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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 6 — Mar. 24, 2014
  • pp: 6905–6918
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Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback

Gaetan Friart, Lionel Weicker, Jan Danckaert, and Thomas Erneux  »View Author Affiliations


Optics Express, Vol. 22, Issue 6, pp. 6905-6918 (2014)
http://dx.doi.org/10.1364/OE.22.006905


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Abstract

The rate equations for a laser with a polarization rotated optical feedback are investigated both numerically and analytically. The frequency detuning between the polarization modes is now taken into account and we review all earlier studies in order to motivate the range of values of the fixed parameters. We find that two basic Hopf bifurcations leading to either stable sustained relaxation or square-wave oscillations appear in the detuning versus feedback rate diagram. We also identify two key parameters describing the differences between the total gains of the two polarization modes and discuss their effects on the periodic square-waves.

© 2014 Optical Society of America

1. Introduction

In a polarization self-modulation regime, the laser output switches regularly between its two natural polarization modes and it can be realized entirely optically by injecting light from one laser polarization mode into the orthogonal one. This requires that the optical feedback be rotated from its original orientation by some means, typically a waveplate [3

3. W. H. Loh, Y. Ozeki, and C. L. Tang, “High-frequency polarization self-modulation and chaotic phenomena in external cavity semiconductor lasers,” Appl. Phys. Lett. 56, 2613–2615 (1990). [CrossRef]

]. By using a Faraday rotator as the rotating element in the external cavity, rather than a waveplate, unidirectional coupling from the dominant, transverse electric (TE) polarization mode to the suppressed, transverse magnetic (TM) mode can be realized. From a practical point of view, polarization-rotated optical feedback (PROF) simplifies the dynamics of the laser because the feedback operates on the weaker TM mode rather than on the main TE mode. For edge-emitting lasers, square-wave self-modulation was found experimentally and simulated numerically using simple rate equations [4

4. A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. 31, 2006–2008 (2006). [CrossRef] [PubMed]

6

6. A. Gavrielides, T. Erneux, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave oscillations in edge-emitting diode lasers with polarization rotated optical feedback,” in Semiconductor Laser and Laser Dynamics II, D. Lenstra, M. Pessa, and I. H. White, eds., Proc. SPIE6184, 255–261 (2006).

]. Another application of the PROF setup has recently been developed to design a random bit generator [7

7. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36, 4632–4634 (2011). [CrossRef] [PubMed]

].

Square-wave oscillations have also been observed for vertical-cavity surface-emitting lasers (VCSELs). VCSELs may operate in a single longitudinal and transverse emission mode. Using the PROF setup the emission of the laser is split into its two linearly polarized components but only one is fed back into the laser after being rotated in the orthogonal polarization direction. It seems, however, that the range of currents for which a PROF successfully leads to square-wave modulation is limited for lasers that have a low dischroism [8

8. J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A 76,043801 (2007). [CrossRef]

10

10. D. W. Sukow, T. Gilfillan, B. Pope, M. S. Torre, A. Gavrielides, and C. Masoller, “Square-wave switching in vertical-cavity surface-emitting lasers with polarization-rotated optical feedback: experiments and simulations,” Phys. Rev. A 86,033818 (2012). [CrossRef]

]. Other experimental studies of square-wave modulations in VCSELs considered cross polarization re-injection in both polarization modes [11

11. S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, “High-frequency polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 63, 3545–3547, (1993). [CrossRef]

, 12

12. H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357, (1998). [CrossRef]

]. These experiments have been interpreted using the standard semiconductor laser rate equations with gain saturation [12

12. H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357, (1998). [CrossRef]

,13

13. M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65,041801 (2002). [CrossRef]

]. The PROF setup considerably facilitates the analysis of the laser rate equations: there exists a nonzero intensity steady state from which we may determine Hopf bifurcations. In the case of cross polarization re-injection in both modes, external cavity modes appear in the bifurcation diagram and their stability needs to be explored numerically [13

13. M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65,041801 (2002). [CrossRef]

].

