Polarization dynamics in semiconductor lasers with incoherent optical feedback David W. Sukow a, Athanasios Gavrielides b, Thomas Erneux c, Michael J. Baracco a, Zachary A. Parmenter a, and Karen L. Blackburn a a Department of Physics and Engineering, Washington and Lee University, 116 N. Main St., Lexington, VA 2445 USA b Air Force Research Laboratory, Directed Energy Directorate AFRL/DELO, 355 Aberdeen Ave. SE, Kirtland AFB, NM 87117 USA c Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine, Code Postal 231, 15 Bruxelles, Belgium ABSTRACT The chaotic dynamics of a semiconductor laser subject to a delayed polarization-rotated optical feedback is investigated theoretically and experimentally. An extension of the usual one-polarization model is derived to account for two orthogonal polarizations of the optical field. The two-polarization model is motivated by observations of lag synchronization in our experiments using polarization-rotated optical feedback and unidirectional injection. Experimental data confirm the predictions of the two-field model. We also show that the two-polarization model can be reduced to the one-polarization model. 1. INTRODUCTION Chaos synchronization has been extensively studied for semiconductor laser (SL) systems because of their interest for optical communication systems. Chaos synchronization on unidirectionally coupled semiconductor lasers has been found in optical injection systems, 1 optical feedback systems, 2, 3 optoelectronic feedback systems, 4, 5 and systems of two mutually coupled semiconductor lasers. 6, 7 In several experiments, it was shown that chaos synchronization can exhibit perfect synchronization, as expected, but driving synchronization as well. If a master laser subject to a delayed optical feedback injects its light into a slave laser of similar wavelength, perfect chaos synchronization between master and slave lasers is possible if the injection and feedback rates are comparable. On the other hand, if the injection rate is much larger than the feedback rate, a driven response is observed where synchronization occurs with the delayed injected signal rather than with the chaotic oscillations of the master. 3 An all-optical system which is dynamically equivalent to the optoelectronic feedback system is the incoherent feedback system. 8 But driving synchronization has never been observed for the optoelectronic feedback system 9 while it was observed for the incoherent feedback system. 1 However, extensive simulations of an incoherent feedback model involving only one polarization field 11 indicate that driving synchronization is not possible. This suggests that the one-polarization model might be too simple to describe the observed synchronization dynamics. In this paper, we formulate a more realistic model where the two polarizations fields are taken into account. We determine the basic properties of the polarization intensities which we then verify experimentally. Specifically, we consider a single mode diode laser subject to incoherent feedback, as described schematically in Fig. 1. It consists of a laser in an external cavity and intracavity devices that rotate the polarization state of the delayed optical feedback, adjust the feedback strength, and sample the output. We find that the steady state intensities consist of a two-polarization state as soon as the feedback is nonzero. In the experiment, the natural polarization is the horizontal polarization and the vertical polarization Please send correspondence to D.W.S. or A.G. D.W.S.: E-mail: sukowd@wlu.edu, A.G.: E-mail: athanasios.gavrielides@kirtland.af.mil.
