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Bifurcations in a semiconductor laser subject to delayed incoherent feedback
Optics Communications 214 (2002) 327–334 www.elsevier.com/locate/optcom
Bifurcations in a semiconductor laser subject to delayed incoherent feedback J.M. Saucedo Solorio a, D.W. Sukow b,*, D.R. Hicks b, A. Gavrielides c b
a Centro de Investigaciones en Optica, Apartado Postal 1-948, 37150 Leon, Guanajuato, Mexico Department of Physics and Engineering, Washington and Lee University, 116 N Main Street, Lexington, VA 24450, USA c AFRL/DELO Directed Energy Directorate, 3550 Aberdeen Avenue, SE Kirtland AFB, NM 87117, USA
Received 7 August 2002; received in revised form 6 November 2002; accepted 7 November 2002
Abstract We examine experimentally and numerically the bifurcation sequence and route into chaos of a semiconductor laser subject to delayed incoherent feedback. We show that the sequence of bifurcations follows a three frequency scenario in which the steady-state of the laser destabilizes by a Hopf bifurcation at the relaxation frequency of the laser. Specifically, we show experimentally and numerically that the Hopf bifurcation can be supercritical or subcritical depending on the length of the delay and the pumping of the laser above threshold. This is followed by a torus bifurcation at the external cavity frequency and further by a tertiary bifurcation at a significantly lower frequency. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.55.PX; 42.60.Mi; 42.65.Sf Keywords: Bifurcation; Semiconductor laser; Incoherent feedback; Delayed feedback
1. Introduction Semiconductor lasers subject to delayed feedback have been studied for the last two decades under various conditions, to understand the complicated dynamics that such systems exhibit. In particular, the semiconductor laser in an external cavity is the archetypical problem that most investigations have focused almost exclusively. This
*
Corresponding author. Tel.: +1-540-458-8881; fax: +1-540458-8884. E-mail address: [email protected] (D.W. Sukow).
system seems to exhibit the richest structure and the most complicated dynamics because both the intensity and the phase of the optical field are subject to delayed feedback. As a result, multiple attractors are possible, stemming from the multiple external-cavity steady-states [1]. Since all of the steady-states destabilize in sequence, multiple coexisting chaotic attractors are possible, as well as global attractors that connect the chaotic ruins of the external cavity modes [2]. With the success of chaotic synchronization [3] such systems have become prime candidates for chaotic communications. A number of experiments have demonstrated the possibility of
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encoding, transmitting, and decoding information by forcing the receiver to synchronize to the chaotic carrier of the transmitter [4]. The primary requirement to achieve such synchronization appears to be that the transmitter and receiver systems be closely matched. This is particularly important in laser systems such as external cavity systems or injection-locked systems, in which the phase of the laser is included in the external interaction, and the detuning between transmitter– receiver must be controlled accurately and minimized. In addition, the linewidth enhancement factor also should be closely matched so that the synchronization remains robust. An alternative is to use or design systems in which the phase of the laser is a free parameter and therefore not involved in determining the systemÕs chaotic dynamics. An example of such a system is the semiconductor laser with delayed optoelectronic feedback [5]. The system requires the electronic detection of the laser intensity, amplification of the signal, and the injection of the amplified signal into the pumping current of the laser. The delay typically arises simply through the propagation time around the feedback loop. Recently there have been extensive studies of this system, its bifurcation sequence, its chaotic dynamics and the ability to synchronize two such laser systems for communication purposes [6]. However, it is apparent that the bandwidth of the electronics is required to be very large and flat, since the chaotic dynamics of the semiconductor laser can span tens of gigahertz. If the bandwidth of the electronics does not match the speed of the optical intensity fluctuations, the dynamics of the system will be dominated by this bandwidth limitation as was found to be the case in [6]. An alternate approach that has been investigated is to consider a semiconductor laser subject to incoherent optical feedback [7]. In this particular case the system consists of a semiconductor laser whose output field is reinjected into the laser after rotation of the polarization to the orthogonal state, providing a delayed feedback that affects only the carriers. In contrast to optoelectronic feedback, this configuration can be implemented using only optical components, and therefore is not subject to electronic bandwidth limitations.
This system has not been studied in extensive detail experimentally because it presents some difficulties in blocking the reinjection of the same polarization radiation. It is the purpose of this paper to examine in detail the bifurcation sequence of the incoherent delayed feedback system. This paper is arranged as follows. In Section 2 we present the pertinent equations and show that the optoelectronic and incoherent feedback systems are expected to show the same bifurcation sequence. We then describe the experiment and present our experimental results. We show in the subsequent discussion that the experimental results are in complete agreement with the analytical and numerical treatment of incoherent optical feedback. 2. Formulation In this section we briefly describe the equations that characterize the system and formulate them in a dimensionless form. A particular format of these equations has been presented in [7], however, we will model our treatment after [8]. The intensity I and the carriers N are determined from dI ¼ 2NI; dt T
dN ¼ P N ð1 þ 2N Þ½I þ gIðt sÞ; dt
ð1Þ ð2Þ
where the time t is normalized to the cavity lifetime, T is the ratio of the carrier lifetime ss ns to the cavity lifetime sp ps, and P ¼ J =Jth 1 is the pumping current above threshold. The last term in Eq. (2) represents the reinjected orthogonal polarization intensity. The delay s ¼ 2L=csp , represents the external optical cavity, and g is proportional to the strength of the external feedback. The phase U of the electric field is a dependent quantity; it follows the dynamics of the carriers and the intensity, and therefore is not required to be considered for the systemÕs dynamics. However, in the computation of numerical spectra, spontaneous emission noise is added to Eq. (1) and the equation for the phase with noise is also included if optical spectra are also required.
