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Optical and Quantum Electronics ISSN 0306-8919 Opt Quant ElectronDOI 10.1007/s11082-013-9762-5

The effect of nonlinear gain on thecharacteristics of an optically injectedVCSEL and cavity solitons

Mansour Eslami, Reza Kheradmand &Ghafour Hashemvand

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Opt Quant ElectronDOI 10.1007/s11082-013-9762-5

The effect of nonlinear gain on the characteristicsof an optically injected VCSEL and cavity solitons

Mansour Eslami · Reza Kheradmand · Ghafour Hashemvand

Received: 17 December 2012 / Accepted: 26 August 2013© Springer Science+Business Media New York 2013

Abstract Using the Lagrange method to fit the curve of maximum gain as a function of car-rier density for an active region consisting of an AlGaAs/GaAs layers sandwiched betweenDBR layers, it is found that the curve is better approximated assuming a quadratic depen-dence on the carrier density. By summarizing all of the calculations into a nonlinear gaincoefficient parameter, β, in the Maxwell–Bloch equations we numerically studied the effectof nonlinear gain on the characteristics of the VCSEL and also on the cavity solitons (CSs)forming in such a device. Particularly, it is shown that with nonlinear gain a wider lockedregion can be achieved along with enhanced sustained relaxation oscillation amplitude. Theswitching on/off time of CSs is modified and there appears a considerable enhancement intheir efficiency and contrast.

Keywords Bistability · Relaxation oscillation · VCSEL · Semiconductor gain ·Cavity soliton

1 Introduction

Cavity Solitons (CSs) are self-localized states of light appearing in the transverse plane of acavity as bright spots sitting on a dark background (Lugiato et al. 2008; Barbay et al. 2011).Experimentally, some properties are attributed to them for instance: CSs are self-localizedstates, independent of the system boundaries (provided that the cavity size is large enough)whose shape and size is fixed by the system parameters and do not depend on the excitationthat gave birth to them; CSs can exist in several (ideally arbitrary) transverse locations of

M. Eslami · R. Kheradmand (B)Photonics Group, Research Institute for Applied Physics and Astronomy,University of Tabriz, Tabriz, Irane-mail: r_kheradmand@tabrizu.ac.ir

G. HashemvandPhotonics Group, Kerman Graduate University of Technology,Kerman, Iran

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the cavity and can be independently manipulated (written, erased, …); CSs can be “moved”or set into motion. The main property of a localized structure is that its correlation lengthis much smaller than the size of the system. Thus, each localized structure will behave asan independent object. Applications of CS in the field of photonics are diverse; particularly,some applications in information technology and in all-optical information storage are of highinterest in this field (Firth and Weiss 2002; Coullet et al. 2004). Due to the fact that they canexist anywhere in the transverse plane of a photonic device, there is a multitude of parallelchannels available in a single device, thus creating the prospect of controllable memoryarrays. Pioneering work putting forward this application of CSs was done in McDonald andFirth (1990). CSs offer many other functional properties in all-optical information processingsuch as, to name a few, spatial information encoding, manipulation and reconfiguration basedon the transverse distribution of light, all-optical delay lines, all optical logic gates (Eslamiand Kheradmand 2012) and soliton force microscopy (see Ackemann 2009 for a thoroughreview).

CS necessitates a cavity whose transverse extent is much larger than the longitudinalextent such that it can host many transverse modes and allow for spatial de-correlationbetween different cavity locations. Therefore, broad-area VCSELs provide a suitable bench-mark to implement CSs in semiconductor material systems. CSs have been theoreticallypredicted and then demonstrated in broad area vertical cavity driven semiconductor lasers(VCSEL) slightly below the lasing threshold (Spinelli et al. 1998; Barland et al. 2002; Hachairet al. 2004). The theoretical model used in those papers is based on standard rate equationsbased on the adiabatic elimination of the material polarization. However, such an approachleads to questionable consequences when the equations are applied to dynamics occurringon the femtosecond time scale (Yao et al. 1995). Moreover, for a laser above threshold and inpresence of diffraction the standard adiabatic elimination of polarization causes unphysicaleffects because unstable transverse wave vectors begin to grow uncontrollably, as a result ofthe fact that adiabatic polarization elimination assumes that the gain width is infinite (Oppoet al. 1991). To overcome these kinds of problems one can use the model introduced byAgrawal and collaborators (Yao et al. 1995) which is based on the density matrix formalismthat leads to a set of effective Maxwell–Bloch equations valid for a semiconductor laser. Themodel includes an equation for the macroscopic material polarization coupled to those forelectric field and carrier density. A slightly different version was applied in (Hachair et al.2006; McIntyre et al. 2010; Prati et al. 2010) to study the formation of CS in a driven VCSELabove threshold. It also led to the prediction of the existence of a long wavelength instabilityassociated with the α factor in a free running VCSEL (Prati and Columbo 2007; Oppo et al.2009).

