Theory of the Shape of Pulses Produced by Transient Parametric Generation of Light

Theory of the optical parametric oscillator in the quadratic and cubic approximations

“…It should be noted that in the previous related works [20,34,36,37,39,40] the relevant amplitude expansion contained only a single quadratic term, namely, the first one according to (14). However, as (14) suggests, the extra quadratic term may significantly contribute to the field amplitude, especially at the nearthreshold oscillation, where the pump amplitude value may be far stronger than the generated amplitudes: A A p j 0 (j = 1, 2).…”

“…In contrast, in the above-threshold generation regime, pump depletion can be significant due to high conversion efficiency. More accurate approximations that account for variations of the field amplitudes have been proposed [33] in order to tackle such important problems as optical pulse generation [34,35], the structure of the radiation modes [36] and fluctuation phenomena in OPOs [37,38]. However, the accuracy of the developed approaches is not always sufficient for the correct estimation of OPO radiation characteristics and, therefore, more accurate approaches are required.…”

“…In the approximation of slowly-varying amplitudes, the set of nonlinear equations governing the three-wave field evolution in the the PDC process reads as (see [34,37]): where β j are the non-linear coupling coefficients dependent on the material properties and the carrier frequencies, the coefficients δ j refer to the linear losses in the medium and u j are the group velocities of the waves (j = 1, 2, p).…”

“…for the amplitude values at the mirror M 1 interface right after reflection. This approximation has been described, for example, in [34]. Obviously, the ratio between τ j and ( ) τ rt has an essential influence on the temporal dynamics of the OPO [8]; for ultrashort pump pulses higher-order terms should be added in (10) to obtain an adequate theory.…”

“…In order to account for higher order terms in the amplitude expansions, whereby to increase accuracy, the following general form of expansion is introduced: are the expansion coefficients to be found, with N being the maximum number of power terms retained in the approximation. By means of substituting the expansion forms (34) into the nonlinear set (2), and grouping together coefficients of equal power with respect to z, one arrives at the following chain of relations:…”