October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
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Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection Romain Modeste Nguimdo,* Mulham Khoder, Jan Danckaert, Guy Van der Sande, and Guy Verschaffelt Applied Physics Research Group, Vrije Universiteit Brussel, 1050 Brussels Belgium *Corresponding author: [email protected] Received August 4, 2014; accepted August 30, 2014; posted September 15, 2014 (Doc. ID 220414); published October 10, 2014 We numerically show the quantitative relation between the chaos bandwidth enhancement and fast phase dynamics in semiconductor lasers with optical feedback and optical injection. The injection increases the coupling between the intensity and the phase leading to a competition between the relaxation oscillation (RO) frequency and the intrinsic response frequency of the phase. For large feedback strengths, it is found that the chaos bandwidth is determined by the intrinsic phase response frequency. For smaller feedback strengths, the system is not chaotic and its bandwidth is determined by the RO frequency. © 2014 Optical Society of America OCIS codes: (250.5960) Semiconductor lasers; (140.1540) Chaos; (060.4785) Optical security and encryption. http://dx.doi.org/10.1364/OL.39.005945
In a typical laser, the bandwidth is largely limited by the relaxation oscillation (RO) frequency. This bandwidth limit typically also persists even when a chaotic signal is generated from such a laser, for example through the use of optical feedback. To support the growing need for high-speed chaos encryption [1,2] and random bit generation [3], much research has been devoted to increasing the bandwidths of semiconductor lasers (SLs). Optical injection is commonly employed to significantly enhance the chaos bandwidths of these devices without modifying their design. This injection can be achieved by injecting a chaotic light (from a master laser) into a slave SL without [4] or with [5] optical feedback. It also can be done by injecting a continuous-wave (CW) light into a slave SL without [6] or with optical feedback [7–9]. In all cases, the chaos bandwidth enhancement depends on the laser’s parameters such as pump current, injection strength, and frequency detuning. In the case of CW optical injection into a slave SL without optical feedback, the bandwidth enhancement can be estimated from theory [10]. However, this configuration is not suitable for generating broadband chaos with a sufficiently large complexity. Such complex broadband chaotic regimes can be obtained in the same systems when at least one of the lasers is subject to feedback [4,5,7–9]. In particular, it is qualitatively shown in [9] that the bandwidth enhancement is a result of beating between the injected field and the chaotic laser field provided that the injected light is detuned toward the edge of the SL’s optical spectrum. Meanwhile, the detuning should not exceed the original chaos bandwidth. However, it is still unclear how the system’s parameters quantitatively change the intrinsic dynamics of the slave laser such that the bandwidth of the chaos is enlarged. In other words, a direct quantitative relationship between slave SL’s parameters and the expansion of the chaotic carrier frequency by a strong optical injection still needs clarification. In this Letter, we provide details about the quantitative relation between the intrinsic phase dynamics response and the bandwidth enhancement in a chaotic SL with CW 0146-9592/14/205945-04$15.00/0
optical injection. Our results also allow a clear understanding of how the injection can suppress the chaotic oscillations in lasers. In our previous work we have shown that, in an SL with optical feedback, the intrinsic phase response frequency (i.e., the dominant peak in the phase spectrum) is typically faster than the RO frequency for large feedback strengths [11]. When there is a strong coupling between the phase and the intensity (through optical injection, for example), the competition between the RO frequency and the intrinsic phase dynamics gives rise, in both the intensity and the phase, to a common resonance frequency which strongly depends on the largest of the two frequencies. We find the condition under which the enhancement of the chaos bandwidth is related to the intrinsic phase response frequency. We consider a quantum well SL operating in a singlelongitudinal mode with delayed optical feedback and CW optical injection. It is modeled by Lang–Kobayashi equations [12] extended to include optical injection. The dynamics of the system can be described in terms of the slowly varying complex electric field amplitude Et Eteiφt and the carrier number inside the active layer Nt [7–9]: 1 _ Et G − γEt ηE T D cosφT D − φt − Ω0 T D 2 (1) kinj E 0 cos−φt Δωt ξE t; ET α _ G − γ η D sinφT D − φt − Ω0 T D φt Et 2 E kinj 0 sin−φt Δωt ξφ t; Et I _ Nt 0 − γ e Nt − GEt2 ; e