In this work we focus on the dependence of material gain on carrier density, which canbe phenomenologically derived by fitting the gain curves calculated experimentally or witha microscopic model. In (Hachair et al. 2006) the fitting curve was linear in the carrierdensity. Here we assume more realistically that the fitting curve is quadratic. Althoughthe coefficient β of the nonlinear term is small, we show that it has important conse-quences on the homogeneous steady state, its stability and the stability domain of CS.Also in this paper we will show, by applying this correction in the gain coefficient of theMaxwell–Bloch equations, that there appears a relatively considerable change in the switch-ing time, or equivalently switching energy, which can play a crucial role in logical operationswhere the speed of switching matters. However, we shall see that this correction in thegain coefficient affects the switching off operation in a different manner. It is also demon-strated that better contrast and efficiency can be achieved for CSs forming in nonlinear gainregime.

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The structure of the present paper is in this sequence: in Sect. 2 the details of the consideredmodel are discussed, the homogeneous solutions and the stability curves are shown in Sect. 3,the results and discussions regarding sustained relaxation oscillations are presented in Sect. 4,Sect. 5 is devoted to the study of switching behavior of CSs as a consequence of nonlineargain and also their properties compared with linear gain. Finally a summary is presented asa conclusion in Sect. 6.

2 The model

The considered system is of the class of broad-area vertical cavity surface emitting laserswhose active region is made of multiple quantum wells whose dynamics is described byeffective Maxwell–Bloch equations (Hachair et al. 2006):

⎧⎨

⎩

∂t E = σ [P + Ei − (1 + iθ)E + i∇2 E]∂t P = [�(D)(1 − iα) + 2iδ(D)][(1 − iα) f (D)E − P]∂t D = −b

[ 12 (E∗ P + E P∗) + D − J − d∇2 D

] (1)

where E and P are the slowly varying envelopes of the electric field and of the effectivemacroscopic polarization, and D is a population variable proportional to the excess of carrierswith respect to transparency. The decay rates σ and b are defined as σ = τd/τp and b = τd/τc,where τp and τc are, respectively, the photon life time and the carrier recombination timeand τd is the dephasing time of microscopic dipoles. Detuning parameter θ , represents thedifference between the cavity longitudinal mode frequency and the frequency of the injectedfield, multiplied by. The amplitude of the holding beam is denoted by Ei , while J is aparameter related to the pump current. Finally, d is the diffusion coefficient for carriers andα is the line width enhancement factor.

The extra equation that appears here with respect to the rate equations is that for theeffective macroscopic polarization P , whose right hand side is multiplied by a complex termcontaining the real functions �(D) and δ(D) that determine the shape of the effective sus-ceptibility and its dependence on the carrier density. For the case of linear approximationwe have the following relations for the parameters �(D) = 3.214D + 0.874/

√1 + α2,

δ(D) = 0.216D − 0.169, = −α + 2δ/� while for quadratic one they change to�(D) = 2.308D + 1.206/

√1 + α2,δ(D) = 0.155D − 0.146, = −α + 2δ/�where

�(D) is associated with gain line width, while δ(D) is the detuning between the referencefrequency and the frequency where the gain is maximum.

The function f (D) contains the dependence of gain on the carrier density. In(Hachair et al. 2006) the simplest choice f (D) = D was made. Since D is defined asD = η(N/N0 − 1) where η is a coefficient that incorporates various laser parameters andN0 is the carrier density at transparency, those choices led to assuming linear gain. To beprecise, for quantum wells the gain is not exactly linear. We used the Lagrange method to fitthe curve of maximum gain as a function of carrier density for an active region consisting ofan AlGaAs/GaAs layers sandwiched between DBR layers. We found that the curve is betterapproximated assuming a quadratic dependence on the carrier density which leads to:

f (D) = D(1 − β D) (2)

with β = 0.125. So, the full set of effective Maxwell–Bloch equations describing our systemwill be:

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Table 1 The fixed values forphysical parameters

Parameter Value

d 0.052

α 4

b 10−4

σ 4 × 10−2

τd 100 fs

τp 2.5 ps

θ −2

⎧⎨

⎩

∂t E = σ [P + Ei − (1 + iθ)E + i∇2 E]∂t P = �(1 + i)[(1 − iα)(1 − β D)DE − P]∂t D = −b

[ 12 (E∗ P + P∗E) + D − J − d∇2 D

] (3)

In Table 1, the values of physical parameters which are considered for the present workare summarized.

