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Backward optical parametric oscillator threshold and linewidth studies

Antoine Godard, Marie Guionie, Jean-Baptiste Dherbecourt, Jean-Michel Melkonian, and Myriam Raybaut

Open Access Open Access

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.

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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]

$$\frac{{\partial {A_b}}}{{\partial z}} - \frac{1}{{{v_{\textit{gb}}}}}\frac{{\partial {A_b}}}{{\partial t}} = - i\kappa {A_p}A_f^*\exp ({i\Delta kz} ),$$
$$\frac{{\partial {A_f}}}{{\partial z}} + \frac{1}{{{v_{\textit{gf}}}}}\frac{{\partial {A_f}}}{{\partial t}} = i\kappa {A_p}A_b^*\exp ({i\Delta kz} ),$$
$$\frac{{\partial {A_p}}}{{\partial z}} + \frac{1}{{{v_{\textit{gp}}}}}\frac{{\partial {A_p}}}{{\partial t}} = i\kappa {A_b}{A_f}\exp ({- i\Delta kz} ),$$
with ${v_{\textit{gj}}}$ the group velocity of wave $j$, where $j = b$, $f$, and $p$ corresponds respectively to the backward, forward, and pump waves, and $\kappa$ the nonlinear coupling coefficient
$$\kappa = \frac{{{d_{{\rm{eff}}}}}}{c}\sqrt {\frac{{{\omega _b}{\omega _f}{\omega _p}}}{{{n_b}{n_f}{n_p}}}} ,$$
with ${d_{{\rm{eff}}}}$ the effective nonlinear coefficient. ${A_b}$, ${A_f}$, and ${A_p}$ are the complex amplitudes related to the real electric fields as follows:
$${E_b}({z,t} )\def\LDeqtab{} = \frac{1}{2}\sqrt {\frac{{{\omega _b}}}{{{n_b}}}} {A_b}({z,t} ){\exp }\left[{i({- {k_b}z - {\omega _b}t} )} \right] + \text{c.c.},$$
$${E_{p,f}}({z,t} ) \def\LDeqtab{}= \frac{1}{2}\sqrt {\frac{{{\omega _{p,f}}}}{{{n_{p,f}}}}} {A_{p,f}}({z,t} )\exp [{i({{k_{p,f}}z - {\omega _{p,f}}t} )} ] + \text{c.c.},$$
where ${n_j}$, ${\omega _j}$, and ${k_j}$ are respectively the refractive index, angular frequency, and wave vector of wave $j$, and $\Delta k$ is the phase mismatch:
$$\Delta k = {k_p} - {k_f} + {k_b} - {K_G},$$
with ${K_G}$ the quasi-phase-matching wave vector. The analysis is limited to plane waves to derive simple formulas. The investigation of finite beam effects, that would require a more elaborate model, possibly without explicit analytical solutions, is beyond the scope of this paper. The proposed model could nonetheless be straightforwardly extended to waveguide structures by using mode propagation constants instead of wave vectors, introducing mode overlap integrals, and, if necessary, adding waveguide losses.
 figure: Fig. 1.

Fig. 1. Schematic of the BWOPO based on a periodically domain-inverted ferroelectric crystal.

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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

$$\frac{{\partial {A_b}}}{{\partial z}} - \frac{1}{{{v_g}}}\frac{{\partial {A_b}}}{{\partial t}} = - i\kappa {A_p}A_f^*,$$
$$\frac{{\partial {A_f}}}{{\partial z}} + \frac{1}{{{v_g}}}\frac{{\partial {A_f}}}{{\partial t}} = i\kappa {A_p}A_b^*,$$
$$\frac{{\partial {A_p}}}{{\partial z}} + \frac{1}{{{v_g}}}\frac{{\partial {A_p}}}{{\partial t}} = i\kappa {A_b}{A_f},$$
where perfect quasi-phase matching is also assumed to investigate the BWOPO. As shown in Section 3.B, this is actually a necessary condition for parametric oscillation.

B. Continuous-Wave Regime

In CW regime, one can discard time dependence and the related partial derivative. The system (5a)–(5c) thus simplifies to

$$\frac{{{\rm{d}}{A_b}}}{{{\rm{d}}z}} \def\LDeqtab{}= - i\kappa {A_p}A_f^*,$$
$$\frac{{{\rm{d}}{A_f}}}{{{\rm{d}}z}} \def\LDeqtab{}= i\kappa {A_p}A_b^*,$$
$$\frac{{{\rm{d}}{A_p}}}{{{\rm{d}}z}} \def\LDeqtab{}= i\kappa {A_b}{A_f}.$$

To solve the above system, we decompose the complex amplitudes as follows:

$${A_j}(z ) = {u_j}(z )\exp [{{\varphi _j}(z )} ],$$
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:
$${u_f}(0 ) = {u_b}(L ) = 0,$$
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:
$$\frac{{{\rm{d}}{u_b}}}{{{\rm{d}}z}} \def\LDeqtab{}= - \kappa {u_p}{u_f},$$
$$\frac{{{\rm{d}}{u_f}}}{{{\rm{d}}z}}\def\LDeqtab{} = \kappa {u_p}{u_b},$$
$$\frac{{{\rm{d}}{u_p}}}{{{\rm{d}}z}} \def\LDeqtab{}= - \kappa {u_b}{u_f}.$$

From Eqs. (9a)–(9c), the “constants of motion” are found to be

$$u_p^2(z ) + u_f^2(z )\def\LDeqtab{} = u_{p0}^2,$$
$$u_p^2(z ) - u_b^2(z ) \def\LDeqtab{}= u_{p0}^2 - u_{{\rm{out}}}^2,$$
$$u_f^2(z ) + u_b^2(z )\def\LDeqtab{} = u_{{\rm{out}}}^2,$$
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:
$$\frac{{{\rm{d}}{u_f}}}{{{\rm{d}}z}} = \kappa \sqrt {({u_{p0}^2 - u_f^2} )\big({u_{{\rm{out}}}^2 - u_f^2} \big)} ,$$
which may be formally integrated as
$$\kappa z = \int_0^{{u_f}(z )} \frac{{{\rm{d}}{u_f}}}{{\sqrt {({u_{p0}^2 - u_f^2} )\big({u_{{\rm{out}}}^2 - u_f^2} \big)}}}.$$

This latter expression has a solution in terms of Jacobi functions:

$${u_f}(z ) = {u_{{\rm{out}}}} {\rm{sn}}\left({\kappa z {u_{p0}}\!\left|\frac{{u_{{\rm{out}}}^2}}{{u_{p0}^2}}\right.} \right),$$
where the Jacobi inverse function sn is defined by [19]
$$\int_0^x \frac{{{\rm{d}}t}}{{\sqrt {({{a^2} - {t^2}} )({{b^2} - {t^2}} )}}} = \frac{1}{a}{\rm{s}}{{\rm{n}}^{- 1}}\left({\frac{x}{b}\!\left|\frac{{{b^2}}}{{{a^2}}}\right.} \right).$$

The pump and backward waves’ spatial evolutions can then be derived by use of (10a) and (10a):

$${u_p}(z ) = {u_{p0}} {\rm{dn}}\left({\kappa z {u_{p0}}\!\left|\frac{{u_{{\rm{out}}}^2}}{{u_{p0}^2}}\right.} \right),$$
and
$${u_b}(z ) = {u_{{\rm{out}}}} {\rm{cn}}\left({\kappa z {u_{p0}}\!\left|\frac{{u_{{\rm{out}}}^2}}{{u_{p0}^2}}\right.} \right),$$
where we use the relations between the squares of the Jacobi functions: ${\rm{d}}{{\rm{n}}^2}({x|m}) = 1 - m\, {\rm{s}}{{\rm{n}}^2}({x|m})$ and ${\rm{c}}{{\rm{n}}^2}({x|m}) = 1 - {\rm{s}}{{\rm{n}}^2}({x|m})$.

