Auto-focusing and self-healing of Pearcey beams

James D. Ring; Jari Lindberg; Areti Mourka; Michael Mazilu; Kishan Dholakia; Mark R. Dennis

doi:10.1364/OE.20.018955

Optics Express
Vol. 20,
Issue 17,
pp. 18955-18966
(2012)
•https://doi.org/10.1364/OE.20.018955

Auto-focusing and self-healing of Pearcey beams

James D. Ring, Jari Lindberg, Areti Mourka, Michael Mazilu, Kishan Dholakia, and Mark R. Dennis

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James D. Ring, Jari Lindberg, Areti Mourka, Michael Mazilu, Kishan Dholakia, and Mark R. Dennis, "Auto-focusing and self-healing of Pearcey beams," Opt. Express 20, 18955-18966 (2012)

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Abstract

We present a new solution of the paraxial equation based on the Pearcey function, which is related to the Airy function and describes diffraction about a cusp caustic. The Pearcey beam displays properties similar not only to Airy beams but also Gaussian and Bessel beams. These properties include an inherent auto-focusing effect, as well as form-invariance on propagation and self-healing. We describe the theory of propagating Pearcey beams and present experimental verification of their auto-focusing and self-healing behaviour.

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Fig. 1 Transverse profile intensities of the Pearcey beam, with parameters x₀ = y₀ = 10⁻⁴ m and where

z_{e} \equiv 2 k y_{0}^{2} \approx 0.251

m and k is the wavenumber for wavelength λ = 500 nm; (a) intensity of the Pearcey function for z = 0 m; (b) the Pearcey beam at z = 0.8z_e m; (c) at z = 0.975z_e m; (d) at z = 1.025z_e m; (e) z = 1.2z_e m; (f) z = 2z_e m. The cusp underlying the Pearcey pattern is shown as a white dashed line in (a). Upon propagation, the cusp - and therefore the shape of the Pearcey pattern - flattens out to a line, then inverts after a singular plane at z = z_e.

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Fig. 2 Transverse intensity of a Pearcey-Gauss beam as z increases for x₀ = y₀ = 0.1 mm, w₀ = 2.0 mm and λ = 500 nm. The scaling and inversion of the pattern is still evident, however, there now exists a small hourglass-shaped focal point that was absent in case of the unmodulated Pearcey beam. The intensities of each image are not on the same scale.

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Fig. 3 Intensity of the Pearcey-Gauss beam (for x₀ = y₀ = 0.1 mm, w₀ = 2.0 mm and λ = 500 nm) at the essential focusing plane; (a) magnification of the essential focus of Fig. 2; the dashed lines correspond to the intensity cross-sections of (c) and (d); (b) intensity of the Fourier distribution of the Pearcey-Gauss beam according to Eq. (9), which mimics the δ-line parabola of Eq. (3); (c) the short-dashed line shows the intensity cross-section of the essential focus in the x-direction; (d) intensity cross-section along the long-dashed line of (a) in the y-direction.

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Fig. 4 Experimental setup; (a) schematic of the experiment, where L_i are lenses, SLM is the spatial light modulator, CCD is the charge coupled device camera, PBS is a polarizing beam splitter; the focal widths of lenses are f₁ = 25 mm, f₂ = 100 cm, f₃ = 680 mm, f₄ = 400 mm and f₅ = 800 mm; (b) image encoded on the SLM. The Fourier transform of the Pearcey-Gauss beam describes a parabola with phase given by Eq. (9). The hue indicates the phase while brightness describes the corresponding intensity. The SLM was used in the standard first-order diffraction configuration.

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Fig. 5 Experimental observation of the Pearcey-Gauss beam for consecutive propagation distances. The collapse of the beam to a point is clearly visible, as well as the predicted inversion. These results agree with the theoretical and numerical predictions. The propagation distances are given, inset in the images.

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Fig. 6 Experimental images of the self-healing of the Pearcey beam from an arbitrary perturbation. The obstacle was cylindrical, with a rectangular projection, and the area blocked is indicated by the white line in (a). It is clear that the Pearcey beam recovers from the initial perturbation and still collapses and inverts after its essential focus.

