We present theoretical and experimental study of a continuous-wave, two-crystal, singly-resonant optical parametric oscillator (T-SRO) comprising two identical 30-mm-long crystals of MgO:sPPLT in a four- mirror ring cavity and pumped with two separate pump beams in the green. The idler beam after each crystal is completely out-coupled, while the signal radiation is resonant inside the cavity. Solving the coupled amplitude equations under undepleted pump approximation, we calculate the maximum threshold reduction, parametric gain acceptance bandwidth and closest possible attainable wavelength separation in arbitrary dual-wavelength generation and compare with the experimental results. Although the T-SRO has two identical crystals, the acceptance bandwidth of the device is equal to that of a single-crystal SRO. Due to the division of pump power in two crystals, the T-SRO can handle higher total pump power while lowering crystal damage risk and thermal effects. We also experimentally verify the high power performance of such scheme, providing a total output power of 6.5 W for 16.2 W of green power at 532 nm. We verified coherent energy coupling between the intra-cavity resonant signal waves resulting Raman spectral lines. Based on the T-SRO scheme, we also report a new technique to measure the temperature acceptance bandwidth of the single-pass parametric amplifier across the OPO tuning range.
©2013 Optical Society of America
The development of coherent optical sources capable of providing multiple wavelengths with wide and arbitrary tuning and high power can be of great interest for a wide range of applications in spectroscopy, microscopy, frequency metrology, short- and long-wavelength nonlinear conversion, and THz generation, amongst many. Optical parametric oscillators (OPOs) have long been recognized as versatile sources of widely tunable radiation in spectral regions inaccessible to lasers [1, 2]. In the conventional configuration, the OPO produces a pair of signal and idler output waves from an input pump wave, where the three wavelengths are subject to energy conservation (ωp = ωs + ωi, where ωp, ωs, and ωi are the pump, signal and idler frequencies, respectively) and phase-matching (kp = ks + ki, where kp, ks and ki are the wave vectors of pump, signal and idler, respectively). Wavelength tuning across broad spectral regions is achieved by varying the phase-match condition to fulfill energy conservation for a new pair of signal and idler waves. On the other hand, due to the coupling of the signal and idler waves through energy conservation and phase-matching, the generation of truly arbitrary signal-idler wavelength pairs with independent tuning control is not possible. This limitation can have consequences, for example, in the generation of closely spaced wavelengths, where the only solution is to tune the OPO to near degeneracy, with the associated difficulties such as broad spectral bandwidth or output power and frequency instabilities .
To overcome the constraints of energy conservation and phase-matching, earlier attempts have included a dual-crystal continuous-wave (cw) OPO  or double-pass-pumping of a single-crystal pulsed OPO , providing two different signal and idler wavelength pairs with arbitrary tuning. In the former scheme , both the crystals are placed in series and the undepleted pump from the first crystal is used as pump for the second crystal. The air gap between the crystals in this scheme is an important parameter to avoid relative phase shift among the interacting waves for efficient parametric generation. In the latter scheme , the undepleted pump after the first pass is used to pump the same crystal in the second pass. However, in both techniques, high pump depletion in the first crystal or the first pass, generating the first signal-idler pair, deteriorates the pump beam quality available in the subsequent crystal or pass. As a result, the overall OPO performance in terms of threshold and output power in the second crystal or second pass, producing the second signal-idler pair, is degraded compared to that in the first crystal or first pass. Additionally, in these schemes, thermal effects and crystal damage issues at higher pump powers are major challenges to overcome. We recently reported a novel and generic approach for the generation of two signal-idler wavelength pairs with truly independent and arbitrary tuning, which are unbound by energy conservation and phase-matching, using a simple design based on a compact four-mirror ring-cavity cw OPO . In this scheme, the two signal-idler wavelength pairs can be independently controlled to provide any arbitrary pair of wavelengths within the available OPO tuning range. Both wavelengths are resonant within the same cavity, thus providing high circulating intensities at the two wavelengths, and the beams have the same optical power and exhibit similar output stability. The cw OPO is based on two identical MgO:sPPLT crystals (30-mm-long), each pumped separately at 532 nm by a single pump laser. We have shown frequency separation of two arbitrary signal wavelengths down to 0.55 THz, demonstrating the potential of the system for tunable THz generation. Moreover, while operating both crystals under the same phase-matching condition, coherent coupling between the circulating signal waves generated by the two crystals results in threshold reduction, enabling OPO operation at reduced pump powers.
Here we present the theoretical framework for such a scheme, explaining the maximum threshold reduction, gain acceptance bandwidth and closest possible arbitrary wavelength pair separation that can be generated, and compare the analysis with experimental results. We have experimentally verified the performance of such a scheme at higher overall pump power, without inducing any crystal damage, resulting total output power of 6.5 W for 16.2 W of green pump power. We have confirmed intra-cavity cw parametric amplification of the resonant signal from one crystal in the other crystal, confirming the generic nature of the scheme, implying that the technique can be deployed with different combinations of crystals and pump wavelengths. We also report a new technique to measure the cw parametric gain acceptance bandwidth of the OPO. For completeness of the theoretical and experimental analysis of the two-crystal OPO, we have included some of the results reported in . The present report is divided into six sections, comprising introduction, theoretical analysis, numerical simulation, experimental configuration, experimental results and conclusions.
