E t (a) (b)

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1 Generation of Ultra-Short Optical Pulses Using Nonlinear Dynamics Final Report of EPSRC Grant GR/L / William J Firth, Gian-Luca Oppo and Giampaolo D'Alessandro ( ) with contributions from A J Scroggie and D Skryabin Department of Physics and Applied Physics, University of Strathclyde, Glasgow ( ) Department of Mathematics, University of Southampton, Southampton January Introduction The project was about the application of techniques of nonlinear dynamics to assist and optimise the design of the devices for the generation of ultra-short optical pulses. In particular we have analysed the use of quadratically nonlinear ( (2) ) materials to generate new, possibly tunable, optical frequencies when excited with a primary pump laser. For example Optical Parametric Oscillators (OPOs) which utilise (2) materials to down convert the input laser frequency, have made enormous experimental progress in recent years and are now commonly produced and utilised. In this nal report we focus on four theoretical and computational descriptions of OPO systems: generation of ultrashort pulses by using OPOs with saturable absorbers and OPOs with asynchronous pumping (see section 2), generation of cavity solitons in OPOs (see section 3), generation of (2) spatial solitons (see section 4) and generation of optical pulses in intracavity second harmonic generation (see section 5). Other related work on group theory applied to eects of polarisation and diraction in nonlinear optics is also briey discussed in section Ultra-short Pulses in OPOs Because of its importance, the system on which we have focused is the synchronously pumped optical parametric oscillator (SPOPO). In this device, a (2) nonlinear crystal is placed inside a cavity and, under the action of a periodic train of pump pulses, generates signal and idler elds. The cavity mirrors reect one of these two elds back to the crystal thus forming a feedback loop that ensures that parametric oscillation can occur even at low pump power. To describe propagation of the elds through the (2) crystal we use the following set of coupled partial dierential equations [@ z + t ] E 1 =?( 1? ik)e 1 + i tt E 1? E 2 E z E 2 =? 2 E 2 + i tt E 2 + E 1 E 3 [@ z + t ] E 3 =? 3 E 3 + i tt E 3 + E 1 E 2 (1) where E 1 denotes the pump, E 2 the resonated eld (signal or idler) and E 3 the third eld (idler or signal). The longitudinal co{ordinate in the direction of propagation is z while t represents time.

2 The i describe losses and k is the phase mismatch between the three elds. The i represent pulse walk{o and the i, group velocity dispersion (GVD). The eect of the cavity is included by means of the boundary conditions E 1 (z = 0; t) = P (t) E 3 (z = 0; t) = 0 E 2 (z = 0; t) = exp(?i)z p R(? t)e 2 (z = 1; )d (2) The input pump eld is denoted by P (t) while R is the combined intensity reection coecient for all cavity mirrors, whose Fourier Transform, R(!), ~ is assumed to have the form of a symmetric bandpass lter. The parameter represents cavity detuning. Equations (1) with boundary conditions (2) describe the propagation of the three elds on one pass through the nonlinear crystal. We can take the elds produced at the exit of the crystal after one such pass and use them, via (2), as the initial conditions for a subsequent pass. By doing this we dene a map [1], \xed point" solutions of which correspond to output pulses which are reproduced identically after each round trip. We have developed both interactive and non{interactive codes to iterate this map: Equations (1) are integrated by means of a split{step spectral method, in which the time derivatives are handled by a Fast Fourier Transform while the remaining, time{independent part of the equations is integrated by means of a fourth order Runge{Kutta technique [2]. The programs have also been checked against a pseudo{spectral code. We nd that the map (Equations (1) and (2)) can exhibit complicated behaviour, such as regimes where there is no stable steady state. A particular complicating factor is the presence of the temporal walk-o terms ( t and t ) which, assuming that P (t) is symmetric, break the t!?t symmetry of the system and, in the picosecond and subpicosecond regime, induce non-trivial structure on the pulses. The nature and magnitude of these terms also frustrate any attempt to treat them as perturbations and hence to simplify the model by means of perturbation theory The use of saturable absorbers [3] To shorten and clean up the pulses we consider a technique which has long been used with success in lasers [4] but which has, up till now, remained untried in OPOs. We consider placing a relatively slow saturable absorber in the SPOPO cavity, the idea being that the leading edge of the pulse will be absorbed, partially saturating the absorber and allowing part of the trailing edge to pass through with decreased absorption [4]. We assume that the absorber is placed in the cavity at some point z =. Because of the fast time scales involved, we consider a semiconductor as the absorber [5, 6, 7] and model its eects by the following equations [5, z 0E =?(1 + t q = 1 "?(q? q 0 )? qjej2 T R I sat Here q is the absorber ground{state population with equilibrium (no{eld) value q 0 ; I sat is the saturation intensity; T R is the absorber recovery time; and is the \linewidth enhancement factor" [6]. # (3)