Square-wave oscillations have also been successfully observed for a ring laser subject to a PROF-like feedback scheme [14

14. L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012). [CrossRef] [PubMed]

]. Here, the counter-and clockwise waves circulating into the ring laser play the role of the two interacting modes.

The laser subject to an optical feedback as well as the recently designed opto-electro oscillators [19

19. M. Peil, M. Jacquot, Y. Kouomou Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic-oscillators,” Phys. Rev. E 79,026208 (2009). [CrossRef]

21

21. L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slow-fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371,20120459 (2013). [CrossRef] [PubMed]

] can be considered as nonlinear oscillators exhibiting damped oscillations that are subject to a delayed feedback. If the delay is sufficiently large, the feedback generates oscillatory instabilities that may either sustain the damped oscillations of the free oscillator or impose a new frequency proportional to the inverse of the delay. As a result, the stability diagram in parameter space typically exhibits two distinct Hopf bifurcations with different frequencies. This can be illustrated by analyzing the simple case of the harmonic oscillator subject to a delayed feedback of the form
x+x=a(x(tτ)x).
(1)
The two parameters a and τ represent the gain and the delay of the feedback. By analyzing the stability of the zero solution [22

22. S.A. Campbell, “Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops,” Dynamics of Continuous, Discrete and Impulsive Systems 5, 225–235 (1999).

, 23

23. F. M. Atay, “Oscillation control in delayed feedback systems,” in Dynamics, Bifurcations, and Control, F. Colonius and L. Grne, eds. (Lect. Notes in Control and Information Sciences 273, 2002), pp. 103–116. [CrossRef]

], we find that two Hopf bifurcation lines delimit the stability domain in the (a, τ) parameter space. Figure 1 shows the two first Hopf bifurcation lines H1 and H2.

Fig. 1 Stability diagram for the zero solution of Eq. (1). The Hopf bifurcation lines H1 and H2 are defined by (2) and (3), respectively. There are no other bifurcations for the chosen ranges of a and τ.

These bifurcations and their frequencies are defined by
H1:a=12(π2τ21)andω1=πτ1,
(2)
H2:τ=2πandω2=1.
(3)
The nature of the two Hopf bifurcation lines is clearly different. H1 admits a frequency inversely proportional to the delay while H2 exhibits the frequency of the free harmonic oscillator and is independent of the feedback strength.

In our laser delayed feedback problem, two distinct frequencies are expected to control the stability diagram, namely, the laser relaxation oscillations (ROs) frequency and the external round-trip frequency defined by
fRO12π2PTandfD12τ,
(4)
respectively. If T ∼ 103, τ ∼ 103, and P ∼ 1, fD << fRO. As we shall demonstrate in this paper, both frequencies are associated with two distinct Hopf bifurcations.

2. Formulation

Table 1. Dimensionless parameters.

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Table 1 is instructive showing similar range of values of the parameters. Except for the two last lines, k ∼ 0.8 and β ∼ 0.1. For all references, α = 2 − 3 and P = 0.5 − 0.6. The delay θ = 103 − 6 × 103 and the ratio of carrier to cavity lifetimes T = 102 − 103 are large. The feedback amplitude η = 3 × 10−2 − 10−1.

We first analyze the steady state solutions of Eqs. (16)(19). The zero intensity solution A1 = A2 = 0 is always unstable if P > 0. We have verified that there exists no pure mode solution with A1 = 0 and A2 ≠ 0. There exists a mixed mode solution A1 ≠ 0 and A2 ≠ 0 if N = 0. We find that the intensities of the two polarization modes are given by
A12=[(kβ)2+(Ω+αkβ)2]P(kβ)2+(Ω+αkβ)2+η2k,
(20)
A22=η2kP(kβ)2+(Ω+αkβ)2+η2k
(21)
and the phase Φ satisfies
tan(Φ)=Ω+αkβkβ.
(22)

3. Hopf stability boundaries

We first concentrate on the emergence of the square-wave oscillations. To this end, we analyze the conditions for a Hopf bifurcation in the limit of large delays and then investigate equations for a nonlinear map that are obtained from Eqs. (16)(19) using the same limit.

