Source: https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-25-20676
Timestamp: 2019-04-19 14:27:11+00:00

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We study the over-focusing of spatial light beams due to self-focusing nonlinearity, in both local and nonlocal nonlinear media. Numerical simulation of both cases reveals a peaked profile, with a near-cusp at the center surrounded by exponentially-decaying tails, at a critical self-focusing power. The profile is a local effect, occurring as diffraction counteracts nonlinearity. Nonlocality, however, is needed to prevent modulation instability of the initial beam and to prevent catastrophic collapse in 2D. The peaked profile remains for weak nonlocality but disappears for wide nonlocal responses. Beyond the critical power for a peaked solution, or for longer propagation distances, competition between nonlinearity and diffraction causes oscillatory collapse-bounce behavior. The numerical results are confirmed by observing these dynamics in a self-focusing glass with a nonlocal, thermal response.
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Fig. 1. Simulation results showing a Gaussian beam propagating in (a)local; (b)weakly nonlocal (w=a/5); and (c) highly nonlocal (w=5a) media. In the local case, the beam narrows and generates a train of solitons; in the highly nonlocal case, the NLS is reduced to a linear harmonic oscillator equation and the beam width and intensity bounce as a function of distance; the weakly nonlocal case is in between the two extremes.
Fig. 2. Numerical results of beam profiles and wavefront forces (a-b) before, (c-d) near and (e-f) after peakon profile in the case of local nonlinearity. Top row: intensity (normalized to peak intensity in each case). Bottom row: derivative of total pressure (quantum pressure and nonlinear index change in Eq. (4)); inset: derivative of nonlinear index change term only. (a) Before the peakon, the Gaussian beam is wider in the center and lower in the tails and (b) focusing nonlinear index change dominates the force and compresses the beam inwards. (c) As the beam propagates, the beam matches the peakon profile exactly while (d) the force of the quantum pressure term becomes comparable to that of the nonlinear index change in the center. (e) Past the peakon, the profile continues to steepen and becomes narrower in the center and higher in the tails while (f) the derivative of the quantum pressure term overtakes the focusing index change near the center, creating a defocusing force near the origin while the rest of the beam continues to focus inwards.
Fig. 3. Simulation results showing peakon profile with increasing nonlocality (a,c,e) in the presence of noise(b,d,f). In the local case (a), the peakon profile is obtained, but is unstable to noise(b); Using a weak nonlocality(c) of w=a/3, the profile is also obtained, albeit with a slightly rounded peak, and is stable to noise(d); with high nonlocality (e-f) of w=2a, the peakon profile is totally lost. Figures are normalized to the maximum intensity in each case.
Fig. 4. Simulation results using different nonlocal kernels — (a) Gaussian, (b) Lorentzian, (c) exp(-|x|), and (d) solving the 2D heat diffusion equation (6) explicitly. Peakon-like profiles are obtained in each case at the same propagation distance, albeit with different input powers required.
Fig. 5. Results of 2+1D simulation of NLS with nonlocal nonlinearity (w=a/5) showing collapse-bounce cycles of an initial Gaussian beam. The beam starts off as a Gaussian (a). As it self-focuses, it narrows (b) and approaches the peakon profile (c), after which it continues to focus into a pedestal-shape profile (d). It then defocuses into a ring-shaped profile (e) before focusing again (f) in an oscillatory manner. The focusing-defocusing bounce cycles continue quasi-periodically throughout the beam propagation.
Fig. 6. Experimental results of self-focusing of a 2D Gaussian beam in a nonlocal nonlinear medium. Shown are experimental output pictures at (a) 200mW, (b) 400mW, (c) 510mW, and (d) 600mW. (e) Cross-sectional profiles of (a-d), individually normalized to peak power.
Fig. 7. Transverse profile of experimentally-observed peakon in Fig. 6(c) showing best fits to hyperbolic secant, Townes and a peakon exp(-|x|) profiles.
Fig. 8. Experimental output pictures for an initial Gaussian beam with power (a) 600mW, (b) 800mW, (c) 1300mW, (d) 1500mW, (e) 1700mW, and (f) 1900mW. The beam focuses, defocuses, and focuses in an oscillatory, quasi-periodic fashion.

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