A semiconductor laser subject to delayed filtered optical feedback can show pure frequency
oscillations with a period of the order to the delay time, while the power remains practically constant. This is remarkable in light of the strong self-phase modulation in semiconductor lasers that couples frequency and power. It turns out that the dynamics of the filter plays an essential role in this behavior, because it changes the instantaneous amount of feedback in response to the instantaneous laser frequency. By using numerical bifurcation techniques we show how frequency oscillations bifurcate in Hopf bifurcatious from the continuous wave solutions known as external filtered modes

Erzgraber, H

Krauskopf, B

Lenstra, D

English

Erzgräber, H

Lenstra, D Daan

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Pure frequency oscillations of semiconductor lasers with filtered optical feedback

A semiconductor laser subject to delayed filtered optical feedback can show pure frequency oscillations with a period of the order to the delay time, while the power remains practically constant. This is remarkable in light of the strong self-phase modulation in semiconductor lasers that couples frequency and power. It turns out that the dynamics of the filter plays an essential role in this behavior, because it changes the instantaneous amount of feedback in response to the instantaneous laser frequency. By using numerical bifurcation techniques we show how frequency oscillations bifurcate in Hopf bifurcatious from the continuous wave solutions known as external filtered modes

Erzgräber, H.

Lenstra, D.

Krauskopf, B.

Pure OAI Repository

A semiconductor laser subject to delayed filtered optical feedback can show pure frequency oscillations with a period of the order to the delay time, while the power remains practically constant. This is remarkable in light of the strong self-phase modulation in semiconductor lasers that couples frequency and power. It turns out that the dynamics of the filter plays an essential role in this behavior, because it changes the instantaneous amount of feedback in response to the instantaneous laser frequency. By using numerical bifurcation techniques we show how frequency oscillations bifurcate in Hopf bifurcations from the continuous wave solutions known as external filtered modes

