## Abstract

Diffraction of the general angle of incidence by dielectric surface-relief gratings is analyzed by use of a differential method. The method is applied to sinusoidal, rectangular, and triangular gratings. In all our results, error in energy conservation is of the order of 10^{−4} to groove depths as deep as 2.5 grating periods. It is also shown that, in conical diffraction mountings, when conditions of incidence upon these gratings are controlled, the gratings function as if a polarizing beam splitter had been cascaded with a rotatable half-wave retardation plate.

© 1994 Optical Society of America

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### Equations (46)

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(1)
$$\mathbf{\text{E}}=\text{\u2211}_{n}\{[{E}_{xn}(y)\widehat{x}+{E}_{yn}(y)\u0177+{E}_{zn}(y)\widehat{z}]\times exp[j({k}_{xn}x+{k}_{z}z)]\},$$
(2)
$$\mathbf{\text{H}}=\sqrt{\frac{{\varepsilon}_{0}}{{\mu}_{0}}}\text{\u2211}_{n}\{[{H}_{\text{x}n}(y)\widehat{x}+{H}_{yn}(y)\u0177+{H}_{zn}(y)\widehat{z}]\times exp[j({k}_{xn}x+{k}_{z}z)]\},$$
(3)
$${k}_{0}=\frac{2\pi}{\mathrm{\lambda}},$$
(4)
$${k}_{xn}={k}_{0}\sqrt{{\varepsilon}_{1}}sin\theta cos\varphi +\frac{2\pi n}{p},$$
(5)
$${k}_{z}={k}_{0}\sqrt{{\varepsilon}_{1}}sin\theta sin\varphi ,$$
(6)
$${E}_{\mathit{\text{ln}}}(y)={A}_{\mathit{\text{ln}}}^{(1)}exp[-j{k}_{yn}^{(1)}y]+{B}_{\mathit{\text{ln}}}^{(1)}exp[j{k}_{yn}^{(1)}y],$$
(7)
$${H}_{\mathit{\text{ln}}}(y)={C}_{\mathit{\text{ln}}}^{(1)}exp[-j{k}_{yn}^{(1)}y]+{D}_{\mathit{\text{ln}}}^{(1)}exp[j{k}_{yn}^{(1)}y],$$
(8)
$${k}_{yn}^{(1)}=\{\begin{array}{l}{({{k}_{0}}^{2}{\varepsilon}_{1}-{{k}_{xn}}^{2}-{{k}_{z}}^{2})}^{1/2}\hspace{0.17em}\text{for}\hspace{0.17em}{{k}_{0}}^{2}{\varepsilon}_{1}-{{k}_{xn}}^{2}-{{k}_{z}}^{2}\ge 0\hfill \\ j{[-({{k}_{0}}^{2}{\varepsilon}_{1}-{{k}_{xn}}^{2}-{{k}_{z}}^{2})]}^{1/2}\hspace{0.17em}\text{for}\hspace{0.17em}{{k}_{0}}^{2}{\varepsilon}_{1}-{{k}_{xn}}^{2}-{{k}_{z}}^{2}<0\hfill \end{array}$$
(9)
$${E}_{\mathit{\text{ln}}}(y)={A}_{\mathit{\text{ln}}}^{(3)}exp[-j{k}_{yn}^{(3)}y],$$
(10)
$${H}_{\mathit{\text{ln}}}(y)={C}_{\mathit{\text{ln}}}^{(3)}exp[-j{k}_{yn}^{(3)}y],$$
(11)
$${k}_{yn}^{(3)}=\{\begin{array}{l}{({{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{xn}}^{2}-{{k}_{z}}^{2})}^{1/2}\hspace{0.17em}\text{for}\hspace{0.17em}{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{xn}}^{2}-{{k}_{z}}^{2}\ge 0\hfill \\ j{[-({{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{xn}}^{2}-{{k}_{z}}^{2})]}^{1/2}\hspace{0.17em}\text{for}\hspace{0.17em}{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{xn}}^{2}-{{k}_{z}}^{2}<0\hfill \end{array}$$
(12)
