## Abstract

The output field of a uniformly illuminated lens contains points of zero intensity on the optical axis and zero-intensity Airy rings in the focal plane, as formed by diffraction. These intensity zeros have been recognized as phase singularities or wave dislocations. Recently it was shown that, under the influence of a perturbation, the axial singularities may transform into rings or disappear and that the Airy rings may split into triplets. Starting from optical diffraction theory, we identify the physical perturbations that can induce such topological transformations. The basic perturbations are phase and amplitude aberrations of the wave front that is incident on the lens; we show that their different natures have consequences for the possible dislocation reactions.

© 1998 Optical Society of America

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### Equations (16)

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(1)
$${u}_{A}=exp\left(+\mathit{ik}\frac{{\rho}^{2}}{2f}-\u220a\frac{{\rho}^{2}}{{a}^{2}}+\mathit{ik}\delta \mathrm{\lambda}\frac{{\rho}^{4}}{{a}^{4}}\right).$$
(2)
$$u(\rho ,z)\propto exp(\mathit{ikz}){\int}_{0}^{1}exp({A}_{1}{t}^{2}+{A}_{2}{t}^{4}){J}_{0}({A}_{3}t)t\mathrm{d}t,$$
(3)
$${A}_{1}\equiv -\u220a-\frac{1}{2}\mathit{ik}\theta {}^{2}z,$$
(4)
$${A}_{2}\equiv -2\pi i\delta ,$$
(5)
$${A}_{3}\equiv k\theta \rho $$
(6)
$$u(\rho =0,z)\propto exp[\mathit{ik}(1-\frac{1}{4}\theta {}^{2})z]\mathrm{sinc}(\frac{1}{4}k\theta {}^{2}z),$$
(7)
$${z}_{n}\equiv \frac{2\mathrm{\lambda}}{\theta {}^{2}}n,\hspace{1em}\hspace{1em}\mathrm{with}\hspace{0.5em}n=\pm 1,\pm 2,\dots ,$$
(8)
$$u(\rho ,z=0)\propto \frac{{J}_{1}(k\theta \rho )}{k\theta \rho},$$
(9)
$$k\theta {\rho}_{m}=3.832,\hspace{0.5em}7.016,\hspace{0.5em}10.173,\dots $$
(10)
$$(m=1,2,3,\dots ).$$
(11)
$$|\u220a|\ll 1,\hspace{1em}\hspace{1em}|\delta |\ll 1,$$
(12)
$$\rho \ll 1/k\theta ,$$
(13)
$$z={z}_{n}+{z}^{\prime},\hspace{1em}\hspace{1em}\mathrm{with}\hspace{0.5em}|{z}^{\prime}|\ll 1/k{\theta}^{2}.$$
(14)
$$u(\rho ,z={z}_{n}+{z}^{\prime})\propto \left[{\rho}^{2}+\frac{2{\mathit{iz}}^{\prime}}{k}-\alpha \right]exp({\mathit{ikz}}^{\prime}),$$
(15)
$$\alpha \equiv -\frac{4}{{k}^{2}\theta {}^{2}}\left[\left(\u220a+\frac{2\delta}{n}\right)+i2\pi \delta \right].$$
(16)
$$u(\rho ,z)\propto \left[{\rho}^{4}+\frac{8\mathit{iz}}{k}{\rho}^{2}-\frac{8}{{k}^{2}}{z}^{2}+\frac{4}{{k}^{2}}{\rho}^{2}+b\left({\rho}^{2}+\frac{2\mathit{iz}}{k}\right)+c\right]exp(\mathit{ikz}),$$