## Abstract

We investigate the diffraction properties of vector synthetic gratings, which consist of two volume index gratings with different grating wave vectors, using coupled-wave analysis. These index gratings can be obtained in photorefractive materials or in Bragg cells with two driving acoustic waves. We consider the case of pseudo phase matching in which a Bragg mismatch exists in the scattering from the individual index gratings while a Bragg condition is satisfied for the scattering from the vector sum of the two index gratings. Analytic expressions for the diffraction efficiency as well as the amplitude of the three waves involved are obtained. The results are presented and discussed.

© 2000 Optical Society of America

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### Equations (41)

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(1)
$$n={n}_{0}+{n}_{a}cos{\mathbf{K}}_{a}\xb7\mathbf{r}+{n}_{b}cos{\mathbf{K}}_{b}\xb7\mathbf{r},$$
(2)
$${E}_{j}={A}_{j}(x)exp[i(\omega t-{\mathbf{k}}_{j}\xb7\mathbf{r})]\hspace{1em}(j=1,2,3),$$
(3)
$${\mathbf{k}}_{j}=\left[\begin{array}{c}{\alpha}_{j}\\ {\beta}_{j}\\ 0\end{array}\right]={k}_{0}\left[\begin{array}{c}cos{\theta}_{j}\\ sin{\theta}_{j}\\ 0\end{array}\right]\hspace{1em}(j=1,2,3),$$
(4)
$$E=[{A}_{1}(x)exp(-i{\alpha}_{1}x-i{\beta}_{1}y)+{A}_{2}(x)exp(-i{\alpha}_{2}x-i{\beta}_{2}y)+{A}_{3}(x)exp(-i{\alpha}_{3}x-i{\beta}_{3}y)]exp(i\omega x).$$
(5)
$${\nabla}^{2}E+{n}^{2}{k}_{0}^{2}E=0,$$
(6)
$${\beta}_{2}-{\beta}_{1}={K}_{\mathit{ay}},\hspace{1em}\hspace{1em}{\beta}_{3}-{\beta}_{2}={K}_{\mathit{by}},$$
(7)
$${\beta}_{3}-{\beta}_{1}={K}_{\mathit{ay}}+{K}_{\mathit{by}},$$
(8)
$${A}_{1}^{\prime}=-i{\kappa}_{12}{A}_{2}exp(-i\mathrm{\Delta}{\alpha}_{a}x),$$
(9)
$${A}_{2}^{\prime}=-i{\kappa}_{21}{A}_{1}exp(i\mathrm{\Delta}{\alpha}_{a}x)-i{\kappa}_{23}{A}_{3}exp(-i\mathrm{\Delta}{\alpha}_{b}x),$$
(10)
$${A}_{3}^{\prime}=-i{\kappa}_{32}{A}_{2}exp(i\mathrm{\Delta}{\alpha}_{b}x),$$
(11)
$$\mathrm{\Delta}{\alpha}_{a}={\alpha}_{2}-{\alpha}_{1}-{K}_{\mathit{ax}},$$
(12)
$$\mathrm{\Delta}{\alpha}_{b}={\alpha}_{3}-{\alpha}_{2}-{K}_{\mathit{bx}},$$
(13)
$${\kappa}_{12}=\frac{\pi {n}_{a}}{\mathrm{\lambda}cos{\theta}_{1}},\hspace{1em}{\kappa}_{21}=\frac{\pi {n}_{a}}{\mathrm{\lambda}cos{\theta}_{2}},$$
(14)
$${\kappa}_{23}=\frac{\pi {n}_{b}}{\mathrm{\lambda}cos{\theta}_{2}},\hspace{1em}{\kappa}_{32}=\frac{\pi {n}_{b}}{\mathrm{\lambda}cos{\theta}_{3}}.$$
(15)
$$cos{\theta}_{1}|{A}_{1}(x){|}^{2}+cos{\theta}_{2}|{A}_{2}(x){|}^{2}+cos{\theta}_{3}|{A}_{3}(x){|}^{2}=\mathrm{const},$$
(16)
$${A}_{1}^{\prime}=-i{\kappa}_{12}{A}_{2}exp(-i\mathrm{\Delta}\alpha x),$$
(17)
$${A}_{2}^{\prime}=-i{\kappa}_{21}{A}_{1}exp(i\mathrm{\Delta}\alpha x)-i{\kappa}_{23}{A}_{3}exp(i\mathrm{\Delta}\alpha x),$$
(18)
$${A}_{3}^{\prime}=-i{\kappa}_{32}{A}_{2}exp(-i\mathrm{\Delta}\alpha x),$$
(19)
$$\mathrm{\Delta}\alpha =\mathrm{\Delta}{\alpha}_{a}={\alpha}_{2}-{\alpha}_{1}-{K}_{\mathit{ax}}.$$
