## Abstract

We present a theoretical model on the effects of mechanical perturbations on the output power instability of singly-resonant optical parametric oscillators (SR-OPOs). Numerical simulations are performed based on real experimental parameters associated with a SR-OPO designed in our laboratory, which uses periodically-poled LiNbO_{3} (PPLN) as the nonlinear crystal, where the results of the theoretical model are compared with the measurements. The out-coupled power instability is simulated for a wide range of input pump powers the SR-OPO oscillation threshold. From the results, maximum instability is found to occur at an input pump power of ~1.5 times above the OPO threshold. It is also shown theoretically that the idler instability is susceptible to variations in the cavity length caused by vibrations, with longer cavities capable of generating more stable output power. The validity of the theoretical model is verified experimentally by using a mechanical vibrator in order to vary the SR-OPO resonator length over one cavity mode spacing. It is found that at 1.62 times threshold, the out-coupled idler suffers maximum instability. The results of experimental measurements confirm good agreement with the theoretical model. An intracavity etalon is finally used to improve the idler output power by a factor of ~2.2 at an input pump power of 1.79 times oscillation threshold.

© 2012 OSA

## 1. Introduction

The capability of optical parametric oscillators (OPOs) for the generation of tunable and high power coherent radiation has been extensively investigated for more than four decades. Singly-resonant OPOs (SR-OPOs) are recognized as promising coherent sources for many applications across the optical spectrum from UV to the mid-infrared. Owing to their wide tunability and high output power, SROs have become viable and highly practical tools for many applications, such as gas tracing, medical diagnostics as well as high resolution remote detection of toxic and explosive species [1–3], which are not attainable with conventional laser sources. Because of homogeneous broadening characteristics of parametric gain and the inherent mode competition associated with the cavity structure, a SR-OPO essentially oscillates in a single frequency [4]. However, the main drawback of SROs is the high oscillation threshold, which necessitates high input pump powers. In contrast, doubly-resonant OPOs (DR-OPOs) exhibit much lower oscillation threshold, but generally suffer from very low passive spectral and amplitude stability [5].

The SR-OPO configuration can also generally provide stable frequency and power stability over a sufficient time-scale necessary for long-term spectroscopy while tuning around the target wavelength. However, small fluctuations in the out-coupled idler power can occur, which are primarily due to the inherent susceptibility of the OPO cavity length to environmental perturbations or gain fluctuations throughout the nonlinear interaction in the crystal. The latter is connected to the variation of non-uniform temperature distribution within the nonlinear medium and amplitude fluctuations of the input pump itself. In combination with a stable pump source and careful temperature control of the nonlinear crystal, as well as the precise control of the cavity length through the use of a piezoelectric transducer, instabilities can be significantly reduced [6]. Depending on the experimental conditions and the nonlinear crystal utilized, induced power instabilities of 1.5% to 20% and frequency fluctuations of 30 MHz to a few GHz have been reported [7–9]. As predicted by Kreuzer [10], for a typical SR-OPO the spectral instability is correlated with the oscillation threshold and the maximum fluctuation occurs when the pump power reaches 4.61 times above threshold. Regardless of the pumping level and scheme, Phillips et al. [11] have shown this value can be moved to lower values around the oscillation threshold. The main reason for the SR-OPO output instability can be explained by energy transfer from the oscillating interacting pump, signal and idler modes to their neighboring side bands associated with the cavity structure. Even though the perfect phase-matching condition is established at higher pump powers, side modes are capable of reaching to oscillation threshold, and hence energy can be alternately exchanged between the main mode and the side modes. This can result in a relatively high instability in the SR-OPO output, which can be improved by using a glass etalon inside the SR-OPO resonator, and hence increasing the optical loss for undesirable side modes. Nonetheless, the etalon has a constant insertion loss, which can lead to a reduction of the SR-OPO output power. However, fluctuations in the output spectrum can be partly due to the variation of the cavity length caused by the external perturbations such as environmental vibrations. This results in the oscillation of a new non-phase-matched resonant signal wavelength and consequently to the reduction of the SR-OPO out-coupled power.