3 Homogeneous stationary state and its stability

The above-mentioned dynamical equations would admit the plane wave stationary statesolutions by considering ∂t = 0 and ∇2 = 0. By doing so we arrive at an equation whichrelates the output intensity |Es |2 to input intensity |Ei |2 as:

|Ei |2 = |Es |2{(1 − Ds + β D2

s )2 + (θ + αDs − αβ D2

s

)2}

(4)

Ds = (1 + |Es |2) −√

(1 + |Es |2

)2 − 4 |Es |2 β J

2 |Es |2 β(5)

The equation to reach the threshold current for such a model can be obtained through settingEi = 0 in the stationary equations of the model and considering the point for which thenon-trivial stationary solutions give |Es | = 0. Thus the condition that determines the laserthreshold reads:

Jth = 1 − √1 − 4β

2β(6)

which leads to the threshold value of Jth = 1.171 for nonlinear gain (β = 0.125) and itreduces to 1 in linear case (β = 0). The details for stationary solutions and stability analysisof them in linear approximation can be found in (Hachair et al. 2006; Prati et al. 2010).

The homogeneous stationary solution exhibits bistability in some parameter region.The bistability curve and linear stability analysis of the homogeneous solutions for bothcases are shown in Fig. 1. Note that in Fig. 1 the ratio J/Jth for both regimes is keptfixed.

Another comparison can be the case in which one keeps the same value of pump current forboth regimes, that is, 1.288 for both linear and quadratic fittings. The homogenous stationarycurves and their instability domains are shown in Fig. 2.

As it is evident from figures above, a part of the upper branch of the homogeneousstationary curve is usually affected by Turing instability, which occurs at a critical value

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Fig. 1 a Bistability curve for pump current of 10 % above threshold for both fittings, i.e. 1.288 for quadraticand 1.1 for linear. b Turing and c Hopf instability domain for the same parameters

kc of the wave–vector and is responsible for pattern formation at a spatial scale of the orderof k−1

c . Figures 1 and 2b show the Turing instability domains for linear and quadratic gainin different pump values. The domain for quadratic gain is shifted towards higher intensityvalues when keeping J/Jth fixed (Fig. 1b) and it can be considered the same for the caseof the same value of pumping current (Fig. 2b). This agrees with Figs. 1 and 2a becausethe intersections of the Turing instability domain with the k = 0 axis are the turning pointsof the stationary curve, and those points are associated with larger intensities for quadraticgain for fixed J/Jth and almost the same when pumped equally (i.e. 1.288). The unstableportion of upper branch is more extended in the case of quadratic gain for fixed ratio ofJ/Jth , which means that the range of input intensities for which spatial patterns can exist ismore extended. Yet, the important quantity is the critical wave–vector, and this differs littlein the two cases. We can then conclude that the pattern forming mechanism is only slightlyaffected by quadratic gain.

Unless the injected field is perfectly frequency matched with the solitary VCSEL (θ = −α)

a portion of the lower branch is also unstable due to the Hopf instability from |Ei |2 = 0 upto the so-called injection locking point

∣∣Ei,L

∣∣2. This instability makes the background for

Cs unstable. Oscillatory CSs have been shown to exist under those conditions, amplitude ofwhich is governed by enhanced relaxation oscillations before locking (Eslami et al.).

The situation is different for Hopf instability domains shown in Fig. 1c. The maximumintensity value of these domains is approximately given by J − Jth , i.e. by the intensity of thesolitary VCSEL. The meaning is clear: locking is achieved only when the intensity emitted

