## Abstract

Experiments and numerical modelling on two different class B lasers that are subjected to external optical light injection are presented. This presentation includes ways of measuring the changes in the laser output, how to numerically describe the systems and how to construct diagrams of the dynamical states in the plane frequency detuning between lasers and injection strength. The scenarios for the semiconductor laser include an area of frequency locking and islands of chaotic behaviour embedded in and mixed with periodic doubling regimes. Using a rate equation model, the largest Lyapunov exponent is calculated as a measure of the stability of equilibriums and the amount of chaos in chaotic regimes. In the solid-state laser case, different dynamical regions were clearly observed. The found boundaries were identified experimentally, using an identification method, and numerically, from bifurcation analysis, as Hopf, saddle-node, period-doubling and torus bifurcations.

## 1. Introduction

Lasers can be classified in many different ways. One straightforward way is based upon the material they are made of. According to such a classification, gas lasers, solid-state lasers, dye lasers and semiconductor lasers, etc. exist. In another classification, the lasers are named as class A, class B and class C lasers depending on how the three relevant variables, polarization, population inversion and electric field, decay in the laser resonator. In this study, we are dealing with class B lasers in which the polarization relaxes faster than the two other variables, and, therefore, two equations are needed to describe the system: one equation for the electric field and the other for the population inversion.

In the case of a single mode laser, the optical spectra show one fixed optical frequency. There is, however, an interplay between the photons in the cavity and population inversion. This is seen as a regular variation in time of the laser output power, and the variation is termed intensity relaxation oscillation. Depending on the lifetimes of the cavity and the involved energy levels for lasing, this variation can be quiet (gas lasers) or more pronounced (semiconductor lasers and solid-state lasers). Also the frequency of the intensity relaxation oscillation of the output varies for different lasers. For semiconductor lasers, the intensity relaxation oscillation frequency is approximately 1 or several gigahertz, whereas a solid-state laser has its relaxation oscillation at a much lower frequency, in the kilohertz or megahertz region. But from a dynamical viewpoint, a free-running laser with only a regular variation of the output power is a bit boring, and usually the laser system is disturbed in order to enrich the behaviour. The disturbance can be performed in different ways, e.g. modulation of the pump, optical feedback from a mirror or, as is done in this case, to direct light from another laser into the laser of interest. Now the dimensions have been increased by one, resulting in a system that exhibits a wealth of different dynamical scenarios.

The specific lasers of interest are of two kinds: an edge-emitting semiconductor laser at 635 nm and a Nd : YVO_{4} solid-state laser, which emits at 1064 nm. For both lasers, the coupling is unidirectional. Both lasers have a non-zero value for the line-width enhancement factor *α*. The *α*-factor is defined through the relation between the variation of the refractive index and gain with respect to the inverse population. For a symmetric gain profile and for a laser at peak gain, the factor is zero. But as soon as the gain profile is asymmetric, and/or the laser is detuned from resonance, the factor is non-zero. Typical values in the case of semiconductor lasers range from 2 to 7, whereas it is small for a solid-state laser. For the present Nd : YVO_{4} laser, *α* is approximately 0.20 (Fordell *et al.* 2005). The value of the *α*-factor is of importance because the kind of dynamics that a certain laser exhibits is tightly connected to this value (Wieczorek *et al.* 1999).

## 2. Measurements and modelling

The experimental set-up with the most important elements is schematically shown in figure 1. The two lasers are unidirectionally coupled, i.e. light from the master laser (ML) is directed into the slave laser (SL) via an optical isolator (OI) that prevents light from propagating back to the ML. An acousto-optic modulator (AOM) serves as a variable beam splitter, and thus it is used to vary the amount of light injected to the SL. The procedure for a fixed frequency detuning between the master and the SL is to continuously record the light output of the SL with a suitable detection device (D) as the amount of light from the ML is varied. The detection of the light of the SL is performed in two ways: (i) as optical spectra using a Fabry–Pérot interferometer or (ii) as a time-series with a detector coupled to a fast oscilloscope. The procedure is then repeated for several different frequency detunings. The decision on which detection scheme to use is mainly based upon how fast the changes are in the SL output under optical injection. Now the intensity relaxation oscillation frequency comes into play: this frequency sets a lower limit for the time-scale of the observed changes. If this frequency is higher than the detection apparatus can handle in recording time-series, one needs to record optical spectra with the Fabry–Pérot interferometer instead.