In this work, for the first time to our knowledge, we describe a theoretical model and present numerical simulations using real practical values that predict the output power instabilities associated with a typical SR-OPO due to the mechanical perturbations. The simulations results have been further compared with experiment measurements obtained using a SR-OPO based on periodically-poled LiNbO_{3} (PPLN), constructed in our laboratory, where the validity of the theoretical model is verified. It is predicted theoretically that output power instabilities due to small perturbations in the cavity length are maximized at a pumping level of ~1.5 times above oscillation threshold and the effects of output mirror reflectivity as well as the SRO-OPO cavity length on output power instabilities been studied and discussed. In all cases, excellent agreement between the theoretical simulations and experimental measurements is obtained, confirming the validity of our theoretical model. The presented analysis provides very useful guidelines for the attainment of highest output stability from SR-OPOs, which can have important implications for many practical applications in spectroscopy, trace gas sensing, and LIDAR measurements, amongst many others.

## 2. Mechanical vibrations: theoretical model

Theoretical investigation of the mechanical vibrations can be modeled in a simple form if the variation of the SR-OPO cavity length is included in the well-known coupled-wave equations of the second-order nonlinear interaction. Principally, for single-pass optical parametric generation, the equations be written as

with*E*are the electric field amplitudes of pump, signal and idler waves;

_{p/s/i}*d*and

_{eff}*Λ*are the effective nonlinear coefficient and grating period of the PPLN crystal, respectively; and

*ω*and

_{p/s/i}*k*are the respective angular frequencies and wave vectors of the interacting waves.

_{s/p/i}It is often instructive to express the field amplitudes in a more useful form of the intensity amplitude, *a _{j}*, defined as

*|a*, provided that the coupled equations can be readily modified into a simplified form as

_{j}|^{2}= I_{j}= 2 ε_{ο}cn_{j}|E_{j}|^{2}*A*are the amplitudes of pump, signal and idler waves, respectively, normalized to the input pump intensity amplitude

_{p/s/i}= a_{p/s/i}/a_{0p}*a*;

_{0p}*β( = ω*is a dimensionless parameter;

_{s}/ω_{p})*ζ( = z/L*is the normalized length of the crystal;

_{c})*σ( = a*is the coupling parameter with

_{0p}L_{c}ω_{p}d_{eff}(2/n_{p}n_{s}n_{i}ε_{ο}c^{3})^{1/2})*n*as the refractive indices of pump, signal and idler waves, respectively; and

_{p/s/i}*Δs = ΔkL*is the normalized phase-mismatch. To solve the modified coupled equations, it is assumed that the intensity of the input pump follows a Gaussian distribution in the

_{c}*xy*plane [12] as

*E*[

_{p}= E_{0p}exp*-*(

*x*)

^{2}+ y^{2}*/w*], where

_{0}^{2}*w*is the pump beam waist. We further assume that the signal and idler waves are equally present in the quantum noise level at the beginning of the interaction. Therefore, to follow the effect of the instabilities imposed on the output power of the generated idler, one has thus to address two experimental issues:

_{0}- I. A constant phase, which is added to the resonant signal wave in each round-trip of the OPO cavity.

As explained by Yang et al [13], in a typical OPO instabilities are mainly due to the idler beam reflection at the output mirror back in to the crystal, which is commonly the case in doubly-resonant OPOs (DR-OPO) with idler as well as signal feedback. In contrast, in an ideal SR-OPO with zero idler feedback, no additional round-trip phase is imposed on the idler by the output mirror. In this regard, a key advantage of the ring cavity configuration used in the experimental SR-OPO in this investigation, as well as those commonly used in other SRO-OPOs [8], is the loss of the idler photons on the four cavity mirrors, thereby rendering the SR-OPO much less sensitive to the round-trip phase. Therefore, in this situation, the stability of the OPO output power cannot be affected by the travelling phase, and thus it can be excluded from consideration.

We can thus assume that no idler remains in the cavity after a round-trip and the signal is the only resonating wave. Thus, the required boundary condition for the electric field of the signal wave is given by

with*m*is an integer number;

*λ*is the wavelength of the standing signal wave;

_{s}*n*is the crystal refractive index for the signal wavelength;

_{s}*L*is the length of the SR-OPO cavity;

*r*is the reflection coefficient of the cavity mirrors at

*λ*; and

_{s}*ϕ*is the round-trip phase experienced by the signal and determined by the cavity length. Note, however, that for a ring cavity geometry

_{RT}*ϕ*is one half of its value in a standing wave cavity.

