Mode-matching,

It was discussed in section II. 2 that the nonlinear coupling within the gain medium was dependent on the exact spatial overlap between the three fields. To obtain maximum benefit from the nonlinear interaction, the beams should be focused to small spot sizes within the nonlinear material to obtain high power densities.

In phase-matching geometries where propagation is significantly removed from a principal crystal axis, the effects of Poynting-vector walk-off

can present a limit to the size of the focused spots desired within the optical cavity. It is then necessary to compare the reduction in threshold from tight focusing, due to the high power densities, with the possible increase in loss due to walk-off caused by the tight spot sizes. These considerations are addressed regularly when operating pulsed OPOs in critical phase-matched geometries.

As discussed earlier, the low pump powers associated with cw lasers require that the effects due to Poynting-vector walk-off over the length of the gain region should be eliminated, or at a level that does not cause a large increase in the threshold of the OPO; see appendix I. It is reasonable to suggest that cw OPOs can only be operated above threshold by focusing the input pump radiation to small spot sizes to obtain high power densities. It was assumed that the optimum focusing of the fields in cw OPOs can be approximated by the analysis of section II. 2, in which the confocal parameters of the three waves were set equal to the length of the nonlinear crystal.

This section summarizes the basic equations that are relevant to the focusing requirements of cw OPOs. The beams considered are assumed to be lowest order fundamental fields (confined most strongly near the optical axis) with a Gaussian radial intensity distribution. In general, higher order transverse modes are suppressed in cw OPOs due to the magnitude of the gains involved compared to the diffraction losses experienced by higher order modes.

The most important mode-matching consideration involves matching the beam parameters of the input pump source to those desired within the OPO cavity for optimum coupling. These matching requirements become more stringent when the OPO cavity is designed to enhance the circulating pum p field, in for example the triply-resonant oscillator. In this case, deviations from the conditions of exact mode-matching can result in decreased coupling of the input pump field into the OPO cavity.

Consider a nonlinear crystal of interaction length, /, situated within a linear, standing-wave, two-mirror, optical cavity with an empty (free-space) cavity length, L. The confocal parameter is defined by b and such that

where zj^ is the Rayleigh range.

Setting the confocal param eter equal to the length of the crystal provides the optimum spot sizes for the three waves. This is because the region from -Zji to +zj^ can be regarded as the focal zone, or the approximate- collimated region. Therefore, in this region (crystal), beam diffraction has less of an effect on the overlap of the three interacting waves over the length of the gain medium. The spot size of the waist (1/e electric field radius) at the centre of the nonlinear crystal is then defined as follows:

12

. [E. 162]

^ iTvrij

The spacing of the m irrors m ust provide these spot sizes, while simultaneously satisfying cavity stability for the resonator. In most cases, this is conveniently provided by placing two curved mirror surfaces either side of the crystal, equidistant from the crystal faces. While this provides a reliable method of satisfying the optimum focusing conditions, other possibilities must be considered. These include direct fabrication of the dielectric reflecting coatings on spherically curved crystal faces, thus forming compact and mechanically stable monolithic resonator configurations. However, this requires accurate polishing of the surfaces and cannot always be provided w ith given crystal grow th procedures. Another variation involves asymmetric resonator designs with one extended optical mirror spacing that allows for the incorporation of intra-cavity components (e.g. within the dual­ cavity resonator). For the remainder of this analysis, the resonators are assumed to be concave symmetric designs, but the method of analysis is valid for the other cavity configurations described above.

The most convenient procedure for location of the cavity mirrors is to propagate the desired Gaussian beam from the waist size at the centre of the nonlinear crystal, over a free-space distance, until the radius of curvature of the beam matches that of the chosen curved mirror surface. The spacing of the mirrors must then satisfy the following relation [78]:

where U is the effective length of the cavity, as defined by

L' = L + (z/ny) . [n. 164]

Therefore, the cavity geometry, as dictated by the mirror curvatures, defines confocal focusing. Further, cavity stability is necessarily achieved since the wavefront curvatures match the mirror curvatures, thereby retracing the same (stable) pattern each time across the cavity. The next step involves ensuring that the pump source is properly mode-matched into this resonator. To provide this, the curvature / spot size of the incident pum p beam (as characterized by the complex q-heam parameter [78]) is matched to that required in the OPO cavity.

Mode-matching can be accomplished by selecting the appropriate lens (or lenses) to transform the input pump beam accordingly. Given the sizes,

laser ^ p ,O P O ' of the input pum p waist and the OPO cavity waist,

respectively, and a lens of focal length, / , the position of this lens to match the two beams can be found by specifying the distances , di^gg^ and dQpQ, of the two waists, either side of the focusing lens, such that [78]

^laser ~ f _Wwp,laser

p,OPO [n. 165 (a)]

^OPO =/ + Wp,OPO

p.laser ■fc [n. 165(b)]

where fc is the minimum, or characteristic, focal length that can be used, and is given by [78]

fc ^ ^ p ,O P O ^ p,laser [E. 166]

Therefore, suitable mode-matching lenses can be selected if the waist sizes and their relative locations are known, either side of the focusing lens.

The above equations can be greatly simplified by assuming that the pump beam is being focused at a point in the far field of the beam divergence profile (i.e. at locations » \zp\). Therefore, assuming that the focusing power

of the lens dominates over the initial divergence, the new waist (W2) is located near to the focal point of the lens, and is of the size

p . 167] where W/ is the spot size at the lens.

Finally, note that equations [II. 165 (a)] and [II. 165 (b)] reduce to the classical Newton object/image expression if f c~0; i.e. the spot sizes are reduced to zero.