3 We integrate Equations (3) using, as an initial condition at z =, the eld obtained after reection from the output mirror. By setting (q = q 0 + noise) at t = 0 we complete the specication of the problem. Equations (3) can then be integrated using a fourth order Runge-Kutta technique [2]. Using parameter values corresponding to the (2) material LBO (Lithium Triborate) we have E E t t (a) (b) Figure 1: Plot of the idler intensity je 2 j 2 as a function of time for (a) the SPOPO without absorber and (b) the SPOPO with absorber. In each case the pulses are stationary solutions of the SPOPO map. The SPOPO parameters are 1 = 0:00659, 3 = 0:00188, 1 =?1:24 10?7, 2 = 9:93 10?8, 3 =?6:64 10?8, 1 = 2 = 3 = 0 and P = 3:25. For panel (b) the absorber parameters are q 0 = 1:12, I sat = 2 and T r = 0:01. carried out sweeps across typical semiconductor parameters. By this means we have demonstrated that compression ratios of the order of 5 are possible (see Figures 1 and 2). Furthermore, we have identied nonlinear resonance phenomena which are unique to the use of saturable absorbers in OPOs and would not occur in laser systems. The maximum in Figure 2(b) is an example: q 0 ' 0:9 produces a value for je 2 j at the entrance to the (2) crystal such that the half{period for energy exchange of the three{wave interaction is equal to the length of the nonlinear crystal. Finally, the general applicability of the technique has been tested by using parameters corresponding to the material Periodically{Poled Lithium Niobate (PPLN) [10] in addition to those corresponding to LBO The use of Asynchronous Pumping Another mechanism that we have examined depends more on intrinsic properties of the OPO. We can introduce an extra degree of freedom into the map by modifying the boundary condition on the pump so that its repetition period is slightly dierent from the round{trip time of the resonated eld (asynchronous pumping): E 1 (z = 0; t) = P (t + nt) (4)

4 7 η Max( E 2 2 ) q 0 q 0 (a) (b) Figure 2: Plots of (a) compression ratio, and (b) peak power of the pulses as a function of q 0, the ground state population in the absence of a eld. The result of each simulation is represented by a diamond. All other parameters are as in Figure 1. where n is the number of round{trips. In order for the OPO to oscillate, the resonated eld must adjust its speed of propagation in the crystal through some nonlinear mechanism to compensate for non{zero values of t. The ability of asynchronous pumping to produce dramatic pulse compression was rst pointed out (but not explained) by Khaydarov and co{workers [11]. Up to twenty{fold compression in an asynchronously pumped, singly{resonant OPO using PPLN was subsequently observed by Hanna and co{workers at the Optoelectronics Research Centre (ORC) at Southampton. In collaboration with ORC we have compared the results of numerical simulations with the output of their experiment, with good quantitative agreement. We have then used detailed observations from numerics, unattainable in the experiment, to identify the major components of the shortening mechanism. The picture is complicated and involves the interplay of various phenomena and a ne{tuning of the time{mismatch t. One important requisite, for example, is back{conversion of the trailing edge of the resonated signal pulse by a more slowly propagating idler. Detailed simulations have been carried out to predict and explain trends which are relevant to the experimental investigation of this method of pulse compression. These include the variation of the maximum compression ratio with pump power and other experimentally accessible parameters. 3. Generation of Domain Walls and Cavity Solitons in OPOs Spatio-temporal eects in OPOs have also been studied. In particular, by generalising a previous model of partial dierential equation used to analyse the onset of pattern formation [12], we have studied the possibility of generating spatially localised structures by using a (2) crystal in an optical cavity. In the case of degenerate operation (i.e. when the frequencies of the idler

