Nonlinear dynamics and polarization bistability in optically injected VCSELs

I. Gatarea , b, J. Buesab, H. Thienpontb, K. Panajotovb, M. Sciamannaa

aSupélec – Ecole Supérieure d’Electricité, Campus de Metz, and Laboratoire Matériaux Optiques,

Photonique et Systèmes (LMOPS), common laboratory between the University of Metz and Supélec, CNRS UMR-7132, 2 Rue Edouard Belin, F-57070 Metz (France)

bDepartment of Applied Physics and Photonics (TW-TONA), Vrije Universiteit Brussel (VUB),

Pleinlaan 2, B-1050 Brussels (Belgium).

ABSTRACT

We investigate experimentally and theoretically nonlinear dynamics and polarization bistability in a vertical-cavity surface-emitting laser (VCSEL) submitted to an external optical injection with orthogonal polarization, i.e., the injected light from a master laser (ML) is linearly polarized and orthogonal to the polarization direction of the free-running VCSEL (slave laser, SL). Depending on the injection power level and the frequency detuning between ML and SL, our experimental results show that the injected VCSEL may undergo complex dynamics including subharmonic resonances and period-doubling route to chaos that are associated to polarization switching (PS). Using continuation method, we carry out a detailed bifurcation analysis and report on qualitatively different dynamics as we change the injection strength and/or the detuning. We show that PS may be strongly influenced by the presence of those dynamics. Our numerical results reveal period doubling dynamics that emerge from a Hopf bifurcation associated to an elliptical polarized injection locking state. Pure frequency-induced polarization bistability, i.e., when the detuning is changed for fixed injection power, is studied theoretically. Our results show that, depending on the level of the fixed injection strength, frequency induced PS may be achieved with or without hysteresis. Keywords: vertical-cavity surface-emitting lasers (VCSEL), optical injection, polarization switching, bistability, laser dynamics.

1. INTRODUCTION

Vertical-cavity surface-emitting lasers (VCSEL) have attracted much attention owing to their numerous advantages with respect to edge-emitting lasers such as low threshold current, high-speed modulation bandwidth and the possibility to fabricate large two-dimensional laser arrays which open novel applications in the area of optical network interconnects1. Unlike edge-emitting lasers, VCSEL may be subject to polarization instabilities which usually lead to polarization switching (PS) between two orthogonal linearly polarized (LP) fundamental modes, when changing the operations conditions such as the bias current or the device temperature.2 Optical injection, i.e., injecting an external light from a master laser into the VCSEL cavity, has been proposed as an efficient method for stabilizing or controlling the VCSEL polarization instabilities by adding an external degree of freedom to the laser system.3 Depending on the injection power strength and the frequency detuning between the master laser (ML) and the VCSEL, PS with or without injection locking of the slave laser to the master laser frequency has been reported experimentally when the injected light is orthogonal to the polarization of the free-running VCSEL.3,4 Furthermore, these experiments showed that bistable PS can be achieved if the injection power is scanned for a fixed frequency detuning. Bistability in semiconductor lasers has attracted much interest due to its potential application in area such as all-optical switching5,6 and signal regeneration.7 Pure-frequency PS bistability in a VCSEL with orthogonal optical injection has also been experimentally demonstrated by Hong et al.8 Here, the frequency detuning of the injected light was scanned for a fixed injection power level. Bistability is found to be strongly associated to the PS point that is located in the low frequency detuning range.

