Square-wave oscillations in edge-emitting diode lasers with polarization-rotated optical feedback

Square-wave oscillations in edge-emitting diode lasers with polarization-rotated optical feedback A. Gavrielides a,t.erneux b,d.w.sukow c, G. Burner c,t.mclachlan c, J. Miller c, and J. Amonette c a Air Force Research Laboratory, Directed Energy Directorate AFRL/DELO, 3550 Aberdeen Ave. SE, Kirtland AFB, NM 87117 USA b Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine, Code Postal 231, 1050 Bruxelles, Belgium c Department of Physics and Engineering, Washington and Lee University, 204 W. Washington St., Lexington, VA 24450 USA ABSTRACT The square-wave response of edge-emitting diode lasers subject to a delayed polarization-rotated optical feedback is studied experimentally and theoretically. Square-wave self-modulated polarization intensities of a period close to twice the delay τ of the feedback gradually appear through a sequence of bifurcations starting with a Hopf bifurcation (Gavrielides et al, Proc. SPIE 6115, to appear, 2006). In Gavrielides et al (submitted, 2006), squarewave solutions were determined analytically from the laser equations in the limit of large τ. A condition on the laser parameters was derived explaining why square-wave oscillations are preferentially observed for sufficiently large feedback strength. In this paper, we concentrate on the relaxation oscillations that always appear at each intensity jump between the plateaus of the square-wave. We show analytically that if the feedback strength is progressively decreased, a bifurcation to sustained relaxation oscillations is possible for one of the two plateaus. 1. INTRODUCTION Using the high sensitivity of diode lasers with respect to weak optical feedback is a well known technique for applications such as sensing and imaging distant objects. The response of the laser depen on many factors, such as the laser architecture and operating conditions, the external cavity length and configuration, and the strength of the feedback. Recently, polarization rotated optical feedback (PROF) has raised considerable attention because the laser system is simpler than the laser subject to conventional optical feedback and is still purely optical. In a PROF set-up, the polarization state of the optical feedback is made to be orthogonal to the laser s natural operating mode. Such systems are of interest from a variety of standpoints. First, their dynamical properties can be investigated in more detail than for conventional optical feedback 1-11 and they also apply for purely optoelectronic feedback systems where the laser field is coupled directly to the carriers via electrical means 11, 12. Second, the effect of optical feedback on Vertical Cavity Surface Emitting Lasers (VCSELs) has been examined in several laboratories. The feedback typically gives rise to switching between the dominant polarization mode and the orthogonal mode that is nearly degenerate due to the cylindrical symmetry of such devices 13-18. Polarization self-modulation has a variety of useful applications stemming from the production of optical pulses at multi-ghz repetition rates without the need for high-speed electronics. But edge-emitting lasers can also be made to exhibit pulse trains generated by polarization self-modulation, even though the losses in the orthogonal mode are typically higher than in VCSELs 19-20. A quarter-wave plate was used as the polarization-rotating element, which allows mutual coupling between the polarization modes as well as multiple cavity round-trips due to reflections from the laser s front facet. In our PROF system, we avoid the multiple round-trip reflections by using a Faraday rotator that only allows the coupling between the natural horizontal polarization mode and the unsupported orthogonal mode. The experimental set-up is described in detail elsewhere 22. Square-wave self-modulated polarization intensities Please send correspondence to A.G. or T.E. A.G.: E-mail: athanasios.gavrielides@kirtland.af.mil, T.E.: E-mail: terneux@ulb.ab.be

Figure 1. Experimental times series of the square-wave self-modulated oscillations. Figures top and bottom show the horizontal and vertical mode intensities, respectively. Figures (a) + (b) and (c) + (d) correspond to τ =2.60 ns and τ =6.92 ns, respectively. oscillating in antiphase were observed experimentally (see Figure 1). The oscillations exhibit a period close to twice the round-trip time in the external cavity typical. We also note fast damped relaxation oscillations (RO) at each jump between the plateaus of the square-wave. Compared to the laser subject to conventional optical feedback, the analysis of the PROF configuration is simpler because the orthogonal mode, even when activated by feedback, cannot influence the natural mode directly via optical injection. In other wor, there is unidirectional coupling from the natural polarization to the orthogonal one. Recent studies of the fundamental dynamics of this system 21, 22, as well as its chaos synchronization properties 23, demonstrate the need for considering the intensities of both polarization modes as independent variables. Rate equations with the two intensities have been successfully used and they reproduce the experimental observations 22. Recently, we examined the laser equations in the limit of large round-trip times. 24 We found that the square-wave regimes are possible provided the feedback strength is sufficiently large. In this paper, we concentrate on the RO and show that they may be sustained if the feedback strength is progressively decreased from a large value. Theplanofthepaperisasfollows. InSection2, weformulate the dimensionless laser equations. In Section 3, we determine the non-zero intensity plateaus of the square-wave and then investigate their stability with respect to the fast RO oscillations. We obtain two stability boundaries for the pure square-wave regimes and the square-wave regimes exhibiting sustained RO. All our predictions are verified by solving numerically the laser rate equations. In Section 4, we summarize and discuss the main result. 2. FORMULATION In order to simulate the square-wave oscillations observed experimentally, we consider rate equations similar to the one used by Heil et al 21. In dimensionless form, these equations are the following three equations for the horizontal and vertical (complex) polarization fiel, E 1 and E 2, respectively, and the carrier density N 22 de 1 dt de 2 dt T dn dt = (1+iα)NE 1, (1) = (1+iα)k(N β)e 2 + ηe 1 (t τ), (2) = P N (1 + 2N)( E 1 2 + E 2 2 ). (3) Eq. (1) describes the horizontal polarization that lases naturally since it has the lowest losses. Eq. (2) describes the vertical polarization with k and β > 0 definedastheratiooftheverticalandhorizontalgaincoefficients and the differential losses, respectively. The last term in Eq. (2) models the reinjection of the delayed and rotated

