Nonlinear dynamics induced by external optical injection in ...

Quantum Semiclass. Opt. 9 (1997) 765–784. Printed in the UK 

PII: S1355-5111(97)82266-X 

Nonlinear dynamics induced by external optical injection 

in semiconductor lasers 

T B Simpson†, JMLiu‡, K F Huang§ and K Tai§ 

† JAYCOR, PO Box 85154, San Diego, CA 92186-5154, USA 

‡ Department of Electrical Engineering, University of California, Los Angeles, CA 90095- 

159410, USA 

§ Department of Electro-Physics, National Chiao Tung University, Hsinchu, Taiwan 

Received 28 February 1997, in final form 12 May 1997 

Abstract. External optical injection of semiconductor lasers changes the resonant coupling 

characteristics between the circulating optical field in the laser cavity and the free carriers (gain 

medium). Changes in the characteristic resonance frequency and damping of the system can 

lead to dynamic instabilities and deterministic chaos and to enhancement of the modulation 

bandwidth. The changes induced depend on key dynamic parameters of the semiconductor laser 

such as the photon and carrier decay rates, gain characteristics and the linewidth enhancement 

factor, and the operating point of the laser determined by the injection current (pump) level, 

circulating optical power, the amplitude of the external optical injection and the frequency offset 

between the master and injected lasers. Here, we describe a mapping of the typical dynamics 

induced in a nearly single-mode semiconductor laser biased well above the threshold for laser 

oscillation as the amplitude and frequency offset of the master laser are changed. We also 

present results on a laser initially biased near threshold in a free-running condition where it 

displays a near-Gaussian optical lineshape. The external optical injection induces spectral holes 

and spikes, as well as spectral shifts. All features that we observe can be recovered in a standard 

coupled equation model of semiconductor laser operation. 

1. Introduction 

The semiconductor laser is important both for its technological applications and as a test 

system for the understanding of many key features of laser operation. Due to the small size 

of the optical cavities that can be constructed with semiconductor material, and the high 

gain of that material, features of the optical spectrum can be related back to fundamental 

quantum noise sources of the system [1]. Semiconductor lasers differ from many other laser 

systems in that the free carriers which control the optical gain of the system also have a 

strong effect on the refractive index, giving it the characteristics of a detuned oscillator [2]. 

This feature leads to an enhancement of the linewidth of a semiconductor laser [3] and also 

has been exploited to induce unstable dynamics and deterministic chaos in semiconductor 

lasers by the addition of an extra external modulation [4] such as a current modulation [5], 

weak optical feedback [6] or external optical injection [7]. 

As a test system for studying nonlinear dynamics in semiconductor lasers, a laser 

undergoing external optical injection has certain advantages. The use of an independent 

laser, rather than feedback from the laser under investigation, adds the freedom to control 

the amplitude and offset frequency of the perturbing optical field. Optical injection, as 

opposed to current modulation, does not require high-speed electronics and bypasses the 

1355-5111/97/050765+20$19.50 c○ 1997 IOP Publishing Ltd 765

766 T B Simpson et al 

electronic parasitic effects that can obscure the intrinsic laser properties. Experimental 

investigations of nonlinear dynamics in a conventional edge-emitting Fabry–Perot laser 

diode biased well above threshold have shown quantitative agreement with a coupled 

equation model for single-mode laser operation [7–10]. Here, we extend that work and 

present the results in the form of a mapping of the observed dynamics of the laser output 

as a function of the amplitude and detuning of the external optical injection. Many features 

of the observed dynamics are expected to appear in a variety of laser structures, while 

some relate to characteristics of the specific laser structure. We also present results on 

optical injection of a vertical-cavity surface-emitting laser (VCSEL) which, in its freerunning 

condition, is biased near threshold with an optical output with a near-Gaussian, 

as opposed to a Lorentzian, spectrum. When the external optical injection destabilizes 

the dynamics of this system, the optical spectrum shows spectral spikes and holes in the 

Gaussian envelope. As before, all of the results can be reproduced with the coupled equation 

model. 

2. Experimental apparatus 

Figure 1 shows the experimental set-up. The output of one laser diode (LD1) at a frequency 

ν 1 was passed through an acousto-optic modulator where a fraction, on the order of 10%, 

of the beam was deflected and shifted in frequency by 80 MHz. The shifted beam at 

ν 1 + 80 MHz was used to probe a second laser (LD2) oscillating at ν 0 = ω 0 /2π. Thus 

the frequency offset f = /2π is given by ν 0 + f = ν 1 + 80 MHz and can be varied 

by varying ν 1 . Both lasers were temperature and current stabilized. To bring them to 

near-degenerate operating frequencies, the temperature was adjusted for gross frequency 

changes and then the current was adjusted for fine tuning of the frequency offset f . Optical 

Figure 1. Schematic diagram of the experimental set-up.

Nonlinear dynamics in semiconductor lasers 767 

isolators were used to avoid mutual injection and to reject back-reflected light from any 

components in the optical path. The use of the acousto-optic modulator improved the 

optical isolation between the two lasers. It could be removed and the full output of LD1 

could be directed toward LD2 when high injection levels were required. The output of the 

injected laser was monitored using both optical and microwave/radio-frequency spectrum 

analysers to measure both the optical field and power spectra. The data presented here were 

taken using the optical spectrum analyser. However, both spectra are necessary for the 

determination of key dynamic parameters required for the comparison of experimental data 

with numerical data [11]. Measuring both spectra simultaneously allowed us to distinguish 

between amplitude and phase modulations of the output of the injected laser. The optical 

spectrum was measured with a Newport SR-240C scanning Fabry–Perot interferometer 

which has a free spectral range of 2 THz and a finesse of approximately 50 000, giving 

an optical frequency resolution of about 40 MHz. The power spectrum was obtained by 

detecting the output of the injected laser with a fast photodiode of 7 GHz frequency response 

and displaying the photodiode signal on the spectrum analyser. In this experiment, the beam 

block B2 was removed to allow the optical spectrum measurement, while the beam block 

B1 in the path of ν 1 was inserted to block off the master laser output at ν 1 from the 

detection system. This allowed the direct measurement of the output from the injected 

laser. 

