The amplitude |T(x, y)|(a) and the phase Φ/π (c) of the transmission function as a function of x (in units of Λ_x) and y (in units of Λ_y). (b) The normalized diffraction intensity I_p(θ₁,θ₂) as a function of sin(θ₁) and sin(θ₂) for the corresponding transmission function shown in panel (a). (d) The two-dimensional gain cross-grating considers the gain and phase modulation. The insets in panels (b) and (d) show the residual parts of the diffraction patterns along sin(θ₁). The parameters are Ω_r = 0.2γ, Ω_s0 = 0.2γ, Δ_r = Δ_p = 80γ, Δ_s = 100γ, Λ_x/λ_p = Λ_y/λ_p = 4, L = 200ξ, and M = N = 5.