Optical parametric oscillation in a random polycrystalline medium

QITIAN RU, NATHANIEL LEE, XUAN CHEN, KAI ZHONG, GEORGIY TSOY, MIKE MIROV, SERGEY VASILYEV, SERGEY B. MIROV, AND KONSTANTIN L. VODOPYANOV* CREOL, College of Optics and Photonics, University of Central Florida, Orlando, Florida 32816, USA College of Precision Instrument and Optoelectronics Engineering, Tianjin University, Tianjin 300072, China IPG Photonics Mid-Infrared Lasers, Birmingham, Alabama 35203, USA Department of Physics, University of Alabama at Birmingham, Alabama 35294, USA *Corresponding author: vodopyanov@creol.ucf.edu


INTRODUCTION
It has commonly been assumed that perfectly regular crystals are paramount for the operation of frequency converters based on quadratic nonlinearity χ (2) , e.g.optical parametric oscillators (OPOs).The necessary phase-matching condition is usually achieved either by a proper crystal orientation (if the crystal has birefringence), or via quasi-phase matching through controlled flipping of crystalline domains.Apparently, as our recent results show, a disordered material consisting of randomly-oriented domains, with the nonlinear wave-coupling coefficient arbitrarily varying between its maximum (dNL) and minimum (-dNL) values, can perform equally well.
Our approach is based on the phenomenon known as random phase matching (RPM) [1][2][3][4].RPM in disordered χ (2)  materials can be described by the random walk (or 'drunken sailor walk') theory that also accounts for diffusion and heat transfer.RPM eliminates the need for orientation patterning and, most importantly, enables 3-wave interactions with extremely large bandwidths.A broadband and flat response in RPM is the result of phase randomization due to arbitrary distribution of the crystalline domains, which eliminates destructive interference.The price to pay, however, is a slow growth of the output signal since the output intensity in RPM scales linearly with the sample length, as opposed to the quadratic dependence for both perfect phase and quasi-phase matching.Although in continuous-wave and nanosecond laser regimes, RPM is much less efficient than the conventional quasi-phase matching process [3].Our finding is that RPM is very well suited for 3-wave processes when short (<100 fs) pump laser pulses are used.

ZNSE SAMPLES
ZnSe, a cubic-symmetry semiconductor with a bandgap of 2.7 eV, is a perfect candidate for three-wave mixing because of its outstanding transparency (0.55 to 20 µm), relatively high 2-nd order nonlinearity (d14= 20 pm/V [5]), high optical damage threshold, and good mechanical properties.Kanner et al. have grown quasi-phase matched (QPM) orientationpatterned ZnSe structures on patterned GaAs substrate and demonstrated second harmonic generation (SHG) and difference frequency generation (DFG) [6].However, because of insufficient quality of these structures, the optical parametric oscillator (OPO) action has not been realized in ZnSe.
In our RPM OPO approach with ZnSe, we took advantage of several important conditions: 1) For short (sub-100-fs) pump pulses used in the experiment, the length of the nonlinear crystal does not need to be larger than ~1 mmthe temporal walk-off length, set by the group velocity dispersion difference between the pump and the OPO wavelength.
2) Large coherence (Lcoh) ≈ 100 µm for the 2.35-µm pumped OPO based on ZnSe 3) High peak intensity that can be achieved with no multi-photon absorption (at least to the 5-th order) 4) Very low group velocity dispersion (GVD) at OPO frequencies (≈ 4.7 µm) -a prerequisite for broadband output 5) The interaction length in ZnSe is only 500 µm (~ 5 coherence lengths) with a tight pump focusing.
Our ZnSe ceramic samples were prepared in the following way: commercial (II-VI Inc.) ZnSe samples grown by CVD, with 11x6x3 mm size, were sealed in quartz ampoules under 10 -5 Torr vacuum and annealed at 900ºC.For nine sets of samples, the annealing time varied from 6 to 10 days with a half-day interval.After annealing, the samples were removed from the ampoules and, together with two untreated samples, were chemically etched and analyzed with a microscope from the viewpoint of grain size.The average grain size of untreated samples was between 50 and 60 μm.The annealed samples have shown a trend of increasing grain size, which reached 100 µm after 8 days of annealing (Figure 1, left).We have used the latter set of samples in our experiments since the grain size was close to the coherence length (Lcoh=102 µm) for our three-wave interaction in ZnSe, which is the optimal condition for RPM [3]. Figure 1, (right) shows the surface of an etched sample after annealing for 8 hours.

SHG MAPPING OF ZNSE SAMPLES
To characterize our ZnSe ceramic samples, we first did XY mapping from the viewpoint of SHG (4.7 -> 2.35 µm) conversion efficiency using a nanosecond λ=4.7-µm source (SHG is an inverse process with respect to OPO, hence SHG efficiency is a direct indicator of the OPO gain that can be achieved).The setup for SHG mapping, based on a nanosecond Normalized SHG conversion efficiency tunable (2.8 -5 µm) OPO, is shown in Figure 2, while Figure 3 illustrates a typical result of SHG mapping of a 5 x 5 mm sample (L=1.5 mm).A histogram reveals a broad distribution related to random variation of the alignment and size of crystalline domains.There are 'hot' spots where SHG efficiency is 2.5-3 times higher than the average (these hot spots were used to achieve the OPO oscillation).In terms of the average (over the histogram) SHG output, we observed the linear dependence on the sample's physical length L, in full accord with RPM theory [3].