Setting $z = L$ in (13), one obtains the implicit equation for BWOPO CW oscillation:

$${\rm{sn}}\left({\kappa L {u_{p0}}\!\left|\frac{{u_{{\rm{out}}}^2}}{{u_{p0}^2}}\right.} \right) = 1.$$

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 $m \to 0$:

$$u_{{\rm{th}}}^{{\rm{CW}}} = \frac{\pi}{{2\kappa L}}.$$

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:

$$I_{{\rm{th}}}^{{\rm{CW}}} = \frac{{{\epsilon_0} c {n_b} {n_f} {n_p} {\lambda _b} {\lambda _f}}}{{32d_{{\rm{eff}}}^2 {L^2}}}.$$

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

$$\eta = u_{{\rm{out}}}^2/u_{p0}^2,$$
and consider that
$${\left({{u_{p0}}/u_{{\rm{th}}}^{{\rm{CW}}}} \right)^2} = {I_p}/I_{{\rm{th}}}^{{\rm{CW}}}.$$

The implicit equation for BWOPO CW oscillation (17) can then be rewritten:

$${\rm{sn}}\left({\left. {\frac{\pi}{2}\sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} } \right|\eta} \right) = 1.$$

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:

$$\frac{\pi}{2}\sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} = K(\eta ),$$
with
$$K(\eta ) = \int_0^1 \frac{{{\rm{d}}x}}{{\sqrt {({1 - {x^2}} )({1 - \eta {x^2}} )}}}.$$

Equation (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 $K(\eta)$:

$$K(\eta ) = \frac{\pi}{2}\sum\limits_{m = 1}^\infty {\left[{\frac{{{{\left({\frac{1}{2}} \right)}_m}}}{{m!}}} \right]^2}{\eta ^m},$$
where Pochhammer’s symbol is defined as ${(a)_n} = a({a + 1})({a + 2}) \ldots ({a + n - 1})$. Equation (25) yields the following first- and second-order approximations:
$$\eta \simeq 4\left({\sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} - 1} \right),$$
and
$$\eta \simeq \frac{8}{9}\left\{{{{\left[{1 + 9\left({\sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} - 1} \right)} \right]}^{1/2}} - 1} \right\}.$$

Conversely, for high conversion efficiencies ($\eta \to 1$), the following approximation may be considered:

$$K(\eta ) \simeq \frac{1}{2}\ln \left({\frac{{16}}{{1 - \eta}}} \right),$$
which yields
$$\eta \simeq 1 - 16\exp \left({- \pi \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right).$$

As 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.

 figure: Fig. 2.

Fig. 2. CW conversion efficiency as a function of pump intensity normalized to the CW threshold intensity. The SROPO has a coupler reflectivity given by (31), which yields the same CW threshold intensity as the BWOPO.

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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

$$u_{{\rm{th,SROPO}}}^{{\rm{CW}}} = \frac{1}{{\kappa L}}\mathop {\cosh}\nolimits^{- 1} \left({\frac{1}{{\sqrt R}}} \right),$$
where $R$ is the output coupler reflectivity at the signal, while the other losses are neglected, and the mirror reflectivity is zero for the idler and the pump. Combining (18) and (30) enables us to determine the coupler reflectivity, ${R_{{\rm{SROPO}}}}$, which yields the same threshold for the BWOPO and SROPO:
$${R_{{\rm{SROPO}}}} = \frac{1}{{\mathop {\cosh}\nolimits^2 ({\pi /2} )}} \simeq 0.16,$$
where we suppose that the product $\kappa L$ is the same for the two OPOs, i.e., we consider the same nonlinear materials and the same wavelengths. We can now compare the conversion efficiencies of both types of OPOs with the same CW threshold using the SROPO implicit Eq. (16) of [14] with $R = {R_{{\rm{SROPO}}}}$. As seen in Fig. 2, the efficiency of the SROPO grows faster just above threshold until it reaches complete pump depletion for ${I_p} \simeq 1.9 I_{{\rm{th}}}^{{\rm{CW}}}$. This optimal normalized pump intensity, which can be evaluated with (24a) in [14], is different from the value of ${\pi ^2}/4 \simeq 2.4$ usually considered in SROPO because of the low cavity finesse with ${R_{{\rm{SROPO}}}} \simeq 0.16$. Then, for input pump intensity higher than this value, part of the signal and idler is converted back into the pump in the nonlinear crystal, and the SROPO efficiency decreases. Such a back-conversion effect is a well-known limit to the conversion efficiency of pulsed SROPOs [16,17,20]. On the other hand, such a detrimental effect is not observed with the BWOPO where the CW conversion efficiency presents a monotonic growth with an asymptotic limit of one for high pumping rates, without the occurrence of back-conversion.

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

$${A_b}({z,t} ) = {a_b}(t )\cos [{\pi z/({2L} )} ],$$
$${A_f}({z,t} ) = {a_f}(t )\sin [{\pi z/({2L} )} ],$$
$${A_p}({z,t} ) = \pi /({2\kappa L} )\sqrt {{I_p}(t )/I_{{\rm{th}}}^{{\rm{CW}}}} ,$$
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:
$${a_b}(t ) + \frac{{2\tau}}{\pi}\frac{{{\rm{d}}{a_b}(t )}}{{{\rm{d}}t}} = \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} a_f^*(t ),$$
$${a_f}(t ) + \frac{{2\tau}}{\pi}\frac{{{\rm{d}}{a_f}(t )}}{{{\rm{d}}t}} = \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} a_b^*(t ).$$

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

$${I_{b,f}}(t ) = {I_{b,f}}(0 )\exp ({t/{\tau _{{\rm{BWOPO}}}}} ),$$
with
$$\frac{1}{{{\tau _{{\rm{BWOPO}}}}}} = \frac{\pi}{\tau}\left({\sqrt {\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}}} - 1} \right).$$

The 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, ${I_{{\rm{noise}}}}$:

$${\tau _{{\rm{bu}}}} = \frac{{\tau {g_{{\rm{Log}}}}}}{{\pi \left({\sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} - 1} \right)}},$$
with
$${g_{{\rm{Log}}}} = \ln \left({\frac{{{I_{{\det}}}}}{{{I_{{\rm{noise}}}}}}} \right).$$

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:

$${\tau _{{\rm{bu,SROPO}}}} = \tau {g_{{\rm{Log}}}}{\left\{{\ln \left[{{\cosh} \left({\frac{\pi}{2}\sqrt {\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}}}} \right)} \right] - {\ln} \left[{{\cosh} \left({\frac{\pi}{2}} \right)} \right]} \right\}^{- 1}}.$$

The latter expression can be approximated with an accuracy better than 10% using the approximation $\cosh (x) \simeq \exp (x)/2$, which yields

$${\tau _{{\rm{bu,SROPO}}}} \simeq \frac{{2\tau {g_{{\rm{Log}}}}}}{{\pi \!\left({\sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}} - 1} \right)}} = 2{\tau _{{\rm{bu}}}}.$$

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.

 figure: Fig. 3.

