We predict that parametric sum-frequency generation of an ultra-short pulse
may result from the mixing of an ultra-short optical pulse with a
quasi-continuous wave control. We analytically show that the intensity, time
duration and group velocity of the generated idler pulse may be controlled in a
stable manner by adjusting the intensity level of the background pump

Baronio, Fabio

Conforti, Matteo

Degasperis, Antonio

Wabnitz, Stefan

Optics Express

English

arXiv

International audienceWe predict that parametric sum-frequency generation of an ultra-short pulse may result from the mixing of an ultra-short optical pulse with a quasi-continuous wave control. We analytically show that the intensity, time duration and group velocity of the generated idler pulse may be controlled in a stable manner by adjusting the intensity level of the background pump

Baronio, F.

Degasperis, A.

Wabnitz, S.

HAL-uB

Parametric frequency conversion of short optical pulses controlled by a CW background

HAL: Hyper Article en Ligne

We predict that parametric sum-frequency generation of
an ultra-short pulse may result from the mixing of an ultra-short optical
pulse with a quasi-continuous wave control. We analytically show that the
intensity, time duration and group velocity of the generated idler pulse
may be controlled in a stable manner by adjusting the intensity level of the
background pump

M. CONFORTI

BARONIO, Fabio

WABNITZ, Stefan

DE GASPERIS, Antonio

Archivio della ricerca- Università di Roma La Sapienza

Parametric frequency conversion ofshort optical pulses controlled by a CWbackgroundMatteo Conforti,1 Fabio Baronio,1∗ Antonio Degasperis,2 and StefanWabnitz31Dipartimento di Elettronica per l’Automazione, Universita` di Brescia, Via Branze 38, 25123Brescia, Italy2Dipartimento di Fisica, Istituto Nazionale di Fisica Nucleare, Universita` “La Sapienza”, P.leA. Moro 2, 00185 Roma, Italy3Institut Carnot de Bourgogne, UMR 5209 CNRS,Universite´ de Bourgogne, 9 Av. A. SavaryBP 46870, 21078 Dijon, Francefabio.baronio@ing.unibs.itAbstract: We predict that parametric sum-frequency generation ofan ultra-short pulse may result from the mixing of an ultra-short opticalpulse with a quasi-continuous wave control. We analytically show that theintensity, time duration and group velocity of the generated idler pulsemay be controlled in a stable manner by adjusting the intensity level of thebackground pump.© 2007 Optical Society of AmericaOCIS codes: (190.5530) Pulse propagation and solitons; (190.7110) Ultrafast nonlinear optics;(190.2620) Frequency conversion; (190.4410) Nonlinear optics, parametric processes.References and links1. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).2. J.A. Armstrong, S.S. Jha, and N.S. Shiren, “Some effects of group-velocity dispersion on parametric interac-tions,” IEEE J. Quantum Electron. QE-6, 123–129 (1970)3. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Pulses (AIP, New York, 1992).4. Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency usinggroup-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990)5. A. Stabinis, G. Valiulis and E.A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,”Opt. Commun. 86, 301–306 (1991).6. Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymiumlaser pulses,” Opt. Lett. 17, 1459–1461 (1992).7. C.Y. Chien, G. Korn, J.S. Coe, J. Squier, G. Mourou and R.S. Craxton, “Highly efficient second-harmonic gen-eration of ultraintense Nd:glass laser pulses,” Opt. Lett. 20, 353–355 (1995).8. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996).9. E. Ibragimov and A. Struthers, “Three-wave soliton interaction of ultrashort pulses in quadratic media,” J. Opt.Soc. Am. B 14, 1472–1479 (1997).10. E. Ibragimov, A. Struthers, D.J. Kaup, J.D. Khaydarov, and K.D. Singer, “Three-wave interaction solitons inoptical parametric amplification,” Phys. Rev. E 59, 6122–6137 (1999).11. E. Ibragimov, A. Struthers, and D.J. Kaup, “Soliton pulse compression in the theory of optical parametric ampli-fication,” Opt. Commun. 