Analysis of polarization properties of shallow metallic gratings by an extended Rayleigh-Fano Theory Page: 3 of 8
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He express the groove profile in a Fourier series
z « f(x) « £ [apcoB(2pirx/d) + b„sin(2pirx/d) ] ■ fp exp(-iklpx), (2)
where f0 = 0 and 1 *A/d , Using Egs. (1) and (2), we expand Maxwell's boundary conditions
on the grating surface to the second order in f(x). Equating the coefficients of exp(-
iklmx) on both sides of the expanded boundary conditions for a given in, we obtain
Gjmlcos^+yScos^)!^ - <G,m+H,m)coa0-Glr£coe«m-i>clinfmsln0
-pJmRp[ <G' Jm,+H',„)cos(p,+G'Jmp0cos*m+ik2 (m-p) fm1,sin(f/p]
“ 0pJm Dpt (®,smi>+H'jm#)cos0p-G'jmpcos0m-ilc2(in-p)fml)sin#p], (3)
Gjm(cos((lm+/>cos0m)Dm - (G1m+H,m)cose+G1mcos((lm-ikIi»fmsine
■pLR*t(G’ »"*+H' l"»> COB«/»_G' irvWVm+ikl (<n-p) f„.pSin<|!p]
-pJm D,I/S(G'Jm.+H'3mp)cos*p+G'J^cos^-ifcl<m-p)f^sin«„), (4)
where fi - n for s component, ft « 1/n for p component. The explicit expressions for G^,
G'imp* (i - 1, 2, 3 ) and H,m in Eqs.(3) and (4) are
z
+ 1st
(Vacuum)
(Grating Material)
+ 1st
G,0 ■ 1- •j*Jg(0)cos20 ,
G1m “ -ikfmcos0 - 3$k2g(m)cos20 , (mpiO) ,
Gjm “ 1- Jj^gtOJcos2^ ,
G'2mP - ikf^coBV, - Jj/^g (m-p )cos2(|ip,
03m = 1- Jsfi2*2g(O)cos20m ,
G'amp - -ikfifp^cos^p - ijk2n2g(m-p)cos2$p ,
Hlm * k^lh{m)Bin6 ,
H'smp * JcMh (m-p )sin#/B ,
H'smp = T^lnh(m-p)sin$p ,
where
(5)
(6)
Fig. 1. Diffraction of a plane wave g(m) * 2 f^fQ,m ,
by a plane grating. The angles are q
taken as positive when measured in h(m) » 2 gf-pfp.m •
the direction indicated by arrows. "
(7)
It is noted that both R,,, and Dm are expressed as a linear combination of two kinds of
terms. One is the functions of quantities associated only with the incident light (A,p),
diffraction order (m), groove profile (d,f„), grating material (h), and grating mounting
(0,V/m). The other is the terms that represent contributions from diffracted waves (both
reflected and refracted) of the other orders. For a mirror, i.e., fc=0, the contributions
from all the diffracted waves vanish, and Egs.(3) and (4) are reduced to Fresnel formulas.
To determine the complex amplitudes R„, and Dm of the diffracted mth order waves from
the grating, we employ an iterative method with the initial values given by
2
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Koike, Masato (Lawrence Berkeley Lab., CA (United States)) & Namioka, Takeshi (Tohoku Univ., Sendai (Japan). Research Inst. for Scientific Measurements). Analysis of polarization properties of shallow metallic gratings by an extended Rayleigh-Fano Theory, article, June 1, 1991; [Berkeley,] California. (https://digital.library.unt.edu/ark:/67531/metadc1069798/m1/3/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.