 
               
                 
                   ± 
                   j 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   o 
                 
                 ⁢ 
                 Sz 
               
             
           
           = 
           
             
               ± 
               
                 W 
                 ⁡ 
                 
                   [ 
                   
                     j 
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         z 
                       
                       ⁢ 
                       
                         
                           ⅇ 
                           
                             
                               ± 
                               j 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               k 
                               o 
                             
                             ⁢ 
                             
                               
                                 K 
                                 z 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   u 
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         X 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             
                               ± 
                               j 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               k 
                               o 
                             
                             ⁢ 
                             
                               K 
                               z 
                             
                             ⁢ 
                             u 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           u 
                         
                       
                     
                   
                   ] 
                 
               
             
             ⁢ 
             
               W 
               
                 - 
                 1 
               
             
           
         
       
     
   
   iii) Expand integrand of equation ii: 
   
     
       
         
           
             
               ⅇ 
               
                 
                   ± 
                   j 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   o 
                 
                 ⁢ 
                 
                   
                     K 
                     z 
                   
                   ⁡ 
                   
                     ( 
                     
                       z 
                       - 
                       u 
                     
                     ) 
                   
                 
               
             
             ⁢ 
             X 
             ⁢ 
             
                 
             
             ⁢ 
             
               ⅇ 
               
                 
                   ± 
                   j 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   o 
                 
                 ⁢ 
                 
                   K 
                   z 
                 
                 ⁢ 
                 u 
               
             
           
           = 
           
             
               k 
               o 
             
             ⁡ 
             
               [ 
               
                 
                   
                     
                       
                         x 
                         00 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             k 
                             zo 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               k 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     k 
                                     z0 
                                   
                                 
                                 + 
                                 
                                   k 
                                   z0 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           u 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         x 
                         
                           0 
                           ⁢ 
                           N 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             k 
                             zo 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               k 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     k 
                                     z0 
                                   
                                 
                                 + 
                                 
                                   k 
                                   zN 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           u 
                         
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     
                       
                         x 
                         mn 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             k 
                             zm 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               k 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     k 
                                     zm 
                                   
                                 
                                 + 
                                 
                                   k 
                                   zn 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           u 
                         
                       
                     
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         x 
                         N0 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             k 
                             zN 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               k 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     k 
                                     zN 
                                   
                                 
                                 + 
                                 
                                   k 
                                   z0 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           u 
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         x 
                         NN 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             k 
                             o 
                           
                           ⁢ 
                           
                             k 
                             zN 
                           
                           ⁢ 
                           z 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             ± 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               k 
                               o 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     k 
                                     zN 
                                   
                                 
                                 + 
                                 
                                   k 
                                   zN 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           u 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
   
   iv) (m, n) th  element of 
   
     
       
         
           
             m 
             ≠ 
             
               n 
               : 
               
                 
                   ± 
                   j 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   o 
                 
                 ⁢ 
                 
                   x 
                   mn 
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       ± 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       o 
                     
                     ⁢ 
                     
                       k 
                       zm 
                     
                     ⁢ 
                     z 
                   
                 
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     z 
                   
                   ⁢ 
                   
                     
                       ⅇ 
                       
                         
                           ± 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             k 
                             o 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   k 
                                   zm 
                                 
                               
                               + 
                               
                                 k 
                                 zn 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         u 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅆ 
                       u 
                     
                   
                 
               
             
           
           = 
           
             
               
                 ± 
                 
                   x 
                   mn 
                 
               
               ⁢ 
               
                 ⅇ 
                 
                   
                     ± 
                     j 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     k 
                     o 
                   
                   ⁢ 
                   
                     k 
                     zm 
                   
                   ⁢ 
                   z 
                 
               
               ⁢ 
               
                 
                   ⅇ 
                   
                     
                       ± 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         k 
                         o 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               k 
                               zm 
                             
                           
                           + 
                           
                             k 
                             zn 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     u 
                   