Fig. 3. Build-up time of BWOPO and SROPO [with ${R_{{\rm{SROPO}}}} = 1/\mathop {\cosh}\nolimits^2 ({\pi /2})$ and ${\tau _{{\rm{cav}}}} = 2\tau$] as functions of the pump intensity normalized to the CW threshold intensity (solid lines: analytic formulas with ${g_{{\rm{Log}}}} = 18$; symbols: numerical simulation). The left axis provides the build-up time, ${\tau _{{\rm{bu}}}}$, normalized to the light propagation time, $\tau$, through the nonlinear medium, while the right axis gives the corresponding values for a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau = 44.3$ ps). The pump intensity of the top axis is evaluated for a typical CW threshold intensity $I_{{\rm{th}}}^{{\rm{CW}}} = 75 {\rm{MW/c}}{{\rm{m}}^2}$.

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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:

$$\bar I _{{\rm{th}}}^{{\rm{peak}}} = I_{{\rm{th}}}^{{\rm{CW}}}{\left({1 + \frac{\tau}{{\pi {{\bar \tau}_p}}}{g_{{\rm{Log}}}}} \right)^2}.$$

The corresponding threshold fluence is thus

$${\bar J _{{\rm{th}}}} = {\bar \tau _p}{\left({1 + \frac{\tau}{{\pi {{\bar \tau}_p}}}{g_{{\rm{Log}}}}} \right)^2}I_{{\rm{th}}}^{{\rm{CW}}}.$$

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 ${\rm{d}}{\bar J _{{\rm{th}}}}/{\rm{d}}{\bar \tau _p} = 0$:

$${\bar \tau _{p{\rm{,opt}}}} = \tau {g_{{\rm{Log}}}}/\pi ,$$
with the corresponding minimum of the threshold fluence:
$${\bar J _{{\rm{th,min}}}} = 4I_{{\rm{th}}}^{{\rm{CW}}} \tau {g_{{\rm{Log}}}}/\pi ,$$
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

$$\frac{{\bar I _{{\rm{th,SROPO}}}^{{\rm{peak}}}}}{{\bar I _{{\rm{th}}}^{{\rm{peak}}}}} \simeq \frac{{\pi {{\bar \tau}_p} + 2\tau {g_{{\rm{Log}}}}}}{{\pi {{\bar \tau}_p} + \tau {g_{{\rm{Log}}}}}}.$$
The ratio of the pulsed threshold peak intensities is thus between one for long pulse durations, where the conditions become close to the CW regime, and two for short pulses. However, this must be balanced by the fact that the considered equivalent SROPO is not representative of a typical SROPO based on a higher finesse cavity whose threshold intensity would actually be lower.

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

$${\bar \eta _{{\rm{pulse}}}} = \frac{{{{\bar \tau}_p} - {\tau _{{\rm{bu}}}}}}{{{{\bar \tau}_p}}}\eta ,$$
where $\eta$ is given by (23) and ${\tau _{{\rm{bu}}}}$ by (36). The value of ${\bar \eta _{{\rm{pulse}}}}$ can then be numerically evaluated by solving (45). If one also approximates $\eta$ by its asymptotic value for strong pumping (29), an analytical formula can be derived for a square temporal pulse profile:
$$\begin{split}{\bar \eta _{{\rm{pulse}}}} &= \left[{1 - \frac{{\tau {g_{{\rm{Log}}}}}}{{{{\bar \tau}_p}\pi \left({\sqrt {\bar I _p^{{\rm{peak}}}/I_{{\rm{th}}}^{{\rm{CW}}}} - 1} \right)}}} \right]\\&\quad\times\left[{1 - 16\exp \left({- \pi \sqrt {\bar I _p^{{\rm{peak}}}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right)} \right].\end{split}$$
The above equation can also be expressed in terms of pump fluence $\bar J = {\bar \tau _p}\bar I _p^{{\rm{peak}}}$.

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.

 figure: Fig. 4.

Fig. 4. BWOPO conversion efficiency as a function of the pulse duration for a square pulse profile and several pump fluences [solid lines: calculation with (45), with ${g_{{\rm{Log}}}} = 18$; dashed lines: approximated expression (46); symbol: numerical simulation]. The bottom axis provides the square pulse duration, ${\bar \tau _p}$, normalized to the light propagation time, $\tau$, through the nonlinear medium, while the top axis gives the corresponding values for a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau \simeq 44.3$ ps). For a typical CW threshold intensity $I_{{\rm{th}}}^{{\rm{CW}}} = 75 \;{\rm{MW/c}}{{\rm{m}}^2}$, the three considered pump fluences are 0.083, 0.166, and $0.665 \;{\rm{J/c}}{{\rm{m}}^2}$. The black dotted line provides the optimum duration and corresponding pulsed efficiency for increasing fluences.

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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.

 figure: Fig. 5.

Fig. 5. Temporal profiles of input pump (dashed black line), depleted pump (blue line), forward pulse (green line), and backward pulse (red line) corresponding to calculated efficiency in Fig. 4 for $J = 200 \tau I_{{\rm{th}}}^{{\rm{CW}}}$ for incident pump pulse durations of (a) ${\bar \tau _p} = 10 \tau$, (b) $50 \tau$, and (c) $100 \tau$.

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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.

 figure: Fig. 6.

Fig. 6. BWOPO conversion efficiency as a function of the pump peak intensity for a square pulse profile and several durations [solid lines: calculation with (45), with ${g_{{\rm{Log}}}} = 18$; dashed lines: approximated expression (46); symbol: numerical simulation). For a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau \simeq 44.3$ ps), the considered durations are 0.22, 0.44, 2.21 and 4.43 ns.

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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

$$\frac{{{\rm{d}}{{\bar \eta}_{{\rm{pulse}}}}}}{{{{\bar \eta}_{{\rm{pulse}}}}}} = \frac{{{\rm{d}}{I_{{\rm{noise}}}}}}{{{I_{{\rm{noise}}}}}}\frac{1}{{\left({\pi {{\bar \tau}_p}/\tau + {g_{{\rm{Log}}}}} \right)\left({\sqrt {\bar I _p^{{\rm{peak}}}/\bar I _{{\rm{th}}}^{{\rm{peak}}}} - 1} \right)}},$$
where (37) and (40) were used for the derivation of the latter expression. As expected, a higher output fluence instability is awaited close to the oscillation threshold. In addition, the shorter is the pulse duration, the higher is the anticipated fluctuation magnitude. This feature is related to the higher relative impact of the build-up time fluctuation on the pulsed efficiency for shorter pulses. For a given pumping ratio and with $\tau = 44.3$ ps, it is thus expected to reduce the fluctuation by typically an order of magnitude by increasing the pulse duration from 200 ps to 5 ns. This trend is confirmed by numerical simulation carried out with a random initial noise for various pulse durations (not shown here). A similar analysis could straightforwardly be carried out to evaluate the influence of pump intensity or pulse duration fluctuations.

