## Abstract

While adaptive optical systems are able to remove moderate wavefront distortions in scintillated optical beams, phase singularities that appear in strongly scintillated beams can severely degrade the performance of such an adaptive optical system. Therefore the detection of these phase singularities is an important aspect of strong-scintillation adaptive optics. We investigate the detection of phase singularities with the aid of a Shack–Hartmann wavefront sensor and show that, in spite of some systematic deficiencies inherent to the Shack–Hartmann wavefront sensor, it can be used for the reliable detection of phase singularities, irrespective of their morphologies. We provide full analytical results, together with numerical simulations of the detection process.

© 2007 Optical Society of America

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### Equations (22)

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(1)
$$\theta (x,y)={\theta}_{\mathrm{C}}(x,y)+\sum _{n}\varphi (x-{x}_{n},y-{y}_{n};{\alpha}_{n},{\beta}_{n}),$$
(2)
$$\varphi (x,y;\alpha ,\beta )=-\frac{i}{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left[\frac{\xi (x+iy)+\zeta (x-iy)}{{\xi}^{*}(x-iy)+{\zeta}^{*}(x+iy)}\right],$$
(3)
$$\xi =\mathrm{cos}(\alpha \u22152)\mathrm{exp}(i\beta \u22152),$$
(4)
$$\zeta =\mathrm{sin}(\alpha \u22152)\mathrm{exp}(-i\beta \u22152),$$
(5)
$${\oint}_{C}\nabla \theta \phantom{\rule{0.1em}{0ex}}\bullet \phantom{\rule{0.2em}{0ex}}\mathrm{d}l=\tau 2\pi ,$$
(6)
$$\frac{\partial {G}_{x}(x,y)}{\partial y}=\frac{\partial {G}_{y}(x,y)}{\partial x}.$$
(7)
$$D(x,y)={\nabla}_{T}\times \mathbf{G}(x,y).$$
(8)
$$\nabla \times \nabla \varphi (x,y,\alpha ,\beta )=\tau 2\pi \delta \left(x\right)\delta \left(y\right),$$
(9)
$$\nabla \times \nabla \theta (x,y)=2\pi \sum _{n}{\tau}_{n}\delta (x-{x}_{n})\delta (y-{y}_{n}).$$
(10)
$${\mathbf{u}}^{m,n}=\frac{{\int}_{H}I\left(\mathbf{u}\right)\mathbf{u}\phantom{\rule{0.2em}{0ex}}{\mathrm{d}}^{2}u}{{\int}_{H}I\left(\mathbf{u}\right)\phantom{\rule{0.2em}{0ex}}{\mathrm{d}}^{2}u}-{\mathbf{u}}_{0}^{m,n},$$
(11)
$${\mathbf{G}}^{m,n}=\frac{{\int}_{\Omega}\nabla \theta \left(\mathbf{x}\right)\phantom{\rule{0.2em}{0ex}}{\mathrm{d}}^{2}x}{{\int}_{\Omega}\phantom{\rule{0.2em}{0ex}}{\mathrm{d}}^{2}x}\approx \frac{k}{f}{\mathbf{u}}^{m,n},$$
(12)
$${D}^{m,n}=\frac{w}{2}({G}_{x}^{m,n}+{G}_{x}^{m,n+1}+{G}_{y}^{m,n+1}+{G}_{y}^{m+1,n+1}-{G}_{x}^{m+1,n+1}-{G}_{x}^{m+1,n}-{G}_{y}^{m+1,n}-{G}_{y}^{m,n}),$$
(13)
$$\nabla \varphi (x,y)=\frac{x\widehat{y}-y\widehat{x}}{{x}^{2}+{y}^{2}}.$$
(14)
$${\mathbf{G}}^{m,n}=\frac{1}{{w}^{2}}{\int}_{-w}^{0}{\int}_{-w}^{0}\frac{x\widehat{y}-y\widehat{x}}{{x}^{2}+{y}^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\mathrm{d}y=(\frac{\pi}{4w}+\frac{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2}{2w})(\widehat{x}-\widehat{y}).$$
(15)
$${D}^{m,n}=4w{G}_{x}^{m,n}=\pi +2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left(2\right)=4.527887.$$
(16)
$$\nabla \varphi (x-{x}_{0},y-{y}_{0})=\frac{(x-{x}_{0})\widehat{y}-(y-{y}_{0})\widehat{x}}{{(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}}.$$
(17)
$$D=\frac{\pi}{2}-2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left(5\right)-2\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left(2\right)+5\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left(2\right)=-0.396641.$$