J.M. Saucedo Solorio et al. / Optics Communications 214 (2002) 327–334
For comparison, in the case of optoelectronic feedback Eq. (1) is unchanged, however, Eq. (2) becomes T
dN ¼ P þ gIðt sÞ N ð1 þ 2N ÞI; dt
ð3Þ
where the second term in the right-hand side of the equations describes the current proportional to the delayed intensity. Here g is the signal amplification, and its sign determines whether positive or negative feedback is desired. The equations can be recast in a form that exhibits the importance of the various terms. From linear stability analysis it is clear that the p relevant ffiffiffiffiffiffiffiffiffiffiffi time scale is the relaxation frequency x ¼ 2P =T , which is of Oð102 Þ small quantity. In addition, the inversion N OðxÞ: Utilizing the following scaling and definitions x ð4Þ N ¼ y; 2 I ¼ P ð1 þ xÞ; ð5Þ s ¼ xt
ð6Þ
we find dx ¼ yð1 þ xÞ ds
ð7Þ
and
dy 2P 2 ¼ x g½1 þ xðs hÞ y 1 þ x ds 1 þ 2P 2P 2 g y½1 þ xðs hÞ; ð8Þ 1 þ 2P where h ¼ xs is the normalized delay and 2 ¼ x
1 þ 2P : 2P
ð9Þ
Eq. (3) transforms to
dy 2P ¼ x þ g½1 þ xðs hÞ 2 y 1 þ x ds 1 þ 2P ð10Þ for the case of the optoelectronic feedback. It is clear that the two differ by the last term of Eq. (8). This term is of order 2 g ¼ Oð4 Þ and for T ! 1, Eq. (8) becomes identical to Eq. (10). Thus, the slow amplitude equations derived in [9] to leading order are identical for the cases of
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optoelectronic and incoherent feedback. It is thus expected that the sequence of bifurcations in the two systems should be very similar. We note that the case of incoherent feedback g > 0 corresponds to the case of negative optoelectronic feedback, as seen in Eqs. (8) and (10). Analysis of the equations indicates that the structure and sequence of the primary bifurcation is not altered for negative or positive optoelectronic feedback [9,10]. However, the specific values of delay at which the bifurcations take place are distinctly different. We find that the destabilizing Hopf bifurcation can be supercritical or subcritical [9] as apfunction ffiffiffiffiffiffiffiffiffiffiffi of normalized delay h ¼ xs, where x ¼ 2P =T is the relaxation frequency of the solitary laser. The frequency of the Hopf bifurcation is close to the relaxation frequency. For the supercritical case, as the feedback rate is increased the Hopf bifurcation is followed by a quasi-periodic bifurcation with a frequency close to the external cavity frequency. This bifurcation in turn is followed by the tertiary bifurcation, at a frequency much smaller than either the relaxation or the external cavity frequency. In the subcritical case, the sequence of bifurcations is similar except that they emerge from an unstable Hopf bifurcation that stabilizes through a limit cycle saddle-node bifurcation. The tertiary frequency arises as the result of the interaction of two oscillating modes. The conditions for three-dimensional tori can be fulfilled if one mode bifurcates supercritically while the other does so subcritically. Near points of degeneracy in delay and feedback a three-dimensional torus can develop. The frequency is sufficiently slow because the system is approaching a homoclinic loop at which the period of the slow amplitude modulation tends to infinity [10,11]. We find that there can be bistability between the steady-state and the Hopf bifurcation or the steady-state and the torus bifurcation. Similar results have been obtained in [10], in which the possibility of a three-dimensional torus formation was analyzed and captured by a numerical series of power spectra. The optoelectronic model was also studied in the vicinity of the Hopf bifurcation by Giacomelli and Politi [12].
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3. Experiment The experiment uses a temperature-stabilized diode laser [(LD), Sharp LT024] operating at a nominal wavelength of k ¼ 778 nm and a measured threshold current Ith ¼ 48:5 mA. As shown in the schematic diagram Fig. 1, the horizontally polarized laser beam is collimated by a high numerical aperture lens [(CL), numerical aperture¼ 0:47] and enters an external cavity formed by a high-reflectivity mirror (HR) that is 32:5 cm distant from the laser. The optical feedback in this experiment is designed to be incoherent, that is, the reinjected optical fieldÕs polarization state is orthogonal to that of the initial beam. This is unlike coherent feedback, where the reinjected light is of the same polarization state as the original beam and therefore can stimulate emission in a supported laser mode. We create the incoherent feedback by a combination of intracavity elements. The beam enters a Faraday rotator whose input polarizer is removed, is rotated in polarization by 45°; and exits through the output polarizer (transmission axis is set at 45° from the horizontal). A rotatable polarizer (POL) in the cavity adjusts the intensity (and thus the feedback strength) before the beam is retroreflected by the end mirror. The polarization of the return beam is forced by the rotatorÕs output polarizer to be oriented at 45° from the horizontal; the beam is then rotated an additional 45° on the return pass, thereby creating a vertically oriented feedback beam that is reinjected into the laser. We note that this configuration prevents any secondary back reflections from the laserÕs front facet from completing a second roundtrip in the external cavity. Thus the system is modeled accurately using only a single delayed
Fig. 1. Schematic of the experimental setup and diagnostics.