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:

$${I_p}(t ) = I_p^{{\rm{peak}}}\exp \big({- 2{t^2}/\tau _p^2} \big).$$

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

$${\bar \tau _p} = {\tau _p}\sqrt {2\ln \left({I_p^{{\rm{peak}}}/I_{{\rm{th}}}^{{\rm{CW}}}} \right)} .$$
The peak power $\bar I _p^{{\rm{peak}}}$ of the equivalent square pulse is defined so that the overall gain experienced by the forward and backward waves is similar to the one provided by the Gaussian pulse when ${I_p}(t) \gt I_{{\rm{th}}}^{{\rm{CW}}}$. From (33), this yields the following relation:
$${\bar \tau _p}\sqrt {\bar I _p^{{\rm{peak}}}} = \sqrt {I_p^{{\rm{peak}}}} \int_{- {{\bar \tau}_p}/2}^{{{\bar \tau}_p}/2} \exp \big({- {t^2}/\tau _p^2} \big){\rm{d}}t,$$
where ${\bar \tau _p}$ is given by (49). One thus obtains
$$\bar I _p^{{\rm{peak}}} = I_p^{{\rm{peak}}}\frac{\pi}{4}{\left\{{\frac{{{\rm{erf}}\left[{{{\bar \tau}_p}/({2{\tau _p}} )} \right]}}{{{{\bar \tau}_p}/({2{\tau _p}} )}}} \right\}^2},$$
with
$${\rm{erf}}(\tau ) = \frac{2}{{\sqrt \pi}}\int_0^\tau \exp ({- {t^2}} ){\rm{d}}t.$$

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

$${J_{{\rm{th}}}} = \sqrt {\frac{\pi}{2}} I_{{\rm{th}}}^{{\rm{peak}}}{\tau _p}.$$
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 $I_{{\rm{th}}}^{{\rm{peak}}} = 6.6 I_{{\rm{th}}}^{{\rm{CW}}}$.
 figure: Fig. 7.

Fig. 7. Threshold fluence of BWOPO as a function of pulse duration for a Gaussian pulse profile. The pump intensity is normalized to the CW threshold intensity [solid blue line: numerical solutions to (40), (49), and (50), with ${g_{{\rm{Log}}}} = 18$; dashed red line: analytic formula (55) for short pulses; dashed green line: analytic formula (57) for long pulses; symbols: numerical simulation]. The left axis provides the build-up time, ${\tau _{{\rm{bu}}}}$, normalized to the light propagation time, $\tau$, through the nonlinear medium, while the right axis gives the corresponding values for a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau \simeq 44.3$ ps).

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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:

$$I_{{\rm{th}}}^{{\rm{peak}}}\left({{{\bar \tau}_p} = 2{\tau _p}} \right) = \frac{4}{{\pi {\rm{er}}{{\rm{f}}^2}\left(1 \right)}}{\left({1 + \frac{\tau}{{2\pi {\tau _p}}}{g_{{\rm{Log}}}}} \right)^2}I_{{\rm{th}}}^{{\rm{CW}}}.$$

We can then use (53) to determine the corresponding threshold fluence:

$${J_{{\rm{th}}}}\left({{{\bar \tau}_p} = 2{\tau _p}} \right) = 2.25 {\tau _p}{\left({1 + \frac{{{g_{{\rm{Log}}}} {n_g} L}}{{2\pi {\tau _p} c}}} \right)^2}I_{{\rm{th}}}^{{\rm{CW}}},$$
with $\tau = {n_g}L/c$, and where we use the numerical evaluation of the factor $2\sqrt 2 /[{\sqrt \pi {\rm{er}}{{\rm{f}}^2}(1)}] = 2.25$. One could notice that the Gaussian pulse threshold fluence provided by (55) is actually close to threshold fluence for a square pulse profile (41): ${J_{{\rm{th}}}}({{{\bar \tau}_p} = 2{\tau _p}}) = 1.125 {\bar J _{{\rm{th}}}}$. Moreover, as seen in Fig. 7, the accuracy of (54) and (55) is the highest close to the pulse duration leading to the minimal threshold fluence, i.e., ${\tau _{p{\rm{,opt}}}} = \tau {g_{{\rm{Log}}}}/({2\pi})$.

For longer pump pulses, a better accuracy is obtained assuming ${\bar \tau _p} = {\tau _p}$. The corresponding approximated expressions are then

$$I_{{\rm{th}}}^{{\rm{peak}}}\left({{{\bar \tau}_p} = {\tau _p}} \right) = \frac{1}{{\pi {\rm{er}}{{\rm{f}}^2}\left({1/2} \right)}}{\left({1 + \frac{\tau}{{\pi {\tau _p}}}{g_{{\rm{Log}}}}} \right)^2}I_{{\rm{th}}}^{{\rm{CW}}},$$
and
$${J_{{\rm{th}}}}\left({{{\bar \tau}_p} = {\tau _p}} \right) = 1.47 {\tau _p}{\left({1 + \frac{{{g_{{\rm{Log}}}} {n_g} L}}{{\pi {\tau _p} c}}} \right)^2}I_{{\rm{th}}}^{{\rm{CW}}},$$
where we use the numerical evaluation of the factor $1/[{\sqrt {2\pi} {\rm{er}}{{\rm{f}}^2}({1/2})}] = 1.47$.

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.

Tables Icon

Table 1. Measured Threshold Peak Intensities of Pulsed BWOPO Reported in the Literature and Corresponding Theoretical Values Determined with the Expressions Derived in This Worka

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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

$$\Delta k({\Delta \omega} ) = k({{\omega _b} + \Delta \omega} ) - k({{\omega _b}} ) - k({{\omega _f} - \Delta \omega} ) + k({{\omega _f}} ).$$
A first-order Taylor expansion yields
$$\Delta k\left({\Delta \omega} \right) = \left({\frac{1}{{{v_{\textit{gb}}}}} + \frac{1}{{{v_{\textit{gf}}}}}} \right)\Delta \omega ,$$
The difference-frequency generation (DFG) bandwidth for backward interaction, defined by the condition $\Delta kL/2 = \pi$, thus reads in terms of frequency, $\nu = \omega /({2\pi})$, rather than angular frequency,
$$\Delta {\nu _{{\rm{BWDFG}}}} = \frac{c}{{2{n_g}L}},$$
where ${n_g}$ is the average group index, ${n_g} = ({{n_{\textit{gb}}} + {n_{\textit{gf}}}})/2$.

One can compare the latter expression with the same definition of the DFG acceptance bandwidth for forward interaction:

$$\Delta {\nu _{{\rm{FWDFG}}}} = \frac{c}{{\Delta {n_g}L}},$$
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

$$\frac{{\Delta {\nu _{{\rm{FWDFG}}}}}}{{\Delta {\nu _{{\rm{BWDFG}}}}}} = \frac{{2{n_g}}}{{\Delta {n_g}}}.$$
In terms of the DFG acceptance bandwidth, backward interaction is thus equivalent to forward interaction in a nonlinear medium with a “giant” group index difference equal to $2{n_g}$. If one considers typical values ${n_g} = 1.9$ and $\Delta {n_g} = 0.02$ [4], the acceptance bandwidth for backward interaction is thus more than 100 times narrower than for forward interaction, as pointed out in early backward DFG experiments [21]. This narrow acceptance bandwidth is very promising to generate Fourier-transform-limited pulses (at both ${\omega _b}$ and ${\omega _f}$) from a backward OPO without any additional spectral filtering, provided that the pump spectrum is also narrow.

