1. Introduction

Optical parametric oscillators (OPOs) can be made resonant for one or both of the generated beams. The doubly resonant OPO (DRO) has the advantage of much lower threshold intensity, at the cost of reduced stability [1]. The reason for this problem is that all three waves are incident on the nonlinear crystal, and the direction of energy transfer in the mixing process depends on the relative phase of the three beams. Thus the output energy from a DRO is sensitive to resonator length changes of a fraction of a wavelength. Stable operation of a DRO requires a single-mode pump source and a very stable OPO cavity [2], and continuous tuning is complicated [3]. The singly resonant OPO (SRO), on the other hand, is robust to length changes because only two waves enter the nonlinear crystal, and the phase of the third wave adapts to optimize conversion. In either type of OPO, the threshold can be reduced by resonant enhancement of the pump wave [4]. This technique requires locking of the pump frequency to a resonance of the OPO cavity. When the pump source is a Q-switched laser with nanosecond pulses the power is usually high enough for efficient operation of a SRO, so this has been the configuration of choice in such applications. However, pure singly resonant operation requires mirror coatings with extremely small reflectance for the nonresonant waves. Making such coatings is difficult, especially if the OPO is designed to be tunable over a substantial bandwidth. Suppression of unwanted reflections can be improved in a ring resonator [5], but a standing wave resonator is often preferred because of its simplicity. Hence it is important to know how such OPOs behave in the presence of unwanted reflections. Falk [1] showed that, for operation near threshold, an idler feedback of only a few percent can lead to large fluctuations in output energy. Yang et. al. [5] generalized this analysis and found that the OPO was more tolerant to idler reflection when pumped higher above threshold. Their analysis was based on a plane-wave model, and the signal was assumed to have negligible output coupling. Neither of these authors considered pump resonance effects, and the pump was taken to have a single frequency. In the present paper, the energy fluctuations in an OPO with residual reflections of both pump and idler are studied under conditions with single- or multi-longitudinal mode pump. The experimental observations are compared with results from a numerical model that includes diffraction, dispersion, multiple transverse and longitudinal modes, pump depletion, and arbitrary reflections of all beams.

The paper is organized as follows: In Section 2 the relevant phase parameters describing the reflections are considered in detail. The experimental setup is described in Section 3, the numerical simulation model in Section 4, and the results are compared in Section 5.

2. Reflection phases

Consider the OPO shown in Figure 1, with a standing wave resonator formed by two mirrors M1 and M2. The signal denotes the beam for which the OPO is designed to be resonant. The pump is coupled in through M1, and the signal is partially coupled out through M2.

 

Figure 1. Model OPO. ω ₁, ω ₂, and ω ₃ denote the angular frequencies of the idler, signal, and pump, respectively, where ω ₃ = ω ₁ + ω ₂.

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Ideally, both mirrors should have 100% transmission for the pump and idler beams, and M1 should be perfectly reflecting for the signal. In reality, both mirrors have at least some reflection for all the beams. The mixing process in the crystal depends on the phase difference Δϕ = ϕ ₃ - ϕ ₂ - ϕ ₁ of the three waves, where ϕ_i denotes the phase of beam i. Let the phase difference of the beams entering and leaving the left end of the crystal be denoted by Δϕ _1,in and Δϕ _1,out, respectively, and similarly with subscript 2 for the right end. The mixing in the forward pass depends on Δϕ _1,in and the mixing in the return pass depends on Δϕ _2,in. The resonance of the pump beam depends on the phase difference Δϕ_p between the reflected and incident pump beams. Define Δϕ _M1 = Δϕ_1,in - Δϕ _1,out and Δϕ _M2 = Δϕ _2,in - Δϕ _2,out. Because the dispersion in air is small Δϕ _M2 is insensitive to the position of M2, so it is determined by the mirror M2 itself. On the other hand, Δϕ _M1 does depend on the position of M1 because the phase of the pump beam incident on the crystal is determined mainly by the external pump beam whereas the phases of the signal and idler beams depend on the position of M1. Moving M1 also changes Δϕ_p, so to control Δϕ _M1 and Δϕ_p independently additional degrees of freedom must be used. Δϕ _M1 = Δϕ _M2 = 0 is equivalent to both the idler and signal being resonant, and the beams can adjust their phases for optimal conversion. On the other hand, if Δϕ _M1 or Δϕ _M2 is nonzero the phases cannot remain optimal for successive round trips. Because of the dispersion in the nonlinear crystal Δϕ _1,in and Δϕ _2,in also depend on the operating frequencies of the OPO. If Δϕ _M1 and Δϕ _M2 are not optimal for the phase matched center frequency, the OPO may be forced to run on an off-center frequency, thus improving the phases but reducing the gain because of phase mismatch.

