Page 339 - Tunable Lasers Handbook

7 Optical Parametric Oscillators 99

In these expressions El is the electric field. 4, is the impedence, v, is the fre-
quency, de is the effective nonlinear coefficient. Ak is the phase mismatch. and j
is the square root of -1. Subscripts 1, 2, and 3 refer to the pump, the signal. and
the idler, respectively. Phase mismatch is the deviation from ideal conservation
of momentum, or

When the idler is initially zero but the signal is not. the coupled equations can be
solved exactly to yield

In this expression, S, is the intensity of the signal, S,, is the initial intensity of
the signal, i is the Ieigth of the nonlinear crystal, and

Although this expression describes the growth of plane waves well. in reality :he
interacting b'eams are not plane naves but are more likely to be Gaussian beams.
When the interacting beams are Gaussian, the gain must be averaged over the
spatial profile of the laser beam.
Two common approximations are available for this expression that demon-
strate the limiting performance of parametric amplification. If the mismatch is
small compared with the gain. that is. if Ak is much smaller than r. this term can
be neglected. In this case

Thus, the signal will enjoy exponential gain as long as the pump is not depleted.
On the other hand if the gain is small compared with the mismatch, that is. if r
is much smaller than Ak, this term can be neglected. In this case,