(2)
(3)
where E T D Et − T D and φT D φt − T D while Gt gm Nt − N 0 ∕1 εEt2 stands for the optical gain, © 2014 Optical Society of America
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ε being the saturation factor. The parameters are the linewidth enhancement factor α, pump current I 0 , electron charge e, photon decay rate γ, electron decay rate γ e , carrier number at transparency N 0 , differential gain gm , loop delay time T D , feedback strength η, injection strength kinj , constant injection field amplitude E 0 , solitary laser angular frequency Ω0 , and frequency detuning between the SL and the injection field Δω. ξE;φ are complex Gaussian white noise terms with zero mean and correlation hξE;φ tξE;φ t0 i βγ e Nδt − t0 . We use the following parameters [11,13]: α 3, γ 200 ns−1 , γ e 1 ns−1 , N 0 1.8 × 108 , gm 10−5 ns−1 , ε 10−7 , T D 4 ns, β 10−6 , Ω0 T D 0, E 0 103 (this is ≈2 times the amplitude of the free-running laser output) and kinj 25 ns−1 , I 0 1.5I th where I th is the threshold of the pump current. With these parameters I th 32 mA and the RO period of the free-running laser τRO ≈ 0.19 ns (corresponding to the RO frequency ≈5.2 GHz). Other parameters, i.e., η and Δω are stated in the figure captions. Although the intensity jEtj2 and the phase φt are coupled through the feedback and the linewidth enhancement factor in the absence of injection (i.e., kinj 0), we have previously shown that jEtj2 and φt can relax at different time scales [11]. These time scales can be revealed by computing time scale identifiers from time series. Out of those, the delayed mutual information, autocorrelation function, and spectrum are robust to noise and therefore are most often used [14,15]. Figure 1 shows the radio-frequency (rf) spectra (a), (c) and the autocorrelation function (b), (d) computed from jEtj2 (a), (b) and φt ∈ 0; 2π (c), (d) considering η 30 ns−1 and η 40 ns−1 . For these values of the feedback strength, the system is chaotic and its dynamics is dominated by two frequency components: the delayinduced oscillation frequency (i.e., T −1 D ) and the intrinsic response frequency of the system. Here, we are interested only in the system’s intrinsic dynamics’ time scales
Fig. 1. Spectra and autocorrelation function computed (a), (b) from intensity jEtj2 and (c), (d) from phase φt. kinj 0 ns−1 . The arrows in the spectra (autocorrelation) indicate the dominant resonance frequencies (fastest intrinsic time scales).
as they are directly connected with the bandwidth of the signal. Thus, we will only show the results in windows allowing for a clear view on the intrinsic time scale, while T −1 D 0.25 GHz is not clearly visible on this scale. For the two values of the feedback strength considered in Fig. 1(a), the RO frequency can be identified by a clear peak frequency located at ≈5.2 GHz. By computing the autocorrelation function from the intensity time series, we can identify τRO as the period of the damped oscillation in the autocorrelation, i.e., τRO ≈ 0.17 ns [Fig. 1(b)]. As expected, τRO is independent of the feedback strength as the same value of τRO is obtained for the two feedback strengths. However, when the same quantifiers are applied to the phase dynamics, the spectra are considerably broader with a peak at ≈12 GHz and ≈17 GHz for η 30 ns−1 and η 40 ns−1 , respectively [Fig. 1(c)]. These results can be understood from mathematical modeling. For kinj 0 ns−1 , the steady states solutions of Eqs. (1)–(3), usually referred to as the external cavity modes, are often seen as the skeleton of the chaotic attractor. Such steady states are typically obtained by assuming E_ 0, φ_ ω, and N_ 0 where ω is a constant angular frequency. It can be straightforwardly demonstrated that the stationary angular frequencies ω satisfy the equation [16]: p ω η 1 α2 sin−tan−1 α − ω − Ω0 T D :
(4)
The peak in the phase spectrum corresponds to the p frequency for which ω ≈ η 1 α2 . Thus, T −1 phase p 2 ω∕2π ≈ η 1 α ∕2π. This suggests that the phase spectrum can be further broadened by increasing the feedback strength. Much faster dynamics can be therefore obtained in the phase as compared to the intensity dynamics. The fast dynamics in the phase is further confirmed in Fig. 1(d) which shows a phase response time T phase ≈ 0.0875 ns and T phase ≈ 0.06 ns for η 30 ns−1 and η 40 ns−1 , respectively. These response times correspond to the inverse of the phase response frequencies identified in Fig. 1(c). The fast dynamics in the phase is essentially because of the nonlinear feedback term E 0 η sinφt − T D − φt∕Et in Eq. (2) which can indeed induce a change in frequency and create fast phase oscillations. We noticed that the phase response time T phase is almost independent of the pump current. The above results suggest that, without the injection, the coupling