Parametric Oscillation

In this section we will discuss the effect of placing the previously discussed parametric amplifier in a resonant cavity. So far we have discussed the gain of a weak signal beam due to the interaction with a strong pump beam. In an oscillator the only input beam is the pump and generation of the signal and idler beams is facilitated by spontaneous parametric emission. Derivations of steady state thresholds are presented to demonstrate the difference between single and double resonance. The requirements that double resonance plays on the stability of the pump frequency and the cavity will be discussed. Calculation of thresholds for a pulsed pump laser are more complicated and presented here is a comparison between the model of Brosnan and Byer [13] and that developed in this work which incorporates time dependence into the calculations of Guha et al [14]. We also discuss the efficiencies that can be expected from these devices.

2.7.1 Spontaneous parametric emission

When a pumped non-linear optical material, i.e. a parametric amplifier, is placed within an optical resonator that possesses a high Q for either the signal or idler, (singly resonant oscillator, SRO) or both (doubly resonant oscillator, DRO) then the resultant device is called an Optical Parametric Oscillator (OPO).

We have seen in the previous sections that under the appropriate conditions of energy conservation, eqn 2.3.2, and momentum conservation, eqn 2.4.8, that if either a signal or idler beam, or both, is present they will experience amplification. (It is only necessary for the beams to lie within the gain bandwidth given by eqn 2.5.2.2 in order to experience some degree of gain.) If the gain received from the parametric interaction is large enough to overcome the cavity losses, both coupling and parasitic, then oscillation will occur. However, before amplification can occur we must have a few signal and/or idler photons present at the crystal to start with. These initial photons are provided by a process variously referred to as spontaneous parametric emission, parametric fluorescence, parametric luminescence and parametric noise [9].

Ch. 2 : Theory of Parametric Interactions

When light from a pump laser is incident on a non-linear crystal, there is a probability that the pump photons will spontaneously split into signal and idler photons. This can be shown quantum mechanically [34,35] to be equivalent to the now accepted view [36] that the process is due to the interaction of the vacuum field fluctuations, equivalent to half a photon per mode, with the pump photons via the non-linear polarisation. This spontaneous parametric emission then provides a source of noise from which the signal and idler beams grow. It is interesting to note the similarities here between an OPO and a laser. Both consist of a gain medium enclosed by a resonator, where, in an OPO, the gain is created by the non-linear polarisation, and in a laser, the gain is provided by a population inversion, (usually![37]). Both also grow from noise, where, in an OPO, the noise is spontaneous parametric emission, whereas, in a laser, the noise is the spontaneous emission from the upper laser level. Spontaneous parametric emission can also be used for determining non-linear coefficients and measuring phase matching curves, where it has a number of advantages over the conventional technique of SHG [36].

2.7.2 CW or steady-state parametric oscillation threshold

The threshold pump power of the oscillator can de defined as the requirement that the single pass gain be sufficient to offset the cavity loss. As the signal and idler interact with the pump in only one direction (unless the pump beam is double passed or even resonated) the gain is only single pass. If we define ai and a2 as the round trip electric

field loss at the signal and idler frequencies respectively then the amplitude requirement for threshold is that the magnitude of each wave is the same after a round trip as it was before the round trip. (We also require the phase relation between the waves to be maintained, that is, the sum of the signal and idler phases is constant when referenced to the phase of the pump for stable single mode pair operation [38]). The relation between the magnitudes requires

E.(z = 0) = (l - fl;).G. £(z = 0) = (l - a;)£■;(z = 4) (2.7.2.1) where j corresponds to signal and idler, and G is the single pass gain. For demonstration purposes, we use here the solution to the coupled amplitude equations for infinite, uniform, plane waves under the assumption of an undepleted pump. The requirements on threshold assuming both signal and idler are resonated are [1 1]

Y ^ E , ( 0 ) = £,(0)coshr4 + i[-^ jE ,£ ;(0 )s in h r4 (2.7.2.2a) — £,(0) = £,(0)cosh n , + if -ÎS. |£ £;(0)sinh n , (2.7.2.2b)

1 - a,. \ Y J

Ch. 2 : Theory of Parametric Interactions gives

cosh n . = 1 + — A A— (2.7.2.3)

2 - a , - a ,

which for low losses at both waves, i.e. a DRO, reduces to

rH I= a,a, (2.7.2.4)

For low loss at only one wave and no feedback at the other a,. = 1, i.e. a SRO, we get

r H l= 2 a , (2.7.2.5)

We see from the above that the DRO threshold differs from that of the SRO by a factor

aJ2. Thus, if the resonator has a high Q for the resonated waves, i.e. low losses, then the DRO threshold may be two orders of magnitude less than that of the equivalent SRO. This assumes that both the signal and idler are resonated simultaneously in the DRO case, which requires a single frequency pump and a stabilised OPO cavity, about which more will be said in the next section. While the first cw DRO OPO was demonstrated as early as 1968 using Ba2NaNb5 0i5 pumped at 532 nm [39], the large

increase in threshold imposed by single resonance has only recently [40] allowed operation of a cw SRO.

2.7.3 Double resonance and the cluster effect

The previous section showed how a lower threshold is obtained by resonating both the signal and idler waves. However, this lower threshold is accompanied by the requirement of a stable, single frequency pump source and also places strict constraints on the stability of the OPO cavity.

The presence of a dispersive medium in the cavity, i.e. the non-linear crystal, means that the axial mode spacings at the signal and idler frequencies are different. Therefore, not every signal cavity resonance will have a corresponding idler resonance such that the two resonance frequencies satisfy the energy conservation condition (2.3.2). If fact, in general, there will be no pair of frequencies where an exact coincidence of resonances occur, though frequency pairs where the matching of resonances are close will occur, and are referred to as ’cluster frequencies’. General consideration of the operating point of a DRO as well as the concept of cluster spacing was first given by Giordmaine and Miller [41]. Fig.2.4 is a reproduction of a diagram from Eckardt et al [42] which expresses particularly well the ideas originally forwarded by Giordmaine and Miller. The difference between signal and idler mode spacings, Ôcoi and ÔC0 2 respectively, is

exaggerated in fig.2.4 for the purpose of illustration. The scales are obtained by considering the mode spectrum for both signal and idler cavity modes, and then

Ch. 2 : Theory o f Paratnetric Interactions

imaging the idler mode spectrum through the degeneracy point so that frequency increases from left to right for the signal, but from right to left for the idler. In this way, by placing the idler scale under the signal scale, signal and idler modes which satisfy energy conservation will coincide vertically. In general there will be some frequency mismatch Aco, which is the frequency shift required of signal or idler to produce exact coincidence. The blow up of a close coincidence, or cluster, shows how the frequency mismatch Aco can be represented by signal, Acoi^ and idler, Aco2, components which are

the respective frequency displacements from the centres of the signal and idler cavity resonances to the frequencies most favourable for parametric oscillation.

axial mode hop cluster jump

Aoa.