The earlier measurements (Eriksson & Lindberg 2001, 2002; Eriksson 2002) were performed on a commercial edge-emitting semiconductor laser (Hitachi HL6313G). That laser had its intensity relaxation oscillations from 1 to 2 GHz, making the recording of time-series of the output power difficult to perform. Hence, the measurements needed to rely upon optical and intensity noise spectra. A typical example of continuously recorded optical spectra placed after each other, while the injection strength is ramped from zero up to a maximum value, is shown in figure 2*a*. In the optical spectra, red corresponds to high intensity whereas blue corresponds to no or little intensity. The frequency for the optical spectra is chosen so that the injection from the ML is at the origin of the *y*-axis. For low injection strengths, i.e. to the left of the picture, the ML oscillates first at its free-running oscillation frequency before it jumps to another longitudinal mode. As the injection strength is further increased, the SL picks up the oscillation frequency of the ML. In other words, frequency locking occurs as indicated in the figure. Thereafter, the light power of the SL is spread out over a large frequency range, and no clear peaks can be seen in the spectra and the laser is in a chaotic regime. However, a careful look reveals tiny peaks in a narrow injection range inside the chaotic regime. In other words, a periodic window appears inside the chaotic regime. For larger injection strengths, the spectra show a periodic structure.

The latest measurements (Valling *et al*. 2005*a*,*b*, 2007) have been performed on a Nd : YVO_{4} laser. For the laser, with length 1 mm and with output power approximately 7–8 mW, the characteristic intensity relaxation oscillation ranges from 3 to 4 MHz. This frequency is low enough to enable time-series measurements of the output power as the injection strength is continuously changed. An example of recorded time-series is seen in the lower part of figure 2. In this case, the injection strength is first ramped up to its maximum value and then down again to zero. During the ramp, the time-series of the SL output is recorded. With such a ramping, it is possible to look for possible hysteresis effects. As a result, the waveforms of the SL for each frequency detuning and for many injection strengths are recorded in one long time trace. In the example shown in the lower part of figure 2, frequency locking occurs at *t*≈0.7 ms. After that, the injection strength is ramped down.

The rate equation model of an injected class B laser (Simpson *et al.* 1994; Fordell & Lindberg 2004) is given by(2.1)(2.2)where *E* is the amplitude of the slowly varying field envelope and *N* is the carrier density. In addition, *α* is the line-width enhancement factor; *γ*_{c} is the decay rate of the cavity; *ω*_{0} is the angular frequency of the optical field; *ω*_{c} is the cold cavity angular resonance field; and *Ω* is the angular frequency detuning between the master and the SL. Moreover, *F* is the Langevin noise source term, and the coupling of the external field *E*_{i} from the ML is given by the coupling rate *η*. The spatial overlap of the optical field and the gain *g* is given by *Γ*, *J* is the current density into the active region of thickness *d*, *R*(*N*) is the carrier recombination rate, *n* is the refractive index and *ϵ*_{0} is the permittivity of vacuum. In the experiments, it is possible to control the coupling of the external field *E*_{i} into the SL, i.e. the injection strength and the frequency detuning *Ω* between the master and SLs. The model can be used on both lasers, if only the appropriate parameter values are used. Most of the parameter values in the equation have been determined experimentally and, as a result, the amount of free parameters is kept at a minimum.

## 3. Results

One aim is to get an overall picture of the dynamics in the optically injected lasers studied. The findings are collected in diagrams showing the different dynamical states in the parameter plane spanned by injection strength and frequency detuning between lasers. In addition, suitable measures are calculated using equations (2.1) and (2.2) in order to be able to compare the experimental diagrams with their numerical versions.

For the semiconductor laser, measurements such as those presented in figure 2*a* were performed for many different detunings. By inspection of such recordings, the coordinates for the three main dynamical behaviours, i.e. frequency locking, chaotic behaviour and periodic behaviour, were picked up. The resulting experimental diagram (Eriksson 2002) is shown in figure 3*a*. In addition to the three main types of output, the experimental diagram included regions where the laser jumps to other modes, or where it is unlocked or where there are regions with frequency pulling. If the largest Lyapunov exponent is computed (Fordell & Lindberg 2004) using equations (2.1) and (2.2) and plotted in a similar diagram, one gets the result presented in figure 3*b*. A negative exponent indicates stable locking (blue colour), zero indicates periodic or aperiodic solutions (turquoise colour) and a positive value (yellow–red colour) indicates chaotic dynamics. By comparing the experiments in figure 3*a* with the calculated Lyapunov exponents in figure 3*b*, the overall agreement between the model and experimental results is good.

Even if the continuously recorded time-series of the output power from the injected Nd : YVO_{4} laser (see figure 2*b*) shows the waveforms, it is not easy to draw detailed conclusions about how the injected light changes the output of the SL. The data need to be presented in a more informative way than as a long time trace for each frequency detuning. A first glance at the time traces indicates that the traces between the free running situation at zero injection strength and the frequency-locking situation constitute smooth sinusoidal oscillations and peaks with varying amplitudes. It is therefore straightforward to just pick the maximum amplitude value of the oscillation or the peak for a certain injection strength. When all the maximum amplitude values are picked, they can be plotted in a diagram spanned by the injection strength and the detuning between the lasers. The results are shown in figure 4*a*. A large amplitude value corresponds to red whereas a small value corresponds to blue. The maxima are normalized with the free running laser power, i.e. the value one (dark blue) in the diagram means that the laser has the same output as in the free running case. The highest peaks are about 12 times higher than the free running laser power. The diagram consists of different regions. One region is the large dark blue triangular-shaped region that opens up as the injection strength is increased. In this region, the SL frequency locks the ML, so it is the frequency-locking region. For negative detuning in the yellow–red region, which is below the frequency-locking region, peaks of different amplitudes can be seen in the output power. Also smaller crescent-shaped regions, where the output consists of peaks, can be seen above and below the frequency-locking region for small injection strengths.