5 and signal elds are the same), dark lines separating regions of opposite phase originate in the transverse distribution of the signal intensity. Such domain walls (DWs) move and coalesce in two transverse dimensions for input pumps close to the signal generation. However, if the input energy exceeds a given threshold, shrinking of a domain of a given phase reaches a minimum size and forms a localised spatial structure. Figure 3 shows the spatial distribution of the signal and pump amplitude for the spatial soliton generated by such a mechanism [13]. This is a novel form of a cavity soliton [14, 15]. Signal field amplitude Pump field amplitude Figure 3: Spatial distribution of signal and pump eld for the novel cavity soliton formed in degenerate OPOs. 4. Soliton Filaments and Bullets in (2) Media In this section we report on conventional (propagating) solitons in (2) media, which have been a subject of huge interest world wide in recent years [16] and op. cit. In relation to the present project, we have shown that modulational instability (MI) arising from group velocity dispersion (GVD) can cause spatial solitons to break up into pulse trains. (This area has beneted immensely from the availability of Dmitry Skryabin as a University/ORS funded PhD student.) We published the rst paper demonstrating the existence of stable optical bullets in (2) media [17]. There we analyse the eects of GVD in de-stabilising self-guided laments, leading to short pulses which are self-conned in all three dimensions. These are termed optical bullets, and represent the ultimate limit to the localisation of light into robust entities useful for applications. In Type II or non-degenerate 3-wave mixing, we showed that the presence of a second phase symmetry gives rise to a new mode of MI [18]. A potentially useful aspect of this is that it leads to a new behaviour in the case of negative GVD (for which there are no optical bullets). Here the MI leads, because of anti-phasing of the signal/idler modes, to enhanced spectral broadening at the sum frequency. This could have applications in situations where short pulses are sought primarily for their spectral breadth.

6 Additionally, this area has led to a number of signicant discoveries. These include the demonstration of the existence of "doughnut solitons" carrying nite angular momentum, and the striking behaviour, just like Newtonian particles, of the daughter solitons generated by their fragmentation [16]; and the rst demonstration of a symmetry-preserving "coalescence" phenomenon in higherorder solitons [19]. The quality of this work led to the award of a three-year BP/RSE Research Fellowship to Dmitry Skryabin, in which he will attempt to combine the interesting spatial properties of cavity solitons with the ultra-fast time-response of (2) media to devise schemes capable of parallel information and image processing at Tbit rates. 5. Optical Pulses in Intracavity Second Harmonic Generation Nonlinear dynamics techniques allowed us to characterise optical pulse generation in a laser with intracavity second harmonic crystal [20]. Its dynamics has been analysed in a model with two modes of one polarization and a third mode orthogonally polarized. In regimes of global mode coupling, anti-phase chaos aects all modes while the intensity maxima of the pulses of the third mode appear locked on a period one dynamics. We have explained this curious feature in terms of periodic decoupling of the modes with orthogonal polarizations and intensity dependent dissipations which force the trajectories onto extremely narrow manifolds. The eect of intensity dependent dissipations on Lyapunov numbers calculated in specic parts of the chaotic attractor has also been characterised [20]. 6. Other Related Work In addition to the main thrust of the grant, we have also generalised our mathematical analysis to include combined eects of polarisation and non-paraxial propagation in spatially extended nonlinear optical systems. For example, by using group theory we have shown analytically that the presence of an aperture inside an optical cavity reduces the invariance of the system from O 2 to D 2. As a consequence, spatial structures with concentric rings of bright or dark spots made up by modes with co-prime azimuthal indexes, that are anomalous if the symmetry is O 2, appear naturally as the result of co-dimension one primary bifurcations [21, 22]. As an example of this type of systems, we have investigated numerically the bifurcations of a ring laser with a metallic duct and a polarizer inside the optical cavity [21, 22]. Finally, during the three years of his appointment, Dr. Scroggie has also continued his previous collaboration with the group on the control of pattern formation and spatiotemporal disorder in nonlinear optics which has resulted in a particularly important publication in Physical Review A.