In the data presented here, two sets of master–injected laser combinations were used. 

The first set consisted of two Spectra Diode Labs SDL-5301-G1 single longitudinal and 

transverse mode GaAs–AlGaAs quantum-well lasers operating at 827.6 nm. These lasers 

are conventional Fabry–Perot edge-emitting lasers with one facet coated for high reflection 

and the output facet coated for a reflection of a few per cent. The threshold current for 

the injected laser was approximately 24 mA and the data presented here were taken at an 

injection current level of 40 mA. The oscillating mode contained approximately 90% of 

the output power of the free-running laser, with the rest spread among many sub-threshold 

longitudinal modes. None of the side modes of the free-running laser contained greater than 

0.5% of the total output power. The master laser, with a threshold current of approximately 

18 mA, was operated at a higher current level, approximately 55–65 mA. The higher 

operating current reduced the magnitude of the noise and increased the frequency and 

damping of the relaxation resonance. Like the injected laser, the master laser had many 

weak sub-threshold longitudinal modes, none strong enough to play a significant role in 

the results described here. The second set consisted of a New Focus Model 6226 tunable 

external cavity laser as the master laser and a vertical-cavity surface-emitting laser (VCSEL) 

as the injected laser. The VCSEL had a threshold for laser oscillation of approximately 

3.9 mA and emission wavelength of approximately 850 nm [14–16]. The output was in a 

single polarization direction and a single longitudinal and spatial mode over the range of 

injection currents discussed here. 

3. Coupled equation model 

Semiconductor lasers have been the subject of extensive modelling. They are an example 

of a class-B laser system where the equation describing the polarization of the gain medium 

can be adiabatically eliminated due to the fast intraband carrier relaxation time [2]. This 

leaves two equations, one for the complex oscillating field within the laser cavity and one 

for the carrier density, to describe the dynamics of the single-mode laser under external


optical injection [11, 17]: 

dA 

=− γ c 

dt 2 A+i(ω 0 − ω c )A + Ɣ 2 (1 − ib)gA + F + ηA i exp(−it) (1) 

dN 

= J dt ed − γ sN − 2ɛ 0n 2 

g|A| 2 (2) 

¯hω 0 

where A is the total complex intracavity field amplitude at the oscillation frequency ω 0 ,γ c 

is the cavity decay rate and ω c is the longitudinal mode frequency of the cold laser cavity. 

Ɣ is the confinement factor which gives the spatial overlap between the active gain volume 

and the optical mode volume, b is the linewidth enhancement factor which relates the 

dependence of the refractive index to changes in the gain, g. The gain is assumed to obey 

a linear dependence on changes in the carrier density, with coefficient g N , and changes in 

the photon density, with coefficient g p , about an appropriately chosen operating point [11]. 

N is the carrier density, J is the injection current density, e is the electronic charge, d is 

the active layer thickness, γ s is the spontaneous carrier decay rate and n is the refractive 

index of the semiconductor medium. Two source terms in the field equation are due to: 

(i) external optical injection where η is the coupling parameter, A i the amplitude of the 

external field and the offset frequency with respect to the free-running frequency of the 

injected laser and (ii) noise. F is the complex Langevin source term for the field noise due 

to spontaneous emission into the mode and external coupling out of the mode [1, 3]: 

〈F(t)〉=0 (3) 

〈F(t)F ∗ (t ′ )〉=R sp δ(t − t ′ ) (4) 

〈F(t)F(t ′ )〉=0 (5) 

where R sp gives the strength of the field source term. The noise source terms in the carrier 

equation are omitted. They are relatively unimportant in determining the optical field spectra 

which we will emphasize. 

4. Semiconductor laser above threshold 

When a semiconductor laser is subjected to injection by a near-resonant external optical 

field, the laser output will exhibit a variety of optical spectra. By comparison between the 

experimentally observed spectra and the calculated spectra, we have been able to categorize 

the dynamics. When the laser is biased well above threshold, the steady-state, free-running 

operating condition is used as the reference point for the gain in the calculations. The 

field amplitude and cavity density variations are normalized with respect to the steadystate 

optical field, A 0 , and carrier density, N 0 , respectively. The equations can then be 

rewritten as three coupled real equations for the normalized field, a, optical phase, φ, and 

normalized carrier density, ñ, in a form which shows explicitly the dependence on specific 

laser parameters which can be experimentally determined in the weak injection limit [8]: 

da 

dt 

dφ 

dt 

dñ 

dt 

= 1 [ ] 

γn γ c 

2 γ s J˜ 

ñ − γ p(2a + a 2 ) (1 + a) + γ c ξ cos(t + φ) + F ′ /|A 0 | (6) 

=− b [ ] 

γn γ c 

2 γ s J˜ 

ñ−γ p(2a+a 2 ) − γ cξsin(t + φ) − F ′′ /|A 0 | 

(7) 

1 + a 

[ 

=−[γ s +γ n (1+a) 2 ]ñ − γ s J˜ 

1 − γ p 

γ c 

(1 + a) 2 ] 

(2a + a 2 ). (8)


Figure 2. Mapping of the locking and stability characteristics of the conventional, edge-emitting 

semiconductor laser with J ˜ = 2 3 as a function of the frequency offset between the locking and 

free-running laser fields and the injection parameter, ξ. The thick full curves are the boundary of 

the injection-locked region. Shown in the interior of the locking region is the Hopf bifurcation 

line (thick broken curve) separating the stable and unstable operating regimes. The positive 

frequency boundary of the locked region is a line of constant output field amplitude at the 

free-running level, A 0 . Also shown are the curves for 1.1A 0 (dotted) and 1.2A 0 (broken). 

