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Moving localized structures in quadratic media with a saturable absorber
Optics Communications 232 (2004) 381–389 www.elsevier.com/locate/optcom
Moving localized structures in quadratic media with a saturable absorber A. Barsella a, C. Lepers a, M. Taki a
a,*
, M. Tlidi
b
Laboratoire de Physique des Lasers, Atomes et Molecules, CNRS UMR 8523, Centre d’Etudes et de Recherches Lasers et Applications, Universite des Sciences et Technologies de Lille, B^atiment P5, 59655 Villeneuve d’Ascq Cedex, France b Universite Libre de Bruxelles, Optique Nonlineaire Theorique, Campus Plaine, CP231, 1050 Bruxelles, Belgium Received 19 May 2003; received in revised form 26 November 2003; accepted 22 December 2003
Abstract Spatial moving pulses are found in an optical resonator with a quadratic medium coupled to an intra-cavity saturable absorber. In the context of frequency degenerate optical parametric oscillators, the saturable absorber is shown to be able to generate large amplitude asymmetric pulses in the transverse section of the signal beam. They arise in a regime where the lasing solution undergoes both Hopf and Turing instabilities. These dynamical pulses have a constant velocity and are stable in a wide domain of parameters. The origin of their transverse velocity stems from the pulse asymmetry. For large enough values of the incident pump beam the moving pulses are destabilized giving rise to a complex spatio-temporal behavior. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.65; 42.60.Mi; 42.65.pc Keywords: Parametric oscillations; The physics of solitons; Spatio-temporal dynamics
1. Introduction Optical parametric oscillations are one of the fundamental mechanisms for the generation of coherent tunable radiation in quadratic media. In large-area devices, the parametrically interacting fields experience diffraction which yields to spontaneous pattern formation in their transverse sec-
*
Corresponding author. Fax: +33-3-20-33-43-40-84. E-mail address: [email protected] (M. Taki).
tion, via symmetry-breaking bifurcations [1]. The transverse structure of the beam emitted by these oscillators is a key factor for their practical use. The structure must be mastered for applications including low-noise measurements and detection [2] or optical coherent information processing [3]. In degenerate optical parametric oscillators (OPOs) the energy conversion process is frequency degenerate and the emerging pattern can be, at the primary instability, either homogeneous or spatially periodic. The formation of a patterned state has been extensively investigated in the mean-field
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.12.078
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A. Barsella et al. / Optics Communications 232 (2004) 381–389
limit [4,5] and beyond this limit [6]. It has also been demonstrated [7,8] that in OPOs, the presence of the walk-off can alter pattern formation leading to the emergence of convective regimes and noisesustained structures. In addition to periodic patterns OPOÕs can exhibit a number of noticeable localized structures (LS) or spatial solitons [9]. These nonlinear solutions are generated in the hysteresis loop involving stable homogeneous states and the inhomogeneous periodical branches of solutions [10]. Intense theoretical research activities have been carried out on this subject [11]. This prediction was confirmed by experiments using a photorefractive material [12], semiconductor micro-resonators [13,14], liquid-crystal light valve with optical feedback [15], and spin-1/2 atomic systems [16]. The LS can exhibit periodic or chaotic oscillations in time [17– 19]. Another type of LS in the form of dark or bright solitary wave has been generated in the regime where the steady state solutions are stable with respect to the modulational instability (often called Turing instability). They are independent of Turing instabilities and their existence is related to phase indetermination in the degenerate optical parametric oscillator [20]. In the vicinity of the codimension-two point where the modulational instability is close to the saddle node bifurcation, the Turing mode and the homogeneous zero mode are liable to interact. The associated pattern selection process may be altered as a consequence of the competition between two processes: the unstable homogeneous zero mode which tends to restore spatial uniformity in the transverse plane and the large unstable Turing wavelength (induced by the coupling between diffraction and vð2Þ nonlinearity) which tends to impose intrinsic periodicity in both transverse directions. The balance between the two processes may stabilize the single bright (or dark) stripe, and its dynamics is characterized by time scaling low [21]. Experimental evidence of the solitary dark solitons in degenerate optical parametric mixing [22] has further stimulated the interest in the transverse single stripe formation. The purpose of this paper is to show that, contrary to what is described in previous studies, robust moving pulses can exist over wide range of
parameters and may therefore dominate the dynamics of OPOÕs. They have large asymmetric amplitudes reflecting the presence of the intracavity saturable absorber and are initiated by the Hopf bifurcation affecting the lasing steady state solution. These moving localized structures have a constant transverse velocity which results from the non-variational effects combined to the pulse asymmetry produced by the saturable absorber. The latter phenomenon arises in the regime of rapid relaxation oscillations when the ratio between the cavity decay rates of the pump and the signal is small. All numerical simulations we have performed, in this regime, show dynamical pulses and we never found them in a stationary state. Furthermore, we have characterized numerically the behavior of the transverse speed with both the diffraction coefficient and the input pump field. So they are different from the stationary dissipative localized structures reported in [17,18] that are found far from threshold and result from the interaction between self-pulsing dynamics and modulational instabilities. These two-dimensional (2D) pulses can be viewed as the spatio-temporal counterpart of the well-known temporal Q-switch pulses appearing in laser systems with a saturable absorber [23–25] or in purely temporal (nondiffractive) OPOs [26,27]. After a brief introduction of the model describing the dynamics of the degenerate OPO, we have performed a linear stability analysis in Section 2. Our investigations on the formation and the stability of the moving pulses and their characteristics, namely, domain of existence and velocity, constitute Section 3. Concluding remarks are summarized in the last section.