Laser Rotator Attenuator Mirror Figure 1: Schematic diagram of a laser with delayed incoherent optical feedback. Figure 2: Steady states of the two polarization emitted radiation from the diode laser under incoherent feedback becomes significant only after the rotated polarization is injected into the laser. In Fig. 2 we show the steady state intensities as functions of the feedback for a laser biased at 64. ma and has a threshold current of 45.6 ma. These data for a single laser subject to incoherent feedback will be useful when we concentrate on the synchronization dynamics for two interacting lasers. The paper is divided into two parts. We first derive the two-polarization model and show analytically that it reduces to the usual one-polarization model plus a specific relation between the two polarization fields. This relation is then verified experimentally by studying the correlation between the two polarization intensities. 2. THEORY 2.1. BASIC INCOHERENT OPTICAL FEEDBACK MODEL The description of the laser subject to incoherent optical feedback is given in terms of the natural lasing field E and the inversion of population N. The polarization of the lasing field is then rotated from the horizontal to the vertical position, delayed, and reinjected back into the laser. The main effect is an depression of the inversion of population. 11 The laser rate equations for E and N are given by de =(1+ia)NE, (1) T dn = P N (1 + 2N)[I + ki(t τ)] (2)
where I = E 2. These equations differ from the standard SL rate equations by the term multiplying I(t τ) in Eq. (2). The dimensionless parameters appearing in these equations are the linewih enhancement factor a, the ratio of the carrier and photon lifetimes T, the pump parameter above threshold P, the feedback injection rate k, and the feedback delay τ. Eqs. (1) and (2) admit the steady state solution N =, I = P 1+k (3) while the phase φ of the laser field is arbitrary. The steady state changes stability at a Hopf bifurcation located at k = k H ω (1+2P) (4) 2P sin(θ) where ω = 2P/T << 1 is the relaxation oscillation frequency of the solitary laser and θ ωτ the normalized delay. 12 The Hopf bifurcation may lead to stable or unstable oscillations and is followed by more complex time-dependent bifurcations. 2.2. EXTENDED INCOHERENT OPTICAL FEEDBACK MODEL The extended model accounts for both polarizations that naturally appear for the SL subject to incoherent feedback. Specifically, the horizontal polarization is dominant and lases because its losses are less than the vertical polarization which is then totally suppressed. However, if the laser is subject to the feedback of the orthogonal polarization as in our experiments, the vertical polarization can overcome the losses and compete for the gain. The extended model equations then consider the evolution of the laser s natural horizontal polarization E 1, the vertical polarization E 2 induced by the feedback, and the inversion of population N. The rate equations have the form de 1 =(1+ia)NE 1, (5) de 2 = iωe 2 +(1+ia)(N β)e 2 + ηe i(ω1τ+φe) E 1 (t τ), (6) T dn = P N (1+2N) [ E 1 2 + E 2 2]. (7) The additional parameters that appear in these equations are the difference of the two polarization frequencies Ω and the phase ω 1 τ resulting from the propagation of the field in the external cavity before reinjection. Since the radiation is rotated from the horizontal to the vertical position and reinjected into the laser, the frequency difference Ω should be strictly zero unless a frequency shift is imposed externally. However, this is no more the case for the transmitter and receiver system that we consider in a synchronization scenario because the dominant polarization may have different frequencies for the two diode lasers. We therefore will keep Ω as a parameter. An additional phase φ e due to the polarization elements in the external cavity is included. Both phases ω 1 τ and φ e can be absorbed in the field phases and do not modify the laser dynamics. The constant β in Eq. (6) is the fractional difference in the losses or the cavities lifetimes for the two polarization fields. It is defined by β τ p2 1 (8) τ p1 where τ pi, i =1, 2 are the lifetimes of the mode cavities respectively. Without the feedback (η =), the steady state solution is N =, E 2 =, E 1 = P (9) and the phases are arbitrary. This steady state is stable. In terms of E i = A i exp(iφ i ), Eqs. (5) (7) are rewritten as da 1 = NA 1, dφ 1 = an, (1)