A similar maximum amplitude diagram can be calculated using the equations (2.1) and (2.2). The result is shown in figure 4*b* plotted with the same linear colour coding as the experimental diagram. The agreement between the measured and calculated maximum amplitude diagrams is good. Hence, the rate equation-based model works very well for the optically injected Nd : YVO_{4} laser, and the model can therefore be used to accurately describe the system numerically.

In nonlinear dynamics, a change in a system, as one parameter is varied, can be classified as a bifurcation. So, an immediate question arises: when there is a change in the maximum amplitude of the output power, can the change be classified as a bifurcation? And would it be possible to construct an experimental bifurcation diagram from the measured time traces that shows the curves where the bifurcation takes place in the parameter plane frequency detuning between lasers versus injection strength?

Bifurcation analysis is an elegant tool for analysis of differential equations to identify changes in the dynamics of a system. In the analysis, mathematical theory together with computational methods is used to track how the steady-state solutions change as one or more parameters are varied. Inspired by the good agreement between the experimental and calculated maximum amplitude diagrams in figure 4*a*,*b*, we performed a bifurcation analysis of the equations (2.1) and (2.2) (Valling *et al.* 2005*b*, 2007) using the software package Matcont (Dhooge *et al.* 2003). The results are shown in figure 4*d*, where the curves are identified as Hopf (H), saddle-node (SN), period-doubling (PD) and torus (T) bifurcations. If this calculated bifurcation diagram is compared with the maximum amplitude diagrams, one observes that there are places for this system where a change in the maximum amplitude coincides with the position of a bifurcation curve.

Then, how does one generate an experimental bifurcation diagram starting from the measured time-series of the output power? One possible method (Valling *et al.* 2007) is based upon different indicators that are determined from the experimental data. When setting up and using the indicators, it turned out that in most cases it was enough to work with the time-series of local maxima instead of working with all measured points. The next step was to specify criteria, as general as possible, for each of the possible bifurcations by a combination of the indicators. The indicators used were: the characteristic frequency, or the period, associated with the maxima, the standard deviation of the amplitude between the maxima, the length of the largest empty interval between maxima and finally a measure of the difference between histograms of all points of neighbouring observation windows. Between two and four criteria together with a few threshold values were defined for identifying each of the different bifurcations. Applying these to the experimental data results in the experimental bifurcation diagram shown in figure 4*c*. With this method, it is possible to detect SN, Hopf (H), PD and torus (T) bifurcations. Comparison with the bifurcation diagram of the rate equation model in figure 4*d* shows excellent agreement for SN, Hopf and PD bifurcations, and torus bifurcations are detected reliably.

## 4. Summary and outlook

The nonlinear response and the precisely adjustable control parameters of optically injected lasers are the reasons behind the suitability of these laser systems in exploring dynamical scenarios. We have looked at two different dynamical scenarios in optically injected lasers: one for an edge-emitting semiconductor laser and another for a solid-state Nd : YVO_{4} laser. We started with tedious data analysis by interpretation by the experimenter of optical spectra to construct diagrams of the dynamics for semiconductor lasers and ended up with a method based upon a direct data analysis of recorded time-series of the output power in order to get the bifurcation diagram. The method uses criteria as general as possible, and therefore the method can be applied to other laser systems as well. The input for the method was long time-series data that were recorded as one parameter was swept controllably while the others remained stable enough throughout a sweep.

Both lasers could successfully be described by a rate equation model for single mode optically injected class B lasers. The agreement between measured and calculated results was good. Most of the parameter values could be determined experimentally with satisfactory accuracy, so, when applying the model, the amount of free parameters were kept at a minimum.

In the future, it would be challenging to extend the ideas of defining criteria for bifurcations on the basis of spectral data. This would also make it possible to generate bifurcation diagrams for semiconductor lasers for which continuously recorded optical spectra have been measured. It would be interesting to take a closer look at how the quantum dot laser behaves when subjected to optical injection. A driving force for such studies is the observed huge variation of the line-width enhancement factor: from very low values up to large values around 60 as the pumping parameter is changed (Dagens *et al.* 2005). It would therefore be possible to study complex dynamics for a parameter range that is by far larger than what has been explored previously.

## Acknowledgments

The authors are grateful for the early measurements by S. Eriksson on the semiconductor lasers and the valuable input of B. Krauskopf concerning bifurcation analysis. The work was supported by the Academy of Finland Project 77582, Magnus Ehrnrooth Foundation, The Finnish Academy of Science and Letters, Finska Vetenskaps-societeten and Svenska Kulturfonden.

## Footnotes

One contribution of 15 to a Theme Issue ‘Experimental chaos I’.

- © 2007 The Royal Society