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:

$$\frac{{{\rm{d}}{A_b}}}{{{\rm{d}}z}} = - i\kappa {A_{p0}}A_f^*\exp ({i\Delta kz} ),$$
$$\frac{{{\rm{d}}{A_f}}}{{{\rm{d}}z}} = i\kappa {A_{p0}}A_b^*\exp ({i\Delta kz} ),$$
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:
$$\begin{split}{A_f}(L ) &= {A_f}(0 ){\big({4{\kappa ^2}{{| {{A_{p0}}} |}^2} + \Delta {k^2}} \big)^{1/2}} {e^{i \Delta k L/2}}\big/\\&\quad\left\{{{{\big({4{\kappa ^2}{{| {{A_{p0}}} |}^2} + \Delta {k^2}} \big)}^{1/2}}} {\cos} \left[\!{{\big({4{\kappa ^2}{{| {{A_{p0}}} |}^2} + \Delta {k^2}} \big)}^{1/2}}L/2\! \right] \right.\\&\quad+\left. i \Delta k \sin \left[{{{\big({4{\kappa ^2}{{| {{A_{p0}}} |}^2} + \Delta {k^2}} \big)}^{1/2}}L/2} \right] \right\}.\end{split}$$
The gain for the forward intensity, ${| {{A_f}(L)} |^2}/{| {{A_f}(0)} |^2} = 1 + {G_{{\rm{BWOPA}}}}$, can be determined to be
$${G_{{\rm{BWOPA}}}} = \frac{{\mathop {\sin}\nolimits^2 \left[{({\pi /2} ){{\left({{I_p}/I_{{\rm{th}}}^{{\rm{CW}}} + \Delta {k^2}{L^2}/{\pi ^2}} \right)}^{1/2}}} \right]{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}}{{\mathop {\cos}\nolimits^2 \left[{({\pi /2} ){{\left({{I_p}/I_{{\rm{th}}}^{{\rm{CW}}} + \Delta {k^2}{L^2}/{\pi ^2}} \right)}^{1/2}}} \right]{I_p}/I_{{\rm{th}}}^{{\rm{CW}}} + \Delta {k^2}{L^2}/{\pi ^2}}},$$
where the incident pump intensity ${I_p}$ is normalized by the BWOPO threshold (19). For $\Delta k = 0$, one can notice that ${G_{{\rm{BWOPA}}}}$ tends to infinity when ${I_p} \to I_{{\rm{th}}}^{{\rm{CW}}}$, which is consistent with the onset of the parametric oscillation. On the other hand, as soon as $\Delta k \ne 0$, there is no value of the pump intensity that makes the denominator in (65) equal to zero, and it is thus not possible to reach oscillation since
$${G_{{\rm{BWOPA}}}} \le \frac{{{\pi ^2}}}{{\Delta {k^2}{L^2}}}\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}}.$$

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:

$${G_{{\rm{BWOPA}}}} \simeq \frac{4}{{{\pi ^2}{{\left({1 - \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right)}^2} + 4\Delta {k^2}{L^2}/{\pi ^2}}}.$$
It is then straightforward to derive the phase-mismatch acceptance, $\Delta {k_{{\rm{BWOPA}}}}$, at half maximum of the BWOPA gain given by (67), which reads
$$\Delta {k_{{\rm{BWOPA}}}} = \frac{{{\pi ^2}}}{{2L}}\left({1 - \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right).$$
The accuracy of the above approximation is better than 10% for ${I_p} \gt 0.35I_{{\rm{th}}}^{{\rm{CW}}}$, compared to the bandwidth derived by numerical calculation with (65).

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)

$$\begin{split}{G_{{\rm{CE}} - {\rm{FWOPA}}}} &= \mathop {\sinh}\nolimits^2 \left({\frac{\pi}{2}} \right) \mathop {\sinh}\nolimits^2 \left[{\frac{\pi}{2}{{\left({\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}} - \frac{{\Delta {k^2}{L^2}}}{{{\pi ^2}}}} \right)}^{1/2}}} \right]\\&\quad\times\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}}\!\left/\!\left({\mathop {\cosh}\nolimits^2 \left({\frac{\pi}{2}} \right){{\left({\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}} - \frac{{\Delta {k^2}{L^2}}}{{{\pi ^2}}}} \right)}^{1/2}}} \right.\right.\\& \quad- \left\{\mathop {\cosh}\nolimits^2 \left[{\frac{\pi}{2}{{\left({\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}} - \frac{{\Delta {k^2}{L^2}}}{{{\pi ^2}}}} \right)}^{1/2}}} \right]\right.\\&\left.\left.\quad\times\frac{{{I_p}}}{{I_{{\rm{th}}}^{{\rm{CW}}}}} - \frac{{\Delta {k^2}{L^2}}}{{{\pi ^2}}} \right\}^2 \right)^2.\end{split}$$
In same way as for the BWOPA gain, one can derive an approximate expansion of (69) for ${I_p}$ close to $I_{{\rm{th}}}^{{\rm{CW}}}$ and for a small phase mismatch:
$$\begin{split}&{G_{{\rm{CE}} - {\rm{FWOPA}}}} \simeq \\&\frac{{4\mathop {\sinh}\nolimits^2 ({\pi /2} )}}{{{{\left\{{\pi \left({1 - \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right) + \left[{\pi /2 - \tanh ({\pi /2} )} \right]\Delta {k^2}{L^2}/{\pi ^2}} \right\}}^2}}}.\end{split}$$

The corresponding phase-mismatch acceptance at half maximum of ${G_{{\rm{CE}} - {\rm{FWOPA}}}}$ is given by

$$\begin{split}\Delta {k_{{\rm{CE}} - {\rm{FWOPA}}}} &= \frac{\pi}{L}{\left[{\frac{{2\left({\sqrt 2 - 1} \right)}}{{1 - ({2/\pi} )\tanh \left({\pi /2} \right)}}} \right]^{1/2}}\\&\quad\times{\left({1 - \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right)^{1/2}}.\end{split}$$
To have an accuracy better than 10% compared to numerical calculation with (69), the pump intensity must satisfy ${I_p} \gt 0.7I_{{\rm{th}}}^{{\rm{CW}}}$.

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

$$\frac{{\Delta {\nu _{{\rm{CE}} - {\rm{FWOPA}}}}}}{{\Delta {\nu _{{\rm{BWOPA}}}}}} = 0.9 \times \frac{{2{n_g}/\Delta {n_g}}}{{{{\left({1 - \sqrt {{I_p}/I_{{\rm{th}}}^{{\rm{CW}}}}} \right)}^{1/2}}}},$$
where the leading factor is numerically evaluated for the sake of readability. The analysis of (72) thus reveals that the BWOPA bandwidth narrows faster than the one of the CE-FWOPA when the pump intensity approaches the OPO threshold, i.e., the ratio of the two bandwidths tends to infinity. This means that the very large reduction of the bandwidth that already occurs at low gain is further increased when the pump intensity approaches the oscillation threshold. Figure 8 illustrates these singular spectral properties for ${I_p} = 0.95I_{{\rm{th}}}^{{\rm{CW}}}$ where one can notice that the gain for the CE-FWOPA is basically flat over the full bandwidth of the BWOPA. These features are confirmed in the following for pulsed parametric oscillation.
 figure: Fig. 8.

Fig. 8. Gain of the BWOPA and CE-FWOPA as functions of the relative frequency mismatch from perfect phase matching [solid lines: formulas (65) and (69); dashed lines: approximated expressions (67) and (70)]. For the calculation, we set ${I_p} = 0.95 I_{{\rm{th}}}^{{\rm{CW}}}$, ${n_g} = 1.9$, $\Delta {n_g} = 0.02$, and $L = 7$ mm. Note the change in scale in the middle section of the $x$ axis.

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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., $2{n_g} \gg \Delta {n_g}$.

Tables Icon

Table 2. Parameters Used for the Numerical Simulation of the Backward OPO and Forward Singly Resonant OPO

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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.

 figure: Fig. 9.

Fig. 9. (a) Temporal profiles of input pump, depleted pump, and forward and backward pulses of the BWOPO calculated by numerical simulation with parameters in Table 2; (b) corresponding spectrum of the forward wave.

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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.

 figure: Fig. 10.

Fig. 10. (a) Temporal profiles of input pump, depleted pump, and signal and idler pulses of the SROPO calculated by numerical simulation with parameters in Table 2; (b) corresponding spectrum of the signal wave.