We start with the characteristic equation for the growth rate λ. Assuming λ = O(θ−1), we find
P+A22(exp(λθ)1)=0
(23)
as θ → ∞. Inserting λ = into Eq. (23) and separating the real and imaginary parts, we find that the first Hopf bifurcation satisfies the conditions
A22=P/2andω=π/θ.
(24)
Using (21), the location of this Hopf bifurcation in the Ω vs η diagram is given by
η2k=(kβ)2+(Ω+αkβ)2.
(25)

Fig. 2 Stability diagram in terms of the frequency detuning Ω versus feedback rate η. The values of the fixed parameters are k = 0.8 (implying β = 0.125), α = 2, P = 0.5, T = 150, and θ = 103. Figure left: The line HRO corresponds to the first Hopf bifurcation of the steady state that leads to fast relaxation oscillations. The line HSW marks the transition to the square-wave oscillations. It is given by (25). Triangles mark parameter values chosen for the numerical simulations. Figure right: blow-up of the region where the two Hopf bifurcation lines are close. They intersect at two codimension 2 bifurcation points characterized by two pairs of distinct imaginary eigenvalues.

We now examine the other Hopf bifurcation (line HRO) which leads to sustained ROs. In order to determine this Hopf bifurcation condition, we insert λ = into the full characteristic equation and separate the real and imaginary parts. An approximation based on the large value of θ is delicate because of the presence of fast changing trigonometric functions of ωθ ( ω~ωRO=2P/T and ωθ = O(θ1/2) if T = O(θ)), in the Hopf bifurcation conditions. However, it is possible to reduce the two Hopf conditions to a single transcendental equation which we solved numerically. The analysis is long and tedious and we omit all details [28

28. G. Friart, “Routes to square-wave oscillations in a semiconductor laser subject to polarization rotated feedback”, MS Thesis in Physics, Université Libre de Bruxelles (2013).

]. The Hopf line is denoted by HRO in Fig. 2. The fast periodic pulsating oscillations were observed experimentally [Fig. 11(a) in [29

29. A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E 81,056209 (2010). [CrossRef]

]].

In summary, two distinct Hopf bifurcation lines control the stability diagram. We now investigate their effects by simulating numerically Eqs. (16)(19). As we progressively increase the feedback amplitude from zero, the basic steady state first exchanges its stability to fast sustained relaxation oscillations. The transition is smooth and the amplitude of the oscillations gradually increases with the feedback amplitude. Above a critical feedback rate, the waveform of the oscillations suddenly change from pulses to much slower square-waves exhibiting a period equal to twice the delay. We have found that this critical rate is close to the second Hopf bifurcation HSW from the steady state. The Hopf bifurcation now admits a frequency close to π/θ. The simulations shown in Fig. 3 illustrate the transition from the RO oscillations to the square-waves. Figures 3(c)–3(e) correspond to the points labeled by c, d, and e in Fig. 2 left. As the feedback rate increases, the oscillations are first harmonic, then grow in amplitude and become square-waves as anticipated from the stability diagram.

Fig. 3 At Ω = −0.2, the first Hopf bifurcation leads to relaxation oscillations. (c) η = 0.078, small amplitude oscillations near the Hopf bifurcation point; (d) η = 0.105, fast relaxation oscillations slightly before the bifurcation of the Period 2 square-wave oscillations; (e) η = 0.12, square-wave oscillations slightly above its bifurcation point.

There exists a small domain of overlap between fast pulsating and square-wave regimes near the HSW critical point. Moreover, solutions combining pulsating and square-wave forms have been found numerically. They were observed experimentally and were called ”complex oscillations” [Figs. 8 and 10 in [29

29. A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E 81,056209 (2010). [CrossRef]

]]. Note that the two Hopf bifurcation lines are both moving to smaller feedback rates if the detuning is negative (but not too large). A negative detuning is thus favourable for the observations of RO and square-wave oscillations.