                 
                 
                   ( 
                   
                     
                       - 
                       
                         k 
                         zm 
                       
                     
                     + 
                     
                       k 
                       zn 
                     
                   
                   ) 
                 
               
             
             ⁢ 
             
               | 
               0 
               z 
             
             ⁢ 
             
               
                 ± 
                 
                   x 
                   mn 
                 
               
               ⁢ 
               
                 
                   
                     
                       
                         ( 
                         
                           
                             ⅇ 
                             
                               
                                 ± 
                                 j 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 o 
                               
                               ⁢ 
                               
                                 k 
                                 zm 
                               
                               ⁢ 
                               z 
                             
                           
                           - 
                         
                       
                     
                   
                   
                     
                       
                         
                           ⅇ 
                           
                             
                               ± 
                               j 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               k 
                               o 
                             
                             ⁢ 
                             
                               k 
                               zn 
                             
                             ⁢ 
                             z 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   ( 
                   
                     
                       k 
                       zm 
                     
                     - 
                     
                       k 
                       zn 
                     
                   
                   ) 
                 
               
             
           
         
       
     
   
   
     
       
         
           
             m 
             ≠ 
             
               n 
               : 
               
                 
                   ± 
                   j 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   o 
                 
                 ⁢ 
                 
                   x 
                   mn 
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       ± 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       o 
                     
                     ⁢ 
                     
                       k 
                       zm 
                     
                     ⁢ 
                     z 
                   
                 
                 ⁢ 
                 
                   
                     ∫ 
                     0 
                     z 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅆ 
                     u 
                   
                 
               
             
           
           = 
           
             
               
                 
                   ± 
                   j 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   k 
                   o 
                 
                 ⁢ 
                 
                   x 
                   mn 
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       ± 
                       j 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       k 
                       o 
                     
                     ⁢ 
                     
                       k 
                       zm 
                     
                     ⁢ 
                     z 
                   
                 
                 ⁢ 
                 u 
               
               ⁢ 
               
                 | 
                 0 
                 z 
               
             
             = 
             
               
                 ± 
                 j 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 k 
                 o 
               
               ⁢ 
               
                 x 
                 mm 
               
               ⁢ 
               
                 ⅇ 
                 
                   
                     ± 
                     j 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     k 
                     o 
                   
                   ⁢ 
                   
                     k 
                     zn 
                   
                   ⁢ 
                   z 
                 
               
               ⁢ 
               z 
             
           
         
       
     
   
   v) Therefore, 
   
     
       
         
           
             
               
                 + 
                 _ 
               
               ⁢ 
               j 
             
             ⁢ 
             
               
                 ∫ 
                 0 
                 z 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       jk 
                       o 
                     
                     ⁢ 
                     
                       
                         K 
                         z 
                       
                       ⁡ 
                       
                         ( 
                         
                           z 
                           - 
                           u 
                         
                         ) 
                       
                     
                   
                 
                 ⁢ 
                 
                   Xe 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     
                       K 
                       z 
                     
                     ⁢ 
                     u 
                   
                 
                 ⁢ 
                 
                   ⅆ 
                   u 
                 
               
             
           
           = 
           
             X 
             × 
             
               K 
               
                 
                   + 
                   _ 
                 
                 ⁢ 
                 
                     
                 
               
             
           
         
       
     
   
   where: 
   
     
       
         
           
             
               { 
               
                 
                   C 
                   ≡ 
                   
                     A 
                     × 
                     
                       B 
                       : 
                       
                         c 
                         mn 
                       
                     
                   
                 
                 = 
                 
                   
                     a 
                     mn 
                   
                   ⁢ 
                   
                     b 
                     mn 
                   
                 
               
               } 
             
             ⁢ 
             
                 
             
             ⁢ 
             and 
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 K 
                 ± 
               
               : 
               
                 k 
                 
                   ± 
                   mn 
                 
               
             
