Order Parameter Equations for Other Nonlinear Resonators

Abstract

An optical parametric oscillator basically consists of a nonlinear χ ⁽²⁾ medium inside a resonator driven by a coherent field of amplitude Ē and frequency ω _L, which propagates along the optical axis of the resonator, parallel to the z axis. The crystal converts the intracavity pump field of frequency ω _L and amplitude A ₀ into two fields of frequency f ₁ ω _L and f ₂ ω _L, and of amplitude A ₁ and A ₂, the signal and idler waves, respectively. Energy conservation requires that f ₁ + f ₂ = 1. Three longitudinal modes of the cavity with frequencies ω ^c_m (m = 0, 1, 2) are assumed to be close to the frequencies f _m ω _L (where f ₀ = 1). Under these conditions, and making some of the usual assumptions of nonlinear optics (the mean-field limit, the paraxial and single-longitudinal-mode approximations), the evolution equations for the pump, signal and idler fields can be written as [1]

$$ \frac{{\partial A_0 }} {{\partial t}} = \gamma _0 \left[ { - \left( {1 + i\omega _0 } \right)A_0 + \bar E - A_1 A_2 + ia_0 \nabla ^2 A_0 } \right], $$

(3.1a)

$$ \frac{{\partial A_1 }} {{\partial t}} = \gamma _1 \left[ { - \left( {1 + i\omega _1 } \right)A_1 + A_0 A_2^ * + ia_1 \nabla ^2 A_1 } \right], $$

(3.1b)

$$ \frac{{\partial A_2 }} {{\partial t}} = \gamma _2 \left[ { - \left( {1 + i\omega _2 } \right)A_2 + A_0 A_2^ * + ia_2 \nabla ^2 A_2 } \right], $$

(3.1c)

where γ _m are the cavity decay rates, $\omega _m = \left( {\omega _m^c - f_m \omega _L } \right)/\gamma _m$ are the detunings and $a_m = c^2 /2\gamma _m f_m \omega _L$ are the diffraction coefficients.