Here, a = (|A| −|A 0 |)/|A 0 |, φ is the phase difference between A and A i and ñ = 

(N −N 0 )/N 0 . The injection parameter, ξ, describing the external optical injection is defined 

as ξ = η|A i |/γ c |A 0 | and the injected power is proportional to ξ 2 . The decay rates γ n and 

γ p are the differential stimulated emission rate and gain saturation rates, formed from the 

steady-state photon density and g N and g p , respectively [11]. J ˜ = (J/ed − γ s N 0 )/γ s N 0 , 

where J/ed is the carrier density injection rate. All the input parameters required to solve 

the set of coupled differential equations numerically for a,φ and ñ can be determined 

experimentally [11]. After numerical integration of the normalized coupled equations, the 

resulting time series are Fourier transformed to calculate the spectra. 

Some key features can be predicted from a perturbation analysis with the set of coupled 

equations which assumes that the changes to the amplitude, phase and carrier density are 

sufficiently small that the three equations can be linearized. Figure 2 shows a typical 

mapping of the predicted locking and stability range based on a linearized analysis [12]. 

The actual values used to calculate the map correspond to the injected SDL laser at the bias 

current of 40 mA, J ˜ = 2 , but the qualitative characteristics are more general. The analysis 

3 

predicts an asymmetric locking range [13] with a bounded range of unstable operation 

within the locking range. The circulating field amplitude increases with increasing injection 

parameter and/or decreasing offset frequency within the locked region. A larger value of b 

increases the low-frequency boundary of the locked operating range and increases the range 

of the unstable dynamics. A larger value of J˜ 

shifts the unstable region to larger values of 

ξ with the low-ξ limit shifting approximately in proportion to J˜ 

or A 2 0 

and the upper limit 

shifting approximately with A 0 [12]. The analysis also predicts that the eigenvalues of the 

coupled equations are strong functions of the operating point. Figure 3 plots the eigenvalues, 

or resonance frequencies of the field-free carrier coupling, as a function of ξ for two 

different cases. The first case is for external optical injection at the free-running frequency 

of the injected laser. This corresponds to a phase offset of arctan b between the master and 

injected laser optical fields in the linearized analysis. For this case the resonance frequencies 

increase monotonically with ξ. In contrast, when the injection frequency is changed as ξ is


Figure 3. Calculated eigenfrequency and measured frequency of the optical spectrum sideband 

peak for the conventional, edge-emitting semiconductor laser with J ˜ = 2 3 as a function of the 

injection parameter, ξ. Two cases are plotted: full curve and squares, injection at the freerunning 

optical frequency; broken curve and triangles, injection corresponding to φ L = 0. The 

former case shows unstable operation for 0.007


Figure 4. Measured optical spectra of the conventional, edge-emitting semiconductor laser with 

J ˜ = 2 3 under optical injection at the free-running optical frequency: (a) free-running operation 

with an expanded scale showing the relaxation resonance sidebands; (b) unstable injection 

locking, limit cycle, at ξ 2 = 1 × 10 −4 with an expanded scale spectrum; (c) period doubling 

at ξ 2 = 3 × 10 −4 with features shown clearly in the expanded scale; (d) chaotic dynamics at 

ξ 2 = 1.3×10 −3 ;(e) limit cycle at ξ 2 = 3.6×10 −3 ;(f) period doubling at ξ 2 = 7.2×10 −3 ;(g) 

transition back to the limit cycle at ξ 2 = 1.7 × 10 −2 and (h) stable operation at ξ 2 = 8.5 × 10 −2 

along with an expanded scale spectrum and the expanded scale sideband spectrum of the master 

laser.


spectra, sharp period-doubling peaks result [8]. In figure 4(d), ξ 2 = 1.3 × 10 −3 , the 

spectrum becomes dominated by a broad pedestal and many secondary peaks develop. 

Relatively little energy remains in the narrow injection spike. At this stage the spectra 

indicated that chaos has fully developed [7]. Because of the high frequencies involved, 

an experimentally measured time series of the laser output could not be obtained. The 

onset of deterministic chaos was confirmed by the observation of strange attractors in 

the phase space of a,φ and ñ from numerically generated time series obtained from the 

coupled equations without the field noise source terms. Calculated spectra qualitatively like 

figure 4(d), with the broad pedestal and irregular secondary peaks, were also consistently 

associated with a positive Lyupanov exponent [18]. Within the region of chaotic dynamics, 

a fraction of the oscillating power, increasing up to 35% as the injection level is increased, 

is shifted from the principal oscillating mode into several of the weak side modes. Over a 

narrow injection range, the broadened spectrum collapses again into an equally spaced set 

of narrow features, as shown in figure 4(e) where ξ 2 = 3.6 × 10 −3 and the principal mode 

regains its full power [7]. Now, however, the separation of the peaks has increased and the 

spectrum is shifted much more strongly to the negative components. At still higher injection 

levels, figure 4(f ) where ξ 2 = 7.2 × 10 −3 , a new, clear period doubling is observed with a 

further increase of the peak separation and relative strengthening of the negative frequency 

components. The strongest spectral feature is clearly frequency-shifted—pushed by the 

resonant injection. The period-doubling peaks then steadily decrease in amplitude as the 

resonance peak monotonically increases while the peak separation continues to increase. 