7 References [1] D.W. McLaughlin, J.V. Moloney and A.C. Newell, Phys. Rev. Lett. 51, 75 (1983) [2] R.H. Hardin and F.D. Tappert, SIAM Rev. 15, 423 (1973) [3] A.J. Scroggie, G. D'Alessandro, N. Langford and G.-L. Oppo, Opt. Comm., to appear (1999) [4] A.E. Siegman, Lasers, University Science Books (1986) [5] F.X. Kartner, I.D. Jung and U. Keller, IEEE Journal of Selected Topics in Quantum Electronics 2, 540 (1996) [6] U. Keller et al., IEEE Journal of Selected Topics in Quantum Electronics 2, 435 (1996) [7] S. Tsuda et al., IEEE Journal of Selected Topics in Quantum Electronics 2, 454 (1996) [8] H. Haug and S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, World Scientic (1993) [9] J. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, Phys. Rev. 127, 1918 (1962). [10] L. Lefort, K. Puech and D.C. Hanna, private communication. [11] J.D.V. Khaydarov, J.H. Andrews, and K.D. Singer, JOSA B 12, 2199 (1995). [12] G.-L. Oppo, M. Brambilla, and L.A. Lugiato, Phys. Rev. A 49, 2028 (1994). [13] G.-L. Oppo, A. J Scroggie and W.J. Firth, Journal of Optics B: Quantum and Semiclassical Optics 1, 133, (1999) [14] W.J. Firth and A.J. Scroggie, Phys. Rev. Lett (1996) [15] W.J. Firth and G.K. Harkness, Asian J. of Physics (due October 1998) [16] W. J. Firth and D. Skryabin, Phys. Rev. Lett. 79, 2450 (1997). [17] D.V. Skryabin and W.J. Firth, Opt. Comm. 148, 79 (1998). [18] D.V. Skryabin and W.J. Firth, Phys. Rev. Lett (1998). [19] D.V. Skryabin and W.J. Firth, Phys. Rev. E 58 R1252 (1998). [20] D.H. Henderson and G.-L. Oppo, Phys. Rev. E (to appear, February 1999) [21] F. Papo, G. D'Alessandro, W. J. Firth and G-L. Oppo, Phys. Rev. Lett. (submitted August 1998) [22] F. Papo, G. D'Alessandro and G.-L. Oppo, Phys. Rev. A, submitted (1999)

