Measurements of dg/dN and dn/dN andtheir dependence on photon energy in

X =1.5/ im InGaAsP laser diodesL.D. Westbrook, B.Eng.. Ph.D.

Indexing terms: Diodes, Lasers

Abstract: The first measurements of the dispersion of dg/dN and dn/dN, the variation in the gain and refractiveindex with injected carrier concentration, are reported for X = 1.5 fxm InGaAsP laser diodes. Particular atten-tion has been paid to the accurate evaluation of the injected carrier concentration N through the direct mea-surement of the carrier lifetime coupled with the use of the ridge-waveguide laser structure, which benefits fromreal-index lateral waveguiding, together with low parallel leakage currents and low parasitic capacitance. Thevalues of dg/dN and dn/dN, at the lasing wavelength, were determined to be 2.7 x 10~1 6cm2 and— 1.8 x 10"20 cm3, respectively.

1 Introduction

The modelling of the behaviour of semiconductor optoel-ectronic components forms an intrinsic part of their devel-opment and subsequent application. This is especially truefor components destined for long-wavelength optical com-munication systems, where the system performance can belimited by that of the devices themselves. Fundamental tothese models are measured data for the device materialitself, and, in semiconductor lasers and laser amplifiers,this means measurements of the gain, carrier lifetime andrefractive index and their dependence on the injectedcarrier density. With this data we can explain and predictsuch properties of the laser as threshold current, periodand damping of the relaxation oscillations, mode spectrabelow and above threshold, amplitude and frequencymodulation characteristics and static and dynamic line-widths. We can also predict the amount of amplification,maximum output power and degree of Instability/multistability in semiconductor-laser optical amplifiers.

Data for InGaAsP laser material at X ~ 1.55 /mi is par-ticularly important because this is the wavelength ofminimum loss in silica-fibre based optical communicationsystems. Calculations of dg/dN based on known data forthe band structure of InGaAsP were performed by Adamsand Osinski [1] and by Dutta [2], who estimated the gain

dg/dN,2

to be 5 x 10"1 6 cm andconstant2.5 x 10"1 6 cm2, respectively, whereas measurements per-formed by Stubkjaer [3], on the other hand, on 1.6 ^mburied heterostructure lasers gave dg/

10~16 cm2. Stubkjaer also estimated dn/dN to10~20 cm3, however, subsequently, Bouley et

have reported a value of dn/x 10 20 cm3 for 1.5 ^m stripe lasers whereas

10-20

dN = 1.2 xbe -0 .7 xal. [4]dN = - 1 .5Mannining [5] reported dn/dN to be —2.8 x 1U""" in1.3 nm InGaAsP broad-area lasers, four times that report-ed by Stubkjaer.

These large variations in the values of dg/dN and dn/dNin the literature underline the need for their proper valuesto be established, and this is the primary purpose of thework described in this paper. My results show that bothdg/dN and dn/dN are, in fact, approximately a factor of 2larger than those reported by Stubkjaer. Moreover, inaddition to measuring dg/dN and dn/dN at the lasing

Paper 4497J (El3), first received 16th September and in revised form 5th December1985The author is with British Telecom Research Laboratories, Martlesham Heath,Ipswich 1P5 7RE, United Kingdom

wavelength only, I report for the first time the dispersionof these parameters in InGaAsP with photon energy.

It seems likely that the discrepancies in the data report-ed in the literature are due to the difficulty in estimatingthe carrier concentration N. Estimation of N requires aknowledge of both the injected current density J and thecarrier lifetime T. Significant errors in J can be introduceddue to undesirable leakage currents in current blockinglayers in buried heterostructure lasers, or due to currentspreading in conventional stripe lasers. Errors in the life-time T, which is usually measured by some form of fast-pulse technique, can be introduced by unwantedcapacitance (often parasitic capacitance) which slows thedriving current pulse while indirect measurements of thecarrier lifetime, such as laser turn-on delay measurements,are also open to various interpretations, depending on thecarrier recombination model chosen.

