Transient analysis of optical bistability ininhomogeneously pumped lasers

M.C. PerkinsR.F. Ormondroyd

Indexing terms: Lasers, Semiconductor lasers, Modelling, Transients

Abstract: Inhomogeneously pumped devices suchas SLEDS and multisegment lasers are known toexhibit absorptive bistability. A new model of thetransient behaviour of the optical output duringswitching is presented for this class of device forthe case of both optical and electrical triggering.The model includes the wavelength dependence ofthe stimulated gain and the spontaneous emissioncharacteristics of the active material. This is par-ticularly necessary because the gain curves are dif-ferent in the pumped and passive segments. Thevariation of the photon flux along the length ofthe laser is also taken into account. Results areobtained for GaAs/GaAlAs devices. Opticallyinduced switching is demonstrated for differentwavelengths of the injected light, and the effect oftriggering the device by optical injection intoeither the pumped or the passive segment is exam-ined. The switch-on time is found to be dependenton the electrical or optical overdrive provided bythe triggering pulse. Large transient changes in thewavelength are seen during switching, and undersome conditions the device can switch off, imme-diately after switching on, due to strong stimu-lated recombination in the regions of gain. Therecovery time from this can be several nanose-conds. It is seen that self-sustained pulsations canbe induced in these devices by optically triggeringthe device with light at the peak superluminescingwavelength injected into the front facet.

List of symbols

zBrXn(z)Poddeg(n(z), X)feerG(n(z), X)

= longitudinal direction= bimolecular recombination rate= wavelength= carrier density= active layer doping density= thickness of active layer= confinement of spontaneous emission= electronic charge= gross optical gain= free carrier absorption coefficient= optical confinement factor= net effective gain after taking into consider-

Paper 5682J (El3), first received 16th June 1986 and in revised form 5thAugust 1987The authors are with the School of Electrical Engineering, University ofBath, Claverton Down, Bath BA2 7AY, United Kingdom

ation confinement factor and free carrierabsorption; where

G(n(z), X) = T • g(n(z), X) -fcc • (2n(z) + p0)

J(z) = current densityP(z, t, X) = time dependent flux in +ve z directionQ(z, t, X) = time dependent flux in — ve z directionRlt R2 = reflectance of facets 1 and 2, respectivelyS(n(z), X) = spontaneous emission, where

S(n(z), X) dX = Br- n(z) • (n(z) + p0)

v = velocity of light in GaAsLm = length of the mth pumped segmentw = width of the buried channel active layer

1 Introduction

Optical bistability in active electro-optic devices is cur-rently of interest [1] because of the potential for fabricat-ing integrated optical regenerators [2] and opticalswitching and logic gates. Harder, Lau and Yariv [3-5]have observed hysteresis and self-pulsations in the roomtemperature characteristics of inhomogeneously pumped(split-stripe) buried channel, d.h. lasers due to saturableabsorption effects, whilst Kawaguchi [6-8] has demon-strated electrically triggered absorptive bistability inmultisegment lasers. The latter has also shown that self-pulsations may be predicted for simple two segmentlasers of the SLED type when the carrier lifetime is short-ened by the addition of nonradiative recombinationcentres into the active layer of the passive segment. Thispaper considers the transient behaviour of the opticaloutput of inhomogeneously pumped GaAs/GaAlAs laserswhen optically and electrically triggered.

The inhomogeneously pumped laser is split into anumber of segments, as shown in the various devices ofFig. 1. Because of the different injection current densities,each segment can have a different average carrier density,and hence gain or loss. Also, below threshold, sponta-neous emission and superluminescence dominate theoptical output. Consequently, because of these twoeffects, it is necessary to include in the model both thelongitudinal variation of the photon density and thewavelength dependence of the gain and spontaneousemission of the active material.

Marcuse and Nash [9] have examined the static char-acteristics of inhomogeneously pumped lasers to explainthe asymmetric output from the facets of proton delin-eated stripe lasers. Their model included the spatialvariation of the photon density along the length of thedevice, but not the wavelength dependence of the gain.

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988 133

Marcuse [10] extended the model to include the wave-length dependence of the gain and spontaneous emission,using a numerical solution to the travelling-wave rateequations, but limited the analysis to the case of a laseramplifier.

ve segment

regrown n-GaAIAs

pumped segment

n-GaAs substrateTi:Au electrodeAu:Ge:Ni electrodeoxide isolation

p'GaAs-p-GaAIAs confininglayern-GaAIAs confining-layer

buried active layer, stripe guiding structure continuedbeneath passive segment

pGaAsp-GaAIAs confininglayer

n-GaAIAs confininglayer

pumped segment

passive segmentpumped segment

passive segment

r pumped segments

Fig. 1 Typical inhomogeneously pumped laser structures which can be

analysed using the method presented in the paper

a Two-segment, SLED type of device with rear facetb Three-segment laser [4, 5]c Multi-segment laser. All require some form of buried beterostructure for guid-ance in pumped and passive segments

In this paper a travelling wave approach is also usedto describe the growth or decay of the circulating opticalflux in the longitudinal direction due to the different gainof each segment. However the method differs consider-ably from that used by Marcuse [10], and is extended tocover the transient case. The analysis is also somewhatdifferent from those found in References 5 and 6 or byBasov [11], Paoli [12], Renner and Carroll [13] andKuznetsov [14]. Most of these authors assume a singlewavelength model to describe the gain in the pumpedand passive segments. They also assume that the photonflux density has a constant photon lifetime. Theseapproximations are valid for the laser, but not for theinhomogeneously pumped device [15]. An importantaspect of the new method is that analytic expressions forthe distribution of optical flux are used, and finite differ-ence or finite element methods are not required to calcu-late the optical flux distribution. This reduces thenumerical computation required considerably.

Inclusion of the wavelength dependence of the stimu-lated gain and spontaneous emission is achieved bydiscretising the gain and spontaneous emission character-istics into wavelength 'slots', as described in Reference 16.Each wavelength slot is sufficiently wide to encompassseveral longitudinal modes. The method of modellingcannot show the longitudinal mode behaviour of the lightonce in the lasing state. Nevertheless, it shows the grosschanges in wavelength which occur as the device switchesto the ON state and towards the steady state, which maybe of importance in some practical applications to whichthis type of switch might be applied.