$${\mathbf{\text{\Phi}}}_{i}={[{A}_{z-N}^{(1)},\dots ,{A}_{zN}^{(1)},{C}_{z-N}^{(1)},\dots ,{C}_{zN}^{(1)}]}^{T},$$
(13)
$${\mathbf{\text{\Phi}}}_{r}={[{B}_{z-N}^{(1)},\dots ,{B}_{zN}^{(1)},{D}_{z-N}^{(1)},\dots ,{D}_{zN}^{(1)}]}^{T},$$
(14)
$${\mathbf{\text{\Phi}}}_{t}={[{A}_{z-N}^{(3)},\dots ,{A}_{zN}^{(3)},{C}_{z-N}^{(3)},\dots ,{C}_{zN}^{(3)}]}^{T},$$
(15)
$${\mathbf{\text{\Phi}}}_{i}={\mathbf{\text{M}}}_{1}{\mathbf{\text{\Phi}}}_{t},$$
(16)
$${\mathbf{\text{\Phi}}}_{r}={\mathbf{\text{M}}}_{2}{\mathbf{\text{\Phi}}}_{t},$$
(17)
$${\mathbf{\text{\Phi}}}_{t}={{\mathbf{\text{M}}}_{1}}^{-1}{\mathbf{\text{\Phi}}}_{i}=\mathbf{\text{T}}{\mathbf{\text{\Phi}}}_{i},$$
(18)
$${\mathbf{\text{\Phi}}}_{r}={\mathbf{\text{M}}}_{2}{{\mathbf{\text{M}}}_{1}}^{-1}{\mathbf{\text{\Phi}}}_{i}=\mathbf{\text{R}}{\mathbf{\text{\Phi}}}_{i}.$$
(19)
$${T}_{n}=\frac{{k}_{yn}^{(3)}}{{k}_{y0}^{(1)}}\frac{{{k}_{0}}^{2}{\varepsilon}_{3}{|{A}_{zn}^{(3)}|}^{2}+{{k}_{0}}^{2}{|{C}_{zn}^{(3)}|}^{2}}{{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{z}}^{2}},$$
(20)
$${R}_{n}=-\frac{{k}_{yn}^{(1)}}{{k}_{y0}^{(1)}}\frac{{{k}_{0}}^{2}{\varepsilon}_{1}{|{B}_{zn}^{(1)}|}^{2}+{{k}_{0}}^{2}{|{D}_{zn}^{(1)}|}^{2}}{{{k}_{0}}^{2}{\varepsilon}_{1}-{{k}_{z}}^{2}}.$$
(21)
$$\varepsilon (x,y)=\text{\u2211}_{n}{a}_{n}(y)exp\left(j\frac{2\pi nx}{p}\right),$$
(22)
$$\frac{1}{\varepsilon (x,y)}=\text{\u2211}_{n}\hspace{0.17em}{b}_{n}(y)exp\left(j\frac{2\pi nx}{p}\right),$$
(23)
$$\text{rot}\hspace{0.17em}\mathbf{\text{E}}\hspace{0.17em}=j\omega \mu \mathbf{\text{H}},$$
(24)
$$\text{rot}\hspace{0.17em}\mathbf{\text{H}}=-j\omega {\varepsilon}_{0}\varepsilon \mathbf{\text{E}},$$
(25)
$$\frac{\partial {E}_{zn}(y)}{\partial y}=j\frac{{k}_{z}}{{k}_{0}}\text{\u2211}_{m}{b}_{n-m}(y){k}_{xm}{H}_{zm}(y)+j\left[{k}_{0}{H}_{xn}(y)-\frac{{{k}_{z}}^{2}}{{k}_{0}}\text{\u2211}_{m}{b}_{n-m}(y){H}_{xm}(y)\right],$$
(26)
$$\frac{\partial {H}_{zn}(y)}{\partial y}=-j\frac{{k}_{z}{k}_{xn}}{{k}_{0}}{E}_{zn}(y)+j\left[\frac{{{k}_{z}}^{2}}{{k}_{0}}{E}_{xn}(y)-{k}_{0}\text{\u2211}_{m}{a}_{n-m}(y){E}_{xm}(y)\right],$$
(27)
$$\frac{\partial {E}_{xn}(y)}{\partial y}=j\left[-{k}_{0}{H}_{zn}(y)+\frac{{k}_{xn}}{{k}_{0}}\text{\u2211}_{m}{b}_{n-m}(y){k}_{xm}(y){H}_{zm}(y)\right]-j\frac{{k}_{z}{k}_{xn}}{{k}_{0}}\text{\u2211}_{m}{b}_{n-m}(y){H}_{xm}(y),$$
(28)
$$\frac{\partial {H}_{xn}(y)}{\partial y}=j\left[-\frac{{{k}_{xn}}^{2}}{{k}_{0}}{E}_{zn}(y)+{k}_{0}\text{\u2211}_{m}{a}_{n-m}(y){E}_{zm}(y)\right]+j\frac{{k}_{z}{k}_{xn}}{{k}_{0}}{E}_{xn}(y).$$
(29)
$$\mathbf{\text{\Psi}}(y)={[{E}_{zn},{H}_{zn},{E}_{xn},{H}_{xn}]}^{T};$$
(30)
$$\partial \mathbf{\text{\Psi}}/\partial y=\mathbf{\text{v}}\hspace{0.17em}\mathbf{\text{\Psi}}.$$
(31)
$${E}_{zn}(0)={A}_{zn}^{(3)},$$
(32)