2. Model and stability analysis We start from the standard description of a degenerate OPO in the mean-field approximation. It includes, in addition to the transverse Laplacian terms (r2? ) accounting for diffraction, the saturable absorber that acts selectively at frequency x. Saturable absorber is modeled by a collection of two level atoms that leads to an intensity-dependent effective absorption coefficient. Assuming fast
A. Barsella et al. / Optics Communications 232 (2004) 381–389
atomic relaxation, the evolution equations are described by the following set of dimensionless partial differential equations [17,18] @t E1 ¼ ð1 þ iD1 ÞE1 þ E1 E2
RE1 1 þ SjE1 j
2
þ ia1 r2? E1 ; ð1Þ
@t E2 ¼ c½ð1 þ iD2 ÞE2 þ
E12
Ei þ
ia2 r2? E2 ;
ð2Þ
where E1 and E2 are the slowly varying amplitudes of the signal field at frequency x and the pump field at frequency 2x, respectively. We have set c ¼ c2 =c1 where c2 and c1 are the decay rates of the pump and the signal. The parameter D1 ¼ ðx1c xÞ=c1 (D2 ¼ ðx2c 2xÞ=c2 ) is the detuning between the frequency of the longitudinal cavity mode and that of the signal (respectively the pump). Ei is the normalized incident pump. R is the unsaturated absorption coefficient of the absorber, and S is the relative saturability of the absorber. a1 and a2 are the diffraction coefficients of the signal and the pump fields, respectively. In order to satisfy the phase-matching condition, the ratio between the diffraction coefficients of the two fields is fixed to a1 =a2 ¼ 2. The homogeneous steady state solutions of Eqs. (1) and (2) are either (i) the trivial solutions E1 ¼ 0 and E2 ¼ Ei =ð1 þ iD2 Þ or (ii) the stationary solutions for which E1 6¼ 0. The latter satisfy the following algebraic equation R Ei2 ¼ I12 þ 2 1 þ D1 D2 I1 1 þ SI1 R 2 þ ð1 þ D22 Þ ð1 þ Þ þ D21 ; ð3Þ 1 þ SI1 2 R I2 ¼ 1 þ þ D21 ; 1 þ SI1 2
with I1;2 ¼ jE1;2 j , and the system is bistable when D1 D2 > ð1 þ RÞ½1 SRð1 þ
D22 Þ:
ð4Þ
This relation generalizes the bistability condition derived in the previous works [26,28]. More precisely, in the absence of the SA [28], it reduces to D1 D2 > 1 whereas, in the presence of a SA but under perfect resonance (D1 ¼ D2 ¼ 0) [26], it simplifies to SR > 1. Let us first perform the linear
383
stability analysis of the trivial solution but in the presence of the SA. It leads to the threshold of pattern forming instability in term of pump gain parameter to be as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð5Þ Eth ¼ 1 þ D22 ð1 þ RÞ þ ½D1 þ a1 k 2 : The above expression indicates simply that the threshold is increased when the SA is present as can be expected. More importantly, D1 is still the key parameter in the formation of patterns meaning that the presence of a SA does not affect the patterned state at the onset of instability sinceffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the most unstable modes have k ¼ kc ¼ D1 =a1 . Hence, when D1 < 0, a set of modes exists that is able to bring the field in resonance and thus the system exhibits structures. For D1 > 0, the system is unable to reach perfect resonance (absolute qffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
minimum threshold i.e. Eth ¼ ð1 þ RÞ 1 þ D22 and it remains in a stable homogeneous state leading to an increase of the OPO threshold emission: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 Eth ¼ ½ð1 þ RÞ þ D1 ð1 þ D2 Þ. To analyze the stability of the lasing steady state solution of Eq. (3), we consider for both fields a linear deviation of the form expðkt þ ik:rÞ. The study of the linearized problem shows that the system exhibits a Turing bifurcation leading to the formation of spatially periodic pattern in the transverse plane. At the bifurcation point the critical wave number is given by D21 þ 2b2 RSI1T ð1 þ bRÞ 2 c þ cðI1T Þ; 8
2kT2 ¼ D1 þ
ð6Þ
with b ¼ 1=ð1 þ SI1T Þ. The threshold I1T satisfies the following relation: 16c½c2 þ 4ðn21 n22 Þ þ 4cn1 I1T 2
¼ ½4ðn21 n22 Þ þ c2 ;
ð7Þ
where n22 ¼ n2þ þ D21 and n1 ¼ n with n ¼ 1 þ ð1 bSI1T ÞbR. Eqs. (6) and (7) are valid only when D1 ¼ 2cD2 . The stability analysis also shows that the system undergoes a Hopf bifurcation leading to the limit cycle (homogeneous self-pulsing). The analytical expression of the bifurcation point I1H is
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cumbersome but can take the following simplified expression when D1 ¼ D2 ¼ 0: 2
2
½ðc þ n Þ n2þ ½ðn þ c2 Þ ð2I1H þ n1 Þ n n2þ ¼ 0: ð8Þ The linear stability analysis of both homogeneous steady states shows that the trivial solution can be destabilized by only a Turing bifurcation. However, the lasing solution can exhibit both types of instability. We shall see in the following section the implication of such an interaction in the dynamics of the localized structures.