OPO EXPERIMENT 4.1 Pump laser
There has been a rapid development, in the last decade, of ultrafast lasers based on transition-metal-doped II-VI semiconductors, among which ZnS and ZnSe doped with Cr 2+ ions (center λ=2.3-2.4 µm) are most frequently used [7][8][9].The advantages of Cr:ZnS/ZnSe lasers include a very broad gain bandwidth that allows producing short (down to few optical cycles) pulses, the absence of excited state absorption, close to 100% quantum efficiency of fluorescence, and convenient pumping by erbium and thulium fiber lasers with a conversion efficiency in excess of 60%.Currently Cr:ZnS/ZnSe lasers enable producing more than 7 W of the average power in the mode-locked regime [8], and up to 1 GW of peak power in the regime of chirped-pulse amplification [9].Cr:ZnS/ZnSe lasers have also proven to be very suitable for pumping mid-IR OPOs based on GaAs [ 10 ].The RPM ZnSe ceramic OPO in our experiment was synchronously pumped by a Kerr-lens mode-locked Cr:ZnS laser with a center wavelength of 2.35 μm, a 62-fs pulse duration, 650-mW average power, and a 79-MHz repetition rate.

OPO operation
The bow-tie ring OPO cavity (Figure 4) was composed of an in-coupling dielectric mirror M1 with a high transmission (>85%) for the 2.35-μm pump and a high reflection (>95%) for 3-8 μm, two gold-coated parabolic mirrors (M2 and M3) with a 30° off-axis angle and a 30-mm apex radius, and five gold-coated flat mirrors (for simplicity only M4 was shown in Figure 4, the other four mirrors were used for folding the beams to reduce the footprint).An uncoated plane-parallel polished L=1.5 mm ZnSe ceramic sample was placed between the two parabolic mirrors at the Brewster's angle (Figure 4, bottom left).A 0.3-mm-thick ZnSe wedge with 1° apex angle was used inside the cavity for variably outcoupling the OPO signal/idler waves.n / spectrum due to negligible GVD of ZnSe in our spectral range (ZnSe has zero GVD at 4.8 µm).The OPO threshold was achieved at 90 mW of average pump power.The output spectrum, measured with a monochromator and a mercury cadmium telluride (MCT) detector, spanned 3-7.5 μm (at −40 dB level) and was centered at the 4.7 μm subharmonic of the pump (Figure 5).The OPO, being a doubly resonant device, is interferometrically sensitive to the cavity length adjustment, accomplished via using a piezo-actuator (Figure 4); this is revealed in the 2D plot in Figure 5(b), which shows the spectrum versus length dependence.The outcoupling of the OPO was tuned by rotating the ZnSe wedge using the Fresnel reflection.At maximum pump, the average OPO power [in two beams from the outcoupler wedge (OC)] was ∼20 mW with 5% outcoupling as shown in Figure 6(a).The output power versus pump power was measured at 5% outcoupling and is shown in Figure 6(b).There is no power roll-over at the maximum pump power, which indicates that the role of multiphoton absorption is negligible.The observed OPO pump depletion was as high as 79%, which indicates that with optimized outcoupling, one can obtain high conversion efficiency, approaching 100%, from such a device.
We also operated the OPO with an off-the-shelf polycrystalline ZnSe sample from II-VI Inc. (L =2 mm, the average grain size 80 μm).It demonstrated similar performance, although with a twice higher (180 mW) pump threshold.Figure 6.Left: the OPO output power versus the outcoupling.Right: the output power of the OPO versus the pump power at 5% outcoupling.

Back of the envelope calculations
In a similar OPO arrangement and with the same pump laser but with an OP-GaAs crystal as the gain element, we achieved the pump threshold of 8 mW [13].What would be the threshold for an ideal QPM ZnSe?Given the fact that a ZnSe nonlinear figure of merit (d 2 /n 4 for Brewster angle operation) is 6.5 times smaller than that of GaAs, we would expect the threshold to be 6.5 times larger, which is 52 mW.In our RPM experiment, we achieved a 90-mW threshold that is less than 2 times that of an 'ideal' QPM case.This is consistent with our RPM model (see Appendix).In fact, given the very tight focusing of the pump (w≈7 µm) and the small associated focal length (≈ 500µm), only ~ 5 ZnSe grains participate in the nonlinear interaction.As our modeling shows, at this small amount of participating grains, there is not a big difference (if one uses 'hot' RPM spots in ZnSe) in nonlinear conversion efficiency between the RPM and QPM cases.