Figure 3. Steady state as a function of feedback strength. The quanities are P =.5, a = 4, Ω =.. We used this value of pump to compare with the experiment in Fig. 1. dφ 2 da 2 Assuming η>, the steady state solution now is given by while the phase difference Φ = φ 1 φ 2 is obtained from =(N β)a 2 + ηa 1 (t τ) cos(φ 1 (t τ) φ 2 ), (11) = Ω+a(N β)+η A 1(t τ) A 2 sin(φ 1 (t τ) φ 2 ), (12) T dn = P N (1+2N)(A2 1 + A2 2 ). (13) A 2 1 = (Ω + aβ) 2 + β 2 P η 2 + β 2 +(Ω+aβ) 2, (14) A 2 2 = η 2 P η 2 + β 2 +(Ω+aβ) 2, (15) N =, (16) tan(φ) = Ω+aβ. (17) β The steady state intensities given by (14) and (15) are shown in Fig. (3). As the feedback strength is increased, the intensity of the horizontal polarization decreases but shares power with the vertical polarization intensity. At the critical feedback rate η c = β 1+a 2, (18) the two intensities become equal. Using (18) with a = 4 and comparing with Fig. (2), we estimate that the fractional loss between the two polarizations is about 13.5%. The first Hopf bifurcation can be determined
Figure 4. Bifurcation diagram calculated numerically from the full problem for two cases. The first bifurcation is for Ω=. The second is for Ω = 1.4. The rest of the parameters are P =.5, T = 1, τ = 1, a = 4, and β =.1. analytically by taking advantage of the large value of T (equivalently, the small value of the relaxation oscillation frequency ω 2P/T). Instead of (4), we now find η 2 H ω (1+2P ) 2P sin(ωτ) [β2 +(Ω+aβ) 2 ]. (19) The direction of bifurcation can be either supercritical or subcritical. The two bifurcation diagrams shown in Fig. (4) for two different values of Ω exhibit a subcritical Hopf bifurcation. As the feedback rate surpasses the Hopf bifurcation point, the laser then jumps to a branch of large amplitude oscillations that overlap part of the branch of stable steady states. The numerical bifurcation diagrams have been obtained by gradually changing the feedback rate first forward and then backward to detect the possible overlap of stable branches of solutions. The bifurcation diagram with the branch of periodic solutions appearing close to η =.14 is for Ω = while the diagram with the branch of periodic solutions appearing close to η =.155 is for Ω = 1.4. 2.3. MODEL REDUCTION Eqs. (5) (7) can be reduced to a system of two equations close to Eqs. (1) (2) by taking advantage of the relatively large value of T and τ (T 1 3 and τ 1 3 ). Introducing the new time s τ 1/2 t and inversion of population Z = τ 1/2 Z into Eqs. (5) (7), we find de 1 ds =(1+ia)ZE 1, (2) τ 1/2 de 2 ds = iωe 2 +(1+ia)(τ 1/2 Z β)e 2 + ηe i(ω1τ+φe) E 1 (s θ), (21) T τ dz ds = P τ 1/2 Z (1 + 2τ 1/2 Z)[ E 1 2 + E 2 2 ] (22)
Figure 5. Time series of the natural horizontal polarization (a) and of the vertical (b). The parameters are P =.5, T = 1, τ = 12, α =2,β =.1, Ω =., and the feedback strength is.132. where θ τ 1/2. If τ 1/2 << β and T = O(τ), we eliminate all τ 1/2 small terms and obtain de 1 ds = (1+ia)ZE 1, (23) = iωe 2 (1 + ia)βe 2 + ηe i(ω1τ+φe) E 1 (s θ), (24) T dz τ ds = P [ E 1 2 + E 2 2 ]. (25)
From Eq. (24), we determine E 2 as and Eqs. (2) and (22) reduce to E 2 = T τ where the feedback rate k is given by η iω+(1+ia)β e i(ω1τ+φe) E 1 (t θ) (26) de 1 ds =(1+ia)ZE 1 (27) dz ds = P [ E 1 2 + k E 1 (s θ) 2 ]. (28) k η 2 [β 2 +(Ω+aβ) 2 ]. (29) With the relaxation oscillation damping terms τ 1/2 Z(1 + 2[ E 1 2 + E 2 2 ]) included, Eqs. (27) and (28) are equivalent to Eqs. (1) and (2). The expressions of the Hopf bifurcation point given by (4) and (19) are therefore identical using the relation (29). Moreover, from (26), we clearly note that the two polarizations are synchronized but shifted relative to each other by the external cavity delay τ. We verify this property from the full laser equations by numerical simulations. We introduce the correlation between two signals as S1 (t)s 2 (t s) C(s) = [ S1 2(t) S2 2. (3) (t)]1/2 The numerical simulations show that the correlation between I 1 and I 2 is.89 and a lag delay of I 1 relative to I 2 of 38.3 as computed from Eq. (3) (the actual delay of the external cavity θ =37.95). Fig. 5(a) shows the time evolution of the dominant horizontal polarization I 1 while Fig. 5(b) represents the time evolution of the vertical polarization I 2. 