Dispersive superfluid-like shock waves in nonlinear optics

Abstract

In most classical fluids, shock waves are strongly dissipative, their energy being quickly lost through viscous damping. But in systems such as cold plasmas, superfluids and Bose-Einstein condensates, where viscosity is negligible or non-existent, a fundamentally different type of shock wave can emerge whose behaviour is dominated by dispersion rather than dissipation.Dispersive shock waves are difficult to study experimentally, and analytical solutions to the equations that govern them have only been found in one dimension (1D). By exploiting a wellknown, but little appreciated, correspondence between the behaviour of superfluids and nonlinear optical materials, we demonstrate an all-optical experimental platform for studying the dynamics of dispersive shock waves. This enables us to observe the propagation and nonlinear response of dispersive shock waves, including the interaction of colliding shock waves, in 1D and 2D. Our system offers a versatile and more accessible means for exploring superfluid-like and related dispersive phenomena.Unlike dissipative shock waves in ordinary gases/fluids, which have a well-defined shock front due to viscosity, dispersive superfluid-like shock waves have an oscillatory front. These oscillations result from two basic, and related, properties of the superfluid state: nonlinearity and coherence. Coherence results from cooling the fluid, so that the constituent particles of the condensate are perfectly correlated, while nonlinearity refers to the inter-particle interactions which make this correlation possible. For different reasons, these two properties also appear in nonlinear optics. While the relationship is well known in condensate community [e.g. nonlinear "atom optics" studies in Bose-Einstein condensates (BEC) 1-3 ], the relationship has been underappreciated from the opposite perspective. Here, we build on previous theoretical 4, 5 and experimental 6, 7 work on superfluid behavior in BEC to examine the optical equivalent of condensate shock waves. We demonstrate basic dispersive, dissipationless shock waves in one and two transverse dimensions, characterize their nonlinear properties, and reveal the nontrivial interactions when two such shocks collide.While dispersive shock waves in optics have been studied previously for temporal pulses in fibers 8-16 , they have not yet been considered in the spatial domain. In this case, the extra dimensional freedom allows consideration of wavefront geometry, which is shown to significantly affect shock propagation and interaction. The particular system considered here is a spatial one in which a continuous optical wave propagates in a nonlinear Kerr-like medium, mainly along the z-axis. To an excellent approximation, the slowly-varying amplitude ψ of such a field can be described by the nonlinear Schrödinger equation:

Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schrödinger equation

“…Under these conditions we have found that the decay instability of the antidark soliton leads to a DSW (see Refs. [20][21][22][23] and references therein); that is, an expanding region filled with fast oscillations which behave asymptotically as solitons. In the present case, starting with a zero-velocity antidark soliton, the decay instability leads to two symmetric DSW fans, as displayed in Fig.…”

“…Moreover, the full characterization of the soliton solutions and their instabilities constitute the starting ground for describing the feature of dispersive shock waves (DSWs, involving multiple solitons in the weakly dispersive regime) [20], which is an active area of research where successful experiments have been recently performed in non-Kerr media under different excitation conditions [21][22][23]. In this respect, here we provide the first prediction of a dispersive shock wave produced directly by the decay of a solitary wave of the CQNLS model.…”

Dispersive shock waves with nonlocal nonlinearity

“…The input intensity profile consists of a Gaussian hump on a constant, lowlevel background, and output profiles are shown as the nonlocal parameter w is increased. As in [3], the expanding wave consists of two repulsive humps whose fronts are characterized by , so that the initial hump is automatically "supersonic." 3 ) In the limit of a response width narrow compared to the input, the nonlinearity can be expanded in a Taylor series, giving…”

“…Recently, the spatial equivalent of dispersion, i.e. diffraction, was used with defocusing nonlinearity to form dispersive shock waves in spatial beams 3 . Unlike dissipative shock waves, which have a thin, well-defined front due to energy absorption (e.g.…”

“…3, we create a hump-on-background input profile by using a Mach-Zehnder interferometer and placing a spherical lens in one of the arms. Both beams are then recombined onto the input face of the cell, and the output face is imaged onto a CCD camera.…”