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High chaotic spiking rate in a closed loop semiconductor laser with optical feedback
Results in Physics 6 (2016) 401–406
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
High chaotic spiking rate in a closed loop semiconductor laser with optical feedback Banaz O. Rasheed a,b, Parekhan M. Aljaff a, Kais A. Al Naimee a,c,⇑, Mahdi H. Al Hasani c, Riccardo Meucci a a
Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy Department of Physics, School of Science, Faculty of Science and Science Education, Sulaimani University, Sulaimani, Iraq c Department of Physics, College of Science, Baghdad, Iraq b
a r t i c l e
i n f o
Article history: Received 29 April 2016 Accepted 15 July 2016 Available online 25 July 2016 Keywords: Chaos Spiking Optical feedback
a b s t r a c t We investigate experimentally and numerically the existence of fast chaotic spiking in the dynamics of a semiconductor laser with ac-coupled optical feedback. The observed dynamics is chaotic in the explored range of both bias current optical feedback strength. The effects of a modulation applied to the bias current are also investigated. We eventually indicate that the observed chaotic dynamics is a good candidate to hide information to satisfy secure communications. Ó 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction
Experimental setup
Semiconductor lasers with optical feedback have proven to be very useful devices in practical applications such as chaos based communication encryption systems and chaotic lidars [1–3]. Two aspects turned out to be essential for the observation of chaos in these systems, that is, the delay in the round trip of the light in the cavity and the nonlinearity in the interaction of the light with the semiconductor medium [4,5]. A semiconductor laser can be stabilized by introducing an external perturbation such as an external optical feedback [6], or optoelectronic feedback [7]. The dynamical behaviour of semiconductor lasers with external optical disturbance was broadly studied [8–10]. Lang and Kobayashi examined feedback-induced instabilities and chaos in semiconductor lasers [11]. The collapse of coherence by increasing the degrees of freedom of the laser from two to three degrees using a delayed optical feedback was investigated by Tromborg and Mork [12]. Many critical parameters, comprising feedback strength, feedback delay, pump current, feedback type and laser nonlinearities affect the dynamics of semiconductor lasers with external cavities [13]. In this work, we investigate the effects of an optical feedback using an optical fibre coupler as a loop mirror. The timescale of the observed dynamics (few ns) is entirely determined by the delay introduced by the optical feedback. In our investigation, we considered the impact of different control parameters such as the bias current and the optical feedback strength and the effect of a modulation applied to the bias current.
We generated chaotic dynamics in our semiconductor laser by introducing a coupled optical feedback. The experimental setup consists of a single-mode semiconductor laser (1550 nm wavelength) with the optical feedback provided by an optical fibre closed loop (Fig. 1). The pigtailed laser output is connected to Variable Optical Attenuator (VOA) which in turn is connected to two 12 directional couplers (12 DC, 90%:10% and 1X2YC, 50%:50%) to form a fibre loop mirror. We realize the loop by connecting the two output branches of the Y-coupler. The reflected light from coupler is split into two parts; the first one provides the optical feedback to the cavity of the semiconductor laser whilst the other is connected to a high-speed InGaAs photodetector with response time <1 ns. The photodetector is connected to a sampling digital storage scope (LeCroy 500 MHz). In our experimental configuration, it is possible to introduce a modulation of the driving current of the semiconductor laser by means of an external sinusoidal function generator.
⇑ Corresponding author at: Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy.
Experimental results and discussion The influence of the bias current After closing the optical feedback loop, the laser output intensity is observed and recorded on a high-speed digital oscilloscope. We performed the temporal and spectral analysis of the chaotic spiking evolution of the laser intensity using the bias current as a control parameter (see Figs. 2–5). We observe that the laser intensity is chaotic in the whole range of the control parameter. Both the
http://dx.doi.org/10.1016/j.rinp.2016.07.006 2211-3797/Ó 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Fig. 1. Experimental setup of a semiconductor laser with optical feedback using two 12 directional couplers. Red lines indicate the optical path, the black lines indicate the electrical connections. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Experimental time series of the laser intensity (a) and its corresponding FFT spectrum, (b) when the bias current of laser is 10 mA.
