The nonlinear dynamics of two semiconductor laser systems: (i) with optical feedback, and (ii) with optical feedback and direct current modulation are evaluated from multi-GHz-bandwidth output power time-series. Animations of compilations of the RF spectrum (from the FFT of the time-series) as a function of optical feedback level, injection current and modulation signal strength is demonstrated as a new tool to give insight into the dynamics. The results are contrasted with prior art and new observations include fine structure in the RF spectrum at low levels of optical feedback and non-stationary switching between periodic and chaotic dynamics for some sets of laser system parameters. Correlation dimension analysis successfully identifies periodic dynamics but most of the dynamical states are too complex to be extracted using standard algorithms.
©2010 Optical Society of America
Semiconductor lasers are mostly inherently stable devices when operated as an isolated device. However, it is well known that they can exhibit complex instabilities and chaos when additional perturbations, such as optical feedback and/or injection current modulation, are introduced .
In this study, experimental output power time-series were recorded from a semiconductor laser with optical feedback (SLWOF), both with and without direct injection current modulation. Sweeps were made of the parameters of interest: injection current, optical feedback level, modulation level and modulation frequency. The fast Fourier transform (FFT) of the highly-sampled (50 ps) output power time-series were used to generate high resolution images displaying compilations of the radio-frequency (RF) spectra. Sequences of these images over large parameter ranges have been generated to show how the system dynamics evolve. Two dimensional maps depicting the root-mean-square (RMS) amplitude of the time-series as a function of varying parameters were also constructed for the two systems studied. Short animations of these FFT images and amplitude map sequences based on fast experimental time-series are a new technique of observing fine detail in the evolution of the dynamics of these systems with changing parameters. Previous analysis has been based on direct observation of optical or RF spectra [2–6], rather than using high speed detectors and oscilloscopes to view the actual output power time-series. It was anticipated that high bandwidth detection and recording of the time-series over large parameter ranges would also allow the application of previously developed chaos characterization algorithms  to determine the level of complexity of the output signal. These algorithms were applied successfully to the output power time series of an optically injected solid-state laser system and dynamical maps of correlation dimension were reported .
Previous studies have shown it is possible to induce instabilities in semiconductor lasers in several ways and the dynamics displayed by laser diode can vary dramatically depending on the setup and system parameters chosen. Direct modulation of the injection current has been shown theoretically to result in period doubling [8–10] as the level of modulation is increased. Experimentally this type of dynamic has been observed [11–13] and has been shown to be a route to chaotic output power fluctuations . In these studies the interesting dynamics are observed for modulation frequencies comparable to the relaxation frequency of the laser.
Optical feedback also has a significant effect on the output of a semiconductor laser. Early studies of feedback effects on semiconductor lasers (with one low reflectance facet) identified five distinct regimes of operation in which the laser is either stabilized or destabilized depending on the strength and phase of the reflected optical field . Weak feedback can either narrow or broaden the laser emission line, depending on the length of the external cavity, whilst higher levels of feedback can result in a dramatic broadening of the linewidth, accompanied by unstable output power fluctuations. This region of operation is known as regime IV or coherence collapse . It was proposed that this region represented a form of deterministic chaos based on experimental optical spectra and numerical simulations [17,18] using the Lang-Kobayashi model , a fact later confirmed both theoretically and experimentally [3,20]. High levels of optical feedback, only achievable with a laser diode with an anti-reflection coating on one end facet , were shown to produce an output that is single mode with narrow linewidth and constant intensity. In this stable state the laser is said to be operating in regime V .
Previous studies have also considered combinations of direct injection current modulation with optical feedback  and also intra-cavity electro-optic modulation of the optical feedback . Direct modulation of the injection current of a semiconductor laser with strong optical feedback has been shown to display a variety of complex dynamical states including amplitude modulation (with frequency modulated spectra), low frequency fluctuations, mode-locking, quasi-periodicity and chaos .