3. EXPERIMENTS The experimental apparatus to study the dynamics of the semiconductor laser is shown in Fig. 6. The laser (LD1, Sharp LT24) is driven into chaos by use of polarization-rotated, delayed optical feedback. It has a nominal wavelength of λ 1 = 78 nm and a solitary threshold of 45.5 ma. For all experimental data presented in this section, LD1 is driven with a current of 62.1 ma and stabilized at a temperature of 16.21 C. Chaos is induced in the laser using delayed optical feedback. The horizontally-polarized beam emerging from LD1 is collimated by a lens (CL1) with numerical aperture of.47, passes through a plate beamsplitter (BS, R = 5%), then enters a Faraday rotator (ROT) whose input polarizer is removed. The beam s polarization rotates 45, and exits through the output polarizer with its transmission axis set at 45. A rotatable linear polarizer (POL) in the external cavity adjusts the beam s intensity (and thus the feedback strength). The beam is retroreflected by a high-reflectivity mirror (HR), and then is rotated an additional 45 on the return pass through the Faraday rotator. This creates a vertically-polarized beam that is reinjected into LD1. The feedback induces dynamical instabilities in LD1, following a route into chaos as described in Solorio et al. 13 The external cavity rounrip length is 8.8cm, with a rounrip time τ ext =2.69 ns. The intensity dynamics of LD1 are sampled immediately after they emerge from the collimating lens, by use of the plate beamsplitter (BS) which directs 5% of the total incident light to the detection arm. This beam is separated into its two constituent linear polarizations when it strikes a polarizing beamsplitter cube. The two resulting polarization-resolved beams strike identical photodetectors (PD1 and PD2). These ac-coupled photodetectors have bandwihs of 8.75 GHz (Hamamatsu C4258-1). Neutral density filters (not shown) attenuate both beams to limit the power incident on the photodetectors. The detected signals are amplified with ac wideband (1 khz - 12 GHz) microwave amplifiers (AMP) with 23 db gain, and are displayed either on an rf spectrum analyzer (Agilent E445B, 13.2 GHz BW) or a high speed digitizing oscilloscope (LeCroy 86, 6 GHz bandwih and 5 ps sampling interval). The system is designed such that the two detection path lengths between the laser and each instrument are equal.
CL1 BS ROT POL HR LD1 AMP PBS PD1 AMP Oscilloscope or RF Analyzer HR PD2 Figure 6: Schematic of the experimental setup for transmitter dynamics Optical power (mw).8.6.4.2 Horizontal, solitary Horizontal, with feedback Vertical, with feedback Vertical, solitary. 3 35 4 45 5 55 Pump current (ma) 6 65 Figure 7: Polarization-resolved threshold measurements for transmitter laser. The simplest demonstration of the effects of polarization-rotated feedback is by measuring the threshold characteristics of the laser under such conditions. This is shown in Fig. 7, which displays the optical power in each polarization as a function of drive current under a constant-strength polarization-rotated feedback. The experimental feedback strength is characterized in terms of the rounrip transmission in the external cavity T ext. Here, T ext =11.3%. In Fig. 7, the natural horizontal laser mode is represented by squares, the vertical by triangles. Filled shapes indicate the solitary laser behavior, and opaque white shapes indicate the data taken when polarization-rotated feedback is present. It is clear that the vertically-polarized mode, which is suppressed in the solitary laser, becomes active in the presence of feedback. Correspondingly, the power in the dominant horizontally-polarized mode is diminished. The total power remains the same at a given current, with and without feedback. A similar comparison can be made by observing the radio-frequency spectra under similar conditions. This is shown in Fig. 8, where four RF spectra represent the power in each polarization mode under solitary operating conditions (Figs. 8a and 8b), and in the presence of polarization-rotated optical feedback (Figs. 8c and 8d). For this data set the transmitter laser pump current is 62. ma and the external cavity transmission T ext =16.9%. As a solitary laser, the horizontal mode displays a typical relaxation oscillation peak, and the vertical mode is indistinguishable from background noise. In the presence of the polarization-rotated feedback, there is power in both polarization states, and the structure of the RF spectrum is indicative of complex dynamics.