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Fig. 3. Experimental time series of laser intensity (a) and its corresponding FFT spectrum, (b) when the bias current of the laser is 15 mA.
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Fig. 4. Experimental time series of the laser intensity (a) and its corresponding FFT spectrum, (b) when the bias current of the laser is 20 mA.
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Fig. 5. Experimental time series of system (a) and its corresponding FFT spectrum, (b) when the bias current of laser diode is 25 mA.
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amplification factor Av (amplification of the detector output voltage) to control the optical feedback. The optical feedback was investigated by keeping constant bias current at a relatively high value (I = 19.5 mA). Figs. 7, 8 and 9 show the time series and the corresponding spectra for increasing values of the optical feedback strength. We observe that the intensity of the chaotic behaviour increases with increasing values of Av. Chaotic signals are characterized by a continuous spectrum over a limited range. Thus, the energy is spread over a wider bandwidth. Fig. 10 shows the bifurcation diagram of the laser output intensity as Av is increased. Also in such a case, the dynamics of the laser is chaotic and its amplitude increases with the amplification factor Av.
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Fig. 6. Bifurcation diagram of chaotic laser intensity as a function of the injected bias current.
spiking rate and amplitude of the chaotic signals increase with increasing injection bias current. Fig. 6 shows the bifurcation diagram obtained by plotting the peaks of the laser output intensity as a function of the bias current. For low values of the bias current the dynamics of the oscillator is chaotic with small amplitudes, for higher values the dynamics remains chaotic but with high amplitudes. The influence of the optical feedback The other control parameter for generating chaotic behaviour in the laser is the optical feedback (OBF). Here we used the voltage
where Ps is the signal power at the given modulating frequency and P n is the value of the power in the absence of the external modulation. The power spectra of the laser output signals for three values of the modulation frequency (80, 90 and 100 MHz) are reported in Figs 11–13. The corresponding SNR values are 56.65, 48.67, and 64.18 dB, respectively.
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The intrinsic dynamic resonance in the photon-carrier interaction determines the modulation response of semiconductor lasers. In order to investigate this effect, the laser has been directly modulated by an external periodic signal (function generator in the experimental set up). The bias current is set at 19.5 mA. To obtain a quantitative measure of this modulation we introduce the signal to noise ratio (SNR) of the power spectrum for each value of the modulating frequency. The SNR is defined as SNR ¼ 10 log PPns ,
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Fig. 7. Time series of the laser intensity (a) and its corresponding FFT, (b) when Av = 0.06.
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Fig. 9. Time series of the laser intensity (a) and its corresponding FFT, (b) when Av = 0.45.
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Fig. 14 shows the maxima of laser intensity as a function of the frequency at a fixed value of its amplitude of 50 mV. We can see that the magnitude of chaotic behaviour is fluctuating with a broad peak around 50 MHz which approximately corresponds to the relaxation frequency of the used semiconductor laser. Dynamical model The static characteristics of semiconductor lasers with optical feedback can be theoretically investigated by the relations
amongst the reflectivity of the internal cavity and external reflector, the gain of the laser medium, and other static parameters. The laser dynamical features are described by differential equations for the electric filed, polarization and population inversion. As a semiconductor laser is a class B laser, the polarization term can be adiabatically eliminated and its effect replaced by the linear relation between the electric field and the polarization. The population inversion in a semiconductor laser is the carrier density N produced by electron-hole recombination. The photon number (which is proportional to the square of the electric field) and the carrier density are frequently used as the variables of the rate equations for semiconductor lasers. The dynamics of a single mode semiconductor laser with optical feedback has been modeled by Lang and Kobayashi [11], which is one of the historical milestone papers on chaotic dynamics of semiconductor lasers. The model is referred to as the Lang–Kobayashi equations, which includes the effect of a time-delayed optical feedback. The Lang–Kobayashi equations for the complex amplitude electric field E(t) and the carrier number N(t) are written as follows [11]:
dEðtÞ 1 ¼ ð1 þ iaÞ½GðtÞ cEðtÞ þ KEðt sf Þ expðxsf Þ dt 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ rEðt sf Þ expðxsf Þ þ 2bNðtÞX dNðtÞ i ¼ cd NðtÞ GðtÞjEðtÞj2 dt q
ð2:1Þ ð2:2Þ
where G(t) is the optical gain that is defined as GðtÞ ¼ g½NðtÞN , i is 1þsjEðtÞj2
the bias current, jEðtÞj2 is the laser intensity or the number of photons inside the cavity, that is I ¼ jEðtÞj2 . In our numerical simu-
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Fig. 11. Time series (a) and corresponding FFT spectrum, (b) at a modulating frequency 80 MHz.
