Collapse Arrest in Instantaneous Kerr Media via Parametric Interactions

We demonstrate, theoretically and experimentally, that a four-wave mixing parametric interaction is able to arrest the collapse of a two-dimensional multicolor beam in an instantaneous Kerr medium. We consider two weak idlers interacting via a third order nonlinearity with two pump beams and we show that a class of collapse-free quasisolitary solutions can be experimentally observed in a normal dispersion Kerr glass. This observation is sustained by rigorous theoretical analysis demonstrating the stability of the observed self-trapped beams.

Figure 1

Self-localized solutions of the normalized system, Eqs. (4). The nonlinear propagation constants are $\beta =1$ and $\mathrm{\Delta }\beta =0$. The relative frequency scaling quantity is $\sigma =0$. The self-localized profiles $p_1$ (thick line, black), $p_2$ (thick dashed line, black), $s_1$ (narrow line, red), $s_2$ (narrow dashed line, red) are shown in the function of the space coordinate $x$ for several values of the phase mismatches parameters ${\alpha }_{(1,2)}$. (a) ${\alpha }_{(1,2)}=0.0625$, (b) ${\alpha }_{(1,2)}=-0.0625$, (c) ${\alpha }_{(1,2)}=\pm 0.0625$, (d) ${\alpha }_{(1,2)}=0.5$, (e) ${\alpha }_{(1,2)}=-0.5$, (e) ${\alpha }_{(1,2)}=\pm 0.5$.

Figure 2

(a) Total photon flux $Q=Q_{P1}+Q_{P2}+Q_{S1}+Q_{S2}$, (b) cross photon flux $J=Q_{P1}-Q_{P2}+3Q_{S2}-3Q_{S1}$, (c) pump $Q_{p1}$, and (d) idler $Q_{s1}$ photon fluxes vs the mismatch parameters ${\alpha }_{(1,2)}$ for nonlinear phase constants $\beta =1$ and $\mathrm{\Delta }\beta =0$. The white line marks the VK threshold, where the critical collapse takes place. The unstable region is enclosed in the section where ${\alpha }_1>0$ and ${\alpha }_2>0$.

Figure 3

Evolution of peak intensity (a) and waist (b) of the pump $p_1$ for the mismatches parameters ${\mathrm{\Delta }}_{(1,2)}$ as calculated from Eq. (4). ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2$ (values in the legend) and the beam input profiles are $p_1(z=0,x,y)=p_2(z=0,x,y)=1.078\exp [-(x^2+y^2)/{1.5}^2]$ and $s_1(z=0,x,y)=s_2(z=0,x,y)=0$.

Figure 4

Sketch of the experimental setup. The two pumps at 1500 and 1720 nm are obtained from an optical parametric amplifier (OPA). Polarizer crystals (PC), half wave plates (HWP), a short pass filter (SPF) are shown together with the energy monitors photodiodes (PD) and the camera (CCD).

Figure 5

Images of the 1500 nm beam collected at the glass sample output. (a),(b),(c),(d) Output for $E_{p1}=0.5$, 0.8, 1.1, and $1.4\mu \mathrm{J}$ input beam energies in the absence of the 1720 nm beam. (e),(f),(g),(h) and (i),(j),(k),(l) Same as (a),(b),(c),(d) for a collinear 1720 nm beam with $E_{p2}=0.3\mu \mathrm{J}$ and $E_{p2}=0.6\mu \mathrm{J}$ input energies, respectively.

Figure 6

Evolution of the 1500 nm output beam size vs input energy. Markers: experimental data for $E_{p2}=0$ (stars, black), $E_{p2}=0.3\mu \mathrm{J}$ (points, green) and $E_{p2}=0.6\mu \mathrm{J}$ (crosses, red) input energies of a collinearly propagating 1720 nm beam. The beam size is extracted with a Gaussian fit of the output images; the accuracy is within the scale of the marker size. Lines: simulations of Eqs. (1) including the temporal dynamics of the pulse. For the simulation, we used Gaussian beams with waists $w_{p1}=20\mu \mathrm{m}$ and $w_{p2}=15\mu \mathrm{m}$ focused at 0 and 2.54 mm from the entrance, respectively, as measured by beam profiling. The pulse duration is 1 ps as measured by the autocorrelation. The propagation length is 6 mm, $\mathrm{\Delta }{k}_1=0.7{\mathrm{mm}}^{-1}$, and $\mathrm{\Delta }{k}_2=4{\mathrm{mm}}^{-1}$. The refractive index is $n=1.76$ and the dispersion coefficient is $50{\mathrm{ps}}^2/\mathrm{km}$. The critical energy is $1.2\mu \mathrm{J}$, calculated as a fitting parameter of the single collapse beam curve. The energy values also include 20% of injection losses.