2. Theoretical analysis
The generic configuration of the two-crystal cw OPO is shown in Fig. 1.The OPO is arranged in a ring resonator with two crystals X1 and X2 of lengths L1 and L2, respectively, located at the two focii of the cavity. The OPO is a singly resonant oscillator (SRO), where only the signal waves generated by the individual crystals are resonant inside the cavity. In the generic design of Fig. 1, the crystals X1 and X2 are pumped by two separate pump beams of power Pp1 and Pp2, respectively, and exit the cavity in a single-pass. However, we will also consider the use of a single pump beam, Pp1, for both crystals X1 and X2, corresponding to the schemes used in earlier reports [4, 5]. Let us consider z as the propagation direction, with the entrance and exit faces of the crystal X1 designated as z1 and z2, respectively, and z3 and z4 as the entrance and exit of the crystal X2 respectively. Using coupled-wave equations , we can investigate the main operating features of the device including parametric gain enhancement, threshold reduction, and gain acceptance bandwidth under two different schemes characterized by different boundary conditions, as described in section 2.1. All results for dual-crystal SRO (D-SRO) and two-crystal SRO (T-SRO) configurations (defined in section 2.1) are compared to that of the conventional single-crystal SRO (S-SRO).
2.1 Coupled-wave equations
To understand the performance of SRO systems, we calculate the parametric gain using Maxwell’s wave equations with nonlinear polarization P(2) = ε0χ(2)A2 as the source term. Throughout the calculations, we consider the pump, signal and idler fields, Ap,s,i, to be plane waves, and for simplicity we also neglect absorption losses and pump depletion. From Maxwell’s equations, one can derive the coupled wave equations describing the evolution of signal and idler field amplitudes within the crystals, as 
Using the treatment of , the signal and idler field amplitudes at the exit of the crystals X1 and X2 can be written as
We analyze two SRO configurations: (a) Dual-crystal SRO (D-SRO), where crystals X1 and X2 are pumped by a single pump beam, Pp1, and the transmitted pump as well as the signal and idler fields generated by X1 are present at the input to X2; and (b) Two-crystal SRO (T-SRO), where X1 and X2 are pumped separately by Pp1 and Pp2, respectively, and only the signal field generated by X1 as well as Pp2 are present at the input to X2, with the idler generated by X1 exiting the cavity in a single pass. In both cases, the signal field at the entrance of crystal X1 is finite, As1(0), and the idler field at the entrance of X1 is zero, Ai1(0) = 0. We neglect pump depletion and all signal losses due to the cavity mirrors.
(a) Dual-crystal SRO (D-SRO)
Both signal and idler fields generated in crystal X1 are the input fields at the entrance of X2, i.e., As2(0) = As1(L1), Ai2(0) = Ai1(L1). The pump for crystal X1, Pp1, is also used to pump crystal X2, i.e. Ap2 = Ap1 (negligible pump depletion). No external pump is provided to crystal X2. The pump and idler fields after crystal X2 are out-coupled, while signal field is resonant inside the cavity to maintain SRO condition.
Under the above boundary conditions and proper substitution of the field amplitudes of the crystal X1 in Eqs. (3) and (4), the signal and idler field amplitudes at the exit of crystal X2 can be represented as
(b) Two-crystal SRO (T-SRO)
The signal field at the exit of crystal X1 is considered as the input signal field at the entrance of crystal X2, i.e., As2(0) = As1(L1). The idler field is growing from zero, i.e., Ai2(0) = 0. A new pump field, Ap2, is incident on X2. The pump and idler fields are out-coupled after each crystal, while signal field is resonant inside the cavity to maintain SRO condition.
2.2 Parametric gain
We define the net power gain of the signal field to beEq. (3) for a single crystal of length L, into Eq. (9), the single-crystal gain, GS, can be represented asEq. (5) and Eq. (7) in Eq. (9), the simplified expression for the double-crystal gain (GD) can be written as
2.3 Parametric gain enhancement
The maximum gain is available for perfect phase-matching, Δk = 0. Comparing Eqs. (11) and (12) with Eq. (10), we can conclude that for two-times longer crystal length, the D-SRO and T-SRO have four- and two-times gain enhancement, respectively, as compared to the single-crystal gain. On the other hand, for the same crystal lengths, the T-SRO has lower gain (by a factor of two) than D-SRO, due to the out-coupling of the idler after crystal X1. In T-SRO, the idler field at the input of crystal X2 starts from zero and adjust its phase depending upon the input phases of the signal and pump fields for optimum energy flow from the pump to the generated signal and idler. As a result, the idler out-coupling after the first crystal in T-SRO overcomes the requirement of constant relative phases among the signal, idler and pump from the output of the first crystal to the input of the second crystal for energy flow from pump to the signal and idler, as required for D-SRO.
2.4 Threshold reduction
The oscillation threshold for a SRO, under perfect phase-matching (Δk = 0), is reached when the single-pass parametric gain experienced by the resonant wave equals its total round-trip loss. In case of the S-SRO, the threshold condition can be represented as (ΓL)2 = αs, under the assumption that ΓL≤1, αs<<1 and αi≈1. Here, αs and αi are the losses of the resonant (signal) and non-resonant (idler) fields respectively. Similarly, the threshold condition for the D-SRO and T-SRO are given by 4(ΓL)2 = αs and 2(ΓL)2 = αs, respectively. Using the formula for Γ, and some simple algebra, the threshold pump power of the D-SRO and T-SRO relative to the S-SRO are found to be, and, respectively, whereis the threshold pump power of the S-SRO. Hence, due to the doubling in the crystal length, the threshold pump power for the D-SRO and T-SRO are reduced by 75% and 50%, respectively, relative to the S-SRO. It is to be noted that the threshold reduction is possible only if the fields generated by crystal X1 are coherently coupled to those generated by crystal X2.When both the crystals are operating under two different phase-matching conditions (Δk1≠Δk2, Δk1 = 0, Δk2 = 0) with two different pair of signal and idler wavelengths, both the D-SRO and the T-SRO behave as two separate S-SROs and their threshold conditions are determined separately.