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Fig. 12. Time series (a) and corresponding FFT spectrum, (b) at a modulating frequency 90 MHz.
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Fig. 13. Time series (a) and corresponding FFT spectrum, (b) at a modulating frequency 100 MHz.
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lations, a set of typical semiconductor laser parameters are considered as a linewidth enhancement factor a ¼ 1:5, transparency carrier
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g ¼ 1:5 10 , gain saturation coefficient s ¼ 5 107 , photon decay rate c ¼ 500 ns1 , carrier decay rate cd ¼ 0:5 ns1 , spontaneous 4
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emission rate b ¼ 1:6 106 ns1 , delay time sf ¼ 10 ns, feedback strength K ¼ 20 ns1 , and injection strength r ¼ 20 ns1 . The initial conditions are: E(0) = 0.001, N(0) = 0.1. Fig. 15 shows the numerical bifurcation diagram as a function of the bias current (the feedback strength is fixed at K = 20 ns1). We can observe different transitions from regular to irregular behaviour as the bias current is increased. Fig. 16 shows the laser intensity bifurcation diagram as a function of the optical feedback strength. In this diagram, we have stable equilibrium solutions with zero intensity up to a value of
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chaos (0.85 ns1). Chaotic behaviour gradually increases with the feedback strength. Conclusions We experimentally and numerically explored the chaotic dynamics of a semiconductor laser with incoherent optical feedback obtained by means of a fibre loop mirror. Both bias current and optical feedback influence the chaotic behaviour which increases with increasing values of the two control parameters. When a modulating frequency is considered, we observe a resonance effect with the intrinsic relaxation oscillation of the laser. This effect could be used to hide information with a high degree of masking. References
Fig. 15. Numerical bifurcation diagram. Laser intensity maxima vs bias current.
Fig. 16. Numerical bifurcation diagram. Laser intensity maxima vs feedback strength.
the feedback strength K = 0.3 ns1. A periodic oscillation appears at K = 0.3 ns1. A further increase in the feedback strength leads to
[1] J. Ohtsubo, Semiconductor lasers stability, instability and chaos, 2013. [2] Guo Y, Wu Y, Wang Y. Method to identify time delay of chaotic semiconductor laser with optical feedback. Chin Optics Lett 2012;10:061901. [3] Theory of optical feedback in semiconductor lasers. Semiconductor lasers. Berlin Heidelberg: Springer; 2008. p. 63–86. [4] van Tartwijk GHM, van der Linden HJC, Lenstra D. Theory of a diode laser with phase-conjugate feedback. Opt Lett 1992;17:1590–2. [5] Haken H. Analogy between higher instabilities in fluids and lasers. Phys Lett A 1975;53:77–8. [6] Junges L, Pöschel T, Gallas JA. Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang–Kobayashi model. Eur Phys J D 2013;67:1–9. [7] Al-Naimee K, Marino F, Ciszak M, Meucci R, Arecchi FT. Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectronic feedback. New J Phys 2009;11:073022. [8] Giudici M, Green C, Giacomelli G, Nespolo U, Tredicce J. Andronov bifurcation and excitability in semiconductor lasers with optical feedback. Phys Rev E 1997;55:6414. [9] Giudici M, Giuggioli L, Green C, Tredicce J. Dynamical behavior of semiconductor lasers with frequency selective optical feedback. Chaos Solitons Fractals 1999;10:811–8. [10] G.P. Agrawal, N.K. Dutta, Long wavelength semiconductor lasers, 1986. [11] Lang R, Kobayashi K. External optical feedback effects on semiconductor injection laser properties. IEEE J Quantum Electron 1980;16:347–55. [12] Tromborg B, Mork J. Nonlinear injection locking dynamics and the onset of coherence collapse in external cavity lasers. IEEE J Quantum Electron 1990;26:642–54. [13] Soriano MC, García-Ojalvo J, Mirasso CR, Fischer I. Complex photonics: dynamics and applications of delay-coupled semiconductors lasers. Rev Mod Phys 2013;85:421–70.