3. Moving localized structures In this section we focus our investigations on the formation and the stability of moving localized structures which are strongly nonlinear. In consequence their dynamics involves nonlinear interactions that cannot be captured by linear analysis. This needs a nonlinear analytical approach to the problem which is far beyond the scope of this paper. We only consider here a numerical study of these solutions by numerically solving the model Eqs. (1) and (2) with periodic boundary condition. The system is operating in a bistable regime, according to the inequality of the expression (4). In addition, we consider a parameter range where (i) the lasing homogeneous state is unstable with respect to both Hopf an Turing bifurcations, (ii) the system exhibits a temporal Q-switching owing to the presence of the saturable absorber. This phenomenon is initiated by Hopf instability of the lasing state and occurs when the ratio c between the decay rates of the pump and the signal is small. In what follows, we show that these LS are different from the ones reported in the previous works (see Section 1), in two respects: (i) they are dynamical LS with a non-vanishing constant velocity in the whole domain of their stability; (ii) the mechanism of their formation results from the coupling between non-variational effects and the presence of the saturable absorber. They have a large amplitude and cannot be described in the frame of weakly nonlinear analysis based on an order parameter description where non-varitional
effects are not taken into account. We fix R ¼ 5, S ¼ 1, c ¼ 0:3, a1 ¼ 2a2 ¼ 2, D1 ¼ D2 ¼ 0:5, and Ei be the control parameter. For this parameters, the threshold associated with Hopf bifurcation is jE1H j ¼ 1:52 for EiH ¼ 4:9 and the one corresponding to Turing instability is jE1T j ¼ 5:8 for EiT ¼ 34:7. At this bifurcation point, the most unstable mode is kT ¼ 1:73: In order to seed the localized structures, we perturb initially the trivial solution, which is always stable for these value of parameters, by a large amplitude asymmetric peak in the transverse direction. Such a perturbation evolves towards the formation of a localized structure which consists of spatial confinement of light in the transverse direction for a fixed time, but move with a constant speed along that direction. An example of a such behavior is illustrated in the typical space–time configuration of the signal field amplitude (see Fig. 1). A cross section along the space and the time coordinates is shown
Fig. 1. Space–time map for the real part of the signal field E1 . The numerical parameters are: D1 ¼ D2 ¼ 0:5, R ¼ 5, S ¼ 1, c ¼ 0:3, and Ei ¼ 5:8. Maxima correspond to white and the mesh number integration is 256 with dx ¼ 0:4. The 1D pulse moves with a speed of 2.73 in units of transverse length times the signal decay rate (c1 ).
6
6
5
5
4
4
|E1|
|E1|
A. Barsella et al. / Optics Communications 232 (2004) 381–389
3
3
2
2
1
1
0
385
0 0
20
40
60
80
100
Space (x)
(a)
0
(b)
20
40
60
80
100
Time (t)
Fig. 2. Separate space and time dependence in the intensity of the propagating signal field E1 . The leftmost figure shows an instantaneous x-dependence of the pulse, while the rightmost one displays the temporal behavior of the pulse for a fixed space location. The numerical parameters are the same as in Fig. 1.
Modulus of the signal field
in the Fig. 2. In fact, for a fixed space location, they exhibit a temporal Q-switch pulsing (Fig. 2(b)) typical of the presence of the SA. Before going further in the investigations of the formation of these LS, let us first check their stability domain by taking the amplitude of the incident pump field as a control parameter. A bistability region between two homogeneous states has been found for 4:7 < Ei < Eth ¼ 6:73 with the set of parameters fixed above. The bistability cycle is plotted in Fig. 3 together with the maximum amplitude of the signal pulse in order to illustrate the limits of the stability domain. In that regime, a set of lo-
4
HSS 2
0 4.65
5.65
Ei
6.65
Fig. 3. Bistability cycle and peak pulse amplitude. The numerical parameters are the same as in Fig. 1.
