Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection

Transcription

1 Optics Communications 215 (2003) Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection T.B. Simpson * Jaycor/Titan, P.O. Box 85154, San Diego, CA , USA Received 13August 2002; received in revised form 11 November 2002; accepted 12 November 2002 Abstract Single-mode semiconductor lasers can exhibit stable, bistable, periodic, quasiperiodic, and chaotic output characteristics when subjected to monochromatic, near-resonant external optical injection. Experimental measurements of the spectral characteristics of a distributed feedback laser diode under external optical injection identify the various nonlinear output characteristics. Single-mode operation is maintained in all cases. The output characteristics are mapped as a function of the strength of the optical injection and the detuning between the injection frequency and the free-running frequency of the unperturbed laser. A comparison of the observed characteristics with the predictions of a coupled rate-equation model shows quantitative agreement over a wide range of injection characteristics. The model shows that the mode hopping previously observed in Fabry Perot laser diodes subject to external optical injection can be explained using a simple coupling of the longitudinal modes through the free carriers of the gain medium. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: Sf; Px; Pq Keywords: Semiconductor lasers with optical injection; Nonlinear dynamics in semiconductor lasers 1. Introduction Lasers made with a semiconductor gain medium are among the most important and widely used today. Their importance lies not only in their many applications, from telecommunications and data/ information processing to environmental sensors * Tel.: ; fax: address: tsimpson@jaycor.com and laser pumping, but also in their usefulness as a system for understanding the fundamental characteristics of lasers. The semiconductor laser is a nonlinear system where quantitative comparisons can be made between experiment and theory. It is well established that a variety of external perturbations such as optical injection from an external source, modulation of the pumping current, external optical feedback from a distant reflector, and electro-optic feedback from a distant photodetector that is added to the bias current, can cause the /02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 136 T.B. Simpson / Optics Communications 215 (2003) output of the laser to become unstable and yield oscillatory or chaotic output [1]. Of the perturbation mechanisms, external optical injection allows the greatest degree of control and most direct access to the fundamental mechanisms that lead to the instability. With external feedback, one is limited to perturbing the laser with its own output. Current modulation and electro-optic feedback can be limited by the fact that semiconductor lasers are circuit elements with inherent capacitance, inductance and resistance. Additional parasitic electronic effects arise when the gain medium is connected into an electronic circuit. The determination of the mechanisms controlling laser performance can involve a deconvolving of the circuit effects with the underlying processes within the laser cavity and gain medium. External optical injection allows one to perturb the laser with an independent optical field. The optical injection bypasses the electronic connections and yields direct information about the laser cavity and gain medium. In addition external optical injection has been proposed and demonstrated as a means to enhance the bandwidth and reduce the noise and distortion characteristics of semiconductor lasers [2], generate tunable, microwave modulation of an optical carrier through double-locked optical output [3], and synchronize chaotic dynamics for possible use in communication systems [4]. Past experimental work on the semiconductor laser subject to external optical injection has established that a range of optical outputs can be produced. For a laser operating at a given bias current, external optical injection can induce stable locked output, multiwave mixing, oscillatory power output due to an undamping of the carrierfield resonance of the system, and chaotic dynamics depending on the amplitude and offset frequency of the external optical field [5]. It has been found that a relatively simple model describing the coupling between the external and circulating optical fields and the free carriers of the gain medium can reproduce the characteristics observed experimentally [6]. Because the key dynamic parameters of the model can be determined experimentally, a quantitative comparison can be made between experimental data and model calculations [7]. Recently, a full comparison of experimental observations and model calculations has been made where excellent agreement between experimental data and model calculations was observed [8]. Previous mappings of the experimentally observed characteristics have been limited by the observation of mode hops to new longitudinal modes [9]. These mode hops tended to occur in regions where the model predicts a rich array of nonlinear dynamics in the laser output. Further, new theoretical tools have been used recently to extract many of the details of the dynamics of the semiconductor laser under optical injection [10]. Equally detailed and precise experimental tools are needed to quantitatively compare experiment and theory. Here, we present experimental details behind the mapping of the nonlinear dynamics of a distributed feedback (DFB) laser diode that maintains single mode operation under external optical injection. The mapping is generated through the analysis of the optical, regenerative amplification, and power spectra of the laser subject to external optical injection. This paper emphasizes the experimental techniques used to generate the mapping and compares the results with calculations based on coupled equations with experimentally determined dynamic parameters. We have refined and augmented past techniques used to generate the spectra and to determine the key dynamical parameters of the coupled equation model. We observe very good quantitative agreement between observed and calculated operating characteristics. This paper is organized as follows. Following this introductory section, the experimental apparatus is described. A section describing the observed spectra, and the new, more complete mapping of the dynamics observed then follows. The next section reviews the model of a nearly single-mode semiconductor laser subject to external optical injection, highlighting some new features. The characterization of the laser using weak current modulation and external optical injection for determining the model parameters is described. This allows a quantitative comparison between the model and the experimental data. The results of model calculations are summarized and show the good agreement with experimentally observed data. Also included are numerical results that