           
           = 
           
             { 
             
               
                 
                   
                     m 
                     ≠ 
                     
                       n 
                       : 
                       
                         ± 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       ⅇ 
                                       
                                         
                                           ± 
                                           j 
                                         
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           k 
                                           o 
                                         
                                         ⁢ 
                                         
                                           k 
                                           zm 
                                         
                                         ⁢ 
                                         z 
                                       
                                     
                                     - 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ⅇ 
                                     
                                       
                                         ± 
                                         j 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         k 
                                         o 
                                       
                                       ⁢ 
                                       
                                         k 
                                         zn 
                                       
                                       ⁢ 
                                       z 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             ( 
                             
                               
                                 k 
                                 zm 
                               
                               - 
                               
                                 k 
                                 zn 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     m 
                     = 
                     
                       n 
                       : 
                       
                         
                           ± 
                           j 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           o 
                         
                         ⁢ 
                         
                           ⅇ 
                           
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               k 
                               o 
                             
                             ⁢ 
                             
                               k 
                               zn 
                             
                             ⁢ 
                             z 
                           
                         
                         ⁢ 
                         z 
                       
                     
                   
                 
               
             
           
         
       
     
   
   iv) Proof follows by substitution of equation v into equation ii. 
   Second proof: Prove: 
   
     
       
         
           
             ∂ 
             
               ( 
               S 
               ) 
             
           
           = 
           
             
               W 
               ⁡ 
               
                 [ 
                 
                   
                     
                       W 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       ∂ 
                       
                         S 
                         2 
                       
                     
                     ⁢ 
                     W 
                   
                   
                     K 
                     s 
                   
                 
                 ] 
               
             
             ⁢ 
             
               W 
               
                 - 
                 1 
               
             
           
         
       
     
   
   Where: 
   
     
       
         
           { 
           
             
               
                 C 
                 ≡ 
                 
                   A 
                   B 
                 
               
               : 
               
                 c 
                 mn 
               
             
             = 
             
               
                 a 
                 mn 
               
               
                 b 
                 mn 
               
             
           
           } 
         
       
     
   
   and
 
 K   s   : k   smn =( k   zm   +k   zn )
 
   Proof: 
   i) ∂(S) 2   =∂(S)S+S∂(S)    
   ii) S=(WK z W −1 ) 
   iii) Substitute equation ii into equation i: W −1 ∂S 2 W=└W −1 ∂(S)W┘K z +K z └W −1 ∂(S)W┘ 
   iv) Define X=W −1 ∂(k o S)W and expand equation iii: 
   
     
       
         
           
             
               W 
               
                 - 
                 1 
               
             
             ⁢ 
             
               ∂ 
               
                 
                   ( 
                   S 
                   ) 
                 
                 2 
               
             
             ⁢ 
             W 
           
           = 
           
             
               
                 XK 
                 z 
               
               + 
               
                 
                   K 
                   z 
                 
                 ⁢ 
                 X 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         x 
                         00 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             k 
                             z0 
                           
                           + 
                           
                             k 
                             z0 
                           
                         
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         x 
                         
                           0 
                           ⁢ 
                           N 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             k 
                             z0 
                           
                           + 
                           
                             k 
                             zN 
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     
                       
                         x 
                         mn 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             k 
                             zm 
                           
                           + 
                           
                             k 
                             zn 
                           
                         
                         ) 
                       
                     
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         x 
                         N0 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             k 
                             zN 
                           
                           + 
                           
                             k 
                             z0 
                           
                         
                         ) 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       
                         x 
                         NN 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             k 
                             zN 
                           
                           + 
                           
                             k 
                             zN 
                           
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
   
   v) From inspection of iv) 
   
     
       
         
           X 
           = 
           
             
               
                 [ 
                 
                   
                     