calized pulse solutions has been shown to be stable in the range 4:9 < Ei < 5:93. These moving localized structures have a large amplitude and a finite wide domain of stability as shown in the bifurcation diagram of the Fig. 3. Note that, for this set of parameters, the domain of stability is inside and smaller than the size of the homogeneous hysteresis loop involving the stable trivial and the lasing solutions. The maximum amplitude of the signal increases with the input pump amplitude and his ratio to the corresponding value of the upper branch stationary solutions is more than 2. This behavior of localized spatial structures is widely encountered for almost all sets of parameters we have considered in our numerical study except for the special case of perfect resonance (D1 ¼ D2 ¼ 0Þ. A striking feature in this configuration is that the moving LS also exist outside of the hysteresis loop when decreasing the input field amplitude. Indeed, we have also performed numerical simulations in this case with the same values for the remaining parameters. Fig. 4 reproduces a similar result of Fig. 3 except that the stability domain can extend out of the bistability domain. As can be shown from the Fig. 4, pulses have been found to be stable for 4:23 < Ei < 5:17, whereas the turning point is reached at Ei ¼ 4:47. They still have large amplitudes but their maxima exhibit a sharp variation for low input pump amplitudes contrary to the non-resonant situation of Fig. 3. This behavior is located outside of the hysteresis loop (4:23 <
Modulus of the signal filed
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A. Barsella et al. / Optics Communications 232 (2004) 381–389
fraction on the LS transverse speed. The result is plotted in Fig. 5 (right) where the dots are the speed values obtained numerically while the dashed curve is a fit function. We have found that the transverse speed follows a scaling law in the form ð1=bÞ where b ’ 2:16. Although this behavior V / a1 recalls the square root dependence typical of a supercritical bifurcation at low values of a1 , we emphasize here that the system is operating in a bistable regime and that the trivial solution is stable. We have also investigated the speed dependence on the input field amplitude. As can be seen from Fig. 5 (right), the LS speed is a nonlinear monotonically increasing function of the input field amplitude. Two qualitatively behaviors characterize the speed variation on the input pump amplitude. In the lower limit of the stability domain the speed increases linearly to asymptotically reach, at the upper boundary limit, an almost stationary value. Beyond this upper limit (Ei ’ 6) the LS experience secondary instabilities which we will now describe. Indeed, if one increases the incident pump amplitude, beyond the stability domain of Fig. 3, moving LS are destabilized as shown in Fig. 6. First, a secondary bifurcation leads to a splitting of the LS. This splitting is initiated by the generation of small satellites (side peaks) which grow until a new localized structure with two peaks is reached but still moving with a constant transverse speed. A typical bifurcating moving two peaks pulse is plotted in Fig. 6(a). The two peaks pulses stability domain is smaller than that of the LS pulses. Moreover, further increase
4
2
HSS
0 4.22
5.22
Ei
Fig. 4. Bistability cycle and peak pulse amplitude showing that the top branch now extends below the turning point. The numerical parameters are the same as in Fig. 1 except for the detunings which correspond to the special case of perfect resonances (i.e., D1 ¼ D2 ¼ 0). Hopf bifurcation point is EiH ¼ 4:47 and E1H ¼ 1:51 while Turing instability is reached for EiT ¼ 27, E1T ¼ 5:1 and kT ¼ 1:67.
Ei < 4:47) while inside it the maxima amplitude variation is similar to that of Fig. 3. Thus, it is obvious from the two figures that these moving pulses do not correspond to LS connecting the lower to the upper branches and the existence of the upper branch is not necessary. However, stable moving pulses can exist only in the parameter domain where the trivial homogeneous solution is stable. To better characterize the behavior of these moving LS, we have investigated the role of dif-
2.9
3.5
2.8
3
2.7 2.6 Speed
Speed
2.5 2
2.5 2.4 2.3
1.5
2.2
1
2.1 2
0.5 0
0.2 0.4 0.6 0.8 a1
1
1.2 1.4 1.6
4.8
5
5.2
5.4 EI
5.6
5.8
6
Fig. 5. Transverse pulse velocity vs. the signal diffraction coefficient a1 (left) and the pump incident field amplitude (right). The numerical parameters are the same as in Fig. 1.
A. Barsella et al. / Optics Communications 232 (2004) 381–389
387
Fig. 6. Transverse pulse instabilities: (a) break up of the pulse; (b) complex spatio-temporal dynamics far from OPO threshold. The numerical parameters are the same as in Fig. 1 except for Ei which have been increased beyond the stability domain (see Fig. 4). (a) Ei ¼ 6:25 and (b) Ei ¼ 6:4:
Fig. 7. Time evolution of the x y distribution of the near field intensity jE1 j2 . The numerical parameters are the same as in Fig. 1. (a) t ¼ 7, (b) t ¼ 17. The 2D pulse moves parallel to the x-axis with a speed of 2.844.
of the incident pump amplitude value gives rise to a complex spatio-temporal dynamics as illustrated in Fig. 6(b). Finally, we have also performed numerical simulations to ensure the stability of these moving LS with respect to two-dimensional perturbations by integrating Eq. (1) in two transverse dimensions (2D) configuration. Fig. 7 shows a typical asymptotic 2D moving LS with a constant transverse speed. However, we have found a slight difference in speeds (see Figs. 1 and 7) of the 1D and 2D pulses.
4. Conclusion Spatial moving localized structures are found in an optical parametric oscillator with a saturable absorber. They dominate the dynamics in the limit of small ratio between the decay rates of the pump and the signal fields. These localized structures arise in a regime of bistability where the lasing stationary solution (upper branch) experiences both Hopf and Turing instabilities. The main feature is that they have a constant transverse
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velocity and are stable in a wide domain of parameters. In this regime, the presence of the saturable absorber leads to self-pulsing phenomenon characterized by large amplitude oscillations that are asymmetric. The latter property combined to non-variational effects are responsible for the existence of the transverse velocity. We have characterized the role of the diffraction in the behavior of the transverse speed. Its dependence on the input field amplitude have been investigated. Further increase of the input amplitude field leads to complex spatio-temporal dynamics initiated by a splitting of the LS. Finally we have shown that the moving LS structures in the form of single stripe persist in the two-dimensional configuration. Because the saturable absorber allows the generation of spatial pulses, the implementation of a saturable absorber into the optical parametric amplification cavities can be of particular relevance for practical applications in which spatial output pulses are required. The investigation of the moving LS in fully two-dimensional problem will be the subject of future work.
Acknowledgements We thank M. Le Berre, E. Ressayre, and A. Tallet for useful discussion. The Centre dÕEtudes et de Recherche Lasers et Applications (CERLA) is supported by the Ministere charge de la Recherche, the Region Nord/Pas de Calais and the Fonds Europeen de Developpement Economique des Re gions. This research was supported in part by the Fonds National de la Recherche Scientifique (Belgium) and the Inter-University Attraction Pole Program of the Belgian government. The support of the European Science Foundation is also acknowledged.