3 T.B. Simpson / Optics Communications 215 (2003) point to a mechanism for the mode hop observed in past experiments on Fabry Perot lasers. We conclude with a brief summary of key results. 2. Experimental apparatus Fig. 1 is a schematic of the experimental apparatus used to perform these measurements. Three DFB lasers under independent temperature and current control are used. All are temperature and current tuned to oscillate at approximately the same wavelength, 1557 nm, and all are coupled to single-mode optical fibers. The laser under investigation, the slave laser, is an Alcatel 1915 LMO DFB laser package that contains no internal optical isolator. The master laser, used as the optical injection source to induce the nonlinear dynamics, and the probe laser, used as a weak optical probe of the slave laser and/or as a local oscillator for heterodyne measurements of the optical output of the slave laser, are both Alcatel 1915 LMI DFB lasers that are packaged with internal optical isolators. Both probe and master lasers are operated at current levels much farther from threshold than the slave laser so that, under the operating conditions used here, their noise characteristics can be ignored and they can be considered to be essentially monochromatic sources. The optical power in all the side modes of the free-running slave laser was below 1% (experimental uncertainty) of the total optical power. Further, under optical injection of the principal mode, the optical power in the side modes never deviated by more than 10% (again, experimental uncertainty) from its freerunning value. Therefore, this laser is effectively a single-mode laser for modeling purposes. For the measurements of the nonlinear dynamics, the laser was operated under dc-current bias. However, a high speed current modulation from a frequency synthesizer was added to the dc bias for determining key laser parameters used in the modeling. This will be discussed further below. The apparatus is fiber-coupled, except for an acousto-optic modulator that is used to generate a replica of the probe output that is frequency shifted by 80 MHz. It was necessary to use angled physical contact (APC) type connectors for any connector not isolated from a laser emitter. The )3 0 to )40 db backscatter from the standard physical contact (PC) connectors produced unacceptable perturbation of the laser output. The master laser is coupled into the slave laser using an optical circulator after having passed through a variable attenuator and a polarization rotator. The latter is used to align the optical polarization of the master with that of the slave. The frequency-shifted replica of the probe laser can also be injected into the slave laser after being coupled into the master laser fiber path with a fiber-optic coupler. Fig. 1. Schematic of the experimental apparatus. Thin solid lines denote connections via single-mode optical fiber while the thick solid lines around the acousto-optic (A/O) modulator denote free-space propagation. The dashed line is a microwave coaxial connection. Arrows denote connections after optical isolation so that propagation is blocked in the opposite direction. The optical circulator permits coupling only from port 1 to port 2 and from port 2 to port 3. All three laser are under independent bias current and temperature control (not shown). A modulation current could also be added to the dc bias of the slave laser.

4 138 T.B. Simpson / Optics Communications 215 (2003) The output of the slave laser is coupled out of the optical circulator where it can be combined with the main output of the probe laser. A fast photodiode is used to detect the optical signal and its output is directed to a microwave spectrum analyzer. Three types of spectra are generated with this apparatus. If the probe laser is turned off, then the detected signal is a measure of the amplitude modulation of the slave laser and the spectrum analyzer measures the amplitude (power) spectrum. If the probe laser is on, but the frequency shifted output of the acousto-optic modulator is blocked, then the probe laser acts as a local oscillator for a heterodyne measurement of the optical spectrum of the slave laser. To generate a spectrum, we fix the frequency of the microwave spectrum analyzer at some convenient frequency, typically 33 MHz, and then ramp the bias current to the probe laser. Ramping the bias current to the probe laser shifts its optical frequency. A lowresolution optical spectrum is generated as the probe laser sweeps across the frequency components of the slave laser output. For these measurements, a spectral resolution of 100 MHz is sufficient, indeed finer resolution was generally not possible due to fluctuations in the optical frequencies of the three lasers. Finally, the frequencyshifted output of the probe laser could be injected simultaneously into the slave laser. If the spectrum analyzer is set to match the 80 MHz frequency offset with a very narrow resolution bandwidth, typically 10 khz as compared to 3MHz with the other measurements, then a spectrum of the regenerative amplification of the probe beam in the slave laser is generated. These three spectra yield complementary information about the output characteristics of the slave laser, both free-running and under external optical injection. 3. DFB laser nonlinear dynamics The fiber-coupled configuration proved to be robust and stable and we were able to make a detailed mapping of the nonlinear dynamics as a function of the detuning and amplitude of the injected optical field. The mapping is made by relating a particular output dynamics to the spectral features displayed at a given injection level and offset frequency [9]. In the optical and regenerative amplification spectra presented here, offset frequency is relative to the free-running optical frequency of the slave laser and signal power as a function of offset frequency is shown on a logarithmic scale. Most of the spectra show data from two scans overlaid. Because there is some jitter during a scan, the spectral features do not precisely overlap. For the results presented here, the slave laser was operated at twice the threshold current where the relaxation resonance frequency of the free-running laser is approximately 4.7 GHz. More details of the operating parameters in relation to the coupled equation model are given below. Fig. 2 shows the transition from free-running output to limit cycle dynamics as the injected signal is increased when the master laser is tuned to the free-running optical frequency of the slave laser. Optical spectra are shown in the left column and regenerative amplification spectra are shown in the right column. Both the free-running optical and regenerative amplification spectra show the weak, broad features associated with the relaxation resonance frequency. At relatively low injection levels, the laser becomes locked to the master signal and the weak probe beam no longer is strongly amplified at the formerly free-running, now locked, optical frequency. As the injected optical signal is increased, the damping on this resonance decreases and the sidebands become stronger until the Hopf bifurcation is passed and the laser output becomes unstable. Note that the sideband components grow in magnitude and sharpen in linewidth, with no pedestal, as the injection level is increased. Because there were no abrupt spectral changes to mark the Hopf bifurcation, criteria had to be developed to define a change in dynamics. We observed that in the regenerative amplification spectra a narrow dip and peak structure developed at the master laser injection frequency. It is clearly visible at the highest injection level and is barely showing at the next lower injection level. The peak and dip reach a maximum in between these two injection levels. Recall that the regenerative amplification spectrum measures the small signal optical gain as a