feedback term in the carrier equation, Eq. (2). The vertically polarized component in the laser emission is negligible with or without the incoherent feedback. We insert a plate beamsplitter [(BS) 30% reflective] into the cavity between the Faraday rotator and the linear polarizer to sample the beam and detect the laser dynamics. The sampled beam is focused by a lens onto a fast ac-coupled photodiode [(PD), Hamamatsu C4258-01, 8:75 GHz bandwidth], after being attenuated by a neutral density filter to limit the power incident on the detector. The photodiode signal is amplified by 23 dB (AMP) and displayed on a rf spectrum analyzer [(RF) Agilent E4405B]. The experimental rf spectra illustrate the bifurcation sequence as a function of optical feedback strength. The feedback level required to destabilize the steady-state is so small that it produces no measurable alteration to the slope efficiency of the light curve. Therefore, we characterize experimental feedback strengths by calculating the reflected optical intensity that reaches the laserÕs front facet. This is effectively the external cavity roundtrip transmission TEC , expressed as a percent of the laserÕs output. For the experimental data shown in Fig. 2, the pump current is held constant at I ¼ 64:38 mA, and the temperature is stable at 15:01 °C. Fig. 2(a) shows the spectrum of the free-running laser in the absence of feedback. A broad feature is apparent, peaked near the relaxation oscillation (RO) frequency slightly below 4 GHz. In Fig. 2(b), the ROs become undamped as weak feedback TEC ¼ 0:05% is applied, and a single sharp peak appears at 3:6 GHz. As the feedback strength is increased to TEC ¼ 0:11%, another frequency appears at 380 MHz as shown in Fig. 2(c). This second frequency is close to the external cavity (EC) fundamental frequency, 460 MHz. This peak is also evident as sidebands about the central RO peak. At even stronger feedback TEC ¼ 0:24%, a weak tertiary bifurcation takes place, identified by the appearance of yet another frequency of 190 MHz. This is clearly evident in Fig. 2(d) as a small peak appearing between the central RO peak and its EC sideband. Further increase of the optical feedback leads to significant growth
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Fig. 2. Experimental power spectra for I ¼ 64:38 mA. (a) Freerunning laser, TEC ¼ 0; (b) limit cycle, TEC ¼ 0:05%; (c) two frequency, TEC ¼ 0:11%; (d) three frequency spectra, TEC ¼ 0:23%.
in the rf power at all frequencies as the dynamics become chaotic. A similar scenario unfolds if the laser is operated under slightly different experimental conditions, at a current of 71:2 mA. In Fig. 3(a) a broad feature now peaks just above 4 GHz, again signifying noise-driven relaxation frequency oscillations in the free-running laser. Upon introducing the feedback TEC ¼ 0:13%, Fig. 3(b), the laser becomes unstable to a quasi-periodic bifurcation, indicating that the limit cycle is born through a subcritical Hopf bifurcation and there is bistability between the steady-state and the torus. A further increase in the feedback TEC ¼ 0:50%, Fig. 3(c), induces a tertiary bifurcation and a third frequency of 210 MHz appears in the rf spectrum of the laser. The system evolves into more complex dynamics at TEC ¼ 1:30%, with the comb of frequencies associated with the EC frequency still apparent, as seen in Fig. 3(d). Finally, chaos is observed for still stronger feedback strength TEC ¼ 3:78%, with Fig. 3(e) displaying a broad spectrum where substantial power is evident at all frequencies.
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Fig. 3. Experimental power spectra for I ¼ 71:2 mA. (a) Relaxation oscillations, TEC ¼ 0%; (b) two frequency, TEC ¼ 0:13%; (c) three frequency, TEC ¼ 0:50%; (d) complex dynamics, TEC ¼ 1:30%; (e) chaotic spectra, TEC ¼ 3:77%.
4. Discussion In this section we present numerical calculations using Eqs. (7) and (8) to further illustrate and interpret the experimental results. The relevant parameters appearing in the equations are taken to be T ¼ 1000, P ¼ 1:7. The time normalization to the cavity lifetime was taken to be cp ¼ 2:4 1011 s1 . The structure and the sequence of bifurcations depend critically on the value of h, and since h ¼ xs, it can be varied by changing either the value of s or P (thus x). In numerics, we vary s to adjust h. However, in the experiment we vary the pumping P since this can be accomplished with greater precision and repeatability than varying the external cavity length. Additional effects that result from changing P (due to the terms in Eq. (8) that explicitly show a dependence on P ) do not contribute to the leading order of bifurcations and are minimal [9]. These are mainly small shifts in the bifurcations as a function of the feedback strength. The reason for this is that the normalized delay s is a number of Oð103 Þ and therefore only
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changes of Oð2 Þ to P are required, to produce changes of Oð1Þ in h. Fig. 4(a) shows the bifurcation diagram for h ¼ 32:94 and Fig. 4(b) for h ¼ 37:90. It is clear in Fig. 4(a) that the steady-state destabilizes by a supercritical Hopf bifurcation followed by a quasiperiodic bifurcation. In addition, a limit cycle that emerges from a saddle-node bifurcation coexists with the steady-state. This limit cycle is almost pulsating since its minima are close to zero. In addition, this limit cycle undergoes a torus bifurcation. Increasing the feedback the laser moves from a steady-state to oscillating intensity, followed by a quasi-periodic intensity. In Fig. 4(b), however, the steady-state is destabilized by a subcritical Hopf bifurcation, and an unstable limit cycle connects the steady-state to the stable limit cycle. Clearly, this limit cycle also undergoes a torus bifurcation. Notice that there are regions of the feedback strength in which the steady-state coexists with a stable limit cycle or with a torus. In this case, as the feedback is increased the laser moves abruptly from the steady-state intensity to a quasi-periodic intensity. The sequences of experi-
Fig. 4. Bifurcation diagrams corresponding to (a) h ¼ 32:94 and (b) h ¼ 39:70.
mental spectra described in Figs. 2 and 3 correspond to the two bifurcation diagrams. As the value of h increases further the subcritical Hopf bifurcation moves to higher values of feedback asymptotically moving to infinity and producing the coexisting limit cycle saddle-node, an example of which is seen in Fig. 4(a). The analytical details of this discussion can be seen in [9]. A series of power spectra computed for the bifurcation diagram of Fig. 4(a) are shown in Fig. 5. Starting at Fig. 5(a) the power spectrum of the steady-state at weak feedback strength of g ¼ 0:021 exhibits a broad maximum at a frequency of about 2.1 GHz that corresponds to the relaxation frequency of laser (relaxation frequency of the solitary laser is 2:21 GHz). Weak modulations at the external cavity frequency can also be seen. In Fig. 5(b) the relaxation frequency becomes undamped and a limit cycle emerges from the Hopf bifurcation as is indicated by the sharp lines at that frequency and its harmonics. This spectrum corresponds to a feedback strength of g ¼ 0:12. With further increase of the feedback to g ¼ 0:155 the torus bifurcation takes place and a new frequency close to the external cavity frequency me ¼ 288 MHz emerges as seen in Fig. 5(c). The
Fig. 5. Numerically computed power spectra for T ¼ 1000, P ¼ 1:7, and h ¼ 32:94 for feedback strength (a) g ¼ 0:021, (b) g ¼ 0:12, (c) g ¼ 0:155, (d) g ¼ 0:167, (e) g ¼ 0:21 and (f) g ¼ 0:31.