More recent work on semiconductor laser instabilities due to optical feedback, both theoretical [23,24] and experimental , has shown the five ‘regimes of feedback’  are an overly simplistic view of a much more complicated system in which any number of parameters can have a significant effect on the dynamics. Dynamic maps are a more appropriate way to display the full range of dynamics that are possible from these systems. Such maps have been employed to show the different dynamical regions for a number of laser systems known to produce complex dynamics. These include maps of semiconductor lasers with varying injection current and optical feedback level [25,26] and optically injected semiconductor lasers with varying injection strength and frequency detuning [5,27–29]. Studies of a laser diode with optical feedback and either direct injection current modulation or electro-optically modulated feedback have used dynamic maps to illustrate different regions of operation for varying modulation power and modulation frequency . In all cases the maps have been produced manually by careful observation of the optical spectra (for the longitudinal mode spectral scale) and/or RF spectra (for the external cavity mode spectral scale) and noting the parameters where dynamical changes occur.
The process of mapping the dynamics of a complex nonlinear system would be much simpler and less tedious if a single measurand that quantifies the dynamics could be calculated from an experimental measurement. Automation of this process would also greatly improve the resolution of the maps by removing the need for an experimenter to manually inspect the system properties for every change in parameter settings. One such measurand, the largest Lyapunov exponent [30,31], which characterizes the sensitivity of a system to initial conditions (i.e. how chaotic the dynamics are) has been calculated from theoretical models on optically injected semiconductor lasers [32,33]. Another quantifier of the level of complexity, the correlation dimension , has been successfully used to map the experimentally recorded dynamics of an optically injected solid-state laser . The correlation dimension method of characterization requires large amounts of highly-sampled time-series data, a condition that can now be met with the availability of fast optical detectors and multi-GHz bandwidth real-time oscilloscopes with long time collection capability. Our previous research used the American Physical Society chaos data analysis software CDA Pro  to estimate the correlation dimension from time series for a SLWOF system . This gave estimates of correlation dimension up to 5. In the protocol now used for correlation dimension analysis  we require a scaling region to be clearly identified before extracting a correlation dimension estimate. This additional condition improves the reliability of dimension estimates and was not implemented when using the CDA Pro software. The result here-in, based on the more stringent protocol, has been that no reliable correlation dimension estimates have been made. Possible explanations for the failure to successfully extract correlation dimensions are given in the discussion.
Improvement in the characterization of these systems is motivated by applications for chaotic laser systems as transmitters and receivers in secure optical communication systems [37,38]. The security of these systems relies on the robust generation of a chaotic output signal. Identification of regions in the parameter space where the laser system is most chaotic and unstable is useful for ensuring the reliability of these communication systems.
The most commonly used theoretical model of a semiconductor laser with optical feedback is the Lang-Kobayashi (LK) model . The LK equations are limited to describing the effect of weak-to-moderate optical feedback on a laser diode, as they only take into account a single roundtrip of the light propagating in the external cavity. They also treat the laser as operating on a single longitudinal mode, whereas in most cases Fabry-Perot cavity lasers are multimode, especially when subject to optical feedback and modulation. Despite this, the LK model successfully describes qualitatively much of what is observed experimentally in SLWOF systems. Quantitative comparisons of experimental time-series and those generated from the LK equations, however, is not feasible due to the nature of chaotic systems, i.e. that similar initial conditions will diverge exponentially as the system evolves. Since many of the parameters for the LK model can only be approximately matched to the experimental system, it is not reasonable to expect quantitative agreement of the resulting time-series. There are other theoretical models available that do take into account multiple longitudinal modes and arbitrary levels of feedback [23,24,39,40], however, due to the complexity of SLWOF systems they are still not at a level at which quantitative comparisons of the resulting time-series can be made with experiments.
Section 2 describes experimental results from the SLWOF system with varying injection current and optical feedback level. Feedback dependent variations in the RF spectra at low optical feedback levels, and intermittent switching between periodic and chaotic dynamics are identified as new observations in this work. Correlation dimension analysis reveals that the chaotic dynamics produced by this system are very complex.