Semiconductor Lasers and Laser Dynamics II, edited by Daan Lenstra, Markus Pessa, Ian H. White,Proc. of SPIE Vol. 6184, 618411, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.662787

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In addition to bistability and injection locking phenomena, semiconductor lasers submitted to external optical injection can exhibit nonlinear dynamics. Nonlinear dynamics induced by optical injection in edge-emitting lasers (EEL) have been a subject of intensive theoretical as well as experimental studies in which periodic regime (limit cycle), wave mixing and period doubling route to chaos have been demonstrated.9-11 Similar nonlinear dynamics have been studied in a VCSEL injected by a light beam with a polarization parallel to the one of the free-running VCSEL.12,13 However, as it has been recently reported in the case of orthogonal optical injection, nonlinear dynamics are found to interplay with polarization switching processes.4 In this paper, we report on experimental study of intensity-induced polarization dynamics in the orthogonal injection configuration and show that PS may be accompanied by complex dynamics including period doubling route to chaos, wave mixing and subharmonic resonances. We also provide a theoretical investigation of optical injection-induced dynamics using continuation method. To this end, we carry out a mapping of different bifurcations that describe a qualitative change in the system dynamics and make a comparison with our experimental results. Finally, pure-frequency polarization bistability induced by orthogonal optical injection is studied theoretically. We complement the experimental results reported by Hong8 and show the existence of large bistable region in the range of positive detuning.

2. EXPERIMENTAL EVIDENCE OF DYNAMICS IN AN OPTICALLY INJECTED VCSEL

2.1 Experimental setup

Figure 1: Experimental setup of orthogonal optical injection in VCSEL. SL: slave laser, ML: master laser, COL: collimator, BS: beam splitter, HWP1-HWP2: half wave plate, ISO1, ISO2: optical isolators, L: lens, P1-P2: polarizers, M: mirror, FC: fiber coupling unit, OF: optical fiber, FP: Fabry-Pérot interferometer, D1: photodiode, Ampl: amplifier, PC: computer, OSA: optical spectrum analyzer, PM: power meter, CTR1 (CTR2): current driver and temperature controller of SL (ML). The VCSEL temperature was fixed at 20°C.

Our experimental investigations of polarization switching and dynamics induced by the variation of the intensity

of the injected power are conducted using the experimental setup shown in figure 1. We use a quantum-well VCSEL emitting around 845 nm as slave laser (SL). It is mounted on a Peltier and its temperature and bias current are controlled by a laser driver CTR1. The injected light is provided by a tunable external cavity diode which is used as a master laser (ML). The tuning range of the ML wavelength is 845 to 855 nm and is controlled by the laser drive CTR2 by acting on the temperature or bias current of ML or even through piezoelectric control. The light emitted by the SL is collimated by a lens (COL) while the injected beam from the ML is focused on SL using another lens (L). The strength of the injected beam is changed using a polarizer P1. A half-wave plate, HWP1, associated with a second polarizer P2 are used to fix the linearly polarized injected beam to be orthogonal to the polarization direction of the VCSEL. An isolator ISO1 with 36-40 dB of attenuation plays the role of preventing ML from instabilities that may be induced by its own feedback effect or from the SL beam and other reflections from the detection branch. A 50/50 beam-splitter and a mirror M are used to align the SL and the ML with the detection branch. A half-wave plate HWP2 allows the choice of the polarization direction in which the measurements are carried out. The second isolator ISO2 with 36-40 dB of attenuation prevents the VCSEL from the light reflected by the fiber-coupling device FC. Spectral measurements are conducted using an optical spectrum analyzer or Fabry-Pérot spectrometer associated with a photodetector D1 and amplifier (Ampl) coupled to a computer (PC). The power emitted by SL or ML is measured using a powermeter PM.

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2.2 Experimental evidence of polarization switching dynamics in VCSEL with orthogonal optical injection

Figure 2: Polarization-resolved optical spectra for ∆υ = 2 GHz and increasing the injection power Pinj: (a) Pinj = 13.2 µW; (b) 23.6 µW; (c) 36.6 µW ; (d) 47.8 µW; (e) 80.3 µW; (f) 136.6 µW ; (g) 204 µW; (h) 351 µW. The black (grey) line represents the vertical (horizontal) polarization. The vertical line shows the ML frequency.