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actual external cavity frequency is 425 MHz. Further increasing the feedback to g ¼ 0:167 a third frequency appears of about 15 MHz; an expanded view around the relaxation frequency peak is shown in Fig. 5(d). Increasing the feedback rate to 0:21 the trajectory leaves this attractor and moves to the larger pulsating attractor; the spectrum of its intensity is shown in Fig. 5(e). The third frequency is seen here clearly and has increased to about half of me , around 125 MHz. Finally Fig. 5(f) shows the power spectrum at g ¼ 0:31 in which the dynamics become considerably more complex. The sequence of power spectra corresponding to the bifurcation sequence depicted in Fig. 4(b) is shown in Fig. 6. The steady-state power spectrum is very similar to that in the previous figure and the sequence of limit cycle Fig. 6(b), torus Fig. 6(c) and the third frequency of 25 MHz appears in Fig. 6(d), at g ¼ 0:10, 0.12, and 0.14. However, because the steady-state destabilizes by a subcritical Hopf bifurcation, the limit cycle is missing in the experimental sequence when the feedback rate is increased. The limit cycle can be attained while decreasing the feedback rate, but the interval over which it exists is too small to be captured experimentally. Increasing the feedback rate to g ¼ 0:231 the trajectory remains in the same attractor and
333
the third frequency increases to 97 GHz as seen in Fig. 6(e). Finally, further increase of the rate leads to the complex dynamics similar to the one depicted in the power spectra in Fig. 5(f). Although at the bifurcation point the tertiary bifurcation frequency can be rather small (of the order of 30 MHz), as the feedback rate is increased the frequency also increases rather quickly. For example, we show in Fig. 6(d) a frequency of about 25 MHz at a feedback rate of g ¼ 0:14. With increase of the feedback to g ¼ 0:231 the frequency increases to 97 MHz. Similarly, attractors with pulsating intensities that coexist with attractors connected to the steady-state can have larger tertiary frequencies (125 MHz) as shown in Fig. 5(e). It is very difficult in the experiment to fine-tune and stabilize the feedback to remain close to the tertiary bifurcation point and on the same attractor, and thus capture the whole frequency range. In conclusion, we have examined the sequence of bifurcations in the incoherent feedback system experimentally and found that it is identical to the sequence found in the case of the optoelectronic feedback [6]. This is verified by the numerical studies that appear to agree quite well with the experimental spectra. We have found supercritical and subcritical Hopf bifurcations, and the sequence of bifurcations and frequencies expected are in very good agreement not only with the numerical results, but also agree very well with the predictions of the analytic results in [9], and also in [11]. Acknowledgements Acknowledgment is made to the W.M. Keck Foundation and the Thomas F. and Kate Miller Jeffress Memorial Trust for the partial support of this research. J.M.S.S. gratefully acknowledges Centro de Investigaciones en Optica for the support as student visitor at AFRL, Directed Energy Directorate, Kirtland AFB. References
Fig. 6. Numerically computed power spectra for T ¼ 1000, P ¼ 1:7, and h ¼ 39:70 for feedback strength (a) g ¼ 0:021, (b) g ¼ 0:10, (c) g ¼ 0:12, (d) g ¼ 0:14, and (e) g ¼ 0:231.
[1] J. Mark, B. Thomborg, J. Mork, IEEE J. Quantum Electron. 28 (1992) 93; J. Mark, B. Thomborg, Phys. Rev. Lett. 65 (1999) 1999.
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[2] A. Hohl, A. Gavrielides, Phys. Rev. Lett. 82 (1999) 1148; T. Heil, I. Fischer, W. Elsasser, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) 413. [3] I. Fischer, Y. Liu, P. Davis, Phys. Rev. A 62 (2000) 1801. [4] G.D. VanWiggeren, R. Roy, Science 279 (1998) 1198; J.-P. Goedgebuer, L. Larger, H. Porter, Phys. Rev. Lett. 80 (1998) 2249. [5] Otsubo, P. Saboureau, J.-P. Foing, P. Schanne, IEEE J. Quantum Electron. 33 (1997) 1582. [6] S. Tang, J.M. Liu, IEEE J. Quantum Electron. 37 (2001) 329; S. Tang, H.F. Chen, J.M. Liu, Opt. Lett. 26 (2001) 1489. [7] K. Otsuka, J.-K. Chern, Opt. Lett. 16 (1991) 1759; T.C. Yen, J.W. Chang, J.M. Lin, R.J. Chen, Opt. Commun. 150 (1998) 159;
[8] [9] [10] [11] [12]
J. Houlihan, G. Huyet, J.G. McInerney, Opt. Commun. 199 (2001) 175; F. Rogister, A. Locquet, D. Pieroux, M. Sciamanna, O. Deparis, P. Megret, M. Blondel, Opt. Lett. 26 (2001) 1486; F. Rogister, D. Pieroux, M. Sciamanna, P. Megret, M. Blondel, Opt. Commun. 207 (2002) 295. D. Pieroux, T. Erneux, K. Otsuka, Phys. Rev. A 50 (1994) 1822. D. Pieroux, T. Erneux, A. Gavrielides, V. Kovanis, SIAM J. Appl. Math. 61 (2000) 966. E.V. Grigorieva, H. Haken, S.A. Kaschenko, Opt. Commun. 165 (1999) 279. D. Pieroux, T. Erneux, T. Luzyanina, K. Engelborghs, Phys. Rev. E 63 (2001) 036211. G. Giacomelli, A. Politi, Physica D 117 (1998) 26.