8 Publications 1. Optical Solitons Carrying Orbital Angular Momentum, W.J. Firth and D. Skryabin, Phys. Rev. Lett (1997) 2. Generation and Stability of Optical Bullets in Quadratic Nonlinear Media, D.V. Skryabin and W.J. Firth, Opt. Comm. 148, 79 (1998) 3. Elimination of Spatiotemporal Disorder by Fourier Space Techniques, G.K. Harkness, G.-L. Oppo, R. Martin, A.J. Scroggie, and W.J. Firth, Phys. Rev. A 58, 2577 (1998) 4. Pulse Compression by Slow Saturable Absorber Action in an Optical Parametric Oscillator, A.J. Scroggie, G. D'Alessandro, N. Langford and G.-L. Oppo, Opt. Comm. (to appear, 1999) 5. From Domain Walls to Localised Structures in Degenerate Optical Parametric Oscillators, G.- L. Oppo, A. J Scroggie and W.J. Firth, Journal of Optics B: Quantum and Semiclassical Optics 1, 133, (1999) 6. Instabilities of Parametric Higher-Order Solitons. Filamentation vs Coalescence, D.V. Skryabin and W.J. Firth, Phys. Rev. E 58, R1252 (1998) 7. Modulational Instability of Solitary Waves in Non-Degenerate Three-Wave Mixing: The Role of Phase Symmetries, D.V. Skryabin and W.J. Firth, Phys. Rev. Lett (1998) 8. Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media, D.V. Skryabin and W. J. Firth, Phys. Rev. E (1998) 9. Diraction-Induced Polarization Eects in Optical Pattern Formation, F. Papo, G. D'Alessandro, W. J. Firth and G-L. Oppo, Phys. Rev. Lett. (submitted August 1998) 10.Anti-phase Chaos and Intensity Dependent Dissipations D.H. Henderson and G.-L. Oppo, Phys. Rev. E (to appear, February 1999) 11. Combined Eects of Polarization and Non-paraxial Propagation on Optical Pattern Formation, F. Papo, G. D'Alessandro and G.-L. Oppo, Phys. Rev. A, submitted (1999) Conference presentations (a selection) 1. Dynamics and Stabilisation of Domain Walls in Optical Parametric Oscillators, G.-L. Oppo, A.J. Scroggie, and W.J. Firth. Oral presentation at IQEC'98, San Francisco, 3-8 May See also the Technical Digest of IQEC'98, page 172 (1998) 2. Beam Quality Characterisation of Nanosecond Optical Parametric Oscillators, S.C. Lyons, G.- L. Oppo, W.J. Firth, J.R.M. Barr, and C. Coia. Poster presentation at CLEO'98, San Francisco,

9 3-8 May See also the Technical Digest of CLEO'98, page 387 (1998) 4. Eects of Diraction-Induced Polarisation on Pattern Formation, F. Papo, G. D'Alessandro, W.J. Firth, and G.-L. Oppo. Oral presentation at "Patterns in Nonlinear Optical Systems", Alicante (Spain), May Stabilisation of Domain Walls in Optical Parametric Oscillators, G.-L. Oppo, A.J. Scroggie, and W.J. Firth. Poster presentation at "Patterns in Nonlinear Optical Systems", Alicante (Spain), May Eects of Diraction-Induced Polarisation on Optical Pattern Formation, F. Papo, G. D'Alessandro, W.J. Firth and G.-L. Oppo. Oral presentation at EQEC'98, Glasgow, September See also the Technical Digest of EQEC'98, page 105 (1998) 7. Pulse Structure and Compression in Singly-Resonant Optical parametric Oscillators, A.J. Scroggie, G. D'Alessandro, G.-L. Oppo, and W.J. Firth. Poster presentation at EQEC'98, Glasgow, September See also the Technical Digest of EQEC'98, page 150 (1998) 8. Domain Walls in Optical Parametric Oscillators: Dynamics and Stabilisation, G.-L. Oppo, A.J. Scroggie, and W.J. Firth. Oral presentation at EQEC'98, Glasgow, September See also the Technical Digest of EQEC'98, page 245 (1998) 9. Beam Quality Characterisation of Nanosecond Optical Parametric Oscillators, S.C. Lyons, G.-L. Oppo, W.J. Firth, J.R.M. Barr, and C. Coia. Poster presentation at CLEO/Europe'98, Glasgow, September See also the Technical Digest of CLEO/Europe'98, page 287 (1998)