Energy Yield of Pulsed Optical Parametric Oscillators: A Rate ...

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 40, NO. 6, JUNE 2004

Energy Yield of Pulsed Optical Parametric Oscillators: A Rate-Equation Analysis Antoine Godard and Emmanuel Rosencher, Senior Member, IEEE

Abstract—In this paper, for the first time, to our best knowledge, single explicit self-contained rate equations describing the dynamic behavior of pulsed optical parametric oscillators are derived, exactly taking into account the parametric interactions between the three waves in the nonlinear crystal, in contrast to most previous approaches which used a linearized expansion of this interaction. Universal expressions using normalized forms are derived which fully include pump depletion and reconstruction effects. Straightforward numerical integration of these equations allow us to study the input energy–output energy characteristics of these pulsed devices. We thus discuss the subthreshold behavior, threshold power, saturation, and build-up time of optical parametric oscillators and find the validity range in which the usual undepleted regime is retrieved as a limit case of the general expressions. Explicit approximated expressions of build-up times are derived which show in particular that the number of round trips needed to reach a CW behavior may be very large. Index Terms—Nonlinear optics, optical parametric oscillators (OPOs), pulsed regime.

I. INTRODUCTION

T

HANKS to recent progresses in nonlinear materials, optical parametric oscillators (OPOs) are emerging as very convenient optical sources for spectroscopic and remote sensing applications [1], [2]. Until recently, most of the papers dealing with OPOs assumed weak parametric interactions, thus assuming small conversion efficiency. Theoretical derivations were then mostly based on low-order Taylor-expansion approximation for the field amplitude variations in the nonlinear crystal. This approximation is no longer valid now that nonlinear materials are becoming so efficient. Recently, Rosencher and Fabre [3] developed a formalism describing continuous wave (CW) behavior of OPOs, taking into account the complete parametric interaction inside the nonlinear crystal, i.e., without resorting to linearized expressions. Nevertheless, though CW OPOs are now routinely operated in many laboratories, pulsed OPOs are still preferred in many applications, mostly because the pump sources needed for OPO operation are more practical for pulsed operation than for CW one. Indeed, for a given quantity of energy (stored in laser pump medium), parametric gain is higher if this energy is delivered in a short time (via -switching, for instance) than in a CW way. Although several theoretical studies have been devoted to the input–output charManuscript received December 29, 2003; revised February 20, 2004. A. Godard is with the Office National d’Études et de Recherches Aérospatiales, F-91761 Palaiseau, France. E. Rosencher with the Office National d’Études et de Recherches Aérospatiales, F-91761 Palaiseau, France and also with the Départment de Physique, Ecole Polytechnique, F-91761 Palaiseau, France (e-mail: [email protected]). Digital Object Identifier 10.1109/JQE.2004.828275

Fig. 1.

Diagram of the ring OPO with the different notations used in this paper.

acteristics of OPO devices in the transient regime, most of them are numerical studies that include diffraction or broad-band spectral effects [4]–[10]. Only a few papers focus on a semi-analytical model of pulsed OPOs without approximation of the field amplitudes, and, to the best of our knowledge, they all rely on iterative methods to obtain the spatio-temporal evolution of the field amplitudes [11], [12]. The purpose of this paper is thus to extend the Rosencher–Fabre [3] formalism to the pulsed case: single explicit self-contained rate equations describing the dynamic behavior of pulsed OPOs are derived, exactly taking into account the parametric interactions between the three waves in the nonlinear crystal. Universal expressions using normalized forms are derived, which fully include pump depletion and reconstruction effects. The geometry of the studied OPO is shown in Fig. 1. For simplicity’s sake, we have supposed a ring configuration. Let be the slowly varying amplitude for the signal, of the different electromagnetic fields ( for the idler, for the pump) in the nonlinear through the crystal, related to the actual electric field , where is relation the linear index of the crystal for the wave of pulsation and wave vector . The normalization factor is is proportional to the chosen so that the intensity (where is the power flux) photon flux , where through the relation is Planck’s constant and the vacuum impedance (377 ). In order to keep things simple, the present model assumes that the different interacting waves are colinear (without birefringent walkoff) plane waves, but it can be straightforwardly extended Gaussian modes, along the to more realistic cases of lines described in [13]–[15]. For the same concern of simplicity, we will assume perfect phase matching in the nonlinear crystal, . i.e., that The effect of nonideal phase matching can also be included in the present model, since the exact solutions of the propagation equations are also known in this case. Finally, since our present paper concentrates on the physical insights gained by a semi-