The spectra in figure 4(g) are observed when ξ 2 = 1.7 × 10 −2 and the injection peak again 

begins to dominate the spectrum. Eventually, the side peaks decrease in magnitude so 

that only a single peak at the injection frequency dominates the spectrum. In figure 4(h) 

ξ 2 = 8.5 × 10 −2 and stable dynamics has been re-established. The peak separation has 

now increased to 10.5 GHz and the side peak is visible in the expanded spectrum. Also 

visible in the expanded spectrum are replicas of the relaxation resonance features of the 

master laser. The identification of these features is confirmed by comparison with the 

sideband spectrum of the master laser, which is also shown. At high injection levels, 

the noise sidebands of the master laser appear in the spectrum of the injection-locked 

slave. These noise features do not reflect the dynamics of the injected laser system. The 

dynamics are reflected in the shifted resonance feature. Note that the strong asymmetry of 

the resonance peaks continues when stable dynamics has been re-established. The weak 

asymmetry in the resonance peaks of the free-running laser has become a strong asymmetry 

in the injection-locked laser. The measured shift in the resonance peaks is compared with 

the calculated eigenfrequencies of the coupled equations in figure 3 and shows very good 

agreement with the full nonlinear calculation and reasonable agreement with the linearized 

treatment. 

Before proceeding to a more general picture of the dynamics induced under external 

optical injection, we distinguish a special type of period-doubling-like spectra, the region 

of subharmonic resonance [10]. Figure 5 shows a set of spectra which illustrates the 

subharmonic resonance. Again, the offset frequency is referenced to the frequency of the 

free-running laser. In these spectra the master laser is offset by ≈5.5 GHz and the injection 

level is varied. A complementary set of spectra, where the injection level is held constant 

while the offset frequency is varied from below to above twice the relaxation resonance 

frequency of the free-running laser, has been presented elsewhere [10]. At low injection 

levels the spectrum consists of narrow, weak sidebands, equally and oppositely offset from 

a main peak with its relaxation resonance sidebands, as shown in figure 5(a). These narrow 

features are due to regenerative amplification of the injected signal and four-wave mixing


Figure 5. Measured optical spectra of the conventional, edge-emitting semiconductor laser 

with J ˜ = 2 3 under optical injection at a frequency offset ≈5.5 GHz: (a) weak injection with 

regenerative amplification and four-wave mixing sidebands; (b) enhanced subharmonic signal at 

ξ 2 = 2.1 × 10 −4 ;(c) subharmonic resonance at ξ 2 = 4 × 10 −4 ;(d) weak period quadrupling at 

ξ 2 = 2.6 × 10 −3 ;(e) period doubling at ξ 2 = 6 × 10 −3 and (f ) limit cycle at ξ 2 = 1.2 × 10 −2 . 

between the injected signal and the central peak [11]. As the injected signal is increased, the 

spectral features near the relaxation resonance frequency narrow and increase, figure 5(b) 

where the injection level is ξ 2 = 2.1 × 10 −4 . Note, however, that these features are halfway 

between the injection feature, or its oppositely shifted partner and the central peak. This 

frequency is slightly less than the relaxation resonance frequency of the free-running laser. 

There is also a small frequency pushing of the central peak. At still higher injection levels, 

figure 5(c) with ξ 2 = 4 × 10 −4 , the subharmonic resonance nature of the dynamics becomes 

clear. The sideband at the injection frequency is now dominated by the feature at half 

the offset. Also, the frequency pushing has stopped and the central peak is slightly pulled 

toward the injected frequency [10]. As the injection level is increased to ξ 2 = 2.6 × 10 −3 , 

figure 5(d), the pushing of the central peak and the dominance of the resonance frequency, 

now significantly larger than the initial offset frequency, are re-established. This is actually 

a period-quadrupling spectrum, with weak, broadened features just discernible above the 

noise level in between some of the sharp spectral features. At higher injection levels the


spectrum follows the progression to period doubling, figure 5(e) with ξ 2 = 6 × 10 −3 , 

and to limit cycle oscillations, figure 5(f ) with ξ 2 = 1.2 × 10 −2 . The frequency pushing 

continues to increase and the pushed feature gets relatively weaker. At very high injection 

levels, ξ 2 > 3–4 × 10 −2 , the laser output begins to smoothly shift to other longitudinal 

modes. 

Using spectra like those of figures 4 and 5, we have constructed a mapping of the 

dynamics for the laser at this pump level as the injection parameter and the frequency 

offset are varied. In the mapping of figure 6, the frequency axis is relative to the freerunning 

frequency of the injected laser. The symbols used are: 4, a perturbation spectrum 

with weak regenerative amplification and four-wave mixing sidebands; S, stable injection 

locking; SR, subharmonic resonance; P1, limit-cycle oscillation; P2, period doubling; P4, 

period quadrupling; chaos, deterministic chaos; M ′ , multiwave mixing with most output 

on another longitudinal mode; hatched regions, principal output on another longitudinal 

mode; thin curves, smooth transition between dynamic regions; thick dotted curves, abrupt 

mode-hop transitions with minor hysteresis; thick broken curves with an arrow, oneway 

mode hops out of mode; thick full curves, abrupt transition to/from a region of 

chaos or multiwave mixing where there is significant power in another longitudinal mode, 

from/to a region with power primarily in the principal mode. The smooth transitions 

represented by the thin curves are approximate. For instance, a peak-to-sideband ratio 

of 10:1 was used for the transition line from stable to unstable dynamics. For small values 

of ξ, the optical injection acts as a perturbation generating weak sidebands at the offset 

frequency, regenerative amplification and equally and oppositely shifted four-wave mixing. 

Figure 6. Mapping of the experimentally observed dynamic regions for a conventional edgeemitting 

semiconductor laser with J ˜ = 2 3 . The dynamics are determined by comparison of 

observed spectra with calculated spectra with and without noise. The symbols are defined in the 

text.


