Abstract
We carry out a theoretical investigation of the backward interaction optical parametric oscillator (OPO), in which one of the generated waves is counterpropagating with respect to the pump. We derive simple, self-consistent analytical formulas for the continuous-wave and pulsed regimes. While consistent with previous works on the continuous-wave regime, our study enables us to extend the analysis to the pulsed regime. In particular, we derive simple expressions of the oscillation build-up time, pulsed threshold, and efficiency, for the first time, to our knowledge. We also investigate the peculiar spectral features of the backward OPO when pumped with a narrow-linewidth pulsed radiation, in particular, the absence of tolerance to phase mismatch and the natural ability to emit two waves with a Fourier-transform-limited spectrum. A comparison with the conventional forward OPO is also carried out to emphasize the unique properties of the backward OPO.
© 2022 Optica Publishing Group
1. INTRODUCTION
Initially proposed by Harris in 1966 [1], the backward-wave optical parametric oscillator (BWOPO) is based on three-wave second-order nonlinear interaction in a nonlinear medium where one generated wave, referred to as the backward wave, propagates in the direction opposite to the incident pump wave and the other generated wave, referred to as the forward wave. In contrast to conventional optical parametric oscillators (OPOs) based on three co-propagating waves where an external feedback has to be provided to the nonlinear medium by an optical cavity, resonant at one or two of the generated waves, oscillation in a BWOPO occurs because of the distributed feedback due to the presence of two counterpropagating parametric waves. Thus, for its operation, the BWOPO does not require to adjust and maintain any fine alignment of cavity mirrors, making the device very simple and reliable.
Despite this striking attractive feature, it took 40 years before its experimental demonstration by Canalias and Pasiskevicius [2]. This long time gap was due to the fact that quasi-phase-matching materials with sub-micrometer poling periods had first to be developed since conventional phase matching would require an extraordinary large birefringence. In [2], the authors also validated the unique spectral properties of the light emitted by the BWOPO, in which the spectral bandwidth of the pump wave is basically transferred to the forward wave, while the backward wave exhibits a narrow spectral linewidth. These spectral properties have then been extensively investigated through experimental and numerical studies in the pulsed regime and exploited to carry out coherent phase transfer from the pump wave to the signal wave [3–6].
Regarding theoretical work, general plane-wave solutions to backward three-wave mixing in the continuous-wave (CW) regime, involving Jacobi elliptic functions, were derived by Meadors in 1969 [7]. Other theoretical studies devoted to the CW BWOPO were then published [8–10]. In particular, [10] contains convenient analytical expressions to determine the CW BWOPO conversion efficiency and oscillation threshold.
Due its high CW oscillation threshold intensity, typically of $50 {-} 100 \; {\rm{MW/c}}{{\rm{m}}^2}$, the BWOPO has been operated only in the pulsed regime so far. In this context, theoretical analysis of the BWOPO threshold and efficiency in the pulsed regime is worth investigating. In particular, even though most previous reported works involved pump pulses in the 10s to 100s of picoseconds duration, the BWOPO is also very promising for operation with longer nanosecond pulses [11], where its unique spectral properties are promising for applications such as remote gas sensing.
Depending on the pulse duration, the pulsed threshold can be significantly higher than the CW threshold. To guide BWOPO design, it would thus be valuable to derive a BWOPO analog of the Brosnan–Byer expression of the pulsed OPO threshold [12]. A first step in this direction was taken in [13], but it was limited to a numerical investigation of the pulsed threshold as a function of the pump pulse duration, and the derivation of an analytical expression of the pulsed threshold was not reported.
In this paper, we carry out the theoretical investigation of the BWOPO by extending the approach we previously applied to conventional OPOs [14–17]. Exactly taking into account the nonlinear interactions between the waves in the nonlinear crystal, we derive universal expressions for the BWOPO in CW and pulsed regimes, which enables us to carry out convenient comparisons with the usual singly resonant OPO (SROPO). The main outcomes are the derivation of the analytical expressions of the BWOPO pulsed threshold and build-up time as well as the optimal pulse duration to minimize threshold energy. We also provide an approximate expression of the pulsed conversion efficiency, which could also be helpful for the design of a BWOPO. We finally carry out a numerical simulation of the BWOPO by finite-difference time-domain (FDTD) simulation to assess the validity of our analytical expressions and further emphasize some distinctive spectral properties of the BWOPO compared to the conventional pulsed OPO.