152, 101–107 (1998).12. S. Fournier, R.L. Lopez-Martens, C. Le Blanc, E. Baubeau, and F. Salin, “Solitonlike pulse shortening in afemtosecond parametric amplifier,” Opt. Lett. 23, 627–629 (1998).13. V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETPLett., 18, 243–245 (1973).14. D. J. Kaup, “The three-wave interaction – a nondispersive phenomenon,” Stud. Appl. Math. 55, 9-44 (1976).#84204 - $15.00 USD Received 19 Jun 2007; revised 16 Jul 2007; accepted 22 Jul 2007; published 11 Sep 2007(C) 2007 OSA 17 September 2007 / Vol. 15,  No. 19 / OPTICS EXPRESS  1224615. K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Japan34, 796–800 (1973).16. Y. Ohsawa and K. Nozaki, “Propagation of solitary pulses in interactions of plasma waves. II,” J. Phys. Soc.Japan 36, 591–595 (1974).17. F. Calogero and A. Degasperis, “Novel solution of the system describing the resonant interaction of three waves,”Physica D 200, 242–256 (2005).18. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wavesolitons,” Phys. Rev. Lett. 97, 093901 (2006).19. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-waveparametric solitons,” Phys. Rev. E 74, 065602(R) (2006).20. A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. 23, 1808-1810 (1998).21. G.M. Gale, M. Cavallari, T.J. Driscoll, and F. Hache,“Sub-20-fs tunable pulses in the visible from an 82-MHzoptical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).1. IntroductionOptical parametric amplification in quadratic nonlinear crystals has been studied since the in-vention of the laser, as it provides a versatile means of achieving widely tunable frequencyconversion [1]. In parametric processes, the effective interaction length of short optical pulsesis limited by temporal walk-off owing to chromatic dispersion, or group velocity mismatch(GVM) [2, 3]. Compression and amplification of ultra-short laser pulses in second harmonicand sum-frequency (SF) generation in the presence of GVM was theoretically predicted [4, 5]and observed in several experiments [6, 7]. The conversion efficiency of generated SF pulsesmay be optimised [8, 9, 10, 11, 12] by operating in the soliton regime [13, 14]. In fact, thetemporal collision of two short soliton pulses in a quadratic nonlinear crystal may efficientlygenerate a short, time-compressed SF pulse [8]. However this SF pulse is unstable: its energydecays back into the two incident pulses after a relatively short distance.In this Paper we consider the parametric SF conversion from the mixing of an ultra-short sig-nal pulse with a quasi-continuous wave (CW) or background pump, in the presence of GVM.Quite surprisingly we find that parametric mixing of these waves may lead to highly efficientgeneration of stable and ultra-short idler pulses. Indeed, we predict that the interaction of anultra-short signal with a CW pump may generate a stable three-wave resonant interaction si-multon (TWRIS) [15, 16], consisting of a locked bright-bright-dark triplet (signal-idler-pump)that propagates with a single nonlinear velocity [17, 18]. We analytically show that the inten-sity, time duration and group velocity of the generated idler pulse may be controlled in a stablemanner [19] by means of simply adjusting the intensity level of the CW background. Althoughwe shall restrict our attention in this work to a travelling-wave interaction geometry, we mayanticipate that our results will have important ramifications in the optimization of the efficiencyof ultrashort pulse optical parametric oscillators [20, 21].2. Three-wave-interaction equationsThe equations describing the quadratic resonant interaction of three waves in a nonlinearmedium read as:∂A1∂ξ + δ1∂A1∂τ = iA∗2A3,∂A2∂ξ + δ2∂A2∂τ = iA∗1A3, (1)∂A3∂ξ + δ3∂A3∂τ = iA1 A2,#84204 - $15.00 USD Received 19 Jun 2007; revised 16 Jul 2007; accepted 22 Jul 2007; published 11 Sep 2007(C) 2007 OSA 17 September 2007 / Vol. 15,  No. 19 / OPTICS EXPRESS  12247withA j = πχ (2)√n jω1ω2ω3n1n2n3ω jE j . (2)Here τ = t/t0, t0 is an arbitrary time parameter; ξ = z/z0, z0 is an unit space-propagationparameter. E j are the slowly varying electric field envelopes of the waves at frequencies ω j, n jare the refractive indexes, χ (2) is the quadratic nonlinear susceptibility, δ j = z0/(v jt0) with v jthe linear group velocities, and j = 1,2,3. We assume that the group velocity v 3 of the wavewith the highest frequency (ω3 = ω1 +ω2) lies between the group velocities of the other waves,i.e. v1 > v3 > v2. With no loss of generality, we shall write the Eqs. (1) in a coordinate systemsuch that δ1 = 0, which implies 0 < δ3 < δ2. Eqs. (1) exhibit the conserved quantitiesU13 = U1 +U3 =12∫ +∞−∞(|A1|2 + |A3|2)dτ, (3)U23 = U2 +U3 =12∫ +∞−∞(|A2|2 + |A3|2)dτ, (4)U = U1 +U2 + 2U3 =12∫ +∞−∞(|A1|2 + |A2|2 + 2|A3|2)dτ. (5)where U1, U2 and 2U3 represent the energies at the frequencies ω1, ω2 and ω3.3. Soliton-based parametric sum-frequency conversionFigure 1 illustrates a typical example of the efficient SF parametric interaction of two shortoptical pulses in the soliton regime [8].−30−15 0 15 30 0102001.53ξτ|A1| −30−15 0 15 30 0102001.53ξτ|A2| −30−15 0 15 30 01020024ξτ|A3| Fig. 1. Sum-frequency parametric interaction of two short optical signals at ω1 and ω2. Thecharacteristic delays are δ1 = 0,δ2 = 2,δ3 = 1.At the crystal input, two isolated pulses A1 and A2 with frequencies ω1 and ω2 propagatewith speeds v1 and v2. Whenever the faster pulse overtakes the slower one, an idler pulse A 3#84204 - $15.00 USD Received 19 Jun 2007; revised 16 Jul 2007; accepted 22 Jul 2007; published 11 Sep 2007(C) 2007 OSA 17 September 2007 / Vol. 15,  No. 19 / OPTICS EXPRESS  12248−1501530 0102002ξτ|A1| −1501530 01020024ξτ|A2| −1501530 0102001.53ξτ|A3| Fig. 2. Sum-frequency parametric interaction of a short pulse at ω2 and a quasi-CW controlat ω1. The characteristic delays are δ1 = 0,δ2 = 2,δ3 = 1.at the SF ω1 +ω2 is generated and propagates with the linear speed v3. Depending on the timewidths and intensities of the input pulses, the duration of the SF pulse is reduced with respect tothe input pulse widths. Correspondingly, the SF pulse peak intensity grows larger than the inputpulse intensities. Figure 1 shows that, eventually, the SF idler pulse decays back into the twooriginal isolated pulses at frequencies ω1 and ω2. Note that the shapes, intensities and widthsof the input pulses are left unchanged in spite of their interaction. As shown in Ref. [8], theabove discussed SF pulse generation process may be analytically described in terms of solitonsolutions of Eqs. (1) [13, 14]. The decay of the SF pulse which is shown in Fig. 1 may bea significant drawback in practical applications, since it implies that a given nonlinear crystallength yields efficient conversion for a limited range of input pulse intensities and time widthsonly.Here we demonstrate that the parametric sum-frequency conversion of an ultra-short sig-nal and a quasi-CW background pump-control may be exploited as a means to reduce or eveneliminate the decay of the generated idler wave. In the presence of GVM, the parametric SFconversion of an ultra-short optical signal and a quasi-CW pump typically leads to the gener-ation of a low-intensity and relatively long idler pulse, whose duration is associated with theinteraction distance in the crystal. This scenario changes dramatically in the soliton regime.Figure 2 illustrates the efficient generation of a stable, ultra-short SF idler pulse from the para-metric SF conversion of a properly prepared ultra-short signal and an arbitrary intensity levelCW background