In this work, therefore, particular attention has beenpaid to the determination of the carrier concentration. Inparticular, we have incorporated the small signal techniquereported by Su et al. [6] for the direct measurement of thecarrier lifetime T. In addition, we have used ridge-waveguide lasers for our measurements (Fig. I), therebyavoiding problems associated with carrier leakage andparasitic capacitance which exist in buried heterostructurelasers (such as used by Stubkjaer [3]) while maintainingthe advantages over broad area and conventional stripelasers of real index waveguiding and negligible currentspreading.

Fig. 1 Ridge-waveguide laser structure

(a) p + — InGaAs(/?) p - InP(c) p - InGaAsP, / = 1.15 /<m(d) u - InGaAsP, / = 1.53 /mi(e) n - InGaAsP, k = 1.15 /im(/) n+ - InP

IEE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986 135

2 Determination of the carrier density

2.1 MethodWe have mentioned that the determination of the injectedcarrier density itself is perhaps the biggest problem in themeasurement of dg/dN and dn/dN. The problem lies withestimating the carrier density from the injected current.Observation of the spontaneous-emission profile belowthreshold for the ridge-waveguide laser demonstrates thatthe electron concentration is not a simple rectangular func-tion of position. The shape of the electron concentrationprofile N(x), and the spontaneous emission profile Sp(x),results from lateral diffusion of carriers once injectedunderneath the ridge (Figure 2a).

ridge width

Fig. 2 Lateral spatial profilesa Carrier densityb Material gainc Guided mode distribution

The nonuniform electron-concentration profile deter-mines the gain profile (Figure 2b), which will also be non-uniform. However, if we write the gain g(N) about somearbitrary electron concentration No as

9(N) = 9(N0) + ̂ j(N- No) (1)

then the total gain experienced by the guided optical modewill be weighted by the guided optical-mode distributionS(x) (Figure 2c). Hence the net gain will be

9net =F_J+«>g(x)S(x)dx

(2)

where the factor F accounts for the fact that only a pro-portion of the optical distribution travels in the activelayer and therefore experience gain.

We can define an average electron concentration Nav

such that

(3)

(4)

9net = n9(NO) + lTr(NaV-No)

hence

$t™ N(x) • S(x) • dx

Likewise, if we write the refractive index n(N) about NQ,

n(N) = n(N0) + ^-(N-N0) (5)

then the net refractive index experienced by the guidedmode is

T dnneff = nefJ{N0) + — (Nav - JV0) (6)

Thus, to evaluate the material parameters dg/dN anddn/dN from measurements of the net gain and refractive-index change in ridge-waveguide laser diodes, we needonly to evaluate Nav.

We achieve this by measuring the near-field emissionprofiles both below and above the lasing threshold,together with the carrier lifetime near x = 0.

The detected spontaneous-emission intensity profileSp(x) is related to the carrier distribution N(x) by

Sp{x) = K •N(x)

(V)

where K is a constant of proportionality, ?r(N) is thecarrier lifetime due to spontaneous emission and isbelieved to depend on the carrier density in the followingmanner:

- = a + bNT.

(8)

The total radiative current is obtained by integrating thespontaneous emission

/ , = edLN(x)

dx (9)

where d is the active layer thickness and L the devicelength.

From eqns. 7 and 9 we may write the ratio of the peakcarrier density N(x = 0) to the radiative current

N(0) _ tr(0) • Sp(Q)

lZ Sp(x) • dx(10)

Not all the injected current produces light, because carriersmay recombine without emitting spontaneous photons.However, provided the internal efficiency of the sponta-neous emission process is not a strong function of position,it is straightforward to show that the ratio of the peakelectron concentration to the total current is

N(0) _ Tne,(0)Sp(0)t% Sp(x) dx (11)

where xnet is the net carrier lifetime including nonradiativerecombination.

Finally, we can see that Nav is given by

Na

It

^ Spjx) - S(x) dx

tZ Sp(x) dx J+S S(x) dx(12)

where it has been assumed that the carrier lifetime isapproximately constant over the central ridge width ( — w/2 > x > w/2), and hence the electron concentration profileover this range of x follows that of the spontaneous emis-sion. This assumption would appear to be born out experi-mentally, as the measured net lifetime xnet(0) varies by~ + 10% in the range 0.5 - 1 x threshold. Thus, Nav can

be obtained by measuring the net carrier lifetime (near

136 IEE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986

x = 0), together with a graphical integration of the near-field intensity profiles.