_{RT}- II. The possibility that resonance frequency for the signal wave changes due to the variation of the OPO cavity length.

This can result in the resonant signal frequency to shift, however slightly, from the peak of the phase matching gain profile. Consequently, because of the phase mismatch term, *Δk*, appearing in the coupled equations the output power of the SR-OPO will tend to change, and as a result power instabilities are induced.

Whenever mechanical vibrations displace the SR-OPO cavity length by *δL*, in addition to the round-trip phase, the wavelength of the signal jumps to a new value, namely

As long as *δL* is assumed to be less than or equal one cavity mode spacing, the established phase-matching condition is no longer maintained for the new resonating signal wavelength, *λ′ _{s}*, and eventually, it is shifted in frequency from the peak of phase-matching curve and thus suffering lower gain, is recognized as the origin of the SR-OPO gain and the output power drops.

Since the round-trip time of the cavity is much shorter than the time-scale between consecutive variations in the cavity length, it can be assumed that the resonant signal waves have enough time to reach the steady-state condition. As a result, to simulate the SR-OPO power instabilities, our strategy is to use the Runge-Kutta 4 numerical method to solve the coupled-wave equations in the presence of the phase-mismatch parameter, *Δk(λ′ _{s})*, while the boundary condition expressed by Eq. (8) is satisfied for consecutive mismatched signal waves generated by new resonant conditions due to the vibrations. A superfast computer was used for the calculation of large amount of the sequences in the MATLAB environment.

## 3. Mechanical vibrations: simulation results

We start by simultaneously solving Eqs. (5)-(7) and Eq. (10), to evaluate the output power instabilities of a SR-OPO schematically illustrated in Fig. 1 .

For further comparison between theoretical and experimental results, our calculation is performed for actual practical values associated with the SR-OPO experimental setup in our laboratory. Therefore, we assume that the input pump source is an e-polarized beam from a Q-switched Nd:YAG laser and nonlinear interaction is accomplished in a 40-mm-long PPLN nonlinear crystal with a grating period of 30.75 μm. This results in the generation of a signal wavelength of ~1735 nm at a crystal temperature of 150 °C, corresponding to a *β* ~0.6. Most recent Sellmeier equations of PPLN reported in Ref [14]. have been used to calculate the momentum mismatch, *Δk*, required for solving the coupled-wave equations, Eqs. (5)-(7). It is further assumed that the Rayleigh range of the focused pump beam is much greater than the crystal length, and hence its beam waist can be assumed to be fixed throughout the interaction length. In the calculation, the beam waist of the pump is kept constant at 50 μm equal to the real experimental value. We define the instabilities, *Δη*, as the relative deviation of the SR-OPO conversion efficiency, *η( = P _{i}/P_{p})*, from its initial value as the cavity length fluctuates between

*L*and

*L + δL*. Therefore, we can write

*Δη*as

Before the vibrations are investigated however, we must note that *Δη* is a reflectivity dependent parameter, which is dictated by the boundary condition expressed in Eq. (8). Figure 2
shows the calculated variation in *Δη* as a function of pump power above threshold, for different signal reflectivity values of the SRO-OPO out-coupling mirror, M_{2}.

As can be seen from the figure, output power instability occurs at different multiples of the oscillation threshold for different reflectivity values of M_{2}, and it tends toward the identical maximum value of 100% for all reflectivities. This is because the cavity length is assumed to change by the simulated vibrations within approximately one cavity mode spacing or less, and the phase-matched signal is instantly forced to jump to the non-phase-matched *λ′ _{s}*. These results have been confirmed by the subsequent experimental results described in Section 4 and presented in Fig. 9. Furthermore, as is clear from Fig. 2, depending on the output mirror reflectivity and length of the SR-OPO cavity varying by

*δL*, the extracted enegry can be converted back to the input pump, provided that the

*e*term in the coupled-wave equations is potentially capable of reaching a negative value at a certain number of times above the threoshld after multiple round-trips, and hence the SR-OPO output power drops to zero, corresponding to 100% of instability.