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GODARD AND ROSENCHER: ENERGY YIELD OF PULSED OPTICAL PARAMETRIC OSCILLATORS: A RATE-EQUATION ANALYSIS

analytical approach, diffraction and group velocity effects that are important in broad-band OPOs are ignored. These effects may be added for a global description of the OPOs at the price of numerical simultations [5]–[10]. With these assumptions in mind, the wave amplitudes evolve within the crystal according to (1a)

In this latter expression, the Jacobi inverse function by

where

is defined

(3) The signal wave is fed back at the cavity entrance by reflection on the cavity mirrors, becoming the input signal wave for the transit step (see Fig. 1), i.e.,

(1b) (1c)

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(4) where reads

is the signal reflectivity. The recurrent equation then

is the nonlinear coupling coefficient given by (5)

and is the material nonlinear susceptibility. In (1a)–(1c), has been omitted [16]–[18]: more the relative phase term . In this way, the comprecisely, we have taken plete depletion effect is taken into account by a change of sign rather than a step of in . This assumption is alin ways valid in singly resonant OPOs (SROs) since, as it is well known, the nonresonant idler phase adapts itself in order to yield the highest possible parametric gain. In doubly resonant OPOs (DROs), this requires independent matching of the signal and idler wave phases, as is obtained in entangled cavity OPOs [3], [19]. In numerical models, the dynamic behavior of, e.g., pulsed SRO is usually obtained in the following way. At each time , the values of the different fields at the OPO input ( , , and ) are propagated to the OPO output by , , ). Then, solving (1a)–(1c), yielding ( describing the optical feedback of the signal wave, the fields , , ) ( are used as the new initial conditions ( is the signal mirror reflectivity). This solution is only numerical and does not gracefully show the main physical constants at work in the OPO, such as time response, and threshold power. The purpose of this paper is to introduce a more convenient theoretical description which allows to describe the output power of OPO directly as a function of the input one, leading to faster calculations and, more importantly, to a more physical description of the dynamic behavior of the OPO including asymptotic behaviors. II. SINGLY RESONANT OPTICAL PARAMETRIC OSCILLATOR Let us consider the evolution of the different wave intensity during one round trip in the singly resonant cavity. The crystal is the distance traveled by light outside the length is and crystal in the cavity. At the round trip, the equation relating to the input pump the signal intensity at the crystal output and is the solution of (1a)–(c) and signal powers and is given by [3] (2)

The increase in signal intensity during a round trip is then

(6) which yields the differential time equation

(7) and are the input pump where and signal intensity. We note that, at steady state, (7) yields directly the pump threshold power (8) . Equation (7) deals with the input signal power which, in We are interested in the signal output intensity the case where no absorption occurs in the mirror, is given by (9) We then make the following change of variables: is the normalized time unit, (respectively ) is the input pump (respectively output signal) power normalized to the power threshold. The final dynamic equation then reads as given in (10a), shown at the bottom of the next page, or, introducing the , we have (10b), shown at the quantum yield bottom of the next page. Equation (10a) is the time differential equation which describes the dynamic behavior of the output signal intensity. We note that this equation is self-contained and has a unique parameter: the mirror signal reflectivity , which makes (10a) and (10b) very universal and useful equations. Clearly, when the input pump power is switched off, the signal power decays to 0 with a characteristic time scale which is nothing else than the signal

