## Abstract

We present a reliable and fast technique to experimentally categorise the dynamical state of optically injected two mode and single mode lasers. Based on the experimentally obtained time-traces locked, unlocked and chaotic states are distinguished for varying injection strength and detuning. For the two mode laser, the resulting experimental stability diagram provides a map of the various single mode and two mode regimes and the transitions between them. This stability diagram is in strong agreement with the theoretical predictions from low-dimensional dynamical models for two mode lasers. We also apply our method to the single mode laser and retain the close agreement between theory and experiment.

© 2014 Optical Society of America

## 1. Introduction

The optically injected laser provides an experimentally accessible window into a dynamical system that produces many interesting phenomena, including locked and unlocked states, as well as periodic and chaotic behaviour [1–10]. The transition between states occurs through complex bifurcation scenarios which include Hopf, saddle node, period doubling and torus bifurcations. A comprehensive understanding of the dynamical features is an important step in the engineering of lasers for telecommunication applications. One of the most information-rich portrayals of a dynamical system is a bifurcation diagram. Obtaining an experimental stability diagram would allow direct and immediate comparison with the theoretical counterpart [11].

In the case of optically injected lasers, the natural bifurcation parameters are injection strength *K* and detuning Δ*ω*. The experimental acquisition of a two-dimensional stability diagram therefore involves characterising the optical output of the laser for varying *K* and Δ*ω*, which then allow us to determine the corresponding dynamical state of the laser [3, 12].

Here we demonstrate an experimental technique, which uses a fast (i.e. within 10*μs*) modulation of the injection strength *K* for fixed Δ*ω* and simultaneously record the time trace of the optical output. This allows for a characterisation of the dynamical state for varying injection strength with high accuracy. The procedure is then repeated for varying values of Δ*ω*. Therefore a complete sweep of the overall *K*-Δ*ω* plane is completed.

We demonstrate the impact of this method for Two Mode and Single Mode lasers. The resulting experimental stability diagrams show parameter regions of different dynamical behaviour in excellent agreement with theoretical rate equation models [2, 13, 14]. We present this paper in the following format; Section 2 explains the experimental technique, Section 3 describes the routines derived to characterise the dynamical behaviour of the laser, and to classify a state as on, off, locked, unlocked or some combination thereof. Section 4 then presents the complete experimental *K*-Δ*ω* stability diagram and discusses the accuracy of the experimental results with their numerical counterparts.

## 2. Experiment

Figure 1(a) depicts the setup for our experiment. The master laser is a low linewidth, single-mode, wavelength tunable laser and the slave laser is a Two Mode semiconductor laser [15,16]. The slave laser is temperature stabilised to within ±1mK and is operated at a normalised pump current
$P=\frac{J-{J}_{\mathit{Thr}}}{2{J}_{\mathit{Thr}}}=0.5$, which is twice threshold and corresponds to where both modes are observed to have equal intensity (two-colour point). *J* is the current supplied to the laser, and *J _{Thr}* is the threshold current of the laser. The spectral output of the free running slave laser is shown in the bottom panel of Fig. 1(b).

Our setup allows for the rapid and accurate data acquisition of optical time-traces as the injection strength is varied. The key step in this experiment is the amplitude modulation of the master laser at fixed detuning. The amplitude is modulated continuously at a max peak of 5.2 mW with a sinusoidal signal of 100kHz, much slower than the relaxation oscillation of the slave laser of around +3 GHz. By utilising a 90:10 beam splitter, 90% of the master laser output is injected into slave laser and the remaining 10% is recorded directly by the oscilloscope. *K* is taken to be the square-root of the amplitude of light injected into the slave laser.

During this modulation the time traces from the slave laser are recorded with a time step Δ*t* = 50ps. The resulting time traces are split into small chunks of length 25ns, which are then analysed using FFT. This time step corresponds to a Nyquist frequency of 10GHz and the time trace length yields a frequency resolution of 50MHz in the power spectra. The value for the bandwidth of the photodiode is 9.5GHz and for the digitizing oscilloscope it is 8 GHz. Electrical spectrum measurements [14] indicate that frequency peaks outside this frequency range are only due to harmonics of fundamental peaks in the detectable bandwidth. As the Nyquist frequency is larger than the bandwidth of the oscilloscope, aliasing effects are avoided.

After a time trace of the injected mode for fixed detuning is recorded, the detuning is increased by a discrete step of 0.1GHz and a new time trace is recorded at this frequency. By repeating this procedure, a total detuning interval from −8 GHz to +8 GHz is covered, which corresponds to the frequency limit of our detection setup. The detuning interval is centred around the long-wavelength primary mode of the dual mode laser and corresponds to a wavelength range of 1546.542nm to 1546.382nm. Finally, after recording all time traces for the injected mode, the injection is returned to −8 GHz where we confirm that amplitude and frequency locking still occurs (i.e. the experiment has not changed) and the time traces of the uninjected mode are recorded for the same detuning range.

Using the above operations, the experiment therefore covers a 500 by 160 grid in the *K*-Δ*ω* parameter space and delivers for every (*K*, Δ*ω*) combination a time trace of the intensities of both modes of 500 points at a sampling rate of 50ps. This output represents the basis for the analysis in the remainder of this paper.

## 3. Dynamical feature extraction

Having collected the data for the *K*-Δ*ω* parameter space, we now proceed to identify the dynamical features of the time traces. A sample of individual time traces for the injected mode is shown in the top row of Fig. 2 and we observe that the average intensity as well as the standard deviation of the intensity vary with the position in the parameter plane. The mean and standard deviation are easily obtained quantities, therefore we determine them for the full *K*-Δ*ω* plane.

This is shown in the top row plots of Fig. 3. The top left panel shows that the injected mode is lasing (red) for large regions of the parameter regime, and non-lasing for a blue region between 0 and 6GHz detuning at low *K* values and a second blue region below −5 GHz. By comparing with the top right hand panel of Fig. 3, we observe that the uninjected mode is complementary to the injected mode and is on in precisely those regions where the injected mode is off and vice versa. In addition we also observe regions where both modes have intermediate intensities (yellow, green and light blue regions). From the plot of the standard deviation of the uninjected mode shown in the bottom right panel of Fig. 3 it is evident that the dynamics of the uninjected mode is not constant in these regions of intermediate intensities suggesting complex dynamical Two Mode behaviour. The standard deviation of the injected mode shown in the bottom left hand panel is also large for large *K* values, where the uninjected mode is off. In this region we expect complex single mode dynamics. The precise value of the standard deviation will depend on the bandwidth of the oscilloscope, but is not critical for the classification of the dynamical states. We therefore observe that already the average intensities and standard deviations of both modes reveal nontrivial dynamical features which depend on injection strength and detuning.

Apart from the intensity average and the standard deviation, another easily accessible quantity is the number of separated peaks in the power spectra of the mode intensities. The power spectra for a number of time traces are shown in the bottom row of Fig. 2. At low *K* (left hand panel at *K* = 0.027) we see that the laser is unlocked, and the only (relatively weak) peak in the power spectrum is due to the beating between injected light and the free running laser. We use an intensity threshold as shown in Fig. 2 to distinguish strong peaks from possible noise. The threshold value for counting the peaks was chosen to be significantly higher than the unavoidable noise background and significantly lower than the maximum oscillation amplitude. The chosen threshold value corresponds to 2% of the maximum peak. Therefore transitions from sub-harmonic peaks are captured even if they are considerably weaker than the fundamental peak.

At *K* = 0.028 the injected mode locks and no peak is present in the lower spectrum. At higher *K* = 0.04 the laser shows undamped relaxation oscillations with a single peak appearing at 5.9GHz. For even higher *K* = 0.047 (right most panel), a number of peaks appear in the power spectrum due to the complex dynamics present in the time trace.

An overview of the number of peaks in the power spectrum of the injected mode at every point in the (*K*, Δ*ω*) plane is shown in Fig. 4. Here the dark blue regions have no peaks above a certain threshold. Most of these regions correspond to states where the injected mode is locked, except for states with a large standard deviations. All other colours correspond to unlocked behaviour. However, only considering the number of peaks present, does not always allow us to distinguish between Single Mode and Two Mode dynamics. For example, the dark blue region around (0.027,+4) is a locked state, where the uninjected mode is on (Two Mode locked state), while in the band shaped dark blue region around (0.043,−2) the uninjected mode is off (Single Mode locked state). In order to distinguish between Two Mode and Single Mode dynamics and to identify other dynamical features, we will now introduce a method, which combines the information from Fig. 3 and Fig. 4.

