## Abstract

In this paper, we investigate the Fraunhofer diffraction of a class of partially coherent light diffracted by a circular aperture. It is shown that by the illumination of partially coherent light of the special spatial correlation function, the anomalous behaviors of the diffraction patterns are found. We find that the decrease of the spatial coherence of the light in the aperture leads to the drastic changes of the diffraction pattern. Specifically, when the light in the aperture is fully coherent, the diffraction pattern is just an Airy disc. However, as the coherence decreases, the diffraction pattern becomes an annulus, and the radius of the annulus increases with the decrease of the coherence. Flattened annuli can be achieved, when the parameters characterizing the correlation of the partially coherent light are chosen with suitable values. Potential applications of modulating the coherence to achieve desired diffraction patterns are discussed.

© 2003 Optical Society of America

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### Equations (9)

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(1)
$$W\left({\overrightarrow{\rho}}_{1},{\overrightarrow{\rho}}_{2}\right)=W\left({\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\right)={s}_{0}\mu \left({\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\right),$$
(2)
$$\mu \left({\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\right)=\frac{1}{1-{\epsilon}_{s}^{2}}\mathrm{Besinc}\left(k\mid {\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\mid b\right)-\frac{{\epsilon}_{s}^{2}}{1-{\epsilon}_{s}^{2}}\mathrm{Besinc}\left(k\mid {\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\mid {\epsilon}_{s}b\right)$$
(3)
$$I(\overrightarrow{r},z)={\left(\frac{k}{2\pi z}\right)}^{2}\underset{A}{\iint}W\left({\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\right)\mathrm{exp}\left\{\mathit{ik}\left({\overrightarrow{\rho}}_{1}-{\overrightarrow{\rho}}_{2}\right).\frac{\overrightarrow{r}}{z}\right\}\text{d}{\overrightarrow{\rho}}_{1}\text{d}{\overrightarrow{\rho}}_{2}$$
(4)
$$I\left(v\right)=2{S}_{0}{\left(\frac{k{a}^{2}}{z}\right)}^{2}\{\frac{1}{1-{\epsilon}_{s}^{2}}{\int}_{0}^{1}C\left(u\right)\mathrm{Besinc}\left[7.664\left(\frac{a}{\overline{L}}\right)u\right]{J}_{0}\left(7.664\mathit{uv}\right)\mathit{udu}-$$
(5)
$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\frac{{\epsilon}_{s}^{2}}{1-{\epsilon}_{s}^{2}}{\int}_{0}^{1}C\left(u\right)\mathrm{Besinc}\left[7.664\left(\frac{a}{\overline{L}}\right){\epsilon}_{s}u\right]{J}_{0}\left(7.664\mathit{uv}\right)\mathit{udu}\}$$
(6)
$$C\left(u\right)=\left(\frac{2}{\pi}\right)\left\{{\mathrm{cos}}^{\mathrm{-1}}\left(u\right)-u{\left(1-{u}^{2}\right)}^{\frac{1}{2}}\right\}.$$
(7)
$$\overline{L}=\frac{3.832}{\mathit{kb}}.$$
(8)
$$v=3.832\mathit{ka}\left(\frac{r}{z}\right)\approx \left(3.832\mathit{ka}\right)\theta ,$$
(9)
$$\theta \approx \frac{r}{z}$$