2.5 Parametric gain acceptance bandwidth
The parametric gain reaches its maximum value under perfect phase-matching. Considering identical crystals with equal lengths (L1 = L2 = L) and under perfect phase-matching (Δk1 = Δk2 = Δk = 0), the gain bandwidths (full width at half maximum, FWHM), BD and BT, of the D-SRO and the T-SRO can be calculated from Eqs. (11) and (12), respectively. Comparing Eqs. (11) and (12) with Eq. (10), one finds that BD = BS/2 and BT = BS, where BS is the gain bandwidth of the S-SRO. It is trivial that the gain bandwidth of the D-SRO will be narrowed by one-half of that of the S-SRO, as the D-SRO has a total crystal length twice (2L) that of the S-SRO, whereas the T-SRO has the same bandwidth as the S-SRO. Given that a narrower gain acceptance bandwidth leads to higher SRO output instability, the D-SRO will thus exhibit increased instability due to small fluctuations in crystal temperature and laboratory environment. On the other hand, in addition to the enhancement of parametric gain due to twice crystal length, the T-SRO has the same bandwidth as the S-SRO, resulting in higher output power and frequency stability compared to the D-SRO.
3. Numerical simulations
The nonlinear crystal used throughout the paper is MgO-doped stoichiometrically grown periodically-poled lithium tantalate (MgO:sPPLT) with a single grating of period, Λ = 7.97 μm. The crystal length of the S-SRO is L = 30 mm, while both the D-SRO and T-SRO have a total crystal length (L = 30 + 30 mm = 60 mm) twice to that of the S-SRO. Both the crystals have effective nonlinearity of deff = 10 pm/V. The refractive index variation with wavelength and temperature is obtained using the Sellmeier equations for the material . We also included thermal expansion of the crystals in the calculations. The pump laser wavelength is 532 nm.
3.1 Parametric gain bandwidth of the S-SRO at different crystal temperatures
Using Eq. (10), we have calculated the parametric gain bandwidth of the S-SRO (with crystal length, L = 30 mm) at three different crystal temperatures across the SRO tuning range (65°C-240°C ), with the results shown in Fig. 2. As evident from Fig. 2(a), the S-SRO has gain bandwidth (FWHM) of Δλs≈1.24 nm (Δλi≈1.98 nm, ΔT = 0.9°C) at crystal temperatures T = T1 = T2 = 87.5°C, corresponding to a signal wavelength of λs = 962.66 nm close to the signal-idler degeneracy at 1064 nm. As evident from Fig. 2(b) and 2(c), operating the S-SRO away from the degeneracy by increasing the crystal temperature results in reduced gain bandwidth of Δλs≈0.66 nm (Δλi≈1.39 nm, ΔT≈0.8°C) and Δλs≈0.41 nm (Δλi≈1.19 nm, ΔT≈0.7°C) at crystal temperatures of T = 139°C (λs = 906.14 nm) and 192°C (λs = 866.41 nm), respectively. This is obvious since the S-SRO signal wavelengths away from degeneracy have higher dispersion, and the phase-matching depends on the refractive index of the material. However, broader values of ΔT near degeneracy make the S-SRO performance in terms of power and frequency stability more immune to moderate temperature instabilities of the crystal, permitting the use of less elaborate and lower-cost temperature controllers. On the other hand, lower values of ΔT (Δλs) facilitate the generation of two arbitrary wavelengths with very close proximity in D-SRO and T-SRO configurations. Figure 2 shows the same normalized values for the maximum gain across the tuning range. However, the absolute value of maximum gain decreases with S-SRO operation away from the degeneracy, and can be simulated with the inclusion of the gain reduction factor. The factor is given by (1-δ2), where δ (0 ≤ δ ≤ 1) is the degeneracy factor defined through 1 + δ = λo/λi, 1-δ = λo/λs, and λo = 2λp is the degenerate wavelength, with λp, λs and λi being the pump, signal and idler wavelengths, respectively .
3.2 Parametric gain enhancement and gain acceptance bandwidth
Inserting the signal field equations for the S-SRO, Eq. (3), for the D-SRO, Eq. (5), and for the T-SRO, Eq. (7), into Eq. (9), we have numerically calculated the variation of the parametric gain as a function of phase-mismatch. The calculations have been performed under the boundary conditions Ai1(0) = 0, initial signal field As1(0), and undepleted pump approximation, with results shown in Fig. 3. We have considered that all the parameters have same values for both the crystals. For the low gain approximation (ΓL~0.1), typical values of the different parameters are pump powers (P1 = P2 = 6 W), pump beam waist radius (wp = wp1 = wp2 = 150 μm), and crystal temperature (T = T1 = T2 = 87.5°C) corresponding signal (idler) wavelength 962.66 nm (1189.18 nm). As evident from Fig. 3, at low gain (ΓL<<1), the parametric gain of the D-SRO (red line) and T-SRO (blue line) are four-times and two-times that of the S-SRO (black line) under perfect phase-matching (ΔkL = 0), as predicted by the analytical formulas given by Eqs. (11) and (12), respectively. At high gain (ΓL≥1), the gain enhancement factor deviates from the predicted values due to the negligible pump depletion approximation in the present analysis. However, solving the coupled wave equations with pump depletion, one can achieve similar results. Comparing the gain acceptance bandwidths (FWHM), it is evident that the D-SRO has a gain bandwidth one-half of that of the S-SRO, whereas the gain bandwidth of T-SRO is equivalent to that of the S-SRO. Despite the reduced gain in the T-SRO as compared to D-SRO, the wider gain bandwidth (larger temperature acceptance bandwidth) of T-SRO provides better tolerance to fluctuations in the crystal temperature, reducing the constraints on the stability of the oven and temperature controller without significant impact on the SRO output stability. On the other hand, as the idler is completely transmitted after the first crystal, it is not necessary to control the relative phases among the interacting waves at the entrance of the second crystal to maintain forward energy flow from the pump to the generated waves. The idler in the second crystal is generated from zero initial power and its phase is automatically adjusted to achieve highest parametric gain in the second crystal. Thus, the T-SRO has all the advantages of an S-SRO including wider gain bandwidth, higher output stability, no requirement of control of the input phases, and additionally higher parametric gain and lower operation threshold. The scheme is also generic and can be used for N number of crystals with proper cavity designs. It is to be noted that the optimum gain enhancement in D-SRO and T-SRO above that in the S-SRO will be possible through coherent energy coupling, but this is only possible if the parametric waves generated in the first crystal interfere constrictively with those generated in the second crystal, and vice versa. This coherent energy coupling occurs when the signal and idler waves generated by both crystals have the same temporal phase and are at same wavelengths under the phase-matching condition (Δk→0). Under such condition, crystals X1 and X2 of D-SRO and T-SRO can be considered as a single crystal of length (L1 + L2). In case of D-SRO, both the signal and idler have to satisfy this phase coherence condition for coherent energy coupling, whereas in the T-SRO only the resonant (signal) wave has to satisfy the condition, since the idler wave is non-resonant and exits the cavity after each crystal. This further demonstrates the flexibility of the T-SRO scheme.