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Equations (12)

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Pe (X, Y) \equiv \int_{- \infty}^{\infty} d s exp [i (s^{4} + s^{2} Y + s X)],

{(\frac{2 y}{3 y_{0}})}^{3} + {(\frac{x}{x_{0}})}^{2} = 0 .

\begin{array}{l} \tilde{Pe} (k_{x} x_{0}, k_{y} y_{0}) & = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x d y Pe (\frac{x}{x_{0}}, \frac{y}{y_{0}}) e^{- i k_{x} x - i k_{y} y} \\ = x_{0} y_{0} e^{i k_{x}^{4} x_{0}^{4}} δ (k_{x}^{2} x_{0}^{2} - k_{y} y_{0}), \end{array}

\begin{array}{l} {Pe}_{beam} (x, y, z) & = \frac{- i k}{2 π z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x^{'} d y^{'} Pe (\frac{x^{'}}{x_{0}}, \frac{y^{'}}{y_{0}}) exp (\frac{i k}{2 z} [{(x - x^{'})}^{2} + {(y - y^{'})}^{2}]) \\ = \frac{1}{{(1 - z / z_{e})}^{\frac{1}{4}}} Pe (\frac{x}{x_{0} {(1 - z / z_{e})}^{\frac{1}{4}}}, \frac{y - z y_{0} / 2 k x_{0}^{2}}{y_{0} {(1 - z / z_{e})}^{\frac{1}{2}}}), \end{array}

{(\frac{2}{3} \frac{y - z y_{0} / 2 k x_{0}^{2}}{y_{0} {(1 - z / z_{e})}^{\frac{1}{2}}})}^{3} + {(\frac{x}{x_{0} {(1 - z / z_{e})}^{\frac{1}{4}}})}^{2} = 0 .

{Pe}_{beam} (x, y, z_{e}) = e^{i π / 4} \sqrt{\frac{π}{y / y_{0} - y_{0}^{2} / x_{0}^{2}}} exp (\frac{- i x^{2} y_{0}}{4 (y x_{0}^{2} - y_{0}^{3})}) .

G (x, y, z) = \frac{1}{(1 + i z / z_{R})} exp (- \frac{x^{2} + y^{2}}{w_{0}^{2} (1 + i z / z_{R})}),

PeG (x, y, z) = \frac{G (x, y, z)}{{[1 - z / ζ (z) z_{e}]}^{\frac{1}{4}}} Pe (\frac{x}{x_{0} ζ (z) {[1 - z / ζ (z) z_{e}]}^{\frac{1}{4}}}, \frac{y - z y_{0} / 2 k x_{0}^{2}}{y_{0} ζ (z) {[1 - z / ζ (z) z_{e}]}^{\frac{1}{2}}}),

\tilde{PeG} (k_{x}, k_{y}) = \frac{w^{2} exp [- w^{2} (k_{x}^{2} + k_{y}^{2}) / 4]}{4 π {(1 + i w^{2} / 4 y_{0}^{2})}^{1 / 4}} Pe (\frac{w^{2} k_{x}}{2 i x_{0} {(1 + i w^{2} / 4 y_{0}^{2})}^{1 / 4}}, \frac{w^{2} (k_{y} - y_{0} / 2 x_{0}^{2})}{2 i y_{0} {(1 + i w^{2} / 4 y_{0}^{2})}^{1 / 2}}),

\begin{array}{r} PeG (x, y_{0}^{3} / x_{0}^{2}, z_{e}) = exp (\frac{- x^{2} - y_{0}^{6} / x_{0}^{4}}{w_{0}^{2} ζ (z_{e})}) \sqrt{\frac{w_{0}}{2 y_{0} ζ {(z_{e})}^{3 / 2}}} [2 Γ (\frac{5}{4}) {}_{0}F_{2} (; \frac{1}{2}, \frac{3}{4}; \frac{w_{0}^{2} x^{4}}{2^{10} ζ {(z_{e})}^{3} x_{0}^{4} y_{0}^{2}}) \\ + \frac{w_{0} x^{2}}{2^{5} ζ {(z_{e})}^{3 / 2} y_{0} x_{0}^{2}} Γ (- \frac{1}{4}) {}_{0}F_{2} (; \frac{5}{4}, \frac{3}{2}; \frac{w_{0}^{2} x^{4}}{2^{10} ζ {(z_{e})}^{3} x_{0}^{4} y_{0}^{2}})], \end{array}

\begin{array}{r} PeG (0, y, z_{e}) = exp (\frac{- y^{2}}{w_{0}^{2} ζ (z_{e})}) \frac{e^{- i π / 4} w_{0}}{4 ζ (z_{e}) y_{0}} \sqrt{\frac{y}{y_{0}} - \frac{y_{0}^{2}}{x_{0}^{2}}} exp [- \frac{w_{0}^{2}}{32 ζ (z_{e})} {(\frac{y}{y_{0}^{2}} - \frac{y_{0}}{x_{0}^{2}})}^{2}] \\ \times K_{\frac{1}{4}} [- \frac{w_{0}^{2}}{32 ζ (z_{e})}] {(\frac{y}{y_{0}^{2}} - \frac{y_{0}}{x_{0}^{2}})}^{2}, \end{array}

U (x, y) = \int_{- \infty}^{\infty} d s exp [i (s^{2 n} + y s^{n} + x s^{m})],

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Equations (12)

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