Mixed-mode dynamics in a semiconductor laser with two optical feedbacks

Transcription

1 Mixed-mode dynamics in a semiconductor laser with two optical feedbacks b D.W. Sukow a, A. Gavrielides b, M.C. Hegg a, and J.L. Wright a adepartment of Physics and Engineering, Washington and Lee University, Lexington, VA 2445 USA Air Force Research Laboratory, Directed Energy Directorate AFRL/DELO, 355 Aberdeen Ave. SE, Kirtland AFB, NM USA ABSTRACT We demonstrate analytically and numerically that multiple mixed external cavity mode solutions are possible for a laser subject to optical feedback from two external cavities. Such solutions exhibit a series of bifurcations and can be easily identified from optical spectra and their frequency content. Similar states have been proposed and analyzed within the framework of the usual Lang-Kobayashi equations describing a semiconductor laser subject to a single optical feedback in short cavities and with moderate pumping. We will present experimental results demonstrating the existence of mixed-mode states in a two-cavity system. We also find that the bifurcation sequence can terminate in low frequency fluctuation states before restabilization on a new maximum power external cavity mode. Keywords: Semiconductor laser, delayed feedback, instabilities, chaos, mixed-mode 1. INTRODUCTION During the last two decades there has been great interest in the dynamics of semiconductor lasers subject to feedback from an external cavity, leading to a large amount of work and publications in this field. There are several reasons that conspire to enhance the dynamical instabilities in such systems. Pulsating intensities are induced even at extremely low feedback rates. The large coupling of the phase of the electric field to the carriers through the linewidth enhancement factor can lead to an oscillating phase that is very clearly evident in the optical spectra of such systems. Unfortunately, the short cavity life time produces dynamics at picosecond timescales and makes it relatively hard to record accurate time series of such dynamics. The usual approach therefore, unless specialized instruments are used to record time series, is to record power and optical spectra whose interpretation can reveal the system s dynamics. Frequency filtered time series can also be used to provide some additional information, especially coupled with numerical simulations from the time-tested Lang-Kobayashi equations. 1 When semiconductor lasers are subjected to external optical feedback, they exhibit strong dynamical instabilities, including coherence collapse 2 and low frequency fluctuations (LFF). 3 Typically the LFF regime is observed when the laser is pumped close to solitary laser threshold; the time series consists of short (picosecond) pulse trains separated by multiples of the external cavity roundtrip time. The envelope of such a train of pulses displays at random intervals drop-out events during which the laser s average intensity is very close to zero. The optical spectrum of the laser becomes very broad and consists of a large number of external cavity modes that appear to lase simultaneously. In addition, the RF spectrum displays a dramatic increase in the low frequency noise in addition to a series of broad peaks at multiples of the external cavity frequency. Because of the large delay and optical feedback there are a large number of external cavity modes (ECMs) generated which correspond to resonances and anti-resonances of the external cavity. A typical LFF trajectory 4 in phase space visits a large number of unstable ECMs on its way to the extreme mode called the maximum gain mode which is usually stable. 5 This mode has recently been observed 6, 7 experimentally but in typical situations Please send correspondence to D.W.S. or A.G. D.W.S.: sukowd@wlu.edu, A.G.: tom@ouzo.plk.af.mil.