3.3 Effect of gain overlap in D-SRO and T-SRO
To study the effects of gain overlap in D-SRO and T-SRO, we calculated the gain for the two configurations as a function of signal wavelength, while maintaining crystal X1 at constant temperature (T1 = 91°C) and varying the temperature, T2, of crystal X2. The results are shown in Fig. 4. The temperature difference between the two crystals is defined as ΔTdiff = T1-T2. For comparison, we have reproduced in Fig. 4 the gain variation of the S-SRO with signal wavelength at ΔTdiff = T1-T2 = 0, similar to the result shown in Fig. 2. As can be seen, the S-SRO (black curve) and T-SRO (blue curve) have similar gain acceptance bandwidth (Δλs≈1.3 nm), whereas D-SRO (red curve) has gain bandwidth one-half to that of the S-SRO, as explained in section 3.1. A small change in the crystal temperature from 91°C to 92°C shifts the gain maximum of the S-SRO from 957.6 nm and 956.3 nm, resulting in a signal wavelength shift of ΔλS-SRO≈1.3 nm (ΔνS-SRO = 0.42 THz), which is equal to the gain acceptance bandwidth (Δλs≈1.3 nm) of the S-SRO, as shown by the black curves in Fig. 4(b). However, for the crystals at slightly different temperatures, T1 = 91°C and T2 = 92°C with ΔTdiff = 1°C, (Δk1≠Δk2, Δk1 = 0, Δk2 = 0), the gain of the individual crystals become overlapped, resulting three main maxima in the D-SRO each separated by ≈0.8 nm (red curve, Fig. 4(b)). The two side maxima correspond to the individual crystal temperatures and the central maximum corresponds to the gain coupling between the crystals, resulting in a wider gain bandwidth of ≈2.1 nm. Consequently, the signal waves generated by the individual crystals in the D-SRO at a wavelength difference of ΔλD-SRO cannot be separated as long as ΔλD-SRO is much greater than the S-SRO gain bandwidth. The side maxima are also shifted by 0.1°C with respect to the main maximum in the S-SRO due to the influence of the secondary maxima of the S-SRO. However, in case of T-SRO, the gain has two maxima corresponding to the individual crystal temperatures with a dip in between, indicating the possibility of operating the T-SRO with two closely spaced wavelengths with difference as small as the gain bandwidth of the individual crystals (S-SRO) without any gain competition, as is the case in the D-SRO. It is also to be noted that both maxima in the T-SRO gain profile exactly match the S-SRO maxima without any shift due to the secondary maxima of the S-SRO, as is the case for the D-SRO. With further increase in the difference in crystal temperature, ΔTdiff = 2°C (T1 = 91°C, T2 = 93°C), the gain of the individual crystals are completely separated in both the D-SRO and T-SRO configurations, as shown in Fig. 4(c), resulting in independent oscillation conditions for the two crystals, thus generating two distinct wavelengths. From this study, it is clear that the generation of two distinct wavelengths with separation, Δλ, can be possible if Δλ is greater than (or equal to) the gain bandwidth of the individual crystals in the D-SRO (T-SRO) configuration.
Based on the above analytical and numerical results, it is clear that the D-SRO has superior performance in terms of higher maximum gain and lower threshold than the T-SRO scheme, but is inferior with regard to gain bandwidth. However, as compared to T-SRO, the D-SRO configuration suffers from disadvantages of cumulative thermal effects as well as optical damage in the individual crystals at higher pump powers, the requirement for the maintenance of constant relative phases among the interacting waves between the two crystals, and other limitations, as summarized in Table 1. Additionally, the pump depletion in the first crystal degrades the performance of the second crystal, making the D-SRO imbalanced when operating at two different wavelengths. Because of its many advantages, we have focused on experimental studies on the T-SRO and have verified the performance of this scheme with regard to the various operating parameters derived in our calculations. The results of these studies are presented in section 5.