This paper is organized as follows. Section 2 presents an analysis of the threshold and the efficiency of the pulsed BWOPO. First, the coupled-wave equations and main assumptions are presented. Second, we derive the exact solutions to the equations in the CW regime using Jacobi elliptic functions. We recover the expressions previously derived in the literature and compare the characteristics of the BWOPO with the conventional SROPO. In particular, we determine the mirror reflectivity to have a SROPO with the same oscillation threshold as a mirrorless BWOPO based on an equivalent nonlinear medium. Then, we derive simple expressions to evaluate the pulsed oscillation threshold and efficiency, which are reported here for the first time, to our knowledge. Section 3 is devoted to an investigation of the spectral properties of the BWOPO. First, we analyze particular phase-matching acceptance that is much narrower than for forward parametric interaction. Second, we investigate the parametric amplification bandwidth. In particular, we show that perfect phase matching is required to be able to reach the BWOPO threshold. Third, numerical calculations are carried out to validate the analytical results and to compare the spectral properties of the BWOPO with the conventional SROPO. The calculation confirms the ability of the BWOPO to naturally deliver single frequency radiation with a Fourier-transform linewidth. The final section summarizes the conclusions.
2. BACKWARD OPO THRESHOLD AND EFFICIENCY
A. Coupled-Wave Equations
The geometry of the studied BWOPO is shown in Fig. 1. We consider a second-order nonlinear material of length $L$ containing a quasi-phase-matching grating to enable efficient interaction with a backward propagating wave. For the theoretical analysis, we start with the nonlinear three-wave-mixing coupled equations in the slowly varying envelope approximation, which reduce in the plane-wave limit to [4,18]
For simplicity’s sake, we consider that the magnitude of the group velocity is the same for the three waves. However, since the backward wave propagates in the opposite direction, its group velocity has an opposite sign compared to the group velocity of the forward and pump waves. As a consequence, the relative group velocity difference is very large given that it is equal to $2{v_g}$, which is dominant compared to other chromatic dispersion effects. Thereby, coupled Eqs. (1a)–(1c) become
B. Continuous-Wave Regime
In CW regime, one can discard time dependence and the related partial derivative. The system (5a)–(5c) thus simplifies to
To solve the above system, we decompose the complex amplitudes as follows:
where amplitude ${u_j}(z)$ and phase ${\varphi _j}(z)$ are both real-valued functions. For the BWOPO, we consider that there is no incident forward or backward wave, leading to the following boundary conditions: and we assume that the relative phase $\Delta \varphi = {\varphi _p} - {\varphi _b} - {\varphi _f}$ has reached the steady-state value that maximizes the coupling between the three waves and thus satisfies $\cos [{\Delta \varphi (z)}] = 1$. Moreover, we take pump depletion into account by a change of sign in ${u_p}(z)$ rather than a step of $\pi$ in $\Delta \varphi (z)$. With this additional assumption, we can now consider the following system that involves only real functions and parameters:From Eqs. (9a)–(9c), the “constants of motion” are found to be
where we use the compact notations ${u_p}(0) = {u_{p0}}$, ${u_b}(0) = {u_f}(L) = {u_{{\rm{out}}}}$. These constants of motion are determined by the boundary conditions. Equation (9b) describing the evolution of the forward wave can then be rewritten as follows:This latter expression has a solution in terms of Jacobi functions:
The pump and backward waves’ spatial evolutions can then be derived by use of (10a) and (10a):
Setting $z = L$ in (13), one obtains the implicit equation for BWOPO CW oscillation:
This expression is equivalent to the one derived by Meadors [7]. The CW oscillation threshold, $u_{{\rm{th}}}^{{\rm{CW}}}$, can then be easily derived using the approximation ${\rm{sn}}({x|m}) \approx \sin (x)$ when $m \to 0$:
The above expression can be conveniently expressed in terms of intensity, $I = {\epsilon_0} c n \langle {E^2}\rangle = {\epsilon_0} c \omega {| A |^2}/2$ (in ${\rm{W/}}{{\rm{m}}^2}$), as follows:
The latter expression of the threshold intensity is consistent with previously reported expressions when one takes the different definitions of the effective nonlinear coefficient into account [1,7,10,13].
One can introduce the photon conversion efficiency, $\eta$, defined as
and consider thatThe implicit equation for BWOPO CW oscillation (17) can then be rewritten:
After some algebra, the latter equation can also be written in terms of a complete elliptic integral of the first kind as follows, which is easier and faster for numerical solving and to derive approximate expressions:
withEquation (23) is similar to the one derived by Ding and Khurgin [10].