Some associated benefits of the multiwavelengthmodel are that:

(a) the transient shift in wavelength of the laser outputduring switching can be shown

(b) the sensitivity of optically triggered switching to thewavelength of the incident light can be foundand

(c) the effect of optical injection into either the passivesegment or the active segment can also be considered.

Using a model which assumed either a single wavelengthor a uniform photon density this would not be possible.

2 Analysis

The types of device which may be modelled using thisapproach are shown in Fig. 1. However, a two segmentstructure, similar to the SLED, is used to illustrate themethod. Extension of the principles to several pumpedsegments is straightforward. It is also assumed that someform of strong lateral index guidance is present along theentire length of the device. This is necessary because thecurved wavefronts of the gain-guided light from theactive segments and the antiguiding nature of the passivesegments cause the light to diverge in the passive seg-ments. The resonator loss then becomes very large. Byassuming index guiding via a buried heterostructurechannel, the problem remains one dimensional along thelongitudinal direction z.

Fig. 2 illustrates the travelling optical flux method forthe two segment device. Light of each wavelength under-

cr

tance

reflec

ront

face

t,

pumped segment

carrier density = nfl

passive segment

1B y

y

—

\ c a r r i e r density = n2

C Q _ _ _ - .

1=0 2=Z, Z=Z2

P is forward travelling flux

Q is backward travelling flux

Fig. 2 Schematic of the simple two-segment laser, illustrating the

fluxes P and Q in each segment

goes gain or loss, depending upon the carrier density ineach segment, and the flux is partially reflected at the twofacets. If the flux at any point within the device (saypoints A, B or C in Fig. 2) undergoes one entire 'round-trip' in the laser cavity then the 'round-trip gain', grt, isthe same for each wavefront. The 'round-trip time' trt

taken for the light to travel around the cavity is alsoindependent of the initial position of the wavefront.

134 IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988

The problem requires the simultaneous solution ofthree nonlinear inhomogeneous differential equations;the carrier conservation equation and the two photonconservation equations for the forward and backwardtravelling fluxes, P and Q, respectively. These are solvedsubject to continuity of optical flux across each segmentboundary, and the boundary conditions at each facet.The carrier conservation equation is linked to the photonconservation equations via the stimulated gain (or loss)coefficients and the spontaneous emission curves of theactive material.

2.1 Stimulated gain and spontaneous emissioncurves

The model requires the stimulated gain and spontaneousemission curves of the active material in a functionalform. There have been a number of papers publishedwhich calculate both these curves for different bandmodels e.g. References 17-19. We are indebted to Dr. W.Liddell for permission to use gain and spontaneous emis-sion curves for GaAs [20]. His computer model assumesparabolic bands and allows for a variety of different bandtail functions. For the purpose of the model presentedhere, the exact form of the gain and spontaneous emis-sion curves is not critical. However, changes in thesecharacteristics will have an effect on the detailed shape ofthe I/L characteristics and the switching transients. Inthis paper, the Kane band tail function [21] is used, withrj = 25 meV. This value was found to be in excellentagreement with experimental results obtained for SLEDdevices obtained by Middlemast et al. [22] and also withthe results of Henry et al. [23]. The actual optical gainand spontaneous emission curves used are shown in Fig.5 and Fig. 4 of Reference 16.

For the purpose of the model, the curves are sampledinto a number of wavelength 'slots' [16] over an appro-priate range and the gain and spontaneous emissionfound as a function of carrier density. Each wavelengthslot has a width of typically 0.5-2.0 nm, depending on theapplication. For the case of the spontaneous emissioncurve it was necessary to normalise the rate of sponta-neous emission using

S(n, A,) = Br • n(n + p0) (1)

where Xt and Xu represent the lower and upper wave-length slots.

To speed up evaluation of this data, each curve wasfitted to a polynomial series for every wavelength. Afourth-order polynomial was found to be adequate to fitthe gain data but an eighth-order polynomial wasrequired to fit the spontaneous emission data.

2.2 Carrier conservation equationAt any point z, the carrier conservation equation is

dtJ{z)

e • d

-B. n(z)(n(z) + p0) (2)

where J is the current density at z. This equation doesnot contain a nonradiative recombination term but thismay be added easily.

If it is assumed that the carrier density distribution isuniform along the length of each segment and the gainand spontaneous emission curves are discretised into

wavelength slots, eqn. 2 becomes

An J •*"

- Br • nm(nm + p0) (3)

where X{ is the ith wavelength slot, m represents the wthsegment of an r segment laser and Pm(Xt) and Qm{X() arethe average photon fluxes over the mth segment, suchthat

PJt, Xd = (1/L J • I L"'P(Z, t, A,) dzJo

(4)

for a segment of length Lm. A similar solution exists for

An approximate solution to eqn. 3 can be found aftert = trt within each segment if the gain and mean photonfluxes change by a negligible amount in this very shorttime interval, which is typically a few picoseconds for a300 nm cavity length and GaAs active material:

njtrt) = nm(0) exp (-Br(nm(0) + po) • trt)

+ [l-exp(-Br(nJ0)

(5)Br • («JO) +

Other, more accurate methods for solving nm(trt) could beused, such as Runge-Kutta, however in the short timeinterval involved, the change in carrier density, and hencegain, will be very small. The steady state carrier density,nm(0) is assumed to exist before the transient in Jm, Pm orQm is applied [16].

Once nm(trt) has been found the new values of sponta-neous emission and stimulated gain may be found fromthe curves. This can then be applied to the photon con-servation equations.