In reality the number of parameters needed to represent the resonator may be even greater because of additional reflections from the end faces of the crystal and from the rear faces of the mirrors. Such reflections give rise to subcavities in the mirrors and in the air gaps. They can be reduced by using wedged mirrors and tilting the crystal.

3. Experimental setup

The OPO corresponds to the one shown in Figure 1. It is based on a 19 mm long, 0.5 mm thick crystal of periodically poled LiNbO₃ (PPLN). The domain period used in these experiments is 28.9μm, which phase matches conversion from 1.064μm to 1.48μm and 3.8 μm at a temperature of 100°C. The resonator consists of two plane mirrors on CaF₂ substrates. The gaps between the mirrors and the ends of the LiNbO₃ crystal are about 2 mm. The mirrors were designed for single pass pump and singly resonant operation for the 1.48 μm wave. The preferred and measured reflectances of the mirrors are shown in Table 1. The mirror substrates are plane, and their rear faces are not coated because of difficulties in obtaining AR coatings for CaF₂. Therefore the effective reflectances are affected by subcavity effects. The LiNbO₃ crystal is broad-band AR coated at both ends, but there are residual reflectances of about 2% at 3.8μm and 1.4% at 1.48μm. The crystal could not be tilted because of the small aperture of the periodically poled stripes. Thus the reflections from the crystal end faces may also contribute to subcavity effects.

Table 1. Design reflectances and actual reflectances of the mirrors. M1 is the pump input mirror, and M2 is the signal output mirror.

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The OPO was pumped by a diode-pumped Nd:YAG laser which could deliver about 30 mJ in 25 ns (FWHM) pulses at 20 Hz repetition rate. The beam quality was good with near field beam diameter times full divergence angle of about 2 mm·mrad for 90% energy content. Only a small part of the available pump energy, up to 660 μJ, was used for pumping the OPO. This pump beam was focused to about 0.3 mm exp(-2) diameter in the PPLN crystal. Normally the pump laser ran on multiple longitudinal modes with a total bandwidth of 15–20 GHz, but it could be injection seeded to run on a single longitudinal mode. The signal and idler energies from the OPO were measured by a pyroelectric joulemeter with a sensitivity of about 10 μJ. The experiments were conducted by recording the range of variation of the output energy for each pump energy. Different phase parameters were sampled by changing the temperature of the nonlinear crystal by a few degrees and noting how the output energy oscillated while the temperature was changing.

4. Simulation model

Because the phase of the idler reflections may force the OPO to run on a longitudinal mode that is not exactly on the phase matched center frequency, and because the multi-mode pump causes temporal modulation of the signal, a model for this OPO needs to include multiple longitudinal signal modes. The model, which has been described previously [6, 7], is based on the beam propagation method for mixing and propagation in the nonlinear crystal. The model takes into account arbitrary transverse beam profiles, diffraction, walk off and other birefringence effects, dispersion, pump depletion, and initiation from spontaneous parametric emission. The mirrors can have arbitrary reflection coefficients for all beams. The model can also handle absorption and thermal effects, but they were not included in the present case because they were negligible.

The pump beam for the OPO simulation was obtained by taking the output beam from a simulation of the pump laser and passing it through the appropriate focusing optics. The resulting fluence distribution at the OPO input was compared to the experimentally measured pump fluence and found to agree well, as shown in Figure 2. While the actual pump fluence could be more accurately measured than simulated, the simulated beam has the advantages of providing phase information and the full temporal evolution of the beam shape, information which is difficult to measure experimentally. The laser simulation assumed a single longitudinal mode. A multi-mode pump was obtained by multiplication of the single frequency signal from the laser simulation with a rapidly varying modulation signal, which was generated by adding up multiple modes with independent complex Gaussian-distributed amplitudes and an expectation spectrum corresponding to the measured spectrum of the laser.

 

Figure 2. (a) Simulated pump fluence incident on the OPO. (b) Actual pump fluence measured by imaging the beam at mirror M1 into a CCD camera. The energy contents in (a) and (b) are equal.