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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]

$$\frac{{\partial {A_i}}}{{\partial z}} + \frac{1}{{{v_{\textit{gi}}}}}\frac{{\partial {A_i}}}{{\partial t}} = i\kappa {A_p}A_s^*\exp ({i\Delta kz} ),$$
$$\frac{{\partial {A_s}}}{{\partial z}} + \frac{1}{{{v_{\textit{gs}}}}}\frac{{\partial {A_s}}}{{\partial t}} = i\kappa {A_p}A_i^*\exp ({i\Delta kz} ),$$
$$\frac{{\partial {A_p}}}{{\partial z}} + \frac{1}{{{v_{\textit{gp}}}}}\frac{{\partial {A_p}}}{{\partial t}} = i\kappa {A_i}{A_s}\exp \left({- i\Delta kz} \right).$$
Considering monochromatic continuous waves and assuming that the pump wave is not depleted, i.e., ${A_p}(z) = {A_{p0}}$, the latter coupled equations reduce to
$$\frac{{{\rm{d}}{A_i}}}{{{\rm{d}}z}} = i\kappa {A_{p0}}A_s^*\exp ({i\Delta kz} ),$$
$$\frac{{{\rm{d}}{A_s}}}{{{\rm{d}}z}} = i\kappa {A_{p0}}A_i^*\exp ({i\Delta kz} ).$$
When there is no input for the idler wave, i.e., ${A_i}(0) = 0$, the gain for the signal intensity after a single pass in the FWOPA, ${| {{A_s}(L)} |^2}/{| {{A_s}(0)} |^2} = 1 + {G_{{\rm{FWOPA}}}}$, can be determined by solving (A2a) and (A2b) to be
$$\begin{split}{G_{{\rm{FWOPA}}}}&= \frac{{4{\kappa ^2}{{| {{A_{p0}}} |}^2}}}{{4{\kappa ^2}{{| {{A_{p0}}} |}^2} - \Delta {k^2}}} \\&\quad\times\mathop {\sinh}\nolimits^2\left[{{{\left({4{\kappa ^2}{{| {{A_{p0}}} |}^2} - \Delta {k^2}} \right)}^{1/2}}L/2} \right].\end{split}$$

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

$${I_s}(t ) = {I_s}(0 ){\left[{R\mathop {\cosh}\nolimits^2 \left({\kappa | {{A_{p0}}} |L} \right)} \right]^{t/{\tau _{{\rm{cav}}}}}},$$
where perfect phase matching was assumed in (A3). The above expression can be rewritten as
$${I_s}(t ) = {I_s}(0 )\exp \big({t/{\tau _{{\rm{SROPO}}}}} \big),$$
with
$$\begin{split}&{\tau _{{\rm{SROPO}}}} =\\& \frac{{{\tau _{{\rm{cav}}}}}}{{\ln (R ) + 2\ln \left\{{\cosh \left[{\mathop {\cosh}\nolimits^{- 1} \left({1/\sqrt R} \right){u_{p0}}/u_{{\rm{th,SROPO}}}^{{\rm{CW}}}} \right]} \right\}}},\end{split}$$
and
$$u_{{\rm{th,SROPO}}}^{{\rm{CW}}} = \frac{1}{{\kappa L}}\mathop {\cosh}\nolimits^{- 1} \left({\frac{1}{{\sqrt R}}} \right).$$

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

$${G_{{\rm{CE}} - {\rm{FWOPA}}}} = \frac{{\left({1 - R} \right){G_{{\rm{FWOPA}}}}}}{{{{\left[{1 - \sqrt {R\left({1 + {G_{{\rm{FWOPA}}}}} \right)}} \right]}^2}}},$$
such that the signal intensity incident on the cavity, $I_s^{{\rm{in}}}$, is related to the output signal intensity, $I_s^{{\rm{out}}}$, by $I_s^{{\rm{out}}}/I_s^{{\rm{in}}} = 1 + {G_{{\rm{CE}} - {\rm{FWOPA}}}}$. Inserting (A3) into (A8), one obtains the following expression:
$$\begin{split}&G_{{\rm{CE}} - {\rm{FWOPA}}} \\&= ({1 - R} )4{\kappa ^2}{| {{A_{p0}}} |^2} \mathop {\sinh}\nolimits^2 \left[{{{\big({4{\kappa ^2}{{| {{A_{p0}}} |}^2} - \Delta {k^2}} \big)}^{1/2}}L/2} \right]\\&\quad\bigg/\left[\big({4{\kappa ^2}{{| {{A_{p0}}} |}^2} - \Delta {k^2}} \big)^{1/2}- \sqrt R \left\{4{\kappa ^2}{{| {{A_{p0}}}s |}^2} \right.\right.\\&\quad\times \mathop{\cosh}\nolimits^2\left.\left.\left[{{{\left({4{\kappa ^2}{{| {{A_{p0}}} |}^2} - \Delta {k^2}} \right)}^{1/2}}L/2} \right] - \Delta {k^2} \right\}^{1/2}\right]^2.\end{split}$$

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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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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Figures (10)
Fig. 1.
Fig. 1. Schematic of the BWOPO based on a periodically domain-inverted ferroelectric crystal.

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Fig. 2.
Fig. 2. CW conversion efficiency as a function of pump intensity normalized to the CW threshold intensity. The SROPO has a coupler reflectivity given by (31), which yields the same CW threshold intensity as the BWOPO.

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Fig. 3.
Fig. 3. Build-up time of BWOPO and SROPO [with ${R_{{\rm{SROPO}}}} = 1/\mathop {\cosh}\nolimits^2 ({\pi /2})$ and ${\tau _{{\rm{cav}}}} = 2\tau$] as functions of the pump intensity normalized to the CW threshold intensity (solid lines: analytic formulas with ${g_{{\rm{Log}}}} = 18$; symbols: numerical simulation). The left axis provides the build-up time, ${\tau _{{\rm{bu}}}}$, normalized to the light propagation time, $\tau$, through the nonlinear medium, while the right axis gives the corresponding values for a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau = 44.3$ ps). The pump intensity of the top axis is evaluated for a typical CW threshold intensity $I_{{\rm{th}}}^{{\rm{CW}}} = 75 {\rm{MW/c}}{{\rm{m}}^2}$.

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Fig. 4.
Fig. 4. BWOPO conversion efficiency as a function of the pulse duration for a square pulse profile and several pump fluences [solid lines: calculation with (45), with ${g_{{\rm{Log}}}} = 18$; dashed lines: approximated expression (46); symbol: numerical simulation]. The bottom axis provides the square pulse duration, ${\bar \tau _p}$, normalized to the light propagation time, $\tau$, through the nonlinear medium, while the top axis gives the corresponding values for a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau \simeq 44.3$ ps). For a typical CW threshold intensity $I_{{\rm{th}}}^{{\rm{CW}}} = 75 \;{\rm{MW/c}}{{\rm{m}}^2}$, the three considered pump fluences are 0.083, 0.166, and $0.665 \;{\rm{J/c}}{{\rm{m}}^2}$. The black dotted line provides the optimum duration and corresponding pulsed efficiency for increasing fluences.

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Fig. 5.
Fig. 5. Temporal profiles of input pump (dashed black line), depleted pump (blue line), forward pulse (green line), and backward pulse (red line) corresponding to calculated efficiency in Fig. 4 for $J = 200 \tau I_{{\rm{th}}}^{{\rm{CW}}}$ for incident pump pulse durations of (a) ${\bar \tau _p} = 10 \tau$, (b) $50 \tau$, and (c) $100 \tau$.