Contents lists available at ScienceDirect
Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics
High chaotic spiking rate in a closed loop semiconductor laser with optical feedback Banaz O. Rasheed a,b, Parekhan M. Aljaff a, Kais A. Al Naimee a,c,⇑, Mahdi H. Al Hasani c, Riccardo Meucci a a
Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy Department of Physics, School of Science, Faculty of Science and Science Education, Sulaimani University, Sulaimani, Iraq c Department of Physics, College of Science, Baghdad, Iraq b
a r t i c l e
i n f o
Article history: Received 29 April 2016 Accepted 15 July 2016 Available online 25 July 2016 Keywords: Chaos Spiking Optical feedback
a b s t r a c t We investigate experimentally and numerically the existence of fast chaotic spiking in the dynamics of a semiconductor laser with ac-coupled optical feedback. The observed dynamics is chaotic in the explored range of both bias current optical feedback strength. The effects of a modulation applied to the bias current are also investigated. We eventually indicate that the observed chaotic dynamics is a good candidate to hide information to satisfy secure communications. Ó 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Introduction
Experimental setup
Semiconductor lasers with optical feedback have proven to be very useful devices in practical applications such as chaos based communication encryption systems and chaotic lidars [1–3]. Two aspects turned out to be essential for the observation of chaos in these systems, that is, the delay in the round trip of the light in the cavity and the nonlinearity in the interaction of the light with the semiconductor medium [4,5]. A semiconductor laser can be stabilized by introducing an external perturbation such as an external optical feedback [6], or optoelectronic feedback [7]. The dynamical behaviour of semiconductor lasers with external optical disturbance was broadly studied [8–10]. Lang and Kobayashi examined feedback-induced instabilities and chaos in semiconductor lasers [11]. The collapse of coherence by increasing the degrees of freedom of the laser from two to three degrees using a delayed optical feedback was investigated by Tromborg and Mork [12]. Many critical parameters, comprising feedback strength, feedback delay, pump current, feedback type and laser nonlinearities affect the dynamics of semiconductor lasers with external cavities [13]. In this work, we investigate the effects of an optical feedback using an optical fibre coupler as a loop mirror. The timescale of the observed dynamics (few ns) is entirely determined by the delay introduced by the optical feedback. In our investigation, we considered the impact of different control parameters such as the bias current and the optical feedback strength and the effect of a modulation applied to the bias current.
We generated chaotic dynamics in our semiconductor laser by introducing a coupled optical feedback. The experimental setup consists of a single-mode semiconductor laser (1550 nm wavelength) with the optical feedback provided by an optical fibre closed loop (Fig. 1). The pigtailed laser output is connected to Variable Optical Attenuator (VOA) which in turn is connected to two 12 directional couplers (12 DC, 90%:10% and 1X2YC, 50%:50%) to form a fibre loop mirror. We realize the loop by connecting the two output branches of the Y-coupler. The reflected light from coupler is split into two parts; the first one provides the optical feedback to the cavity of the semiconductor laser whilst the other is connected to a high-speed InGaAs photodetector with response time <1 ns. The photodetector is connected to a sampling digital storage scope (LeCroy 500 MHz). In our experimental configuration, it is possible to introduce a modulation of the driving current of the semiconductor laser by means of an external sinusoidal function generator.