Approximate expressions can be derived for BWOPO operation close to the oscillation threshold by use of the following expansion of $K(\eta)$:
Conversely, for high conversion efficiencies ($\eta \to 1$), the following approximation may be considered:
which yieldsAs seen in Fig. 2, the first-order approximation, (26), is valid only very close to the oscillation threshold, while the second-order approximation, (27), provides a reasonable agreement (better than 10%) for ${I_p} \lt 1.4 I_{{\rm{th}}}^{{\rm{CW}}}$. Conversely, the accuracy of asymptotic expression (29) is better than 10% for ${I_p} \gt 1.6 I_{{\rm{th}}}^{{\rm{CW}}}$. As discussed in the next section, this latter approximation is actually convenient to evaluate the efficiency in the pulsed regime where the involved peak intensity is often high enough.
Let us now carry out a comparison with the more conventional SROPO based on the forward nonlinear interaction among three co-propagating waves while the feedback is provided by an optical cavity resonant at the signal wave. Using the same formalism as the one derived here [14], the CW oscillation of the SROPO reads
C. Pulsed Regime
In addition to the steady-state conversion efficiency, another important effect for the overall efficiency in the pulsed regime is the buildup of the oscillation from quantum noise during which the pump intensity is not significantly depleted. To derive an analytic expression of the build-up time, we have to further simplify the coupled-wave equations (5a)–(5c), which involve space and time partial derivatives. First, we neglect pump depletion during buildup, which enables us to reduce the system to (5a) and (5b) with the pump amplitude as a driving term. Second, we assume that the characteristic time scale of the pulses is longer than the propagation delay through the nonlinear crystal, $\tau = L/{v_g}$, so that space and time profiles can be factorized as separated functions with the longitudinal profiles determined with CW-regime solutions. With these two assumptions, the amplitudes of the three waves read
where the longitudinal profiles correspond to CW solutions at threshold. Inserting amplitude profiles (32a)–(32c) in coupled equations (5a) and (5b) and carrying out integration over $z$, the following set of two coupled equations involving only time derivatives is obtained:For a step-wise pump temporal profile with the pump intensity switched on to a constant value, ${I_p}$, at $t = 0$, the solution in terms of intensity, ${I_{b,f}} \propto {| {{a_{b,f}}} |^2}$, is
withThe oscillation build-up time is defined as the time to reach a detectable intensity level, ${I_{{\det}}}$, starting from the quantum noise characterized by the equivalent intensity level, ${I_{{\rm{noise}}}}$:
The duration of the build-up time is thus proportional to ${g_{{\rm{Log}}}}$ whose value depends on the detection sensitivity, mode volume, and other experimental parameters. Conversely to the CW regime, the threshold definition in pulsed regime is thus somewhat arbitrary. For the calculations presented in the following, we set ${g_{{\rm{Log}}}} = 18$, as in [16]. However, the value of ${g_{{\rm{Log}}}}$ is straightforwardly adjustable to be more relevant for other conditions. Moreover, one could note that a variation of the noise or detection level over more than an order of magnitude only induces a 10% variation of the ${g_{{\rm{Log}}}}$ value.
The comparison with the SROPO can be carried on for the build-up time. In the case of a SROPO with the same CW threshold as the BWOPO, i.e. $R = {R_{{\rm{SROPO}}}}$, with ${R_{{\rm{SROPO}}}}$ defined in (31), and with cavity mirrors directly located on the crystal facets so that the cavity round-trip time ${\tau _{{\rm{cav}}}}$ is equal to $2\tau$, the SROPO build-up time can be determined from (A6) (see Appendix A) as follows:
The latter expression can be approximated with an accuracy better than 10% using the approximation $\cosh (x) \simeq \exp (x)/2$, which yields
Whatever the input pump intensity, the build-up time of the BWOPO is thus typically two times shorter than the one of the considered “equivalent” SROPO. This is indeed confirmed in Fig. 3 where we also indicate estimated values for OPO setups based on typical parameters ($I_{{\rm{th}}}^{{\rm{CW}}} = 75 \;{\rm{MW/c}}{{\rm{m}}^2}$ and $\tau = 44.3 \;{\rm{ps}}$). In Fig. 3, the build-up times calculated respectively with formulas (36) and (38) are also compared with FDTD numerical simulations, where the coupled nonlinear equations (5a)–(5c) are solved as explained in Section 3.C, except that we inject here a constant intensity instead of a random noise for an easier evaluation of the oscillation threshold criterion and consider here a step-wise pump temporal profile. The value of the build-up time provided by the analytic evaluation of the BWOPO is higher than the one obtained by numerical calculation. Nonetheless, except for peak intensities close to the CW threshold where there is a factor of two between numerical and analytical evaluations, the difference between the two values reduces rapidly to be less than 25% for ${I_p} \gt 2I_{{\rm{th}}}^{{\rm{CW}}}$ despite the relatively strong assumptions made to derive the analytic expression.