In Section 3, the direct injection current modulated SLWOF system is analyzed as the optical feedback level, modulation power and modulation frequency are varied. It will be shown that the system is capable of producing a variety of dynamical outputs. Correlation dimension analysis is used to produce a sequence of maps showing regions of periodic dynamics.
2. Semiconductor laser with optical feedback
2.1 Experimental setup
The laser used was a Fabry-Perot semiconductor device Access Pacific Model APL 830-40 emitting at 830 nm, as used in a number of studies of synchronized chaotic semiconductor laser systems [41–43]. The laser was driven by an ultra-low-noise current source (ILX-Lightwave LDT-3620) and temperature controlled to a precision of 0.01K (ILX Lightwave LDT-5412). Optical feedback was provided by a high reflectance mirror at a distance of approx. 30 cm. Roundtrip time of the cavity was measured as 2.05 ns from autocorrelation of the output power time-series, giving an external cavity frequency of f ext = 488 MHz. A variable attenuator was used to control the level of optical feedback. A beamsplitter was used to sample the laser beam with one beam sent to an optical power meter (Anritsu ML9001A) and another coupled to a fast, 12 GHz bandwidth photodetector (New Focus 1554-B). An optical isolator (Optics For Research IO-5-NIR-HP) was used to ensure no back reflection from the fiber coupled detector occurred. The setup can be seen in Fig. 1 .
The level of feedback was controlled by varying the optical density (O.D.) of the variable attenuator (New Focus 5215). This was varied between O.D. = 0.08 to 2.2 with 23 discrete
feedback levels. Alignment was optimized at each level by maximizing the average output power at threshold. This was to ensure the system had a common alignment for all attenuator settings. At each feedback level, the injection current to the laser was swept from just below threshold up to 40 mA in steps of 0.1 mA. The laser threshold decreases with increasing optical feedback so the starting point of the current sweep varied from 15 mA with maximum feedback, to 29 mA with minimum feedback, where the laser is essentially a free-running device. A 20 Giga-sample per second digital oscilloscope with 4 GHz bandwidth was used to capture the output power time-series at each injection current, for each feedback level, giving 3523 individual time-series to be analyzed. A local projective noise reduction process was performed on all of the captured time-series prior to analysis. This involved applying three iterations of the ghkss algorithm from the TISEAN package .
2.2 Results and discussion
A fast Fourier transform (FFT) was performed on each of the captured time-series from every injection current level. These were then used as slices to make up a color coded image of the RF spectra (up to 5 GHz) as a function of injection current. One of these images was generated for each feedback level and the sequence of images is displayed in the animation Media 1. It shows the injection current and optical feedback level at which the laser becomes unstable. A sweep from low to high levels of optical feedback initially reveals some low power, low frequency features in the RF spectra images. Increasing feedback firstly enhances these frequency features, whilst pulling them towards multiples of the external cavity frequency, and then causes temporal instabilities (coherence collapse). At high levels of optical feedback the region of instability is reduced to a smaller range of injection currents.
Some key frames from this animation depicting the RF spectra for feedback attenuation levels O.D. = 2.2, 1.9, 1.2 and 0.7 are shown in Figs. 2(a) -2(d) (Media 1) respectively. These frames were chosen as being representative of the different operational states of the laser system. The type of operation seen in Fig. 2(a) (Media 1) suddenly transitions to that seen in Fig. 2(b) (Media 1), between feedback attenuation levels O.D. 2.1 and 2.0. This type of operation then changes suddenly to that seen in Fig. 2(c) (Media 1) as the feedback attenuation is dropped from O.D. 1.8 to 1.7. Further decrease in attenuation from O.D. = 1.7 to 0.7 causes a gradual change from the type of dynamic seen in Fig. 2(c) (Media 1) to that in Fig. 2(d) (Media 1). For feedback attenuation levels of less than 0.7, stable CW laser operation is observed and the associated frames are without frequency features.