First we demonstrate our method for determining the dynamical states of the laser for the case of Δ*ω* = +2.7GHz. In the middle and lower panel of Fig. 5 the time traces of the injected and uninjected modes are shown during one upward sweep of *K* and the top panel shows a power spectrum density plot. For low *K* = 0.027 we observe a low intensity peak in the FFT, which is below our cut-off point for peak intensity. However, the standard deviation of the injected mode is non-zero as indicated by a non-zero standard deviation at *K* = 0.027. For this Two Mode unlocked state we adopt a light-grey colour as shown in the background of FFTs in Fig. 2 and the middle and lower panels of Fig. 5. For *K* between 0.028 and 0.036 the standard deviation in the injected mode decreases, but both modes remain on. This is a Two Mode locked state which we indicate with a light green colour. Then for *K* > 0.036 the state unlocks and the relaxation oscillation frequency appears in the power spectrum. This unlocked state is again indicated by a light grey colour. For higher *K* values between 0.046 and 0.049 additional peaks are picked up in the power spectrum, and we denote them with orange and red colours. At *K* = 0.047, both modes are on with a significant standard deviation, and contains three peaks in the power spectrum (see the lower right hand panel in Fig. 2). We identify this as a Two Mode chaotic state, and denote this point by a light orange colour. In the following, points with more than one peak in the power spectrum are coloured from yellow to red, for small and large standard deviation, respectively. This yields a useful indicator of chaotic behaviour. At even higher *K* > 0.05 the uninjected mode vanishes and we enter a regime of Single Mode dynamics, which is indicated by a dark shading.

## 4. Stability diagram

Applying the technique demonstrated in Fig. 5 for all Δ*ω* allows us to build a *K*-Δ*ω* stability diagram which identifies the dynamical state of the two colour laser at given parameter values. The result is presented in the top left panel of Fig. 6. We observe two regions of Two Mode locked states (light green), one at positive detuning (Δ*ω* between 0 and 6GHz) and one at negative detuning (Δ*ω* < −6GHz). This latter region of Two Mode locked states was partly identified previously in [13]. Both Two Mode locked states are bounded by regions of Two Mode dynamical states (light grey). The top Two Mode dynamical region shows a distinctive triangular shape. Two bubbles of Two Mode complex behaviour (light orange/ light red) extend at two of the sides of this triangle. At larger *K* values the dynamics becomes single mode (dark shading), including a prominent bubble of complex behaviour (dark orange and red) at the centre of the stability diagram. In the top right corner a further bubble of complex Single Mode dynamics (dark red) exists. We observe a large dark green band of Single Mode locked states below 1GHz and a large dark grey region of single mode unlocked states and there is a further region of Single Mode complex dynamics around Δ*ω* = −4GHz below the band of Single Mode locked states.

After generating the experimental stability diagram for a Two Mode laser at *P* = 0.5, our aim is now to put this result in a broader context, firstly by comparing with a theoretical model and secondly by comparing with a number of different experimental configurations. For the theoretical comparison we use the following dimensionless model for Two Mode lasers for the amplitude of the uninjected mode |*E*_{1}|, the complex slowly varying field for the injected mode *E*_{2} and the population inversion *n* [13]:

*g*

_{1/2}=[1 +

*ε*(|

*E*

_{1/2}|

^{2}+

*β*|

*E*

_{2/1}|

^{2})]

^{−1}. The parameters for the numerical simulations are

*α*= 2.6,

*P*= 0.5,

*T*= 800,

*β*= 2/3, and

*ε*= 0.01. The experimental parameters have been established by our previous work [13, 14]. The bifurcation parameters are the injection strength

*K*and the frequency detuning Δ

_{T}*ω*. As previously mentioned, the parameter

*K*in (2) and the experimental injection strength of the master laser

_{T}*K*cannot be quantitatively compared, because the experimental coupling efficiency is not known. Integrating this model numerically and calculating the Lyapunov exponents (LEs) [17] allows us to distinguish between fixed points (all LEs less than zero), limit cycles (one LE equals zero), tori (two LEs equal to zero) and chaotic states (one or more LE positive).