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                          Erzgraber, H., Lenstra, D., & Krauskopf, B. (2005). Pure frequencyoscillations of semiconductor lasers with filtered optical feedback.Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.Pure frequency oscillations of semiconductorlasers with filtered optical feedbackH. Erzgra¨ber1,2, D. Lenstra1,3 and B. Krauskopf2,11Natuurkunde en Sterrenkunde, Vrije Universiteit Amsterdam, The Netherlands2Department of Engineering Mathematics, University of Bristol, UK3COBRA Research Institute, Technische Universiteit Eindhoven, The NetherlandsA semiconductor laser subject to delayed filtered optical feedback can show pure fre-quency oscillations with a period of the order to the delay time, while the power remainspractically constant. This is remarkable in light of the strong self-phase modulation insemiconductor lasers that couples frequency and power. It turns out that the dynamicsof the filter plays an essential role in this behavior, because it changes the instantaneousamount of feedback in response to the instantaneous laser frequency. By using numericalbifurcation techniques we show how frequency oscillations bifurcate in Hopf bifurcationsfrom the continuous wave solutions known as external filtered modes.The extreme sensitivity of semiconductor lasers (SLs) to external perturbation has let to avast amount of studies investigating the onset of instabilities caused by feedback in SLs,and still there are many open questions. A lot of these studies are focusing on the origin ofinstabilities and chaotic dynamics that degrade the performance of the system [1, 2, 3, 4].On the other hand, ideas have been developed to utilize chaotic dynamics for applications,such as secure communications [5]. This asks for control of the feedback.We investige a SL that receives filtered optical feedback; see the sketch in Fig. 1. Afraction of the light emitted by the laser is traveling through a feedback loop, where aFabry-Perot filter gives control over the feedback light by means of the filter detuning andthe filter width; see also Refs. [6, 7]. In particular, we show the important role of thefeedback phase on the structure and stability of the system.This system can be modeled by the set of rate equations for the complex envelope of thelaser field E and the filter field F and the laser inversion N given bydEdt = (1+ iα)N(t)E(t)+κF(t,τ) (1)TdNdt = P−N(t)− (1+2N(t))|E(t)|2 (2)dFdt = ΛE(t− τ)e−iCp +(i∆−Λ)F(t) . (3)Here α = 5 is the self-phase modulation parameter, κ the feedback rate, T = 100 theelectron decay rate, P = 3.5 the pump rate, Cp the feedback phase, ∆ = 0.007 the filterdetuning and Λ =−0.007 the filter width (HWHM). In Eqs. (1)–(3) time t is measured inunits of the photon lifetime; see Refs. [7, 8] for more details of the modelThe basic solution of Eqs. (1)–(3) are CW states, with a frequency deviation ωs from thesolitary laser frequency at threshold. The amplitudes of the laser field Rs and the filteredfeedback field As and the inversion Ns of laser are constant in time. The feedback fieldmay have a constant phase shift φ. We call these solutions external filtered modes (EFMs):E(t) = Rseiωst ,N(t) = Ns ,F(t) = Aseiωst+iφ . (4). .LaserE(t), N(t)Ω0, α, P, TFilterF (t)Λ, ∆IsolatorE(t)κF (t, τ)Figure 1: Sketch of the system.Even though the delay time τ is long the number of EFMs is small, because of the rel-atively small linewidth of the filter. However, one should note that the distance betweenEFMs is still approximately given by the roundtrip time in the feedback loop. Sincethe number of EFMs is small one can trace each EFM individually. We use DDE-BIFTOOL [9, 10] for the numerical continuation of the EFMs.In Fig. 2 the EFMs are plotted as a function of the feedback rate κ for six different valuesof the feedback phase Cp. Because of the small number of EFMs the feedback phaseCp has a vital influence on the structure and stability of the EFMs as can be seen fromFig. 2. The branches of EFMs are plotted in black, where thin curves indicate unstableEFMs and thick curves stable EFMs. Apart from the solitary laser modes that existsfor all feedback rates κ, EFMs are born in pairs in saddle-node bifurcation (+), one ofwhich may be stable. EFMs destabilize in Hopf bifurcations (∗). As Cp is changed thepositions of these bifurcations change, and so does the structure and stability of the EFMs.The dark gray (light gray) curves in Fig. 2 are the saddle-node (Hopf) bifurcations in the(κ,Ns)-plane parametrized by the feedback phase Cp. As Cp is changed the saddle-nodebifurcations (+) and the Hopf bifurcation (∗) move along these curves.Especially for small values of κ there are several stable EFMs. Up to four EFMs canbe stable simultaneously, depending on the feedback rate κ and the feedback phase Cp.Moreover, EFMs can undergo Hopf bifurcations in which periodic orbits are created.In this case the EFM still exists after the bifurcation, only its stability (more generallythe number of unstable eigendirections) has changed. (This is in contrast to a saddle-node bifurcation where a pair of EFMs annihilate each other and disappear.) If the Hopfbifurcation is supercritical the created periodic orbit is stable.There are two