Specific features of the stability diagram have been checked. The black dots in Fig. 2 left indicate the observation of the RO instability and they correctly match the Hopf line HRO. Figure 2 right indicates that near Ω = −0.15, the second Hopf line HSW appears before the first Hopf line HRO. Consequently, the bifurcation to square-waves appears first as we increase the feedback rate and if the bifurcation is supercritical, we may expect a gradual change from nearly sinusoidal to square-waves close to HSW. This is exactly what we observe numerically in Fig. 4. In Fig. 4(a), the oscillations are of small amplitude and nearly sinusoidal but the period is already close to 2 (two delays). They have been obtained numerically very close to HSW [red triangle labelled by a in Fig. 2]. If we slightly increases the feedback rate, square-wave oscillations clearly appear [Fig. 4(b) and red triangle labelled by b in Fig. 2]. Note the damped RO oscillations on the upper plateaus.

Fig. 4 At Ω = −0.15, the first Hopf bifurcation corresponds to a bifurcation to Period 2 square-waves. (a) η = 0.125 small amplitude oscillations very close to the bifurcation point (b) η = 0.13 fully developed square-waves. The values of the other parameters are the same as in Fig. (2).

4. Discussion

In this paper, we combine asymptotic and numerical techniques to explore the bifurcation possibilities of a semiconductor laser with a PROF setup. We have included the frequency detuning into our analysis, which is a parameter that is most often present experimentally and that plays a crucial role, often in combination with the α factor and the gain/loss difference. We found that two basic Hopf bifurcations leading to stable solutions appear in parameter space. Except for a small range of detuning, the bifurcation to sustained ROs always appears before the bifurcation to the square-wave oscillations as we increase the feedback rate from zero. Close to the bifurcation to the square-waves, regimes involving both RO and square-wave oscillations have been found numerically. For other ranges of the fixed parameters, square-waves with rapidly sustained ROs on the top of one of the two plateaus have been found numerically but not observed experimentally yet.

Two parameters defined in (11) measure small differences between the total gains of the two polarization modes. They are k, the ratio of the differential gain coefficients and β, the dimensionless cavity loss parameter of the passive TM mode with respect to the TE mode. Assuming equal cavity decay rate for both the TE and TM modes, only k controls the differences between the total gains. We have investigated the limit k → 1 (β = (1 − k)/(2k) → 0) both numerically and analytically [28

28. G. Friart, “Routes to square-wave oscillations in a semiconductor laser subject to polarization rotated feedback”, MS Thesis in Physics, Université Libre de Bruxelles (2013).

]. We found that the 2θ–periodic square-wave progressively degrades both in form and stability. Specifically, the 2θ-periodic square-wave becomes asymmetric and exhibit more than 2 plateaus. Figure 5 gives a typical example of this behaviour. The total period becomes larger than 2θ and the periodic regime is highly sensitive to noise. Our results suggest that the total gains of the two polarization modes cannot be too close for a successful generation of stable and robust square-waves. This is in agreement with recent work on VCSELs with the PROF setup [9

9. M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87,013827 (2013). [CrossRef]

] for which square-waves regimes are not observed for lasers with a low dischroism.

Fig. 5 Asymmetric square-waves. The values of the parameters are P = 0.5, α = 2, T = 150, θ = 2000, k = 0.9, Ω = 0, and η = 0.4.

A. Appendix: Dimensionless equations

Heil et al [26

26. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A 68,033811 (2003). [CrossRef]

], Takeuchi et al [25

25. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. 17, 144–151 (2010). [CrossRef]

], and Fischer et al [27

27. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19,1501610 (2013). [CrossRef]

] simulated their experiments by solving numerically rate equations. In this appendix, we formulate their equations in the same dimensionless form and evaluate the values of the dimensionless parameters.

4.1. Equations by Heil et al [26

26. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A 68,033811 (2003). [CrossRef]

]

The optical fields in [26

26. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A 68,033811 (2003). [CrossRef]

] are defined as TE,TM(t) = ETE,TM(t) exp(TE,TMt). The rate equations are given by
dETEdt=12(1+iα)(G1(nn0)γp)ETE,
(33)
dETMdt=12(1+iα)(G2(nn0)γp)ETM+γETE(tτ)exp(iC),
(34)
dndt=Jγsn(nn0)(G1|ETE|2+G2|ETM|2)
(35)
where C = ωTEτ. The values of the parameters are listed in Table 2.