2.2 ExperimentTwo ridge-waveguide lasers were used in our investiga-tion: both operated at X — 1.53 ^m with a thresholdcurrent of 32 mA. The ridge dimensions were 5 ̂ m wideby 190 \im. long. These devices were bonded to copperheatsinks and maintained at 20 ± 0.05°C using a Peltiercooler. In addition, nearly all the measurements were per-formed under pulsed bias conditions so as to eliminatecomplications due to temperature effects.

The carrier lifetime rnef(0) was measured using the tech-nique developed by Su et al. [6] to estimate the recombi-nation constant in GaAs and I = 1.3 /an InGaAsP laserdiodes. This technique is to be preferred to that more oftenused; i.e. of measuring the laser turn-on delay. This isbecause (i) the technique measures the lifetime directly andis not therefore open to different interpretations of theresults depending on which carrier recombination model isfavoured, and (ii) being a small signal technique, it is lesssusceptable to errors due to charging of the diffusioncapacitance. A schematic of the experimental apparatus isshown in Fig. 3. The laser was biased either by a DC

biaspulse

computer controlledsampling oscilloscope

y

T"IT.

PIN* detector

Fig. 3 Apparatus for the measurement of the carrier lifetime T

current or by a current pulse. A short 2 mA pulse with arise time of < 200 ps was then superimposed on the bias,and the carrier lifetime was taken to be the rise time of theresulting spontaneous emission.

A lifetime measurement is shown in Fig. 4 for one of thedevices biased at 20 mA. In this case the carrier lifetimewas estimated to be 1.9 ns. Measurements were performedwith both a DC bias and 500 ns bias pulses, but both setsof data agreed to within 5%.

The variation in carrier lifetime with injected current isgiven in Fig. 5. The variation in rnef(0) between 15 and30 mA is ^ 2 + 0.2 ns.

The spontaneous and guided-mode intensity profiles

81 2 3 A 5 6 7

time.nsFig. 4 Spontaneous-emission risetime measurement for Ibias = 20 mA

were measured by scanning the laser in front of a lens/detector arrangement. The positional resolution of the

-o

10 15 20bias current.mA

25 30

Fig. 5 Carrier lifetime against bias current for two ridge waveguidelasers

stepper motor drive was 0.1 fim, and the optical resolutionwas estimated to be ^ 1 /mi. The spontaneous near-fieldprofile for one of the devices biased at 20 mA is shown inFig. 6 (outer solid line). The guided-mode profile, obtainedfor the same laser above threshold, is also shown in thisFigure (inner solid line). Profiles were measured at several

1.0

0.9

0.8

>>S 0.7cQj

c 0.6

•I 0.5a

0.3

0.2

0.1

- 1 0 - 5 0 5 10distance, pm

Fig. 6 Near-field intensity profiles measured below (outer) and above(inner) the losing threshold

Eqn. 13

different bias levels, although little variation was observed.The value of Nav was then calculated as a function of

current by integration of these intensity profiles asrequired by eqn. 12. However it is interesting to comparethis technique with the more approximate method for cal-culating Nav described by Hakki [7], who assumed thatthe carrier lifetime was independent of N for all x. Solvingthe carrier continuity equation he obtained

N(x) =

It ( cosh (x/Ld)

edwL\ exp (w/2Ld)

sinh (w/Ld) exp ( - |x\/Ld) \x\> w/2edwL

(13)

IEE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986 137

where w is the ridge (stripe) width. The fitting parameter Ld

is the carrier diffusion length. Hakki describes a methodfor the evaluation of Ld from the value of the normalisedspontaneous-emission intensity at x = vv/2. Using thismethod on the spontaneous-emission profile in Fig. 6 weobtained Ld = 1.2 /im. However, we obtained a betteroverall fit using Ld = 1.7 fim (broken line in Fig. (6)).

