7 Optical Parametric Oscillators 301
18
15
t-
"~ 10
(.9
FIGURE 1
o Experimental points
- Theoretical model
o l= 20 mm
I I I I
1 O0 200 300 400
(Ep/~'p)l/2in (W)1/2
Average gain of 3.39%tm HeNe laser as a function of pump power.
In a laser amplifier, energy is stored in the laser material for long time intervals,
on the order of 100 Its. During this time interval, spontaneous emission can
deplete the stored energy, thus reducing the gain. In an optical parametric ampli-
fier, energy is not stored in the nonlinear material. In addition, gain is only pre-
sent while the pump pulse traverses the nonlinear crystal, a time interval on the
order of 10 ns or less. As such, ASE does not detract from the gain significantly.
3. PARAMETRIC OSCILLATION
Whereas parametric amplification occurs at any pump level, parametric
oscillation exhibits a threshold effect. The threshold of a parametric oscillator
can be determined for either pulsed or cw operation of the device. In a cw para-
metric oscillator, threshold will occur when gain exceeds losses in the resonator
even though the time interval required to achieve steady state may be relatively
long. In a pulsed parametric oscillator, on the other hand, gain may exceed the
losses with no measurable output. In these cases, the pump pulse may become
powerful enough to produce a net positive gain. However, before the generated
signal reaches a measurable level, the pump power falls below the level at which
positive gain is achieved. Consequently, to describe this situation both an instan-
taneous threshold and an observable threshold are defined. Pulsed gain is shown
in Fig. 2 with a threshold set by the losses in the parametric oscillator resonator.
Although an observable threshold depends on the detection system, it remains a
useful concept. As the signal grows below observable threshold, it will enjoy
~~ Norman P. Barnes
1.0
0.8
c
0.6
-g
N
0.4
Z
0.2
I I
0 0.5 1.0 1.5
FIGURE 2
Normalized Time (t/tp)
Pulsed gain as a function of time showing instantaneous threshold.
exponential gain. Because of this large gain, the difference between an observ-
able threshold that produces 1.0 or 10.0 ~tJ is relatively small.
In the cw parametric oscillator, a mode gain can be determined under
threshold conditions. Because the pump beam will not be significantly depleted
at threshold, the longitudinal variation of the pump beam may be neglected.
Because the product of two Gaussian beams is another Gaussian beam, interact-
ing beams will generate a nonlinear polarization, which is also a Gaussian. If the
electric fields at wavelengths
~i and
~,j interact, they will generate a nonlinear
polarization at wavelength ~'k, which will have a spatial variation characterized
by a beam radius given by
1 = 1_[_+ 1 (16)
w
kg
Note that the generated nonlinear polarization does not necessarily have the
same spatial variation as the incident field at ~'k" Because of the potential mis-
match between the incident electric field and the generated electric field, the gain
coefficient will have an additional term to account for this effect [6]. Including
this term in the gain expression yields
222 ( )2
(FI)2_ 8rt
de I P1 8__ w, w2w3
-- 2 2 W2W3
n~ n 2 ~3~2~Ceo rt w~w 2 + wlw3
+ 2 9 (17)

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