Stabilization of vortex beams in Kerr media by nonlinear absorption

Miguel A. Porras, Márcio Carvalho, Hervé Leblond, and Boris A. Malomed

Phys. Rev. A 94, 053810 – Published 7 November 2016

Abstract

We elaborate a solution for the problem of stable propagation of transversely localized vortex beams in homogeneous optical media with self-focusing Kerr nonlinearity. Stationary nonlinear Bessel-vortex states are stabilized against azimuthal breakup and collapse by multiphoton absorption, while the respective power loss is offset by the radial influx of the power from an intrinsic reservoir. A linear stability analysis and direct numerical simulations reveal a region of stability of these vortices. Beams with multiple vorticities have their stability regions too. These beams can then form robust tubular filaments in transparent dielectrics as common as air, water, and optical glasses at sufficiently high intensities. We also show that the tubular, rotating, and specklelike filamentation regimes, previously observed in experiments with axicon-generated Bessel beams, can be explained as manifestations of the stability or instability of a specific nonlinear Bessel-vortex state, which is fully identified.

Received 30 June 2016
Revised 10 October 2016

DOI:https://doi.org/10.1103/PhysRevA.94.053810

Physics Subject Headings (PhySH)

Angular momentum of light Beam instabilities Multiphoton or tunneling ionization & excitation Optical solitons Spontaneous symmetry breaking

Solitons

Nonlinear DynamicsAtomic, Molecular & Optical

Authors & Affiliations

Miguel A. Porras¹, Márcio Carvalho¹, Hervé Leblond², and Boris A. Malomed^3,4

¹Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Rios Rosas 21, 28003 Madrid, Spain
²Laboratoire de Photonique d'Angers, EA 4464, Université d'Angers, 2 Boulevard Lavoisier, 49000 Angers, France
³Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel
⁴Laboratory of Nonlinear-Optical Informatics, ITMO University, 197101 Saint Petersburg, Russia

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Issue

Vol. 94, Iss. 5 — November 2016

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Images

Figure 1
(a) For $M = 4$ , BVBs with $s = 0$ , 1, and 2 exist at $b_{0} < b_{0, max}$ . The dependence of $b_{0, max}$ on $α$ defined in Eq. (4), and relating Kerr nonlinearity, nonlinear absorption, and cone angle, is shown in the plot. (b) The dimensionless intensity profile $| \tilde{A} |^{2}$ of the nonlinear BVB with $s = 1, α = 1$ , and $b_{0} = 2$ (solid curve), compared to the intensity profile $b_{0}^{2} J_{s}^{2} (ρ)$ of the linear Bessel vortex with the same radial profile at $ρ \to 0$ (dashed curve).
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Figure 2
(a,b) Dimensionless growth rates of unstable perturbation modes versus $α$ , for values of $s, b_{0}$ , and $m$ indicated in the panels. (c,d) The largest growth rate computed with step size $h = 0.2$ and increasing numbers of points in the numerical grid, $N = 1000$ (circles), 2000 (squares), 4000 (up triangles), 8000 (down triangles), and 16 000 (rhombuses), for values of $α$ at which the computed growth rates significantly depend on $N$ . For all perturbation winding numbers $m$ , the growth rates at large $α$ quickly converge to positive values as $N$ increases, thus representing a genuine instability of the underlying BVB. Instead, for smaller values of $α$ the growth rates approach zero as $N$ increases, implying that the BVBs are stable in this case. The insets additionally display the growth rates as functions of $N$ for particular values of $α$ , indicating a decay of $\sim 1 / N$ .
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Figure 3
For $M = 4, s = 1$ , and $b_{0} = 1.6$ , the transverse intensity distributions, $| \tilde{A} |^{2}$ , of the initially perturbed vortex with (a) $α = 1.6$ at propagation distance $ζ = 300$ , (b) $α = 2.2$ at $ζ = 100$ , (c) $α = 2.8$ at $ζ = 61.25$ , and (d) $α = 4$ at $ζ = 15$ . The scaled coordinates are $(ξ, η) = \sqrt{2 k | δ |} (x, y)$ . The numbers of fragments into which the unstable vortices split are exactly predicted by the linear-stability analysis.
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Figure 4
For $M = 4, s = 1$ , and $b_{0} = 1.2$ , the transverse intensity distributions, $| \tilde{A} |^{2}$ , of the initially perturbed BVB with $α = 3$ at propagation distances (a) $ζ = 41.25$ and (b) $ζ = 91.25$ , and with $α = 6$ at (c) $ζ = 15$ and (d) $z = 100$ . These results demonstrate the secondary breakup of fragments produced by the primary instability of the vortices.
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Figure 5
For $M = 4$ , dimensionless growth rates of the most unstable azimuthal modes of BVBs (solid curves) and of their counterparts with the same parameters in the absence of absorption (dashed curves). Panels (a) and (b) display the results for $s = 2$ and 3, respectively. Inset in panel (a): Dimensionless radial intensity profiles of the indicated stable (black curve) and unstable (gray dashed curve) BVBs in media with and without absorption, respectively.
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Figure 6
Transverse intensity distributions, $| \tilde{A} |^{2}$ , at the indicated propagation distances for $α = 1$ (left), $α = 3$ (center), and $α = 6$ (right), in the presence of the four-photon absorption, produced by axicons illuminated by vortex-carrying Gaussian beams with $ρ_{0} = 400$ and $s = 1$ , but different amplitudes: $b_{G} = 0.0402$ (left), 0.0341 (center), and 0.030 (right), yielding Bessel amplitudes $b_{B} = 1.11$ , 0.94, and 0.829, respectively. The three nonlinear BVBs with $s = 1$ and $| b_{in} |$ , matching these values of $b_{B}$ in samples with the corresponding values of $α$ and $M$ , have the same parameter $b_{0} = 1.2$ , which can be identified using the method of Ref. [29].
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