Bifurcations in a semiconductor laser subject to delayed incoherent feedback J.M. Saucedo Solorio a, D.W. Sukow b,*, D.R. Hicks b, A. Gavrielides c b
a Centro de Investigaciones en Optica, Apartado Postal 1-948, 37150 Leon, Guanajuato, Mexico Department of Physics and Engineering, Washington and Lee University, 116 N Main Street, Lexington, VA 24450, USA c AFRL/DELO Directed Energy Directorate, 3550 Aberdeen Avenue, SE Kirtland AFB, NM 87117, USA
Received 7 August 2002; received in revised form 6 November 2002; accepted 7 November 2002
Abstract We examine experimentally and numerically the bifurcation sequence and route into chaos of a semiconductor laser subject to delayed incoherent feedback. We show that the sequence of bifurcations follows a three frequency scenario in which the steady-state of the laser destabilizes by a Hopf bifurcation at the relaxation frequency of the laser. Specifically, we show experimentally and numerically that the Hopf bifurcation can be supercritical or subcritical depending on the length of the delay and the pumping of the laser above threshold. This is followed by a torus bifurcation at the external cavity frequency and further by a tertiary bifurcation at a significantly lower frequency. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.55.PX; 42.60.Mi; 42.65.Sf Keywords: Bifurcation; Semiconductor laser; Incoherent feedback; Delayed feedback
1. Introduction Semiconductor lasers subject to delayed feedback have been studied for the last two decades under various conditions, to understand the complicated dynamics that such systems exhibit. In particular, the semiconductor laser in an external cavity is the archetypical problem that most investigations have focused almost exclusively. This
*
Corresponding author. Tel.: +1-540-458-8881; fax: +1-540458-8884. E-mail address: [email protected] (D.W. Sukow).
system seems to exhibit the richest structure and the most complicated dynamics because both the intensity and the phase of the optical field are subject to delayed feedback. As a result, multiple attractors are possible, stemming from the multiple external-cavity steady-states [1]. Since all of the steady-states destabilize in sequence, multiple coexisting chaotic attractors are possible, as well as global attractors that connect the chaotic ruins of the external cavity modes [2]. With the success of chaotic synchronization [3] such systems have become prime candidates for chaotic communications. A number of experiments have demonstrated the possibility of
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 1 8 4 - 3
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J.M. Saucedo Solorio et al. / Optics Communications 214 (2002) 327–334
encoding, transmitting, and decoding information by forcing the receiver to synchronize to the chaotic carrier of the transmitter [4]. The primary requirement to achieve such synchronization appears to be that the transmitter and receiver systems be closely matched. This is particularly important in laser systems such as external cavity systems or injection-locked systems, in which the phase of the laser is included in the external interaction, and the detuning between transmitter– receiver must be controlled accurately and minimized. In addition, the linewidth enhancement factor also should be closely matched so that the synchronization remains robust. An alternative is to use or design systems in which the phase of the laser is a free parameter and therefore not involved in determining the systemÕs chaotic dynamics. An example of such a system is the semiconductor laser with delayed optoelectronic feedback [5]. The system requires the electronic detection of the laser intensity, amplification of the signal, and the injection of the amplified signal into the pumping current of the laser. The delay typically arises simply through the propagation time around the feedback loop. Recently there have been extensive studies of this system, its bifurcation sequence, its chaotic dynamics and the ability to synchronize two such laser systems for communication purposes [6]. However, it is apparent that the bandwidth of the electronics is required to be very large and flat, since the chaotic dynamics of the semiconductor laser can span tens of gigahertz. If the bandwidth of the electronics does not match the speed of the optical intensity fluctuations, the dynamics of the system will be dominated by this bandwidth limitation as was found to be the case in [6]. An alternate approach that has been investigated is to consider a semiconductor laser subject to incoherent optical feedback [7]. In this particular case the system consists of a semiconductor laser whose output field is reinjected into the laser after rotation of the polarization to the orthogonal state, providing a delayed feedback that affects only the carriers. In contrast to optoelectronic feedback, this configuration can be implemented using only optical components, and therefore is not subject to electronic bandwidth limitations.
This system has not been studied in extensive detail experimentally because it presents some difficulties in blocking the reinjection of the same polarization radiation. It is the purpose of this paper to examine in detail the bifurcation sequence of the incoherent delayed feedback system. This paper is arranged as follows. In Section 2 we present the pertinent equations and show that the optoelectronic and incoherent feedback systems are expected to show the same bifurcation sequence. We then describe the experiment and present our experimental results. We show in the subsequent discussion that the experimental results are in complete agreement with the analytical and numerical treatment of incoherent optical feedback. 2. Formulation In this section we briefly describe the equations that characterize the system and formulate them in a dimensionless form. A particular format of these equations has been presented in [7], however, we will model our treatment after [8]. The intensity I and the carriers N are determined from dI ¼ 2NI; dt T
dN ¼ P N ð1 þ 2N Þ½I þ gIðt sÞ; dt
ð1Þ ð2Þ
where the time t is normalized to the cavity lifetime, T is the ratio of the carrier lifetime ss ns to the cavity lifetime sp ps, and P ¼ J =Jth 1 is the pumping current above threshold. The last term in Eq. (2) represents the reinjected orthogonal polarization intensity. The delay s ¼ 2L=csp , represents the external optical cavity, and g is proportional to the strength of the external feedback. The phase U of the electric field is a dependent quantity; it follows the dynamics of the carriers and the intensity, and therefore is not required to be considered for the systemÕs dynamics. However, in the computation of numerical spectra, spontaneous emission noise is added to Eq. (1) and the equation for the phase with noise is also included if optical spectra are also required.