1. Square Temporal Profile
For a square temporal profile of duration ${\bar \tau _p}$, one can derive from (36) the pulsed peak intensity to reach a detectable level of generated radiation at the trailing edge of the pump pulse:
The corresponding threshold fluence is thus
The latter expression provides a first estimation of the threshold fluence for a given pulse duration. Expression (41) has a minimum for pulse duration ${\bar \tau _{p{\rm{,opt}}}}$ that satisfies ${\rm{d}}{\bar J _{{\rm{th}}}}/{\rm{d}}{\bar \tau _p} = 0$:
with the corresponding minimum of the threshold fluence: which corresponds to a peak power of four times the CW threshold power over a duration of $4.7\tau$.The pulsed threshold peak intensity can be derived in the same way for the equivalent SROPO from (38) or its approximation (39). The ratio between pulsed threshold peak intensities of the BWOPO and its equivalent SROPO can then be written as
As in [16], we can derive an approximate expression of the pulsed conversion efficiency. For this purpose, we assume that the temporal profile of the output divides into two distinct temporal phases in the same manner as in [20]. The first one is the BWOPO build-up phase, during which the pump is assumed constant and whose duration is ${\tau _{{\rm{bu}}}}$. The second one is the steady-state regime where all the intensities are known and are identical to the CW regime solutions derived in Section 2.B. We can hence write
Figure 4 shows the pulsed conversion efficiency evaluated with expression (45) and its approximation (46) as a function of the pulse duration for constant pump fluences. The agreement between the two expressions is excellent for pulse duration leading to peak intensities that are high enough for (29) to be accurate, i.e., better than 10% for ${\bar \tau _p} \lt 0.62 \bar J /I_{{\rm{th}}}^{{\rm{CW}}}$. The estimation remains nevertheless reasonable for longer pulse lengths.
Analytic evaluation of the pulsed efficiency is also compared with results obtained from numerical simulations in Fig. 4. The largest discrepancy is observed at short pulse durations where the assumption of a characteristic time scale longer than $\tau$, used to derive the analytic formulas, is not valid. The agreement remains nevertheless quite good despite the pulse profiles shown in Fig. 5(a) where a steady state is far from being reached during the pulse duration. On the other hand, as expected, an excellent agreement is obtained in Fig. 4 for pulses lengths corresponding to Figs. 5(b) and 5(c) where transient modulations only occur during a relatively short duration.
For a given pump fluence, the optimum of efficiency results from a balance between the minimization of the energy lost during the build-up phase during which the optimal peak power is four times the CW threshold and the maximization of steady-state conversion efficiency, which requires the highest possible peak power. For the pump fluences considered in Fig. 4, the corresponding optimal pump peak powers are respectively 4.07, 4.71, and $6.37 \times I_{{\rm{th}}}^{{\rm{CW}}}$.
Figure 6 presents the BWOPO efficiency as a function of the pump peak intensity for various pulse durations. This illustrates the effect of the build-up time on the oscillation threshold and efficiency. It can also be noticed that the BWOPO operation is expected to be close to the CW limit for pulse durations longer than typically $100 \tau$, which corresponds to a few nanoseconds for typical crystal parameters.
