Azimuthal and radial modulation of double-four-wave mixing in a coherently driven graphene ensemble

We investigate in detail the azimuthal and radial modulation (i.e., the azimuthal order l_j and radial order p_j with j = 1, 2) of double-four-wave mixing (double-FWM) by use of two higher-order Laguerre-Gaussian (LG) beams in a Landau quantized graphene ensemble. A pair of weak probe pulses in the graphene ensemble interacts with two LG beams and thus two vortex FWM fields with the opposite vorticity are subsequently generated. In combination with numerical simulations, we reveal that (i) there appear l₁ + l₂ periods of phase jumps in the phase profiles under any conditions; (ii) p + 1 concentric rings emerge in the intensity profile and the strength is mainly concentrated on the inner ring when the two LG beams have the same radial orders (i.e., p₁ = p₂ = p); (iii) there are p raised narrow rings occurring in the phase profile in the case of p₁ = p₂ = p and l₁ ≠ l₂, and the raised narrow rings would disappear when p₁ = p₂ and l₁ = l₂; (iv) p_max + 1 concentric rings appear in the intensity profile, meanwhile, |p₁ - p₂| convex discs and p_min raised narrow rings emerge in the phase diagram in the case of p₁ ≠ p₂, here p_max = max(p₁, p₂) and p_min = min(p₁, p₂). Moreover, the two generated FWM fields have the same results, and the difference is that the phase jumps are completely opposite. These findings may have potential application in graphene-based nonlinear optical device by using LG beams with adjustable mode orders.