between the intensity and the phase is not strong enough to impose a dependence on their response times. With optical injection (i.e., kinj ≠ 0), Eqs. (1)–(2) provide an additional coupling between the amplitude and the phase through the terms E 0 cos−φt Δωt and E 0 sin−φt Δωt∕Et in the intensity and the phase, respectively. However, this coupling between intensity and phase will only substantially increase if the injection strength is well chosen. On the one hand, a small injection strength will have a weak effect. On the other hand, a very large injection strength will force the slave laser to evolve to the same dynamical behavior as the master (frequency locking) meaning that E 0 ∕E ≈ constant and φ_ Δω leading to cos−φt Δωt ≈ 1 and sin−φt Δωt ≈ 0. Under these
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
conditions, the coupling between the intensity and the phase provided by the injection is suppressed. For kinj 25 ns−1 and Δω 0 GHz (all other parameters being unchanged), we compute again the rf spectra and the autocorrelation function both from the intensity [Figs. 2(a) and 2(b)] and the phase [Figs. 2(c) and 2(d)] considering different feedback strengths. Contrary to Fig. 1(a), it is clear from Fig. 2(a) that the bandwidth of the signal is increasingly broadened as the feedback strength is increased. While a clear peak frequency still emerges at τ−1 RO ≈ 5.2 GHz, the spectra in all cases broaden up to about the intrinsic phase response frep ≈ η 1 α2 ∕2π. The frequency τ~ −1 quency, i.e., τ~ −1 RO RO is indicated in Fig. 2(a) by the solid arrows while τ−1 RO is indicated by the dashed arrows. These results are further confirmed in Fig. 2(b) in which the period of the damped oscillations (indicated by the solid arrows) is found to be close to the value of τ~ RO derived from the rf spectra. Thus, the bandwidth is now proportional to the inverse of the phase response time T phase and consequently to the feedback strength. This is because of the interplay between the intensity and the phase dynamics. The remarkable fingerprint of the phase dynamics in the presence of the CW injection is mainly because the phase response is faster so that the dynamics in the intensity is determined by this fast phase time scale T phase . The dependence between the intensity and the phase response times can be further evidenced in the phase spectra shown in Fig. 2(c). It is seen that the peaks in Fig. 2(c) are the same as τ~ −1 RO in Fig. 2(a), and they both correspond to T −1 observed in the phase spectrum phase without the injection in Fig. 1(c). This confirms that the dynamics both in the intensity and the phase is dominated by the fast intrinsic phase time scale in the presence of the injection. By way of further illustration,
Fig. 2. Same as in Fig. 1 for kinj 25 ns−1 and Δω 0 GHz. Solid arrows indicate the dominant resonance frequency (or fastest intrinsic time scale) while dashed arrows indicate other resonance frequencies (or intrinsic time scales).
Fig. 3. Optical (b) kinj 25 ns−1 .
spectra
for
(a)
kinj 0 ns−1
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and
only the signature of the T phase is identifiable in Fig. 2(d) although the phase and the intensity are coupled. In Fig. 3, we show the optical spectra (which depends on both intensity and phase) without injection (a) and with optical injection (b) for two different values of the feedback strength. For kinj 0 ns−1 , the optical spectrum is not peaked around 0 GHz, but the maxip mum is shifted toward ≈η 1 α2 ∕2π. With the injection (e.g., kinj 25 ns−1 ), a new peak emerges at zero frequency (for Δω 0). This peak corresponds to a new frequency excited by the injection light. The separation between the two strongest peaks in Fig. 3(b) corresponds withpthe chaos bandwidth observed in Fig. 2(a), −1 i.e., τ~ RO ≈ η 1 α2 ∕2π. While τ−1 RO is the RO frequency for a free-running laser, this RO frequency will change because of optical injection. We define τ¯ −1 RO as the new RO frequency in the presence of the injection, but for η 0. Then, we gradually increase the feedback strength η to appreciate the change in the width of the intensity’s rf spectrum. Figure 4 shows, for different injection strengths, the dependence of the intensity response frequency (for a slave SL with injection) on the feedback strength for Δω 0. Instead of τ−1 RO ≈ 5.2 GHz found for the free-running laser, it can be noticed that the injection shifts the RO frequency to τ¯ −1 ¯ −1 ¯ −1 RO ≈ 9 GHz, τ RO ≈ 13 GHz and τ RO ≈ 20 GHz for −1 −1 kinj 25 ns , kinj 50 ns and kinj 100 ns−1 , respectively, when the values of the feedback strength are small. More precisely, the dynamics both in the intensity
Fig. 4. Peak rf frequency τ~ −1 RO of the laser’s intensity with optical feedback and CW optical injection as a function of the feedback strength for different injection strengths. Δω 0 GHz.
