## Abstract

It is shown that branch points present in a turbulence-distorted optical field can be visualized as peaks and valleys of a certain potential function. Peaks correspond to positive branch points and valleys correspond to negative ones, thus allowing one to study the formation, movements, and merging of branch points. A closed-form formula is given for the potential in terms of wave-front-sensor measurements; branch points appear as logarithmic singularities that are easy to detect visually through computer-generated images. In fact, the branch-point potential is obtained by means of a single matrix multiplication. An electrostatic analogy is given, as well as a proof that the continuous part of the wave front does not change the location of the potential singularities. Applications can be found in adaptive optics, in the airborne laser system, in speckle or coherent imaging, and in high-bandwidth laser communication.

© 1999 Optical Society of America

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### Equations (15)

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(1)
$$\mathbf{curl}\mathbf{g}(\mathbf{x})=2\pi \left[\sum _{i=1}^{{N}_{+}}\delta (\mathbf{x}-{\mathbf{x}}_{i}^{+})-\sum _{j=1}^{{N}_{-}}\delta (\mathbf{x}-{\mathbf{x}}_{j}^{-})\right]\stackrel{\u02c6}{\mathbf{z}},$$
(2)
$$\mathrm{div}({\mathbf{Rot}}_{-\pi /2}\mathbf{g})=(\mathbf{curl}\mathbf{g})\cdot \stackrel{\u02c6}{\mathbf{z}}.$$
(3)
$$\mathbf{grad}V=-{\mathbf{Rot}}_{-\pi /2}\mathbf{g}={\mathbf{Rot}}_{\pi /2}\mathbf{g}.$$
(4)
$${\int}_{\mathcal{S}}{\mathrm{d}}^{2}\mathbf{x}\Vert \mathbf{grad}V(\mathbf{x})-\mathbf{R}(\mathbf{x}){\Vert}^{2},$$
(5)
$$\sum _{\mathbf{k}}\Vert i\mathbf{k}{\tilde{V}}_{\mathbf{k}}-{\tilde{\mathbf{R}}}_{\mathbf{k}}{\Vert}^{2},$$
(6)
$$\tilde{{V}_{\mathbf{k}}^{\mathrm{sol}}}\equiv -i\frac{\mathbf{k}\cdot {\tilde{\mathbf{R}}}_{\mathbf{k}}}{\Vert \mathbf{k}{\Vert}^{2}}.$$
(7)
$$i\mathbf{k}\cdot {\tilde{\mathbf{R}}}_{\mathbf{k}}=2\pi \left[\sum _{i=1}^{{N}_{+}}exp(-i\mathbf{k}\cdot {\mathbf{x}}_{i}^{+})-\sum _{j=1}^{{N}_{-}}exp(-i\mathbf{k}\cdot {\mathbf{x}}_{j}^{-})\right],$$
(8)
$$\tilde{{V}_{\mathbf{k}}^{\mathrm{sol}}}=-2\pi \left[\sum _{i=1}^{{N}_{+}}\frac{exp(-i\mathbf{k}\cdot {\mathbf{x}}_{i}^{+})}{\Vert \mathbf{k}{\Vert}^{2}}-\sum _{j=1}^{{N}_{-}}\frac{exp(-i\mathbf{k}\cdot {\mathbf{x}}_{j}^{-})}{\Vert \mathbf{k}{\Vert}^{2}}\right].$$
(9)
$$\mathbf{R}(\mathbf{x})=-\frac{2\pi}{\Vert \mathbf{x}\Vert}\stackrel{\u02c6}{\mathbf{r}}=-2\pi \mathbf{grad}log\Vert \mathbf{x}\Vert .$$
(10)
$${V}^{\mathrm{sol}}(\mathbf{x})=-2\pi log\Vert \mathbf{x}\Vert .$$
(11)
$$\tilde{{V}_{\mathbf{k}}^{\mathrm{sol}}}=-\frac{2\pi}{\Vert \mathbf{k}{\Vert}^{2}}.$$
(12)
$${V}^{\mathrm{sol}}(\mathbf{x})\equiv {\mathrm{grad}}^{+}({\mathbf{Rot}}_{\pi /2}\mathbf{g})(\mathbf{x})=-2\pi \left(\sum _{i=1}^{{N}_{+}}log\Vert \mathbf{x}-{\mathbf{x}}_{i}^{+}\Vert -\sum _{j=1}^{{N}_{-}}log\Vert \mathbf{x}-{\mathbf{x}}_{j}^{-}\Vert \right),$$
(13)
$${\mathbf{A}}^{+}=\underset{m\to 0}{\mathrm{lim}}({\mathbf{A}}^{\u2020}\mathbf{A}+{m}^{2}{)}^{-1}{\mathbf{A}}^{\u2020},$$
(14)
$${\mathbf{V}}^{\mathrm{sol}}\equiv {\mathbf{A}}^{+}{\mathbf{R}}_{\pi /2}\mathbf{g},$$
(15)
$${\mathbf{R}}_{\pi /2}^{-1}\mathbf{A}{\mathbf{V}}^{\mathrm{sol}}.$$