Table 2. Values of the parameters

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Introducing the new variables
s=γpt,N=12[G1γp1(nn0)1],Y1=γs1G12ETE,Y2=γs1G22ETM
(36)
into Eqs. (33)(35), we find
dY1ds=(1+iα)NY1,
(37)
dY2ds=(1+iα)k(Nβ)Y2+ηkY1(sθ)exp(iC),
(38)
TdNds=PN(1+2N)(|Y1|2+|Y2|2)
(39)
where
T=γpγs,θ=γpτ,k=G2G1,β=12(G1G21),
(40)
P=G1γp1γs12(JJth),Jth=(n0+1G1γp1)1γs1.
(41)
Using the two first columns of Table 2, we compute the values of the dimensionless parameters (two last columns in Table 2).

4.2. Takeuchi et al equations [25

25. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. 17, 144–151 (2010). [CrossRef]

]

The optical fields in [25

25. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. 17, 144–151 (2010). [CrossRef]

] are defined as TE,TM(t) = ETE,TM(t) exp(−TE,TMt). The rate equations are now formulated as
dETEdt=12(1iα)G1(nnthTE)ETE,
(42)
dETMdt=12(1iα)G2(nnthTM)ETM+γETE(tτ)exp[i(Δt+C)],
(43)
dndt=Jnτs(nn0)(G1|E1|2+G2|E2|2)
(44)
where Δ = ωTEωTM and C = ωTEτ. The values of the parameters are listed in Table 3.

Table 3. Values of the parameters. Questions marks indicate unclear or non documented values.

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4.3. Fischer et al equations [27

27. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19,1501610 (2013). [CrossRef]

]

The equations used by Hicke, Fischer and their collaborators [27

27. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19,1501610 (2013). [CrossRef]

] are given by
dE||dt=12(1+iα)(g||(NNT)γ||)E||,
(53)
dEdt=12(1+iα)(g(NNT)γ)E+κE||(tτec),
(54)
dNdt=IeγeN(NNT)(g|||E|||2+g|E|2).
(55)
The values of the parameters are listed in Table 4.

Table 4. Values of the parameters.

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Introducing the new variables
s=γ||t,n=12[g||γ||1(NNT)1],Y1=γe1g||2E||,Y2=γe1g2E,
the evolution equations become
dY1ds=(1+iα)NY1,
(56)
dY2ds=(1+iα)k(Nβ)Y2+ηkY1(sθ),
(57)
TdNds=PN(1+2N)(|Y1|2+|Y2|2)
(58)
where
T=γ||/γe,k=g/g||,β=(1k)/(2k),η=κ/γ||,
(59)
P=12g||γ||1γe1(IeIthe),andIthe=γe(N++1g||γ||1).
(60)
The values of the dimensionless parameters are listed in the last two columns in Table 4.

Acknowledgments

The authors benefited from fruitful discussions with I. Fischer and A. Gavrielides. TE acknowledges the support of the F.N.R.S. (Belgium). GF and LW acknowledge the support of the Belgian F.R.I.A. for PhD scholarships. This work benefited from the support of the Belgian Science Policy Office under Grant No IAP-7/35 “photonics@be”.

References and links

1.

K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991). [CrossRef] [PubMed]

2.

D.-L. Cheng, T.-C. Yen, J.-W. Chang, and J.-K. Tsai, “Generation of high-speed single-wavelength optical pulses in semiconductor lasers with orthogonal-polarization optical feedback,” Opt. Commun. 222, 363–369 (2003). [CrossRef]

3.

W. H. Loh, Y. Ozeki, and C. L. Tang, “High-frequency polarization self-modulation and chaotic phenomena in external cavity semiconductor lasers,” Appl. Phys. Lett. 56, 2613–2615 (1990). [CrossRef]

4.

A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. 31, 2006–2008 (2006). [CrossRef] [PubMed]

5.

A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-waveforms in edge-emitting diode lasers subject to polarization rotated optical feedback,” in Physics and Simulation of Optoelectronic Devices XIV, M. Osinski, F. Henneberger, and Y. Arakawa, eds., Proc. SPIE6115, 60–69 (2006).