2 Optical Power (normalized) (g) (f) 11.4 % 1.4 % (e) 8.35 % (d) 6.42 % (c) 1.58 %.44 %. % Frequency shift (GHz) Threshold reduction for second cavity Figure 1. Optical spectra of a semiconductor laser subject to two optical feedbacks. The first cavity feedback is keep constant at 7.5% threshold reduction while the second feedback is indicated next to each trace. Reproduced from Ref. 1. the LFF trajectory dominates, either because the maximum gain mode has a small basin of attraction, or for a number of other possible reasons such as spontaneous emission noise. The use of a second external optical feedback was originally proposed as novel idea to stabilize the a chaotic laser in the coherence collapse regime. 8 Later it was adapted and suggested that it could also be used to control LFF and stabilize the laser in the maximum gain mode. 9 Indeed, this was successfully demonstrated experimentally. 1 Fig. 1 shows a series of optical spectra of a semiconductor laser subject to two optical feedbacks in which the first feedback is kept constant and corresponds to a threshold reduction of 7.5%. There is zero feedback from the second cavity for trace and the laser is undergoing LFF with the trajectory being limited to essentially seven external cavity modes as can be surmised from the number of broad lines in the spectrum. For a feedback corresponding to trace with threshold reduction I 2 =.44%, the laser has stabilized in the maximum gain mode. Further increase of the feedback from the second cavity leads again to LFF dynamics in trace (c) with eventual stabilization in trace (f). However, a distinct new type of spectrum is indicated in traces (d) and (g) in which there are two dominant frequencies appearing separated by a frequency spacing corresponding to several external cavity modes. Such a spectrum does not correspond to the usual Hopf bifurcation spectrum in which there are three dominant peaks, but rather to a new possible periodic bifurcation in which the laser appears to be lasing simultaneously in two external cavity modes. Such combinations are called mixed-mode solutions as suggested by Tager 11 et al. and analyzed in detail by Erneux et al SINGLE DELAY SOLUTIONS The Lang-Kobayashi equations 1 have been used extensively to describe a semiconductor laser subject to delayed optical feedback. For weak to moderate feedback, single delay equations have been used successfully to describe the dynamics of such systems. The dimensionless rate equations for the field E and the inversion N are: de dt =(1+ia)NE + ηe(t τ)e iωτ (1) T dn dt = P N (1 + 2N) E 2 (2)

3 ωτ 1-2 A η 2 η 2 Figure 2. Frequency of the steady states as a function of feedback η. The corresponding electric field amplitude of the steady states as a function of η. The parameters appearing in Eqs. (1) and (2) are: η is the feedback strength; τ is the delay time, P is the pumping above threshold, a is the linewidth enhancement factor, ω is the frequency of the solitary laser, and T is the ratio of the carrier lifetime to the cavity lifetime. The time t is normalized to the cavity lifetime as well as all quantities that refer to time. Gain saturation effects have not been included in this treatment as they do not modify the results appreciably and tend to obscure the analytical expressions. The basic solutions of these equations are single frequency solutions, or external cavity modes, and are given by: E = A s e iωt, N = N s (3) where A s, N s, and ω are constants. The ECM frequency then satisfies the following transcendental equation: ωτ = ητ 1+a 2 sin(ωτ + ω τ + tan 1 ) (4) and the quantities A s, N s, are obtained from ω by: N s = η cos(ωτ + ω τ), A s = P N s. (5) 1+2N s Analyzing Eq. (4) we note that for a fixed value of η there are an odd number of solutions and their number increases as η increases. Approximately half of the solutions are stable nodes called modes, while the other half are unstable saddles called anti-modes. A plot of these solutions is shown in Fig. 2. as a function of the normalized feedback strength η. For a cavity life time of γ =1/τ p = sec 1 and from the experimental conditions we estimate the various parameters appearing in the equations to be: τ = 533, P =.11. We estimate that the laser has a linewidth enhancement factor a =4, the ratio of the carrier lifetime to the photon lifetime T = 42, and we arbitrarily set ω τ = In Fig. 2a the frequencies of the ECMs are shown as a function of feedback strength and the emergence of the ECMs from saddle-node bifurcations is clearly noted. In Fig. 2b the electric field amplitude of the ECMs is plotted for the same conditions. We also note that in Fig. 2b there are distinct crossing of the amplitude of a node and the adjacent saddle for specific values of η. At such crossings the intensity and the inversion are the same for the saddle and the nearby node, leading to the possibility that a combination of the form E = A 1 e iω1t + A 2 e iω2t (6)