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As explained in Section 2 the total effective reflectances are affected by subcavity effects, and they depend on several unknown phase parameters. Because it would be extremely time consuming to search the resulting parameter space, we have simply used the measured mirror reflectances from Table 1 and neglected the subcavity resonances. In the worst cases the effective reflectances can deviate significantly from the reflectances in the table. Nevertheless the simulated and experimental results agree well, as we will show in the following section. One reason for this is probably that the worst case conditions are unlikely to happen because there are so many reflections that contribute. Simulations were performed for the optimal phase parameters Δϕ _M1 = Δϕ _M2 = Δϕ_p = 0, and for various other combinations of phases. The range of variation of the output energy over several simulation runs was recorded for each set of phases. In a DRO with a single pass pump the energy would have been lowest for a total phase shift of π, e.g. Δϕ _M1 = π and Δϕ _M2 = 0. In the present example, on the other hand, the phases are also affected by mixing in the return pass, and the lowest energy was obtained with Δϕ _M1 = Δϕ _M2 = Δϕ_p = π.

A further uncertain parameter is the nonlinear coefficient d ₃₃. This has been carefully measured for second harmonic generation (SHG) of 1.06μm and 1.3 μm [8], but it was found in the same reference that it does not obey Miller’s rule. For the lack of a better estimate, we nevertheless used Miller’s rule with the measured values for SHG at 1.06μm or 1.3μm, leading to d ₃₃ = 20pm/V or d ₃₃ = 17pm/V respectively, for the present mixing process. The former value turned out to give the best match with experimental results, and this was used in the simulations presented below. The resonator loss was neglected because it is much smaller than the output coupling.

Simulations were performed with two different spatial resolutions: 32 × 32 transverse points on a 30 × 30 μm grid or 64 × 128 points on a 15 × 10 μm grid, with the longest matrix side along the width of the periodically poled region of the crystal. The signal energies calculated with the two resolutions differed by ¡3% at low energies and up to about 8% at the highest energies. The resolution was not increased further because the run time would have been very long. The agreement between the two resolutions that were tried, and with the experimental results, makes us confident that the accuracy of the model is within a few percent. The number of longitudinal modes depended on the situation being simulated. For optimal reflection phases the OPO oscillated near the phase matched center frequency and the model spectrum could be narrow. For other reflection phases the OPO was forced to operate off the center frequency and the model spectrum had to be wider to accommodate the dominating modes.

5. Results

Figure 3 shows the experimental and simulated input-output curves for multi-mode and single-mode pumping. The curves for single-mode pumping show large energy fluctuations just above threshold. For stronger pumping the relative magnitude of the fluctuations is reduced, but it is still substantial. This agrees qualitatively with the analysis of Yang [5]. With the multi-mode pump, the energy fluctuations are always smaller, and at high energy the relative fluctuations are small. The reason for the reduced energy fluctuations is that the phase of the multi-mode pump varies randomly, therefore the OPO becomes less sensitive to the phases of the reflected waves. This was noted by Hovde et. al. [9], and the difference in spatial beam quality with single- and multi-longitudinal mode pumping observed by Haub et. al. [10] may be related to the same effect. The simulation results agree reasonably well with the experimental results, but there are discrepancies that can be attributed to at least three sources: (i) Only a single parameter (the temperature) was changed in the experiments, therefore it is not likely that the extreme phase conditions were sampled. (ii) The value of the nonlinear coefficient d ₃₃ is uncertain. (iii) Subcavity effects were neglected in the simulations.

 

Figure 3. Signal energy vs pump energy. The solid lines represent minimum and maximum energy observed in the experiments, and the dashed lines represent the simulations. (a) Single-mode pump. (b) Multi-mode pump.

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The pulse to pulse energy fluctuations have contributions from changes in the resonator (e.g. phase changes because of vibration or temperature change), from fluctuations in the pump pulses, and from the quantum noise initiating the OPO. The latter contribution would lead to signal pulse fluctuations even if the parameters of the OPO and the pump pulses were constant. It is not possible to separate these contributions in the experiments, but in the simulations they can be studied separately. Figure 4 shows the simulated range of energy variation for an OPO with multi-mode pump and fixed resonator parameters. These fluctuations are due to the randomness of the pump pulses and to the quantum noise in the OPO. Comparison with several simulations with a fixed pump pulse indicates that pump pulse fluctuations are the major source of the energy fluctuations seen in Figure 4.

 

Figure 4. Range of variation of the simulated signal energy for an OPO with fixed phase conditions Δϕ _M1 = Δϕ _M2 = Δϕ_p = 0 and a multi-mode pump.