                       W 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       ∂ 
                       
                         
                           ( 
                           
                             
                               k 
                               o 
                             
                             ⁢ 
                             S 
                           
                           ) 
                         
                         2 
                       
                     
                     ⁢ 
                     W 
                   
                   
                     K 
                     s 
                   
                 
                 ] 
               
               ⁢ 
               
                   
               
               ⁢ 
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 { 
                 
                   
                     
                       C 
                       ≡ 
                       
                         A 
                         B 
                       
                     
                     : 
                     
                       c 
                       mn 
                     
                   
                   = 
                   
                     
                       a 
                       mn 
                     
                     
                       b 
                       mn 
                     
                   
                 
                 } 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   K 
                   s 
                 
                 : 
                 
                   k 
                   smn 
                 
               
             
             = 
             
               ( 
               
                 
                   k 
                   zm 
                 
                 + 
                 
                   k 
                   zn 
                 
               
               ) 
             
           
         
       
     
   
   Therefore, 
   
     
       
         
           
             ∂ 
             S 
           
           = 
           
             
               WXW 
               
                 - 
                 1 
               
             
             = 
             
               
                 W 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         W 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         ∂ 
                         
                           S 
                           2 
                         
                       
                       ⁢ 
                       W 
                     
                     
                       K 
                       s 
                     
                   
                   ] 
                 
               
               ⁢ 
               
                 W 
                 
                   - 
                   1 
                 
               
             
           
         
       
     
   
   Third proof: Prove: 
   
     
       
         
           
             ∂ 
             
               ⅇ 
               
                 
                   
                     + 
                     _ 
                   
                   ⁢ 
                   
                     jk 
                     o 
                   
                 
                 ⁢ 
                 Sz 
               
             
           
           = 
           
             
               
                 + 
                 _ 
               
               ⁢ 
               
                 jk 
                 o 
               
             
             ⁢ 
             
               
                 ∫ 
                 0 
                 z 
               
               ⁢ 
               
                 
                   ⅇ 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     
                       S 
                       ⁡ 
                       
                         ( 
                         
                           z 
                           - 
                           u 
                         
                         ) 
                       
                     
                   
                 
                 ⁢ 
                 
                   ∂ 
                   
                     Sⅇ 
                     
                       
                         + 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Su 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   u 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   Wilcox 
                   ⁡ 
                   
                     ( 
                     1967 
                     ) 
                   
                 
               
             
           
         
       
     
   
   Proof: 
   i) Factor e ±jk     o     Sz  from integral in 1) and rewrite: 
   
     
       
         
           
             
               je 
               
                 
                   
                     + 
                     _ 
                   
                   ⁢ 
                   
                     jk 
                     o 
                   
                 
                 ⁢ 
                 Sz 
               
             
             ⁢ 
             
               ∂ 
               
                 e 
                 
                   
                     
                       + 
                       _ 
                     
                     ⁢ 
                     
                       jk 
                       o 
                     
                   
                   ⁢ 
                   Sz 
                 
               
             
           
           = 
           
             
               
                 + 
                 _ 
               
               ⁢ 
               
                 jk 
                 o 
               
             
             ⁢ 
             
               
                 ∫ 
                 0 
                 z 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     Su 
                   
                 
                 ⁢ 
                 
                   ∂ 
                   
                     Se 
                     
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Su 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   u 
                 
               
             
           
         
       
     
   
   ii) Fundamental Theorem of Calculus implies: 
   
     
       
         
           
             
               ∂ 
               
                 ∂ 
                 z 
               
             
             ⁢ 
             
               ( 
               
                 
                   e 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     Sz 
                   
                 
                 ⁢ 
                 
                   ∂ 
                   
                     e 
                     
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Sz 
                     
                   
                 
               
               ) 
             
           
           = 
           
             
               
                 + 
                 _ 
               
               ⁢ 
               
                 jk 
                 o 
               
             
             ⁢ 
             
               e 
               
                 
                   