control.In Fig. 2 we injected in the quadratic nonlinear crystal the short signal at frequency ω 2,along with a delayed and relatively long pump-control pulse at frequency ω 1. Initially, the twopulses propagate uncoupled; as soon as the faster pulse starts to overlap in time with the slowerquasi-CW control, their nonlinear mixing generates a short SF idler pulse. The sum-frequencyprocess displayed in Fig. 2 can be analytically explained and explored in terms of stable TWRISsolutions [18]. In the notation of Eqs. (1), the TWRIS solution reads as#84204 - $15.00 USD Received 19 Jun 2007; revised 16 Jul 2007; accepted 22 Jul 2007; published 11 Sep 2007(C) 2007 OSA 17 September 2007 / Vol. 15,  No. 19 / OPTICS EXPRESS  12249A1 = {1+ 2pb∗|b|2 + a2 [1− tanh[B(−τ + δξ )]]}iag1 exp(iq1τ1)g(δ2− δ3)A2 =2pa√|b|2 + a2g2g(δ2− δ3)exp[i(q2τ2 + χτ +ωξ )]cosh[B(−τ + δξ )] ,A3 =−2pb∗√|b|2 + a2g3g(δ2− δ3)exp[−i(q3τ3 − χτ−ωξ )]cosh[B(−τ + δξ )] , (6)whereb = (Q−1)(p+ ik/Q), r = p2− k2−|a|2,Q = 1p√12[ r +√r2 + 4k2 p2 ],B = p[δ2 + δ3−Q(δ2− δ3) ]/(δ2 − δ3),δ = 2δ2δ3/[δ2 + δ3−Q(δ2− δ3) ],χ = k[δ2 + δ3− (δ2− δ3)/Q ]/(δ2− δ3),ω = −2kδ2δ3/(δ2− δ3),τn =−τ + δnξqn = q(δn+1 − δn+2), gn = |(δn − δn+1)(δn − δn+2)|−1/2g = g1 g2 g3 , n = 1,2,3 mod (3). (7)For a given choice of the characteristic linear group velocities, we are left with the four realindependent parameters p,a,k,q. The parameter p is associated with the re-scaling of the waveamplitudes, and of the coordinates τ and ξ . Whereas a measures the amplitude of the CWbackground in wave A1 (namely a√δ2δ3). The value of k is related to the soliton wavenumber.The parameter q simply adds a phase shift which is linear in both τ and ξ (see [18] for parameterdetails).At the input, the properly prepared short pulse at frequency ω 2 and with a speed v2 is a stablesingle component TWRIS (6) with parameters p > 0,k,q,a = 0. The background control inthe interaction region can be modeled with A1(τ) = Ce−iγτ . When this faster pulse, pre-delayedwith respect to the slower quasi-CW pump at frequency ω1, overtakes the background (at τ = 0,in Fig. 2), their collision leads to the generation of a short idler pulse at the SF ω 3. Additionally,a dip appears in the quasi CW-control; whereas the intensity, duration and propagation speedof the input wave at frequency ω2 are modified. Indeed, the signal-pump interaction generatesa new stable TWRIS (6), with parameters p,k,q,a, moving with the locked nonlinear velocityv = z0/(t0δ ), where δ is given in (7). It is remarkable that we may analytically predict theparameters p,k,q,a of the generated TWRIS from the corresponding parameters of the inputsingle wave TWRIS and the complex amplitude of the pump control. This can be achieved bysupposing that the input TWRIS adiabatically (i.e., without emission of radiation) reshapes intoa new TWRI simulton after its collision with the quasi-CW pump at a given point in time (say,at τ = 0). Under this basic hypothesis, the conservative nature of the three-wave interactionpermits us to suppose that: i) the energy U23 (4) of the input TWRI soliton is conserved in thegenerated TWRI simulton; ii) the phase of the ω2 frequency components of the input TWRIsoliton and of the generated TWRI simulton is continuous across their time interface (i.e.,at τ = 0); iii) the amplitude and phase of the control pump coincide with the correspondingvalues of the asymptotic plateau of the generated TWRI simulton component at frequency ω 1.By imposing the above three conditions, after some straightforward calculations we obtain thefollowing relations that relate the parameters of the incident and of the transmitted TWRISp = p, a = |C|/√δ2δ3, q = γ, k = k +(q−q)(δ2− δ3)/2. (8)#84204 - $15.00 USD Received 19 Jun 2007; revised 16 Jul 2007; accepted 22 Jul 2007; published 11 Sep 2007(C) 2007 OSA 17 September 2007 / Vol. 15,  No. 