Horizontal polarization Vertical polarizatio n RF Power (dbm ) RF Power (dbm ) -3-4 -5-6 -7-3 -4-5 -6-7 (a) 2 4 6 8 (c) 2 4 6 8 Frequency (GHz) -3-4 -5-6 -7-3 -4-5 -6-7 (b) 2 4 6 8 (d) 2 4 6 8 Frequency (GHz) Solitary Laser With Feedback Figure 8: Polarization-resolved RF measurements for transmitter laser. Laser power (normalized) 2 1-1 -2 2 4 6 Time (ns) 8 1 12 Figure 9: Polarization-resolved time series for transmitter laser. It is even more interesting to analyze the complex transmitter dynamics in the time domain, which reveals the nature of the synchronization relationship between the polarization modes. These time series are displayed in Fig. 9. With the experimental setup described in Fig. 6, we use the fast oscilloscope to acquire simultaneous data sets from both linear polarizations of the transmitter laser. The external cavity transmission T ext =16.9%. The time series have lengths of 4 points with a 5 ps sampling interval for a total duration of 2 ns. We then determine the maximum cross-correlation coefficient S between the pair using the time-shifted function S( t) = P 1(t)P 2 (t t) [ P1 2(t) P 2 2 (31) (t) ]1/2 where P 1 and P 2 are the two powers to be compared, in this case the two linear polarizations of the transmitter laser, and t is the shift between the two time series at which the cross-correlation is calculated. We obtain a maximum correlation between the two polarizations of S =.94 when t =2.8 ns. This time is close to but slightly longer than the external cavity rounrip time τ ext =2.69 ns. In Fig. 9, the thick grey and thin black curves represent the horizontal and vertical polarizations of the transmitter laser, respectively. For ease of visual comparison, we scale and shift the axes for each time series.
Vertical power (normalized) 4 2-2 -4-4 -2 2 4 Horizontal power (normalized) Figure 1: Synchronization plot for transmitter laser. Both waves are scaled vertically by subtracting the mean and normalizing the standard deviation to unity. The leading wave (natural polarization state) has also been shifted back in time by t = 2.8 ns so the synchronized behavior appears at the same times for both waves. As is expected for a high cross-correlation value, the similarity between waves is striking. An alternate view of the dynamics can be obtained by plotting the two time series against each other in a synchronization plot. In Fig. 1, the same time series as plotted in Fig. 9 are shown against each other, with the power of the vertical polarization plotted against that of the horizontal. The full 4-point sets are shown, with scaling and time-shifts the same as in the previous figure. The data map out a neat diagonal, confirming the synchronization. From the experimental data shown, it is clear that a semiconductor laser subject to polarization-rotated feedback has two active polarization modes: the naturally-supported state, and the orthogonal state activated by the feedback. Furthermore, if chaotic dynamics are induced, the orthogonal state is well-synchronized with the natural state, at a time lag very close to the external cavity rounrip time. Thus the total laser output becomes an unusual composite that fluctuates both in intensity and polarization. 4. SUMMARY In summary, we have presented theoretical, numerical, and experimental results that elucidate the nature of the dynamics and synchronization properties of semiconductor lasers subjected to polarization-rotated delayed optical feedback. The extended two-polarization model predicts that the two polarizations will be synchronized but shifted relative to each other by the external cavity delay. These predictions are confirmed by experimental data. Furthermore, it is demonstrated that by using appropriate simplifications the extended two-polarization model can be reduced to the usual one-polarization incoherent feedback model. ACKNOWLEDGMENTS This material is based upon work supported by the U.S. National Science Foundation under CAREER Grant No. 239413. Acknowledgment also is made to the W.M. Keck Foundation for the partial support of this research. T.E. acknowledges the support of the Fonds National de la Recherche Scientifique (FNRS, Belgium) and the InterUniversity Attraction Pole program of the Belgian government. REFERENCES 1. H.F. Chen and J.M. Liu, Open-loop chaotic synchronization of injected-locked semiconductor lasers with gigahertz range modulation, IEEE J. Quantum Electron. 36, 27-34 (2). 2. J. Ohtsubo, Chaotic synchronization and chaotic signal masking in semiconductor lasers with optical feedback, IEEE J. Quantum Electron. 38, 1141-1154 (22).
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