6.

A. Gavrielides, T. Erneux, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave oscillations in edge-emitting diode lasers with polarization rotated optical feedback,” in Semiconductor Laser and Laser Dynamics II, D. Lenstra, M. Pessa, and I. H. White, eds., Proc. SPIE6184, 255–261 (2006).

7.

N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36, 4632–4634 (2011). [CrossRef] [PubMed]

8.

J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A 76,043801 (2007). [CrossRef]

9.

M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87,013827 (2013). [CrossRef]

10.

D. W. Sukow, T. Gilfillan, B. Pope, M. S. Torre, A. Gavrielides, and C. Masoller, “Square-wave switching in vertical-cavity surface-emitting lasers with polarization-rotated optical feedback: experiments and simulations,” Phys. Rev. A 86,033818 (2012). [CrossRef]

11.

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, “High-frequency polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 63, 3545–3547, (1993). [CrossRef]

12.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357, (1998). [CrossRef]

13.

M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65,041801 (2002). [CrossRef]

14.

L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012). [CrossRef] [PubMed]

15.

D. W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81,025206 (2010). [CrossRef]

16.

C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: Theory and experiment,” Phys. Rev. A 84,023838 (2011). [CrossRef]

17.

M. Sciamanna, M. Virte, C. Masoller, and A. Gavrielides, “Hopf bifurcation to square-wave switching in mutually coupled semiconductor lasers,” Phys. Rev. E 86,016218 (2011). [CrossRef]

18.

C. Masoller, M. Sciamanna, and A. Gavrielides, “Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers,” Phil. Transac. Roy. Soc. A 371,20120471 (2013). [CrossRef]

19.

M. Peil, M. Jacquot, Y. Kouomou Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic-oscillators,” Phys. Rev. E 79,026208 (2009). [CrossRef]

20.

L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square-waves of time delayed systems,” Phys. Rev. E 86,055201 (2012). [CrossRef]

21.

L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slow-fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371,20120459 (2013). [CrossRef] [PubMed]

22.

S.A. Campbell, “Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops,” Dynamics of Continuous, Discrete and Impulsive Systems 5, 225–235 (1999).

23.

F. M. Atay, “Oscillation control in delayed feedback systems,” in Dynamics, Bifurcations, and Control, F. Colonius and L. Grne, eds. (Lect. Notes in Control and Information Sciences 273, 2002), pp. 103–116. [CrossRef]

24.

Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaotic dynamics in semiconductor lasers subjected to polarization-rotated optical feedback,” Appl. Phys. Lett. 93,181105 (2008). [CrossRef]

25.

Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. 17, 144–151 (2010). [CrossRef]

26.

T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A 68,033811 (2003). [CrossRef]

27.

K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19,1501610 (2013). [CrossRef]

28.

G. Friart, “Routes to square-wave oscillations in a semiconductor laser subject to polarization rotated feedback”, MS Thesis in Physics, Université Libre de Bruxelles (2013).

29.

A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E 81,056209 (2010). [CrossRef]

OCIS Codes
(140.5960) Lasers and laser optics : Semiconductor lasers
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 17, 2014
Revised Manuscript: March 2, 2014
Manuscript Accepted: March 2, 2014
Published: March 18, 2014

Virtual Issues
Physics and Applications of Laser Dynamics (2014) Optics Express

Citation
Gaetan Friart, Lionel Weicker, Jan Danckaert, and Thomas Erneux, "Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback," Opt. Express 22, 6905-6918 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6905