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The validity of the model is further tested by comparing observed and simulated power curves, as shown in Figure 5. The measured power traces in Figure 5(a) and (b) have been calibrated to match the measured energies. The measured pump pulses appear slightly too long because of the fall-time of the detector, and this also reduces their peak power because of the scaling. Note that the modulation pattern of the multi-mode pulses varies randomly, so the experimental and simulated pulses in Figure 5(a) and (c) can only be expected to agree qualitatively. The most rapid modulations in the simulation cannot be seen in the experiment because the oscilloscope had a bandwidth of 500 MHz. The small modulation of the pump pulse in Figure 5(b) indicates that the injection seeding of the pump laser was not perfect.

We also compared the measured and simulated beam profiles. It would have been desirable to compare the signal and idler beams, but suitable cameras for measuring these wavelengths were not available. However, the pump beam is also strongly modified by passing through the OPO, so inspection of it can serve as a useful test of the model. Figure 6 shows simulated and measured fluence of the output pump beam for several different pump energies. The model predicts that the pump beam is strongly focused by pump depletion and, for high energy, back conversion. Although the experimental results do not agree exactly, they do show the same effect qualitatively. The deviations from the very sharp beams predicted by the simulation can be attributed to various experimental imperfections: The real pump beam differs slightly from the simulated pump beam, there may be small alignment errors in the OPO, the coatings on the PPLN crystals were imperfect, and the fine details may not be imaged sharply onto the camera. Finally, the simulation model cannot be trusted to give accurate results for the extremely narrow beams at the highest energies.

 

Figure 5. Pump and signal powers vs time. In each graph the pump is the upper trace and the signal is the lower trace. (a) Experimental, multi-mode pump. (b) Experimental, single-mode pump. (c) Simulation, multi-mode pump. (d) Simulation, single-mode pump. The pump energy was 190 μJ.

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In addition to testing the model the focusing of the pump is interesting in its own right. Self focusing and self phase modulation because of cascaded second order nonlinearity has received much attention [11, 12], but in the present case the focusing is not due to cascading: It can be verified by simulation of single pass parametric amplification that the pump beam becomes convergent before there is back conversion. The intensity dependent phase shift for divergent beams in frequency mixing crystals has been studied theoretically [13]. The mechanism is that the pump is depleted by diverging signal and idler beams, and the resulting transverse phase variation of the driving polarization leads to focusing of the pump. Subsequently the sharp pump beam generates similarly shaped signal and idler beams. This focusing has important practical consequences for the operation of the OPO. We did indeed observe damage on the crystal faces after the OPO was operated at the highest pump energy.

The results shown in Figure 6 are for a multi-mode pump. The output pump beams for a single-mode pump are qualitatively similar, but they tend to be somewhat less peaked. This is probably because the focusing is enhanced by the spikes of the multi-mode pump.

Note that cascaded operation, i.e. generation of a second pair of signal and idler beams pumped by the primary signal beam, has been observed in an OPO similar to our [14]. The mirrors in that OPO had approximately the same reflectance for the primary and the secondary signal wavelengths. We did not look for secondary waves, which would be phase matched at 2.1 μm and 5.1μm, but the mirrors in our OPO had 90% or greater output coupling for both these wavelengths, so parasitic oscillation should not occur. Simulations including the secondary waves confirm that they do not have enough gain to reach threshold.

 

Figure 6. Simulated (a-e) and measured (f-j) output pump beams for different pump energies: (a,f) 110 μJ, (b,g) 230 μJ, (c,h) 320 μJ, (d,i) 410 μJ, (e,j) 640 μJ. The pump was multi-mode. The pump was measured by imaging the beam at mirror M2 into a CCD camera.

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6. Conclusion

We have shown that an OPO with non-ideal mirrors displays much smaller pulse to pulse energy fluctuations when it is pumped by a multi-longitudinal-mode pump than when it is pumped by a single-mode pump. The reason for this is that the phase variation of the multi-mode pump reduces the sensitivity to the phases of the reflected signals. In either case, the relative fluctuations decrease for higher pump energies, but if small fluctuations are a prime concern spurious reflections must be carefully minimized. The energy fluctuations of the non-ideal OPO have been successfully simulated by a numerical model that includes multiple longitudinal modes and allows arbitrary reflectance of all three beams.

Self focusing of the pump beam was observed in the experiments and the simulations. The resulting high peak fluences limit the energy which the OPO can handle, and simulation of this effect is important to identify the safe operating regime of the OPO.