## Abstract

The coupling between two gratings of arbitrary spatial frequency and relative fringe orientation, recorded in the same thick emulsion, is examined. Both direct coupling of the incident energy into the diffracted waves of each grating and the cross coupling of energy through diffracted waves are considered. The analysis is for a read-out wave at arbitrary angle of incidence. The angular-selectivity curves for the diffracted waves are computed for varying relative strengths of the two gratings and also for varying degrees of coupling, determined by the relative fringe slants of the two gratings. The necessary Bragg-angle difference to decouple the effects of the two gratings is considered. Experimental results for double gratings recorded in dichromated gelatin are given to support the theory.

© 1975 Optical Society of America

Full Article |

PDF Article
### Equations (9)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$E(x,z)={E}_{1}\{1+{r}_{1}\hspace{0.17em}\text{cos}[2\pi {f}_{1}(x-{x}_{1}(z))]\}+{E}_{2}\{1+{r}_{2}\hspace{0.17em}\text{cos}[2\pi {f}_{2}(x-{x}_{2}(z))]\},$$
(2)
$${T}_{A}(x,z)={e}^{jmE(x,z)}.$$
(3)
$${T}_{A}(x,n)=K\sum _{q=-\infty}^{\infty}{(j)}^{q}{J}_{q}({b}_{1})\hspace{0.17em}\text{exp}\{j2\pi q{f}_{1}[x-{x}_{1}(n)]\}\times \sum _{p=-\infty}^{\infty}{(j)}^{p}{J}_{p}({b}_{2})\hspace{0.17em}\text{exp}\{j2\pi p{f}_{2}[x-{x}_{2}(n)]\},$$
(4)
$$j{J}_{0}({b}_{2}){J}_{1}({b}_{1})\hspace{0.17em}\text{exp}[-j2\pi {f}_{1}{x}_{1}(n)],$$
(5)
$${}_{n}H=\left(\begin{array}{cccc}{e}^{j{\alpha}_{0}}{J}_{0}({b}_{10}){J}_{0}({b}_{20})& j{e}^{j{\alpha}_{1}}{J}_{0}({b}_{21}){J}_{1}({b}_{11}){e}^{j{\delta}_{1}(n)}& j{e}^{j{\alpha}_{2}}{J}_{0}({b}_{12}){J}_{1}({b}_{22}){e}^{j{\delta}_{2}(n)}& 0\\ j{e}^{j{\alpha}_{0}}{J}_{0}({b}_{20}){J}_{1}({b}_{10}){e}^{-j{\delta}_{1}(n)}& {e}^{j{\alpha}_{1}}{J}_{0}({b}_{11}){J}_{0}({b}_{21})& 0& j{e}^{j{\alpha}_{3}}{J}_{0}({b}_{13}){J}_{1}({b}_{23}){e}^{j{\delta}_{2}(n)}\\ j{e}^{j{\alpha}_{0}}{J}_{0}({b}_{10}){J}_{1}({b}_{20}){e}^{-j{\delta}_{2}(n)}& 0& {e}^{j{\alpha}_{2}}{J}_{0}({b}_{12}){J}_{0}({b}_{22})& j{e}^{j{\alpha}_{3}}{J}_{0}({b}_{23}){J}_{1}({b}_{13}){e}^{j{\delta}_{1}(n)}\\ 0& j{e}^{j{\alpha}_{1}}{J}_{0}({b}_{11}){J}_{1}({b}_{21}){e}^{-j{\delta}_{2}(n)}& j{e}^{j{\alpha}_{2}}{J}_{0}({b}_{22}){J}_{1}({b}_{12}){e}^{-j{\delta}_{1}(n)}& {e}^{j{\alpha}_{3}}{J}_{0}({b}_{13}){J}_{0}({b}_{23})\end{array}\right),$$
(6)
$$\begin{array}{l}{\alpha}_{l}=\frac{2\pi}{\mathrm{\lambda}}\mathrm{\Delta}z\hspace{0.17em}{n}_{0}\sqrt{1-{\mathrm{\lambda}}^{2}\hspace{0.17em}{F}_{l}^{2}}\\ {\delta}_{i}(n)=-2\pi \hspace{0.17em}{F}_{l}n\hspace{0.17em}\text{tan}{\mathrm{\Phi}}_{i},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}i=1,2\\ {b}_{il}=\frac{2\pi \mathrm{\Delta}\hspace{0.17em}z{n}_{i}}{\sqrt{1-{\mathrm{\lambda}}^{2}\hspace{0.17em}{F}_{l}^{2}}},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}i=1,2\end{array}$$
(7)
$$\begin{array}{l}{F}_{0}={f}_{i},\\ {F}_{1}={f}_{i}-{f}_{1},\\ {F}_{2}={f}_{i}-{f}_{2},\\ {F}_{3}={f}_{i}-{f}_{1}-{f}_{2}.\end{array}$$
(8)
$${\mathbf{A}}_{0}=\left(\prod _{n=1}^{N}{}_{n}H\right){\mathbf{A}}_{i}$$
(9)
$${\mathbf{A}}_{i}=\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),$$