## Abstract

In a recent paper [J. Opt. Soc. Am. A **16**, 1108 (1999)] Logofǎtu *et al*. demonstrated by experimental and numerical evidence that the 0th-order cross-polarization (*s* to *p* and *p* to *s*) reflection coefficients of isotropic, symmetrical, surface-relief gratings in conical mount are identical. Here an analytical proof is given to show that the observed identity is merely a manifestation of the electromagnetic reciprocity theorem for the 0th-order diffraction of symmetrical gratings. The above result is further generalized to bianisotropic gratings, to the 0th-order cross-polarization transmission coefficients, and to the $m\mathrm{th}$-order reflection and transmission coefficients when the wave vector of the incident plane wave and the negative of the wave vector of the $m\mathrm{th}$ reflected order are symmetrical with respect to the plane perpendicular to the grating grooves.

© 2000 Optical Society of America

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### Equations (13)

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(1)
$$|\stackrel{\u02c6}{\mathbf{y}}\xb7{\mathbf{k}}_{i}^{a}|({\mathbf{E}}_{i}^{a}\xb7{\mathbf{E}}_{d}^{b})=|\stackrel{\u02c6}{\mathbf{y}}\xb7{\mathbf{k}}_{d,l}^{a}|({\mathbf{E}}_{i,l}^{b}\xb7{\mathbf{E}}_{d,l}^{a}),\hspace{1em}\hspace{1em}l=T,B.$$
(2)
$$\mathbf{D}=\tilde{\mathit{\u03f5}}\xb7\mathbf{E}+\tilde{\xi}\xb7\mathbf{H},$$
(3)
$$\mathbf{B}=\tilde{\zeta}\xb7\mathbf{E}+\tilde{\mu}\xb7\mathbf{H}.$$
(4)
$$\tilde{\mathit{\u03f5}}={\tilde{\mathit{\u03f5}}}^{\mathrm{T}},\hspace{1em}\hspace{1em}\tilde{\mu}={\tilde{\mu}}^{\mathrm{T}},\hspace{1em}\hspace{1em}\tilde{\xi}=-{\tilde{\zeta}}^{\mathrm{T}},$$
(5)
$${\oint}_{S}\mathrm{d}\mathbf{S}\xb7({\mathbf{E}}^{a}\times {\mathbf{H}}^{b}-{\mathbf{E}}^{b}\times {\mathbf{H}}^{a})=0,$$
(6)
$$\stackrel{\u02c6}{\mathbf{s}}=(\mathbf{k}\times \stackrel{\u02c6}{\mathbf{y}})/|\mathbf{k}\times \stackrel{\u02c6}{\mathbf{y}}|,\hspace{1em}\hspace{1em}\stackrel{\u02c6}{\mathbf{p}}=\stackrel{\u02c6}{\mathbf{s}}\times \mathbf{k}/|\mathbf{k}|.$$
(7)
$${r}_{0\mathit{sp}}=-{r}_{0\mathit{ps}},$$
(8)
$${t}_{0\mathit{sp}}={t}_{0\mathit{ps}},$$
(9)
$$sin\theta cos\varphi =-\frac{m\mathrm{\lambda}}{2d},$$
(10)
$${r}_{\mathit{msp}}={r}_{\mathit{mps}},$$
(11)
$${t}_{\mathit{msp}}=-{t}_{\mathit{mps}}$$
(12)
$${t}_{\mathit{msp}}=-(-1{)}^{m}{t}_{\mathit{mps}},$$
(13)
$${t}_{\mathit{mss}}={t}_{\mathit{mpp}}=0,\hspace{1em}\hspace{1em}\mathrm{mod}(m,2)=1.$$