CONCLUSION
We demonstrated the world's first OPO based on RPM [14].The gain element was a 1.5-mm-thick ZnSe ceramic sample with the average grain size near 100 µm, placed at the Brewster's angle.The bow-tie ring-cavity OPO was synchronously pumped by a Kerr-lens mode-locked Cr:ZnS laser (λ=2.35µm) and operated in the subharmonic mode near degeneracy.Its oscillation threshold was 90 mW and the output spectrum spanned 3-7.5 μm.The observed pump depletion of 79% indicates that with an optimized outcoupling, one can obtain conversion efficiency approaching 100%.In our view, RPM in ZnSe and similar readily accessible random polycrystalline ceramics (ZnS, ZnTe, ZnO, CdSe, GaP, GaN) opens a new avenue in nonlinear optics and represents a viable novel route for the generation of few-cycle pulses and multi-octave frequency combs.

APPENDIX: MODELING OF THE RPM PROCESS
Although there were several attempts to describe analytically the process of frequency conversion in random polycrystalline materials [1][2][3], we found it necessary to develop our own model since several assumptions in these works do not apply to our case.For example, the beam radius in all our experiments is much smaller than the average grain size (as opposed to [3]) and we have small ~ 1 mm interaction length, which is dictated by the use of femtosecond pulses.Also, previous analytical studies did not discuss polarization effects.Here we make an attempt to describe RPM using the example of SHGan inverse process with respect to OPOand compare our theory with experimental findings.
In our modeling of the RPM process we made the following assumptions: • The orientation of grains is random with all possible direction and has no correlation between adjacent grains.
• We have normal grain-size distribution along the light path.
The nonlinear susceptibility of purely random orientation of a grain was computed by the transformation rule for third rank tensors.Assuming the input electric field oscillates along the x axis (laboratory frame), the output polarizations of both x and y are functions of  11 ′ and  21 ′ respectively.
(2)  ( One million iterations of random rotation were performed to yield the distribution of nonlinear coefficient in a single crystal as shown in Figure 7. Figure 7.The normalized effective nonlinear coefficient of a single crystal with random orientation for the same output polarization (left) and orthogonal polarization (right).The maximum value of 1.155 on the left agrees with the value when the beam polarization is parallel to <111> [15].
In the simulation of RPM in SHG, 1D structure of polycrystalline material was considered and the approach developed in Ref. 16 was applied.The effective nonlinear coefficient follows the distribution in Figure 7 and the grain size follows the normal distribution with an average value of coherence length  coh and a standard deviation of  = 30 %.The SH intensity has a variation due to random value of nonlinear coefficient and grain size.Figure 8 shows the distribution of SH power for different crystal lengths, while Figure 9 plots RPM SHG efficiency (only parallel output field component with respect to the pump polarization is counted) that is normalized to the QPM efficiency, as a function of crystal length.(For the QPM case we used the fixed nonlinear coefficient d14).1,000 sets of random structure are tested (shown as dots) for each length.The averaged values are shown as a solid line.
The main conclusion is that our Monte Carlo calculations confirm the fact that at short interaction lengths RPM efficiency may be close to that of QPM, as discussed in the previous section 4.3.For example, at L/Lcoh=5, RPM efficiency at hot spots can be within a factor of two as compared to that of the pure QPM case.

Figure 1 .
Figure 1.Commercial CVD-grown ceramic ZnSe samples were sealed in quartz ampoules under 10 -5 Torr vacuum and annealed at 900ºC.The annealed samples demonstrated an apparent trend of grain size increase from 50 to 100 µm over a period of 8 days

Figure 2 .
Figure 2. Setup for ZnSe ceramic characterization via second harmonic generation.

Figure 3 .
Figure 3.A typical result of SHG mapping (pixel size 50 x 50 µm, sample size 5 x 5 mm).Left: false-color map of SHG efficiency distribution.There are 'hot' spots where SHG efficiency is 2.5-3 times higher than the average.Top right: SHG signal variation from pixel to pixel.Bottom right: a histogram that reveals a broad distribution related to random variation of the alignment and size of crystalline domains.

Figure 4 .
Figure 4. Setup of the bow-tie ring OPO cavity.Bottom right: the photo of the ZnSe ceramic sample placed at the Brewster's angle.The OPO was operating in a doubly resonant frequency-divide-by-2 mode at degeneracy.In addition to lowering the pump threshold, this arrangement provides other advantages[11,12]: (i) phase-and frequency-locking to the pump lasera precondition for creating precision mid-IR frequency combs, and (ii) the possibility of achieving an extreme broadband

Figure 5 .
Figure 5. Top: the OPO output spectrum showing a continuous spectral span of 3-7.5 μm.Bottom: 2D spectrum where y axis shows the cavity length detuning.

Figure 8 .
Figure 8. Histogram of the total SH power with different crystal lengths: 0.5, 1.0 and 1.5mm.

Figure 9 .
Figure 9. RPM efficiency normalized to QPM efficiency (only parallel output field component with respect to the pump polarization is counted) versus normalized crystal length.For each crystal length, a Monte Carlo simulation was performed using 1000 sets of random crystal structure (dots) to obtain the average value (solid line).Nonlinear coefficient of  14 is used in the QPM condition.