In order to investigate the dynamics induced by orthogonal optical injection, we bias the VCSEL with a drive

current of 2.105 mA and the device temperature is fixed at 20°C. For this current, the free-running VCSEL emits in horizontal polarization and below the region of bistable polarization switching of the LI curve characteristic.4 In figure 2, we show an experimental result of complicated dynamics that accompanies polarization switching induced by optical injection. The frequency detuning between the ML and the SL is fixed at 2 GHz which is close to the frequency splitting between the two orthogonal polarized modes of the free-running VCSEL. Figure 2(a) corresponds to the injection locking regime, i.e., the VCSEL is locked to the ML frequency after a switch to the vertical polarization. As the injection power is increased, the injection locking is exited through a Hopf bifurcation that is detected by the appearance of a limit cycle dynamic [see the two peaks on both sides of the main peak at the ML frequency in figure 2(b)]. The two small peaks in figure 2 (b) are separated from the main peak by a frequency splitting close to the frequency of the undamping relaxation oscillations (RO) of the free-running VCSEL (The measured value of the RO of the free-running VCSEL is 3.8 GHz). If the injection power is increased further, harmonics of the limit cycle are excited as shown in figure 2 (c). By still increasing the injection power, the VCSEL undergoes a period doubling dynamics [Figure 2 (d)] leading to chaos which is characterized by the broadening of the VCSEL spectra with a large pedestal [Figure 2 (e)]. The presence of this chaotic attractor involves both orthogonal polarization modes, even if the vertically polarized mode is the dominating one. As it is shown in figure 2(f) and 2(g), for higher injection strengths, chaos is exited by a reverse period doubling bifurcation mechanism. For still strong injection levels, a second period doubling is again detected [Figure 2 (h)] which in turn evolves to a period one dynamics as we reach the maximum of the injection power.

In figure 3, we describe a scenario of the VCSEL dynamics for a detuning of 10 GHz. The injection power corresponding to figure 3 (a) is such that polarization switching is already achieved. The VCSEL emits in vertical polarization with a strong peak at the VCSEL frequency. Two side lobes are located on both sides of the VCSEL peak with a frequency splitting of 4 GHz which is close to the RO frequency. The strong excitation of the two side lobes that correspond to the RO undamping if found when the frequency splitting between ML and SL is close to the double of RO frequency. This type of dynamics is called subharmonic resonance and was also reported in edge-emitting lasers.14 The other peak, which is located in the lower frequency range, is a result of wave mixing dynamic. As the injection strength is increased, other subharmonic resonances are also resolved [see figure 3(b) and 3(c)]. However, it is worth noting that an increase in the injection power leads to an increase in the frequency offset between the RO undamping and the VCSEL frequency.15 For higher injection levels, the subharmonic resonance disappears and the VCSEL dynamics is dominated by a limit cycle, period one dynamics [Figure 3 (d)].

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Figure 3: Polarization-resolved optical spectra of the VCSEL subject to optical injection at a frequency detuning of 10 GHz: (a) Pinj = 29 µW; (b) 70 µW; (c) 251 µW ; (d) 800 µW. (a), (b) and (c) shows subharmonic resonances with wave mixing and (d) represent a limit cycle dynamics (period-one). Only vertical polarization is represented. The vertical line shows the ML frequency.

3. THEORY

3.1 Model for optical injection in VCSEL

A model for the solitary VCSEL has been developed by San Miguel et al.16 In this model, the spin sublevels of the conduction and valence bands of a quantum-well VCSEL have been considered in order to account for the polarization properties of the laser electric field. Using this model, optical injection can be modeled by adding the effect of the external field. The rate equations for a VCSEL operating in the fundamental mode and submitted to orthogonal optical injection are:

(1 )( ) ( )xx y x p x a x

dEi NE inE E i E E

dtκ α γ ω γ= + + − − + ∆ − (1)

(1 )( ) ( )yy x y p y a y inj inj

dEi NE inE E i E E E

dtκ α γ ω γ κ= + − − + − ∆ + + (2)

22 * *(1 ) ( )x y y x x ydN N E E in E E E Edt

γ µ⎡ ⎤= − + + − + −⎢ ⎥⎣ ⎦ (3)