5 T.B. Simpson / Optics Communications 215 (2003) Fig. 2. Spectra with injection at the free-running optical frequency showing the transition from stable to limit cycle dynamics. The left column contains the optical spectra and the right column consists of the regenerative amplification spectra. The Injection Field is proportional to the square root of the injected power. The free-running regenerative amplification spectrum has been corrected for background noise and the variation of local oscillator power during the scan so that it can be compared to the model calculation, shown as a thin line.

6 140 T.B. Simpson / Optics Communications 215 (2003) function of offset frequency. A dip and peak around an injection feature indicates that the perturbing probe is having a strong effect on the spectral feature locked to the strong signal from the master laser. Because it is occurring as the optical spectrum is showing a change from weakly damped relaxation resonances to simultaneous oscillation at several optical frequencies, we interpreted the maximum in the peak and dip as an indicator of the Hopf bifurcation point. Associated with this transition were sidebands approximately db below the resonant, locked feature in the optical spectrum. Therefore, we used the criterion that a spectral feature exhibited in the optical spectra had to be no more than 20 db below the strongest spectral feature for its associated dynamics to be the characteristic dynamics of the system. This choice is made to have a consistent quantitative measure relating the changes in the observed spectra to the dynamics. This criterion can be used to analyze the dynamics associated with more complex spectra. Fig. 3shows optical and amplitude spectra in the left and right columns, respectively, at increasing optical injection levels when the master laser is offset by 2 GHz. At the weakest injection levels, the optical spectrum shows side peaks due to regenerative amplification of the master laser signal and four-wave and multiwave mixing between the master laser and unlocked slave laser frequencies, as shown in the Injection Field ¼ 0.14 spectra. The multiwave mixing becomes more pronounced as the injection level is increased and mixing peaks appear as sharp spectral features in the amplitude modulation spectra. Simultaneously, there is enhancement of the broad features associated with the relaxation resonance frequency of the freerunning laser, particularly in the amplitude spectra where it quickly becomes the dominant spectral component, as shown in the Injection Field ¼ 0.33 spectra. The dominance of the multiwave mixing components in the optical spectra and their relative weakness in the amplitude spectra show that these features arise primarily from phase modulations of the laser field, consistent with Adler-type frequency pulling towards locking [11]. However, here the frequency pulling is accompanied by a destabilization of the relaxation resonance. This is not the case for an offset frequency of )2 GHz where the enhancement of the relaxation resonance features is considerably less and a transition to stable, locked operation is observed. As the injection level is further increased, the multiwave mixing features are pulled to the injection frequency and the features associated with the relaxation resonance sharpen and strengthen. In the locked output, shown in the Injection Field ¼ 0.41 spectra, the frequency-pulled multiwave mixing components have disappeared and there are narrow, strong spectral components with pedestals at the relaxation resonance/oscillation frequency. The spectrum shows a laser that is only marginally stable, or marginally unstable. As the injection level is further increased these pedestals broaden to show weak spectral features around the strongest spectral peaks. These new features are clearly evident in the Injection Field ¼ 0.52 spectra and indicate that another incommensurate frequency is present in the dynamics. This is strong evidence for the existence of quasiperiodic dynamics that have been predicted from numerical calculations [12]. As the injection power is further increased, the weak features about each strong spectral peak become relatively stronger, broader and shift in offset as the injection power is increased Eventually, the broad continuum grows until only a few spectral features are evident. This spectrum indicates that a transition to chaotic dynamics has probably occurred. Past calculations have shown that spectra like those at the Injection Field ¼ 0.77 are associated with chaotic dynamics [13]. Therefore, the laser has made a transition to chaotic dynamics through a quasiperiodic route, as opposed to the period doubling route [13], or abrupt saddle-node bifurcation [8], previously observed. The amplitude spectrum also shows some evidence for period-three dynamics as well. Other spectra with significant broadening also show different periodic motion. The spectra at an Injection Field ¼ 1.02 show broadened period-two features. At still higher injection levels, the laser makes a transition back to strong, narrow spectral features as shown in the Injection Field ¼ 1.3spectra. Again, the narrow features have a pedestal, the remnants of the higher-order periodic features

7 T.B. Simpson / Optics Communications 215 (2003) Fig. 3. Spectra with injection at +2 GHz relative to the free-running optical frequency showing a range of output characteristics. The left column contains the optical spectra and the right column consists of the amplitude (power) spectra. The amplitude spectra signal is relative to the background noise level. The Injection Field is proportional to the square root of the injected power.