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photons lifetime in the cavity. The term on the right-hand side is the nonlinear driving term which describes the pumping of the nonlinear cavity. Equations (10a) or (10b) are easily numerically solved with any standard mathematical software. In order to gain physical insight, we first treat the case of a square pump impulse: the pulse duration is in the normalized in the nonnormalized one) and the pump power is form ( during this period. The average output constant and equal to . This is a relepower is given by vant quantity since it converges to a steady-state value when the pulse duration tends to infinity. Fig. 2(a) shows the variation of the average normalized signal output power as a function of the round-trip number for different normalized input for a reflectivity . pump powers When the pump power is a few times above threshold (e.g., ), the parametric gain is low, the OPO rise time is slow, and it takes more than 500 round trips for the OPO to converge to a steady-state value. This has been numerically observed by Brunner et al. [4], but, thanks to our approach, a literal expression can be derived for the number of round trips necessary to reach the steady state. Indeed, when the pump duration is short, the signal wave has no time to build up in the OPO cavity so that . We can thus use the approxithe yield is small, i.e., mation when [20], so that (10b) can be approximated by the following expression:

, this point is ( , of round trips: for ). Such a point exists for all reflectivities, but we have not found a mathematical proof of the existence of such a point. characFig. 2(c) shows a logarithmic plot of the teristics for low values of the pumping ratio: this figure clearly reveals the arbitrary character of the OPO threshold definition in the pulsed regime, particularly for short pump duration [13]. Moreover, it also reveals an exponential behavior of the characteristics at a pumping ratio in the low-yield regime. This result can be straightforwardly explained: in the low-yield regime, the signal builds up from noise in the OPO cavity so that

(11) The normalized build-up time in the cavity is then given by

(15a)

(13) where is then

is the quantum noise signal power. The average power

(14) Consequently, the described by

versus

relationship is

or

(12a)

(15b)

From this last equation, it is clear that the build-up time constant depends strongly on the pump power . A straightforward expansion of (12a) near threshold ( with ) yields

near threshold. The agreement between this simple model and the results of numerical simulations shown in Fig. 2(c) is excellent. The same results are shown for a Gaussian pulse in Fig. 3 as a function of the which traces the mean output energy mean input power . The same conclusions can then be drawn from Fig. 2, the main one being that an equally slow convergence toward an adiabatic behavior of the OPO is observed for characteristics for square low pumping. Note that the and Gaussian pump pulses are very similar for low values of pumping power [see Figs. 2(b) and 3(b)]. is averaged with respect to One should note that, since the pulse duration, it must be multiplied by the duty cycle to obtain the average power produced by the OPO. Furthermore, it is valid only when the time interval between two pulses is longer so that there is no overlapping between a pulse and the than following one.

(12b) which can be directly derived from (1a)–(1c) in the undepleted pump approximations. In Fig. 2(b), we now present the input–output characteristics of an SRO for different curves should be pump powers. Experimental compared to this model to determine OPO ideality. As in Fig. 2(a), the overshoot in pulsed OPO yield for short pulses is clearly observed since higher yields are obtained in the pulsed regime than in the CW one. This figure numerically shows also the existence of an constant yield point where all the curves intersect, for which the OPO yield is independent on the number

(10a) (10b)


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III. DOUBLY RESONANT OPTICAL PARAMETRIC OSCILLATOR As far as the DRO is concerned, we shall concentrate on the more illustrative case: the balanced DRO, for which the signal and idler mirror reflectivities are equal to . We recall that, in that case, the power conservation along the wave propagation reads [3] (16) where is the input pump power. The evolution of the signal wave is then given by [3, eq. (36a)]

(17)

This formulation allows complete pump depletion to be taken into account by allowing to be negative. Equation (17) integrates trivially to yield (18) This latter equation is easily solved for

as follows: (19)

Using (16), the signal power at the crystal exit is then given by (20) which, after some algebra, can be written as (21)

At threshold, the self-consistency condition for (i.e., ) yields the value of the DRO threshold, i.e., (22) The time evolution of the signal wave power is then obtained in the same way as for the SRO in (4), writing (again with )

Fig. 2. Average yield hY i = hP =P i of an SRO pumped by temporal step impulses as a function of pulse duration (in round-trip number) for different pumping ratios X = P =P . The inset shows an example of transient behavior (a). Average yield of an SRO pumped by temporal step impulses as a function of pumping ratio for different pump pulse duration (b). Logarithmic enlarged vision of (b) near threshold (c). The signal mirror reflectivity is R = 0:9.