As the injection parameter is increased, various dynamic instabilities develop. At negative 

detunings the locked region resembles the predictions of the mapping from the linearized 

analysis. There is a narrow region of stable injection locking at lower values of ξ which 

opens up for ξ 0.1. Also at negative offsets there is a range of unstable dynamics. 

Within this range, as ξ is increased or the detuning is reduced to move away from the 

narrow region of stable dynamics, there is a period-doubling progression, largely obscured 

by noise, to chaotic dynamics. Within the range of chaotic dynamics a varying fraction 

of the optical output begins to shift to other modes. One edge of this region of chaotic 

dynamics is an abrupt transition where power returns to the original longitudinal mode as 

the laser returns to oscillatory dynamics. There is a small hysteresis, not shown, associated 

with this transition. 

Some features of the mapping reflect specific characteristics of the laser under 

investigation, a conventional, edge-emitting, Fabry–Perot laser diode, where the oscillating 

mode is always in competition with other longitudinal modes which are close to threshold. 

Because of this, the external optical signal can induce a mode hop and certain regions of the 

detuning–injection plane become inaccessible. For instance, there is an abrupt mode hop 

near the locking–unlocking boundary at negative detunings which has a small hysteresis, not 

shown, associated with it. For more negative offsets, the mode hop is one way and the laser 

does not re-establish operation on the original longitudinal mode when recrossing at this 

boundary. Analytical studies of the locking–unlocking boundary at negative detunings have 

shown that there is a region of bistability associated with the locking–unlocking transition 

[19]. The bistability results from competing attractors representing locked and unlocked 

solutions for the coupled equations [20]. Associated with the stable, locked solution is 

a reduced average carrier density relative to N 0 , while the average carrier density of the 

unlocked solution is greater than N 0 . The increased average carrier density increases the 

average gain of the side modes and causes changes in the refractive index. When the 

oscillation threshold for one of the near-threshold longitudinal modes is crossed and the 

nonlinear dynamics suppresses the effective gain of the principal mode, the laser undergoes 

a mode hop. In lasers where the non-oscillating side modes are more strongly suppressed 

than is the case for this laser, the single-mode model predicts that one would observe 

the bistability and no mode hops. Past work using distributed feedback laser diodes clearly 

showed the bistability without mode hops [21] and we have observed similar behaviour with 

the VCSEL when operated well above threshold and under single-mode operating conditions. 

Similarly, in regions of chaotic dynamics, the calculated average carrier density is often 

close to, or greater than, N 0 . Strong mode competition which results in a large fraction 

of the output power shifting to other longitudinal modes, is, therefore, not unexpected in 

the Fabry–Perot laser. We do not observe any power shift from the principal mode when 

the VCSEL has output spectra indicating chaotic dynamics. The single-mode model used 

here, does not accurately describe the dynamics associated with the mode hop induced by 

external injection. However, it does uncover a key physical mechanism, refractive index 

changes and increased gain in the side modes due to an average carrier density greater than 

the free-running value. 

Most of the dynamic regions and transitions are directly recovered in the single-mode 

model of semiconductor laser operation [7–10]. The value of the linewidth enhancement 

factor, b, is a critical parameter in determining the limits of the range of nonlinear dynamics 

and whether that range will include regions of chaotic dynamics. From the mapping it can be 

seen that the second region of period-doubling dynamics along the = 0 line is associated 

with a second region of chaotic dynamics at positive offset frequencies. The positive 

detuning region is more complicated because boundaries between different types of nonlinear


dynamics are difficult to draw. For instance, when the offset is approximately equal to the 

free-running relaxation resonance frequency, the distinction between the perturbation, fourwave-mixing 

type, spectrum and the limit cycle spectrum blurs into one of qualitative 

difference. There are no clear features in the spectra which distinguish between locked and 

unlocked operation. The various unstable dynamics show no qualitative distinction as they 

cross the line where the linearized analysis divides locked and unlocked operation. Within 

the nominally locked operating regime, the various spectral features of the optical spectrum 

clearly show broadening with respect to the narrow feature at the injection frequency and 

are, therefore, not locked to the injection source. 

Also associated with injection at positive offset frequencies is the general trend of 

frequency pushing of the originally free-running spectral feature. One anomally of the 

subharmonic resonance is the slight pulling of the original oscillation peak in that region. 

For optical injection at positive offset frequencies, the excitation has a tendency to push the 

original oscillation peak to negative offset frequencies with the degree of pushing increasing 

as the offset frequency of the excitation is decreased or the injection parameter is increased. 

Figure 7 plots the frequency pushing for the injection offset at 5.5 GHz. More details of 

the weak frequency pulling effect around the subharmonic resonance are given elsewhere 

[10]. Here, we concentrate on the general frequency pushing trend. The pushing effect 

highlights a key point which is central to the asymmetry of the mapping. The external 

optical injection shifts the Fabry–Perot oscillation frequency of the laser cavity by changing 

the steady-state carrier density. The excitation follows the new Fabry–Perot frequency along 

the zero phase offset line in the stable operating region. When the optical excitation is off of 

the Fabry–Perot resonance, there is a new dynamical frequency in the coupled field–carrier 

system. The shifting of the resonance oscillation frequency with excitation level is due to the 

competition between the original relaxation resonance and its associated damping, and the 

frequency difference between the injected signal and this new resonance and its associated 

phase offset, to control the response. The system restabilizes within the injection-locked 

operating region when the Fabry–Perot frequency shifts and associated damping are large 

compared to the original system resonances [12]. In this region of stable operation under 

strong injection, where linearized dynamics again prevail, the modulation characteristics of 

the semiconductor laser are strongly modified. Bandwidth enhancement, associated with 

Figure 7. Frequency pushing of the initially free-running spectral feature as a function of 

injection with the master laser frequency offset by 5.5 GHz: experimental data, squares and 

dotted curve; full nonlinear equations, full curve; eigenvalues of linearized equations, broken 

curve. Frequency pulling in the subharmonic resonance region causes the abrupt dip in the 

experimental data and nonlinear calculation curves.


the enhanced resonance frequencies, beyond the theoretical limit of the free-running laser 

has been calculated [12] and observed recently [22]. 