The impact of quantum noise fluctuations on the pulsed conversion efficiency stability can be evaluated by differentiation of (45) with respect to ${I_{{\rm{noise}}}}$. After some algebra, one can write
2. Gaussian Pulse Profile
To derive the pulsed threshold formula, we chose the same convention as Brosnan and Byer [12] to define the Gaussian pulse profile:
Also similar to Brosnan and Byer, we introduce an equivalent square pulse profile whose duration ${\bar \tau _p}$ corresponds to the time during which the instantaneous power is higher than the CW threshold power, i.e., ${I_p}(t) \gt I_{{\rm{th}}}^{{\rm{CW}}}$. One obtains
The equivalent-square-pulse intensity at the pulsed oscillation threshold $\bar I _{{\rm{th}}}^{{\rm{peak}}}$ has to satisfy (40), and the corresponding threshold peak intensity of the Gaussian pulse $I_{{\rm{th}}}^{{\rm{peak}}}$ is related to $\bar I _{{\rm{th}}}^{{\rm{peak}}}$ through (51). For a given pulse duration ${\tau _p}$, it is thus possible to numerically evaluate the threshold peak intensity of the Gaussian pulse by solving the closed system provided by (40), (49), and (51). The corresponding threshold fluence of the Gaussian pulse is then given by
The threshold fluence from (53) is plotted in Fig. 7 as a function of the Gaussian pulse duration. The minimal fluence is obtained for a duration of typically $3\tau$, a fluence of $25\tau I_{{\rm{th}}}^{{\rm{CW}}}$, and a corresponding peak power of $I_{{\rm{th}}}^{{\rm{peak}}} = 6.6 I_{{\rm{th}}}^{{\rm{CW}}}$.However, the latter evaluation of the threshold fluence requires to numerically solve several equations, and a more straightforward way to estimate the threshold peak intensity and fluence for the Gaussian pulse would be more convenient for practical use. For this purpose, as in Brosnan–Byer’s analysis, we assume that ${\bar \tau _p} = 2{\tau _p}$, which enables us to derive a simple expression of the threshold peak intensity:
We can then use (53) to determine the corresponding threshold fluence:
For longer pump pulses, a better accuracy is obtained assuming ${\bar \tau _p} = {\tau _p}$. The corresponding approximated expressions are then
Threshold fluences determined by numerical simulation are also shown in Fig. 7. A very good agreement with approximate expression (57) is obtained for pulse durations longer than typically $30\tau$. For a pulse shorter than $10\tau$, it is more accurate to use approximate expression (55) with a relative difference of typically 20% down to a duration shorter than $\tau$, where the assumptions made to derive expressions (53) and (55) are too invalid. For such short pulse durations, the threshold fluence determined by numerical simulation grows more slowly than the one provided by the expressions when the duration shortens. The discrepancy remains nonetheless still reasonable (less than a factor of two).
To further assess the relevance of our model, we compare experimental values of the peak intensity at threshold, reported in the literature for two different pulse durations (47 ps and 13 ns), with $I_{{\rm{th}}}^{{\rm{peak}}}$, determined by solving the closed system provided by (40), (49), and (51), and its approximations $I_{{\rm{th}}}^{{\rm{peak}}}({{{\bar \tau}_p} = 2{\tau _p}})$ and $I_{{\rm{th}}}^{{\rm{peak}}}({{{\bar \tau}_p} = {\tau _p}})$ respectively given by expressions (54) and (56). As shown in Table 1, a good agreement (better than 5%) between experimental values and $I_{{\rm{th}}}^{{\rm{peak}}}$ is obtained for both pulse durations assuming ${d_{{\rm{eff}}}} = 8$ pm/V and with ${g_{{\rm{Log}}}} = 18$. As expected, $I_{{\rm{th}}}^{{\rm{peak}}}({{{\bar \tau}_p} = 2{\tau _p}})$ is more accurate than $I_{{\rm{th}}}^{{\rm{peak}}}({{{\bar \tau}_p} = {\tau _p}})$ for the shortest pulse duration and conversely for the longest pulse duration. The threshold expressions provided by our model might thus be useful for the practical design of BWOPOs.
3. LINEWIDTH
A. Phase-Matching Acceptance Bandwidth
As shown by Canalias and Pasiskevicius [2], the BWOPO exhibits peculiar spectral properties with the generation of a backward wave with a very narrow spectrum, while the spectral content of the pump is transferred to the forward wave. For a narrow Fourier-transform-limited pump, it is thus expected that both backward and forward waves have narrow spectral linewidths.