The injection current at each feedback level was swept from just below threshold up to 40 mA. The start of each current sweep can be identified in Fig. 2 (Media 1) by the boundary of the dark blue (−80 dB), sub-threshold region. The lasing threshold can be seen to decrease as the optical feedback level increases (attenuation decreases).
At minimum optical feedback (maximum feedback attenuation O.D. = 2.2) the laser operates similar to a free-running laser diode. The spectrum in Fig. 2(a) (Media 1) shows a very broad relaxation oscillation frequency (f RO) envelope, indicated by the curve overlayed on Fig. 3 (the additional frequency features within the RO envelope will be discussed shortly). The centre frequency of the RO peak is observed to shift upwards to a maximum of 1.3 GHz at 40 mA, scaling as the square root of the injection current minus the threshold current as expected [45,46]. Closer inspection of the FFT images for feedback attenuation O.D. = 2.2 and 2.1 reveals structure within the boundary of the broad relaxation oscillation peak.
The FFT image for low feedback (O.D. = 2.2) has been rescaled to highlight five readily visible peaks that are seen within the RO envelope (Fig. 3). Three of these peaks, labeled 1, 2 and 3, are at 400, 800 and 1200 MHz respectively. The external cavity frequency is 488 MHz. The relative amplitudes of these three peaks vary with the injection current. At low injection currents (less than 33 mA) the first frequency peak (centre frequency ~400 MHz) has the highest amplitude. For currents 33 mA to 37 mA the second peak (centre frequency ~800 MHz) is largest and for injections above 37 mA the third peak (centre frequency ~1.2 GHz) is dominant. It would seem that as the relaxation oscillation frequency increases with injection current it becomes resonant with each of the frequencies in turn.
These features show up as very small amplitude (RMS amplitude ~3% of the DC level) periodic modulation of the laser output power. The origin of the frequencies to which the relaxations become resonant is likely an oscillation associated with the cables, connectors and/or packaging, similar to the parasitic frequency identified in another semiconductor laser device . In this case, the optical feedback is amplifying this ‘noise’ frequency of the laser, since running the laser as an isolated device (no optical feedback) results in a FFT image that shows just the RO envelope, without the peaks seen in Fig. 3 which are masked by the broad spectrum noise. This shows that the noise of the laser may affect the nonlinear dynamics produced by the SLWOF system in ways that have not previously been proposed. These specific frequencies, characteristic of the device and its packaging, introduce additional frequencies to be drivers of nonlinear dynamics in addition to the relaxation oscillation frequency and external cavity frequency; another complication which is not accounted for in the current theoretical models.
Increasing the level of optical feedback (decreasing the level of feedback attenuation from O.D. = 2.1 to 2.0) caused these small peaks within the RO envelope to increase by about 20 dB and also causes them to be pulled toward the external cavity frequency. Higher harmonics of these frequency peaks were also observed at higher injection currents. For feedback attenuation levels O.D. = 2.0, 1.9 and 1.8, similar RF spectra frames were observed in which sharp peaks occurred for most injection currents above threshold (see Fig. 2(b) (Media 1) as an example). Again, the amplitude of the peaks was enhanced by the near coincidence with the relaxation oscillation frequency.
Looking at the time domain within this feedback region, the laser output displayed low amplitude power fluctuations. Switching between stable, quasi-periodic and chaotic dynamics was observed. In some cases the switching between periodic and chaotic output power variations occurred even when the injection current was kept constant, as seen in Fig. 4 . The dominant frequency seen in the periodic section is 440 MHz corresponding to the first peak seen in Fig. 2(b) (Media 1).
After observing this type of switching experimentally, theoretical simulations generated by the Lang-Kobayashi equations  for a semiconductor laser with low levels of optical feedback have been completed and parameter values which generate this type of behavior have been identified. A sample time-series showing this switching behavior in noise-free simulations from this model is shown in Fig. 5 .
These regions where the output power switches between unpredictable chaotic and predictable periodic signals need to be avoided if the laser system is to be implemented as a chaotic transmitter in a secure communication scheme. Such non-stationary dynamics also compromise robust chaos data analysis, such as the computing of the correlation dimension. This type of dynamic was frequently observed over a range of injection currents at low optical feedback levels (O.D. = 1.8 – 2.0). We believe it is the first time it has been reported.