2.3 Photon conservation equationsThe photon conservation equations for the forward andreverse travelling fluxes, P and Q, respectively, for the ithwavelength slot may be expressed as [24-26]

1 dP(z, t, A,-) t dP(z, t, A,)

v dt dz

= G(n(z), AJ • P(z, t, A,) + 5 • S(n(z), X{) (6)

v dt dz

= G(n(z), Xd • Q(z, t, Xd + d • S(n(z), A,) (7)

If the gain, G(n(z), X{), and the spontaneous emission,S(n(z), X^, are independent of time and space then ananalytic solution to eqns. 6 and 7 may be found. The firstcondition requires that there must be negligible change incarrier density, and hence g(n(z), A,) within k round triptimes of the optical flux, where k is a relatively smallinteger. The purpose of k is to provide a scaling factor sothat the time axis may be coarsened. For high timeresolution applications, k would be set equal to one andthe minimum time period examinable would then be trt.

The second condition requires that the carrier densityis considered uniform over some appropriate length. It ispossible to split each pumped or passive segment intosmaller regions where the uniform n[z) approximationwould certainly hold. However, it has been shown [15,

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988 135

16] that there is little error in assuming a uniform carrierdensity within an entire segment. In this case eqns. 6 and7 may be applied to each segment separately, subject tothe condition that the flux must be continuous acrosseach segment boundary, and that P and Q are related atthe facets by

and

P(0, Xd = Rt • fi(0, A,) (8)

A solution to eqn. 6 for the mth segment of an r segmentlaser, having the uniform carrier density, n(z) — nm in themth segment is

Pm(z, t, Xt) =f((v -t-z), A,-) • e x p (GJLXd

-(3 • S(Xd/GJLXd) (9)

where/((y • t — z), A,) is an arbitrary function; Gm(A,) =G(nm, A,) and SJA,) = S(nm, A,).

A similar solution exists for Q(z, t, A,). These solutionsrepresent travelling optical fluxes in the positive andnegative z direction which undergo loss or gain, depend-ing on Gm(A,), to which spontaneous emission is added.

The round trip gain for a single wavelength slot, A,, ofthis r segment laser is simply

- exp G2(Xt)L

(10)

and for k round trips the gain is:

grt(Xf = (R, • R2)k • exp G2(X()L

(11)

After k round trips time, k • trt, the flux in the positivedirection will be given by

P(z , k-trt, Xd = P{z, 0 , A,-) + (12)

where spon(Xi) is the spontaneous emission added to thephoton flux in the ith wavelength slot after k round trips.A similar expression may be derived for Q{z, k • trt, A,) inthe negative z direction.

2.4 Spon taneous emissionThis is particularly important in the inhomogeneouslypumped laser because spontaneous radiation and super-luminescence are dominant before the device switches tothe lasing state. In this paper it is necessary to calculatethe transient change in the longitudinal distribution ofthe spontaneous emission, and this is an essentially differ-ent problem from the one studied by Marcuse [10] forthe laser amplifier. In both cases, however, spontaneousradiation generated within the device is amplified (orattenuated) as it propagates longitudinally through theregions of gain (or loss).

Assuming that current is applied to all segments attime t = 0. At t = 0 + , each segment of the device hasmarkedly different levels of spontaneous emission andoptical gain because of the different levels of pumping.The problem is to find the distribution of spontaneousradiation at the next time step which occurs after k roundtrips, i.e. k • trt. There are three contributions to the buildup of spontaneous light during this period. In order toexplain these contributions the transient change in theoptical flux, P, crossing the boundary between the mthand (m + l)th segments is considered.

At t = 0 + the spontaneous radiation at the boundarywill be due to the wavefront actually generated at the

boundary. A short time interval later this wavefront willhave travelled into the (m + l)th segment as part of its'round-trip'. At the same time, the light reaching theboundary will be due to a wavefront which at t = 0 orig-inated further inside the mth segment. Consequently, thiswavefront has experienced gain, or loss, depending on thelevel of pumping. At t = LJv, where Lm is the length ofthe mth segment, the light appearing at the boundary isdue to a wavefront originating at the boundary of the(m — 1) and m segments. This represents the maximumoutput due to photons originating from within the mthsegment, and light at the boundary due purely tophotons spontaneously generated in the mth segmentreaches a 'steadystate' value. For t > LJv, light reachesthe boundary due to spontaneous radiation initially gen-erated in the (m — l)th segment. This also undergoes gainwithin the mth segment and adds to the spontaneousradiation generated within the mth segment. After onecomplete round trip, the light reaching the boundary isnow the wavefront originally generated at the boundaryat t = 0, but this has now been amplified by the round-trip gain of the device. This process of amplification con-tinues for all the wavefronts generated in the initial singleround trip until k round trips have elapsed. This consti-tutes one component of the transient change in the fluxdistribution.

However, for each round trip of the original flux, newspontaneously generated photons are adding to it, andthey undergo round trip gain also. This constitutes thesecond component of the transient.

Finally at t = k • trt it is necessary to calculate the newdistribution of the spontaneous emission along the lengthof the device. The above arguments are also appliedsimultaneously to the flux Q travelling in the negativedirection.

The analysis of the transient changes in the longitudi-nal distribution of spontaneous emission is detailed in theappendix. From this, the spatial distribution of the totalphoton flux travelling in the positive z direction of thefirst segment of a two segment laser, P l 5 due to the spon-taneous radiation, obtained after k round trips is givenby

x {[( / iW • exp (G2(Xi)L2)

+ f2{Xd)R2 • exp (

1 •exp(G1(A,.)z)

x ri — Q (AVn

where/^A,) is given by

3^ • [ e x p ( G 1 ( A i ) L 1 ) - l ]

and/2(A,)is

/2(A.) = d • [exp 1]

(13)

(14)

(15)2(Aj)

The carrier conservation equation (eqn. 2) only requiresthe average flux in each segment, found using eqn. 4.Thus, after k round trips the average forward travelling

136 1EE PROCEEDINGS, Vol. 135, Pi. J, No. 2, APRIL 1988

flux P in region 1 is

Pi(k • trt, Xt) =

x { [ ( / A ) • exp (G2(A,.)L2)

+f2(Xi))R2exp(G2(Xi)L2)

<5 • S^A;) (exp (GiiX^Li — 1)

x[l- (16)

The total output power from either facet may now becalculated using eqn. 13. The light output from the frontfacet is simply

Gi(0, (17)i = Aj

Similar expressions must be then found for the averageflux P2 in region 2 and for flux Q and Q. For the moregeneral r segment laser, functions f3(Xt) through to /r(A,)need to be defined, and the number of terms in eqns. 13and 16 increase accordingly. This is straightforward,albeit tedious.