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Fig. 5. (a) Spectra and (b) autocorrelation computed from intensity jEtj2 for different values of the detuning considering kinj 25 ns−1 and η 40 ns−1 . Solid arrows indicate the dominant resonance frequency (or fastest intrinsic time scale).
and the phase remain dominated by τ¯ −1 RO for η ≲ p −1 2 2π¯τRO ∕ 1 α so that both the phase and the intensity relax at the RO frequency τ¯ −1 RO . We find in such a case that the dynamical behavior of the system is either stable or periodic, but not chaotic. This explains why an increase of the injection strength (which accordingly leads to the increase of τ¯ −1 RO ) can typically suppress chaotic oscillations and even lead to injection locking as has been pointed out in some previous works (see Ref. [8]). Note that it may still be possible to induce chaotic poscillations by properly choosing Δω. For η ≳ 2π¯τ−1 ∕ 1 α2 , the RO dominates the dynamics both phase frequency T −1 phase in the intensity and in the phase. The system is chaotic, and the chaos bandwidth both in the intensity and the phase is determined by the phase response time. −1 The rf and the phase spectra peak at τ~ −1 RO T phase ≈ p η 1 α2 ∕2π and the chaos bandwidth is broader as shown in Figs. 2(a) and 2(c). The chaos bandwidth therefore clearly depends on the feedback strength η and the linewidth enhancement factor α. These results also suggest that the chaotic oscillations typically result from the interaction between the intensity and the fast phase dynamics. In previous works, the frequency detuning between the injection laser and the slave laser has been found as one of the crucial parameters for bandwidth enhancement in slave lasers [4,6–9]. To investigate how such a detuning can modify the relationship between the chaos bandwidth and the feedback strength, we fix kinj 25 ns−1 and η 40 ns−1 and compute the same quantifiers as in Fig. 2 for different frequency detuning values. The results are shown in Fig. 5 both for (a) the spectra and (b) the autocorrelation function. For Δω 40 GHz, it is seen that the chaos bandwidth is further expanded while it is symmetrically reduced for Δω −40 GHz. We have checked through the computation of the optical spectra (not shown here) that positive Δω increases the shift between the injection light frequency and the peak
frequency in the optical spectrum while negative Δω narrows such a shift. In conclusion, we have discussed the quantitative relationship between the chaos bandwidth enhancement and the fast phase dynamics in semiconductor lasers (SLs) with optical feedback and optical injection. As the phase response frequency depends on the feedback strength, a clear relationship between the bandwidth enhancement and the feedback strength has been shown. Our results have shown that the bandwidth enhancement in SL with optical feedback and injection can be quantitatively predicted even for chaotic regimes once the system parameters are known. As a final remark, our results are also qualitatively in agreement with the experimental results in [9] obtained with different laser’s parameters. We acknowledge the Research Foundation Flanders (FWO) for project support, the Research Council of the VUB, and the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 photonics@be. References 1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005). 2. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, Phys. Rev. Lett. 107, 034103 (2011). 3. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, Rev. Mod. Phys. 85, 421 (2013). 4. A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, IEEE J. Quantum Electron. 39, 1462 (2003). 5. S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, IEEE Photon. Technol. Lett. 24, 1753 (2012). 6. E. K. Lau, L. J. Wong, X. Zhao, Y.-K. Chen, C. J. C. Hasnain, and M. C. Wu, J. Lightwave Technol. 26, 2584 (2008). 7. Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, Opt. Lett. 28, 319 (2003). 8. A. B. Wang, Y. C. Wang, and H. C. He, IEEE Photon. Technol. Lett. 20, 1633 (2008). 9. A. B. Wang, Y. C. Wang, and J. F. Wang, Opt. Lett. 34, 1144 (2009). 10. E. K. Lau, H. K. Sung, and M. C. Wu, IEEE J. Quantum Electron. 44, 90 (2008). 11. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, Opt. Express 22, 8672 (2014). 12. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980). 13. K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013). 14. R. M. Nguimdo, M. C. Soriano, and P. Colet, Opt. Lett. 36, 4332 (2011). 15. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, Opt. Lett. 37, 2541 (2012). 16. J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