Further increase in the feedback level (attenuation to O.D. = 1.7) again caused a sudden change in the operational state of the system as it enters what appears to be coherence collapse. The sharp peaks in the frequency spectrum seen at lower feedback levels became much broader for nearly all injection currents above threshold (Fig. 2(c) (Media 1)). For feedback attenuation levels from O.D. = 1.7 to 1.1 a gradual change in the dynamics was observed, as the frequency peaks become slightly broader and continue to trend towards the external cavity frequency and its higher harmonics with the increasing optical feedback. The laser output power time-series in this region appear quite chaotic with a very broad 20 dB enhancement over the background noise. A typical time-series and corresponding frequency spectrum, for this region is shown in Fig. 6 .
Increasing the optical feedback still further (attenuation level to O.D. = 1.0) the reduction of the laser threshold current becomes noticeable, as seen in Fig. 7(a) , which shows the averaged output power of the laser as a function of injection current and optical feedback level. This is accompanied by a significant increase in RMS amplitude of the chaotic power fluctuations shown in Fig. 7(b). These larger amplitude instabilities take the form of low-frequency fluctuations (LFF) and can be identified by the increase in power of frequency components less than 400 MHz in the RF spectra images. The RF spectral peaks increase in amplitude and begin to narrow. Significant power (greater than −45 dB) also appears at frequency components corresponding to higher harmonics of the external cavity frequency at this level of optical feedback.
The amplitude of the output power instabilities continued to increase, as the feedback level was increased. However, the band of injection currents over which the large amplitude instabilities occurred within the range measured was reduced as seen in the RMS amplitude map in Fig. 7(b). This can also be identified in the map in Fig. 7(a) as a drop of ~13% in average output power when the instabilities occur. For injection currents just above threshold the laser operated in stable CW mode and only became unstable at the higher injection currents. The current at which the laser became unstable increased with increasing optical feedback. The map in Fig. 2(d) (Media 1) corresponds to feedback attenuation O.D. = 0.7 and shows the laser in stable CW operation for all injection currents except for the small band from 36.5 mA to 39.5 mA, where it became unstable.
A typical output power time-series and frequency spectrum from a high optical feedback region (attenuation O.D. = 0.7) are shown in Fig. 8 . The LFF can be seen in the time series as the irregular, large amplitude power jumps modulated by much higher frequency fluctuations. The FFT reveals significant power (−30 dB) at multiples of the external cavity frequency beyond the nominal bandwidth limit of the oscilloscope (4 GHz).
For the highest optical feedback levels (attenuation less than O.D. = 0.7) the output power is practically constant as the laser operates CW over the whole injection current range. The laser threshold continues to decrease as the optical feedback increases but no instabilities were observed. This region is typical of an external cavity laser under strong optical feedback operating in Regime V .
2.3 Correlation dimension analysis
In the fields of chaos and nonlinear dynamics fractal dimensions are often used to quantify the level of complexity displayed by a chaotic system. The correlation dimension is one type of fractal dimension generally used to quantify experimental time-series data since the algorithm (developed by Grassberger and Procaccia ) is suited to making dimension estimates from time-series data of a single system variable, such as laser output power. In prior work the output power fluctuations of a semiconductor laser with optical feedback was determined to have a correlation dimension between 2 and 3 . However it is possible this value was an underestimate caused by temporal correlations and scaling region identification issues. A much more reliable technique, which dealt with these issues, was established in . The algorithm itself, all the necessary considerations/protocols for its use, as well as the previously developed automated system for generating maps from thousands of experimental time-series, are explained in detail in .