(23) We then make the same change of variables as for the SRO: is the normalized time unit, , , and are the

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input pump power and the signal output power at the entrance and the output of the cavity normalized to the power threshold. The normalized form of differential equation is then (24) with (25) This latter expression is somewhat similar to (10a) with another right-hand driving term. Near threshold , the signal power is increasing with time as (26) , i.e., above threshold, which makes that is, as soon as , an exponential sense. Once again, for a step input asymptotic behavior can be derived with a time constant (27a)

or, near threshold

, as (27b)

Following the same line of arguments as in the SRO case, we thus expect a characteristics given by

(28)

Fig. 3. Average yield hY i = hP =P i of an SRO pumped by Gaussian temporal impulses as a function of pulse duration (in round-trip number) for different pumping ratios hX i = hP =P i. The inset shows an example of transient behavior. (a) Average yield of an SRO pumped by Gaussian temporal impulses as a function of pumping ratio for different pump pulse duration (b). Logarithmic enlarged vision of (b) near threshold (c). The signal mirror reflectivity is R = 0:9.

Here again, this expression is in excellent agreement with the numerical simulation near threshold. Equation (24) can be trivially numerically solved and Fig. 4 shows the calcu. Since the threshold power is about lated behavior for ), the 40 times lower for a DRO than for an SRO (for scale of the abscissa axis in Fig. 4 is larger than the one in Fig. 2 curves. so as to recover the overall shape of the Equations (12b) and (27b) allow to compare the build-up time constants in SRO and DRO configurations (see also Table I where the normalized differential time equations are summarized for the two configurations). Since for all values of reflectivities , one finds that the SRO build-up times are smaller than the DRO ones for the same pumping level (normalized to their respective threshold power), a result already obtained through numerical resolution by Brunner et al. [4]. Finally, in Figs. 2–4, it should be noted that the density of . This density should seed photons is arbitrarily taken as have been taken equal to one photon per mode, but in that case no universal curves could have been presented. Of course, a change in this arbitrary number would lead to a shift of the


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TABLE I NORMALIZED DIFFERENTIAL TIME EQUATIONS OF PULSED OPOS IN TWO CONFIGURATIONS: SROS AND BALANCED DROS

IV. CONCLUSION We have developed a formalism adapted to the description of pulsed optical parametric oscillators without resorting to the usual mean field approximation. We obtained a self-contained, autonomous rate equation describing the time evolution of the parametric signal even in the case of heavy pump depletion. These expressions are written in a normalized universal form, with a single parameter (i.e., the mirror reflectivity) which allows a graceful convergence of the pulsed signal toward already charobtained continuous expressions. Moreover, the acteristics of the OPO devices are obtained by straightforward characteristics of practical devices should integration. be compared with these abacus to determine the OPO ideality. Explicit approximated expressions of build up times are derived which show in particular that the number of round trips needed to reach a continuous wave behavior may be very large. Expovs characteristics is also demonnential behavior of the strated near threshold, which could be a useful way to determine the real pulsed OPO threshold.

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[6] Fig. 4. (a) Average yield of a DRO pumped by temporal step impulses as a function of pumping ratio for different pump pulse duration. (b) Logarithmic enlarged vision of (a) near threshold. The signal mirror reflectivity is R = 0:9.

characteristics and particularly in a shift of the apparent threshold (see [13]). Nevertheless, the versus plot in the subthreshold regime is a very useful plot which could be used to check OPO ideality and determine the real pulsed OPO threshold.

[7] [8] [9]