4 is an approximate solution of the system of Eqs. (1) and (2). Indeed for T, Eq. (2) is approximated with dn = (7) dt leading to N = constant, where N denotes the leading order approximation and is the value of the inversion at such crossings. A detailed perturbation analysis leads to a description of A i (η) and ω i (η), i =1, 2 about the leading order values at such crossings and can be found in Ref. 12. These calculations and analysis in the case of a single cavity system have indicated the possibility of a particular kind of a Hopf bifurcation in which two ECMs, a mode and an antimode, form a mixed mode of the type indicated by Eq. (6). Such a Hopf bifurcation is indicated by an oscillating intensity I = E = A A A 1 A 2 cos( t + φ). (8) at a frequency = ω 1 ω 2 equal to the difference in the frequency of a mode and an antimode. Such Hopf bifurcations were analyzed in Ref. 11 for the case of short cavities in an attempt to produce high microwave frequency sources. 3. DUAL DELAY SOLUTIONS The extension of the Lang-Kobayashi equations to describe the dual external optical delay is straightforward. Eq. (1) is now modified to read de dt =(1+ia)NE + η 1E(t τ 1 )e iωτ1 + η 2 E(t τ 2 )e iωτ2 (9) while Eq. (2) remains the same. The parameters appearing in Eq. (9) are: η 1,η 2 are the feedback strengths and τ 1,τ 2 are the two delay times for the two cavities, respectively, while the rest of the parameters remain the same. The steady state solutions are again described by Eq. (3) where the frequencies of the ECMs are determined from the slightly more complicated expression ωτ 1 = 1+a 2 [η 1 τ 1 sin(ωτ 1 + ω τ 1 + tan 1 ) + η 2 τ 1 sin(ωτ 2 + ω τ 2 + tan 1 )]. (1) The steady state field amplitudes and inversions are obtained from N s = η 1 cos(ωτ 1 + ω τ 1 ) η 2 cos(ωτ 2 + ω τ 2 ), A s = P N s. (11) 1+2N s Using the same parameters as for the single cavity case and retaining the value of the delay of the first cavity as before we take: τ 1 = 533, τ 2 = 45, η 1 =.24, P =.11. We keep the linewidth enhancement factor value at a =4, and the ratio of the carrier lifetime to the photon lifetime T = 42. For all our subsequent work we regard η 2 as our bifurcation parameter and examine the behavior of the laser as the strength of the feedback of the second cavity is increased. We arbitrarily fix the feedback phase of the two cavities to be ω τ 1 = ω τ 2 = 1.45, since the relative phase is not determined experimentally and it does not affect the calculations and conclusions. The values of the delays and of the rest of the parameters have been estimated to correspond to the experimental conditions of the system to be described in the subsequent section. Fig. 3 shows the effects of the second cavity on the steady state solutions of the system as computed from Eqs. (1) and (11). The feedback of the first cavity is held fixed at η 1 =.24. In Fig. 3a we plot the frequency of the steady states as a function of η 2, while Fig. 3b shows the amplitude of the electric field. Clearly, there are a large number of steady states possible, some surviving from the first cavity and others being destroyed through saddle-node bifurcations. There are also a large number of states being created as the feedback rate is increased. In addition, there are a large number of crossings; however, in this case not only saddle-node crossings

5 ωτ A η 2 η 2 Figure 3. Frequency of the steady states as a function of feedback η2 with the feedback from the first cavity kept constant at η1 =.24. The corresponding electric field amplitude of the steady states as a function of η2. are possible, but also node-node and saddle-saddle crossings. These crossings are not necessarily between an adjacent saddle and node as in the case of the single external cavity, but crossings of modes separated by a number of external cavity frequencies are now possible. 4. EXPERIMENTAL OBSERVATIONS The experiment uses a temperature-stabilized diode laser (SDL 54L-G1) that emits at a nominal wavelength of λ = 88 nm. The solitary laser threshold is ma, and for all experimental results described in this paper, the pump current is maintained at ma. Fig. 4 shows a schematic of the experimental setup. The laser beam passes through a collimating lens and a polarizing beamsplitter (PBS) whose transmission axis is parallel to the polarization axis of the laser beam. The reflected component of the polarizing beam splitter is used for diagnostics and it is detected with a photodetector [PD (8.75 bandwidth)]. The photodetector signal goes to an ISO FP DSO HR GR RF POL HR NPBS PBS CL LD PD AMP DSO Figure 4: Schematic of the experimental setup