The theoretical stability diagram is shown in the upper right panel of Fig. 6. We find that this diagram mirrors astonishingly well the overall shape and location of both regions of Two Mode locked states (light green) identified in the experimental diagram shown in the top left panel of Fig. 6. In addition, the band of Single Mode locked states (dark green) agrees with the theoretically obtained Single Mode fixed points at negative detuning. The top row of Fig. 6 also shows that the regions of Two Mode (light grey) and Single Mode (dark grey) dynamics in the left upper panel are consistent with the stable limit cycles found theoretically in the upper right panel. The remaining dynamics in the theoretical model are the tori and chaotic states, which are identified with a blue and yellow/red colouration, respectively. Experimentally we did not attempt to distinguish between tori and chaotic behaviour, however, the complex dynamical regions in the upper left panel of Fig. 6 correspond strongly with the collective tori and chaotic regions in the central lower panel of Fig. 6. Unavoidably, we experimentally slightly overestimate the band of Single Mode locked states at low *K*. This is explained by the experimental difficulty of distinguishing between Single Mode locked and Single Mode unlocked states. This issue does not arise for Two Mode states, as our state identification routine can use the behaviour of the uninjected mode as a reference point.

Having established qualitative agreement for the Two Mode laser at *P* = 0.5, we now examine the Single Mode laser at *P* = 0.5 shown in the lower left and right hand columns of Fig. 6. By its nature, there are no Two Mode states present in the Single Mode laser and we now distinguish between locked (green), unlocked (grey) and complex (red/yellow) states. In the experimental stability diagram we observe a band of locked states and two prominent bubbles of complex dynamics. These features are in agreement with the corresponding theoretical stability diagram in the lower right panel. Again a slight overestimation of the band of locked states occurs at low *K*. One of the more striking results of the Two Mode/Single Mode laser comparison is the persistence of states identified as Single Mode states in the Two Mode laser, that remain Single Mode states in the Single Mode device. For example, all states in the Single Mode band of locked states in the Two Mode laser are still locked states in the Single Mode laser. However, states which are Two Mode in the Two Mode laser in general change their character in the Single Mode laser, for example a region of Two Mode unlocked states near the centre of the top central panel becomes part of the complex bubble in the top left panel. This behaviour is reproduced theoretically in the lower right panel, and is due to the presence of the invariant manifold |*E*_{1}| = 0 in the system of equations (1)–(3).

The final configuration explored via our technique is the Two Mode laser at a lower pump current *P* = 0.2. The respective experimental and theoretical stability diagrams are shown in the left and right hand panels of Fig. 7. Experimentally, we observe that all dynamical regions identified for *P* = 0.5 are still present at *P* = 0.2, however, the boundaries shrink to smaller *K* and Δ*ω* values. For example, the upper bound of the Two Mode locked state at positive detuning has decreased from 6GHz to 4GHz. In addition the distance between the Two Mode and Single Mode locked states at negative detuning becomes smaller and almost closes at Δ*ω* ≈ −6GHz. These experimental trends are in striking agreement with the theoretical results shown in the lower right panel of Fig. 6, where the same dramatic shrinking of the domain boundaries to smaller values of *K _{T}* and Δ

*ω*is observed.

## 5. Conclusion

In conclusion we have experimentally obtained a two-dimensional stability diagram of an optically injected Two Mode laser for varying injection strengths and detunings. Our method is based on a fast modulation of the injection strength and an analysis of the resulting time traces of the individual modes. The experimental stability diagram agrees quite well with the theoretical model, and thereby confirms the existence, shape and location of many dynamical phenomena. Locked and unlocked states, and complex behaviour have been clearly identified.

The agreement between theory and experiment is mirrored in our results for the Single Mode laser. This validates our technique as device independent and opens new opportunities for other optically injected dynamical systems.

## Acknowledgments

This work was supported by Irish Research Council (Grant No. RS/2012/514) and Science Foundation Ireland (Grant No. 09/SIRG/I1615).

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