Page 356 - Tunable Lasers Handbook

31 6 Norman P. Barnes

where El is the electric field of the interacting wave and wl is the beam radius.
For ordinary waves. the expression for the electric field is similar but the bire-
fringence angle is zero.
In the case of a singly resonant oscillator, an effective length for the nonlin-
ear crystal can be calculated using the preceding expressions. As an example.
consider the case where the signal is resonant. In this case, the beam radius of
the nonresonant idler “v3 is given by

With this nonresonant beam radius, the integral can be evaluated to obtain an
effective length le for the nonlinear crystal:

Here, erf(x) is the error function and I,, is a parameter that depends on the beam
radii of the pump beam and signal beam as well as birefringence.
In general, the parameter I,, is sensitive to which beams are ordinary and
extraordinary as well as which waves are resonant and nonresonant. If the pump
beam is an extraordinary beam and the signal and idler are both ordinary beams
while the signal is resonant. l,,, can be expressed as [21]

If the pump beam and the resonant wave are extraordinary waves, the expression
for l,,, becomes [8]

For other combinations of ordinary and extraordinary beams as well as resonant
and nonresonant waves, the parameter lw can be calculated using the same
approach.
Because birefringence is needed to effect phase matching, but the birefrin-
gence angle eventually limits the effective length of the nonlinear crystal. it is of
interest to explore methods of achieving the former while minimizing the latter.
One method of reaching this end is phase matching at 90” to the optic axis. If this