The previous research showed the successful application of correlation dimension analysis using data from an optically injected Nd:YVO4 solid-state laser  (which also exhibits complex output power fluctuations under certain operational conditions). The dimension maps generated using data from the SLWOF system in this study did not contain enough dimension estimates to convey any meaningful information about the dynamics over the parameter range studied. When a dimension estimate cannot be obtained by the algorithm it means either there is too much noise in the data, or the complexity of the signal is such that the value of the correlation dimension is too large for the algorithm to measure. Our results appear to indicate the latter is the case here. Firstly, no distinction can be made in terms of correlation dimension, between time-series with large RMS amplitude, and therefore a large signal-to-noise ratio, and those with lower amplitude. If it was noise that was compromising the analysis then we would expect to generate some dimension estimates for data sets with high signal-to-noise, however, this is not the case. Secondly, when the correlation dimension analysis is applied to the noise free, simulated chaotic data from the Lang-Kobayashi equations again no estimate is obtained. This implies that the dynamics of the semiconductor laser with optical feedback are very complex with a correlation dimension higher than is measurable with standard algorithms. In the prior work with the optically injected solid-state laser, the complex power fluctuations were on a much slower time-scale (maximum frequency features ~30 MHz) compared to the current work with a semiconductor laser (several GHz). It is possible under-sampling of the much higher frequency output power dynamics of the semiconductor laser may mean that some deterministic structure is not captured in the time-series which would also contribute to a correlation dimension estimate being unattainable.
In the following section on the directly modulated SLWOF system, we successfully map out regions of periodicity using the correlation dimension algorithm. The reason such a map was not obtained for the un-modulated system is that when periodic dynamics do occur, generally at low optical feedback levels, they are of such low amplitude that the signal-to-noise is not sufficient for the algorithm to correctly identify a limit cycle with correlation dimension equal to 1.
3. Directly modulated semiconductor laser with optical feedback
3.1 Experimental setup
The experimental setup was identical to that shown in Fig. 1, except that an AC component was added to the DC injection current to the laser. A signal generator provided a sinusoidal modulation of the 30.75 mA DC injection current via a bias-T. In this experiment the modulation frequency (f mod) was swept from 480 MHz to 580 MHz in steps of 1 MHz and the modulation level was varied between −4 dBm and + 6 dBm in steps of 1 dBm. The external cavity round trip time was 2 ns, giving an external cavity frequency of f ext = 500 MHz. The optical feedback level was controlled by varying the optical density of the attenuator from O.D. = 0.08 to 1.2. The maximum optical feedback attenuation was reduced to O.D. = 1.2 in this experiment due to data collection limitations. The range of feedback levels was selected to highlight the most interesting dynamics. The dynamics for lower levels of optical feedback (O.D. up to 2.2) were not significantly different to those seen at O.D. = 1.2. A total of 14 300 output power time-series were captured at 20 Giga-samples per second for analysis. Once again, all time series were processed with the local projective noise reduction algorithm ghkss from the TISEAN package .
3.2 Results and discussion
Fast Fourier transforms were again generated from each experimental time-series. The resulting RF spectra as a function of modulation frequency for every feedback level and modulation level are depicted in the color coded image sequence in Media 2. Key frames from this animation sequence (Media 2) are shown in Fig. 9 and Fig. 10 . The animation allows identification of the parameter settings that cause unstable laser output. The animation also shows there is competition between the external cavity frequency and the driving modulation frequency as to which dominates the dynamics.
As with using optical feedback alone, modulating the injection current of the semiconductor laser with optical feedback also caused chaotic fluctuations of the output power over the range of parameters used. With the lowest level of optical feedback (O.D. = 1.2) and low power modulation (−4 dBm), the laser showed unstable operation for all modulation frequencies, showing LFF and broad peaks at multiples of f ext (see Fig. 10(a) (Media 2)). Under the same feedback conditions without injection current modulation, the system operated with stable CW output.
Increasing the power of the modulation up to + 6 dBm caused the LFF to disappear and reduced the effect of the external cavity on the dynamics as the maps showed standard modulation spectra for all modulation frequencies where the peaks in the RF spectra occur at multiples of f mod (see Fig. 10(b) (Media 2)). Under this strong modulation the RF images also show intermittent suppression on the FM spectra for some values of f mod.