8 142 T.B. Simpson / Optics Communications 215 (2003) Fig. 3. (continued)

9 T.B. Simpson / Optics Communications 215 (2003) from the region of more complex dynamics. Further increase in the injected power eventually yields a simple limit cycle spectra, dominated by two strong optical field components, as shown in the Injection Field ¼ Note that now the characteristic frequency of the limit cycle has increased well above the free-running relaxation resonance frequency. At these high injection levels the resonance frequency is a function of the injection level and offset frequency of the master laser. At still higher injection levels, the laser can be expected to reestablish stable, locked operation. However, we feared that we might damage the laser and did not attempt to observe this transition at this offset frequency. We did observe it at offset frequencies between )4 and 1.5 GHz. The clear observation of spectra consistent with quasiperiodic motion in the spectral characteristics of Fig. 3confirms theoretical predictions and numerical calculations [12]. This dynamics occurs under optical injection conditions where the previously studied Fabry Perot lasers were induced to hop to a different longitudinal mode [9]. As we will show below in the numerical calculations, the quasiperiodic dynamics occurs with a slight decrease in the average power level. This allows one of the other longitudinal modes of a conventional edge-emitting, Fabry Perot laser to become the dominant mode. With a DFB laser, like the one used here, the loss for each of the side modes is sufficiently enhanced so that none achieve significant power. Therefore, only single-mode output is observed. We have combined spectra such as those in Figs. 2 and 3to produce a mapping of the observed dynamics. While spectra that show stable or periodic dynamics can be interpreted in a straightforward manner, the quasiperiodic and chaotic spectra are more difficult to distinguish. The presence of spontaneous emission noise and, potentially, multiple competing attractors means that broadening is not enough to distinguish between periodic motion on multiple attractors, quasiperiodic motion with broadening due to noise and true chaotic dynamics. Therefore, sophisticated modeling tools are required to determine the details of the dynamics. Here, we will not attempt to resolve dynamics of the complex spectra that show that stable or periodic dynamics no longer occurs. Results are given in a companion paper [14]. Fig. 4 is a mapping of the observed dynamics based on interpretation of spectra like those described above. The abscissa of the map is proportional to the square root of the injected optical power, the injected field amplitude, and the ordinate is the offset frequency of the master laser with respect to the free-running frequency of the slave laser. Boundaries of the regions of stable and periodic dynamics are shown. Diamonds mark the saddle-node boundary between stable locked and unlocked dynamics, squares mark the Hopf bifurcation boundary between stable locked and limit cycle dynamics, triangles mark the boundaries of regions of period two dynamics and circles mark the boundary of period four dynamics. The three Fig. 4. Mapping of the observed dynamics based on the experimentally measured spectra of the DFB laser under optical injection when biased at twice the threshold current. The injection field is proportional to the square root of the injected optical power. The full diamonds mark the saddle node bifurcation between stable and unlocked locked operation while the open diamonds mark the unlocking locking transition in a region of bistability (see text). Squares mark the Hopf bifurcation between stable and limit cycle operation. The triangles bound regions of period two operation. Within the period two regions are regions of complex dynamics marked by the shaded lines and crosses and a region of period four operation that is bounded by the circles. At injection levels below the saddle node bifurcation line and at low offset frequencies, multiwave mixing and Adler-type frequency pulling to locking are observed in the lightly shaded regions.

10 144 T.B. Simpson / Optics Communications 215 (2003) regions with shaded lines mark the areas where more complex dynamics are observed. The two positive offset regions were observed in previous mappings of the dynamics of a Fabry Perot laser diode under external optical injection. However, a significant amount of optical power was transferred to other longitudinal modes in the past measurements, and mode hopping obscured the negative offset region. With the DFB laser, the power, >99%, remained in the principal mode. There is also a region of bistability between locked and unlocked dynamics that is observed at negative offsets and relatively high injection levels. This bistable region has been observed previously with DFB lasers [15], and explained as being due to a crossing of the saddle node bifurcation by a torus bifurcation [16]. The open diamonds mark the transition from unlocked to locked output characteristics at the torus bifurcation while the full diamonds marking the saddle-node bifurcation show the locking to unlocking transition. The mapping displays all the regions of stable, bistable, periodic, quasiperiodic and chaotic dynamics. Unlike previous measurements, there was no mode hop or optical power transfer to side modes under optical injection that obscured the single-mode dynamics. 4. Coupled equation model The basic set of coupled equations for the complex optical field, A, of a single-mode semiconductor laser that is oscillating at free-running optical frequency x 0, subject to an injected optical field, A i, at optical frequency x 1, and is coupled to a carrier density, N, can be written as da h dt ¼ c i c 2 þ iðx 0 x c Þ A þ C gðn; jajþa 2 þ ga i exp½ iðx 1 x 0 ÞtŠ; ð1þ dn dt ¼ J ed c sn 2e 0n 2 gðn; jajþjaj 2 ; ð2þ hx 0 gðn; jajþ Re½gðN; jajþš: ð3þ Here, c c is the photon decay rate from the laser cavity, x c is the cold cavity resonance frequency, C is the confinement factor that relates the overlap of the oscillating mode with the gain medium, g is the injection coupling rate, J =ed is the carrier density injection rate due to the injection current density J, c s is the carrier spontaneous decay rate, and e 0 n 2 is the non-resonant (real) dielectric constant of the semiconductor gain medium. When free-running, the laser oscillates at an optical frequency x 0 with a steady-state field amplitude A 0 and a steady-state carrier density N 0. The complex gain, gðn; jajþ, is assumed to vary about the free-running, steadystate operating point as a linear function of the carrier density and circulating intensity: gðn; jajþ ¼ ð1 ibþgðn 0 ; A 0 Þ þð1 ibþ og ðn N on 0 Þ N0 ;A 0 þð1 ib 0 Þ og ðjaj 2 A 2 ojaj 2 0 Þ: ð4þ N0 ;A 0 Here, b is the linewidth enhancement factor, the ratio of the imaginary to real parts of the complex gain due to carrier density changes, and b 0 is the ratio due to saturation effects [17]. These ratios relate the changes in carrier density to refractive index changes in the system. The optical resonance frequency changes with carrier density. In a simple two-level system b 0 ¼ b, and on resonance b ¼ 0. The semiconductor laser operates with the characteristics of a detuned oscillator, b 6¼ 0. However, the two-level model is, at best, an approximation for the two-band interaction of the semiconductor laser. Theoretical work has suggested that the two b parameters should be different, and that b 0 0 for typical values of semiconductor laser operation [18]. As we show below, two different parameters, both non-zero, allow us to match details of our experiment with the model, details that cannot be otherwise readily accounted for. Therefore, this is a key modification to the model that we have previously used. We make one additional modification. In past work using conventional Fabry Perot cavity, edge emitting lasers it has been observed that a mode hop to another longitudinal mode is induced under some external optical injection conditions and that significant power can be emitted by other longitudinal modes under other conditions. Therefore, we add an equation describing a non-resonant