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Moving localized structures in quadratic media with a saturable absorber A. Barsella a, C. Lepers a, M. Taki a
a,*
, M. Tlidi
b
Laboratoire de Physique des Lasers, Atomes et Molecules, CNRS UMR 8523, Centre d’Etudes et de Recherches Lasers et Applications, Universite des Sciences et Technologies de Lille, B^atiment P5, 59655 Villeneuve d’Ascq Cedex, France b Universite Libre de Bruxelles, Optique Nonlineaire Theorique, Campus Plaine, CP231, 1050 Bruxelles, Belgium Received 19 May 2003; received in revised form 26 November 2003; accepted 22 December 2003
Abstract Spatial moving pulses are found in an optical resonator with a quadratic medium coupled to an intra-cavity saturable absorber. In the context of frequency degenerate optical parametric oscillators, the saturable absorber is shown to be able to generate large amplitude asymmetric pulses in the transverse section of the signal beam. They arise in a regime where the lasing solution undergoes both Hopf and Turing instabilities. These dynamical pulses have a constant velocity and are stable in a wide domain of parameters. The origin of their transverse velocity stems from the pulse asymmetry. For large enough values of the incident pump beam the moving pulses are destabilized giving rise to a complex spatio-temporal behavior. Ó 2004 Elsevier B.V. All rights reserved. PACS: 42.65; 42.60.Mi; 42.65.pc Keywords: Parametric oscillations; The physics of solitons; Spatio-temporal dynamics
1. Introduction Optical parametric oscillations are one of the fundamental mechanisms for the generation of coherent tunable radiation in quadratic media. In large-area devices, the parametrically interacting fields experience diffraction which yields to spontaneous pattern formation in their transverse sec-
*
Corresponding author. Fax: +33-3-20-33-43-40-84. E-mail address: [email protected] (M. Taki).
tion, via symmetry-breaking bifurcations [1]. The transverse structure of the beam emitted by these oscillators is a key factor for their practical use. The structure must be mastered for applications including low-noise measurements and detection [2] or optical coherent information processing [3]. In degenerate optical parametric oscillators (OPOs) the energy conversion process is frequency degenerate and the emerging pattern can be, at the primary instability, either homogeneous or spatially periodic. The formation of a patterned state has been extensively investigated in the mean-field
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.12.078
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A. Barsella et al. / Optics Communications 232 (2004) 381–389
limit [4,5] and beyond this limit [6]. It has also been demonstrated [7,8] that in OPOs, the presence of the walk-off can alter pattern formation leading to the emergence of convective regimes and noisesustained structures. In addition to periodic patterns OPOÕs can exhibit a number of noticeable localized structures (LS) or spatial solitons [9]. These nonlinear solutions are generated in the hysteresis loop involving stable homogeneous states and the inhomogeneous periodical branches of solutions [10]. Intense theoretical research activities have been carried out on this subject [11]. This prediction was confirmed by experiments using a photorefractive material [12], semiconductor micro-resonators [13,14], liquid-crystal light valve with optical feedback [15], and spin-1/2 atomic systems [16]. The LS can exhibit periodic or chaotic oscillations in time [17– 19]. Another type of LS in the form of dark or bright solitary wave has been generated in the regime where the steady state solutions are stable with respect to the modulational instability (often called Turing instability). They are independent of Turing instabilities and their existence is related to phase indetermination in the degenerate optical parametric oscillator [20]. In the vicinity of the codimension-two point where the modulational instability is close to the saddle node bifurcation, the Turing mode and the homogeneous zero mode are liable to interact. The associated pattern selection process may be altered as a consequence of the competition between two processes: the unstable homogeneous zero mode which tends to restore spatial uniformity in the transverse plane and the large unstable Turing wavelength (induced by the coupling between diffraction and vð2Þ nonlinearity) which tends to impose intrinsic periodicity in both transverse directions. The balance between the two processes may stabilize the single bright (or dark) stripe, and its dynamics is characterized by time scaling low [21]. Experimental evidence of the solitary dark solitons in degenerate optical parametric mixing [22] has further stimulated the interest in the transverse single stripe formation. The purpose of this paper is to show that, contrary to what is described in previous studies, robust moving pulses can exist over wide range of
parameters and may therefore dominate the dynamics of OPOÕs. They have large asymmetric amplitudes reflecting the presence of the intracavity saturable absorber and are initiated by the Hopf bifurcation affecting the lasing steady state solution. These moving localized structures have a constant transverse velocity which results from the non-variational effects combined to the pulse asymmetry produced by the saturable absorber. The latter phenomenon arises in the regime of rapid relaxation oscillations when the ratio between the cavity decay rates of the pump and the signal is small. All numerical simulations we have performed, in this regime, show dynamical pulses and we never found them in a stationary state. Furthermore, we have characterized numerically the behavior of the transverse speed with both the diffraction coefficient and the input pump field. So they are different from the stationary dissipative localized structures reported in [17,18] that are found far from threshold and result from the interaction between self-pulsing dynamics and modulational instabilities. These two-dimensional (2D) pulses can be viewed as the spatio-temporal counterpart of the well-known temporal Q-switch pulses appearing in laser systems with a saturable absorber [23–25] or in purely temporal (nondiffractive) OPOs [26,27]. After a brief introduction of the model describing the dynamics of the degenerate OPO, we have performed a linear stability analysis in Section 2. Our investigations on the formation and the stability of the moving pulses and their characteristics, namely, domain of existence and velocity, constitute Section 3. Concluding remarks are summarized in the last section.