Hakki also approximated the guided-mode intensityS(x) by cos2 (nx/w), in which case Nav simplifies to [7]

/_ r _ / _ ( 2 L » • sinh (w/2Ld)

edwL V ((nw/2Ld)2 + 1)

(14)

It is interesting to note that Hakki's approximate formula(eqn. 14) for Ld = 1.7 ^m agrees with the more accurategraphical technique described above to within 5%. Thevalue of Nav for Ld = 1.2 fim (Ld estimated using Hakki'smethod), though, is approximately 30% too large.

3 Gain

3.1 Improvements to the Hakki-Paoli techniqueSeveral techniques have been reported for the measure-ment of the gain coefficient g in semiconductor laserdiodes. These have included the injection of an externalprobe signal [8], the use of segmented electrical contacts[9] and, by inference from the spontaneous emission spec-trum [10], optical pumping methods have also beenreported [11], although these are difficult to relate to thecase of electrical injection. The most widely used tech-nique, however, is that due to Hakki and Paoli [12], inwhich the gain is obtained from the Fabry-Perot reson-ances in the laser emission spectrum below threshold. Thewide application of the Hakki-Paoli technique is probablya result of its experimental simplicity compared with theother techniques, and it is a variation of the Hakki-Paolitechnique designed to give greater sensitivity which is usedhere.

The form of the laser emission spectrum below thresh-old results from spontaneous emission, coupled into theguided mode, experiencing gain (or loss) together withseveral partial reflections before being emitted. The spon-taneous emission, which is distributed along the cavity inthe real device, can be lumped into a single effective sourceexternal to the Fabry-Perot cavity. The power outputfrom the other side of the Fabry-Perot resonator is thenthe product of the input (spontaneous emission) and thetransmission of the resonator. The transmission itself isgiven by

T =(1 - R)2 exp (GL)

((1 - R exp (GL))2 + 4R exp (GL) sin2 (2nneffL/X))

(15)

where neff is the effective refractive index and G is the netgain experienced by the guided mode which takes intoaccount internal losses (due to scattering etc.) and the factthat only a finite fraction of the wave travels in the activelayer and experiences gain. The ratio r, defined as themaximum transmission (at 2nneff L/X = mn) over theminimum transmission (at 2nneffL/X = (2m + l)n/2), isthen

r =(1 + R exp (GL)):

(1-R exp (GL))2

and hence the mode gain G is

G =1

In r- 1r+ 1

- In (R)

(16)

(IV)

The Hakki-Paoli analysis assumes that the spectrometerhas an infinitely narrow passband. Using monochromatorsat their best resolution has the drawback, however, ofreducing the experimental signal-to-noise ratio. The effectof opening up the monochromator slits is to increase theoverall sensitivity, but at the expense of errors in theapparant max/min ratio. We have developed a simplemodification to the basic Hakki-Paoli technique whichallows the use of wide monochromator slit widths with noloss of accuracy in the determination of the mode gain. Infact, a bonus of the new technique (besides increasedsignal-to-noise ratio) is an added degree of confidence inthe results which was not previously available. This arisesbecause the technique described below uses spectra mea-sured for more than one known resolution (slit width)setting, three in our case.

Spectral measurements were performed using a Hilger1 metre monochromator with a minimum resolution of0.3 A. First the monochromator pass band was calibratedin terms of the laser mode spacing. The laser spectra werethen measured for three values of passband width corre-sponding to 0.2, 0.4 and 0.6 of a mode spacing. These widepassbands resulted in significantly better signal-to-noiseratios at the detection equipment, which was particularlyimportant in our measurements as all the emission spectrawere obtained under pulsed conditions, where the averagedetected power is low, thereby eliminating errors due tothermal effects.