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[10] A. V. Smith, D. J. Armstrong, M. C. Phillips, J. G. Russell, and G. Arisholm, “Degenerate type I nanosecond optical parametric oscillators,” J. Opt. Soc. Amer. B, vol. 20, pp. 2319–2328, 2003. [11] T. Debuisschert, “Nanosecond optical parametric oscillators,” Quantum Semiclass. Opt., vol. 9, pp. 209–219, 1997. [12] K. D. Shaw, “Spatio-temporal evolution of the intra-cavity fields in a pulsed doubly resonant optical parametric oscillator,” Opt. Commun., vol. 144, pp. 134–160, 1997. [13] S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron., vol. QE-15, pp. 415–431, June 1979. [14] C. Schwob, P. F. Cohadon, C. Fabre, M. A. M. Marte, R. H. A. Gatti, and L. Lugiato, “Transverse effects and mode couplings in OPOs,” Appl. Phys. B, vol. 66, pp. 685–699, 1998. [15] S. Schiller, K. Schneider, and J. Mlynek, “Theory of an optical parametric oscillator with resonant pump and signal,” J. Opt. Soc. Amer. B, vol. 16, pp. 1512–1524, 1999. [16] Y. R. Shen, The Principles of Nonlinear Optics. New York: Wiley, 1984. [17] J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev., vol. 127, pp. 1918–1939, 1962. [18] R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron., vol. QE-15, pp. 432–444, June 1979. [19] B. Scherrer, I. Ribet, A. Godard, E. Rosencher, and M. Lefebvre, “Dual-cavity doubly resonant optical parametric oscillators: demonstration of pulsed single-mode operation,” J. Opt. Soc. Amer. B, vol. 17, pp. 1716–1729, 2000. [20] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1970.

Antoine Godard was born in Caen, France, in 1975. He received the degree of engineering from the Ecole Supérieure d’Optique, Orsay, France, in 1998, and the Ph.D. degree from the University of Paris XI, Orsay, France, in 2003. His Ph.D. dissertation dealt with the stabilization of an extended-cavity semiconductor laser by use of self-organizing filtering, particularly focusing on the mode-coupling phenomena in the laser gain medium. He is currently a Postdoctoral Researcher with the French aerospace research agency Office National d’Etudes et de Recherches Aérospatiales (ONERA), Palaiseau, France, where he is a member of the Optical and Plasma Diagnostics Unit in the Physics, Instrumentation and Sensing Department. His research involves laser applications and nonlinear optics with a special interest in optical parametric oscillators.


Emmanuel Rosencher (M’93–SM’94) was born in Paris, France, in 1952. He graduated from the Ecole Polytechnique, Paris, France, in 1975 and from the Ecole Nationale Supérieure des Télécommunications, Paris, France in 1977. He received the Doctorat in Applied Mathematics from the Université de Paris IX, Paris, France, in 1978 and the Habilitation Degree in Physics from the Université de Grenoble, Grenoble, France, in 1986. From 1978 to 1988, he was with the Centre National d’Etudes des Télécommunications (CNET) in Grenoble. In 1984, he discovered the transistor effect in the monolithic Si–CoSi –Si heterostructure and studied the related metal base transistor devices as well as the quantum transport in ultrathin epitaxial metal film until 1988. In 1988, he joined the Laboratoire Central de Recherches de THOMSON-CSF, Orsay, France, as the head of the Physics laboratory. In 1989, he discovered the giant optical nonlinear effects in asymmetrical quantum wells. In 1998, he joined the Office National d’Etudes et des Recherches Aerospatiales (ONERA), Palaiseau, France, where he is currently the Director of the Physics Branch (420 people). He is a pioneer in the physics of intersubband transitions in semiconductor quantum wells, optical nonlinearities in semiconductors, and optical parametric oscillators. He has published more than 250 papers in the field of semiconductor heterostructures, nonlinear optics, and laser physics with 140 papers in refereed international reviews and 70 invited talks. He also holds 15 patents. He is Associate-Professor of Physics at the Ecole Polytechnique and writes also physics textbooks (Optoelectronics, Cambridge University Press in French, English and Russian) as well as books for the popularization of Science (La Puce et l’Ordinateur, Flammarion, translated in Italian, Spanish, Portuguese,. . .). Dr. Rosencher is a Fellow of the Optical Society of America and the Institute of Physics (IoPF96) and a member of the French and American Physics Societies. He was the recipient of the 1991 Prix Foucault (Physique Appliquée) from the Société Française de Physique, the 2000 Montgolfier Award (Arts Physiques) from the Société d’Encouragement de l’Industrie Nationale, the 2001 Arnulf-Françon Prize from the Société Française d’Optique, and the 2003 Grand Prix de Physique Appliquée from the Société Française de Physique, and he is Chevalier de l’Ordre National du Mérite.