2. Model and stability analysis We start from the standard description of a degenerate OPO in the mean-field approximation. It includes, in addition to the transverse Laplacian terms (r2? ) accounting for diffraction, the saturable absorber that acts selectively at frequency x. Saturable absorber is modeled by a collection of two level atoms that leads to an intensity-dependent effective absorption coefficient. Assuming fast
A. Barsella et al. / Optics Communications 232 (2004) 381–389
atomic relaxation, the evolution equations are described by the following set of dimensionless partial differential equations [17,18] @t E1 ¼ ð1 þ iD1 ÞE1 þ E1 E2
RE1 1 þ SjE1 j
2
þ ia1 r2? E1 ; ð1Þ
@t E2 ¼ c½ð1 þ iD2 ÞE2 þ
E12
Ei þ
ia2 r2? E2 ;
ð2Þ
where E1 and E2 are the slowly varying amplitudes of the signal field at frequency x and the pump field at frequency 2x, respectively. We have set c ¼ c2 =c1 where c2 and c1 are the decay rates of the pump and the signal. The parameter D1 ¼ ðx1c xÞ=c1 (D2 ¼ ðx2c 2xÞ=c2 ) is the detuning between the frequency of the longitudinal cavity mode and that of the signal (respectively the pump). Ei is the normalized incident pump. R is the unsaturated absorption coefficient of the absorber, and S is the relative saturability of the absorber. a1 and a2 are the diffraction coefficients of the signal and the pump fields, respectively. In order to satisfy the phase-matching condition, the ratio between the diffraction coefficients of the two fields is fixed to a1 =a2 ¼ 2. The homogeneous steady state solutions of Eqs. (1) and (2) are either (i) the trivial solutions E1 ¼ 0 and E2 ¼ Ei =ð1 þ iD2 Þ or (ii) the stationary solutions for which E1 6¼ 0. The latter satisfy the following algebraic equation R Ei2 ¼ I12 þ 2 1 þ D1 D2 I1 1 þ SI1 R 2 þ ð1 þ D22 Þ ð1 þ Þ þ D21 ; ð3Þ 1 þ SI1 2 R I2 ¼ 1 þ þ D21 ; 1 þ SI1 2
with I1;2 ¼ jE1;2 j , and the system is bistable when D1 D2 > ð1 þ RÞ½1 SRð1 þ
D22 Þ:
ð4Þ
This relation generalizes the bistability condition derived in the previous works [26,28]. More precisely, in the absence of the SA [28], it reduces to D1 D2 > 1 whereas, in the presence of a SA but under perfect resonance (D1 ¼ D2 ¼ 0) [26], it simplifies to SR > 1. Let us first perform the linear
383
stability analysis of the trivial solution but in the presence of the SA. It leads to the threshold of pattern forming instability in term of pump gain parameter to be as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð5Þ Eth ¼ 1 þ D22 ð1 þ RÞ þ ½D1 þ a1 k 2 : The above expression indicates simply that the threshold is increased when the SA is present as can be expected. More importantly, D1 is still the key parameter in the formation of patterns meaning that the presence of a SA does not affect the patterned state at the onset of instability sinceffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the most unstable modes have k ¼ kc ¼ D1 =a1 . Hence, when D1 < 0, a set of modes exists that is able to bring the field in resonance and thus the system exhibits structures. For D1 > 0, the system is unable to reach perfect resonance (absolute qffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
minimum threshold i.e. Eth ¼ ð1 þ RÞ 1 þ D22 and it remains in a stable homogeneous state leading to an increase of the OPO threshold emission: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 2 Eth ¼ ½ð1 þ RÞ þ D1 ð1 þ D2 Þ. To analyze the stability of the lasing steady state solution of Eq. (3), we consider for both fields a linear deviation of the form expðkt þ ik:rÞ. The study of the linearized problem shows that the system exhibits a Turing bifurcation leading to the formation of spatially periodic pattern in the transverse plane. At the bifurcation point the critical wave number is given by D21 þ 2b2 RSI1T ð1 þ bRÞ 2 c þ cðI1T Þ; 8
2kT2 ¼ D1 þ
ð6Þ
with b ¼ 1=ð1 þ SI1T Þ. The threshold I1T satisfies the following relation: 16c½c2 þ 4ðn21 n22 Þ þ 4cn1 I1T 2
¼ ½4ðn21 n22 Þ þ c2 ;
ð7Þ
where n22 ¼ n2þ þ D21 and n1 ¼ n with n ¼ 1 þ ð1 bSI1T ÞbR. Eqs. (6) and (7) are valid only when D1 ¼ 2cD2 . The stability analysis also shows that the system undergoes a Hopf bifurcation leading to the limit cycle (homogeneous self-pulsing). The analytical expression of the bifurcation point I1H is
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cumbersome but can take the following simplified expression when D1 ¼ D2 ¼ 0: 2
2
½ðc þ n Þ n2þ ½ðn þ c2 Þ ð2I1H þ n1 Þ n n2þ ¼ 0: ð8Þ The linear stability analysis of both homogeneous steady states shows that the trivial solution can be destabilized by only a Turing bifurcation. However, the lasing solution can exhibit both types of instability. We shall see in the following section the implication of such an interaction in the dynamics of the localized structures.