                     + 
                     _ 
                   
                   ⁢ 
                   
                     jk 
                     o 
                   
                 
                 ⁢ 
                 Sz 
               
             
             ⁢ 
             
               ∂ 
               
                   
               
               ⁢ 
               
                 Se 
                 
                   
                       
                   
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     Sz 
                   
                 
               
             
           
         
       
     
   
   iii) Complete proof by showing that the right hand side of ii) is always equal to the left hand side: 
   iv) Expand right hand side and simplify: 
   
     
       
         
           
             
               ∂ 
               
                 ∂ 
                 z 
               
             
             ⁢ 
             
               ( 
               
                 
                   e 
                   
                     
                         
                     
                     
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Sz 
                     
                   
                 
                 ⁢ 
                 
                   ∂ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     
                       
                           
                       
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                         ⁢ 
                         Sz 
                       
                     
                   
                 
               
               ) 
             
           
           = 
           
             
               
                 ∂ 
                 
                   ∂ 
                   z 
                 
               
               ⁢ 
               
                 e 
                 
                   
                       
                   
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     Sz 
                   
                 
               
               ⁢ 
               
                 ∂ 
                 
                     
                 
                 ⁢ 
                 
                   e 
                   
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                     
                     ⁢ 
                     Sz 
                   
                 
               
             
             + 
             
                 
             
             ⁢ 
             
               
                 e 
                 
                   
                     
                       + 
                       _ 
                     
                     ⁢ 
                     
                       jk 
                       o 
                     
                   
                   ⁢ 
                   Sz 
                 
               
               ⁢ 
               
                 ∂ 
                 
                   ∂ 
                   z 
                 
               
               ⁢ 
               
                 ( 
                 
                   ∂ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Sz 
                     
                   
                 
                 ) 
               
             
           
         
       
     
   
   v) Exchange order of 
           ∂     ∂   z           
and ∂ differentiation in last term:
 
   
     
       
         
           
             
               ∂ 
               
                 ∂ 
                 z 
               
             
             ⁢ 
             
               ( 
               
                 
                   e 
                   
                     
                         
                     
                     
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                       
                       ⁢ 
                       Sz 
                     
                   
                 
                 ⁢ 
                 
                   ∂ 
                   
                     e 
                     
                       
                           
                       
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                         ⁢ 
                         Sz 
                       
                     
                   
                 
               
               ) 
             
           
           = 
           
             
               
                 
                   + 
                   _ 
                 
                 ⁢ 
                 
                   jk 
                   o 
                 
               
               ⁢ 
               
                 Se 
                 
                   
                     
                       + 
                       _ 
                     
                     ⁢ 
                     
                       jk 
                       o 
                     
                   
                   ⁢ 
                   Sz 
                 
               
               ⁢ 
               
                 ∂ 
                 
                   e 
                   
                     
                         
                     
                     
                       
                         + 
                         _ 
                       
                       ⁢ 
                       
                         jk 
                         o 
                       
                       ⁢ 
                       Sz 
                     
                   
                 
               
             
             ⁢ 
             
               + 
               _ 
             
             ⁢ 
             
               
                 jk 
                 o 
               
               ⁢ 
               
                 e 
                 
                   
                     
                       + 
                       _ 
                     
                     ⁢ 
                     
                       jk 
                       o 
                     
                   
                   ⁢ 
                   Sz 
                 
               
               ⁢ 
               
                 ∂ 
                 
                   ( 
                   
                     Se 
                     
                       
                           
                       
                       
                         
                           + 
                           _ 
                         
                         ⁢ 
                         
                           jk 
                           o 
                         
                         ⁢ 
                         Sz 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
             
               ∂ 
               
                 ∂ 
                 z 
               
             
             ⁢ 
             
               ( 
               
                 
                   e 
                   
                     
                         
                     
                     
                       
                         
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   vi) From v), right hand side of ii) must always be equal to left hand side. 
   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.