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Fig. 6.
Fig. 6. BWOPO conversion efficiency as a function of the pump peak intensity for a square pulse profile and several durations [solid lines: calculation with (45), with ${g_{{\rm{Log}}}} = 18$; dashed lines: approximated expression (46); symbol: numerical simulation). For a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau \simeq 44.3$ ps), the considered durations are 0.22, 0.44, 2.21 and 4.43 ns.

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Fig. 7.
Fig. 7. Threshold fluence of BWOPO as a function of pulse duration for a Gaussian pulse profile. The pump intensity is normalized to the CW threshold intensity [solid blue line: numerical solutions to (40), (49), and (50), with ${g_{{\rm{Log}}}} = 18$; dashed red line: analytic formula (55) for short pulses; dashed green line: analytic formula (57) for long pulses; symbols: numerical simulation]. The left axis provides the build-up time, ${\tau _{{\rm{bu}}}}$, normalized to the light propagation time, $\tau$, through the nonlinear medium, while the right axis gives the corresponding values for a typical nonlinear crystal with $L = 7$ mm and ${n_g} = 1.9$ (i.e., $\tau \simeq 44.3$ ps).

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Fig. 8.
Fig. 8. Gain of the BWOPA and CE-FWOPA as functions of the relative frequency mismatch from perfect phase matching [solid lines: formulas (65) and (69); dashed lines: approximated expressions (67) and (70)]. For the calculation, we set ${I_p} = 0.95 I_{{\rm{th}}}^{{\rm{CW}}}$, ${n_g} = 1.9$, $\Delta {n_g} = 0.02$, and $L = 7$ mm. Note the change in scale in the middle section of the $x$ axis.

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Fig. 9.
Fig. 9. (a) Temporal profiles of input pump, depleted pump, and forward and backward pulses of the BWOPO calculated by numerical simulation with parameters in Table 2; (b) corresponding spectrum of the forward wave.

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Fig. 10.
Fig. 10. (a) Temporal profiles of input pump, depleted pump, and signal and idler pulses of the SROPO calculated by numerical simulation with parameters in Table 2; (b) corresponding spectrum of the signal wave.

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Tables (2)

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Table 1. Measured Threshold Peak Intensities of Pulsed BWOPO Reported in the Literature and Corresponding Theoretical Values Determined with the Expressions Derived in This Worka

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Table 2. Parameters Used for the Numerical Simulation of the Backward OPO and Forward Singly Resonant OPO

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Equations (99)