^{iΔkz}It is also evident that for the highest output reflectivity of 99.9%, as used in our experiments, the peak instability is displaced toward the higher pumping ratio of 1.5 times threshold. On the other hand, for reflectivities higher than 95%, the instabilities are monotonically moved towards the lower multiples of pump power above threshold. Indeed, the instability peak width increases while the reflectivity of the M_{2} mirror is raised up to 99.9%. One physical explanation is that at higher reflectivities more signal photons are recirculating inside the cavity, and hence the SR-OPO gain is less sensitive to the variation in pumping level above threshold. On the other hand, at lower M_{2} reflectivities this sensitivity is increased, owing to the reduction of the gain-to-loss ratio. The calculated variation of *Δη* for input pump powers beyond 3 times above threshold, for the different reflectivity values of M_{2}, are shown in Fig. 3
, confirming this behavior.

The plot confirms that for powers above 3 times threshold, the SR-OPO output power instabilities are below 0.4%, and as the input power increases up to 10 times the threshold, output instability rapidly declines to negligible values towards ~0.1% for all reflectivities of M_{2}. The oscillating behavior in the declining instability curve is related to the non-zero wave-vector mismatch associated with the new resonating signal wavelength, *λ′ _{s}*, appearing in the coupled-wave equations. However, we believe that the idler output power instability is mainly due to mechanical vibrations of the experimental setup, whereas other sources such as pump frequency jitter and noise arising from thermal fluctuations have minor effect and can be neglected. As discussed earlier, the effect of vibrations on the length of the SR-OPO resonator can be considered through calculation of the new resonance condition for the signal wave, which in turn changes the required phase-matching condition throughout the coupled-wave equations, Eqs. (5)-(7). In Fig. 4
, the instability parameter,

*Δη*, is computed for different resonator lengths constrained by mechanical vibrations and the input pump is simultaneously scanned over the multiples of the required SR-OPO threshold, while

*δL*varies by less than one cavity mode spacing for each length.

In order to compare the theory and experiment, calculation is performed for the out-coupling mirror reflectivity of 99.9%. As can be seen from the figure, the maximum instability again peaks at the same value of ~1.5 times above threshold for all cavity lengths, and declines to negligible values with increasing pumping ratios. In addition, the instabilities are suppressed more rapidly for longer cavity lengths. This can be correlated again with the obtained experimental results described in Section 4 and presented in Fig. 11, where the SR-OPO output power fluctuates between its maximum value and zero owing to the jumping between *λ _{s}* and

*λ′*modes. Figure 5 indicates the same calculation at higher values of pump power above threshold to provide better interpretation. It is evident from Fig. 5 that the contribution of mechanical vibrations to the output instability is more significant in this case compared with the results shown in Fig. 3, but with the similarity in the oscillation behavior. Furthermore, for shorter cavity lengths the idler power is more unstable than for longer cavities, which again can be associated with mechanical vibrations and subsequent cavity length variation, provided that the non-phase-matched signal wavelength,

_{s}*λ′*, is more likely resonating inside the resonator. As a result, it can be concluded that the ouput of the ring-cavity SR-OPO with longer resontor length is more stable than that with a shorter cavity. In other words, the output power stability can be enhanced by minimizing the small deviations in

_{s}*δL*, which in turn implies the reduction in axial mode spacing of the resonant signal wave (or free-spectral range of the cavity), with the SR-OPO is operating well above the threshold. Figure 6 illustrates the variation of the instability with

*δL*for different output mirror reflectivity values, while the input pump power is kept constant at 3 times above threshold.

The OPO instability is highly susceptible to the deviation amplitude, *δL*. However, as *δL* is increased the output instability rises strongly, with the highest increase for the maximum output mirror reflectivity of 99.9%. In Fig. 7
, the output power instability is calculated for various ratios of pump power above threshold while *δL* is scanned over the same range and the reflectivity of the out-coupling mirror, M_{2}, is assumed 99.9% in the simulation.