6 6 Optical power (normalized) (g) (f) (e) (d) (c) 5.74% 5.45% 4.97% 4.92% 4.42% 2.18%.% Threshold reduction due to second cavity Frequency shift (GHz) Figure 5: Experimental optical spectra as a function of feedback from the second cavity. amplifier [AMP; 23 db gain] whose output then feeds into a digital storage oscilloscope [DSO; LeCroy 9384M; 1 GHz bandwidth] and a radiofrequency spectrum analyzer [RF (Agilent E445 B)]. The transmitted beam of the polarizing beamsplitter strikes a diffraction grating (GR). The zeroth order beam of the diffraction grating travels to a Fabry-Perot interferometer [OSA (Newport SR-24C; FSR 11 GHz; Finesse > 13,)]; the Faraday Isolator (ISO) protects the system from unwanted feedback from the Optical Spectrum Analyzer. The first order beam of the diffraction grating leads to the two external cavities, which are formed by a non-polarizing beamsplitter (NPBS) and two 99% reflective mirrors. The diffraction grating allows frequency selectivity of the optical feedback, narrowing the cavity bandwidth to approximately 5 GHz. Both cavities are aligned to force the laser to oscillate in the same solitary longitudinal mode. The path lengths of the two external cavities are L 1 =19cmandL 2 = 16 cm. Two rotatable polarizers (POL) adjust the feedback strengths of the external cavities. The fractional threshold reductions I =(I th I)/I th characterizes the feedback strengths. The experimental data consist of optical spectra and RF spectra, and are presented in Figs. 5 and 6 respectively. Figure 5 displays a series of optical spectra as the feedback from the second cavity is increased; successive spectra are offset vertically for clarity. The vertical axis is scaled in normalized units of optical power such that the height of the peak in trace is set to one. The horizontal axis is the frequency shift as measured relative to the first ECM that becomes active when feedback from the first cavity alone is applied. The initial optical spectrum Fig. 5, trace is the reference state where I 2 =.%. At this point, the threshold reduction due to the first cavity feedback is I 1 =2.4%. Three ECMs of the first external cavity are active at this feedback level. As the feedback from the second cavity increases, characterized by a threshold reduction of I 2 =2.18%, the laser stabilizes on one ECM Fig. 5 trace. The RF spectrum Fig. 6a is flat indicating the laser is in a steady state. As the feedback strength of the second cavity increases to I 2 =4.42% the optical spectrum in Fig. 5 trace (c) acquires a small sideband located 3.78 GHz from the original frequency line. The RF spectrum Fig. 6b confirms this, showing that the intensity has developed a high frequency oscillation. This behavior of the optical and RF spectra is entirely consistent with a bifurcation to a mixed mode solution. Further increasing the feedback to I 2 =4.92%, the sideband grows in amplitude in the optical spectrum Fig. 5 trace (d). A small increase in the second feedback strength to I 2 =4.97% leads to a quasiperiodic bifurcation. In the optical spectrum Fig. 5, trace (e), small sidebands appear about the two main peaks, with the second frequency close to that of the ECM spacing. The corresponding RF spectrum Fig. 6c shows the associated set of frequencies; the narrowness of the lines confirms the interpretation of quasiperiodicity.

7 RF Power (dbm) (c) Frequency (GHz) 4 6 (d) 8 1 Figure 6. Experimental RF spectra. steady state corresponding to I2 = 2.18%. Periodic state corresponding to I2 = 4.42%. (c) Quasiperiodic state at I2 = 4.92%. (d) LFF state at I2 = 4.97%. Finally, as the feedback strength increases further, I 2 =5.45%, the laser displays characteristics consistent with a LFF state. The optical spectrum clearly shows the laser oscillating on multiple ECMs Fig. 5, trace (f) and in the RF spectrum, Fig. 6d, the linewidth of the lines shows considerable broadening associated with a increase in the low-frequency noise. Time series data (not shown) confirm this identification. As the feedback strength increases, I 2 =5.74%, the laser restabilizes onto a different ECM Fig. 5, trace (g). It was found experimentally that sequences of stable, periodic, quasiperiodic and LFF dynamics can repeat a number of times for stronger feedback from the second cavity, but for ECMs with more negative frequencies. Such modes and crossings become available as the feedback increases. 5. NUMERICAL CALCULATIONS The numerical calculations were performed using the full LK equations Eqs. (2) and (9) with the parameters already described in the Dual Delay Solutions section. The feedback from the first cavity is fixed at η 1 =.24, consistent with the threshold reduction maintained in the experiment. Under this condition the trajectory is in a LFF state. Figure 7 shows a numerical bifurcation diagram as a function of η 2 superimposed on the steady state solutions. For convenience we focus on the region around the crossing at η 2 =.36 and only this portion of the steady states of Fig. 3a is included. The numerical steady state overlaps the analytical steady state until a Hopf bifurcation occurs just before the ECM crossing, consistent with the analysis and numerical results of Ref. 12. Further, this oscillating state is destabilized by a secondary quasiperiodic bifurcation leading to eventually LFF. We will now reproduce a series of optical and RF spectra calculated as the feedback from the second cavity is increased from zero and concentrate at the mode-antimode crossing indicated by the diamond. Fig. 8a shows the optical spectrum at η 2 =.31. Clearly, the laser is stabilized in a steady state and is located on an ECM with optical frequency 11. GHz relative to the solitary laser optical frequency. Naturally the RF spectrum is entirely flat and therefore it is not shown. Fig. 8b shows the optical spectrum at η 2 =.33 after the original steady state has undergone a Hopf bifurcation to oscillating intensity. The spectrum has acquired a second frequency located at about 15 GHz and corresponds to the frequency of another ECM. The RF spectrum (not shown) has a single line at approximately 3.7 GHz corresponding to the beat of the two modes. This frequency