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Fig. 2 a Bistability curve for pump current of 1.288 which is equal to 10 % above threshold for quadraticfitting and, virtually to 29 % above threshold for linear fitting. b Turing and c Hopf instability domain for thesame parameters

by the locked injected VCSEL is larger than the intensity of the solitary VCSEL. Therecannot exist an injected VCSEL which emits continuously a spatially homogeneous beamwith intensity smaller than the one that would be produced by a VCSEL pumped at the samelevel and without injection. In the situation considered here of VCSEL 10% above threshold(J = 1.1Jth) the intensity associated with the injection locking point is then approximatelyequal to 0.1Jth , which means 0.1 for linear gain and 0.117 for quadratic gain. The relevantpoint is that the increase of the injection locking point due to quadratic gain is smaller thanthe increase of the input intensity of the first turning point. As a consequence, the stableportion of the lower branch is more extended in presence of quadratic gain, and this meansthat the range of injected intensities for which stable CS with stable background exist is alsomore extended. This point is clearly shown in Fig. 3 where locked and unlocked regionsare sketched for input intensity and pump current. It can be easily verified that the extentof the locked area (stable background for the CS) for quadratic gain is roughly 1.8 fold incomparison with the linear gain, meaning that in quadratic gain one has a wider region ofstable CS.

4 Sustained relaxation oscillations due to optical injection

It is well-known that any small perturbation in the intensity will induce oscillations around|Es |2 with a certain frequency. The amplitude of these relaxation oscillations decreases

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Fig. 3 Hopf instability region in terms of the two control parameters J, |Ei |2, for Linear (a) and nonlinear(b) gain. The end point of J axis is where the Hopf bifurcation reaches saddle-node

exponentially in semiconductor lasers. In fact, relaxation oscillations describe a physicalprocess in which energy contained inside the cavity is periodically exchanged between thenumber of photons and the number of the carries.

Among the consequences of applying an external light into the cavity is the occurrenceof locking. This injection locking occurs via Hopf bifurcation when stable periodic orbit(sustained relaxation oscillations) shrinks onto unstable stationary point making it stable(Wieczorek et al. 2006, 2005, 1999). A detuning parameter, which is the difference betweenfrequency of master laser and that of the cavity resonance, θ , is introduced when injectionis applied. This detuning is the origin of beating between the injected light and the fieldinside the cavity which leads to the sustained relaxation oscillations in the output light. Theamplitude of these oscillations tends to reduce as it approaches the locking point and it hasits highest value in the middle of the Hopf region, this is because by increasing the intensityof injected field the detuning gains more beating energy. The same holds for the phase ofthe output field and it undergoes a decreasing oscillations as it reaches the locking point(Wieczorek et al. 2006; Eslami et al.).

In Fig. 4 the amplitude of these sustained oscillations in the intensity, population andphase of the output light is depicted for linear and nonlinear gain regimes. It is clear thatthe amplitude of sustained relaxation oscillations is enhanced when considering quadraticgain for fixed J/Jth ratio. Regarding phase oscillations, it is seen that there is only a shifttowards higher injected intensities which means that the system in nonlinear gain loses itsphase oscillation in the higher holding beam intensities. System in linear gain regime whenpumped 28.8 % above threshold shows remarkably higher oscillation amplitudes since it iswell known that by pumping far beyond threshold the system takes on a turbulent behavior.By approaching the locking point amplitude of oscillations decreases, however, for lineargain pumped 28.8% above threshold no reduction is observed since whole lower branch ofstationary solution is in the Hopf region.

5 The effects on the CSs

The effect of nonlinear gain on the switching on/off performance of CSs is to be discussed inthis section. Since carriers play an undeniable role in the Maxwell–Bloch model for describingVSCEL type lasers and with respect to the fact that the newly introduced coefficient for gaingives rise to a more close-knit relationship between gain and the carrier population, the

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Fig. 4 Difference of maximum and minimum intensity (a), population (b) and phase (c) of the output lightwhich occurs as a consequence of Hopf instability

switching characteristics of the associated CSs would be affected when passing from lineargain regime to the nonlinear one. This issue is numerically studied by coherent switchingmethod which uses a Gaussian pulse in phase with the holding beam.

The switching pulse is introduced to the simulation through using a function of the formbelow:

Ei (x, y) = Eiinj exp

(

− (i − iloc)2 + ( j − jloc)

2

2(w)2 + iϕπ

180

)

(7)

where Eiinj , w, ϕ are the amplitude, width and phase of the injected Gaussian pulse, respec-tively. (i, j) denotes the grid points and (iloc, jloc) is where the pulse is applied. The compar-ison is made between two different pump currents, the first case compares switching on/offof CSs when both linear and quadratic regimes have been pumped 10 % above threshold asin Fig. 1, and the second is that of the same pump current 1.288 for both regimes.