(18)
$$\nabla \theta (x,y)=\frac{(x\widehat{y}-y\widehat{x})C}{{x}^{2}(1+A)-2yxB+{y}^{2}(1-A)},$$
(19)
$$A=\mathrm{sin}\left(\alpha \right)\mathrm{cos}\left(\beta \right),$$
(20)
$$B=\mathrm{sin}\left(\alpha \right)\mathrm{sin}\left(\beta \right),$$
(21)
$$C=\mathrm{cos}\left(\alpha \right),$$
(22)
$${D}^{m,n}=\frac{(\mu +1)({A}_{m}-B)}{2{A}_{m}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\mu +1)B-(\nu -1){A}_{m}}{(\mu +1)C}\right]-\frac{(\mu +1)({A}_{m}+B)}{2{A}_{m}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\mu +1)B-(\nu +1){A}_{m}}{(\mu +1)C}\right]+\frac{(\mu -1)({A}_{m}+B)}{2{A}_{m}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\mu -1)B-(\nu -1){A}_{m}}{(\mu -1)C}\right]-\frac{(\mu -1)({A}_{m}-B)}{2{A}_{m}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\mu -1)B-(\nu +1){A}_{m}}{(\mu -1)C}\right]+\frac{(\nu +1)({A}_{p}-B)}{2{A}_{p}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\nu +1)B-(\mu -1){A}_{p}}{(\nu +1)C}\right]-\frac{(\nu +1)({A}_{p}+B)}{2{A}_{p}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\nu +1)B-(\mu +1){A}_{p}}{(\nu +1)C}\right]+\frac{(\nu -1)({A}_{p}+B)}{2{A}_{p}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\nu -1)B-(\mu -1){A}_{p}}{(\nu -1)C}\right]-\frac{(\nu -1)({A}_{p}-B)}{2{A}_{p}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{(\nu -1)B-(\mu +1){A}_{p}}{(\nu -1)C}\right]-\frac{(\mu +1)B}{{A}_{m}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\nu {A}_{m}-(\mu +1)B}{(\mu +1)C}\right]+\frac{(\mu -1)B}{{A}_{m}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\nu {A}_{m}-(\mu -1)B}{(\mu -1)C}\right]-\frac{(\nu +1)B}{{A}_{p}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\mu {A}_{p}-(\nu +1)B}{(\nu +1)C}\right]+\frac{(\nu -1)B}{{A}_{p}}\phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\mu {A}_{p}-(\nu -1)B}{(\nu -1)C}\right]+\mu \phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\mu B-(\nu +1){A}_{m}}{\mu C}\right]-\mu \phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\mu B-(\nu -1){A}_{m}}{\mu C}\right]+\nu \phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\nu B-(\mu +1){A}_{p}}{\nu C}\right]-\nu \phantom{\rule{0.2em}{0ex}}\mathrm{arctan}\left[\frac{\nu B-(\mu -1){A}_{p}}{\nu C}\right]+\frac{(\mu +1)C}{4{A}_{m}}\{\mathrm{ln}[2(\mu +1)(\nu -1)B-{(\mu +1)}^{2}{A}_{p}-{(\nu -1)}^{2}{A}_{m}]+\mathrm{ln}[2(\mu +1)(\nu +1)B-{(\mu +1)}^{2}{A}_{p}-{(\nu +1)}^{2}{A}_{m}]-2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}[2(\mu +1)\nu B-{(\mu +1)}^{2}{A}_{p}-{\nu}^{2}{A}_{m}]\}-\frac{(\mu -1)C}{4{A}_{m}}\{[2(\mu -1)(\nu +1)B-{(\mu -1)}^{2}{A}_{p}-{(\nu +1)}^{2}{A}_{m}]+\mathrm{ln}[2(\mu -1)(\nu -1)B-{(\mu -1)}^{2}{A}_{p}-{(\nu -1)}^{2}{A}_{m}]-2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}[2(\mu -1)\nu B-{(\mu -1)}^{2}{A}_{p}-{\nu}^{2}{A}_{m}]\}+\frac{(\nu +1)C}{4{A}_{p}}\{\mathrm{ln}[2(\mu +1)(\nu +1)B-{(\mu +1)}^{2}{A}_{p}-{(\nu +1)}^{2}{A}_{m}]+\mathrm{ln}[2(\mu -1)(\nu +1)B-{(\mu -1)}^{2}{A}_{p}-{(\nu +1)}^{2}{A}_{m}]-2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}[2\mu (\nu +1)B-{\mu}^{2}{A}_{p}-{(\nu +1)}^{2}{A}_{m}]\}-\frac{(\nu -1)C}{4{A}_{p}}\{\mathrm{ln}[2(\mu -1)(\nu -1)B-{(\mu -1)}^{2}{A}_{p}-{(\nu -1)}^{2}{A}_{m}]+\mathrm{ln}[2(\mu +1)(\nu -1)B-{(\mu +1)}^{2}{A}_{p}-{(\nu -1)}^{2}{A}_{m}]-2\phantom{\rule{0.2em}{0ex}}\mathrm{ln}[2\mu (\nu -1)B-{\mu}^{2}{A}_{p}-{(\nu -1)}^{2}{A}_{m}]\},$$