Experimental output power time-series, phase space diagrams and frequency spectra from sections of the maps in Fig. 10 (Media 2) corresponding to f mod = 500 MHz are shown in Fig. 11 . The phase space diagrams are constructed by plotting the output power time-series against the same time-series delayed by 2 data points (100 ps). For low modulation power (−4 dBm) the laser output shows regular power spikes with frequency f mod but with unpredictable amplitudes (Fig. 11(a-c)). Increasing the modulation level to + 6 dBm (Fig. 11(d-f)) appeared to stabilize the amplitude of the periodically spiking output power and the RF spectrum showed much sharper peaks and significantly more power at higher multiples of f mod.
Increasing the optical feedback (O.D. = 0.9 – 0.6) whilst keeping the modulation level low (−4 dBm) stabilized the laser output over an increasing range of modulation frequencies, see f mod = 480 – 520 MHz in Fig. 9(a) (Media 2) for example. In these stable regions, the laser output power was simply modulated at the driving frequency f mod. By increasing the modulation level, the size of the stable region was reduced, as in Fig. 9(b) (Media 2) where the range of modulation frequencies that give a stable output can be seen to have shrunk to f mod = 498 – 508 MHz and another small band at f mod = 482 – 486 MHz. The increase in modulation power also resulted in the output power time-series displaying frequency components at higher harmonics of f mod. A further increase of the modulation level to + 6 dBm produced a combination of external cavity and modulation frequency peaks to appear in the FFT images (see Fig. 9(c) (Media 2)).
As the optical feedback level was increased further, the external cavity begins to dominate the observed dynamics. Higher modulation levels were required to disturb the laser from stable regime V operation and induce instabilities. At the highest optical feedback level (O.D. = 0.08), the laser output power was stable, modulating at the driving frequency, over the whole frequency range up to a modulation level of + 5 dBm, at which point a region of instability occurs at a f mod ≈530 MHz (see Fig. 9(d) (Media 2)). It is unclear why the laser destabilizes around this particular modulation frequency. Output power time-series showing the stable and unstable regions for this strong optical feedback region are shown in Fig. 12 .
Typical laser operation with strong optical feedback is shown in Fig. 12(a-c) where a sinusoidal output power modulation occurs for all f mod except for modulation levels greater than + 5 dBm and modulation frequencies surrounding 530 MHz, where the output becomes chaotic as seen in Fig. 12(d-f). The chaotic dynamics seen here are similar to that seen in the un-modulated SLWOF system with moderate optical feedback and high injection current (Fig. 8). The slightly sharper external cavity peaks in the directly modulated SLWOF system imply that the chaos produced by the un-modulated system is somewhat more complex.
With the exception of very low feedback levels O.D. > 0.9 and low power modulation (less than + 2 dBm), where power spikes with fluctuating amplitudes occur, temporal dynamics appear mostly stable over the range of parameters used for values of f mod ≈f ext. This is in agreement with previous work on a SLWOF system which also identified this region to be the most temporally stable under direct injection current . This is despite the APL laser used in this experiment having more technical noise than the laser used in .
Maps of the RMS amplitude of the output power time-series fluctuations for different feedback levels are shown as a sequence in Media 3. They reveal additional information that, when combined with the RF spectra sequences, provide a more detailed picture of the laser dynamics. The largest amplitude fluctuations correspond to conditions where the laser output power is unstable. Some key frames from the map sequence are shown in Fig. 13 .
For low feedback levels (Fig. 13(a) (Media 3)) the RMS amplitude is largest for f mod close to f ext = 500 MHz. As optical feedback was increased the frequency at which the largest amplitude oscillations occur shifts towards f mod ≈530 MHz, corresponding to the frequency where unstable dynamics first appear under increased modulation power (Fig. 13(d) (Media 3)).