In Fig. 5 switching on behavior of CSs in linear gain is compared with nonlinear one bytwo scales, one is the minimum required amplitude for writing beam to have a successfulswitching on operation and the other is its minimum duration versus increasing holding beamintensities, which clearly show that switching on operation in linear fitting is faster than thatof nonlinear fitting. Roughly speaking, for holding beam amplitude of 0.55 (a.u.) a CS can beswitched on 2.1 times faster in linear regime than in quadratic. Faster switching on in linearregime is also the case when considering the amplitude of writing beam where one needs35 % less power for injection duration of 0.6 ns in linear gain regime.

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Fig. 5 Comparison between linear and nonlinear gain regimes in terms of a minimum required injection timewhen the writing beam amplitude is fixed to 1 (a.u.) and b minimum required amplitude for writing beamwhen the injection time is fixed to 0.6 ns

Another comparison can be made keeping fixed the pump current for both regimes at1.288. This equals to 10 % above threshold for quadratic fitting, and virtually to 29 % abovethreshold for linear gain. It is seen that when we compare the switching on time for thesame pump current the results are different from when we keep the J/Jth ratio fixed (10 %above threshold for both). Here it is evident that switching on time is longer for linear gain,particularly in higher holding beam intensities. Increase in the needed injection time forhigher pumps is a well-established property irrelevant of gain considerations. So, the shortlength of injection time for quadratic gain in comparison with linear one happens due to farbeyond threshold pumping of the system in linear fitting, 29 % above threshold, while it isjust 10 % for quadratic gain.

Switching off time shows relatively different behavior, since the required injection timeincreases with increasing holding beam intensity, in contrast to what we had for switching onprocess where one is faced with decreased amount of switching power when having higherholding beams. When comparing switching off performance in linear and nonlinear gainregimes, interesting results emerge. As above, firstly we compare the switching off time ofthe fixed J/Jth ratio. Figure 6 shows that switching off the CSs in linear regime takes longerthan that in nonlinear regime.

Another comparison can be made in a similar way to switching on process, for the samepump current, fixed to 1.288 for both regimes. As it is seen from Fig. 6, in this case theswitching off time is shorter for linear gain and the reason goes, to some extent, similarly asthe switching on time for the same fixed pump current.

The effect of nonlinear gain can also be studied in terms of the contrast (ξ) and efficiency(η) of the written CSs. In order to investigate these parameters, the two ratios ξ = C S

background

and η = C S|Ei |2 are defined, where the mean value of CS intensity and that of the background

intensity are used for intensity contrast and in the case of efficiency the mean value of theCS intensity is divided by the corresponding holding beam intensity. Figure 7a shows thatthe contrast of CSs for the same amount of pump current (i.e. 1.288) is by far better for thecase of nonlinear gain while it shows the same behavior but shifted towards higher injectedintensities for the case of fixed J/Jth ratio.

For the case of efficiency parameter, it seems more reasonable to compare the results forthe same value of pump current for both fittings rather than fixed J/Jth ratio, since the total

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Fig. 6 Comparison of switching off time when the amplitude of writing beam is fixed to 1 (a.u.) (a) andthe minimum required amplitude for writing beam when the injection time is fixed to 0.6 ns (b) in linear andnonlinear gain regimes

Fig. 7 The ratio of CS intensity to that of background (a) and CS intensity to the corresponding holding beamintensity (b) for different pump values in linear and nonlinear gain regimes

energy injected to the cavity is made up of both holding beam and pump current. For the sameholding beam values, it is seen from Fig. 7b that quadratic gain exhibits higher efficiencyvalues.

6 Conclusion

Making use of the Lagrange method to fit the curve of maximum gain as a function of carrierdensity, it is shown that by considering nonlinear gain, which is a better approximation thanlinear, Turing instability region in the upper branch of bistability curve is more extended,which corresponds to wider range of input intensities for which pattern forming mechanismis available. As to Hopf instability, it is demonstrated that there is a wider locked region inquadratic fitting, although unlocked region itself is also broadened but is not more than lockedarea. It is also shown that nonlinear gain provides enhanced relaxation oscillations in termsof amplitude of oscillations in Hopf region. Some effects on the switching characteristics ofCSs are also studied and shown that in nonlinear gain switching on time is virtually 2.1 times

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longer than that in linear gain, when keeping J/Jth ratio fixed. This fact is the opposite ofwhat happens in switching off operation, in which it takes longer for linear gain. Properties ofCSs are also altered as a result of nonlinear gain; simulations show that the intensity contrastand efficiency of written CSs are enhanced when working in quadratic fitting.

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