3. Moving localized structures In this section we focus our investigations on the formation and the stability of moving localized structures which are strongly nonlinear. In consequence their dynamics involves nonlinear interactions that cannot be captured by linear analysis. This needs a nonlinear analytical approach to the problem which is far beyond the scope of this paper. We only consider here a numerical study of these solutions by numerically solving the model Eqs. (1) and (2) with periodic boundary condition. The system is operating in a bistable regime, according to the inequality of the expression (4). In addition, we consider a parameter range where (i) the lasing homogeneous state is unstable with respect to both Hopf an Turing bifurcations, (ii) the system exhibits a temporal Q-switching owing to the presence of the saturable absorber. This phenomenon is initiated by Hopf instability of the lasing state and occurs when the ratio c between the decay rates of the pump and the signal is small. In what follows, we show that these LS are different from the ones reported in the previous works (see Section 1), in two respects: (i) they are dynamical LS with a non-vanishing constant velocity in the whole domain of their stability; (ii) the mechanism of their formation results from the coupling between non-variational effects and the presence of the saturable absorber. They have a large amplitude and cannot be described in the frame of weakly nonlinear analysis based on an order parameter description where non-varitional
effects are not taken into account. We fix R ¼ 5, S ¼ 1, c ¼ 0:3, a1 ¼ 2a2 ¼ 2, D1 ¼ D2 ¼ 0:5, and Ei be the control parameter. For this parameters, the threshold associated with Hopf bifurcation is jE1H j ¼ 1:52 for EiH ¼ 4:9 and the one corresponding to Turing instability is jE1T j ¼ 5:8 for EiT ¼ 34:7. At this bifurcation point, the most unstable mode is kT ¼ 1:73: In order to seed the localized structures, we perturb initially the trivial solution, which is always stable for these value of parameters, by a large amplitude asymmetric peak in the transverse direction. Such a perturbation evolves towards the formation of a localized structure which consists of spatial confinement of light in the transverse direction for a fixed time, but move with a constant speed along that direction. An example of a such behavior is illustrated in the typical space–time configuration of the signal field amplitude (see Fig. 1). A cross section along the space and the time coordinates is shown
Fig. 1. Space–time map for the real part of the signal field E1 . The numerical parameters are: D1 ¼ D2 ¼ 0:5, R ¼ 5, S ¼ 1, c ¼ 0:3, and Ei ¼ 5:8. Maxima correspond to white and the mesh number integration is 256 with dx ¼ 0:4. The 1D pulse moves with a speed of 2.73 in units of transverse length times the signal decay rate (c1 ).
6
6
5
5
4
4
|E1|
|E1|
A. Barsella et al. / Optics Communications 232 (2004) 381–389
3
3
2
2
1
1
0
385
0 0
20
40
60
80
100
Space (x)
(a)
0
(b)
20
40
60
80
100
Time (t)
Fig. 2. Separate space and time dependence in the intensity of the propagating signal field E1 . The leftmost figure shows an instantaneous x-dependence of the pulse, while the rightmost one displays the temporal behavior of the pulse for a fixed space location. The numerical parameters are the same as in Fig. 1.
Modulus of the signal field
in the Fig. 2. In fact, for a fixed space location, they exhibit a temporal Q-switch pulsing (Fig. 2(b)) typical of the presence of the SA. Before going further in the investigations of the formation of these LS, let us first check their stability domain by taking the amplitude of the incident pump field as a control parameter. A bistability region between two homogeneous states has been found for 4:7 < Ei < Eth ¼ 6:73 with the set of parameters fixed above. The bistability cycle is plotted in Fig. 3 together with the maximum amplitude of the signal pulse in order to illustrate the limits of the stability domain. In that regime, a set of lo-
4
HSS 2
0 4.65
5.65
Ei
6.65
Fig. 3. Bistability cycle and peak pulse amplitude. The numerical parameters are the same as in Fig. 1.
