## Abstract

Speckle suppression in a two-diffuser system is examined. An analytical expression for the speckle space–time correlation function is derived, so that the speckle suppression mechanism can be investigated statistically. The grain size of the speckle field illuminating the second diffuser has a major impact on the speckle contrast after temporal averaging. It is shown that, when both the diffusers are rotating, the one with the lower rotating speed determines the period of the speckle correlation function. The coherent length of the averaged speckle intensity is shown to equal the mean speckle size of the individual speckle pattern before averaging. Numerical and experimental results are presented to verify our analysis in the context of speckle reduction.

© 2013 Optical Society of America

Full Article |

PDF Article
### Equations (18)

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

(1)
$$\u3008I({\mathit{r}}_{1},t)I({\mathit{r}}_{2},t+\tau )\u3009=\u3008A({\mathit{r}}_{1},t){A}^{*}({\mathit{r}}_{1},t)A({\mathit{r}}_{2},t+\tau ){A}^{*}({\mathit{r}}_{2},t+\tau )\u3009,$$
(2)
$${\mu}_{2}({\mathit{p}}_{1},{\mathit{p}}_{2},{\omega}_{1}\tau ,{\omega}_{2}\tau ,v)=\frac{\u3008I({\mathit{p}}_{1},t)I({\mathit{p}}_{2},t+\tau )\u3009-\u3008I({\mathit{p}}_{1},t)\u3009\u3008I({\mathit{p}}_{2},t+\tau )\u3009}{\u3008I({\mathit{p}}_{1},t)\u3009\u3008I({\mathit{p}}_{2},t+\tau )\u3009}=\frac{{\mathrm{\Gamma}}_{12}({\mathit{p}}_{1},{\mathit{p}}_{2},{\omega}_{1},{\omega}_{2},\tau )}{{\mathrm{\Gamma}}_{0}}=\frac{4{v}^{2}}{{[1+v-(1-v)\mathrm{cos}({\omega}_{1}\tau -{\omega}_{2}\tau )]}^{2}}\text{\hspace{0.17em}}\mathrm{exp}\{-\frac{2v{({\mathit{p}}_{2}-{\mathit{p}}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{2}\tau +\mathfrak{R}{\mathit{p}}_{1}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\omega}_{2}\tau )}^{2}}{{\rho}_{2}^{2}[1+v-(1-v)\mathrm{cos}({\omega}_{1}\tau -{\omega}_{2}\tau )]}\}\times \mathrm{exp}\{-\frac{2(1-v)[1-\mathrm{cos}({\omega}_{1}\tau -{\omega}_{2}\tau )]({\mathit{p}}_{1}^{2}+{\mathit{p}}_{2}^{2})}{{\rho}_{2}^{2}[1+v-(1-v)\mathrm{cos}({\omega}_{1}\tau -{\omega}_{2}\tau )]}\},$$
(3)
$$v=\frac{{\zeta}^{2}}{({\rho}_{1}^{2}+{\zeta}^{2})}=\frac{1}{N+1},$$
(4)
$${\rho}_{1}=\frac{\lambda {d}_{i}}{\pi {a}_{0}},$$
(5)
$${\rho}_{2}=\frac{\lambda {d}_{o}}{\pi {a}_{0}}.$$
(6)
$${\mu}_{1}({\mathit{p}}_{1},{\mathit{p}}_{2},{\omega}_{2}\tau )=\frac{\u3008I({\mathit{p}}_{1},t)I({\mathit{p}}_{2},t+\tau )\u3009-\u3008I({\mathit{p}}_{1},t)\u3009\u3008I({\mathit{p}}_{2},t+\tau )\u3009}{\u3008I({\mathit{p}}_{1},t)\u3009\u3008I({\mathit{p}}_{2},t+\tau )\u3009}\phantom{\rule{0ex}{0ex}}=\mathrm{exp}[-\frac{{({\mathit{p}}_{2}-{\mathit{p}}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{2}\tau +\mathfrak{R}{\mathit{p}}_{1}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\omega}_{2}\tau )}^{2}}{{\rho}_{2}^{2}}].$$