Hopf bifurcation denoted by HRO and HFO that lead to stable periodic orbits.HRO leads to stable relaxation oscillations with a frequency ωRO ≈√2PT . It is well knownthat external perturbations, such as feedback, may result in an undamping of relaxationoscillations. However, it is a special property of FOF that there is another type of Hopfbifurcation HFO which leads to the so-called frequency oscillations [3]. Figure 3 shows atime series where the characteristic time scale of the frequency oscillations is ωFO ≈ 2piτ .The remarkable property of frequency oscillation is that the intensity of the laser is almostconstant in time; only the optical frequency of the laser oscillates as shown Fig. 3. Thisis only possible because of the presence of the filter which changes the amount of feed-back according to its lineshape. The structure and stability of the frequency oscillationsdepends on the feedback phase as well, which is the subject to ongoing research...0 0.01 0.02 0.03−0.00500.0050 0.01 0.02 0.03−0.00500.0050 0.01 0.02 0.03−0.00500.0050 0.01 0.02 0.03−0.00500.0050 0.01 0.02 0.03−0.00500.0050 0.01 0.02 0.03−0.00500.005(a)HFOHROCp = 0Ns(b)Cp = 1/6piHFOHRO(c)Cp = 2/6piNsHFOHRO(d)Cp = 3/6piHFOHRO(e)Cp = 4/6piNsκHFOHRO(f)Cp = 5/6piκHFOHROFigure 2: EFMs in the (κ,Ns)-plane for six different values of the feedback phase Cpindicated in the panels. Branches of EFMs are plotted in black, stable parts are thick andunstable parts are thin. Saddle-node bifurcations are indicated by pluses (+) and Hopfbifurcations by stars (∗). Saddle-node lines (dark gray) and Hopf lines (light gray) areparametrized by Cp.30 32 34 36 38 4001t[×τ]Frequnecy [GHz]Figure 3: Time series of frequency oscillations; time is in units of the delay time τ.In conclusion, we discussed the structure and stability of the external filtered modes for asemiconductor laser with coherent filtered feedback in dependence on the feedback phaseand the feedback rate. In particular, it turns out that the feedback phase plays a vital rolefor the dynamics of the system. This is mainly due to the small number of external filteredmodes. We showed that, in addition to the well known relaxation oscillations, the systemcan exhibit frequency oscillations. For frequency oscillations the intensity of the laser isalmost constant and only the laser frequency oscillates. Interesting further studies mayinclude the influence of the detuning and the linewidth of the filter.References[1] T. Sano. Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor laserswith optical feedback. Phys. Rev. A, 50:2719–2726, 1994.[2] K. Green and B. Krauskopf. Bifurcation analysis of a semiconductor laser subject to non-instantaneousphase-conjugate feedback. Opt. Commun., 231:383–393, 2004.[3] A. P. A. Fischer, M. Yousefi, D. Lenstra, M. W. Carter, and G. Vemuri. Experimental and theoreticalstudy of semiconductor laser dynamics due to filtered optical feedback. Phys. Rev. Lett., 92:023901–023904, 2004.[4] S. M. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra. The dynamical complexity of opticallyinjected semiconductor lasers. Physics Reports, 416(1-2):1–128, 2005.[5] I. Fischer, Y. Liu, and P. Davis. Synchronization of chaotic semiconductor laser dynamics on subn-anosecond time scales and its potential for chaos communication. Phys. Rev. A, 62:011801(R), 2002.[6] M. Yousefi, D. Lenstra, and G. Vemuri. Nonlinear dynamics of a semiconductor laser with filteredoptical feedback and the influence of noise. Phys. Rev. E, 67:046213, 2003.[7] K. Green and B. Krauskopf. Mode structure of a semiconductor laser subject to filtered optical feed-back. Opt. Commun., to appear, 2005.[8] D. Lenstra, G. Vemuri, and M. Yousefi. Generalized optical feedback: Theory. In D.M. Kane and K.A.Shore, editors, Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers,pages 55–80, New York, 2005. Wiley.[9] K. Engelborghs, T. Luzyanina, and G. Samaey. DDE-BIFTOOL v. 2.00 user manual: a Matlab pack-age for bifurcation analysis of delay differential equations. Technical report TW-330. Department ofComputer Science, K.U. Leuven, Leuven, Belgium, 2001.[10] B. Krauskopf. Bifurcation analysis of lasers with delay. In D.M. Kane and K.A. Shore, editors,Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, pages 147–183,New York, 2005. Wiley.

Antimode-dynamicsand chaotic itinerancy in the coherence collapse of semiconductorlasers with optical feedback.

Bifurcation analysis of lasers with delay.

Bifurcationanalysis ofa semiconductorlaser subjectto non-instantaneous phase-conjugatefeedback.

DDE-BIFTOOL v. 2.00 user manual: a Matlab package for bifurcation analysis of delay differential equations.

Experimental and theoretical study of semiconductor laser dynamics due to ﬁltered optical feedback.

Generalizedoptical feedback: Theory.

Mode structure of a semiconductor laser subject to ﬁltered optical feedback.

Nonlinear dynamics of a semiconductor laser with ﬁltered optical feedback and the inﬂuence of noise.

Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication.

Thedynamicalcomplexityofoptically injected semiconductor lasers. Physics Reports,

Pure frequency oscillations of semiconductor lasers with filtered optical feedback

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