As evident in the figure, again the idler output power instability increases strongly with the deviation in *δL* for all pump powers above threshold. In the special case of 3 times above threshold, the instabilities are stronger, since the pump power level is closer to the instability peak at 1.5 times above the threshold (see Figs. 4 and 5), but the fluctuations can be suppressed to <0.2% for input pump powers >7 times above threshold.

## 4. Experimental results

The experimental setup of our SR-OPO apparatus used to verify the simulation results presented in the previous section is shown in Fig. 8
. A Q-Switched Nd:YAG laser delivering up to 4W of average power in single-mode output with pulses of 170 ns duration at 10 kHz repetition rate is used as pump source for the SR-OPO based on PPLN as the nonlinear crystal. The crystal was 40-mm-long, as used in our modeling, 10-mm-wide and 0.5-mm-thick. The utilized grating period of the crystal was Λ = 30.75 μm and its faces were AR-coated at all the interacting wavelengths (R<1%). To generate the experimental data, a commercial oven was used to vary the crystal temperature from 100°C to 200°C with a stability of ± 0.1 °C. The crystal was then placed at the beam waist of a symmetric bow-tie ring resonator consisting of four mirrors, all antireflection (AR)-coated over the range 2200-5000 nm and coated for high reflectivity for signal wavelengths over 1450-2100 nm. The two spherical mirrors, M_{1} and M_{2}, with radius of curvature of *r* = −20 cm were placed at a distance of 22 cm, while the two plane mirrors, M_{3} and M_{4}, were 25 cm apart to form SR-OPO resonator of ~90-cm length. The required pump beam polarization was maintained using a combination of a half-wave plate and a polarizing beam-splitter (PBS) cube before entry into the SR-OPO cavity. The Q-switched Nd:YAG laser beam was focused into the PPLN crystal using a lens with a focal length of *f* = 20 cm, providing an optimum spot size of about *w*_{0} = 50 μm at the center of the crystal, corresponding to a focusing parameter of *ξ*~1.2. The temperature of the crystal was kept constant at 150 ± 0.1°C to generate a signal wavelength of ~1735 nm, providing a dimensionless parameter, *β*~0.6, as it was assumed in the simulation. The generated idler output power was measured using a thermal detector (Thorlabs PM100 & DCMM) after a 2-mm-thick germanium plate used as cut-off filter.

To verify the simulated results presented earlier, we introduce the standard deviation of the out-coupled idler power as a measurable parameter of SR-OPO instability, defined as

where*N*is the number of measurements, with

*P*and $\overline{P}$ the instantaneous and average output power of the idler radiation, respectively. For increased accuracy and maximum experimental consistency for comparison with the modeling results, output power measurements were performed over an interval of 10 minutes at sampling rate of 17 Hz, corresponding to a sample acquisition of about

_{m}*N*= 10

^{4}.

In order to measure the effect of mechanical vibrations on the instability of the out-coupled idler power, and hence to correlate them with the simulated results, a 12000 rpm electrical vibrator was placed on the optical table close to the SR-OPO setup. The device was capable of varying the SR-OPO resonator length over a small range (few tenths of μm) in order to induce instabilities in the out-coupled idler power. The idler output power was then simultaneously monitored while the Nd:YAG pump power was gradually increased to about 4 W, corresponding to 4.5 times above experimentally measured threshold of the SR-OPO. The increase in pump power was performed by fixing the Nd:YAG output at its maximum value and varying the polarization ratio of the pump beam using the combination of the half-wave plate and a PBS cube. The results of power instability measurements are illustrated in Fig. 9 .

As can be seen from the plot, the out-coupled idler output power instability peaks when pumping at 1.62 times above the SR-OPO threshold, and is also identical for both with and without the vibrator. Although in both cases, the instability curves exhibit similar trend, the power fluctuation in the presence of vibration is stronger for all input pump powers, as intuitively expected. At pump powers beyond 1.62 above threshold, the output power instabilities are drastically reduced in both cases.