The equations above, representing the transient carrierdensity and optical flux distributions, reduce to thesteady state equations derived in Reference 16 as t -> 00,if the device is well behaved and stable (i.e. does notexhibit self sustained oscillations).

3 Modelling procedure

A two segment laser is modelled as an example. Thesteady state initial conditions are first set up using thetechnique detailed in Reference 16. This involves apply-ing an electrical pre-bias to each electrode and includingthe effect of steady-state optical injection into either ofthe facets. The coupled rate equations are solved using aMiiller technique at this stage. The problem is organisedso that the light is the dependent variable, from which thecorresponding single value of current is calculated. Thisis necessary because of the two-valued nature of the lightoutput when the device switches regeneratively. For thetransient problem, each device can be prebiased to apoint on the I-L characteristics just below threshold. Theequations developed above, describing the transientbehaviour of n, P and Q, may then be applied from thissteady state condition for step or pulse changes in eitherthe current density or optical injection level to switch thedevice into the lasing state.

The gain and spontaneous emission curves are splitinto a number of wavelength slots over a range of wave-lengths from 780 to 840 nm, with each slot typically 0.5-2.0 nm wide. Because a wavelength slot is likely toencompass a number of longitudinal modes, the methodof modelling cannot show the modal behaviour of theoutput once in the lasing state. Nevertheless, the resultantspectrum is representative of the optical power in the lon-gitudinal mode.

A factor governing the accuracy of the model is thechange of photon flux after a step change in current. Themodel assumes that the changes in flux result as a conse-quence of changes in gain, which is determined by the

IEE PROCEEDINGS, Vol. 135, PL J, No. 2, APRIL 1988

loss in the passive region. If rapid optical/optical orcarrier/optical interactions occur in less than k round-triptimes they cannot be detected using this model.

3.1 Electrically triggered switchingA small step change of current in segment 1 is used toinitiate switching. The carrier conservation equation issolved numerically using the steady state values ofaverage photon flux P and Q and the new current level, toobtain the new average carrier density in each segmentafter k round trips time. From this, the correspondinggain and spontaneous emission can be found for all X(

and the new value of round-trip gain found using eqn. 11.The new value of average flux can then be found for bothregions after k round trips time using equation 16. This isthen used in the carrier conservation equation, which issolved to find «1(2/ctrr) and n2(2ktrt), and so on.

3.2 Optically triggered s witchingExternal injection of light at either, or both, facets isaccomplished by considering each facet to be equivalentto a segment boundary. External optical injection at thefront facet, say, is just the flux PO(XX), where Xx is thewavelength of the external light source. Continuity of fluxacross this boundary is assumed in the normal way, andthe light actually injected into the pumped segment isJ'oOU ' (1 ~ R-i)- Depending on the wavelength of theinjected light, either absorption occurs and the segmentbecomes optically pumped, or the light undergoes ampli-fication in the region of gain and becomes absorbed inthe passive segment. The net effect in both cases is thatthe cavity loss is reduced.

A similar effect occurs for injection into the rear facet.For a two segment laser the rear segment would not beelectrically pumped and because of this there would beloss at all wavelengths and only absorption of theinjected light would occur. In this case the externallyinjected light would be, Q2(XX), and the amount injectedinto the passive segment, Qz(Xx) • (1 — R2).

Switching occurs for both cases if the optical pumpingincreases the round trip gain sufficiently to overcome thecavity losses.

4 Results

Table 1 details the electrical and optical parameters usedin the analysis. For all the results presented, the timeinterval was set to a single round trip time (i.e. k = 1), toensure maximum temporal resolution. For the case of

Table 1

= 0.75

w = 1 //md = 0.3 fjm

electrically induced switching, 31 wavelength slots wereused covering the range 780-840 nm, but for opticallyinduced switching 151 wavelength slots were used overthe same range. The transient response of each of the fol-lowing devices was obtained.

4.1 LaserFig. 3a shows the static I/L characteristics of a 200 jumlong laser using the model. The device was pre-biased to

137

a current of 2.82 mA, to give an optical output of 10 4

mW/facet. Fig. 4 shows the optical transient to a stepchange in electrode current to 7.96 mA (corresponding to

10

10

.•§> io2

10*

b inhomogeneously pumpedlaser with 10pm passivesegment and 200|jm activesegment

"a 200um laser

Hi-

c inhomogeneously pumpedlaser with 60um passivesegment and 200um activesegment

6 7 8 9

current.mA

10 11 12 13

Fig. 3 I/L characteristics of various devices

a 200 fim long laserb Two-segment device with a 200 /jm pumped segment and a 10 nm passivesegmentc Two-segment device with a 200 fim pumped segment and a 60 /im passivesegment.

10

| 10°

1

I 1 0

I fo2

S1 -310

410

time.ns6

Fig. 4 Optical transient from the front facet of the laser after beingelectrically switched into the on state from an off-state pre-bias current of2.82 mA to a new current of 7.96 mA

a Transient change in the emitted spectrumb Transient of the total light output

a steady state output power of 1 mW at the front facet).The results show the damped oscillatory response andthe sharpening of the spectrum as the device begins tolase. Fig. 4b shows the transient of the total light output.

4.2 Inhomogeneously pumped device with shortpassive segment

This device has the same length pumped segment asabove, but has a 10 fim passive segment. The modelignores the effect of diffusion of carriers from the activeinto the passive segment and also current spreading. Forthis geometry these approximations may lead to a signifi-cant error, nevertheless, the model gives an insight intothe dynamic behaviour of a laser which has undergonedamage or ageing. The static I/L characteristic of thisdevice is shown in Fig. 3b. The device is pre-biased at2.91 mA, as shown, with an optical output of 10 ~4 mWat the front facet.