The same correlation dimension algorithm used for the SLWOF data was also applied to the directly modulated SLWOF data. Again, no dimension estimates were able to be generated for the unstable chaotic regions, indicating once again that the dynamics are very complex and beyond the analysis capabilities of the correlation dimension algorithm. However, the periodic regions where the output was modulated at f mod showed up on the maps as correlation dimension equal to 1, correctly representing a limit cycle in the phase space. A sequence of maps showing the regions where periodic behavior can be seen in the second map in Media 3. A sample frame is shown in Fig. 14 , the periodic regions are blue compared to the black unstable regions.
The map sequence in Media 3 shows that for low levels of optical feedback, the periodic regions are confined to the diagonal of the map from low modulation level (−4 dBm) and low f mod (480 MHz) to high modulation level ( + 6 dBm) and high f mod (580 MHz). As optical feedback is increased, the periodic regions spread to eventually dominate most of the map for strong optical feedback (O.D. = 1.2).
Experimental output power time-series measurements have been used to create high density sequences of RF spectrum and RMS amplitude maps. These have been utilized to identify regions of instability displayed by semiconductor lasers with optical feedback, both with and without direct injection current modulation.
In the case of the SLWOF with no direct modulation, previously unseen dynamics were observed in the RF spectrum for very low levels of optical feedback. As the feedback level is increased, these small peaks within a broad relaxation oscillation envelope increase in amplitude and higher harmonics appear in the RF spectrum. Further increase in feedback causes the peaks to broaden and shift upwards in frequency towards the external cavity frequency and its harmonics. Even stronger levels of feedback result in the appearance of large amplitude, low frequency fluctuations as well as significant power at higher frequency multiples of the external cavity, indicative of the coherence collapse regime. Along with a reduction in the threshold injection current, very high levels of feedback have a stabilizing effect on the output power of the laser over the whole range of injection currents investigated.
Switching between chaotic and periodic dynamics was observed in experimental time series even when there was no change in operating parameters. This is supported by the evidence of switching in simulations based on Lang-Kobayashi equations for a semiconductor laser with optical feedback. Applications where robust chaotic output is required, such as in secure communication systems, need to take care to avoid these regions.
Direct modulation of the injection current introduces another degree of freedom to the system and, as expected, instabilities were observed. For low feedback levels the output power can be destabilized by low power injection modulation over the whole range of modulation frequencies. Increased modulation power generates a spiky, periodic output with the dominant frequency at f mod but with features at higher harmonics of f mod.
Moderate feedback levels result in the RF spectra showing broad peaks at the external cavity frequency and higher harmonics, except for modulation at frequencies close to f ext, where the output power remains modulated at f mod. Increasing the power of the injection modulation reduces the amount of detuning of f mod from f ext required to destabilize the laser.
Strong optical feedback allows the laser to display stable modulation at f mod, except under large modulation powers, where the output becomes comparable with the chaotic output seen in the SLWOF system with strong optical feedback and high injection currents.
One of the benefits of recording the output power time-series from these systems is the possibility that the complexity can be quantified using techniques based on nonlinear chaos analysis. The level of complexity within the unstable regions of operation of the both SLWOF and directly-modulated-SLWOF systems is shown to be greater than is measurable with standard correlation dimension algorithm , based on analysis of experimental time-series with different signal-to-noise ratios and simulated time-series from the LK model. The analysis did allow the identification of regions of periodicity within the parameter range used. Future work to research algorithms that can measure correlation dimensions greater than that achievable with the standard algorithms may allow the very complex data here-in to be reanalyzed. Applying the methodology to a semiconductor laser with optical injection, a less complex system than the SLWOF system, will be valuable.
The observation that resonant noise peaks of the semiconductor laser can be enhanced by optical feedback and proximity to the relaxation oscillation frequency and/or the external cavity frequency warrants further investigation of the effect on the nonlinear dynamics.
This research was supported by the Australian Research Council. Experimental work was performed at the School of Electronic Engineering, Bangor University and J. P. Toomey would like to thank his hosts during this time, M. W. Lee and K. A. Shore. J. P. Toomey is supported by an Australian Postgraduate Award PhD scholarship.
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