22 * *( ) ( )s x y y x x ydn n n E E iN E E E Edt

γ γ ⎡ ⎤= − − + + −⎢ ⎥⎣ ⎦ (4)

Where Ex, Ey represent the slowly varying amplitudes of the orthogonal linearly polarized (LP) components of

the electric fields. The model considers two population inversions. N is the total population inversion between the conduction and the valence bands while n accounts for the population difference between carrier densities with opposite spins. Several parameters are considered: κ represent the decay rate of the electric field in the cavity, γ is the decay rate of the total carrier population inversion, and γs is a phenomenological modeling parameter that accounts for microscopic processes which lead to the mixing of carriers with opposite spin values.17 α is the linewidh enhancement factor also present in the modeling of edge-emitting lasers and µ is the normalized injection current (µ takes the value 1 at threshold). The effect of linear amplitude anisotropy that considers the fact that the two LP modes may have different gain-to-loss ratio is modeled through γa, while γp represents linear phase anisotropy or birefringence of the medium leading to frequency splitting between the two LP modes. The external optical injection is considered through κinj which is the optical injection rate, Einj which is the amplitude of the injected field and the frequency detuning ∆ω between master and the solitary VCSEL. The frequency detuning is defined as the difference between the master laser frequency ωinj and an intermediate frequency ωint which is an average between the frequencies of the two orthogonal LP modes

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corresponding to x-and y-polarization directions, i.e. 2/)(int yx ωωω += . Our numerical simulations have been conducted using the following choice for the VCSEL’s parameters: 50=sγ ns-1, 300== injκκ ns-1, 30=pγ rad.ns-1, 5.0=aγ ns-1,

3=α , 5.1=µ . These conditions lead to a free-running VCSEL emitting with a LP mode in the horizontal direction (x direction).

3.2 Nonlinear dynamics and bifurcation study of optically injected VCSEL

Figure 4: (Left) Mapping of different bifurcations showing qualitative change in the VCSEL dynamics in the injection parameter plane, i.e., frequency detuning vs. amplitude of the injected electric filed, Einj. Labeling: SN: Saddle-Node bifurcation; H: Hopf bifurcation; PD: Period Doubling bifurcation; and T: Torus bifurcation. The superscript are used to distinguish different bifurcations of the same type and the subscript x and y specify which polarization direction is involved in the corresponding bifurcation mechanism. The mapping does not report on stability of the different bifurcation boundaries. (Right) bifurcation diagram representing the maxima and minima of the intensities in the x-(black) and y-(gray) linearly polarized modes when the injected field is swept from 0 to 0.035 and for a detuning of 42 rad/ns. The VCSEL parameters are the same as presented in section 3.1.

In order to better understand the mechanisms that underlie nonlinear dynamics in VCSEL subject to orthogonal optical injection, we carry out a bifurcation study using continuation method. Using AUTO 97©18 continuation software package, we are able to detect and follow the bifurcation points that characterize qualitative changes in the system dynamics for both stable and unstable solutions. Figure 4, left-hand side, shows the mapping of bifurcation boundaries in the plane frequency detuning-injection field. SNy accounts for Saddle-Node bifurcation, and H1

y, H2y represent Hopf

bifurcations of the y linearly polarized (LP) mode. SNy and H1y delimit a zone of injection locking of the y polarized

VCSEL mode to the ML frequency. In the range of negative detuning, another injection-locking state that involves both x- and y-polarization modes is found. A Saddle-Node bifurcation SNxy and two Hopf bifurcation boundaries, H1

xy and H2

xy, are associated to this particular two-mode injection-locking state. A stability analysis and a detailed description of the different bifurcations has been recently reported.19 In addition to the bifurcation of the steady state injection locking solutions, we are able to identify fist-(1PDy), second-(2PDy) period doubling, and even a torus (2Ty) bifurcations boundaries. Period doubling bifurcations have been detected when the system is driven towards positive detuning from H1