The transient behaviour of the device to a step incurrent to 8.34 mA (corresponding to 1 mW steady stateat the front facet) is shown in Fig. 5. One striking feature

10

o -2£ 1 0g>

-3.10-i,10

0 1 2 3 A 5 6 7time.ns

b

Fig. 5 Optical transient from the front facet of the two-segment device,with 10 iim passive segment after being electrically switched into the onstate from an off-state pre-bias current of 2.91 mA to a new current of8.34 mA

a the transient change in the emitted spectrum andb the transient of the total light output

is the initial transient which differs from the laser tran-sient of Fig. 4. The spectral peak during this initial tran-sient is slightly different from the steady state lasingwavelength. As the device begins to lase the strong stimu-lated recombination depletes the gain in the pumpedregion and this gives the large drop in light output afterthe initial peak. There then follows the usual under-

138 IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988

damped response towards the steady state. The responseis more oscillatory than the response of the laser.

4.3 Inhomogeneously pumped laser with a 60passive segment

The steady state characteristics of the device are shown inFig. 3c. The device is pre-biased, as shown, (mid-waybetween the upper and lower threshold currents) to givean output of 6.3 x 10~3 mW at the front facet for aninjection current of 10.3 mA. Figs. 6-9 show the transient

10

3101

1

io H

10

I ioi

•j? 10--710.

time.nsb

Fig. 6 Optical transient from the front facet of the pre-biased two-segment device, with 60 urn passive segment after being electricallyswitched into the on state from an off-state pre-bias current of 10.3 mA toa new current of 12 mA

a Transient change in the emitted spectrumb Transient of the total light output

behaviour to different values of step current. In each case,curve b shows the transient of the total light output fromthe front facet, calculated using eqn. 17

The transient response is similar to that obtained forthe short passive region, but the effects are exaggerated.The major difference is that below threshold the devicewith the longer passive region behaves similarly to asuperluminescing diode, emitting light of a relativelyshort wavelength. This wavelength is greatly attenuatedin the passive region, and as a result the initial lasingfrequency can be seen to peak at a much longer wave-length than the peak of superluminescence. For thisgeometry, the shift in wavelength is approximately 8 nm.There is also a slight shift in lasing wavelength of about0.5 nm as the device settles to the steady state due to theslight changes in carrier density within the two regions asthe steady state is approached.

Also apparent is the rapid turn-off after the device hasstarted to lase. This is more pronounced than for the

device with the shorter passive region, and has a longerrecovery time, but the explanation is similar for bothdevices. The effect is similar to Q switching in lasers. Here

Fig. 7 Optical transient from the front facet of the pre-biased two-segment device, with 60 nm passive segment after being electricallyswitched into the on state from an off-state pre-bias current of 10.3 mA toa new current of 13 mAa Transient in the emitted spectrumb Transient of the total light output

the cavity gain is controlled by the non-linear absorptionregion of the passive segment. The explanation is asfollows:

Just before the device switches, the gain in the pumpedsegment is very large, almost compensating for the loss inthe passive segment. However, when the device isswitched by increasing the gain further, the loss in thepassive segment suddenly reduces. There is an excess ofgain and a very high density of photons is produced. Thisis shown on the transients by the very fast growth of theinitial lasing peak. This has two consequences. First, thepassive region is being optically pumped at wavelengthswhere the optical loss at this (transient) lasing wavelengthis becoming less. The carrier lifetime of the passive regionis likely to be rather long and so this region can bethought of as a reservoir of carriers. Consequently, oncethis region has been optically pumped to transparency, itstays in this state for a relatively long time. At the sametime the large increase in photon density within the res-onator leads to the carrier density of the pumpedsegment being depleted very quickly. The net effect is thatthe overall gain falls by so great an extent that it dropsbelow that normally required to sustain lasing, and thereis a rapid loss of light leaving the resonator.

It should be noted that as a result of the depletion ofcarriers in the pumped segment, and the resulting shift in

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988 139

the gain curves, the light output due to super-luminescence decreases around the initial peak in thetransient. The spontaneous emission from the pumped

(a) the initial delay before the device starts lasingdecreases, as expected

(b) the depth of the 'turn off' in light output increasesquite markedly

10

$ 1E 10

| 10

I -38 1 0

time/is

Fig. 8 Optical transient from the front facet of the pre-biased two-segment device, with 60 \im passive segment after being electricallyswitched into the on state from an off-state pre-bias current of 10.3 mA toa new current of 14 mAa Transient change in the emitted spectrumb Transient of the total light output

segment then falls to a very low value, indicative of thefall in carrier density in this segment. This is seen in thecurves of the transient. Remembering that the passivesegment is still moderately transparent to light at lasingwavelengths, the carrier density in the pumped segmentincreases through the relatively slow injection of carriers,and lasing is possible once again when the gain in thepumped segment has increased sufficiently. However,there is now less loss to overcome in the passive region,and so a much reduced carrier density is required in thepumped region to maintain the resonator round-trip gainclose to unity (this is illustrated by the spontaneous emis-sion staying at a very low level). Consequently, the 'excessgain' is much less when lasing is re-established after theinitial switch-off transient.

The effect of the length of the passive layer may beexplained as follows. For long passive regions the level oflight required to optically pump the region must be con-siderably higher than for short passive regions. Conse-quently, at the lasing condition, the excess gain, andhence photon density, must be larger and the resultingdepletion of gain in the pumped segment much deeper,which means that the time taken to re-establish the lasingcondition must be longer.

As the step change in current is increased, it is seenthat:

10

0J10 1

I -1o 10-1a

£ -5.? 10"

100 2 Z, 6 8

time.nsb

Fig. 9 Optical transient from the front facet of the pre-biased two-segment device, with 60 \im passive segment after being electricallyswitched into the on state from an off-state pre-bias current of 10.3 mA toa new current of 15 mAa Transient change in the emitted spectrumb Transient of the total light output

(c) the recovery time back to the subsequent stable,but underdamped, lasing condition initially increases, butat higher step currents reduces.