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References

  1. K. Otsuka, J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991). [CrossRef] [PubMed]
  2. D.-L. Cheng, T.-C. Yen, J.-W. Chang, J.-K. Tsai, “Generation of high-speed single-wavelength optical pulses in semiconductor lasers with orthogonal-polarization optical feedback,” Opt. Commun. 222, 363–369 (2003). [CrossRef]
  3. W. H. Loh, Y. Ozeki, C. L. Tang, “High-frequency polarization self-modulation and chaotic phenomena in external cavity semiconductor lasers,” Appl. Phys. Lett. 56, 2613–2615 (1990). [CrossRef]
  4. A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. 31, 2006–2008 (2006). [CrossRef] [PubMed]
  5. A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, J. Amonette, “Square-waveforms in edge-emitting diode lasers subject to polarization rotated optical feedback,” in Physics and Simulation of Optoelectronic Devices XIV, M. Osinski, F. Henneberger, Y. Arakawa, eds., Proc. SPIE6115, 60–69 (2006).
  6. A. Gavrielides, T. Erneux, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, J. Amonette, “Square-wave oscillations in edge-emitting diode lasers with polarization rotated optical feedback,” in Semiconductor Laser and Laser Dynamics II, D. Lenstra, M. Pessa, I. H. White, eds., Proc. SPIE6184, 255–261 (2006).
  7. N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36, 4632–4634 (2011). [CrossRef] [PubMed]
  8. J. Mulet, M. Giudici, J. Javaloyes, S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A 76,043801 (2007). [CrossRef]
  9. M. Marconi, J. Javaloyes, S. Barland, M. Giudici, S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87,013827 (2013). [CrossRef]
  10. D. W. Sukow, T. Gilfillan, B. Pope, M. S. Torre, A. Gavrielides, C. Masoller, “Square-wave switching in vertical-cavity surface-emitting lasers with polarization-rotated optical feedback: experiments and simulations,” Phys. Rev. A 86,033818 (2012). [CrossRef]
  11. S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, K. Kojima, “High-frequency polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 63, 3545–3547, (1993). [CrossRef]
  12. H. Li, A. Hohl, A. Gavrielides, H. Hou, K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357, (1998). [CrossRef]
  13. M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A 65,041801 (2002). [CrossRef]
  14. L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express 20, 22503–22516 (2012). [CrossRef] [PubMed]
  15. D. W. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81,025206 (2010). [CrossRef]
  16. C. Masoller, D. Sukow, A. Gavrielides, M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: Theory and experiment,” Phys. Rev. A 84,023838 (2011). [CrossRef]
  17. M. Sciamanna, M. Virte, C. Masoller, A. Gavrielides, “Hopf bifurcation to square-wave switching in mutually coupled semiconductor lasers,” Phys. Rev. E 86,016218 (2011). [CrossRef]
  18. C. Masoller, M. Sciamanna, A. Gavrielides, “Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers,” Phil. Transac. Roy. Soc. A 371,20120471 (2013). [CrossRef]
  19. M. Peil, M. Jacquot, Y. Kouomou Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic-oscillators,” Phys. Rev. E 79,026208 (2009). [CrossRef]
  20. L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, L. Larger, “Strongly asymmetric square-waves of time delayed systems,” Phys. Rev. E 86,055201 (2012). [CrossRef]
  21. L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, L. Larger, “Slow-fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A 371,20120459 (2013). [CrossRef] [PubMed]
  22. S.A. Campbell, “Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops,” Dynamics of Continuous, Discrete and Impulsive Systems 5, 225–235 (1999).
  23. F. M. Atay, “Oscillation control in delayed feedback systems,” in Dynamics, Bifurcations, and Control, F. Colonius, L. Grne, eds. (Lect. Notes in Control and Information Sciences 273, 2002), pp. 103–116. [CrossRef]
  24. Y. Takeuchi, R. Shogenji, J. Ohtsubo, “Chaotic dynamics in semiconductor lasers subjected to polarization-rotated optical feedback,” Appl. Phys. Lett. 93,181105 (2008). [CrossRef]
  25. Y. Takeuchi, R. Shogenji, J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. 17, 144–151 (2010). [CrossRef]
  26. T. Heil, A. Uchida, P. Davis, T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A 68,033811 (2003). [CrossRef]
  27. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. 19,1501610 (2013). [CrossRef]
  28. G. Friart, “Routes to square-wave oscillations in a semiconductor laser subject to polarization rotated feedback”, MS Thesis in Physics, Université Libre de Bruxelles (2013).
  29. A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E 81,056209 (2010). [CrossRef]

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