11 T.B. Simpson / Optics Communications 215 (2003) optical mode that is coupled to the others only through the carrier density and saturation effects. Eqs. (1) and (2) can then be recast in a convenient form, making use of change of variables and introducing some useful parameters. Because our primary interest is in how external optical injection modifies the optical field, we use the steady-state, free-running condition of a semiconductor laser under dc bias above threshold in a single optical mode as our starting point. The optical injection field is characterized by the frequency detuning f ¼ X=2p ¼ðx I x 0 Þ=2p and the normalized dimensionless injection parameter n ¼ gja i j=c c ja 0 j. For some of our characterization measurements current modulation with a modulation amplitude J m at a modulation frequency f m ¼ X m =2p is applied to the laser. Therefore, the total timedependent current density injected to the laser is JðtÞ ¼J 0 ð1 þ m cosðx m tþþ, where m ¼ J m =J 0 is defined as the modulation index. By expressing the field amplitude and the carrier density of the injection-locked semiconductor laser as A ¼ A 0 ð1 þ aþe iu and N ¼ N 0 ð1 þ ~nþ, respectively, the characteristics of its modulation response can be numerically simulated using the following coupled equations: " da dt ¼ c c c n 2 c s ~J ~n c # p ð2a þ a 2 þ w 2 Þ ð1 þ aþ c c þ c c n cos½xt þ uðtþš; ð5þ " # du dt ¼ bc c c n 2 c s ~J ~n b0 c p ð2a þ a 2 þ w 2 Þ b c c c cn sin½xt þ uðtþš; ð6þ 1 þ a ( d~n dt ¼ c ~ J s J m ~ þ 1 cosðx m tþ ~J " # þ 1 þ c n ~J c s ~J ½ð1 þ aþ2 þ w 2 Š ~n þ 1 c ) p ½ð1 þ aþ 2 þ w 2 Š ð2a þ a 2 þ w 2 Þ ; c c ð7þ " dw dt ¼ c c c n 2 c s ~J ~n c # p ð2a þ a 2 þ w 2 Þ l w; ð8þ c c c c where c s is the spontaneous carrier relaxation rate, c n is the differential carrier relaxation rate, c p is the nonlinear carrier relaxation rate, and b is the linewidth enhancement factor. The parameter ~J ¼ðJ 0 =ed c s N 0 Þ=c s N 0, is the differential injection current density above threshold, normalized to the threshold current density. It is often referred to as the pump parameter. Eq. (8) is added to model the side mode amplitude, w, with the additional parameter describing the gain defect of this secondary mode, l. This formulation assumes that the secondary mode is weak under free-running operation due to a small, relative to c c, reduction of the steady-state gain or loss rate. Further, the coupling between modes occurs because of the mutual dependence on the gain medium, without any significant coupling of the modal fields. As is shown below, this model reproduces key findings for laser diodes that operate in a single spatial and polarization mode under free-running operation when they are subject to external optical injection and/or current modulation. Several important features emerge from a linearized analysis of the model equations. In this case, the optical injection and the current modulation are weak, causing only small deviations from the steady-state, free-running condition. The variables a, ~n, and/ are assumed to be small and there will be no significant excitation of the weak side mode. The analysis for the case when b ¼ b 0 has been published elsewhere [7,19]. The amplitude modulation spectrum of the freerunning laser due to a weak optical probe does not depend on the value of b or b 0 and can be written as: X 2 þðc a 2 r c p Þ 2 ¼ X 2 þ X 2 2 c 2 r þ X 2 c 2 c n2 ; r ð9þ where a ¼ a½maxš is the normalized Fourier amplitude of the oscillation at X, X 2 r ¼ c cc n þ c s c p and c r ¼ c s þ c n þ c p are the frequency and damping of the relaxation resonance. For the case when b 6¼ b 0 there is a significant prediction for the optical sidebands generated by a sinusoidal modulation current. The equation for the modulation sidebands is