Assuming that quasi-phase matching is satisfied for angular frequencies ${\omega _p}$, ${\omega _b}$, and ${\omega _f}$ with ${\omega _p} = {\omega _b} + {\omega _f}$, the magnitude of the phase mismatch, $\Delta k({\Delta \omega})$, for frequencies ${\omega ^\prime _b} = {\omega _b} + \Delta \omega$ and ${\omega ^\prime _f} = {\omega _f} - \Delta \omega$ is given by
One can compare the latter expression with the same definition of the DFG acceptance bandwidth for forward interaction:
where $\Delta {n_g}$ is the group index difference between signal and idler, $\Delta {n_g} = | {{n_{\textit{gs}}} - {n_{\textit{gi}}}} |$.For similar crystal lengths, the ratio of the two bandwidths is thus given by
B. Optical Parametric Amplifier Bandwidth
To derive the optical parametric amplifier (OPA) gain spectrum, we solve the coupled-wave equations in CW regime under the undepleted pump approximation, where we now consider imperfect phase matching:
where ${A_p}(z) = {A_p}(0) = {A_{p0}}$, and $\Delta k = {k_p} - {k_f} + {k_b} - {K_G}$. We assume that there is no input for the backward wave, i.e., ${A_b}(L) = 0$, and that there is an input for the forward wave. After solving the system (63) under these assumptions, one obtains the expression of the output forward amplitude:This feature is a very particular characteristic of the BWOPO that differs from the conventional forward OPO, where the small-signal gain can be larger than the cavity loss for phase-mismatched interactions if the pump intensity is high enough.
When ${I_p}$ approaches $I_{{\rm{th}}}^{{\rm{CW}}}$ and for a small phase mismatch, (65) can be approximated by the following expression:
As for threshold and efficiency, we can compare the backward OPA bandwidth with its forward interaction analog. For this purpose, we consider the sub-threshold SROPO or cavity-enhanced forward OPA (CE-FWOPA) with an equivalent mirror reflectivity ${R_{{\rm{SROPO}}}}$ given by (31) leading to the same CW oscillation threshold as the BWOPO. The corresponding CE-FWOPA gain reads (see Appendix A)
The corresponding phase-mismatch acceptance at half maximum of ${G_{{\rm{CE}} - {\rm{FWOPA}}}}$ is given by
Including the group index scaling factor provided by (62), the ratio of the two OPA bandwidths (71) and (68), expressed in terms of frequency bandwidth, is given by
C. Pulsed OPO Bandwidth
To extend our investigation of the spectral bandwidth to the case of pulsed OPOs, we carry out FDTD numerical simulations for both the BWOPO and conventional SROPO under the plane-wave approximation, i.e., we numerically solve system (1a)–(1c) for the BWOPO and system (A.1a)–(A1c) for the SROPO. The considered simulation parameters are detailed in Table 2. In both cases, quasi-phase matching is assumed at the carrier frequencies. In the case of the BWOPO, comparable results are obtained with identical group velocities for the three waves, which confirms the relevance of the assumption made for the analytic analysis, i.e., $2{n_g} \gg \Delta {n_g}$.
For the conventional forward SROPO, since we consider $\Delta {n_g} = 0.02$, we have $\Delta {\nu _{{\rm{FWDFG}}}} = 2.1$ THz for a crystal length $L = 7$ mm. We still consider a SROPO with an equivalent cavity reflectivity given by (31) to have the same CW threshold. We also assume the shortest possible cavity length for the SROPO with mirrors directly on the nonlinear medium ends, i.e., at $z = 0$ and $z = L$.
In both cases, we consider a broadband noise ($\Delta {\nu _{{\rm{noise}}}} = 40$ THz) characterized by an electric field with random amplitude and phase both with zero mean and an equivalent intensity of half a photon per time unit and surface unit.
As shown in Fig. 9, the pulses emitted by the BWOPO exhibit smooth temporal profiles with the emission of a narrow Fourier-transform-limited spectrum for the forward wave. The spectrum of the backward wave is similarly narrow. To investigate the stability of the output fluence and spectrum, the simulation is carried out 50 times with the same parameters (only the initial random noise changes from pulse to pulse). This leads to relative fluctuation of 0.4% (standard deviation) of the conversion efficiency of the BWOPO without any observable variations of the forward and backward central wavelengths and spectral linewidths. Whatever the spectral distribution of the initial noise, only the forward and backward wavelengths at perfect phase matching oscillate. The only observable variations concern the value of the peak spectral intensity. In the time domain, this is reflected in variations of the pulse build-up time.
On the other hand, as seen in Fig. 10, the SROPO delivers pulses with strong temporal modulations, whose features are consistent with bandwidth and group-velocity effects previously reported in the literature [22]. These modulations display a periodic pattern with a period corresponding to cavity round-trip time. The resulting signal spectrum is thus multimode with a bandwidth of about 800 GHz. As expected, the build-up time of the SROPO is longer than the one of the BWOPO.