horizontal polarization from the external cavity. We shall consider k fixed and use β as the parameter that controls the different losses for the two modes. Finally, Eq. (3) describes the evolution of the carrier density which depen on the intensities of both polarization fiel. All parameters are dimensionless. In addition to k and β, the laser parameters include the linewidth enhancement factor α, the ratio T of the carriers lifetime to the cavity lifetime, and the pump parameter above threshold P. The feedback is controlled by its strength η and its delay τ. The range of parameters leading to square-wave regimes appears to be fairly broad and only requires small values of β. The values of the laser parameters representative of a SDL semiconductor laser 4 are T =150, α =2, and P =0.5. These equations have been studied systematically 22 for different values of k =0.3 1, β =3 9, η =0.1, and τ =10 2 10 4. By gradually increasing η, the non-zero intensity steady state transfers its stability to sustained relaxation oscillations that are progressively modulated by a 2τ periodic square-wave. Above a critical value of η, the square wave oscillations dominate and only exhibit decaying relation oscillations at the beginning of each square wave. In all our simulations, a small amount of noise of amplitude 10 6 has been introduced in the right hand sides of Eqs. (1) and (2). Without noise, square-wave oscillations with more than two plateaus and with period larger than 2τ are possible but they have never been seen experimentally. 3. LARGE DELAY ANALYSIS After introducing E j = A j exp(iφ j )(j =1, 2) and the new time s t/τ into Eqs. (1)-(3), we obtain: τ 1 da 1 τ 1 da 2 τ 1 dφ = NA 1 (4) = k(n β)a 2 + ηa 1 (s 1) cos(φ) (5) = αk(n(s 1) N + β) η A 1(s 1) A 2 sin(φ) (6) τ 1 T dn = P N (1 + 2N)(A 2 1 + A 2 2) (7) where Φ(s) =φ 1 (s 1) φ 2 (s). The delay τ is typically large in the experiments (τ 10 3 ) which motivates to eliminate all the τ 1 terms in the left hand sides. The resulting algebraic equations are given by NA 1 = 0, (8) k(n β)a 2 + ηa 1 (s 1) cos(φ) = 0, (9) α [N(s 1) k(n β)] A 2 ηa 1 (s 1) sin(φ) = 0, (10) P N (1 + 2N)(A 2 1 + A 2 2) = 0. (11) These equations relate the values of the dependent variables at time s to their values at time s 1. Eq. (8)is satisfied if either N =0or A 1 =0. We anticipate that the square-wave solution exhibits two successive stages characterized by one of these two conditions. From Eqs. (8)-(11), we find (1) N =0,A 1 (s 1) = 0, A 2 =0and A 1 = P (12) for Stage (1) and (2) A 1 =0,N(s 1) = 0, A 1 (s 1) = P (13) for Stage (2). The values of Φ, Nand A 2 for Stage (2) satisfy the following equations tan(φ) = α, (14) P k(n β)a 2 + η 1+α 2 = 0, (15) A 2 2 P N = 0. (16) 1+2N