y. In figure 4, right-hand side, we present a bifurcation diagram showing the evolution of the maxima and minima of the intensities of x- and y-LP modes (using numerical integration method). The magnitude of the injected field, Einj, is swept from 0 to 0.035 and the frequency detuning is fixed at 42 rad/ns. This allows exploring the zone delimited by the first period doubling bifurcation curve (1PDy). The VCSEL emits x-LP light in the free-running operation (Einj= 0). As the magnitude of the injection field is increased, the stationary state bifurcate to a wave mixing solution at around Einj= 0.009, which also coincide with the onset of the y LP mode. As the injection field is increased further, polarization switching is achieved at about Einj= 0.013, i.e. when 1PDy is crossed, the VCSEL undergoes period doubling dynamics route to chaos. Chaotic dynamics that involves both x- and y-LP modes is found for Einj ranging from about 0.016 to

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0.02. This numerical result is in agreement with our experimental result presented in section 2.2, figure 2, where PS accompanied by period doubling route to chaos has been reported. For much stronger injection strengths, chaos evolves to periodic solutions of high period. A periodic window, which corresponds to a regime with low period, is also found around Einj= 0.025.

Figure 5: (Left) Detailed mapping of different bifurcations showing qualitative change in the VCSEL dynamics that develop from the Elliptically Polarized Injection-Locking (EPIL) state in the injection parameter plane, i.e., frequency detuning vs. amplitude of the injected electric field, Einj. Labeling: SN: Saddle-Node bifurcation; H: Hopf bifurcation; PD: Period Doubling bifurcation; and T: Torus bifurcation. The superscript are used to distinguish different bifurcations of the same type and the subscript x and y specify which polarization direction is involved in the corresponding bifurcation mechanism. The mapping does not report on stability of the different bifurcation boundaries. (Right) bifurcation diagram representing the maxima and minima of the intensities in the x-(black) and y-(gray) linearly polarized modes when the injected field is swept from 0 to 0.06 and for a detuning of -28.72 rad/ns. The VCSEL parameters are the same as presented in section 3.1.

Figure 5, left-hand side, presents a close view on the Elliptically Polarized Injection-Locking (EPIL) state and

the associated dynamics. In this regime, both the x- and y-LP modes are locked to the master laser frequency. The EPIL, which corresponds to a thin zone located in the range of negative detuning, is delimited by a Saddle-Node bifurcation (SNxy) and a Hopf bifurcation (H1

xy). We unveil dynamics that underlie the EPIL state and show that this particular injection locking state is exited through period doubling dynamics from H1

xy. Bifurcation curves of those period doublings are denoted 1PDxy, 2PDxy, and 3PDxy for the fist, second and third period doubling respectively. We investigate also how polarization switching is affected by the simultaneous presence the EPIL regime, the associated period doublings, and the injection-locking state in y-LP mode that is delimited by SNy. PSon represents the PS boundary when the injection field is increased from the free-running regime (Einj= 0) until PS is achieved. Whereas PSoff represents the PS boundary when Einj is decreased after SNy has been crossed. We show in figure 5, right-hand side, a sample of the bifurcation scenario for a detuning of -28.72 rad/ns. The evolution of the maxima and minima of the x-(black) and y-(grey) LP modes is depicted when Einj is swept from 0 to 0.06. As the injection strength is increased to about Einj= 0.02, The EPIL regime is destabilized through a Hopf bifurcation (H1

xy) and a periodic solution is found. This periodic solution is maintained over a relatively large range of injection strength values until the first period doubling bifurcation point 1PDxy is crossed. Then, a period doubling to chaos is revealed over a small range of Einj as also shown in the mapping. Since this chaotic attractor is close to SNy, by slightly increasing the injection strength, PS accompanied by injection locking of the VCSEL to the ML is achieved. It is worth noting that when SNy is crossed, the x-LP mode is suppressed. In such a way, we unveil rich bifurcation picture that describes how the VCSEL evolves from a two-mode injection locking state (EPIL) to a y-LP injection locked state. Furthermore, by superposing PS boundaries that have been obtained by numerical simulations, we also show that the presence of a torus bifurcation may influence the PS process by causing discontinuities in the PS boundary.