4. Experimental configuration
The experimental configuration of the T-SRO is shown in Fig. 5. The setup has been slightly modified from the earlier arrangement  to generalize the T-SRO scheme. The T-SRO is configured in a compact symmetric ring cavity comprising four concave mirrors, M1-M4, of the same radius of curvature (r = 10 cm). All mirrors are highly reflecting (R>99%) for the signal (890-1000 nm), while highly transmitting (T = 75%-95%) for the idler (1100-1400 nm) and pump (T = 97% at 532 nm). In the present study, we have used two identical 30-mm-long MgO:sPPLT nonlinear crystals (X1 and X2) containing a single grating period, Λ = 7.97 μm . The crystals are housed in separate ovens with temperature stability of ± 0.1°C. Due to different heating configurations [10, 11] used for crystal X1 (closed top) and X2 (open top), the crystals have a temperature offset of ~2.5°C (T1 = T2-2.5°C, where T1 and T2 are the temperatures of crystal X1 and X2, respectively). The crystals are placed at the two focii of the cavity, with X1 between the mirrors M1 and M2 and X2 between M3 and M4. The crystal faces are AR-coated for the signal (R<0.5%) and pump (R<0.5% at 532 nm), with varying transmission for the idler across the tuning range (T = 85-99%). The crystals are pumped separately with pump, P1, for X1, and pump, P2, for X2. Both pumps can be derived either from a single input laser or two different lasers. In the present experiment, we have used a single pump laser and divided into two pump beams, P1 and P2, using a beam-splitter. The input powers to X1 and X2 are controlled using two separate power attenuators comprising a half-wave-plate (λ/2) and a polarizing beam-splitter cube (PB1 for X1 and PB2 for X2). The second half-wave-plate (λ/2) in each pumping arm is used to yield the correct input polarization relative to the crystal orientation for perfect phase-matching. Using two identical lenses, L1 and L2 (f = 15 cm), the pump beams are focused to the same beam radius of wop~31 μm at the center of X1 and X2 , corresponding to a focusing parameter, ξ = 1. The focusing parameter is defined as ξ = l/bp, where l is the length of the crystal, and bp = kwop2 is the confocal parameter of the pump, with k = 2πnp /λp, where np, λp, and wop are the refractive index, wavelength, and waist radius of the pump beam inside the crystal, respectively. The T-SRO cavity provides a signal beam waist wos~41 μm at 900 nm, resulting in optimum spatial overlap in both crystals (bs = bp) . Both mirrors, M2 and M4, transmit the pump and idler radiation generated in the respective crystals, X1 and X2. The output idler beams are separated from the residual pump and the leaked signal beams using the mirrors, M5. The total optical length of the cavity is 56 cm, corresponding to a free-spectral range of 535 MHz. The fundamental pump source is a 10 W, frequency-doubled, cw diode-pumped Nd:YVO4 laser at 532 nm .
5. Results and discussions
5.1 Parametric gain bandwidth across the tuning range of the T-SRO
We have experimentally determined the parametric gain bandwidth by measuring the idler power generated by crystal X2 as a function of temperature across the tuning range, with the results shown in Fig. 6. For a pump power of P1 = 5 W, above the S-SRO threshold , crystal X1 generated signal (idler) wavelengths of 961 nm (1191.7 nm), 913 nm (1274.8 nm) and 880 nm (1345.2 nm), at temperatures T1 = 87.5°C, 139°C and 192°C, respectively. The discrepancy between the measured wavelengths (Fig. 6) and the calculated values (Fig. 2), as mentioned in section 3.1, is attributed to the limited accuracy of the Sellmeier equations for MgO:sPPLT , which has also been reported earlier . The measured idler power generated by X1 varies from 1.5 W to 1.0 W, depending on the crystal temperature away from degeneracy, similar to our previous reports .
Now, if the crystal X2 is pumped with a power of P2 = 1 W, while the pump power to X1 is maintained at P1 = 5 W, one should not expect any idler power generation from X2. However, if we vary T2 towards T1, the idler output from X2 begins to rise from zero and reaches a maximum when T2 = T1 (corrected for the offset between the crystal temperatures by subtracting 2.5°C from the experimentally measured T2), with a corresponding FWHM acceptance bandwidth, ΔT. The generation of idler output from X2, when P2 is clearly well below the S-SRO threshold, is due to seeded parametric generation in X2 by the signal from X1, which occurs when the phase-matched wavelengths in the two crystals come into close proximity. The temperature acceptance bandwidth for single-pass parametric amplification decreases with the increase in phase-matching temperature, measuring to be ΔT = 3°C, 2.6°C and 2°C at 87.5°C, 139°C and 192°C, respectively, with corresponding signal (idler) wavelength acceptance bandwidths of 4.29 nm (6.6 nm), 2.2 nm (4.29 nm) and 1.2 nm (2.6 nm). This is obvious, since the increase in crystal temperature results in the generation of shorter wavelengths with higher dispersion, and hence narrower phase-matching bandwidth. However, the experimentally measured acceptance bandwidths are typically three times wider than the theoretical values in Fig. 2. This discrepancy can be attributed to the fact that the theoretical curves are calculated taking the approximation of the sinc function, which is valid in the low-gain limit. Similar broadening has also been reported previously  even when the optical parametric generator was pumped near threshold. On the other hand, the bandwidth broadening in the present case may also be attributed to the bandwidth of the intra-cavity seed signal radiation generated by crystal X1 at a pump power above threshold, and also the pump power to X2. To verify the contribution of the pump power to crystal X2 in the broadening of the temperature acceptance bandwidth, we measure the idler power as a function of X2 temperature for two different pump powers P2 = 1 W and 4.6 W, while pumping X1 at a constant input power P1 = 5 W. The results are shown in Fig. 7. As evident from the plot, the temperature acceptance bandwidth of the parametric amplifier increases from ΔT = 2°C at P2 = 1 W to ΔT = 2.2°C at P2 = 4.6 W, due to the higher parametric fluorescence bandwidth resulting from higher pump power. It is also evident that with the increase in the pump power to crystal X2, the peak of the idler power has a shift of ~1°C to lower temperatures. This shift can be attributed to pump absorption and crystal heating effects in X2. As a result, the crystal temperature increases from its set value, T2, thus requiring a reduction in the set temperature to obtain exact phasing-matching relative to the seed wavelength generated by crystal X1. This effect is also evident in our earlier report .