19 / OPTICS EXPRESS  12250As an example, in Fig. 2 the input TWRI soliton at frequency ω 2 is described by Eqs. (6)with p = 1.3,k = 0,q = 0,a = 0, and the background control amplitude with C = 1.7,γ = 0.After the collision with the CW background, the above equations predict that the generatedTWRIS is again described by Eqs. (6), with p = 1.3,k = 0,q = 0, and a = 1.2. The accuracy ofthis prediction is well confirmed by its comparison with the numerical solutions of the TWRIEqs. (1). Indeed, Fig. 3 compares the numerical with the analytical evolutions (along the crys-tal length ξ ) of the energy, the pulse duration and the velocity of the idler and signal pulseswhich correspond to the case shown in Fig. 2. We performed further extensive numerical simu-lations, which confirmed the general validity of the above described adiabatic transition modelfor TWRIS generation upon collision with a CW background.0 20012ξEnergySignal Idler 0 2024ξdurationSignal Idler0 2001ξv  to/zoIdler Signal Fig. 3. Numerical evolution (lines) and theoretical predictions (circles) of energy, pulseduration and velocity of idler (red) and signal (blue) waves reported in Fig.2.Indeed, by increasing or decreasing the CW background amplitude |C| in the range[0, p√δ2δ3], we observed that stable TWRISs with different velocity, duration and energy dis-tributions may be adiabatically shaped. The important consequence of this result is that, bymeans of Eqs. (6)–(8), we may analytically predict and control the characteristics of the gen-erated idler pulse (namely, its velocity, time duration and energy) simply as a function of theintensity level of the CW pump. Moreover, we would like to emphasize that the stability of thewhole conversion process is ensured by the underlying stability of the generated TWRIS [19].We would like to point out that the observation of the above described sum–frequency shortpulse generation phenomena appears to be readily achievable in nonlinear optical experiments.For instance, let us consider the eee interaction of three-waves with carrier wavelengths ofλ1 = 1.55μm, λ2 = 3.4μm, λ3 = 1.064μm in a 2cm (8cm) long periodically poled bulk LithiumNiobate crystal with 28μm periodicity. In this case, the parametric mixing of a 100 f s (1ps)incident pulse with a quasi–CW (say, with a 3ps (30ps) time duration) control pulse leads tothe generation of an ultrashort sum-frequency pulse of approximately the same time width ofthe incident short pulse whenever the field intensities of the two input pulses are of the order ofa few hundreds of MW/cm2 (or a few MW/cm2, respectively).4. ConclusionsIn conclusion, we demonstrated the parametric SF conversion of an ultra-short pulse from themixing of an ultra-short optical pulse with a quasi-continuous wave control in quadratic non-linear crystals in the presence of GVM. We analytically showed that the intensity, time durationand group velocity of the generated pulses may be controlled in a stable manner by simplyadjusting the intensity level of the background pump.AcknowledgementsThe present research of A.D. and S.W. is within the French CNRS project GDR 3073.#84204 - $15.00 USD Received 19 Jun 2007; revised 16 Jul 2007; accepted 22 Jul 2007; published 11 Sep 2007(C) 2007 OSA 17 September 2007 / Vol. 15,  No. 19 / OPTICS EXPRESS  12251

A. DEGASPERIS

Archivio istituzionale della ricerca - Università di Brescia

Matteo Conforti

Fabio Baronio

Antonio Degasperis

Stefan Wabnitz

Crossref

Parametric Frequency Conversion of Short Optical Pulses Controlled by a
  CW Background

Parametric Frequency Conversion of Short Optical Pulses Controlled by a CW Background

Abstract

Similar works

Full text

Available Versions

HAL-uB

HAL: Hyper Article en Ligne

Archivio della ricerca- Università di Roma La Sapienza

Archivio istituzionale della ricerca - Università di Brescia

Crossref