$${H}_{zn}(0)={C}_{zn}^{(3)},$$
(33)
$${E}_{xn}(0)=-\frac{{k}_{xn}{k}_{zn}{A}_{zn}^{(3)}-{k}_{0}{k}_{yn}^{(3)}{C}_{zn}^{(3)}}{{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{zn}}^{2}},$$
(34)
$${H}_{xn}(0)=-\frac{{k}_{0}{\varepsilon}_{3}{k}_{yn}^{(3)}{A}_{zn}^{(3)}+{k}_{xn}{k}_{zn}{C}_{zn}^{(3)}}{{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{zn}}^{2}},$$
(35)
$${E}_{zn}(d)={A}_{zn}^{(1)}exp[-j{k}_{yn}^{(1)}d]+{B}_{zn}^{(1)}exp[j{k}_{yn}^{(1)}d],$$
(36)
$${H}_{zn}(d)={C}_{zn}^{(1)}exp[-j{k}_{yn}^{(1)}d]+{D}_{zn}^{(1)}exp[j{k}_{yn}^{(1)}d],$$
(37)
$$\frac{\partial {E}_{zn}(d)}{\partial y}=-j{k}_{yn}^{(1)}{A}_{zn}^{(1)}exp[-j{k}_{yn}^{(1)}d]+j{k}_{yn}^{(1)}{B}_{zn}^{(1)}exp[j{k}_{yn}^{(1)}d],$$
(38)
$$\frac{\partial {H}_{zn}(d)}{\partial y}=-j{k}_{yn}^{(1)}{C}_{zn}^{(1)}exp[-j{k}_{yn}^{(1)}d]+j{k}_{yn}^{(1)}{D}_{zn}^{(1)}exp[j{k}_{yn}^{(1)}d].$$
(39)
$${A}_{z0}^{(1)}=cos\delta cos\theta sin\phi -sin\delta cos\phi ,$$
(40)
$${C}_{z0}^{(1)}=\sqrt{{\varepsilon}_{1}}(cos\delta cos\phi +sin\delta cos\theta sin\phi ),$$
(41)
$$\left[\begin{array}{l}{A}_{zn}^{(3)}\\ {C}_{zn}^{(3)}\end{array}\right]=\left[\begin{array}{cc}{t}_{1}& {t}_{2}\\ {t}_{3}& {t}_{4}\end{array}\right]\hspace{0.17em}\left[\begin{array}{l}{A}_{z0}^{(1)}\\ {C}_{z0}^{(1)}\end{array}\right].$$
(42)
$${t}_{i}=|{t}_{i}|exp(j{\tau}_{i})\hspace{0.17em}(i=1,2,3,4)$$
(43)
$${T}_{n}=Pcos(2\delta -{\delta}_{0})+R,$$
(44)
$$\begin{array}{ll}P\hfill & =\frac{{k}_{yn}^{(3)}}{{k}_{y0}^{(1)}}\frac{{{k}_{0}}^{2}}{{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{z}}^{2}}\sqrt{{{P}_{1}}^{2}+{{P}_{2}}^{2}}\sqrt{{{Q}_{1}}^{2}+{{Q}_{2}}^{2}},\hfill \\ R\hfill & =\frac{{k}_{yn}^{(3)}}{{k}_{y0}^{(1)}}\frac{{{k}_{0}}^{2}}{{{k}_{0}}^{2}{\varepsilon}_{3}-{{k}_{z}}^{2}}\frac{{cos}^{2}\theta {sin}^{2}\phi +{cos}^{2}\phi}{2}\hfill \\ \hfill & \times [{\varepsilon}_{3}{|{t}_{1}|}^{2}+{|{t}_{3}|}^{2}+{\varepsilon}_{1}({\varepsilon}_{3}{|{t}_{2}|}^{2}+{|{t}_{4}|}^{2})],\hfill \\ {P}_{1}\hfill & ={\varepsilon}_{3}{|{t}_{1}|}^{2}+{|{t}_{3}|}^{2}-{\varepsilon}_{1}({\varepsilon}_{3}{|{t}_{2}|}^{2}+{|{t}_{4}|}^{2}),\hfill \\ {P}_{2}\hfill & =2\sqrt{{\varepsilon}_{1}}[{\varepsilon}_{3}|{t}_{1}|\hspace{0.17em}|{t}_{2}|cos({\tau}_{1}-{\tau}_{2})+|{t}_{3}|\hspace{0.17em}|{t}_{4}|cos({\tau}_{3}-{\tau}_{4})],\hfill \\ {Q}_{1}\hfill & =\frac{{cos}^{2}\theta {sin}^{2}\phi -{cos}^{2}\phi}{2},\hfill \\ {Q}_{2}\hfill & =cos\theta sin\phi cos\phi ,\hfill \\ {\delta}_{0}\hfill & ={tan}^{-1}\left(\frac{{P}_{2}{Q}_{1}-{P}_{1}{Q}_{2}}{{P}_{1}{Q}_{1}+{P}_{2}{Q}_{2}}\right).\hfill \end{array}$$
(45)
$$m=\frac{2p\sqrt{{\varepsilon}_{1}}sin\theta cos\varphi}{\mathrm{\lambda}};$$
(46)
$$tan{\delta}_{E}=-\frac{cos\theta}{tan\varphi}.$$