4.4 Optically induced switching in aninhomogeneously pumped laser

This is illustrated for the case of a device with a 200 fj.mpumped segment and a 60 fxm passive segment. Thestatic light input-output characteristics from the frontfacet of this device is shown in Figs. 10a and 106 forinjected light of several wavelengths. Fig. 10a is for injec-tion into the pumped segment through the front facet,and Fig. 106 is for injection into the passive segmentthrough the rear facet. In both cases the device is electri-cally pre-biased by a current of 9.27 mA, which is belowthe lower (i.e. OFF state) threshold current. The threewavelengths chosen represent; (i) a wavelength shorterthan the superluminescing wavelength, (ii) the peaksuperluminescing wavelength and (Hi) the lasing wave-length. Attention is drawn to the output limiting actionof the device when light of the lasing wavelength isinjected into the front facet of the device, as shownexperimentally by Kawaguchi [7]. Even for light ofsuperluminescing wavelengths some limiting is seen.

To achieve the transient response, the device is bothelectrically and optically pre-biased at the appropriatewavelength, as shown in Fig. 11. Im represents the optical

140 IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988

pre-bias level, and is chosen to be the geometric meanbetween the upper and lower threshold values of lightinput. This is because of the very wide range of lightinput level between the upper and lower thresholds.

10light input power, mW

10 10

tight input power, mW

b

Fig. 10 Optical input-output characteristics of a two-segment laserwith a 200 pm pumped segment and a 60 urn passive segment, electricallypre-biased by a current of 9.27 mAfor three characteristic wavelengthsinjecteda into the pumped segment from the front facetb into the passive segment from the rear facet

Transient changes in the total light output from thefront facet of the device have been obtained for the fol-lowing cases of optical injection: (a) at the lasing andsuperluminescing wavelengths into the rear facet for twodifferent optical injection levels, and (b) at the super-luminescing and short wavelengths into the front facet,also for two different power levels. The two power levelsused to initiate switching are marked on Fig. 11, and are

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988

determined using the following expressions

U = h XU = h (18)

light input/output charcteristicfor Xx and electrical pre-bias

light input (log scale)

Fig. 11 Schematic diagram illustrating how the optical pre-bias levelIm and the two step changes of optical injection level, I3 and / 4 are definedfor each wavelength, at a particular electrical pre-bias level

4.4.1 Light at the lasing wavelength injected into thepassive segment: The transient for this condition isshown in Fig. 12. At this wavelength, much of the inputlight is transmitted through the device, and appears atthe front facet within half a round trip time. The switchon transient then follows. The response is well damped,more so for the lower level of injection. A feature ofthe results for both input levels is the very long switch ontime. These particular results should be treated withcaution, however, because the analysis lumps together theeffects of each longitudinal mode in the device by the useof the finite width wavelength slot, and the phase reson-ance condition is not considered. In practice the responsewill be a combination of absorptive and dispersivebistability. Such problems do not exist for injection atother wavelengths.

4.4.2 Light at the peak superluminescence wavelengthinjected into the passive segment: These transients areshown in Fig. 13 for both optical injection levels. Theyshow the normal underdamped response. The turn ontime is shorter than the previous example. However, thisis strongly dependent upon the injection level. The tran-sients are similar to those obtained for a step change incurrent, shown in Figs. (6-9). This is not surprising sincethe optical injection at this wavelength is largelyabsorbed in the passive region, whereupon the carrierdensity in this region is increased.

4.4.3 Light at the peak superluminescing wavelengthinjected into the pumped segment: These transientsare shown in Fig. 14. A feature is the self-sustained pulsa-tions for the lower of the two injection levels, whichbecome damped at the higher injection level. The reasonfor this is discussed below.

4.4.4 Light at a short wavelength injected into thepumped segment: These transients, shown in Fig. 15,are similar to Fig. 13. Here, light is strongly absorbed inthe pumped segment, and this increases the gain.

141

4.5 Discussion of resultsThe transient response of the device to either electricallytriggered or optically triggered switching is very similar,

10.2 -

i .10

i_- o| 10oQ.

g 10 -

cn

,62

To3

10 20 30 40 50 60time.ns

a

70 80

The observation of self-sustained oscillations, due tooptical injection of light at the superluminescing wave-length, is of particular interest, and the dependence of

2 - ,10

10

I io°

a"5 -1o 102:cn

- 2

10

-310

10 12

time.nsa

2

10 1

110

10 ~

5a.

° 10cn

-210

-310

10 20 30 40 50time.ns

b

60 70 80

Fig. 12 Transient in the total light output from the front facet of atwo-segment laser with a 200 /im pumped segment and a 60 nm passivesegment, electrically pre-biased by a current of 9.27 mA and optically pre-biased with light of 824 nm wavelength at a power of 8.96 p.W into therear facet due to an increase in the optical injection power to

a IOOJAV

h 224 /iW

and optical triggering does not induce a faster response.The reason for this is that the effect of the optical injec-tion is simply to optically pump the device to a largergain, which is precisely the same as electrical inducedswitching. Wavelength has an effect simply in terms ofthe extent to which the injected light is absorbed in eachregion. It should be noted that increasing the length ofthe passive region generally reduces the damping of thetransient.

10

10 -

(D 0

i 1 0

Q.

