Self-bound droplets of light with orbital angular momentum

Niclas Westerberg, Kali E. Wilson, Callum W. Duncan, Daniele Faccio, Ewan M. Wright, Patrik Öhberg, and Manuel Valiente

Phys. Rev. A 98, 053835 – Published 21 November 2018

Abstract

Systems with competing attractive and repulsive interactions have a tendency to condense into droplets. This is the case for water in a sink, liquid helium, and dipolar atomic gases. Here we consider a photon fluid which is formed in the transverse plane of a monochromatic laser beam propagating in an attractive (focusing) nonlocal nonlinear medium. In this setting we demonstrate the formation of the optical analog of matter-wave droplets and study their properties. The system we consider admits droplets that carry orbital angular momentum. We find bound states possessing liquidlike properties, such as bulk pressure and compressibility. Interestingly, these droplets of light, as opposed to optical vortices, form due to the competition between long-range $s$ -wave (monopole) and $d$ -wave (quadrupole) interactions as well as diffraction.

Received 8 February 2018

DOI:https://doi.org/10.1103/PhysRevA.98.053835

Physics Subject Headings (PhySH)

Angular momentum of light Nonlinear optics

Atomic, Molecular & OpticalNonlinear Dynamics

Authors & Affiliations

Niclas Westerberg^1,*, Kali E. Wilson¹, Callum W. Duncan¹, Daniele Faccio^1,2,3, Ewan M. Wright^1,3, Patrik Öhberg¹, and Manuel Valiente^1,†

¹Institute of Photonics and Quantum Sciences, Heriot-Watt University, SUPA, Edinburgh EH14 4AS, United Kingdom
²School of Physics & Astronomy, University of Glasgow, SUPA, Glasgow G12 8QQ, United Kingdom
³College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA

^*nkw2@hw.ac.uk
^†M.Valiente_Cifuentes@hw.ac.uk

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Issue

Vol. 98, Iss. 5 — November 2018

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Images

Figure 1
(a) Normalized intensity of a $p$ -wave droplet at $P = 1$ W input power and an equal superposition of $ℓ = \pm 1$ orbital angular momentum. (b) Corresponding normalized nonlocal interaction potential $Δ n$ .
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Figure 2
(a) Pseudoenergy surface of the droplet as a function of the droplet size $ξ$ and $| δ |$ as given by $H_{*}$ for $P = 2 W$ . Note the global minimum at $| δ | = 1$ and at the finite size $ξ = ξ_{*}$ . (b) Pseudoenergy per unit power (blue solid line) and binding energy contribution (green dashed line) for $P = 2 W$ and $| δ | = 1$ as a function of bulk density (peak power). (c) Corresponding inverse compressibility $K^{- 1}$ .
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Figure 3
(a) Evolution of the radius $〈 r 〉$ as predicted by analytical theory (red dashed line) and direct numerical simulation (blue solid line) with initial conditions $ξ_{0} = 70 μ m$ , $| δ | = 0.9$ , and $P = 2 W$ . (b) Similarly, numerical evolution of $〈 {cos}^{2} ϕ 〉$ [blue solid (lower) line] and $〈 {sin}^{2} ϕ 〉$ [red shaded (upper) line]. In the inset, droplet rotation can be seen more clearly at longer propagation distance.
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Figure 4
Evolution snapshots for an initial power $P = 15 W$ , radial size $ξ_{0} = 70 μ m$ , $| δ | = 0.05$ , and (a) $z = 0$ m, (b) $z = 0.05$ m, (c) $z = 0.075$ m, (d) $z = 0.11$ m, (e) $z = 0.13$ m, and (f) $z = 3$ m. Observe the initial collapse in (b), followed by a violent rebound in (c). This collapse-rebound cycle continues (less violently) until the system has redistributed most of its pseudoenergy into $p$ -wave droplet binding energy, such as in (f). Excess pseudoenergy emerges as surface vibrations (see Supplemental Material, Video 2 [62]).
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Figure 5
(a) Scaled index change $Δ n / γ P$ versus $r / σ$ according to the long-wavelength approximation (A4) (dashed line), according to the Snyder-Mitchell approximation (A5) (solid line with crosses), and the exact index change (solid line). (b) Exact solution for $Δ n (r)$ (solid line), the long-wavelength approximation (dashed line), and the Snyder-Mitchell approximation (solid line with crosses). Notice that the different approximate refractive-index profiles both give similar integrands. One should note that a $p$ -wave droplet is expected to be localized at radii $r ≳ 0.0035 σ$ , where the long-wavelength approximation better approximates the distributed loss model. The inset shows the crossover region.
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Figure 6
Pseudochemical potential surface of the droplet as a function of the variational parameter $δ$ as given by Eq. (B6) from the Snyder-Mitchell model with $P = 1 W$ . As can be seen, this energy landscape is qualitatively different from the one seen in Fig. 2. This is also qualitatively different from numerical evidence.
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