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Fast phase response and chaos bandwidth enhancement in semiconductor lasers subject to optical feedback and injection Romain Modeste Nguimdo,* Mulham Khoder, Jan Danckaert, Guy Van der Sande, and Guy Verschaffelt Applied Physics Research Group, Vrije Universiteit Brussel, 1050 Brussels Belgium *Corresponding author: [email protected] Received August 4, 2014; accepted August 30, 2014; posted September 15, 2014 (Doc. ID 220414); published October 10, 2014 We numerically show the quantitative relation between the chaos bandwidth enhancement and fast phase dynamics in semiconductor lasers with optical feedback and optical injection. The injection increases the coupling between the intensity and the phase leading to a competition between the relaxation oscillation (RO) frequency and the intrinsic response frequency of the phase. For large feedback strengths, it is found that the chaos bandwidth is determined by the intrinsic phase response frequency. For smaller feedback strengths, the system is not chaotic and its bandwidth is determined by the RO frequency. © 2014 Optical Society of America OCIS codes: (250.5960) Semiconductor lasers; (140.1540) Chaos; (060.4785) Optical security and encryption. http://dx.doi.org/10.1364/OL.39.005945
In a typical laser, the bandwidth is largely limited by the relaxation oscillation (RO) frequency. This bandwidth limit typically also persists even when a chaotic signal is generated from such a laser, for example through the use of optical feedback. To support the growing need for high-speed chaos encryption [1,2] and random bit generation [3], much research has been devoted to increasing the bandwidths of semiconductor lasers (SLs). Optical injection is commonly employed to significantly enhance the chaos bandwidths of these devices without modifying their design. This injection can be achieved by injecting a chaotic light (from a master laser) into a slave SL without [4] or with [5] optical feedback. It also can be done by injecting a continuous-wave (CW) light into a slave SL without [6] or with optical feedback [7–9]. In all cases, the chaos bandwidth enhancement depends on the laser’s parameters such as pump current, injection strength, and frequency detuning. In the case of CW optical injection into a slave SL without optical feedback, the bandwidth enhancement can be estimated from theory [10]. However, this configuration is not suitable for generating broadband chaos with a sufficiently large complexity. Such complex broadband chaotic regimes can be obtained in the same systems when at least one of the lasers is subject to feedback [4,5,7–9]. In particular, it is qualitatively shown in [9] that the bandwidth enhancement is a result of beating between the injected field and the chaotic laser field provided that the injected light is detuned toward the edge of the SL’s optical spectrum. Meanwhile, the detuning should not exceed the original chaos bandwidth. However, it is still unclear how the system’s parameters quantitatively change the intrinsic dynamics of the slave laser such that the bandwidth of the chaos is enlarged. In other words, a direct quantitative relationship between slave SL’s parameters and the expansion of the chaotic carrier frequency by a strong optical injection still needs clarification. In this Letter, we provide details about the quantitative relation between the intrinsic phase dynamics response and the bandwidth enhancement in a chaotic SL with CW 0146-9592/14/205945-04$15.00/0
optical injection. Our results also allow a clear understanding of how the injection can suppress the chaotic oscillations in lasers. In our previous work we have shown that, in an SL with optical feedback, the intrinsic phase response frequency (i.e., the dominant peak in the phase spectrum) is typically faster than the RO frequency for large feedback strengths [11]. When there is a strong coupling between the phase and the intensity (through optical injection, for example), the competition between the RO frequency and the intrinsic phase dynamics gives rise, in both the intensity and the phase, to a common resonance frequency which strongly depends on the largest of the two frequencies. We find the condition under which the enhancement of the chaos bandwidth is related to the intrinsic phase response frequency. We consider a quantum well SL operating in a singlelongitudinal mode with delayed optical feedback and CW optical injection. It is modeled by Lang–Kobayashi equations [12] extended to include optical injection. The dynamics of the system can be described in terms of the slowly varying complex electric field amplitude Et Eteiφt and the carrier number inside the active layer Nt [7–9]: 1 _ Et G − γEt ηE T D cosφT D − φt − Ω0 T D 2 (1) kinj E 0 cos−φt Δωt ξE t; ET α _ G − γ η D sinφT D − φt − Ω0 T D φt Et 2 E kinj 0 sin−φt Δωt ξφ t; Et I _ Nt 0 − γ e Nt − GEt2 ; e
(2)
(3)
