## Abstract

An expression derived for hologram exposures made along the straight-line portion of an H–D curve of a photographic plate gives the first-order transmittance of a hologram made of several object points exposed simultaneously (conventional holograms). This expression is compared with a similar expression derived previously for holograms made of several object points exposed sequentially (synthetic holograms). Theory and experiments show the effect of the nonlinearity of the photographic process on the contrast of the reconstruction of conventional holograms. Synthetic and conventional holograms are studied theoretically and experimentally to determine how the total amount of light in the reconstruction image depends upon the number of object points when the total amount of light in the object is constant. It is shown that the reconstructed image formed by a conventional hologram contains more light than the image formed by a synthetic hologram of the same number of object points. Both synthetic and conventional holograms are also studied to determine the ratio of reference-beam illuminance to object-beam illuminance that maximizes the amount of light in the reconstructed image.

© 1969 Optical Society of America

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### Equations (10)

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(1)
$$TA={(E/{E}^{\prime})}^{-\gamma /2},$$
(2)
$$A=R+\text{\u2211}_{n=1}^{N}{A}_{n},$$
(3)
$$\begin{array}{ll}{E}_{H}=t(AA*)\hfill & =t(R+\text{\u2211}_{n=1}^{N}{A}_{n})\hspace{0.17em}(R+\text{\u2211}_{n=1}^{N}{A}_{n})*\hfill \\ \hfill & =t({I}_{R}+I+R*\text{\u2211}_{n=1}^{N}{A}_{n}+R\text{\u2211}_{n=1}^{N}{A}_{n}*+\text{\u2211}_{n=1}^{N}\text{\u2211}_{\begin{array}{l}m=1\\ m\ne n\end{array}}^{N}{A}_{n}{A}_{m}*),\hfill \end{array}$$
(4)
$$|{T}_{A}|={({E}^{\prime}C)}^{\gamma /2}{[1+C(a+b+d)]}^{-\gamma /2},$$
(5)
$$\begin{array}{ccc}C={[{E}_{0}+t({I}_{R}+I)]}^{-1},& a=tR*\text{\u2211}_{n=1}^{N}{A}_{n},& b=tR\end{array}\text{\u2211}_{n=1}^{N}{A}_{n}*,$$
(6)
$$d=t\text{\u2211}_{i=1}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne i\end{array}}^{N}{A}_{i}{A}_{j}*.$$
(7)
$${{I}_{K}}^{\prime}={|D|}^{2}{({E}^{\prime}C)}^{\gamma}{(\frac{1}{2}\gamma )}^{2}{C}^{2}{t}^{2}{I}_{R}{I}_{K}{(1-U+V-S+\text{higher}\u2010\text{order terms})}^{2},$$
(8)
$$\begin{array}{ccc}U=(\frac{1}{2}\gamma +1)Ct\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}{I}_{n}& \text{and}& {I}_{n}={|{A}_{n}|}^{2},\end{array}$$
(9)
$$V=(\frac{1}{2}\gamma +1)\hspace{0.17em}(\frac{1}{2}\gamma +2){C}^{2}{t}^{2}[{I}_{R}(\frac{1}{2}{I}_{K}+\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}{I}_{n})+{I}_{K}\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}{I}_{n}+\frac{3}{2}\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne k\\ j\ne n\end{array}}^{N}{I}_{n}{I}_{j}],$$
(10)
$$\begin{array}{ll}S\hfill & =(\frac{1}{2}\gamma +1)\hspace{0.17em}(\frac{1}{2}\gamma +2)(\frac{1}{2}\gamma +3){C}^{3}(\frac{1}{6}){t}^{3}({I}_{R}(3\text{\u2211}_{n=1}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\end{array}}^{N}{I}_{n}{I}_{i}\hfill \\ \hfill & +6{I}_{K}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne k\end{array}}^{N}{I}_{j}+9\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\\ i\ne k\end{array}}^{N}{I}_{n}{I}_{i})+\text{\u2211}_{n=1}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne k\end{array}}^{N}{I}_{n}{I}_{i}{I}_{j}\hfill \\ \hfill & +2\text{\u2211}_{n=1}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne i\\ j\ne n\end{array}}^{N}{I}_{n}{I}_{i}{I}_{j}+3\text{\u2211}_{\begin{array}{l}i=i\\ i\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne k\\ j\ne i\end{array}}^{N}\text{\u2211}_{\begin{array}{l}n=1\\ n\ne j\\ n\ne k\end{array}}^{N}{I}_{i}{I}_{j}{I}_{n}\hfill \\ \hfill & +\text{\u2211}_{n=1}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\\ i\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne k\\ j\ne n\end{array}}^{N}{I}_{n}{I}_{i}{I}_{j}+\text{\u2211}_{i=1}^{N}\text{\u2211}_{\begin{array}{l}n=1\\ n\ne i\\ n\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne n\\ j\ne i\\ j\ne k\end{array}}^{N}{I}_{i}{I}_{n}{I}_{j}\hfill \\ \hfill & +\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne i\\ j\ne n\end{array}}^{N}{I}_{n}{I}_{i}{I}_{j}+2\text{\u2211}_{\begin{array}{l}n=1\\ n\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}i=1\\ i\ne n\\ i\ne k\end{array}}^{N}\text{\u2211}_{\begin{array}{l}j=1\\ j\ne k\\ j\ne i\\ j\ne n\end{array}}^{N}{I}_{n}{I}_{i}{I}_{j}),\hfill \end{array}$$