J.M. Saucedo Solorio et al. / Optics Communications 214 (2002) 327–334
For comparison, in the case of optoelectronic feedback Eq. (1) is unchanged, however, Eq. (2) becomes T
dN ¼ P þ gIðt sÞ N ð1 þ 2N ÞI; dt
ð3Þ
where the second term in the right-hand side of the equations describes the current proportional to the delayed intensity. Here g is the signal amplification, and its sign determines whether positive or negative feedback is desired. The equations can be recast in a form that exhibits the importance of the various terms. From linear stability analysis it is clear that the p relevant ffiffiffiffiffiffiffiffiffiffiffi time scale is the relaxation frequency x ¼ 2P =T , which is of Oð102 Þ small quantity. In addition, the inversion N OðxÞ: Utilizing the following scaling and definitions x ð4Þ N ¼ y; 2 I ¼ P ð1 þ xÞ; ð5Þ s ¼ xt
ð6Þ
we find dx ¼ yð1 þ xÞ ds
ð7Þ
and
dy 2P 2 ¼ x g½1 þ xðs hÞ y 1 þ x ds 1 þ 2P 2P 2 g y½1 þ xðs hÞ; ð8Þ 1 þ 2P where h ¼ xs is the normalized delay and 2 ¼ x
1 þ 2P : 2P
ð9Þ
Eq. (3) transforms to
dy 2P ¼ x þ g½1 þ xðs hÞ 2 y 1 þ x ds 1 þ 2P ð10Þ for the case of the optoelectronic feedback. It is clear that the two differ by the last term of Eq. (8). This term is of order 2 g ¼ Oð4 Þ and for T ! 1, Eq. (8) becomes identical to Eq. (10). Thus, the slow amplitude equations derived in [9] to leading order are identical for the cases of
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optoelectronic and incoherent feedback. It is thus expected that the sequence of bifurcations in the two systems should be very similar. We note that the case of incoherent feedback g > 0 corresponds to the case of negative optoelectronic feedback, as seen in Eqs. (8) and (10). Analysis of the equations indicates that the structure and sequence of the primary bifurcation is not altered for negative or positive optoelectronic feedback [9,10]. However, the specific values of delay at which the bifurcations take place are distinctly different. We find that the destabilizing Hopf bifurcation can be supercritical or subcritical [9] as apfunction ffiffiffiffiffiffiffiffiffiffiffi of normalized delay h ¼ xs, where x ¼ 2P =T is the relaxation frequency of the solitary laser. The frequency of the Hopf bifurcation is close to the relaxation frequency. For the supercritical case, as the feedback rate is increased the Hopf bifurcation is followed by a quasi-periodic bifurcation with a frequency close to the external cavity frequency. This bifurcation in turn is followed by the tertiary bifurcation, at a frequency much smaller than either the relaxation or the external cavity frequency. In the subcritical case, the sequence of bifurcations is similar except that they emerge from an unstable Hopf bifurcation that stabilizes through a limit cycle saddle-node bifurcation. The tertiary frequency arises as the result of the interaction of two oscillating modes. The conditions for three-dimensional tori can be fulfilled if one mode bifurcates supercritically while the other does so subcritically. Near points of degeneracy in delay and feedback a three-dimensional torus can develop. The frequency is sufficiently slow because the system is approaching a homoclinic loop at which the period of the slow amplitude modulation tends to infinity [10,11]. We find that there can be bistability between the steady-state and the Hopf bifurcation or the steady-state and the torus bifurcation. Similar results have been obtained in [10], in which the possibility of a three-dimensional torus formation was analyzed and captured by a numerical series of power spectra. The optoelectronic model was also studied in the vicinity of the Hopf bifurcation by Giacomelli and Politi [12].
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3. Experiment The experiment uses a temperature-stabilized diode laser [(LD), Sharp LT024] operating at a nominal wavelength of k ¼ 778 nm and a measured threshold current Ith ¼ 48:5 mA. As shown in the schematic diagram Fig. 1, the horizontally polarized laser beam is collimated by a high numerical aperture lens [(CL), numerical aperture¼ 0:47] and enters an external cavity formed by a high-reflectivity mirror (HR) that is 32:5 cm distant from the laser. The optical feedback in this experiment is designed to be incoherent, that is, the reinjected optical fieldÕs polarization state is orthogonal to that of the initial beam. This is unlike coherent feedback, where the reinjected light is of the same polarization state as the original beam and therefore can stimulate emission in a supported laser mode. We create the incoherent feedback by a combination of intracavity elements. The beam enters a Faraday rotator whose input polarizer is removed, is rotated in polarization by 45°; and exits through the output polarizer (transmission axis is set at 45° from the horizontal). A rotatable polarizer (POL) in the cavity adjusts the intensity (and thus the feedback strength) before the beam is retroreflected by the end mirror. The polarization of the return beam is forced by the rotatorÕs output polarizer to be oriented at 45° from the horizontal; the beam is then rotated an additional 45° on the return pass, thereby creating a vertically oriented feedback beam that is reinjected into the laser. We note that this configuration prevents any secondary back reflections from the laserÕs front facet from completing a second roundtrip in the external cavity. Thus the system is modeled accurately using only a single delayed
Fig. 1. Schematic of the experimental setup and diagnostics.
