Page 344 - Tunable Lasers Handbook

304 Norman P. Barnes

device, laser induced damage and birefringence will limit the minimum size of
the resonant beam radii.
Given the expressions for the gain, threshold can be defined by equating the
gain and the losses. For cw operation, threshold will occur when [4]

cosh (rZ) = 1 + a,a,
2-a2-a3 ’

where a, is the round trip field loss at the signal wavelength and a, is the round
trip fieldloss at the idler wavelength. In the singly resonant case and under small
gain, a, is near unity and a3 is near zero. Under these circumstances, the thresh-
old for ;he singly resonant signal becomes approximately

A similar expression exists for the situation where the signal is resonant. Again
under the small-gain approximation but in the doubly resonant situation where
both effective reflectivities are close to unity, the approximate expression for
threshold becomes

By employing a doubly resonant parametric oscillator, the threshold can be
reduced substantially since a2 can be an order of magnitude smaller than 2.0.
An observable threshold can be defined for pulsed parametric oscillators.
An instantaneous threshold for a pulsed parametric oscillator is similar to the
threshold for the cw case just defined. To define the observable threshold. Fig.
2 can be utilized. At time rl, a net positive gain exists. At this time, the signal
and the idler begin to evolve from the zero point energy. At time t, the pump
power decreases to a point where the net gain is no longer positive. In the
interim, as the signal and idler evolve, they are initially too small to be
observed. For an observable threshold to be achieved, the power level in the
resonator must increase essentially from a single circulating photon to a level
that is amenable to measurement. To accomplish this, the gain must be on the
order of exp(33).
Observable threshold depends on the time interval over which a net positive
gain exists as well as how much the pump power exceeds the pump power
required for threshold. For a circular pump beam, the observable threshold can
be approximated by a closed-form expression [8]. In this approximation, a gain
coefficient can be defined as