5.2 Generation of dual-wavelengths with closest possible proximity
To compare the theoretical calculations in section 3.3 with measurements, we reproduced the experimental results on the closest possible pair of distinct resonant signal wavelengths obtained previously . In the experiment, we varied the crystal temperatures towards each other, while recording the distinct wavelengths. The results are shown in Fig. 8. At T1 = 90°C and T2 = 94°C, two distinct signal wavelengths with a separation, Δλ~1.76 nm, corresponding to a frequency difference, Δν~0.55 THz, were observed.
The wavelength separation between the two distinct signal outputs is greater than the spectral acceptance bandwidth (~1.3 nm) of the 30-mm-long MgO:sPPLT crystals. Further variation in the crystal temperatures produces gain overlap of the two SROs, thus resulting in a single resonant wavelength. Reducing the spectral acceptance bandwidth of the MgO:sPPLT crystal, one can in principle generate two distinct signals with frequency difference below 0.55 THz. This can be achieved by using longer crystals (currently limited to 30 mm), or by operating the two SROs far from degeneracy, where the signal wavelengths have higher dispersion, thus resulting in narrower phase-matching bandwidth.
5.3 Generation of arbitrary dual-wavelengths
From the theoretical study and experimental results on the temperature acceptance bandwidth in sections 3.1 and 5.1, it is clear that both the crystals in the T-SRO can be operated as two separate S-SROs to generate two arbitrary pairs of signal-idler output beams, while sharing the same optical cavity. If both the crystals are pumped with equal pump power, the generated output from the two crystals can have same power. The arbitrary output waves will be distinct, provided their wavelength separation is greater than the gain acceptance bandwidth of the individual crystals. To verify the generation of dual-wavelength radiation with arbitrary values, we adjust the individual crystal temperatures and measure the signal wavelengths from the two crystals. The results are shown in Fig. 9. To compare the theoretical predictions with experimental measurements, we have reproduced earlier results reported previously .
For dual-wavelength generation, we operate the T-SRO by pumping both crystals with equal input powers (~5 W), well above the S-SRO threshold , so that each crystal can operate as an independent S-SRO. As a result, at T1 = 77°C and T2 = 110°C, the T-SRO generates two resonant signal waves at 939.4 nm and 971.4 nm (idler at 1226.7 nm and 1176.1 nm, respectively) with a frequency difference of 10.5THz. If we increase T1 and decrease T2, for different combinations of (T1, T2) resonant signal waves with frequency difference of 7 THz (82°C, 104°C) and 1.66 THz (88°C, 96°C) are generated. Further increase in T1 to 91°C and decrease in T2 to 93.5°C results in both crystals generating a single wavelength (955 nm), but with an additional spectral component. Such additional spectral components in MgO:PPLN crystals have been attributed speculatively either to the stimulated Raman lines [15, 16] or to cascaded phase-matched optical parametric processes . Despite the ambiguities in terms of the actual processes, all previous reports have a common finding in the appearance of such additional spectral peaks at higher intra-cavity powers. In the present experiment, the additional component, which has a wavelength shift of ~17 nm (~204 cm−1)  from the resonant signal wavelength, is attributed to stimulated Raman emission.
5.4 Verification of coherent energy coupling and Raman effect
Typically, the stimulated Raman lines appear at intra-cavity powers greater than the Raman threshold power. When both crystals are operating away from each other, there is no evidence of the additional spectral components, since the intra-cavity power of each of the two circulating signal waves at a given pump power is below the Raman threshold power. Once both the crystals operate at the same signal wavelength, coherent-coupling between the resonant signal waves increases the total intra-cavity power, leading to the generation of Raman lines. Under coherent energy coupling, the signal generated by one crystal is amplified in the other crystal in presence of the pump power, and vice versa, resulting in an increase of the overall signal power. In this case, the T-SRO behaves as an S-SRO with a crystal length equal to the sum of the lengths of the two crystals. If the pump beam to one crystal is blocked, the other crystal still operates, maintaining OPO oscillation. However, in this case, there was no signature of additional spectral lines despite the longer total crystal length. To verify coherent energy coupling between the signal waves generated by the two crystals, we pumped X1 with a power of P1 = 5.2 W, while pumping X2 with a power of P2 = 1 W, and measured the signal spectra. The results are shown in Fig. 10. When P1 to crystal X1 is blocked, no oscillation is observed, since P2 to crystal X2 is well below the S-SRO threshold . On the other hand, when we block P2 to X2, crystal X1 still generates signal and idler radiation, since the pump power to X1 is well above the S-SRO threshold. However, we did not observe any additional spectral lines, which can be attributed to the insufficient intra-cavity power. When both pump beams are unblocked, we observe signal spectra with additional peaks at different sets of temperatures (T1, T2) for the two crystals, as shown in Fig. 10(a) and Fig. 10(b). Although the individual crystals cannot generate additional peaks when operating separately, when they oscillate simultaneously and under the same phase-matching condition, the signal radiation generated by X1 is amplified in X2, in the presence of P2, and vice versa. As a result, the total intra-cavity signal power increases from its initial value, thus generating the additional spectral peaks. This observation verifies coherent energy coupling in the T-SRO, which can is also confirmed by the threshold reduction effect reported in , and reproduced in section 5.6 for completeness.
From Fig. 10(a) and Fig. 10(b), it is evident that for two different signal wavelengths of 922 nm (T1 = 125.5°C, T2 = 128°C) and 912 nm (T1 = 139°C, T2 = 141.5°C), and for any other signal wavelength across the entire SRO tuning range (850-1430 nm) , the additional spectral line is always red-shifted from the main signal peak by ~17 nm (~204 cm−1), independent of the signal wavelength. This observation confirms that the additional spectral line is only due to the Raman effect, as the excitation of the phonon mode is the same for any excitation energy, and not due to the cascaded parametric process as reported for MgO:PPLN crystal . The constant wavelength shift of ~17 nm (~204 cm−1) is a characteristics of the medium, here the MgO:sPPLT crystal . The difference in the crystal temperatures T1 and T2 to generate same wavelength is due to the difference in crystal heating configurations, as noted earlier.