The difference between the idealised Hakki-Paoli mea-surement and the modified technique can be seen sche-matically in Fig. 7. On the left, the max/min ratio r for the

wavelengtha

wavelengthb

Fig. 7 Fabry-Perot resonances in the laser emission spectrum belowthresholda Measurements with infinitely narrow bandpass, r — /i,//i2

b Finite bandpass width, r' = AJA2

Hakki-Paoli technique as calculated from eqn. 16 is givenby hjh2. On the right, the detected max/min ratio r', whenusing finite passband widths, is AJA2 • The passband is, toa good approximation, rectangular (the distortions at theedges of the passband being much diluted) allowing thecalculation of r'( = A1/A2) by integration of eqn. 15 forvarious passband widths.

The result of this calculation is shown in Fig. 8. Fig. 8permits the direct conversion of the measured max/minratio for a given passband width into the mode gain G inthe form R • exp (GL), where R is the facet reflectivity andL the device length, and this graph is all that is needed tointerpret the experimental data. The ratio begins on theleft-hand side of the graph (zero passband width) at thevalue of r given by eqn. 16 for the conventional Hakki-Paoli technique, decreasing to 1 at a passband width equalto the mode spacing.

The additional confidence in the results for R exp (GL)mentioned earlier results from the use of more than onepassband width, as all the values of r' for the different slit

138 IEE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986

widths should all lie on the same line (constantR exp {GL}). That this is so in the actual measured

Using this technique, we anticipate that errors due tononguided spontaneous emission emitted from the end of

0.4 0.5 0.6 0.7normalised passband width

0.8 0.9 1.0

Max/min ratio r' against normalised spectrometer passband width for different values of the single pass gain R exp (GL)

max/min ratios can be seen in Fig. 9, which shows thevariation in max/min ratios with slit width for threecurrent values at X = 1.54 ^m (the other values ofR exp (GL) present in Fig. 8 are omitted for clarity). Mea-surements for the remaining wavelengths followed thesame pattern as those shown in Fig. 9: max/min ratiosfrom spectra for monochromator passbands of 0.2, 0.4 and0.6 times the laser mode spacing were plotted on Fig. 8and the value of R exp (GL) read off from the nearest line.

R.exp(G.L)

0.1 0.8 0.9 1.00.2 0.3 0.4 0.5 0.6 0.7normalised passband width

Fig. 9 Measured max/min ratio r' against normalised spectrometer pass-band width for three different currents

the laser and collected by the lens optics, resulting in toosmall a value of the max/min ratio r', can be detected, asthis excess spontaneous emission will increase with thepassband width and result in deviation of the measuredratio r' with slit width from the curves in Fig. 8.

3.2 Confinement factorIn order to obtain the material gain g from the mode gainG it is first necessary to calculate the wave confinementfactor F, which is the proportion of the guided opticalwave (in the direction normal to the plane) travelling in theactive layer and is hence the proportion of the guidedwave which experiences gain. The layer structure of ourridge-guide lasers is slightly more complicated than that of

1.0

0.9

0.8

0.7

£ 0.4c^ 0 . 3ou

0.2

0.1

buffer layerthickness(ijm)0.15

0.1 0.50.2 0.3 0.4active layer thickness , pm

Fig. 10 Confinement factor against active layer thickness

n, = 3.52, n2 = 3.27, n3 = 3.17

IEE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986 139

1.3 fim lasers because of the presence of 1.15 fim bufferlayers. Therefore the calculation of F, which is crucial tothe accuracy of g, is set out in the Appendix. The variationin the confinement factor with active layer width for ourfive-layer 1.53 ^m InGaAsP laser structure is shown inFig. 10 for buffer layer thicknesses of 0.15 /im. At ouractive layer thickness of 0.15 ^m therefore, we calculateT ~ 0.3.

3.3 ResultsThe measured results for the net material gain g — OL-JT areshown in Fig. 11 for one of the two lasers used. The netgain is plotted as a function of photon energy hv for fivedifferent values of injected carrier density. The value of theinternal cavity loss a,/F can be estimated from a compari-son of Fig. 11 with the form of the calculated gain spectra

200N=1.63x1018cm-1

150

100

50

0'eo

u" -508~

£ -100

o>

-150

-200

-250

U 5.Ax1017

\

0.79 0.830.80 0.81 0.82energy , eV

Fig. 11 Measured net material gain g — xJV against photon energy

of Stern [13] for GaAs, or Dutta [2] and Osinski andAdams [14] for InGaAsP, to be roughly ~ 50 cm"1 (i.e.a,; ~ 15 cm"1).