12 146 T.B. Simpson / Optics Communications 215 (2003) n 1 þðb b 0 2 o Þðc a 2 m ¼ p =X m Þ þ b 2 c 2 c X 2 m þ c2 n ð ~J þ 1Þ 2 2 m X2 r þ X 2 4 ~J 2 : m c2 2 r ð10þ Eq. (10) gives the power ratio of the optical sideband to the central peak under weak injection as would be measured with an optical spectrum analyzer. The effects of b 6¼ b 0 are contained in the term in the numerator that disappears when the two parameters are assumed to be equal. Two key modifications to the modulation spectrum due to this term are an asymmetry in the sidebands, with the positively offset sidebands enhanced when b > b 0, and an X 2 dependence to the modulation spectra near zero offset frequency. The unphysical divergence of the latter prediction is a consequence of the linearization used in the weak injection analysis. Eq. (10) predicts that the oppositely shifted sidebands power ratio will peak at a value that depends only on b with a frequency that depends on b, b 0, and c p. There will also be modifications to the regenerative amplification spectrum of a weak optical probe. These spectra will follow the equation 88 ja r j 2 < < 2 ja 0 j ¼ 2 :: X2 þ X2 r 2 þ b0 Xc p " þ Xc r Xc # 9 2 p 2 bx2 r 2 ðb0 bþðc r c p Þc = p 2 ;, n X 2 þ Xrr 2 2 o 9 = þ X 2 c 2 c 2 c r ; X 2 n2 ; ð11þ where the equation gives the power ratio of the optical sideband at the optical injection frequency to the central peak as would be measured by an optical spectrum analyzer. Unlike Eq. (10), Eq. (11) has a relatively weak dependence on the parameter b 0 and predicts spectra similar to the previously published b ¼ b 0 case. Because c n and c p are proportional to the freerunning power, the key rate parameters, c c ; c s ; c n, and c p, can be determined by using Eq. (9) to fit the amplitude modulation spectra of the laser under weak optical injection at several bias currents. When optical injection is used as a probe, the measurement is not corrupted by circuit characteristics of the laser. With the rates known, Eq. (10) can then be used to fit the spectrum of the ratios of the oppositely shifted sidebands under weak current modulation to determine b and b 0. Because we are only interested in the ratios of the sidebands, the measurement also is not sensitive to circuit parasitics. Eq. (11) can be used as a check on the fitting. Therefore, all of the parameters necessary to compare experimental results with model can be determined from spectra of the weakly perturbed system. 5. DFB laser characterization The DFB laser under study exhibited single mode laser operation under all excitation conditions investigated here. Under free-running operation, the optical power emitted in all of the side modes was at least 20 db below the optical power in the principal mode and each side mode power was more than 30 db below the principal mode power. Under external optical injection the total power in all of the side modes remained at least 20 db below the principal mode power and increased from the free-running levels by no more than )30 db relative to the principal mode power. Therefore, the principal mode spectra contained the key information about the characteristics and dynamics of the laser. For characterization, the freerunning laser is subjected to weak optical injection or weak current modulation and the resulting spectra are compared with the weak signal equations above. A typical fitting comparison is shown in the free-running regenerative amplification spectrum in Fig. 2. Fig. 5 plots the relaxation resonance frequencies, f r ¼ X r =2p, resonance decay frequencies, g r ¼ c r =2p, and saturation decay frequencies, g p ¼ c p =2p, for the laser at a series of bias currents. The figure also gives the linear least squares fits to the data for g r and g p, and the least squares fit assuming a power law dependence, y ¼ ax b, for f r. The relaxation resonance frequency, f r, should scale with the square root of the current level above threshold while the other two rates should scale linearly because the output power was measured to scale linearly.

13 T.B. Simpson / Optics Communications 215 (2003) Table 1 Parameter values determined from experimental measurements Linewidth enhancement factor (b) 2.4 Ratio factor for nonlinear gain (b 0 ) )1.5 Cavity photon decay rate ðc c Þ s 1 Carrier spontaneous decay rate ðc s Þ s 1 Differential gain relaxation rate ðc n Þ 1:7 ~J 10 9 s 1 Nonlinear gain relaxation rate ðc p Þ 5:4 ~J 10 9 s 1 Threshold current 5.4 ma Fig. 5. Key frequencies measured from the spectra of the 1915 LMO as a function of the current level above threshold. The relaxation resonance frequency (diamonds) is plotted along with a power-law least squares fit. The resonance decay (squares) and gain saturation decay (triangles) are plotted along with least squares fits. All data are given in frequencies in this chart, as opposed to the angular rates given elsewhere in the text. Fig. 6 plots the sideband ratio under weak current modulation and the fit to the data using Eq. (11). There is some scatter in the data and a relatively poor fit at higher modulation frequencies. This was determined to be due to the effects of a weak feedback signal from one of the FC/APC connectors. Feedback, being an optical injection, leads to modification to the asymmetry that peaks near the relaxation resonance frequency. Jitter in the laser optical frequency caused the measured signal to be modified by feedback that was randomly in and out of phase with the circulating field, causing the data scatter. However, at the peak of the asymmetry, which is the key point for the determination of b and b 0, the effects of feedback are relatively small and the data can be reasonably well fit. Note that the data are very poorly fit by the curve with b 0 ¼ 0, at variance with theoretical predictions and consistent with recently published data [17]. The parameters determined from the data are summarized in Table 1. Accuracy for the first two parameters is estimated to be 15% and 50%, respectively, for the decay rates 10%, and 0.1 ma for the threshold current. 6. Model calculations of nonlinear dynamics Fig. 6. Asymmetry of the optical sidebands generated under weak current modulation with the 1915 LMO laser biased at twice the bias current. The ratio of the positive frequency offset sideband power to the negative offset sideband is plotted as a function of modulation frequency. The diamonds are experimental data while the solid curve is calculated using Eq. (10), the relevant rate parameters determined from Fig. 2, and b ¼ 2:4 and b 0 ¼ 1:5, the upper and lower dot-dashed curves were calculated by changing the value of b to 2.1 and 2.7, respectively, and the dotted curve was calculated with b ¼ 2:4 and b 0 ¼ 0. Computer programs were written to numerically integrate the four coupled nonlinear differential equations (5) (8). To achieve consistent results, the time step had to be ð8c c Þ 1, or somewhat shorter. Runs using double precision gave essentially the same results as those using single precision though, of course, specific details changed in the region of chaotic dynamics. We can compare the observed dynamics with calculated dynamics by relating the injected field to the injection parameter, n, that we use in the calculations. Note that the locking unlocking, saddle-node bifurcation line shows a nearly linear