feedback term in the carrier equation, Eq. (2). The vertically polarized component in the laser emission is negligible with or without the incoherent feedback. We insert a plate beamsplitter [(BS) 30% reflective] into the cavity between the Faraday rotator and the linear polarizer to sample the beam and detect the laser dynamics. The sampled beam is focused by a lens onto a fast ac-coupled photodiode [(PD), Hamamatsu C4258-01, 8:75 GHz bandwidth], after being attenuated by a neutral density filter to limit the power incident on the detector. The photodiode signal is amplified by 23 dB (AMP) and displayed on a rf spectrum analyzer [(RF) Agilent E4405B]. The experimental rf spectra illustrate the bifurcation sequence as a function of optical feedback strength. The feedback level required to destabilize the steady-state is so small that it produces no measurable alteration to the slope efficiency of the light curve. Therefore, we characterize experimental feedback strengths by calculating the reflected optical intensity that reaches the laserÕs front facet. This is effectively the external cavity roundtrip transmission TEC , expressed as a percent of the laserÕs output. For the experimental data shown in Fig. 2, the pump current is held constant at I ¼ 64:38 mA, and the temperature is stable at 15:01 °C. Fig. 2(a) shows the spectrum of the free-running laser in the absence of feedback. A broad feature is apparent, peaked near the relaxation oscillation (RO) frequency slightly below 4 GHz. In Fig. 2(b), the ROs become undamped as weak feedback TEC ¼ 0:05% is applied, and a single sharp peak appears at 3:6 GHz. As the feedback strength is increased to TEC ¼ 0:11%, another frequency appears at 380 MHz as shown in Fig. 2(c). This second frequency is close to the external cavity (EC) fundamental frequency, 460 MHz. This peak is also evident as sidebands about the central RO peak. At even stronger feedback TEC ¼ 0:24%, a weak tertiary bifurcation takes place, identified by the appearance of yet another frequency of 190 MHz. This is clearly evident in Fig. 2(d) as a small peak appearing between the central RO peak and its EC sideband. Further increase of the optical feedback leads to significant growth
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Fig. 2. Experimental power spectra for I ¼ 64:38 mA. (a) Freerunning laser, TEC ¼ 0; (b) limit cycle, TEC ¼ 0:05%; (c) two frequency, TEC ¼ 0:11%; (d) three frequency spectra, TEC ¼ 0:23%.
in the rf power at all frequencies as the dynamics become chaotic. A similar scenario unfolds if the laser is operated under slightly different experimental conditions, at a current of 71:2 mA. In Fig. 3(a) a broad feature now peaks just above 4 GHz, again signifying noise-driven relaxation frequency oscillations in the free-running laser. Upon introducing the feedback TEC ¼ 0:13%, Fig. 3(b), the laser becomes unstable to a quasi-periodic bifurcation, indicating that the limit cycle is born through a subcritical Hopf bifurcation and there is bistability between the steady-state and the torus. A further increase in the feedback TEC ¼ 0:50%, Fig. 3(c), induces a tertiary bifurcation and a third frequency of 210 MHz appears in the rf spectrum of the laser. The system evolves into more complex dynamics at TEC ¼ 1:30%, with the comb of frequencies associated with the EC frequency still apparent, as seen in Fig. 3(d). Finally, chaos is observed for still stronger feedback strength TEC ¼ 3:78%, with Fig. 3(e) displaying a broad spectrum where substantial power is evident at all frequencies.
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Fig. 3. Experimental power spectra for I ¼ 71:2 mA. (a) Relaxation oscillations, TEC ¼ 0%; (b) two frequency, TEC ¼ 0:13%; (c) three frequency, TEC ¼ 0:50%; (d) complex dynamics, TEC ¼ 1:30%; (e) chaotic spectra, TEC ¼ 3:77%.
4. Discussion In this section we present numerical calculations using Eqs. (7) and (8) to further illustrate and interpret the experimental results. The relevant parameters appearing in the equations are taken to be T ¼ 1000, P ¼ 1:7. The time normalization to the cavity lifetime was taken to be cp ¼ 2:4 1011 s1 . The structure and the sequence of bifurcations depend critically on the value of h, and since h ¼ xs, it can be varied by changing either the value of s or P (thus x). In numerics, we vary s to adjust h. However, in the experiment we vary the pumping P since this can be accomplished with greater precision and repeatability than varying the external cavity length. Additional effects that result from changing P (due to the terms in Eq. (8) that explicitly show a dependence on P ) do not contribute to the leading order of bifurcations and are minimal [9]. These are mainly small shifts in the bifurcations as a function of the feedback strength. The reason for this is that the normalized delay s is a number of Oð103 Þ and therefore only
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changes of Oð2 Þ to P are required, to produce changes of Oð1Þ in h. Fig. 4(a) shows the bifurcation diagram for h ¼ 32:94 and Fig. 4(b) for h ¼ 37:90. It is clear in Fig. 4(a) that the steady-state destabilizes by a supercritical Hopf bifurcation followed by a quasiperiodic bifurcation. In addition, a limit cycle that emerges from a saddle-node bifurcation coexists with the steady-state. This limit cycle is almost pulsating since its minima are close to zero. In addition, this limit cycle undergoes a torus bifurcation. Increasing the feedback the laser moves from a steady-state to oscillating intensity, followed by a quasi-periodic intensity. In Fig. 4(b), however, the steady-state is destabilized by a subcritical Hopf bifurcation, and an unstable limit cycle connects the steady-state to the stable limit cycle. Clearly, this limit cycle also undergoes a torus bifurcation. Notice that there are regions of the feedback strength in which the steady-state coexists with a stable limit cycle or with a torus. In this case, as the feedback is increased the laser moves abruptly from the steady-state intensity to a quasi-periodic intensity. The sequences of experi-
Fig. 4. Bifurcation diagrams corresponding to (a) h ¼ 32:94 and (b) h ¼ 39:70.
mental spectra described in Figs. 2 and 3 correspond to the two bifurcation diagrams. As the value of h increases further the subcritical Hopf bifurcation moves to higher values of feedback asymptotically moving to infinity and producing the coexisting limit cycle saddle-node, an example of which is seen in Fig. 4(a). The analytical details of this discussion can be seen in [9]. A series of power spectra computed for the bifurcation diagram of Fig. 4(a) are shown in Fig. 5. Starting at Fig. 5(a) the power spectrum of the steady-state at weak feedback strength of g ¼ 0:021 exhibits a broad maximum at a frequency of about 2.1 GHz that corresponds to the relaxation frequency of laser (relaxation frequency of the solitary laser is 2:21 GHz). Weak modulations at the external cavity frequency can also be seen. In Fig. 5(b) the relaxation frequency becomes undamped and a limit cycle emerges from the Hopf bifurcation as is indicated by the sharp lines at that frequency and its harmonics. This spectrum corresponds to a feedback strength of g ¼ 0:12. With further increase of the feedback to g ¼ 0:155 the torus bifurcation takes place and a new frequency close to the external cavity frequency me ¼ 288 MHz emerges as seen in Fig. 5(c). The
Fig. 5. Numerically computed power spectra for T ¼ 1000, P ¼ 1:7, and h ¼ 32:94 for feedback strength (a) g ¼ 0:021, (b) g ¼ 0:12, (c) g ¼ 0:155, (d) g ¼ 0:167, (e) g ¼ 0:21 and (f) g ¼ 0:31.