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∂ A b ∂ z − 1 v gb ∂ A b ∂ t = − i κ A p A f ∗ exp ⁡ ( i Δ k z ) ,
∂ A f ∂ z + 1 v gf ∂ A f ∂ t = i κ A p A b ∗ exp ⁡ ( i Δ k z ) ,
∂ A p ∂ z + 1 v gp ∂ A p ∂ t = i κ A b A f exp ⁡ ( − i Δ k z ) ,
κ = d e f f c ω b ω f ω p n b n f n p ,
E b ( z , t ) = 1 2 ω b n b A b ( z , t ) exp [ i ( − k b z − ω b t ) ] + c.c. ,
E p , f ( z , t ) = 1 2 ω p , f n p , f A p , f ( z , t ) exp ⁡ [ i ( k p , f z − ω p , f t ) ] + c.c. ,
Δ k = k p − k f + k b − K G ,
∂ A b ∂ z − 1 v g ∂ A b ∂ t = − i κ A p A f ∗ ,
∂ A f ∂ z + 1 v g ∂ A f ∂ t = i κ A p A b ∗ ,
∂ A p ∂ z + 1 v g ∂ A p ∂ t = i κ A b A f ,
d A b d z = − i κ A p A f ∗ ,
d A f d z = i κ A p A b ∗ ,
d A p d z = i κ A b A f .
A j ( z ) = u j ( z ) exp ⁡ [ φ j ( z ) ] ,
u f ( 0 ) = u b ( L ) = 0 ,
d u b d z = − κ u p u f ,
d u f d z = κ u p u b ,
d u p d z = − κ u b u f .
u p 2 ( z ) + u f 2 ( z ) = u p 0 2 ,
u p 2 ( z ) − u b 2 ( z ) = u p 0 2 − u o u t 2 ,
u f 2 ( z ) + u b 2 ( z ) = u o u t 2 ,
d u f d z = κ ( u p 0 2 − u f 2 ) ( u o u t 2 − u f 2 ) ,
κ z = ∫ 0 u f ( z ) d u f ( u p 0 2 − u f 2 ) ( u o u t 2 − u f 2 ) .
u f ( z ) = u o u t s n ( κ z u p 0 | u o u t 2 u p 0 2 ) ,
∫ 0 x d t ( a 2 − t 2 ) ( b 2 − t 2 ) = 1 a s n − 1 ( x b | b 2 a 2 ) .
u p ( z ) = u p 0 d n ( κ z u p 0 | u o u t 2 u p 0 2 ) ,
u b ( z ) = u o u t c n ( κ z u p 0 | u o u t 2 u p 0 2 ) ,
s n ( κ L u p 0 | u o u t 2 u p 0 2 ) = 1.
u t h C W = π 2 κ L .
I t h C W = ϵ 0 c n b n f n p λ b λ f 32 d e f f 2 L 2 .
η = u o u t 2 / u p 0 2 ,
( u p 0 / u t h C W ) 2 = I p / I t h C W .
s n ( π 2 I p / I t h C W | η ) = 1.
π 2 I p / I t h C W = K ( η ) ,
K ( η ) = ∫ 0 1 d x ( 1 − x 2 ) ( 1 − η x 2 ) .
K ( η ) = π 2 ∑ m = 1 ∞ [ ( 1 2 ) m m ! ] 2 η m ,
η ≃ 4 ( I p / I t h C W − 1 ) ,
η ≃ 8 9 { [ 1 + 9 ( I p / I t h C W − 1 ) ] 1 / 2 − 1 } .
K ( η ) ≃ 1 2 ln ⁡ ( 16 1 − η ) ,
η ≃ 1 − 16 exp ⁡ ( − π I p / I t h C W ) .
u t h , S R O P O C W = 1 κ L cosh − 1 ⁡ ( 1 R ) ,
R S R O P O = 1 cosh 2 ⁡ ( π / 2 ) ≃ 0.16 ,
A b ( z , t ) = a b ( t ) cos ⁡ [ π z / ( 2 L ) ] ,
A f ( z , t ) = a f ( t ) sin ⁡ [ π z / ( 2 L ) ] ,
A p ( z , t ) = π / ( 2 κ L ) I p ( t ) / I t h C W ,
a b ( t ) + 2 τ π d a b ( t ) d t = I p / I t h C W a f ∗ ( t ) ,
a f ( t ) + 2 τ π d a f ( t ) d t = I p / I t h C W a b ∗ ( t ) .
I b , f ( t ) = I b , f ( 0 ) exp ⁡ ( t / τ B W O P O ) ,
1 τ B W O P O = π τ ( I p I t h C W − 1 ) .
τ b u = τ g L o g π ( I p / I t h C W − 1 ) ,
g L o g = ln ⁡ ( I det I n o i s e ) .
τ b u , S R O P O = τ g L o g { ln ⁡ [ cosh ( π 2 I p I t h C W ) ] − ln [ cosh ( π 2 ) ] } − 1 .
τ b u , S R O P O ≃ 2 τ g L o g π ( I p / I t h C W − 1 ) = 2 τ b u .
I ¯ t h p e a k = I t h C W ( 1 + τ π τ ¯ p g L o g ) 2 .
J ¯ t h = τ ¯ p ( 1 + τ π τ ¯ p g L o g ) 2 I t h C W .
τ ¯ p , o p t = τ g L o g / π ,
J ¯ t h , m i n = 4 I t h C W τ g L o g / π ,
I ¯ t h , S R O P O p e a k I ¯ t h p e a k ≃ π τ ¯ p + 2 τ g L o g π τ ¯ p + τ g L o g .
η ¯ p u l s e = τ ¯ p − τ b u τ ¯ p η ,
η ¯ p u l s e = [ 1 − τ g L o g τ ¯ p π ( I ¯ p p e a k / I t h C W − 1 ) ] × [ 1 − 16 exp ⁡ ( − π I ¯ p p e a k / I t h C W ) ] .
d η ¯ p u l s e η ¯ p u l s e = d I n o i s e I n o i s e 1 ( π τ ¯ p / τ + g L o g ) ( I ¯ p p e a k / I ¯ t h p e a k − 1 ) ,
I p ( t ) = I p p e a k exp ⁡ ( − 2 t 2 / τ p 2 ) .
τ ¯ p = τ p 2 ln ⁡ ( I p p e a k / I t h C W ) .
τ ¯ p I ¯ p p e a k = I p p e a k ∫ − τ ¯ p / 2 τ ¯ p / 2 exp ⁡ ( − t 2 / τ p 2 ) d t ,
I ¯ p p e a k = I p p e a k π 4 { e r f [ τ ¯ p / ( 2 τ p ) ] τ ¯ p / ( 2 τ p ) } 2 ,
e r f ( τ ) = 2 π ∫ 0 τ exp ⁡ ( − t 2 ) d t .
J t h = π 2 I t h p e a k τ p .
I t h p e a k ( τ ¯ p = 2 τ p ) = 4 π e r f 2 ( 1 ) ( 1 + τ 2 π τ p g L o g ) 2 I t h C W .
J t h ( τ ¯ p = 2 τ p ) = 2.25 τ p ( 1 + g L o g n g L 2 π τ p c ) 2 I t h C W ,
I t h p e a k ( τ ¯ p = τ p ) = 1 π e r f 2 ( 1 / 2 ) ( 1 + τ π τ p g L o g ) 2 I t h C W ,
J t h ( τ ¯ p = τ p ) = 1.47 τ p ( 1 + g L o g n g L π τ p c ) 2 I t h C W ,
Δ k ( Δ ω ) = k ( ω b + Δ ω ) − k ( ω b ) − k ( ω f − Δ ω ) + k ( ω f ) .
Δ k ( Δ ω ) = ( 1 v gb + 1 v gf ) Δ ω ,
Δ ν B W D F G = c 2 n g L ,
Δ ν F W D F G = c Δ n g L ,
Δ ν F W D F G Δ ν B W D F G = 2 n g Δ n g .
d A b d z = − i κ A p 0 A f ∗ exp ⁡ ( i Δ k z ) ,
d A f d z = i κ A p 0 A b ∗ exp ⁡ ( i Δ k z ) ,
A f ( L ) = A f ( 0 ) ( 4 κ 2 | A p 0 | 2 + Δ k 2 ) 1 / 2 e i Δ k L / 2 / { ( 4 κ 2 | A p 0 | 2 + Δ k 2 ) 1 / 2 cos [ ( 4 κ 2 | A p 0 | 2 + Δ k 2 ) 1 / 2 L / 2 ] + i Δ k sin ⁡ [ ( 4 κ 2 | A p 0 | 2 + Δ k 2 ) 1 / 2 L / 2 ] } .
G B W O P A = sin 2 ⁡ [ ( π / 2 ) ( I p / I t h C W + Δ k 2 L 2 / π 2 ) 1 / 2 ] I p / I t h C W cos 2 ⁡ [ ( π / 2 ) ( I p / I t h C W + Δ k 2 L 2 / π 2 ) 1 / 2 ] I p / I t h C W + Δ k 2 L 2 / π 2 ,
G B W O P A ≤ π 2 Δ k 2 L 2 I p I t h C W .
G B W O P A ≃ 4 π 2 ( 1 − I p / I t h C W ) 2 + 4 Δ k 2 L 2 / π 2 .
Δ k B W O P A = π 2 2 L ( 1 − I p / I t h C W ) .
G C E − F W O P A = sinh 2 ⁡ ( π 2 ) sinh 2 ⁡ [ π 2 ( I p I t h C W − Δ k 2 L 2 π 2 ) 1 / 2 ] × I p I t h C W / ( cosh 2 ⁡ ( π 2 ) ( I p I t h C W − Δ k 2 L 2 π 2 ) 1 / 2 − { cosh 2 ⁡ [ π 2 ( I p I t h C W − Δ k 2 L 2 π 2 ) 1 / 2 ] × I p I t h C W − Δ k 2 L 2 π 2 } 2 ) 2 .
G C E − F W O P A ≃ 4 sinh 2 ⁡ ( π / 2 ) { π ( 1 − I p / I t h C W ) + [ π / 2 − tanh ⁡ ( π / 2 ) ] Δ k 2 L 2 / π 2 } 2 .
Δ k C E − F W O P A = π L [ 2 ( 2 − 1 ) 1 − ( 2 / π ) tanh ⁡ ( π / 2 ) ] 1 / 2 × ( 1 − I p / I t h C W ) 1 / 2 .
Δ ν C E − F W O P A Δ ν B W O P A = 0.9 × 2 n g / Δ n g ( 1 − I p / I t h C W ) 1 / 2 ,
∂ A i ∂ z + 1 v gi ∂ A i ∂ t = i κ A p A s ∗ exp ⁡ ( i Δ k z ) ,
∂ A s ∂ z + 1 v gs ∂ A s ∂ t = i κ A p A i ∗ exp ⁡ ( i Δ k z ) ,
∂ A p ∂ z + 1 v gp ∂ A p ∂ t = i κ A i A s exp ⁡ ( − i Δ k z ) .
d A i d z = i κ A p 0 A s ∗ exp ⁡ ( i Δ k z ) ,
d A s d z = i κ A p 0 A i ∗ exp ⁡ ( i Δ k z ) .
G F W O P A = 4 κ 2 | A p 0 | 2 4 κ 2 | A p 0 | 2 − Δ k 2 × sinh 2 ⁡ [ ( 4 κ 2 | A p 0 | 2 − Δ k 2 ) 1 / 2 L / 2 ] .
I s ( t ) = I s ( 0 ) [ R cosh 2 ⁡ ( κ | A p 0 | L ) ] t / τ c a v ,
I s ( t ) = I s ( 0 ) exp ⁡ ( t / τ S R O P O ) ,
τ S R O P O = τ c a v ln ⁡ ( R ) + 2 ln ⁡ { cosh ⁡ [ cosh − 1 ⁡ ( 1 / R ) u p 0 / u t h , S R O P O C W ] } ,
u t h , S R O P O C W = 1 κ L cosh − 1 ⁡ ( 1 R ) .
G C E − F W O P A = ( 1 − R ) G F W O P A [ 1 − R ( 1 + G F W O P A ) ] 2 ,
G C E − F W O P A = ( 1 − R ) 4 κ 2 | A p 0 | 2 sinh 2 ⁡ [ ( 4 κ 2 | A p 0 | 2 − Δ k 2 ) 1 / 2 L / 2 ] / [ ( 4 κ 2 | A p 0 | 2 − Δ k 2 ) 1 / 2 − R { 4 κ 2 | A p 0 s | 2 × cosh 2 ⁡ [ ( 4 κ 2 | A p 0 | 2 − Δ k 2 ) 1 / 2 L / 2 ] − Δ k 2 } 1 / 2 ] 2 .
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