8 Max(A), Min(A) x1-3 η 2 Figure 7. Numerical bifurcation diagram around the mode-antimode crossing at η2 =.36 for fixed feedback from the first cavity η1 =.24. is close to four times the external cavity frequency of the first cavity and is in very good agreement with the experimental measurements. Increasing the feedback further, the periodic state destabilizes though a secondary quasiperiodic bifurcation. This is shown in Fig. 9 in which the optical and RF spectra are exhibited for η 2 =.355. In the optical spectrum, sidebands appear about the ECM frequencies. The frequency of these sidebands corresponds to the relaxation frequency of the laser. However, since the laser is biased close or below to the solitary laser threshold this frequency is very close to the external cavity frequency as discussed in Lythe 13 et al. Finally in Figs. 1a, 1b, and 1c we show the numerical time series, the optical and the RF spectra for η 2 =.37. The laser at this value of feedback is in a LFF state. The optical spectrum indicates that the trajectory moves essentially over the previous two ECMs, however, now a higher frequency mode is also included. The RF spectrum looks very similar to that of Fig. 8b, however the lines now are considerably broadened. This 12 1 E(ν) ν (GHz) Figure 8. Optical spectrum of the steady state at η2 =.31. Optical specrum of the periodic state at η2 =.33.

9 Optical RF E(ν) I(ν) ν (GHz) ν (GHz) 5 6 Figure 9: Optical spectrum and RF spectrum of the quasiperiodic state at η2 = A(t) t 15 2x1 3 3 E(ν) I(ν) ν (GHz) (c) ν (GHz) Figure 1: Time series, Optical spectrum and (c) RF spectrum of the LFF state at η2 =.37. characteristic sequence of bifurcations is very close to that observed experimentally. 6. SUMMARY In summary, we have presented experimental results and numerical calculations of dynamical states of diode lasers subject to two delayed optical feedbacks. These states originate from novel Hopf bifurcations involving two dinstict ECMs at values of the feedback strength at which the intensity and inversion of the two modes are degenerate. Spectral data show a sequence of bifurcations that evolves through stability, periodicity, quasiperiodicity and LFF before restabilization on a new ECM. Additional data gathered show that this sequence is repeated at increasing stronger feedback from the second cavity.

10 ACKNOWLEDGMENTS 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. REFERENCES 1. R. Lang and K. Kobayashi, IEEE J. Quant. Electron. QE-16, 347 (198). 2. D. Lenstra, B. H. Verbeek, and A. J. den Boef, IEEE J. Quant. Electron. QE-21, 674 (1985). 3. C. Rich and C. Voumard, J. App. Phys. 48, 283 (1977). 4. T. Sano, Phys. Rev. A 5, 2719 (1994). 5. A. M. Lavine, G. H. M. van Tartwijk, D. Lenstra, and T. Erneux, Phys. Rev. A 52, R3436 (1995). 6. A. Hohl, and A. Gavrielides, Phys. Rev. Lett. 82, 1148 (1999). 7. T. Heil, I. Fischer, and W. Elsäßer, Phys. Rev. A 58, R2672 (1998). 8. Y. Liu, and J. Otsubo, IEEE J. Quant. Electron. 33, 1163 (1997). 9. F. Rogister, P. Mégret, O. Deparis, M. Blondel, and T. Erneux, Opt. Lett. 24, 1218 (1999). 1. F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, Opt. Lett. 25, 88 (2). 11. A. A. Tager, and K. Petermann, IEEE J. Quant. Electron. 3, 1553 (1994). 12. T. Erneux, F. Rogister, A. Gavrielides, and V. Kovanis, Opt. Commun. 183, 467 (2). 13. G. Lythe, T. Erneux, A. Gavrielides and V. Kovanis, Phys. Rev. A 55, 4443 (1997).