5. Semiconductor laser near threshold 

The threshold for laser oscillation is a prototype for the disorder-to-order transition that 

occurs in a wide range of physical systems [23]. It has a strong analogy with phase 

transitions of systems in thermal equilibrium [24]. At such transition points, the fluctuations 

induced by noise sources play a critical role. Noise in lasers near threshold has been studied 

extensively, but most analyses have concentrated on laser systems where the refractive index 

of the gain medium is relatively unaffected by changes in the optical gain [25]. Here, we 

present data starting from a free-running condition which shows a striking change in the 

lineshape of the semiconductor laser. While the lineshape is Lorentzian below and well 

above the threshold region of laser oscillation, it is near Gaussian at and just above the 

threshold. The Gaussian lineshape reflects the fluctuations of the cavity resonance frequency 

due to the carrier density fluctuations. Injection of an external optical field induces spectral 

holes and spikes as well as an overall shift in the Gaussian lineshape. The typical optical 

spectra of a semiconductor laser undergoing deterministic chaos is observed as well as a 

transition back to linear dynamics and a Lorentzian lineshape of the field-noise-induced 

spectrum at high injection levels. We observed these features in a VCSEL. However, the 

key point is not that the laser structure is very different from the conventional edge-emitting 

laser described above; it is that this laser had significantly larger values of b and γ c . The 

coupled equation model reproduces the Gaussian lineshape just above threshold for the 

free-running laser. It also yields the spectral holes and spikes, the shift of the Gaussian 

lineshape and the transition back to a Lorentzian lineshape when the semiconductor laser is 

subject to external optical injection. 

Well below and well above the threshold for laser oscillation the lineshape of a 

semiconductor has been well studied and is well understood [3]. At an injection current 

of 3.8 mA, below the oscillation threshold, the laser exhibits a Lorentzian lineshape 

determined by the difference between the gain and γ c , as shown in figure 8(a). As has 

been observed previously, the linewidth of a semiconductor laser then begins to broaden 

as the injection current is increased [26] due to the linewidth enhancing effects of b [3]. 

We also observe that this broadening is accompanied by a change in the lineshape from 

Lorentzian to near Gaussian. This is shown in figure 8(b) where the injection current 

to the VCSEL is 4.1 mA. As the injection current is increased further, the Lorentzian 

lineshape, accompanied by relaxation resonance sidebands, reappears. This is the typical 

optical spectrum of a semiconductor laser biased above threshold. The spectrum for an 

injection current of 4.4 mA is shown in figure 8(c). In the wings the laser lineshape 

falls off more rapidly than the Lorentzian curve. This feature, along with the relaxation 

resonances, has been observed previously in semiconductor lasers and explained by pointing 

out that the linewidth enhancement effect, acting through carrier density fluctuations, is not 

instantaneous [3]. Ultimately, for large offset frequencies, Lorentzian wings should reappear 

on the lineshape, reflecting only field noise sources acting directly on the optical phase. It 

has been established previously in lasers where changes in the gain are not accompanied 

by significant changes in the refractive index that the lineshape always remains Lorentzian 

[25]. Past work on the lineshape of semiconductor lasers near threshold has shown non- 

Lorentzian lineshapes, though this point has not been emphasized, but these lineshapes were 

not so clearly Gaussian as was observed in our laser [26].


Figure 8. Measured optical spectra of a VCSEL as 

the bias current is varied: (a) 3.8 mA, just below the 

oscillation threshold; (b) 4.1 mA, above the oscillation 

threshold and (c) 4.4 mA, approximately 10% above the 

oscillation threshold. Also shown are Lorentzian (full 

curve) and Gaussian (broken curve) curves fit to match 

the spectra full width at half maximum. 

Figure 9. Calculated optical spectra (full curve) and 

histograms of the Fabry–Perot resonance frequency 

variation based on fluctuations in the carrier density 

(squares) using the experimentally determined VCSEL 

parameters for different values of the normalized bias 

current parameter, . (a)=−0.02, (b) = 0.1 and 

(c) = 0.2. 

We have calculated the optical spectra for the VCSEL using the coupled equation model. 

To emphasize the generic nature of the observed phenomena, we use a simplified form of 

the model. Calculated optical spectra of the free-running laser are plotted in figure 9. Near 

threshold, where the circulating optical field is small, it is more convenient to use as the 

reference point the injection current, J 0 , and the carrier density, N 0 , where Ɣg(N 0 ) = γ c 

when no optical field is present. Coupled equations for the two quadrature components of the 

optical field then replace the equations for amplitude and phase used for the above-threshold 

case. Near threshold, we can ignore the contributions of γ n and γ p to the damping due to 

the small value of the field. We retain the important contributions of the field dependence 

to the cavity resonance frequency shift and to the field–carrier resonant coupling. If we 

make the assumptions that γ n = γ s J˜ 

and that cavity losses are due to output coupling only,


then the coupled equations take on a particularly simple form: 

dE 1 

= 1 2 

dt 

γ c(E 1 + bE 2 )ñ + A i cos(t) + F 1 (9) 

dE 2 

=− 1 2 

dt 

γ c(bE 1 − E 2 )ñ − A i sin(t) + F 2 (10) 

dñ 

= γ s ( −ñ) − 2ɛ 0n 2 ( 

g N E 

2 

dt 

¯hω 1 + E 2 ) 

2 . (11) 

0 

Here, E 1 and E 2 and F 1 and F 2 are the quadrature field and field noise components, 

respectively. The experimentally accessible control parameter is the injection current, 

which is normalized for numerical calculation in the parameter = (J − J 0 )/ed/γ s N 0 . 

