Page 359 - Tunable Lasers Handbook

7 Optical Parametric Osci!Iators 3 19

Under the assumption of radial crystal symmetry and lateral heat extraction.
the phase mismatch can be approximated as a function of radial position, that is,

for a circular and a Gaussian beam profile, respectively. In these expressions arc
and a,,? can be defined as

(56)

In these expressions, an,/dT is the variation with temperature of the refractix e
index II~ at wavelength hl, pa is the average absorption coefficient, Pa is the aver-
age poNer. and Xc is the thermal conductivity. With the mismatch known as a
function of the radial position, the conversion efficiency can be integrated over
the cross section of the nonlinear crystal.
To explore this elTect, a simple example can be investigated that illuminates
the salient features. Effects of phase mismatch on parametric generation, under the
low conversion efficiency approximation, can be described in terms of a sin:(x)k?
function. A relatile efficiency qR can be defined as the fractional decrease in the
conversion efficiency caused by the effects of cq stal heating. Integrating this over
the cross section of the nonlinear crystal yields

Evaluation of this integral is straightforward using numerical techniques. Refer-
ring back to the expressions for Ak. it can be seen that there are two contribu-
tions, a zeroth-order term that does not depend on the average power and another
term that does. The zeroth-order term represents the residual phase mismatch in
the absence of average power heating effects. For cases where there is no aver-
age power heating effects, the residual phase mismatch is minimized. However,
with average power heating effects, this term can be optimized fiir maximum
efficiency.