## Abstract

The process of producing a hologram of an object that is transmitting or reflecting diffuse, partially coherent, quasimonochromatic light is described mathematically. The discussion shows how the degree of coherence between the reference beam and the beam illuminating the object affects the reconstruction. The types of image degradation resulting from the use of partially coherent light are outlined. The application of holography to the measurement of second-order spatial coherence is suggested and a possible experiment is described.

© 1966 Optical Society of America

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### Equations (18)

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(1)
$$I(P)=\u3008{V}_{P}(t)\hspace{0.17em}{V}_{P}*(t)\u3009,$$
(2)
$${V}_{P}(t)={V}_{RP}(t)+\sum _{m}{V}_{mP}(t).$$
(3)
$$I(P)=\sum _{m}\u3008{V}_{mP}(t)\hspace{0.17em}{V}_{mP}*(t)\u3009+\sum _{m}\sum _{n\ne m}\u3008{V}_{mP}(t)\hspace{0.17em}{V}_{nP}*(t)\u3009+\u3008{V}_{RP}(t)\hspace{0.17em}{V}_{RP}*(t)\u3009+\sum _{m}\u30082\hspace{0.17em}\text{Re}{V}_{mP}(t)\hspace{0.17em}{V}_{RP}*(t)\u3009.$$
(4)
$$I(P)={I}_{o}(P)+{I}_{B}(P)+2\hspace{0.17em}\sum _{m}\u3008\text{Re}{V}_{mP}(t)\hspace{0.17em}{V}_{RP}*(t)\u3009,$$
(5)
$${V}_{mP}(t)={A}_{m}(t-{s}_{mP}/c){e}^{j(\omega t-{s}_{mP}k+{\varphi}_{m})},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}k=\omega /c$$
(6)
$${V}_{RP}(t)={A}_{RP}(t-s/c){e}^{j(\omega t-sk)},$$
(7)
$$\begin{array}{c}\u3008\text{Re}{V}_{mP}(t)\hspace{0.17em}{V}_{RP}*(t)\u3009=\u3008\text{Re}(1/{s}_{mP}){A}_{m}(t-{s}_{mP}/c){A}_{RP}(t-s/c)*\times \text{exp}\{j[\omega (s-{s}_{mP})/c+{\varphi}_{m}]\}\\ =\text{Re}\{\u30081/{s}_{mP}){A}_{m}(t){A}_{RP}(t-{\tau}_{mP})*\u3009{e}^{j(\omega {\tau}_{mP}+{\varphi}_{m})}\},\\ {\tau}_{mP}=(s-{s}_{mP})/c.\end{array}$$
(8)
$${\gamma}_{mRP}({\tau}_{mP})=\frac{\u3008{A}_{m}(t){A}_{RP}(t-{\tau}_{mP})*\u3009{e}^{j(\omega {\tau}_{mP}+{\varphi}_{m})}}{{[{{s}_{mP}}^{2}{I}_{m}(P){I}_{R}(P)]}^{{\scriptstyle \frac{1}{2}}}}$$
(9)
$${I}_{R}(P)=\u3008{A}_{RP}(t-{\tau}_{mP}){A}_{RP}(t-{\tau}_{mP})*\u3009$$
(10)
$$\u3008\text{exp}j[\text{arg}{A}_{m}(t)-\text{arg}{A}_{RP}(t-{\tau}_{mP})]\u3009=M{e}^{j{\theta}_{mRP}},$$
(11)
$$\u3008{A}_{m}(t){A}_{RP}(t-{\tau}_{mP})*\u3009={A}_{m}{A}_{RP}M{e}^{j{\theta}_{mRP}}.$$
(12)
$$\begin{array}{l}\u3008\text{Re}{V}_{mP}(t)\hspace{0.17em}{V}_{RP}*(t)\u3009=\text{Re}\{{[{I}_{m}(P){I}_{R}(P)]}^{{\scriptstyle \frac{1}{2}}}{\gamma}_{mRP}({\tau}_{mP})\}\\ ={[{I}_{m}(P){I}_{R}(P)]}^{{\scriptstyle \frac{1}{2}}}\mid {\gamma}_{mRP}({\tau}_{mP})\mid \times \hspace{0.17em}\text{cos}(\omega {\tau}_{mP}+{\varphi}_{m}+{\theta}_{mRP}).\end{array}$$
(13)
$$I(P)={I}_{0}(P)+{I}_{B}(P)+2{[{I}_{R}(P)]}^{{\scriptstyle \frac{1}{2}}}\hspace{0.17em}\sum _{m}{[{I}_{m}(P)]}^{{\scriptstyle \frac{1}{2}}}\times \mid {\gamma}_{mRP}({\tau}_{mP})\mid \text{cos}(\omega {\tau}_{mP}+{\varphi}_{m}+{\theta}_{mRP}).$$
(14)
$$\begin{array}{l}\sum _{m}{[{I}_{R}(P){I}_{m}(P)]}^{{\scriptstyle \frac{1}{2}}}\mid {\gamma}_{mRP}({\tau}_{mP})\mid \times \text{cos}(\omega t\pm \omega {\tau}_{mP}\pm {\varphi}_{m}\pm {\theta}_{mRP})=\sum _{m}\frac{{[{I}_{R}(P)]}^{{\scriptstyle \frac{1}{2}}}}{{s}_{mP}}{A}_{m}\mid {\gamma}_{mRP}({\tau}_{mP})\mid \times \text{cos}(\omega t\mp {s}_{mP}k\pm [{\varphi}_{m}+sk+{\theta}_{mRP}])\\ =U{(P,t)}_{\pm},\end{array}$$
(15)
$${I}_{m}(P)=(1/{{s}_{mP}}^{2})\u3008{A}_{m}(t){A}_{m}(t)*\u3009=(1/{{s}_{mP}}^{2}){{A}_{m}}^{2}$$
(16)
$${[{I}_{m}(P)]}^{{\scriptstyle \frac{1}{2}}}=(1/{s}_{mP}){A}_{m}.$$
(17)
$$W(P,t)=\sum _{m}\frac{1}{{s}_{mP}}{A}_{m}\hspace{0.17em}\text{cos}(\omega t-{s}_{mP}k+{\varphi}_{m}),$$
(18)
$$\mid {\gamma}_{mRP}({\tau}_{mP})\mid \hspace{0.17em}\cong \hspace{0.17em}\mid {\gamma}_{mRP}(O)\mid \hspace{0.17em}\cong \hspace{0.17em}\mid {\gamma}_{m{m}_{1}}(O)\mid ,$$