5.5 Intra-cavity parametric amplification
We also studied the intra-cavity parametric amplification process in the T-SRO. The results of this study are shown in Fig. 11. In the experiments, crystal X1 is pumped with a power of P1 = 5W, above the S-SRO threshold. The generated idler is transmitted through the output mirror M2 (see Fig. 5), while the signal radiation seeds the crystal X2. In the absence of any pump power to X2, no idler power is present at the output mirror M4. However, increasing the pump power to X2 from zero, and under the same phase-matching condition as X1, the idler power generated by X2 rises linearly with the pump power, P2. At crystal temperatures of T1 = 77.5°C and T2 = 80°C (corresponding signal at λs = 972 nm, idler at λi = 1175.5 nm), the idler power measured at the output of mirror M4 increases from zero to a maximum of 1.5 W at the highest available pump power of P2 = 4.5 W. The slope efficiency is 33.3% and there is no sign of saturation. Similarly at other crystal temperatures, T1 = 192°C and T2 = 194.5°C (λs = 880 nm, λi = 1345.2 nm) further away from degeneracy, a maximum idler power of 1.2 W is generated for a pump power of P2 = 4.25 W at slope efficiency of 26.4%. The reduction in the maximum idler power and slope efficiency is attributed to the reduction in the overall gain further away from degeneracy. The pump depletion in both the cases has a maximum of ~80%.
To verify the effect of the seed signal power on the idler generation from X2, we measured the idler power generated by X2 as a function of the pump power, P2, for different signal seed powers. The results are shown in Fig. 12. For different pump powers, P1 = 4.5, 5.5, 6 and 6.5 W, to crystal X1, at T1 = 139°C, the intra-cavity signal power can be estimated from the leaked signal power from one of the high reflectors to be 55, 60, 70 and 78 W, respectively, although a precise estimate requires an output coupler. At each seed power, we varied the pump power, P2, and measured the idler slope efficiency, as shown in Fig. 12. The idler slope efficiency for the crystal X2 at T2 = 141.5°C is measured to be almost constant at ~25%, similar to our earlier report , irrespective of the signal seed power. As the seed power is already sufficiently high for all pump powers to X2, it has no effect on the slope efficiency of the idler. The effect of the seed on idler slope efficiency can become significant when the seed power is comparable to the input pump power, which can be the case when the crystal X1 is operated at threshold. Again the maximum pump depletion is typically ~80%.
5.6 Threshold reduction
Intra-cavity parametric amplification and the appearance of stimulated Raman spectral lines verify coherent energy coupling in the T-SRO when the two crystals are operating under the same phase-matching condition. This will, in turn, reduce the T-SRO operation threshold, as predicted in the theoretical analysis of this scheme in section 2.4. To compare the threshold reduction estimated by theoretical analysis with experimental results, we have reproduced the power scaling for both the S-SRO and T-SRO in Fig. 13. For the T-SRO both the crystals were pumped at equal powers, whereas for the S-SRO one crystal was pumped while the other crystal was kept inside the cavity on purpose to obtain similar cavity mode condition as the T-SRO. At crystal temperatures, T1 = 91°C and T2 = 93.5°C, the S-SRO has a threshold of 3.17 W, slightly higher than our previous report , attributed to the additional loss due to the coating of the second crystal. The maximum total output power of the S-SRO (idler plus leaked-out signal) is >2.52 W for a pump power of >9.5 W. The output power is nearly saturated at higher pump powers due to the thermal effects, as discussed in our earlier report . We observed Raman peak at a pump power of 7.4 W. In case of T-SRO, we have twice the crystal length as compared to S-SRO, and hence intuitively expect a 75% reduction in threshold. However, the T-SRO has an operating threshold of 1.94 W, representing a 39% threshold reduction with respect to S-SRO, and has a higher total output power (>2.81 W) without any sign of saturation. In the T-SRO, the signal of one crystal is amplified in the other crystal and the idler is completely transmitted after each crystal, resulting in partial reduction in operational threshold. However, the sacrifice in the optimum threshold reduction (75%) is compensated by the less concomitant difficulties in resonating both the signal and idler in the same cavity, in addition to preserving the relative phases among the interacting waves from one crystal to the other as necessary in doubly-resonant OPOs and D-SROs. Moreover, the generated idler waves in the T-SRO automatically adjust their phase depending on the initial phases of the pump and signal at the input to each crystal, thus avoiding the need for phase control elements.
It can also be seen from Fig. 13 that the Raman shift appears at lower pump power (6.1 W) as compared to the S-SRO. The threshold reduction and the appearance of Raman peak at lower pump power confirm coherent coupling between the resonant signals of the individual crystals and the T-SRO behaves as a S-SRO with longer crystal length. In addition, the distribution of pump power reduces thermal effects and potential damage to the crystals, while improving the overall performance of the SRO in terms of higher output power and lower operation threshold. The T-SRO scheme thus offers the possibility of operation at relatively high pump powers (>10 W), where crystal damage is a major challenge to overcome. The lower threshold reduction (39%) from S-SRO to T-SRO in Fig. 13, compared to the theoretically estimated value (50%) can be attributed to imperfect mode overlap, crystal absorption and coating loss.