The most important feature of the measurements shown in Fig. 9, however, is the significant increase in the maximum instability while the SR-OPO resonator is vibrating over very small range of microns. This can be compared to the theoretical results deduced in Fig. 2, where the maximum power instability of the idler wave occurs at 1.5 times above the threshold, assuming the signal reflectivity of the out-coupled mirror, M_{4}, as 99.9%, the same reflectivity as in the designed experimental SR-OPO. The obtained results indicate very good agreement between experiment and theory. Similar results have been reported in related publications [11, 13],but to our knowledge none of the earlier reports have discussed the occurrence of a maximum point of the instability.

In order to enhance the SR-OPO output power stability, we used an uncoated etalon inside the cavity and recorded the out-coupled idler power instability for a range of pump powers up to 4 times above threshold. The intracavity etalon was 1-mm-thick fused silica with refractive index of 1.44 (FSR = 104 GHz) and transmittance of 85% (finesse = 1.43). The insertion loss resulting from the etalon was kept on its minimum value during the measurement by adjusting the angle of rotation at which the peak of etalon transmission coincided with the phase-matched signal and idler wavelengths corresponding to the maximum idler output power. The results of the instability measurements are plotted in Fig. 10 .

As can be observed from the plot, the significant feature of using the intracavity etalon is the remarkable decrease in the idler output instability by a factor of about 2.2 when the input Nd:YAG power is 1.79 times above the SR-OPO threshold. In fact, the etalon acts as an interferometric filter to suppress some of undesirable side modes located in the neighborhood of the resonant signal mode at the peak of the phase-matching curve. The measurements thus confirm that the use of an intracavity etalon not only enhances the frequency stability of the SR-OPO, as is well known, it also offers the great advantage of substantially improving the output power stability of the device, which is very important for the applications requiring infrared sources with long-term stability such as trace detection of very low concentration species in medicine. Figure 11
shows the idler power fluctuations over 10 minutes for the Nd:YAG pumping level of 1.62 and 3 times the threshold. As can be seen from Fig. 11, when the SR-OPO operates far above threshold, power instability of less than 5% is measured. However, near the maximum instability when pumping at 1.62 times above threshold, as shown in Fig. 10, power fluctuations around the mean value are very strong and reach a maximum value of nearly 100%. The main reason for the SR-OPO power fluctuations is the alternative jumping between the phase-matched and non-phase-matched signal wavelengths, *λ _{s}* and

*λ′*, respectively, during the measurement. This can be compared with the presented theoretical model, which for a reflectivity of 99.9% for the out-coupling mirror, M

_{s}_{2}, predicts maximum instability of 100% at 1.5 times the threshold, as shown in Fig. 2 and Fig. 4. This again underlines the excellent agreement between the predicted theory and experimental results obtained in this work, confirming the validity of our numerical model.

## 5. Conclusions

The effects of the mechanical perturbations on the output power instability of SR-OPOs have been theoretically modeled and discussed, and experimentally investigated. Using numerical study of the coupled-wave equations with real boundary conditions of an experimental SR-OPO in our laboratory, we have shown that the out-coupled idler wave exhibits power instability, which is dependent on the input pump power level relative to oscillation threshold, with maximum instability occurring at 1.5 times above threshold. Using the theoretical model, we have also studied the effects of cavity length and output mirror reflectivity on the SRO-OPO power instability, as well as perturbations to the cavity length over distances comparable to a cavity axial mode for a fixed length of the SR-OPO resonator. We have further compared the results of the theoretical simulations with experimental measurements using a PPLN-based SR-OPO constructed in our laboratory. We have investigated the SR-OPO idler output power instability for different levels of input Nd:YAG pump power relative to oscillation threshold, where we have found excellent agreement between the theoretical model and experimental data. We also deployed a mechanical vibrator to induce variations in the SR-OPO resonator length up to one cavity mode spacing and investigated its influence on the output power stability. It was experimentally observed that the maximum instability occurs at 1.62 times above the threshold, again confirming very good consistency between the theoretical model and experimental results. It is further observed that out-coupled power instability can be reduced by a factor of 2.2 if an intracavity etalon is used inside the SR-OPO resonator. To the best of our knowledge, the output power instability of a SR-OPO due to mechanical perturbations has been theoretically modeled and numerically simulated in this work, for the first time, and experimental results have confirm the excellent validity of the presented model.

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