where E T D Et − T D and φT D φt − T D while Gt gm Nt − N 0 ∕1 εEt2 stands for the optical gain, © 2014 Optical Society of America
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ε being the saturation factor. The parameters are the linewidth enhancement factor α, pump current I 0 , electron charge e, photon decay rate γ, electron decay rate γ e , carrier number at transparency N 0 , differential gain gm , loop delay time T D , feedback strength η, injection strength kinj , constant injection field amplitude E 0 , solitary laser angular frequency Ω0 , and frequency detuning between the SL and the injection field Δω. ξE;φ are complex Gaussian white noise terms with zero mean and correlation hξE;φ tξE;φ t0 i βγ e Nδt − t0 . We use the following parameters [11,13]: α 3, γ 200 ns−1 , γ e 1 ns−1 , N 0 1.8 × 108 , gm 10−5 ns−1 , ε 10−7 , T D 4 ns, β 10−6 , Ω0 T D 0, E 0 103 (this is ≈2 times the amplitude of the free-running laser output) and kinj 25 ns−1 , I 0 1.5I th where I th is the threshold of the pump current. With these parameters I th 32 mA and the RO period of the free-running laser τRO ≈ 0.19 ns (corresponding to the RO frequency ≈5.2 GHz). Other parameters, i.e., η and Δω are stated in the figure captions. Although the intensity jEtj2 and the phase φt are coupled through the feedback and the linewidth enhancement factor in the absence of injection (i.e., kinj 0), we have previously shown that jEtj2 and φt can relax at different time scales [11]. These time scales can be revealed by computing time scale identifiers from time series. Out of those, the delayed mutual information, autocorrelation function, and spectrum are robust to noise and therefore are most often used [14,15]. Figure 1 shows the radio-frequency (rf) spectra (a), (c) and the autocorrelation function (b), (d) computed from jEtj2 (a), (b) and φt ∈ 0; 2π (c), (d) considering η 30 ns−1 and η 40 ns−1 . For these values of the feedback strength, the system is chaotic and its dynamics is dominated by two frequency components: the delayinduced oscillation frequency (i.e., T −1 D ) and the intrinsic response frequency of the system. Here, we are interested only in the system’s intrinsic dynamics’ time scales
Fig. 1. Spectra and autocorrelation function computed (a), (b) from intensity jEtj2 and (c), (d) from phase φt. kinj 0 ns−1 . The arrows in the spectra (autocorrelation) indicate the dominant resonance frequencies (fastest intrinsic time scales).
as they are directly connected with the bandwidth of the signal. Thus, we will only show the results in windows allowing for a clear view on the intrinsic time scale, while T −1 D 0.25 GHz is not clearly visible on this scale. For the two values of the feedback strength considered in Fig. 1(a), the RO frequency can be identified by a clear peak frequency located at ≈5.2 GHz. By computing the autocorrelation function from the intensity time series, we can identify τRO as the period of the damped oscillation in the autocorrelation, i.e., τRO ≈ 0.17 ns [Fig. 1(b)]. As expected, τRO is independent of the feedback strength as the same value of τRO is obtained for the two feedback strengths. However, when the same quantifiers are applied to the phase dynamics, the spectra are considerably broader with a peak at ≈12 GHz and ≈17 GHz for η 30 ns−1 and η 40 ns−1 , respectively [Fig. 1(c)]. These results can be understood from mathematical modeling. For kinj 0 ns−1 , the steady states solutions of Eqs. (1)–(3), usually referred to as the external cavity modes, are often seen as the skeleton of the chaotic attractor. Such steady states are typically obtained by assuming E_ 0, φ_ ω, and N_ 0 where ω is a constant angular frequency. It can be straightforwardly demonstrated that the stationary angular frequencies ω satisfy the equation [16]: p ω η 1 α2 sin−tan−1 α − ω − Ω0 T D :
(4)
The peak in the phase spectrum corresponds to the p frequency for which ω ≈ η 1 α2 . Thus, T −1 phase p 2 ω∕2π ≈ η 1 α ∕2π. This suggests that the phase spectrum can be further broadened by increasing the feedback strength. Much faster dynamics can be therefore obtained in the phase as compared to the intensity dynamics. The fast dynamics in the phase is further confirmed in Fig. 1(d) which shows a phase response time T phase ≈ 0.0875 ns and T phase ≈ 0.06 ns for η 30 ns−1 and η 40 ns−1 , respectively. These response times correspond to the inverse of the phase response frequencies identified in Fig. 1(c). The fast dynamics in the phase is essentially because of the nonlinear feedback term E 0 η sinφt − T D − φt∕Et in Eq. (2) which can indeed induce a change in frequency and create fast phase oscillations. We noticed that the phase response time T phase is almost independent of the pump current. The above results suggest that, without the injection, the coupling between the intensity and the phase is not strong enough to impose a dependence on their response times. With optical injection (i.e., kinj ≠ 0), Eqs. (1)–(2) provide an additional coupling between the amplitude and the phase through the terms E 0 cos−φt Δωt and E 0 sin−φt Δωt∕Et in the intensity and the phase, respectively. However, this coupling between intensity and phase will only substantially increase if the injection strength is well chosen. On the one hand, a small injection strength will have a weak effect. On the other hand, a very large injection strength will force the slave laser to evolve to the same dynamical behavior as the master (frequency locking) meaning that E 0 ∕E ≈ constant and φ_ Δω leading to cos−φt Δωt ≈ 1 and sin−φt Δωt ≈ 0. Under these