1b2-

io3

10 12

time.nsb

Fig. 13 Transient in the total light output from the front facet of atwo-segment laser with a 200 \an pumped segment and a 60 fxm passivesegment, electrically pre-biased by a current of 9.27 mA and optically pre-biased with light of 804 nm wavelength at a power of 157 pW into therear facet due to an increase in the optical injection power to

a 634 (iV/b 1.01 raW

this on the input injection level illustrates the nonlin-earity of the original problem. The self-sustained oscil-lations occur because of two competing mechanisms.Light at the superluminescence wavelength is amplifiedby the gain in the pumped segment, depleting the carriersslightly. This light is then completely absorbed in thepassive segment, pumping the passive region to transpar-ency for light at the lasing wavelength. Lasing is nowpossible as the round trip gain overcomes the loss at thelasing wavelength. The light in the cavity builds up,

142 1EE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988

thereby depleting the carriers in the region of gain. Thesuperluminescence is not now amplified to the sameextent, and does not pump the passive region. Conse-

10%

10 -

10

-110

-2

io i

-3

10-

10 -I

I 10Q.

a

I To'cn

<b3i

10I12 16

time.nsa

10 12 U 16

time.nsb

Fig. 14 Transient in the total light output from the front facet of atwo-segment laser with a 200 nm pumped segment and a 60 nm passivesegment, electrically pre-biased by a current of 9.27 mA and optically pre-biased with light of 804 nm wavelength at a power of 37.7 [iW into thefront facet due to an increase in the optical injection power to

a 73.5 /iW/>91.6/AV

quently, the round trip gain falls for the lasing frequency,and the light output with it. The gain in the pumpedregion now increases. The input light is still present,however, and the above process is repeated.

The light externally injected into the device acts as anoptical control of the self-sustained optical oscillation,and this may well have interesting systems applications.It will be appreciated that the mechanism for opticallycontrolled self oscillation is subtly different from thatcausing transient turn-off, discussed in Section 4.3. For

light of shorter wavelength, this light is purely absorbedin the appropriate segment, and there are no longer twowavelengths competing for gain. This also explains why

10

10 -

1 ,0°a

" -1

r 10

10

io3

time.nsa

10 12 U 16

time.nsb

Fig. 15 Transient in the total light output from the front facet of atwo-segment laser with a 200 nm pumped segment and a 60 nm passivesegment, electrically pre-biased by a current of 9.27 mA and optically pre-biased with light of 784 nm wavelength at a power of 1.02 mW into thefront facet due to an increase in the optical injection power to

a 8.99 mWb 18.6 mW

the self-pulsations stop if the optical injection level is toohigh. In this case, there is significant depletion in theregion of gain, and this becomes the dominant effect.

5 Conclusions

The paper has presented a new model of the transientbehaviour of absorptive bistability in inhomogeneouslypumped lasers which includes the wavelength dependenceof the gain and spontaneous emission curves of the activelayer. The results show the increasingly underdampedresponse of the transient as the length of the passive layer

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988 143

is increased. The initial switch-on time is seen to bedependent upon the size of the step current, as expected.In particular, for devices with a passive segment, there isa shift in wavelength as the device switches to the onstate, which is more marked for devices with longerpassive regions owing to the transient change in carrierdensity in each segment, and the consequent effect on thegain and spontaneous emission. It is also seen that thedevice effectively switches off immediately after initiallystarting to lase. The recovery time from this is severalnanoseconds, which lengthens the overall switch on timeconsiderably.

Optically induced switching has also been demon-strated, and the switching time is found to be similar tothe electrically switched case. Optically controlled self-sustained oscillations are reported, and the stability ofthese is found to be dependent upon the wavelength andmagnitude of the injected light.

6 Acknowledgments

The authors wish to thank Dr. W. Lidell for provision ofthe gain and spontaneous emission data for GaAs, and toProfessor T.E. Rozzi, Dr. J. Sarma and Dr. I. Middlemastfor their many helpful comments.

7 References

1 GIBBS, H.M., McCALL, S.L., and VENKATESAN, T.N.C.: 'Opti-cally bistable devices: the basic components of all-optical systems?',Opt. Eng., 1980,19, pp. 463-468

2 NAKAI, T., OGASAWARA, N , and ITO, R.: 'Optical bistability insemiconductor laser amplifiers', Jpn. J. Appl. Phys., 1983, 22, pp.L31O-L312

3 HARDER, C.H., LAU, K.Y., and YARIV, A.: 'Bistability and pulsa-tions in CW semiconductor lasers with a controlled amount ofsaturable absorption', Appl. Phys. Lett., 1981, 39, pp. 382-384

4 LAU, K.Y., HARDER, C.H., and YARIV, A.: 'Dynamical switchingcharacteristic of a bistable injection laser', Appl. Phys. Lett., 1982,40, pp. 198-200

5 HARDER, C.H., LAU, K.Y., and YARIV, A.: 'Bistability and pulsa-tions in semiconductor lasers with inhomogeneous current injec-tion', IEEE J. Quantum Electron., 1982, QE-18, pp. 1351-1361

6 KAWAGUCHI, H.: 'Optical bistable-switching in semiconductorlasers with inhomogeneous excitation', IEE Proc. I, 1982, 129, (4),pp. 141-147

7 KAWAGUCHI, H.: 'Optical input-output characteristics forbistable semiconductor lasers', Appl. Phys. Lett., 1982, 41, pp.702-704

8 KAWAGUCHI, H.: 'Optical bistable-switching and chaos in asemiconductor laser with a saturable absorber', Appl. Phys. Lett.,1984, 45, pp. 1264-1266

9 MARCUSE, D., and NASH, F.R.: 'Computer model of an injectionlaser with asymmetrical gain distribution', IEEE J. Quantum Elec-tron., 1982, QE-18, pp. 30-43

10 MARCUSE, D.: 'Computer model of an injection laser amplifier',IEEE J. Quantum Electron., 1983, QE-19, pp. 63-73

11 BASOV, N.G.: 'Dynamics of injection lasers', IEEE J. QuantumElectron., 1968, QE-4, pp. 855-864

12 PAOLI, T.L.: 'Saturable absorption effects in the self-pulsing(AlGa)As injection laser', Appl. Phys. Lett., 1979, 34, pp. 652-655

13 RENNER, D., and CARROLL, J.E.: 'Transients in injection lasers— phase plane analysis and effects of absorbing regions', Solid-stateand Electron Devices, 1979, 3, pp. 224-232

14 KUZNETSOV, M.: 'Theory of bistability in two-segment diodelasers', Opt. Lett., 1985,10, pp. 399-401

15 PERKINS, M.C., ORMONDROYD, R.F., and ROZZI, T.E.: 'Theeffect of photon lifetime on absorptive bistability in inhomoge-neously pumped lasers', Electron. Lett., 1985, 21, pp. 857-858