(20)
$${A}_{2}^{\u2033}-i\mathrm{\Delta}\alpha {A}_{2}^{\prime}+{\kappa}^{2}{A}_{2}=0,$$
(21)
$${\kappa}^{2}={\kappa}_{12}{\kappa}_{21}+{\kappa}_{23}{\kappa}_{32}.$$
(22)
$${A}_{2}(x)=exp\left(i\frac{\mathrm{\Delta}\alpha}{2}x\right)({C}_{2}sin\mathit{sx}+{C}_{4}cos\mathit{sx}),$$
(23)
$${s}^{2}={\kappa}^{2}+{\left(\frac{\mathrm{\Delta}\alpha}{2}\right)}^{2}.$$
(24)
$${{A}^{\prime}}_{2}(0)=-i{\kappa}_{21}{A}_{1}(0)-i{\kappa}_{23}{A}_{3}(0).$$
(25)
$${A}_{1}(x)={A}_{1}(0)-i{\kappa}_{12}{\int}_{0}^{x}{A}_{2}({x}^{\prime})exp(-i\mathrm{\Delta}\alpha {x}^{\prime})\mathrm{d}{x}^{\prime},$$
(26)
$${A}_{3}(x)={A}_{3}(0)-i{\kappa}_{32}{\int}_{0}^{x}{A}_{2}({x}^{\prime})exp(-i\mathrm{\Delta}\alpha {x}^{\prime})\mathrm{d}{x}^{\prime}.$$
(27)
$${A}_{1}(0)=1,\hspace{1em}\hspace{1em}{A}_{2}(0)=0,\hspace{1em}\hspace{1em}{A}_{3}(0)=0,$$
(28)
$${A}_{2}(x)=-i\frac{{\kappa}_{21}}{s}exp\left(i\frac{\mathrm{\Delta}\alpha}{2}x\right)sin\mathit{sx}.$$
(29)
$${A}_{1}(x)=\frac{{\kappa}_{21}{\kappa}_{12}}{s}exp\left(-i\frac{\mathrm{\Delta}\alpha}{2}x\right)\frac{scos\mathit{sx}+(i/2)\mathrm{\Delta}\alpha sin\mathit{sx}}{{\kappa}^{2}}+\frac{{\kappa}_{23}{\kappa}_{32}}{{\kappa}^{2}},$$
(30)
$${A}_{3}(x)=\frac{{\kappa}_{32}{\kappa}_{12}}{s}exp\left(-i\frac{\mathrm{\Delta}\alpha}{2}x\right)\frac{scos\mathit{sx}+(i/2)\mathrm{\Delta}\alpha sin\mathit{sx}}{{\kappa}^{2}}-\frac{{\kappa}_{32}{\kappa}_{21}}{{\kappa}^{2}}.$$
(31)
$${\eta}_{s}=\frac{cos{\theta}_{3}|{A}_{3}(L){|}^{2}}{cos{\theta}_{1}|{A}_{1}(0){|}^{2}},$$
(32)
$${\eta}_{s}=\frac{cos{\theta}_{3}}{cos{\theta}_{1}}{\left|\frac{{\kappa}_{32}{\kappa}_{21}}{{\kappa}^{2}}\right|}^{2}{\left|exp\left(-i\frac{\mathrm{\Delta}\alpha}{2}L\right)(cos\mathit{sL}+i\frac{\mathrm{\Delta}\alpha}{2s}sin\mathit{sL})-1\right|}^{2}.$$
(33)
$${\eta}_{s}=\frac{cos{\theta}_{3}}{cos{\theta}_{1}}{\left|\frac{{\kappa}_{32}{\kappa}_{21}}{{\kappa}^{2}}\right|}^{2}(1-cos\kappa L{)}^{2}\hspace{1em}(\mathrm{\Delta}\alpha =0).$$
(34)
$$\kappa L=(2n+1)\pi \hspace{1em}\hspace{1em}(n=0,1,2,3,4\dots ),$$
(35)
$${\eta}_{smax}=1-{\left(\frac{{n}_{a}^{2}cos{\theta}_{3}-{n}_{b}^{2}cos{\theta}_{1}}{{n}_{a}^{2}cos{\theta}_{3}+{n}_{b}^{2}cos{\theta}_{1}}\right)}^{2}.$$
(36)
$${\eta}_{s}=\frac{cos{\theta}_{3}}{cos{\theta}_{1}}{\left|\frac{{\kappa}_{32}{\kappa}_{21}}{{\kappa}^{2}}\right|}^{2}\left\{2-2cos\mathit{sL}cos\frac{\mathrm{\Delta}\alpha}{2}L-\left[1-{\left(\frac{\mathrm{\Delta}\alpha}{2s}\right)}^{2}\right]{sin}^{2}\mathit{sL}-2\frac{\mathrm{\Delta}\alpha}{2s}sin\mathit{sL}sin\frac{\mathrm{\Delta}\alpha}{2}L\right\}.$$
(37)
$${\eta}_{s}=\frac{cos{\theta}_{3}}{cos{\theta}_{1}}{\left|\frac{{\kappa}_{32}{\kappa}_{21}}{{\kappa}^{2}}\right|}^{2}4$$
(38)
$$\frac{\mathrm{\Delta}\alpha L}{2}=m\pi \hspace{1em}\hspace{1em}(m=0,1,2,3\dots ),$$
(39)
$$\mathit{sL}=(2n+1)\pi +m\pi \hspace{1em}(n=0,1,2,3\dots ).$$
(40)
$$\kappa L=\sqrt{(2n+1)(2n+1+2m)}\pi $$
(41)
$$(n=0,1,2,3,\dots ,m=0,1,2,3,\dots ).$$