β 0.15 0.10 no SW η c 5 η H stable SW unst. SW 0 0 2 4 6 8 0.10 η Figure 2. The stabily diagram of the square-wave (SW) oscillations for large delay. The straight line η = η c (β) separate theregionwherestage(2)oftheswadmitsanegativen. The line η = η H (β) correspon to Hopf bifurcation points obtained numerically from Eqs. (18)-(20). The three black squares denote the values of (η, β) used for our numerical simulations of the full laser equations. Eqs. (15) and (16) lead to a condition for the square-wave regime. The non-zero intensity plateau for A 1 always equal P but the non-zero intensity plateau for A 2 depen on the value of η and is larger than P. The constant N is always negative allowing A 1 =0tobestable during this stage (from Eq. (4), we have A 1 A 1 (0) exp(ns) 0 because N<0). There is a critical feedback rate where A 2 equals P and N =0 given by η = η c βk (1 + α 2 ). (17) Below this critical value, N may become positive and the stability of A 1 =0cannot be guaranteed. We observe numerically complex fast oscillating regimes. Because the delay is large, the fast relaxation oscillations have the time to decay for each plateau. We wonder, however, if this is always the case. We examine the stability of each stage of the square wave. For Stage (1), we find the single mode laser rate equations which always admit damped RO. But for Stage (2), the equations are more complicated. From Eqs. (4)-(7) with (13), we need to solve τ 1 da 2 1 dφ τ τ 1 T dn = k(n β)a 2 + η P cos(φ), (18) = P αk( N + β) η sin(φ), A 2 (19) = P N (1 + 2N)(A 2 1 + A 2 2). (20) These equations are ordinary differential equations which admit a non-zero A 2 solution of Eqs. (15) and (16). An analysis of these equations indicates a change of stability as the feedback strength is below a critical value η = η H. It correspon to a Hopf bifurcation to sustained oscillations. We have found numerically that this Hopf bifurcation is always supercritical for the values of P =0.5, k =1/3, α =2,T=150and 0 < β < 6 (the value of τ is not relevant since it can be eliminated in Eqs. (18)-(20) by redefining the time variable). Figure 2 summarizes the different stability regions for the square wave solutions. In Figure 3 and Figure 4, we show the numerical solution for the amplitude of the fiel obtained from the laser equations (4)-(7) for a case of unstable and a case of stable square-wave, respectively. Figure 5 illustrates the case where a square-wave regime is no more possible (η < η c ) and where fast highly pulsating oscillations are observed.

E 1 E 2 0.7 0.5 0.3 0.1 1.0 t/τ Figure 3. Long time numerical simulation of Eqs. (4)-(7). The values of the parameters are P =0.5, α =2,T=150, k =1/3, τ = 4000, η =4, and β =3. Sustained relaxation oscillations for amplitude of the vertical polarization field is observed. E 1 E 2 0.7 0.5 0.3 0.1 1.2 1.0 t/τ Figure 4. Long time numerical simulation of Eqs. (4)-(7). The values of the parameters are the same as in Figure 3 except that η =6 is larger. The square-wave solution is stable and only exhibit damped relaxation oscillations.

E 1 1.0 0.9 0.7 0.5 0.3 0.1 t/τ Figure 5. Long time numerical simulation of Eqs. (4)-(7). The values of the parameters are the same as in Figure 3 except that β =6 is larger. A stable square-wave solution is no more possible and the laser exhibits strongly pulsating oscillations. 4. DISCUSSION In summary, we have found that square-wave self-modulated oscillations of a period close to twice the round-trip time naturally appear for edge-emitting diode lasers subject to PROF. The main condition is that the feedback strength is sufficiently large to compensate the effect of the differential losses. We also found that the RO that appear after each jump between plateaus of the square-wave may be sustained but only during the time interval where the vertical polarization admits a non-zero plateau. All our analytical predictions have been verified by numerical simulations of the laser rate equations. In our numerical simulations, we took a large value of τ and two values of the feedback strength that are not close to the Hopf bifurcation point η = η H in order to clearly differentiate sustained and damped RO. If η is close to η H, transient RO are too long for the time interval of the plateau (0 <t<τ) and the actual bifurcation point could be at a larger value of the feedback strength than the value estimated in the limit of large delays. ACKNOWLEDGMENTS This material is based upon work supported by the U.S. National Science Foundation under CAREER Grant No. 0239413. T.E. acknowledges the support of the Fon National de la Recherche Scientifique (FNRS, Belgium) and the InterUniversity Attraction Pole program of the Belgian government. REFERENCES 1. K. Otsuka and J.-K. Chern, Opt. Lett. 16, 1759 (1991). 2. D.W.Sukow,A.Gavrielides,T.Erneux,M.J.Baracco,Z.A.Parmenter,andK.L.Blackburn,Phys.Rev. A 72, 043818(2005);D.W.Sukow,A.Gavrielides,T.Erneux,M.J.Baracco,Z.A.Parmenter,andK.L. Blackburn, Proc. PIE 5722, 256 (2005). 3. T. Heil, A. Uchida, P. Davis, and T. Aida, Phys. Rev. A 68, 033811 (2003). 4. T. B. Simpson, and J. M. Liu, J. App. Phys. 73, 2587 (1993); J. M. Liu, and T. B. Simpson, IEEE J. Quant. Electron. 30, 957 (1994). 5. C. L. Tang, J. Opt. B: Quant. Semiclass. Opt. 10, R51 (1998). 6. W. H. Loh, Y. Ozeki, and C. L. Tang, App. Phys. lett. 56, 2613 (1990). 7. W. H. Loh, A. T. Schremer, and C. L. Tang, IEEE Phot. Tech. Lett. 2, 467 (1990).K. Otsuka and J.-K. Chern, Opt. Lett. 16, 1759 (1991) 8. T. C. Yen, J. W. Chang, J. M. Lin, and R. J. Chen, Opt. Commun. 150, 159 (1998) 9. J. Houlihan, G. Huyet, and J. G. McInerney, Opt. Commun. 199, 175 (2001)

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