A convenient normalization for the equations can be made by defining R sp = γ c F0 

2 and 

normalizing the field equations with respect to F 0 . Values for the parameters are derived 

from previously reported data [16] with b = 5.5,γ c =5.5×10 11 s −1 ,γ s =9×10 −3 γ c and 

2ɛ 0 n 2 

g N F0 2 ¯hω = 10−6 γ c . (12) 

0 

After numerically integrating the normalized, coupled equations, the resulting time series 

are Fourier transformed to determine the spectra. Calculated optical spectra are plotted in 

figure 9. Using the simplified equations, we can recover the transition from Lorentzian to 

near Gaussian and back to Lorentzian as , or the injection current, is increased. Due to the 

more simplified model, the agreement is relatively less quantitative than was achieved with 

the edge-emitting laser described above. However, the key changes in the spectral profiles 

are clearly reproduced. Lorentzian wings are calculated for the near-Gaussian lineshape 

at offset frequencies beyond 15 GHz. If, however, the linewidth enhancement factor is 

reduced to b = 3, or the strength of the noise power, R sp , is reduced by a factor of 2, the 

Gaussian lineshape of the intermediate value in figure 9(b) disappears. In either of these 

cases the lineshape remains essentially Lorentzian, with the relaxation resonance sidebands 

above threshold, throughout. We should point out that a change in the noise power is 

also equivalent to a simultaneous scaling of γ s and g N because of the intimate relationship 

between R sp and γ c [27]. 

Unlike the optical spectrum, the amplitude (intensity) spectrum of the laser field and 

the spectrum of carrier density fluctuations are not influenced by a change in the linewidth 

enhancement factor. Similarly, for the range of values considered here, they show only 

minor changes in shape, accompanied by an overall scaling in strength, due to a change 

in the amplitude of the noise source. The calculated spectra for the intensity and carrier 

density follow the expected profile from a linearized analysis [17], except at low frequencies 

due to the incomplete amplitude pinning near threshold. The amplitude of the laser field 

and the carrier density are mutually coupled, but are independent of the phase of the field. 

While the phase fluctuations do not couple back to these two quantities, they are strongly 

influenced by them through the linewidth enhancement factor and the dependence of the 

gain on the carrier density. However, the near-Gaussian lineshape is not a result of nonlinear 

dynamics. All three equations can be linearized and the same lineshape will appear. In 

a semiconductor laser, the Lorentzian lineshape occurs when the relaxation resonances are 

well outside of the central linewidth so that the resonant peaks separated by the relaxation 

resonance frequency overlap weakly [3, 28]. Near threshold this approximation will fail. In 

cavities characterized by a fast photon decay rate, with a correspondingly large field-noise 

source term and a large linewidth enhancement factor like the VCSEL studied here, strong 

deviations from the Lorentzian lineshape can be expected.


The source of the near-Gaussian lineshape can be determined by analysing the calculated 

time series for the carrier density equation. The carrier density fluctuations are proportional 

to refractive index changes through the dependence of the gain on the carrier density. These 

fluctuations lead to instantaneous variations in the cavity resonant frequency. A histogram of 

the frequency variations is also shown in figure 9. It is generated by counting the number of 

times the frequency (carrier density) value falls within a frequency bin interval and plotting 

the count as a function of frequency. This produces a spectrum with values proportional to 

the fraction of time that the particular frequency bin is the instantaneous resonant frequency. 

Below threshold, the spectrum of resonance frequencies is relatively narrow. As threshold 

is approached, and the amplitude fluctuations of the laser field become suppressed through 

coupling with the carrier density, broadening sets in. Initially, the broadening is asymmetric, 

as shown in figure 9(a), reflecting the increased coupling of the field amplitude fluctuations 

to carrier density fluctuations for carrier densities closer to the threshold level. Above 

threshold, the broadening of resonance frequency fluctuations quickly increases to a level 

where it saturates with further increases in the pump level, as shown in figures 9(b) and (c). 

The distribution of resonance frequencies is dissimilar to the laser lineshape in figures 9(a) 

and (c), but follows the same Gaussian profile in figure 9(b). At this pump level, it is the 

range of carrier density fluctuations that determines the laser lineshape and not the usual 

phase diffusion fluctuations. The instantaneous cavity resonance frequency fluctuations 

are caused by the same field noise source terms as the phase diffusion fluctuations, but 

usually they are not a significant factor in determining the lineshape because the linewidth 

of each of the relaxation resonance features that forms the overall lineshape is less than the 

resonance frequency. Therefore, the normal phase diffusion term reasserts its dominance 

above threshold even though the carrier density fluctuations persist. Only when the different 

relaxation resonance components show substantial overlap is there strong deviation from a 

central Lorentzian lineshape. 

When the free-running Gaussian lineshape laser is subjected to external optical injection 

several spectral changes occur. As before, we first concentrate on resonant injection. The 

first observable change as the injection level is increased is a shifting of the spectrum 

to lower optical frequencies. This reflects the drop in the overall carrier density due 

to the higher circulating field and enhanced stimulated emission. As the injection level 

is increased further, the spectrum shows a new spike at negative offset frequencies and 

a dip at positive offset frequencies in addition to the overall shift and a spike at the 

injection frequency, as shown in figure 10(a). The free-running spectrum is also shown for 

reference. The spike and dip are oppositely offset by 1 GHz—approximately the relaxation 

resonance frequency. As the excitation level is increased further, spikes and dips appear 

at multiples of the original offset. At higher excitation levels, the spectra are similar to 

the chaotic dynamics spectra discussed above, figure 10(b). Finally, at very high excitation 

levels, the spectral structure has shifted away from the excitation peak and assumed a 

more Lorentzian lineshape, figure 10(c). Calculated spectra show the progression from 

shift with spike and dip, figure 11(a), to chaotic-like, figure 11(b), to shifted Lorentzian, 

figure 11(c). 

