Observation of Bessel-beam self-trapping

We predict and observe Bessel-beam optical self-trapping in a strongly self-focusing medium. Nonlinearity leads to solutions with a signature pulsing along the propagation direction, a breather that we observe in a photorefractive potassium-lithium-tantalate-niobate crystal. The nonlinear construct is formed as an interference pattern locks together with a diffracting wave.

Figure 1

Bessel-beam breather self-trapping. Numerical 2+1D simulated nonlinear propagation of an initial $\lambda =532$ nm fundamental BB with $k_r=0.28\mu {\mathrm{m}}^{-1}$ and a Gaussian $\sigma =60\mu \mathrm{m}$ envelope for different input peak intensities $I_p/I_s$, with $n_0=2.3$ and $\mathrm{\Delta }{n}_0=5.6\times {10}^{-4}$. (a) $x,z$ section of the intensity distribution (the distribution is symmetric for rotations around the propagation axis) with $I_p/I_s=0.04$ and (b) corresponding $k_x, z$ section of the transverse spatial spectrum distribution, where the ring structure leads to a propagation-invariant double spot. (c) BB dynamics for $I_p/I_s=0.4$ and (d) corresponding evolution of the spatial spectrum, with the characteristic generation of low-frequency components and the appearance of a second ring. (e) A breathing self-trapped solution at $I_p/I_s=4.4$ and (f) corresponding spatial spectrum. The original BB is locked into an oscillating solution that is now localized in the transverse plane and periodic along propagation. (g) Peak intensity $I_p\left(z\right)/I_p$ for the various conditions.

Figure 2

(a) BB breather-breather collision for an input launch angle of $\theta =100$ mrad. (b) Nonlinear propagation with $\sigma =120\mu \text{m}$ in an exponential nonlinearity ($\mathrm{\Delta }{n}_0=5.65\times {10}^{-4}, I/I_s=4.4$).

Figure 3

(a) Experimental setup. The input BB is generated using a liquid-crystal SLM. The laser beam from a doubled Nd:YAG laser at 532 nm is expanded (lenses L1 and L2) to impinge as a plane wave onto the SLM that, using a polarizer (POL), transmits a ring. The transmitted light is then focused and Fourier transformed onto the input of a photorefractive KLTN crystal, generating the input BB. $\mathrm{\Delta }{n}_0$ is fixed through the $x$-directed bias electric field $E_0=V/L_x$ and the Peltier-controlled sample temperature $T$. CAM1 records the intensity distribution, while CAM2 the spatial spectrum. (b),(c) Transverse intensity distribution at input (b) and output (c) facets of the crystal with negligible diffraction for the $L_z=1.8$ mm propagation.

Figure 4

Observation of a Bessel-beam breather. Transverse profile for (a) experiments and (b) numerical simulations. Experimental central lobe complete oscillation and partial one for (c) $I_{d1}$ and (d) $I_{d2}$, where $I_{d2}\simeq 10I_{d1}$ is achieved using an illuminator. A complete oscillation is also reported for numerical simulation (e). The intensity reported in plots (c),(d), and (e) is spatially averaged on a 6 $\mu \text{m}$ radius circular region around the geometric central lobe. The agreement between observations and numerical simulations not only validates the quasi-steady-state model, for which $I_s\propto t$, but also the general physical picture of a breathing self-trapped state. In turn, differences in the fine details in the $x$-directed profiles are due to the fact that anisotropy is not considered in the model (see text).

Figure 5

Bessel-beam breathing spectrum. The intensity of the ring is progressively reduced and a central lobe appears (a) in agreement with numerical simulations (b) (see text).

Figure 6

(a) Long propagation ($L_z=300$ mm) of a Bessel-Gauss beam with $k_r=0.28\mu {\text{m}}^{-1}$ and $\sigma =60\mu \text{m}$. The numerical simulations are performed with the exponential nonlinearity with $I/I_s=4.4$ and $\mathrm{\Delta }{n}_0=4.87\times {10}^{-4}$. (b) Breathing period ${\mathrm{\Lambda }}_z$ versus $I_p/I_s$. The gray area indicates values of $I_p/I_s$ that give rise to the breathing soliton without Bessel self-trapping. The values of ${\mathrm{\Lambda }}_z$ are obtained averaging on more than three periods, disregarding the first four oscillations.

Figure 7

$L_z=100$ mm propagation of Bessel-Gauss with $k_r=0.28\mu {\text{m}}^{-1}$ and $\sigma =60\mu \text{m}$ evolving according to the saturable Kerr nonlinearity law. The other parameters of this simulation ($I/I_s=4.4, \mathrm{\Delta }{n}_0=4.87\times {10}^{-4}$) are the same as in Fig. 6.