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conditions, the coupling between the intensity and the phase provided by the injection is suppressed. For kinj 25 ns−1 and Δω 0 GHz (all other parameters being unchanged), we compute again the rf spectra and the autocorrelation function both from the intensity [Figs. 2(a) and 2(b)] and the phase [Figs. 2(c) and 2(d)] considering different feedback strengths. Contrary to Fig. 1(a), it is clear from Fig. 2(a) that the bandwidth of the signal is increasingly broadened as the feedback strength is increased. While a clear peak frequency still emerges at τ−1 RO ≈ 5.2 GHz, the spectra in all cases broaden up to about the intrinsic phase response frep ≈ η 1 α2 ∕2π. The frequency τ~ −1 quency, i.e., τ~ −1 RO RO is indicated in Fig. 2(a) by the solid arrows while τ−1 RO is indicated by the dashed arrows. These results are further confirmed in Fig. 2(b) in which the period of the damped oscillations (indicated by the solid arrows) is found to be close to the value of τ~ RO derived from the rf spectra. Thus, the bandwidth is now proportional to the inverse of the phase response time T phase and consequently to the feedback strength. This is because of the interplay between the intensity and the phase dynamics. The remarkable fingerprint of the phase dynamics in the presence of the CW injection is mainly because the phase response is faster so that the dynamics in the intensity is determined by this fast phase time scale T phase . The dependence between the intensity and the phase response times can be further evidenced in the phase spectra shown in Fig. 2(c). It is seen that the peaks in Fig. 2(c) are the same as τ~ −1 RO in Fig. 2(a), and they both correspond to T −1 observed in the phase spectrum phase without the injection in Fig. 1(c). This confirms that the dynamics both in the intensity and the phase is dominated by the fast intrinsic phase time scale in the presence of the injection. By way of further illustration,
Fig. 2. Same as in Fig. 1 for kinj 25 ns−1 and Δω 0 GHz. Solid arrows indicate the dominant resonance frequency (or fastest intrinsic time scale) while dashed arrows indicate other resonance frequencies (or intrinsic time scales).
Fig. 3. Optical (b) kinj 25 ns−1 .
spectra
for
(a)
kinj 0 ns−1
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only the signature of the T phase is identifiable in Fig. 2(d) although the phase and the intensity are coupled. In Fig. 3, we show the optical spectra (which depends on both intensity and phase) without injection (a) and with optical injection (b) for two different values of the feedback strength. For kinj 0 ns−1 , the optical spectrum is not peaked around 0 GHz, but the maxip mum is shifted toward ≈η 1 α2 ∕2π. With the injection (e.g., kinj 25 ns−1 ), a new peak emerges at zero frequency (for Δω 0). This peak corresponds to a new frequency excited by the injection light. The separation between the two strongest peaks in Fig. 3(b) corresponds withpthe chaos bandwidth observed in Fig. 2(a), −1 i.e., τ~ RO ≈ η 1 α2 ∕2π. While τ−1 RO is the RO frequency for a free-running laser, this RO frequency will change because of optical injection. We define τ¯ −1 RO as the new RO frequency in the presence of the injection, but for η 0. Then, we gradually increase the feedback strength η to appreciate the change in the width of the intensity’s rf spectrum. Figure 4 shows, for different injection strengths, the dependence of the intensity response frequency (for a slave SL with injection) on the feedback strength for Δω 0. Instead of τ−1 RO ≈ 5.2 GHz found for the free-running laser, it can be noticed that the injection shifts the RO frequency to τ¯ −1 ¯ −1 ¯ −1 RO ≈ 9 GHz, τ RO ≈ 13 GHz and τ RO ≈ 20 GHz for −1 −1 kinj 25 ns , kinj 50 ns and kinj 100 ns−1 , respectively, when the values of the feedback strength are small. More precisely, the dynamics both in the intensity
Fig. 4. Peak rf frequency τ~ −1 RO of the laser’s intensity with optical feedback and CW optical injection as a function of the feedback strength for different injection strengths. Δω 0 GHz.