The statistics over 50 pulses show relative fluctuations of the conversion efficiency of 0.3% (standard deviation), which is comparable to the case of the BWOPO. The most striking difference compared to the BWOPO actually concerns the pulse to pulse variations of the spectrum. Indeed, as expected for a SROPO [18], we observe large variations of the power partition between the emitted modes, which are related to the spectral distribution of the initial noise.
This study confirms the very different spectral properties of the BWOPO and the SROPO. While the pump pulse features are the same for the two OPOs and the nonlinear parameters are identical, the BWOPO naturally emits a Fourier-transform-limited spectrum while the SROPO delivers a broadband multimode spectrum. To obtain a Fourier-transform-limited emission from the SROPO, one would need to implement more complex cavity schemes with intracavity spectral filters or based on injection seeding of a narrow-linewidth radiation [23]. On the other hand, the BWOPO delivers a narrow linewidth without any cavity or additional spectral filter.
One should nevertheless keep in mind that our approach considers plane waves in an ideal quasi-phase matching nonlinear medium with a monochromatic pump. It would be interesting to extend this work by analyzing the effects of an imperfect quasi-phase matching period, while it is expected that a large or non-Fourier-transform-limited pump spectrum would mainly alter the spectrum of the forward wave [4–6]. Another useful outlook would be to develop a more elaborate model to take into account finite beam effects that might alter the BWOPO spectral and spatial profiles. Dedicated experimental studies would also be essential to validate the theoretical results.
4. CONCLUSION
In this paper, we have presented a theoretical investigation of the BWOPO in CW and pulsed regimes. By exactly taking into account the parametric interaction between the three waves in the nonlinear crystal, our approach has enabled us to establish steady-state solutions that are consistent with the literature. Then, we have developed an approach adapted to the description of the pulsed BWOPO oscillation buildup. As the main outcome of this analysis, we have derived an analytic expression of the BWOPO build-up time. Owing to this expression, we have been able to derive a BWOPO analog of the Brosnan–Byer threshold fluence formula for the conventional SROPO. These expressions have then been successfully compared with experimental threshold values reported in the literature. We have also derived an approximate expression of the conversion efficiency in the pulsed regime. These expressions can be useful to design a pulsed BWOPO without resorting to extensive numerical simulations. The developed formalism has also been exploited to carry out informative comparisons between the BWOPO and SROPO. In particular, we show that a SROPO with a coupler reflectivity of about 16% has the same CW oscillation as the BWOPO based on a similar nonlinear material. This equivalent SROPO has however a build-up time that is typically two times longer than the BWOPO.
We have also studied the spectral properties of the BWOPO. A striking result is that the BWOPO oscillation threshold can be reached only at perfect quasi-phase matching and that the subthreshold gain bandwidth is typically several hundred times narrower than for forward-wave parametric amplification. Numerical simulations of the BWOPO and SROPO pumped by a Fourier-transform-limited nanosecond Gaussian pump pulse have confirmed that the spectral properties of both OPOs are very different. While the SROPO delivers a broadband multimode spectrum, the BWOPO emission is naturally Fourier transform limited. This unique feature is very promising to implement differential absorption lidar emitters based on BWOPOs for remote gas sensing.
APPENDIX A: BUILD-UP TIME AND GAIN BANDWIDTH OF THE SROPO
1. SINGLE-PASS FWOPA GAIN
The coupled nonlinear equations for forward parametric interaction read [23]
2. SROPO BUILD-UP TIME
Let us consider a SROPO characterized by a cavity round-trip time, ${\tau _{{\rm{cav}}}}$, and a coupler reflectivity, $R$, while the other cavity losses are negligible. Considering that the pump is not depleted during the oscillation build-up time, the intracavity signal intensity temporal evolution can approximated by
3. CAVITY-ENHANCED FWOPA GAIN SPECTRUM
We now consider the CE-FWOPA (or sub-threshold SROPO). Assuming that the signal frequency is a cavity eigenfrequency, the gain of the CE-FWOPA can be written as
Funding
Agence Nationale de la Recherche (ANR-10-LABX-0039-PALM, ANR-16-CE01-0009); Horizon 2020 Framework Programme (821868).
Acknowledgment
The authors are grateful to Prof. Valdas Pasiskevicius for critical reading of the paper.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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