From phase to amplitude modulation in transverse nonlinear optics

Control of spatial light modulation is an important and useful technique in optical applications ranging from the manipulation of small particles to lithography. We show here that an input phase (amplitude) modulated beam can be effectively converted to an amplitude (phase) modulated output in a frequency converter such as the resonant optical parametric oscillator (OPO). This peculiar conversion property is, however, riot restricted to the OPO but applies to a wide variety of broad area nonlinear cavities. The slowly varying amplitudes of the pump A₀(x,t) and signal A₁(x,t) fields of a degenerate OPO at resonance, in the mean-field approximation, are described by: ∂_tA₀ = Γ[-A₀ + Ee ^μcos(kx) - A₁ ²] + i∂_x ²A₀ ∂_tA₁ = -A₁ + A₀A₁* + 2i∂_x ²A ₁ (1) where Γ is the ratio of the pump and signal cavity decay rates, the parameter n is the (complex) amplitude of the input modulations and is such that |μ| < < 1, while E is the un-modulated amplitude of the external pump field. Perturbative calculations show that full phase (amplitude) to amplitude (phase) modulation conversion from the pump to the signal field is achieved for a modulation wavevector k_c given by k_c ² = √Γ(E-1) (k_c ² = √ΓE)