⇑ Corresponding author at: Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy.
Experimental results and discussion The influence of the bias current After closing the optical feedback loop, the laser output intensity is observed and recorded on a high-speed digital oscilloscope. We performed the temporal and spectral analysis of the chaotic spiking evolution of the laser intensity using the bias current as a control parameter (see Figs. 2–5). We observe that the laser intensity is chaotic in the whole range of the control parameter. Both the
http://dx.doi.org/10.1016/j.rinp.2016.07.006 2211-3797/Ó 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Fig. 1. Experimental setup of a semiconductor laser with optical feedback using two 12 directional couplers. Red lines indicate the optical path, the black lines indicate the electrical connections. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Experimental time series of the laser intensity (a) and its corresponding FFT spectrum, (b) when the bias current of laser is 10 mA.
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Fig. 3. Experimental time series of laser intensity (a) and its corresponding FFT spectrum, (b) when the bias current of the laser is 15 mA.
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Fig. 4. Experimental time series of the laser intensity (a) and its corresponding FFT spectrum, (b) when the bias current of the laser is 20 mA.
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Fig. 5. Experimental time series of system (a) and its corresponding FFT spectrum, (b) when the bias current of laser diode is 25 mA.
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amplification factor Av (amplification of the detector output voltage) to control the optical feedback. The optical feedback was investigated by keeping constant bias current at a relatively high value (I = 19.5 mA). Figs. 7, 8 and 9 show the time series and the corresponding spectra for increasing values of the optical feedback strength. We observe that the intensity of the chaotic behaviour increases with increasing values of Av. Chaotic signals are characterized by a continuous spectrum over a limited range. Thus, the energy is spread over a wider bandwidth. Fig. 10 shows the bifurcation diagram of the laser output intensity as Av is increased. Also in such a case, the dynamics of the laser is chaotic and its amplitude increases with the amplification factor Av.
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Fig. 6. Bifurcation diagram of chaotic laser intensity as a function of the injected bias current.
spiking rate and amplitude of the chaotic signals increase with increasing injection bias current. Fig. 6 shows the bifurcation diagram obtained by plotting the peaks of the laser output intensity as a function of the bias current. For low values of the bias current the dynamics of the oscillator is chaotic with small amplitudes, for higher values the dynamics remains chaotic but with high amplitudes. The influence of the optical feedback The other control parameter for generating chaotic behaviour in the laser is the optical feedback (OBF). Here we used the voltage
where Ps is the signal power at the given modulating frequency and P n is the value of the power in the absence of the external modulation. The power spectra of the laser output signals for three values of the modulation frequency (80, 90 and 100 MHz) are reported in Figs 11–13. The corresponding SNR values are 56.65, 48.67, and 64.18 dB, respectively.
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The intrinsic dynamic resonance in the photon-carrier interaction determines the modulation response of semiconductor lasers. In order to investigate this effect, the laser has been directly modulated by an external periodic signal (function generator in the experimental set up). The bias current is set at 19.5 mA. To obtain a quantitative measure of this modulation we introduce the signal to noise ratio (SNR) of the power spectrum for each value of the modulating frequency. The SNR is defined as SNR ¼ 10 log PPns ,
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Fig. 7. Time series of the laser intensity (a) and its corresponding FFT, (b) when Av = 0.06.
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Fig. 8. Time series of laser intensity (a) and its corresponding FFT, (b) when Av = 0.1.
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Fig. 9. Time series of the laser intensity (a) and its corresponding FFT, (b) when Av = 0.45.
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Feedback strength Av Fig. 10. Bifurcation diagram of the laser intensity vs the amplification factor of feedback strength Av.