14 148 T.B. Simpson / Optics Communications 215 (2003) Fig. 7. Bifurcation diagrams showing extrema of the optical field as a function of the injection parameter for an offset frequency of 2 GHz. In (a) the gain defect of the side mode is large so that the laser operates in a single longitudinal mode. In (b) the gain defect is small and significant power leaks into the side mode when the circulating power is below the free-running value. In (c) the relative circulating power level in the injected mode is plotted for single mode, diamonds, and multimode, open squares, operation. dependence with injected field, as expected from both a linearized and full nonlinear analysis. Using this line of data as a reference, the injection parameter, n, can be related to the measured injection signal. This is the only free variable necessary to make the connection between experimentally measured and calculated data. In Fig. 7 we show representative bifurcation diagrams generated from the calculated time series. The diagram plots extrema in the field amplitude as a function of the injection parameter assuming an offset frequency of 2 GHz for the master laser with respect to the free-running optical frequency of the slave. A stable output consists of a single point per injection parameter value. Limit cycle oscillations are indicated by two points, period-two oscillations by four points, etc., and quasiperiodic and chaotic dynamics show a large set of points. We used the experimentally measured parameters from Table 1 for the calculations. For single-mode calculations, a relatively large gain defect, 0:1c c, was used to suppress any build-up of the optical field in the side mode. If this value is reduced to 0:01c c, or less, then there is evidence of power leaking out of the principal mode and a strong change in the dynamics. In Fig. 7, two cases are shown, one with the gain defect for the side mode set to 0:1c c, and the other where the gain defect was reduced to a value of 0:001c c. Fig. 7 also shows the power in the principal mode, comparing the cases with and without the participation of the side mode. The total power from the laser is essentially identical in both cases. When the gain defect of the side mode is sufficiently small, power is transferred and the dynamics are changed. This occurs in and near regions of complex dynamics and the locking unlocking (saddle node) transition, in agreement with earlier data on Fabry Perot lasers. The calculated data from the bifurcation diagrams are combined into a mapping of the dynamics that is shown in Fig. 8(a). At low injection levels, the laser is unlocked and undergoing Adlertype frequency pulling and, simultaneously, destabilization of the relaxation resonance so that the laser locks as it makes a transition to a limit cycle mode of operation. The agreement between theory and experiment is excellent, showing that the model reproduces the experimentally observed

15 T.B. Simpson / Optics Communications 215 (2003) characteristics. We should note that we were never able to observe periodic dynamics of period greater than six, a weak period three oscillation in a strong period two feature, and were never able to observe period doubling beyond period four. The DFB laser always broke out of the resonance condition back into the more general quasiperiodic, or chaotic, condition. The bifurcation diagrams do show higher periodic dynamics, but these regions cover a very narrow range of injection levels. Possibly, the inherent noise in the real semiconductor laser system is sufficient to make these regions indistinguishable from more complex dynamics. Alternatively, the noise induces hops between multiple attractors that may occupy different regions of phase space at these excitation levels [10]. In Fig. 8(b) the regions where mode hops alter the bifurcation diagrams are shown. Comparing this to the previously published map of a Fabry Perot laser in [9], there is qualitative agreement. 7. Discussion and conclusions Fig. 8. Calculated mappings of the nonlinear dynamics using the rate equation model and the experimentally determined parameters listed in Table 1. The symbols are the same used in Fig. 5 with the lightly shaded region covering the range of multiwave mixing and Adler-type frequency pulling and the striped regions the complex dynamics. In (a) single mode operation is maintained by using a large gain defect for the side mode, while in (b) the gain defect is small and the darkly shaded regions show where significant power has been transferred to the side mode. The use of a DFB laser diode has permitted us to map out the nonlinear dynamics of a semiconductor laser under external optical injection and quantitatively compare the results with the singlemode, rate-equation model. The agreement, both qualitative and quantitative, is extraordinary and makes this system a very good candidate for investigating nonlinear dynamics in a technologically relevant physical system. Relative to our previous work, the work here shows that, in general, the saturation effects on the complex gain have a different proportionality constant than does the differential gain component, in other words, b 0 6¼ b. The influence of non-lasing side modes can be included, qualitatively at least, by the addition of an additional field amplitude equation, with this equation coupled to the original set through the carrier density equation. When such an equation is included with a gain defect that is on the order of 10 3 of the gain of the principal mode, we observe leakage of a significant fraction of the laser output to the side mode. The leakage typically occurs when the injection laser induces output characteristics with an average output power below the free-running power of the principal mode. This often happens in regions of quasiperiodic or chaotic output. This level of gain defect is consistent with the observed side mode characteristics of the Fabry Perot lasers previously investigated. In the DFB laser, all side modes are sufficiently suppressed that there is no appreciable leakage out of the principal mode and we were able to produce a mapping of the regions of nonlinear dynamics