5.7 Power scaling and high power operation of the T-SRO
To verify the performance of the T-SRO at higher pump powers, we measured the total output power as a function of the input pump power >10 W, with the results shown in Fig. 14. To extract the signal radiation, we set the crystal temperatures at T1 = 70.5°C and T2 = 73°C, corresponding to a signal (idler) wavelength of 980 nm (1163.7 nm), where both the mirrors, M2 and M4, have a small transmission of 0.4% at the signal wavelength. At other signal wavelengths, all mirrors have reflectivity of >99.9%. As evident from Fig. 13, the S-SRO output power saturates at higher pump powers (>10 W), and roll-off in the pump depletion occurs, as reported earlier , due to the thermal effect arising from the higher pump powers. Using a green laser of maximum power 18 W, we further increased the pump power to the S-SRO and measured the idler output power. The idler power increases with the pump power in a similar manner to the results shown in Fig. 13. However, for pump powers beyond 10 W, we observe crystal damage. In repeated trials, we have observed similar bulk damage in the crystal, which can be attributed to thermal effects arising from the focusing of the high pump power into a small beam waist radius (w0~30 μm) at the centre of the crystal. Such crystal damage places an upper limit to the input pump power for the S-SRO under the given experimental conditions of ~10 W. We, however, expect that high-power operation of the S-SRO can be obtained using loose focusing, but at the expense of increased threshold. On the other hand, since in the T-SRO we divide the input power into two crystals, the pump power in each crystal is below the damage threshold, even for a total input power of 18 W. As evident from Fig. 14, the total output power of the T-SRO (the sum of the powers at the exit of mirror M2 and M4) increases with the increase in pump power at a slope efficiency of 40%, providing a maximum total power of 6.5 W for 16.2 W of pump power. The corresponding maximum signal and idler powers are 2.2 W and 4.3 W, respectively, which is the highest idler power ever reported using MgO:sPPLT crystal pumped in the green.
It should also be noted that although in the T-SRO the thermal dephasing effect is also distributed with the division of the pump power into the two crystals, thermal effects arising from the higher intra-cavity power, as also reported earlier , can still be a major challenge to overcome. As also evident from the Fig. 14, the idler power increases with the pump power while there is no significant increase in the out-coupled signal power. The saturation of the intra-cavity signal power can be attributed to the thermal dephasing effect in the nonlinear crystals at higher intra-cavity power and can, in principle, be lowered by reducing the intra-cavity power using higher output coupling. The increase in operation threshold, due to higher output coupling and loose focusing of the pump beam to avoid crystal damage, can be compensated by the threshold reduction effect in the T-SRO. There is no evidence of saturation in the output power, indicating that the T-SRO can be operated at even a higher power. Given this potential, using a single output coupler for the signal, one could possibly extract sufficient signal power in a single signal beam for extra-cavity THz wave generation.
5.8 Power across the tuning range
We also measured the total output power as a function of the temperature T1 of crystal X1 across the tuning range, with the results shown in Fig. 15. For each value of T1, we have adjusted the temperature T2 of crystal X2, so that both crystals are operating under coherent coupling condition at the same signal and idler wavelengths. At a total pump power of 16.2 W (with P1 = P2 = 8.1 W), the total output power from the T-SRO varies from 5.52 W at T1 = 64°C to 3.53 W at T1 = 127°C, with a maximum of 6.5 W at T1 = 70.5°C. The reduction in the total output for crystal temperatures away from degeneracy can be attributed to the reduced signal power (few milliwatts) due to the high reflectivity (R>99.9%) of the output mirrors, M2 and M4, higher reflection loss at the crystal coating, and also the gain reduction factor, as reported earlier . Although we have measured the output power up to 127°C to verify the high power performance of the T-SRO without any sign of crystal damage even at lower crystal temperatures, one can in principle extend its operation across the entire tuning range of 850-1430 nm . The pump depletion of both crystals varies by ~70-80% across the tuning range. The signal (idler) wavelength varies from 991 nm (1148.6 nm) to 916 nm (1269 nm) for a change in crystal temperature from 64°C to 127°C.
In conclusion we have theoretically and experimentally analyzed the performance of a two-crystal SRO. Solving the coupled wave equations under undepleted pump approximation, we calculated the maximum threshold reduction of 50%, temperature acceptance bandwidth (FWHM) of 0.9°C, 0.8°C and 0.7°C at crystal temperatures 87.5°C, 139°C and 192°C, respectively, and closest possible arbitrary dual-wave generation with a wavelength separation of 1.39 nm at 91°C. We have measured a lower threshold reduction of 39% compared to the theoretically predicted value, which can be attributed to the additional losses due to the crystal coating, and material absorption. Based on the T-SRO scheme, we also report a new technique to measure the temperature acceptance bandwidth of the single-pass parametric amplifier of 3°C, 2.6°C and 2°C at 87.5°C, 139°C and 192°C, respectively, across the OPO tuning range. The discrepancy between the experimental and theoretical temperature acceptance bandwidth can be attributed to the bandwidth of the intra-cavity seed signal radiation generated by crystal X1 at pump power above threshold and also the pump power to crystal X2. Although the T-SRO has two identical crystals, the parametric gain in this device has an acceptance bandwidth equivalent to a single crystal. Experimentally, the closest signal wavelength (frequency) separation under arbitrary dual-wavelength operation has been observed to be 1.76 nm (0.55 THz), which also verifies the broadening of the temperature acceptance bandwidth. Due to the division of pump in two crystals, the T-SRO can tolerate a higher total pump power, reducing the risk of optical damage and thermal loading in the crystal. We also experimentally verified the high power performance of such scheme, where we obtained a total output power of 6.5 W for 16.2 W of green power at 532 nm. The intra-cavity parametric amplification of the resonant signal of one crystal in the other crystal shows the possibility of using different combinations of crystals and pump radiation. We verified the coherent energy coupling between the intra-cavity resonant signal fields resulting Raman spectral emission.
This research was supported by the Ministry of Science and Innovation, Spain, through project OPTEX (TEC2012-37853) and by the European Office of Aerospace Research and Development (EOARD) through grant FA8655-12-1-2128.
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