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actual external cavity frequency is 425 MHz. Further increasing the feedback to g ¼ 0:167 a third frequency appears of about 15 MHz; an expanded view around the relaxation frequency peak is shown in Fig. 5(d). Increasing the feedback rate to 0:21 the trajectory leaves this attractor and moves to the larger pulsating attractor; the spectrum of its intensity is shown in Fig. 5(e). The third frequency is seen here clearly and has increased to about half of me , around 125 MHz. Finally Fig. 5(f) shows the power spectrum at g ¼ 0:31 in which the dynamics become considerably more complex. The sequence of power spectra corresponding to the bifurcation sequence depicted in Fig. 4(b) is shown in Fig. 6. The steady-state power spectrum is very similar to that in the previous figure and the sequence of limit cycle Fig. 6(b), torus Fig. 6(c) and the third frequency of 25 MHz appears in Fig. 6(d), at g ¼ 0:10, 0.12, and 0.14. However, because the steady-state destabilizes by a subcritical Hopf bifurcation, the limit cycle is missing in the experimental sequence when the feedback rate is increased. The limit cycle can be attained while decreasing the feedback rate, but the interval over which it exists is too small to be captured experimentally. Increasing the feedback rate to g ¼ 0:231 the trajectory remains in the same attractor and
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the third frequency increases to 97 GHz as seen in Fig. 6(e). Finally, further increase of the rate leads to the complex dynamics similar to the one depicted in the power spectra in Fig. 5(f). Although at the bifurcation point the tertiary bifurcation frequency can be rather small (of the order of 30 MHz), as the feedback rate is increased the frequency also increases rather quickly. For example, we show in Fig. 6(d) a frequency of about 25 MHz at a feedback rate of g ¼ 0:14. With increase of the feedback to g ¼ 0:231 the frequency increases to 97 MHz. Similarly, attractors with pulsating intensities that coexist with attractors connected to the steady-state can have larger tertiary frequencies (125 MHz) as shown in Fig. 5(e). It is very difficult in the experiment to fine-tune and stabilize the feedback to remain close to the tertiary bifurcation point and on the same attractor, and thus capture the whole frequency range. In conclusion, we have examined the sequence of bifurcations in the incoherent feedback system experimentally and found that it is identical to the sequence found in the case of the optoelectronic feedback [6]. This is verified by the numerical studies that appear to agree quite well with the experimental spectra. We have found supercritical and subcritical Hopf bifurcations, and the sequence of bifurcations and frequencies expected are in very good agreement not only with the numerical results, but also agree very well with the predictions of the analytic results in [9], and also in [11]. Acknowledgements Acknowledgment is made to the W.M. Keck Foundation and the Thomas F. and Kate Miller Jeffress Memorial Trust for the partial support of this research. J.M.S.S. gratefully acknowledges Centro de Investigaciones en Optica for the support as student visitor at AFRL, Directed Energy Directorate, Kirtland AFB. References
Fig. 6. Numerically computed power spectra for T ¼ 1000, P ¼ 1:7, and h ¼ 39:70 for feedback strength (a) g ¼ 0:021, (b) g ¼ 0:10, (c) g ¼ 0:12, (d) g ¼ 0:14, and (e) g ¼ 0:231.
[1] J. Mark, B. Thomborg, J. Mork, IEEE J. Quantum Electron. 28 (1992) 93; J. Mark, B. Thomborg, Phys. Rev. Lett. 65 (1999) 1999.
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[2] A. Hohl, A. Gavrielides, Phys. Rev. Lett. 82 (1999) 1148; T. Heil, I. Fischer, W. Elsasser, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) 413. [3] I. Fischer, Y. Liu, P. Davis, Phys. Rev. A 62 (2000) 1801. [4] G.D. VanWiggeren, R. Roy, Science 279 (1998) 1198; J.-P. Goedgebuer, L. Larger, H. Porter, Phys. Rev. Lett. 80 (1998) 2249. [5] Otsubo, P. Saboureau, J.-P. Foing, P. Schanne, IEEE J. Quantum Electron. 33 (1997) 1582. [6] S. Tang, J.M. Liu, IEEE J. Quantum Electron. 37 (2001) 329; S. Tang, H.F. Chen, J.M. Liu, Opt. Lett. 26 (2001) 1489. [7] K. Otsuka, J.-K. Chern, Opt. Lett. 16 (1991) 1759; T.C. Yen, J.W. Chang, J.M. Lin, R.J. Chen, Opt. Commun. 150 (1998) 159;
[8] [9] [10] [11] [12]
J. Houlihan, G. Huyet, J.G. McInerney, Opt. Commun. 199 (2001) 175; F. Rogister, A. Locquet, D. Pieroux, M. Sciamanna, O. Deparis, P. Megret, M. Blondel, Opt. Lett. 26 (2001) 1486; F. Rogister, D. Pieroux, M. Sciamanna, P. Megret, M. Blondel, Opt. Commun. 207 (2002) 295. D. Pieroux, T. Erneux, K. Otsuka, Phys. Rev. A 50 (1994) 1822. D. Pieroux, T. Erneux, A. Gavrielides, V. Kovanis, SIAM J. Appl. Math. 61 (2000) 966. E.V. Grigorieva, H. Haken, S.A. Kaschenko, Opt. Commun. 165 (1999) 279. D. Pieroux, T. Erneux, T. Luzyanina, K. Engelborghs, Phys. Rev. E 63 (2001) 036211. G. Giacomelli, A. Politi, Physica D 117 (1998) 26.