GUILLERMO LEON OSORIO ESPECIALISTAS
10. PLAN DE IMPLEMENTACIÓN
The first calculations on OPO efficiency were by Siegman [45], even before the demonstration of the first working device. These calculations predicted the limiting of the pump exiting the cavity to the threshold gain level, in a similar manner that the gain saturates in a laser. This was also the first realisation of what are called power dependent reflections of the pump, which are due to the signal and idler interacting on the return journey through the non-linear crystal and converting back into the pump frequency via sum frequency generation. These power dependent reflections only occur for DROs and limit the maximum conversion efficiency to 50%. If these phenomena can be removed, which is the case for a ring cavity where the signal and idler are not phase matched on the return journey and don't reproduce the pump, then the theoretical conversion efficiency, under the appropriate conditions, can reach 100%. The theoretical maximum efficiency for a SRO is also 100%. These comments apply to the case of uniform, plane waves.
The conversion efficiency for a DRO with power dependent reflections under the conditions of steady state operation is [1 1]
7? = ^ ^ t ^ = ^ ( V iV - l) (2.9.1) where N is the number of times steady state threshold of the pump. It can be seen that the maximum efficiency is 50 %, and occurs for N=4. By eliminating the power dependent back reflections, e.g. by using a ring resonator, the efficiency is increased to
100 % for N=4.
The conversion efficiency for a SRO in steady state can reach 100 %, and is governed by the implicit relation
Ch. 2 : Theory of Parametric Interactions
where p is a gain parameter given by . For Ak=0, the SRO reaches 100 %
efficiency for pumping ( n /lŸ times threshold. As mentioned above, this theory only applies to the case of uniform, plane waves.
When the transverse nature of the pump is taken into consideration, a better qualitative agreement between theory and experiment is obtained. This analysis was done by Bjorkholm [46] under the assumption of pulsed pumping of a plane-plane resonator such that the resonator transverse modes were not established during the pump pulse. By integrating radially over the beam, the pump power transmitted can be calculated by setting the output power equal to threshold where the input pump intensity is greater than threshold, and leaving the output power equal to the input power in the regions where the intensity is insufficient to overcome threshold. In this way the maximum efficiency of a DRO without power dependent reflections is 82 % for N=4, and for a SRO the maximum efficiency is 71 % for pumping 6.5 times threshold. These efficiencies are power conversion efficiencies.
Despite the fact that Bjorkholm's work applied to pulsed pumping the above power efficiencies only apply once steady state has been reached. In chapter 5 we show how energy conversion efficiencies can be calculated, again for steady state, and also derive an empirical relation between conversion efficiency and the number of times the device is pumped above the pulsed threshold. The conversion efficiencies under pulsed pumping are seen to approach the steady state case for pumping sufficiently above threshold.
2.10 Summary
In this chapter we have considered some of the general theory which describes the parametric interaction, and tried to give more detail about the particular theory which is applicable here. The non-linear susceptibility convention adopted was presented leading to the coupled amplitude equations describing transfer of power between waves of different frequency. This lead to the concept of parametric amplification and the effect that momentum or phase mismatch had on this. The plane wave theory was extended to allow for focused gaussian beams. The factors affecting parametric oscillation threshold both in steady state and for pulsed pumping were discussed, including single and double resonance, leading to the development of a model for calculating pulsed thresholds for SROs. This theory forms the foundation for the rest of the work, allowing choice of a non-linear material (ch. 3), requirements for a pump laser (ch. 4), and design and operation of an all-solid-state OPO (ch. 5).
Ch. 2 : Theory of Parametric Interactions
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