calized pulse solutions has been shown to be stable in the range 4:9 < Ei < 5:93. These moving localized structures have a large amplitude and a finite wide domain of stability as shown in the bifurcation diagram of the Fig. 3. Note that, for this set of parameters, the domain of stability is inside and smaller than the size of the homogeneous hysteresis loop involving the stable trivial and the lasing solutions. The maximum amplitude of the signal increases with the input pump amplitude and his ratio to the corresponding value of the upper branch stationary solutions is more than 2. This behavior of localized spatial structures is widely encountered for almost all sets of parameters we have considered in our numerical study except for the special case of perfect resonance (D1 ¼ D2 ¼ 0Þ. A striking feature in this configuration is that the moving LS also exist outside of the hysteresis loop when decreasing the input field amplitude. Indeed, we have also performed numerical simulations in this case with the same values for the remaining parameters. Fig. 4 reproduces a similar result of Fig. 3 except that the stability domain can extend out of the bistability domain. As can be shown from the Fig. 4, pulses have been found to be stable for 4:23 < Ei < 5:17, whereas the turning point is reached at Ei ¼ 4:47. They still have large amplitudes but their maxima exhibit a sharp variation for low input pump amplitudes contrary to the non-resonant situation of Fig. 3. This behavior is located outside of the hysteresis loop (4:23 <
Modulus of the signal filed
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A. Barsella et al. / Optics Communications 232 (2004) 381–389
fraction on the LS transverse speed. The result is plotted in Fig. 5 (right) where the dots are the speed values obtained numerically while the dashed curve is a fit function. We have found that the transverse speed follows a scaling law in the form ð1=bÞ where b ’ 2:16. Although this behavior V / a1 recalls the square root dependence typical of a supercritical bifurcation at low values of a1 , we emphasize here that the system is operating in a bistable regime and that the trivial solution is stable. We have also investigated the speed dependence on the input field amplitude. As can be seen from Fig. 5 (right), the LS speed is a nonlinear monotonically increasing function of the input field amplitude. Two qualitatively behaviors characterize the speed variation on the input pump amplitude. In the lower limit of the stability domain the speed increases linearly to asymptotically reach, at the upper boundary limit, an almost stationary value. Beyond this upper limit (Ei ’ 6) the LS experience secondary instabilities which we will now describe. Indeed, if one increases the incident pump amplitude, beyond the stability domain of Fig. 3, moving LS are destabilized as shown in Fig. 6. First, a secondary bifurcation leads to a splitting of the LS. This splitting is initiated by the generation of small satellites (side peaks) which grow until a new localized structure with two peaks is reached but still moving with a constant transverse speed. A typical bifurcating moving two peaks pulse is plotted in Fig. 6(a). The two peaks pulses stability domain is smaller than that of the LS pulses. Moreover, further increase
4
2
HSS
0 4.22
5.22
Ei
Fig. 4. Bistability cycle and peak pulse amplitude showing that the top branch now extends below the turning point. The numerical parameters are the same as in Fig. 1 except for the detunings which correspond to the special case of perfect resonances (i.e., D1 ¼ D2 ¼ 0). Hopf bifurcation point is EiH ¼ 4:47 and E1H ¼ 1:51 while Turing instability is reached for EiT ¼ 27, E1T ¼ 5:1 and kT ¼ 1:67.
Ei < 4:47) while inside it the maxima amplitude variation is similar to that of Fig. 3. Thus, it is obvious from the two figures that these moving pulses do not correspond to LS connecting the lower to the upper branches and the existence of the upper branch is not necessary. However, stable moving pulses can exist only in the parameter domain where the trivial homogeneous solution is stable. To better characterize the behavior of these moving LS, we have investigated the role of dif-
2.9
3.5
2.8
3
2.7 2.6 Speed
Speed
2.5 2
2.5 2.4 2.3
1.5
2.2
1
2.1 2
0.5 0
0.2 0.4 0.6 0.8 a1
1
1.2 1.4 1.6
4.8
5
5.2
5.4 EI
5.6
5.8
6
Fig. 5. Transverse pulse velocity vs. the signal diffraction coefficient a1 (left) and the pump incident field amplitude (right). The numerical parameters are the same as in Fig. 1.
A. Barsella et al. / Optics Communications 232 (2004) 381–389
387
Fig. 6. Transverse pulse instabilities: (a) break up of the pulse; (b) complex spatio-temporal dynamics far from OPO threshold. The numerical parameters are the same as in Fig. 1 except for Ei which have been increased beyond the stability domain (see Fig. 4). (a) Ei ¼ 6:25 and (b) Ei ¼ 6:4:
Fig. 7. Time evolution of the x y distribution of the near field intensity jE1 j2 . The numerical parameters are the same as in Fig. 1. (a) t ¼ 7, (b) t ¼ 17. The 2D pulse moves parallel to the x-axis with a speed of 2.844.
of the incident pump amplitude value gives rise to a complex spatio-temporal dynamics as illustrated in Fig. 6(b). Finally, we have also performed numerical simulations to ensure the stability of these moving LS with respect to two-dimensional perturbations by integrating Eq. (1) in two transverse dimensions (2D) configuration. Fig. 7 shows a typical asymptotic 2D moving LS with a constant transverse speed. However, we have found a slight difference in speeds (see Figs. 1 and 7) of the 1D and 2D pulses.
4. Conclusion Spatial moving localized structures are found in an optical parametric oscillator with a saturable absorber. They dominate the dynamics in the limit of small ratio between the decay rates of the pump and the signal fields. These localized structures arise in a regime of bistability where the lasing stationary solution (upper branch) experiences both Hopf and Turing instabilities. The main feature is that they have a constant transverse
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velocity and are stable in a wide domain of parameters. In this regime, the presence of the saturable absorber leads to self-pulsing phenomenon characterized by large amplitude oscillations that are asymmetric. The latter property combined to non-variational effects are responsible for the existence of the transverse velocity. We have characterized the role of the diffraction in the behavior of the transverse speed. Its dependence on the input field amplitude have been investigated. Further increase of the input amplitude field leads to complex spatio-temporal dynamics initiated by a splitting of the LS. Finally we have shown that the moving LS structures in the form of single stripe persist in the two-dimensional configuration. Because the saturable absorber allows the generation of spatial pulses, the implementation of a saturable absorber into the optical parametric amplification cavities can be of particular relevance for practical applications in which spatial output pulses are required. The investigation of the moving LS in fully two-dimensional problem will be the subject of future work.
Acknowledgements We thank M. Le Berre, E. Ressayre, and A. Tallet for useful discussion. The Centre dÕEtudes et de Recherche Lasers et Applications (CERLA) is supported by the Ministere charge de la Recherche, the Region Nord/Pas de Calais and the Fonds Europeen de Developpement Economique des Re gions. This research was supported in part by the Fonds National de la Recherche Scientifique (Belgium) and the Inter-University Attraction Pole Program of the Belgian government. The support of the European Science Foundation is also acknowledged.
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