16 PERKINS, M.C., ORMONDROYD, R.F., and ROZZI, T.E.:'Analysis of absorptive bistable characteristics of multi-segmentlasers', IEE Proc. J, 1986,133, (4), pp. 283-292

17 LASHER, G., and STERN, F.: 'Spontaneous and stimulated recom-bination radiation in semiconductors', Phys. Rev., 1964, 133, pp.A553-A563

18 STERN, F.: 'Effect of band tails on stimulated emission of light insemiconductors', Phys. Rev., 1966,148, pp. 186-194

19 STERN, F.: 'Calculated spectral dependence of gain in excitedGaAs', J. Appl. Phys., 1976, 47, pp. 5382-5386

20 LIDDELL, W.: Ph.D. thesis University of Bath, October, 198621 KANE, E.O.: 'Thomas-Fermi approach to impure semiconductor

band structure', Phys. Rev., 1966,131, pp. 79-8822 MIDDLEMAST, I., SARMA, J., and KAMBAYASHI, T.: 'A com-

prehensive study and characterisation of superluminescent lightemitting diodes', IEEE, TELD '82, Ottowa, Canada, 1982

23 HENRY, C.H., LOGAN, R.A., and MERRIT, F.R.: 'Measurementof gain and absorption spectra in AlGaAs buried heterostructurelasers', J. Appl. Phys., 1980, 51, pp. 3042-3050

24 PANTELL, R.H., and PUTOFF, H.E.: 'Fundamentals of quantumelectronics' (John Wiley and Sons, New York, 1969)

25 HASUO, S., and OHMI, T.: 'Spatial distribution of the light inten-sity in the injection laser', Jpn. J. Appl. Phys., 1974, 13, pp. 1429—1434

26 WHITEAWAY, J.E.A., and THOMPSON, G.H.B.: 'Optimisationof power efficiency of (GaAl)As injection lasers operating at highpower levels', IEE J. Solid-state & Electron Dev., 1977, 1, (3), pp.81-88

8 Appendix

To calculate the transient change in the distribution ofspontaneous emission spon(Xi) it is necessary to find thearbitrary function, f((v • t — z), A,) in eqn. 12, for eachsegment. At t = 0 for the mth segment, eqn. 9 becomes

Pm(z, 0, A,-) = / ( - z , A,-) • exp

- {S •

hence

z)

(19)

x exp(-Gm(Ai) • z) (20)

and

Pm(z, t, A,) = Pm((z -vt), 0, A,) • exp (Gm(^) • vt)

+ {exp (GM -vt) - 1} • 6J;^i)

(21)

Using the initial condition that the spontaneous emissionis zero for t ^ 0 then

spm(z, t, A,-) = {exp - 1} (22)

This represents the initial growth of the spontaneousemission in the mth segment due to photons generated inthe mth segment. Thus, the first contribution to the spon-taneous emission, that of photon flux which originated insegment m over the time period, (LJv) ^ t ^ 0 is

spam(z, t, A,) = {exp (Gm(X{) • vt - 1} • w '

valid for Zm ^ z ^ vt ^ Zm_x where Zm is the actual dis-tance of the mth segment boundary from the front facet.spajz, t, A,) may be used as an initial condition for the

144 IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988

complementary function, and eqn. 9 used to calculatehow the flux propagates into the m + 1 segment. Withsome manipulation it may be shown that after k roundtrips, the distribution of spontaneous emission within themth segment due to spontaneous emission originating inthe mth segment is

i s :

spam(z, ktrt, Xt) = (GM

x (Zm - z)) - 1] • grt{X?

x exp (Gm{Xi) • (z — Zm)) (23)

valid for Zm ^ z ^ Zm_l5 where Ld is the total length ofthe device. The second mechanism where spontaneousemission contributes to the total flux after k round tripsis as a result of spontaneous emission which originated atk • trt ^ t ^ LJv. The significance of this time is that thespontaneous emission from the mth segment is constant.For each round trip of the original flux new photons aregenerated spontaneously, and these undergo round tripgain also. After the k round trips of the original photons,the contribution to the flux from these new photons willbe:

[0A)k"~' + 0 M T 2 + • • • + gM)2 +

x exp (Gm(Xt) • (z - Z J )

To simplify, let:

<5 • Sm

fM =

x [exp (Gm(A,-)Lm - 1] (24)

- • [exp (Gm(Xi)Lm) - 1] (25)

The total flux due to this second mechanism must takeinto account the contribution of flux from the other seg-ments as well as the contribution of spontaneous emis-sion from the negative z direction (i.e. from Q). For a twosegment laser the spontaneous emission in the firstsegment after k round trips due to this second mechanism

spbx{z, ktrt,

x exp

x exp

exp

• exp (G

A,) • L2)

J +/1(AI)}

z)-^(A,)(z - ZJ)

R

(26)

A similar expression exists for the spontaneous emissionspb2{X^ in the second segment.

The third contribution to the spontaneous emission isfrom spontaneous emission in the first segment due to thespatial variation of the spontaneous emission at the endof the time interval ktrl. This is

spc^z, ktrt, X{) = [exp - 1] (27)

a similar expression exists for the distribution of sponta-neous emission in all the other segments.

The total positive travelling flux in segment 1 due tothese three spontaneous contributions is

sptx{z, ktrt, X{) = spa^z, ktrt, X()

spbx{z, ktrt, A,) , ktrt, X{) (28)

Similar expressions may be derived for the spontaneousradiation in the reverse direction of segment 1, and forboth directions in segment 2, to give sqt^z, ktrt, A,),spt2(z, ktrt, Xt) and sqt2(z, ktrt, Xt) respectively. As thenumber of segments is allowed to increase, the complex-ity of the spontaneous term increases due to the extrasources of spontaneous radiation and gain. After somemanipulation the total photon flux, P, travelling in thepositive direction after k round trips, for a two segmentlaser is eqn. 13 of the main text. A similar expressionmust be derived for the total photon flux, Q, travelling inthe negative direction.

IEE PROCEEDINGS, Vol. 135, Pt. J, No. 2, APRIL 1988 145