We have previously presented the shift of the Fabry–Perot resonance feature due to 

optical injection when this laser was initially biased at 3.9 mA, at the laser threshold [16]. 

There, the initial deviation from Lorentzian was not large, but the spectra clearly showed 

a narrowing and a progression to a more Lorentzian lineshape as the injected power was 

increased. This effect can also be seen as the laser frequency is tuned across the resonance 

at strong injection levels at the higher operating current described by the spectra presented 

here. Figure 12 plots this trend under strong external injection. For large positive detunings


Figure 10. Measured optical spectra of the VCSEL 

biased at 4.1 mA and subjected to external optical 

injection. The ratio of injected power is: (a) 1;(b) 

7; (c) 940. The free-running spectrum is also shown 

for reference. 

Figure 11. Calculated optical spectra of the VCSEL 

biased at = 0.1 and subjected to external optical 

injection. The ratio of injected optical power is: (a) 1; 

(b)4;(c) 100. 

the Gaussian lineshape is retained but there is a shift of the feature. As the detuning 

is decreased the feature narrows, with increased amplitude. For injection close to the 

edge of the locking range, near −24 GHz, the emission feature becomes weak, indicating 

stable, injection-locked operation. The smooth progression of the shift of the Fabry–Perot 

resonance feature is summarized in figure 13. For positive detunings, the shift decreases with 

increasing offset as the injection undergoes less amplification in the cavity. For negative 

detunings the shift is greater than, but approaching, the magnitude of the injection offset as 

the offset decreases to more negative values. At the edge of the locking range this smooth 

progression abruptly halts and the laser displays a weaker, more complex spectrum. The 

shift of the cavity resonance frequency and the associated damping are caused by the reduced 

carrier density under strong external injection. The dynamics are strongly influenced by the 

magnitude of the shift and damping.


Figure 12. Spectra of the VCSEL under strong optical injection as the detuning is varied: 

12 GHz (squares), −12 GHz (diamonds) and −24 GHz (triangles). Each spectrum has two 

principal features, the regeneratively amplified injected signal and the regeneratively amplified 

spontaneous emission at the shifted Fabry–Perot resonance peak of −8.5, −19.5 and −29.5 GHz, 

respectively. The −12 GHz offset excitation spectrum also shows a weak four-wave mixing 

peak at −4.5 GHz. 

Figure 13. Measured pushing of the Fabry–Perot resonance as a function of the detuning of the 

injected signal. The diamonds are for an injected power approximately an order of magnitude 

stronger than the squares. The full curve is a slope of one line at negative offsets and slope of 

zero line at positive offsets to aid the eye. 

6. Conclusions 

The coupled equation model for a single-mode semiconductor laser successfully reproduces 

the characteristic spectra that are observed in nearly single-mode semiconductor lasers 

subject to external optical injection. When a full set of key dynamic parameters, 

b, γ c ,γ s ,γ n ,γ p , J,ξ ˜ and the strength of the noise source, is known then there is good 

quantitative agreement, as has been achieved in our studies of a conventional, edge-emitting 

Fabry–Perot laser diode. Even a simplified model can recover key features, as was shown 

in our discussion of a VCSEL biased near threshold. The model is quite general and should 

apply to a variety of laser structures. While missing multimode phenomena, such as mode 

hops, it accurately predicts the observed dynamics when semiconductor laser operation is 

nearly single mode. Lasers which typically oscillate simultaneously in several longitudinal 

modes, as is common for Fabry–Perot laser diodes operating in the 1.3 µm and 1.5 µm 

wavelength regions, or on multiple transverse modes, will require additional field equations


for each mode and attention to the coupling between the modes and with the carrier density. 

We have emphasized the importance of the frequency shift of the cavity resonance frequency 

induced by the external optical injection. Because the free carriers influence both the gain 

and the cavity frequency, through b, the shift, phase offset and damping associated with 

the reduced gain compete with the initial carrier–field resonant coupling frequency and 

damping to control the dynamics. An asymmetric mapping, as a function of the offset 

frequency of the master laser, results from the frequency shift. Enhanced system resonance 

frequencies, accompanied by stable or unstable dynamics, are observed at larger injection 

levels. At very high injection levels, when the resonance offset and damping associated with 

the external injection dominate the other characteristic frequencies, the laser re-establishes 

stable operation over a large fraction of the expected injection-locked operating range. Both 

the range of chaotic dynamics, and the larger range of unstable dynamics, are bounded as 

functions of the strength and detuning of the external injection. 

Acknowledgments 

The authors would like to thank Drs C Clayton, A Gavrielides and V Kovanis for many 

useful discussions. Drs Gavrielides and Kovanis performed many of the calculations 

establishing the correlation between measured spectra and dynamics which have been 

referenced here. Dr Kovanis also performed the calculations of the Lyupanov exponents 

needed to verify the correspondence between measured spectra and chaotic dynamics. We 

would also like to thank Dr T Day for the loan of the New Focus tunable laser. This 

work was supported by the Phillips Laboratory, AF Materiel Command, under contract 

no F29601-94-C-0166 and the Army Research Office under contract DAAH04-96-C-0038. 

The content of the information in this manuscript does not necessarily reflect the position 

or policy of the Government and no official endorsement is inferred. 

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