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Fig. 5. (a) Spectra and (b) autocorrelation computed from intensity jEtj2 for different values of the detuning considering kinj 25 ns−1 and η 40 ns−1 . Solid arrows indicate the dominant resonance frequency (or fastest intrinsic time scale).
and the phase remain dominated by τ¯ −1 RO for η ≲ p −1 2 2π¯τRO ∕ 1 α so that both the phase and the intensity relax at the RO frequency τ¯ −1 RO . We find in such a case that the dynamical behavior of the system is either stable or periodic, but not chaotic. This explains why an increase of the injection strength (which accordingly leads to the increase of τ¯ −1 RO ) can typically suppress chaotic oscillations and even lead to injection locking as has been pointed out in some previous works (see Ref. [8]). Note that it may still be possible to induce chaotic poscillations by properly choosing Δω. For η ≳ 2π¯τ−1 ∕ 1 α2 , the RO dominates the dynamics both phase frequency T −1 phase in the intensity and in the phase. The system is chaotic, and the chaos bandwidth both in the intensity and the phase is determined by the phase response time. −1 The rf and the phase spectra peak at τ~ −1 RO T phase ≈ p η 1 α2 ∕2π and the chaos bandwidth is broader as shown in Figs. 2(a) and 2(c). The chaos bandwidth therefore clearly depends on the feedback strength η and the linewidth enhancement factor α. These results also suggest that the chaotic oscillations typically result from the interaction between the intensity and the fast phase dynamics. In previous works, the frequency detuning between the injection laser and the slave laser has been found as one of the crucial parameters for bandwidth enhancement in slave lasers [4,6–9]. To investigate how such a detuning can modify the relationship between the chaos bandwidth and the feedback strength, we fix kinj 25 ns−1 and η 40 ns−1 and compute the same quantifiers as in Fig. 2 for different frequency detuning values. The results are shown in Fig. 5 both for (a) the spectra and (b) the autocorrelation function. For Δω 40 GHz, it is seen that the chaos bandwidth is further expanded while it is symmetrically reduced for Δω −40 GHz. We have checked through the computation of the optical spectra (not shown here) that positive Δω increases the shift between the injection light frequency and the peak
frequency in the optical spectrum while negative Δω narrows such a shift. In conclusion, we have discussed the quantitative relationship between the chaos bandwidth enhancement and the fast phase dynamics in semiconductor lasers (SLs) with optical feedback and optical injection. As the phase response frequency depends on the feedback strength, a clear relationship between the bandwidth enhancement and the feedback strength has been shown. Our results have shown that the bandwidth enhancement in SL with optical feedback and injection can be quantitatively predicted even for chaotic regimes once the system parameters are known. As a final remark, our results are also qualitatively in agreement with the experimental results in [9] obtained with different laser’s parameters. We acknowledge the Research Foundation Flanders (FWO) for project support, the Research Council of the VUB, and the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 photonics@be. References 1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 438, 343 (2005). 2. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, Phys. Rev. Lett. 107, 034103 (2011). 3. M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, Rev. Mod. Phys. 85, 421 (2013). 4. A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, IEEE J. Quantum Electron. 39, 1462 (2003). 5. S. Y. Xiang, W. Pan, N. Q. Li, B. Luo, L. S. Yan, X. H. Zou, L. Zhang, and P. Mu, IEEE Photon. Technol. Lett. 24, 1753 (2012). 6. E. K. Lau, L. J. Wong, X. Zhao, Y.-K. Chen, C. J. C. Hasnain, and M. C. Wu, J. Lightwave Technol. 26, 2584 (2008). 7. Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, Opt. Lett. 28, 319 (2003). 8. A. B. Wang, Y. C. Wang, and H. C. He, IEEE Photon. Technol. Lett. 20, 1633 (2008). 9. A. B. Wang, Y. C. Wang, and J. F. Wang, Opt. Lett. 34, 1144 (2009). 10. E. K. Lau, H. K. Sung, and M. C. Wu, IEEE J. Quantum Electron. 44, 90 (2008). 11. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, Opt. Express 22, 8672 (2014). 12. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980). 13. K. Hicke, M. A. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, IEEE J. Sel. Top. Quantum Electron. 19, 1501610 (2013). 14. R. M. Nguimdo, M. C. Soriano, and P. Colet, Opt. Lett. 36, 4332 (2011). 15. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, Opt. Lett. 37, 2541 (2012). 16. J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