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3.3 Pure frequency-induced bistability in VCSEL with orthogonal optical injection

Figure 6: Pure frequency polarization bistabilty induced by orthogonal optical injection. Solid (dotted) line represents the evolution of the averaged optical intensity when the frequency detuning is decreased from right to left (increased from left to right) and for fixed injection strength level. Pinj/P0 = 0.05 %: (a1) x-polarized mode, (a2) y-polarization direction; Pinj/P0 = 0.15 %: (b1) x-polarized mode, (b2) y-polarized mode; Pinj/P0 = 1.5 %: (c1) x-polarized mode, (c2) y-polarized mode. The injected power Pinj has been normalized to P0, the output intensity of the free-running VCSEL.

In figure 6, we present pure frequency-induced polarization bistability. The frequency detuning between master laser and the VCSEL is scanned for different levels of the injection power in the orthogonal optical injection configuration. The results have been obtained using the model and the VCSEL parameters presented in section 3.1. The intensities of the x-LP and the y-LP modes averaged in time (over the last 50 ns of a simulation time of 200 ns) are presented in figure 6. The frequency detuning is first scanned from the largest positive detuning to the largest negative detuning value (solid line) and then scanned in the reverse direction (dashed line). For low injection power level, Pinj/P0 = 0.05 %, bistability is associated only to the left-hand side PS point [see figure 6(a1) and 6(a2)]. In addition to polarization bistability, we also identify a narrow region of complex dynamic for which bistability is also found. By performing a bifurcation study around this zone, we have checked that it corresponds to the presence of a chaotic attractor located in between the two PS points. We have also found that this chaotic zone is achieved through period doubling cascade as the frequency detuning is scanned. For moderate injection strength (Pinj/P0 = 0.15 %), no such dynamics are present in between the two PS points. Moreover, as shown in figure 6(b1) and 6(b2) in the two polarization directions respectively, bistability is now associated to the two PS points. We also find that, for the latter injection level, the larger hysteresis is the one associated to the PS in the positive detuning range. For much higher injection strength [see figure 6(c1) and 6(c2)], the frequency splitting between the two PS points increases, polarization hysteresis keeps wider for the PS point located in the positive detuning range and that of the PS in negative detuning tends to shrink. This complements previous experimental results where large frequency-induced polarization bistability was associated to the left-hand side PS point.8 We also notice that, in between the two PS points, the negative slope of the averaged intensity the of y-LP mode increases with the injection strength as shown in figure 6(b2) and 6(c2).

4. CONLUSIONS

In conclusion, we have experimentally shown that, depending on the frequency detuning and the injection power strength, polarization switching induced in VCSEL by orthogonal optical injection may be accompanied by complicated dynamics. Around a detuning value of 2 GHz, which is close to the frequency splitting of the two orthogonal fundamental modes of the VCSEL, we have identified period doubling route to chaos when the injection power is increased. Above this region of complicated dynamics, and for a detuning of 10 GHz, we have found that the injected

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VCSEL may undergo subharmonic dynamics. By using continuation method and a model based on the spin flip approach that accounts for the VCSEL polarization properties, we have performed a bifurcation analysis and have mapped the VCSEL dynamics in the plane detuning vs. injection power. In the range of positive detuning, we have shown a period doubling bifurcation scenario that agrees qualitatively with our experimental results. Finally, we have presented a numerical result that shows the influence of the injection power on pure frequency polarization bistability. In addition to previous experimental result on bistability associated to the left hand side PS, our numerical show that, for strong injection power levels, large zone of hysteresis can be also associated to the right hand side PS when scanning the detuning from negative to positive detuning values.

ACKNOWLEDGEMENTS

This work has been supported by the IAP “Photon” Network of the Belgian government, as well as FWO-Flanders, GOA and OZR of the Vrije Universiteit Belgium. The authors also acknowledge the support of COST 288 European Action.

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