16 150 T.B. Simpson / Optics Communications 215 (2003) displayed by a semiconductor laser under optical injection. This map bears a strong similarity to previous mappings of dynamics of a Fabry Perot cavity, edge-emitting laser. The nonlinear dynamics is more tightly bounded. This is expected, because the laser used here has a smaller linewidth enhancement factor and larger ratio of the nonlinear gain factor to the differential gain factor. From this map it is clear that all regions of periodic dynamics with period two or larger have an associated region of complex dynamics. The higher period oscillatory dynamics represent resonances of the competing frequencies initially associated with the offset between the master and slave laser optical frequencies and the relaxation resonance frequency. Due to the influence of the carrier density on the refractive index as well as the gain, both the offset and dynamic resonance frequencies change as the optical injection is increased. It is the interplay between these two resonant frequencies that forms the overall basis for the global dynamics. Here, we have not rigorously compared experiment and theory in the region of complex dynamics. There is not an exact matching of spectra but overall, qualitative consistency between model and experiment. Indeed, the consistency is sufficiently good that the data and model can be compared to demonstrate that different transitions to and from chaotic dynamics predicted by theory can be seen in the experimental results. We have described the spectral evidence for the quasiperiodic route to chaos here. More sophisticated theoretical tools than those used here are required to make the connection explicit. These results are given in a companion paper [14]. Together, these works demonstrate that the agreement between model and experimental data is so good that the semiconductor laser subject to external optical injection is an excellent model system for testing the predictions of nonlinear dynamics theories and models. Acknowledgements The author has benefited from many useful discussions with Dr. Athanaseos Gavrielides, Prof. Bernd Krauskopf, Prof. Jia-ming Liu, and Dr. Sebastian Wieczorek while pursuing this work. Dr. Wieczorek first calculated the approximate boundary of the region of bistability at negative detunings for this laser. I would also like to thank Dr. Michel Nizette and Prof. Thomas Erneux for a very careful reading of the manuscript and many constructive comments. The work was supported, in part, by the Army Research Office under contract DAAG5598C0038. References [1] J. Sacher, D. Baums, P. Panknin, W. Els asser, E.O. G obel, Phys. Rev. A 45 (1992) [2] T.B. Simpson, J.M. Liu, IEEE Photon. Technol. Lett. 9 (1997) 1322; X.J. Meng, D.T.K. Tong, T. Chau, M.C. Wu, IEEE Photon. Technol. Lett. 10 (1998) 1620; G. Yabre, H. de Waardt, H.P.A. van dem Boom, G.D. Khoe, IEEE J. Quantum Electron. 36 (2000) 385. [3] T.B. Simpson, F. Doft, IEEE Photon. Technol. Lett. 11 (1999) 1476; T.B. Simpson, Opt. Commun. 170 (1999) 93. [4] H.F. Chen, J.M. Liu, IEEE J. Quantum Electron. 36 (2000) 27. [5] T.B. Simpson, J.M. Liu, A. Gavrielides, V. Kovanis, P.M. Alsing, Phys. Rev. A 51 (1995) 4181; S. Eriksson, A.M. Lindberg, Quantum Semiclassical Opt. 14 (2002) 149. [6] S. Wieczorek, B. Krauskopf, D. Lenstra, Opt. Commun. 172 (1999) 279. [7] J.M. Liu, T.B. Simpson, IEEE J. Quantum Electron. 30 (1994) 957. [8] S. Wieczorek, T.B. Simpson, B. Krauskopf, D. Lenstra, Phys. Rev. E 65 (2002) (R). [9] T.B. Simpson, J.M. Liu, K.F. Huang, K. Tai, Quantum Semiclassical Opt. 9 (1997) 765. [10] B. Krauskopf, S. Wieczorek, D. Lenstra, Appl. Phys. Lett. 77 (2000) [11] A.F. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986 (Chapter 29). [12] B. Krauskopf, N. Tollenaar, D. Lenstra, Opt. Commun. 156 (1998) 158; M. Nizette, T. Erneux, A. Gavrielides, V. Kovanis, Proc. SPIE 3625 (1999) 679. [13] V. Kovanis, A. Gavrielides, T.B. Simpson, J.M. Liu, Appl. Phys. Lett. 67 (1995) [14] S. Wieczorek, T.B. Simpson, B. Krauskopf, D. Lenstra, Opt. Commun. 215 (2003) 125. [15] R. Hui, A. DÕOttavi, A. Mecozzi, P. Spano, IEEE J. Quantum Electron. 27 (1991) [16] V. Kovanis, T. Erneux, A. Gavrielides, Opt. Commun. 159 (1999) 177;