## Abstract

An equivalence theorem is formulated that provides conditions under which planar sources of different states of spatial coherence will generate optical fields that have identical far-zone intensity distributions. As an example, a partially coherent source whose linear dimensions are large compared with the correlation length of the light across the source is described that will generate a field whose far-zone intensity distribution is identical with that of a Gaussian laser beam.

© 1978 Optical Society of America

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### Equations (12)

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(1)
$${J}_{\omega}(\mathbf{s})={(2\pi k)}^{2}{\text{cos}}^{2}\phantom{\rule{0.3em}{0ex}}\theta {\tilde{W}}^{(0)}(k{\mathbf{s}}_{\perp},-k{\mathbf{s}}_{\perp};\omega )\phantom{\rule{0.2em}{0ex}}.$$
(2)
$${\tilde{W}}^{(0)}({\mathbf{f}}_{1},{\mathbf{f}}_{2};\omega )=\frac{1}{{(2\pi )}^{4}}\iint {W}^{(0)}({\mathbf{r}}_{1},{\mathbf{r}}_{2};\omega )\times \text{exp}[-i({\mathbf{f}}_{1}\cdot {\mathbf{r}}_{1}+{\mathbf{f}}_{2}\cdot {\mathbf{r}}_{2})]\phantom{\rule{0.2em}{0ex}}{\text{d}}^{2}{r}_{1}{\text{d}}^{2}{r}_{2},$$
(3)
$${\mathbf{f}}_{1}=-{\mathbf{f}}_{2}=k{\mathbf{s}}_{\perp}.$$
(4)
$$\left|k{\mathbf{s}}_{\perp}\right|\le \left|k\mathbf{s}\right|=k,$$
(5)
$${\mu}^{\phantom{\rule{0.1em}{0ex}}(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{1},{\mathbf{r}}_{2};\omega )=\frac{{W}^{(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{1},{\mathbf{r}}_{2};\omega )}{\sqrt{{I}^{\phantom{\rule{0.1em}{0ex}}(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{1};\omega )}\sqrt{{I}^{\phantom{\rule{0.1em}{0ex}}(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{2};\omega )}},$$
(6)
$${I}^{(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{j};\omega )={W}^{(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{j},{\mathbf{r}}_{j};\omega )\phantom{\rule{0.1em}{0ex}},(j=1,2)$$
(7)
$${{I}_{L}}^{(0)}\phantom{\rule{0.1em}{0ex}}(\mathbf{r};\omega )={A}_{L}\phantom{\rule{0.3em}{0ex}}\text{exp}\phantom{\rule{0.1em}{0ex}}(-{r}^{2}/2{{\sigma}_{L}}^{2})\phantom{\rule{0.1em}{0ex}},$$
(8)
$${{\mu}_{L}}^{(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{1},{\mathbf{r}}_{2};\omega )\equiv {{g}_{L}}^{(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{1}-{\mathbf{r}}_{2};\omega )=1$$
(9)
$$J(\mathbf{s})={J}^{\phantom{\rule{0.1em}{0ex}}(0)}\phantom{\rule{0.3em}{0ex}}{\text{cos}}^{2}\phantom{\rule{0.3em}{0ex}}\theta \phantom{\rule{0.3em}{0ex}}\text{exp}\phantom{\rule{0.2em}{0ex}}[-2{(k{\sigma}_{L}\phantom{\rule{0.1em}{0ex}})}^{2}{\text{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta ]\phantom{\rule{0.2em}{0ex}},$$
(10)
$${{g}_{Q}}^{(0)}\phantom{\rule{0.1em}{0ex}}({\mathbf{r}}_{1}-{\mathbf{r}}_{2};\omega )=\text{exp}\phantom{\rule{0.1em}{0ex}}[-{({\mathbf{r}}_{1}-{\mathbf{r}}_{2})}^{2}/8{{\sigma}_{L}}^{2}]$$
(11)
$${{I}_{Q}}^{(0)}\phantom{\rule{0.1em}{0ex}}(\mathbf{r};\omega )={({\sigma}_{L}/{\sigma}_{Q})}^{2}{A}_{L}\phantom{\rule{0.3em}{0ex}}\text{exp}\phantom{\rule{0.2em}{0ex}}(-{r}^{2}/2{{\sigma}_{Q}}^{2})\phantom{\rule{0.1em}{0ex}},$$
(12)
$${\sigma}_{Q}\gg 2{\sigma}_{L},$$