Fig. 14 shows the maxima of laser intensity as a function of the frequency at a fixed value of its amplitude of 50 mV. We can see that the magnitude of chaotic behaviour is fluctuating with a broad peak around 50 MHz which approximately corresponds to the relaxation frequency of the used semiconductor laser. Dynamical model The static characteristics of semiconductor lasers with optical feedback can be theoretically investigated by the relations
amongst the reflectivity of the internal cavity and external reflector, the gain of the laser medium, and other static parameters. The laser dynamical features are described by differential equations for the electric filed, polarization and population inversion. As a semiconductor laser is a class B laser, the polarization term can be adiabatically eliminated and its effect replaced by the linear relation between the electric field and the polarization. The population inversion in a semiconductor laser is the carrier density N produced by electron-hole recombination. The photon number (which is proportional to the square of the electric field) and the carrier density are frequently used as the variables of the rate equations for semiconductor lasers. The dynamics of a single mode semiconductor laser with optical feedback has been modeled by Lang and Kobayashi [11], which is one of the historical milestone papers on chaotic dynamics of semiconductor lasers. The model is referred to as the Lang–Kobayashi equations, which includes the effect of a time-delayed optical feedback. The Lang–Kobayashi equations for the complex amplitude electric field E(t) and the carrier number N(t) are written as follows [11]:
dEðtÞ 1 ¼ ð1 þ iaÞ½GðtÞ cEðtÞ þ KEðt sf Þ expðxsf Þ dt 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ rEðt sf Þ expðxsf Þ þ 2bNðtÞX dNðtÞ i ¼ cd NðtÞ GðtÞjEðtÞj2 dt q
ð2:1Þ ð2:2Þ
where G(t) is the optical gain that is defined as GðtÞ ¼ g½NðtÞN , i is 1þsjEðtÞj2
the bias current, jEðtÞj2 is the laser intensity or the number of photons inside the cavity, that is I ¼ jEðtÞj2 . In our numerical simu-
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Fig. 11. Time series (a) and corresponding FFT spectrum, (b) at a modulating frequency 80 MHz.
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Fig. 12. Time series (a) and corresponding FFT spectrum, (b) at a modulating frequency 90 MHz.
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Fig. 13. Time series (a) and corresponding FFT spectrum, (b) at a modulating frequency 100 MHz.
Intensity Maxima (a.u.)
0,16
lations, a set of typical semiconductor laser parameters are considered as a linewidth enhancement factor a ¼ 1:5, transparency carrier
0,12
number
N ¼ 1:5 108 ,
differential
gain
coefficient
g ¼ 1:5 10 , gain saturation coefficient s ¼ 5 107 , photon decay rate c ¼ 500 ns1 , carrier decay rate cd ¼ 0:5 ns1 , spontaneous 4
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emission rate b ¼ 1:6 106 ns1 , delay time sf ¼ 10 ns, feedback strength K ¼ 20 ns1 , and injection strength r ¼ 20 ns1 . The initial conditions are: E(0) = 0.001, N(0) = 0.1. Fig. 15 shows the numerical bifurcation diagram as a function of the bias current (the feedback strength is fixed at K = 20 ns1). We can observe different transitions from regular to irregular behaviour as the bias current is increased. Fig. 16 shows the laser intensity bifurcation diagram as a function of the optical feedback strength. In this diagram, we have stable equilibrium solutions with zero intensity up to a value of
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chaos (0.85 ns1). Chaotic behaviour gradually increases with the feedback strength. Conclusions We experimentally and numerically explored the chaotic dynamics of a semiconductor laser with incoherent optical feedback obtained by means of a fibre loop mirror. Both bias current and optical feedback influence the chaotic behaviour which increases with increasing values of the two control parameters. When a modulating frequency is considered, we observe a resonance effect with the intrinsic relaxation oscillation of the laser. This effect could be used to hide information with a high degree of masking. References
Fig. 15. Numerical bifurcation diagram. Laser intensity maxima vs bias current.
Fig. 16. Numerical bifurcation diagram. Laser intensity maxima vs feedback strength.
the feedback strength K = 0.3 ns1. A periodic oscillation appears at K = 0.3 ns1. A further increase in the feedback strength leads to
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