(7)
$$C=\sqrt{\frac{N+M+1}{NM}},$$
(8)
$${\sigma}_{c}={\rho}_{2}=\frac{\lambda {d}_{o}}{\pi {a}_{0}}.$$
(9)
$${I}_{f}(x,y,z)=\frac{1}{8{\lambda}^{2}{z}^{2}}{\left[\pi {d}_{p}^{2}\text{\hspace{0.17em}}\mathrm{exp}\left(\frac{{s}_{x}^{2}+{s}_{y}^{2}}{4{r}_{0}^{2}}\right)\right]}^{2}{\left[\frac{{J}_{1}\left(\frac{k{d}_{p}}{2z}\sqrt{{x}^{2}+{y}^{2}}\right)}{\left(\frac{k{d}_{p}}{2z}\sqrt{{x}^{2}+{y}^{2}}\right)}\right]}^{2}\{1+\gamma \text{\hspace{0.17em}}\mathrm{cos}[\frac{k}{z}({s}_{x}x+{s}_{y}y)]\},$$
(10)
$$\gamma =\mathrm{exp}[-\frac{({s}_{x}^{2}+{s}_{y}^{2})}{2{\sigma}_{c}^{2}}].$$
(11)
$$U(\mathit{r},t)=\mathrm{exp}(-\frac{{\mathit{r}}^{2}}{{w}_{0}^{2}})\mathrm{exp}\left(i\frac{k{\mathit{r}}^{2}}{2R}\right),$$
(12)
$$\u3008{A}^{*}({\mathit{p}}_{2},{t}_{2})A({\mathit{p}}_{1},{t}_{1})\u3009=\iint \iint {U}^{*}({\mathit{r}}_{2},{t}_{2})U({\mathit{r}}_{1},{t}_{1})\u3008\mathrm{exp}[i\varphi ({\mathit{r}}_{2})-i\varphi ({\mathit{r}}_{1}^{\prime})]\u3009{T}^{*}({\mathit{r}}_{2},{\mathit{p}}_{2})T({\mathit{r}}_{1},{\mathit{p}}_{1}){\mathrm{d}}^{2}{r}_{1}{\mathrm{d}}^{2}{r}_{2},$$
(13)
$$T(\mathit{r},\mathit{p})=\mathrm{exp}[-\frac{i\pi}{\lambda}(\frac{{\mathit{r}}^{2}}{{d}_{i}}+\frac{{\mathit{p}}^{2}}{{d}_{o}})]\mathrm{exp}[-{(\frac{\mathit{r}}{{\rho}_{1}}+\frac{\mathit{p}}{{\rho}_{2}})}^{2}],$$
(14)
$${\mathit{r}}_{1}^{\prime}={\mathit{r}}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{2}\tau -\mathfrak{R}{\mathit{r}}_{1}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\omega}_{2}\tau ,$$
(15)
$$\mathfrak{R}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right),$$
(16)
$$\left(\begin{array}{c}{\xi}_{1}^{\prime}\\ {\eta}_{1}^{\prime}\end{array}\right)=\left(\begin{array}{c}{\xi}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{1}\tau -{\eta}_{1}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\omega}_{1}\tau \\ {\xi}_{1}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\omega}_{1}\tau +{\eta}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}{\omega}_{1}\tau \end{array}\right).$$
(17)
$$\u3008\mathrm{exp}[i\varphi ({\mathit{r}}_{2})-i\varphi ({\mathit{r}}_{1}^{\prime})]\u3009=\delta ({\mathit{r}}_{2}-{\mathit{r}}_{1}^{\prime}).$$
(18)
$${U}^{*}({\mathit{r}}_{2},{t}_{2})U({\mathit{r}}_{1},{t}_{1})=\mathrm{exp}(-\frac{{\mathit{r}}_{2}^{2}}{{r}_{0}^{2}})\mathrm{exp}(-\frac{{\mathit{r}}_{1}^{2}}{{r}_{0}^{2}}).$$