RCWA derivation steps in Soifer diffraction grating analysis

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Krzysztow
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Hi,

Recently I'm trying to understand some explanation of RCWA, which introduces to computational analysis of diffraction grating efficiences. For explanation I use Soifer book, but cannot understand or don't agree with it. I tried to understand the derivation, not to write it straightforward into Matlab but understand as well. And I found that my derivation doesn't agreefully with the book. Can You explain to me:
1) how one can get from the set of equations 3.138 & 3.139 to the set 3.140 & 3.141. I got something for 3.140 like in equation (*).
2) where did He get set 3.150 from? It seems as if assumption (**) or (3.127) is not valid anymore. However then He shouldn't write 3.152.
If assumption (3.127) is valid for modulation zone, then result for coefficients is given by (***)

Please, help me. It's not I want anyone to do my work. I just dwell for too long on it. Thank in advance!
 
Attachments
  • Textbook page 177, Diffraction by Transmission Dielectric Gratings: equation 3.103 for the relative error, section 3.3 text, and Figure 3.5 showing a dielectric grating geometry with regions labelled epsilon = 1 and epsilon = 2 and wave vector k.
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  • Scanned textbook page 178 titled Design of DOE Using Electromagnetic Theory, showing equations 3.105 to 3.110 on TM-polarization and Rayleigh expansions with Helmholtz equation 3.108.
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  • Printed textbook page 179, titled Diffraction by Transmission Dielectric Gratings, with equations 3.111 to 3.117 on the continuity of tangential field components and Fourier expansion of quasi-periodic functions.
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More attachements
 
Attachments
  • Scanned textbook page 180, Design of DOE using electromagnetic theory, showing equations 3.118 to 3.122 for Fourier expansions of k squared and 1 over k squared, and the resulting system of differential equations for a binary grating.
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  • Photographed textbook page 181 showing a diagram of diffraction by a dielectric binary grating with zones 1, 2 and 3 and labelled coordinates x1 to x2K, followed by printed equations (3.123) and (3.124) and text on boundary conditions.
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  • Photo of a printed textbook page, page 162, titled Design of DOE Using Electromagnetic Theory, showing equations 3.127 to 3.132 for field expansions and boundary-condition matching in a diffraction grating analysis.
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7th, 8th and 9th pages
 
Attachments
  • Photo of a printed textbook page, page 185, titled Diffraction by Transmission Dielectric Gratings, showing equations 3.147 to 3.153 with boundary conditions and expressions for field coefficients.
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  • Photo of a printed textbook page, page 184, showing a derivation of diffraction grating field coefficients with numbered equations 3.142 to 3.146 and summation expressions for H and E field components.
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  • Photo of a printed textbook page titled Diffraction by Transmission Dielectric Gratings, page 183, showing equations 3.132 to 3.141 with summation expressions and a matrix system for the Rayleigh coefficients.
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10th and 11th and (*)
 
Attachments
  • Photographed page of a textbook on diffraction gratings showing numbered equations 3.154 to 3.160, including the matrix exponential form and the BC boundary condition matrix, with dense printed body text.
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  • Photo of a printed textbook page, page 187, titled Diffraction by Transmission Dielectric Gratings, showing equations 3.161 to 3.164 on the BC matrix, the matrix form for binary gratings, and the start of section 3.3.2 TE-Polarization.
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  • Equation (*): T = exp(-i beta_0 a)(H_01 - (k_0^2 / (i beta_p)) H_11)^(-1)(I + delta_{p-j} beta_0 / beta_p)
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(**) and (***)
 
Attachments
  • Equation (**): H_m(y) = sum from j = -N to N of C_j^+ psi_mj^-(y) plus sum from j = -N to N of C_j^+ psi_mj^+(y).
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  • Labeled equation (***) with a left brace over two lines: C_p^+ = T_p · μ_p and C_p^- = T_p · γ_p.
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