5 r

en"O

0.79 0.80 0.81 0.82 0.83 0.84energy, eV

Fig. 12 Measured gain constant dg/dN against photon energy for thetwo devices

140

The rate of change of gain with injected carrier densityis shown in Fig. 12 also as a function of the photonenergy, for both devices. The lasing energy in bothlasers was hv = 0.81 eV (1.53 /jm) at which energydg/dN = 2.7 x 10~16cm2.

It is common to express the gain near to threshold as aparabola

g(N • hv) = a,(N - No) - a2[hv - (Eo No))]

(18)

Here, the carrier concentration No is that which wouldresult in zero material gain, No ~ 9 x 1017 cm3. The otherparameters are: ax = dg{peak)/dN = 2.7 x 10~16 cm2, a2,which describes the width of the parabola, is determined tobe 4 x 105 cm"1 eV"2, and a3 the shift in the peak gainenergy with electron concentration dEJdN is 1.4 x10"2OeVcm3. These values compare with Stubkjaersvalues (Table 1 of Reference 3) of a\ = 1.2 x 10"16 cm2,a'2 = 1.5 x 105cm"1eV-2anda'3 = 5.9 x 10-19eVcm3.

Alternatively, in terms of wavelength,

g(NX) = b^N - No) - b2\_k - {k0 - b3(N - JV0))]2 (19)

when

l = a1= 2.7 x 10"16 cm2, b2 = 0.15 cm"1 nm" 1 nm"2

and

= -2.7 x 10"17 nmem3.

4 Refractive-index change with injected carriers

The refractive-index change due to the injected carriers canbe obtained experimentally from the shift in the wave-length of the Fabry-Perot resonances. The resonance con-dition is

= 2neffL (20)

Rewriting the effective refractive index about its value atwavelength Ao

dneff (21)

and substituting eqn. 21 into eqn. 20 and differentiatingwith respect to the carrier concentration N we obtain

dneff h dX

dN = I ' IN

where the group index n is defined as

X dn.n = neff 1 -

leffdk

(22)

(23)

Finally, we must take into account the wave confinementfactor F which dilutes the change in refractive index. Thus

hdk_

kdN(24)

The group index was determined in the usual way from thelongitudinal mode spacing dk using the relation

k:

n = 2Lbk(25)

The group index h is shown in Fig. 13 over the energyrange 0.79-0.833 eV. It is interesting to note that thegroup index increases monotonically over the energy

1EE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986

range, in contrast to the measurements of Manning et al.[5] and Van Der Ziel et al. [15], who reported a disconti-nuity in h for GaAs lasers on the low wavelength side of

4.0

3.9

3.8

3.7

1C3.6

3.5

3.4

13

3.2

3.1

3.0

0.79 0.80 0.81 0.82 0.83photon energy, eV

Q84

Fig. 13 Variation in the group refractive index with photon energy{two devices)

the lasing wavelength. This they attributed to a change inthe sign of the dispersive component dn/dk near to theband edge.

The variation in dn/dN obtained from the longitudinalmode shifts with current is shown in Fig. 14, also as a

Or

-i -

ou

-2

0.79 0.80 0.81energy

0.82

.eV

0.83 0.84

Fig. 14 Carrier induced refractive-index change dn/dN against photonenergy (two devices)

function of the photon energy. The mean value for the twodevices for dn/dN at the lasing wavelength is~ —1.8 x 10~20 cm3, which is similar to that reported by

Bouley ( — 1.5 x 10~20 cm3) but approximately twice thatmeasured by Stubkjaer (-0.7 x 10"2 0 cm3). We note alsothat dn/dN varies slowly with photon energy when com-pared with the variation of dg/dN in Fig. 12.

Finally, in Fig. 15 we plot the quantity a defined as [16]

4TT dn/dN<x =

15

14

13

12

11

10

9

V 8

I*6

5

4

3

2

1

X dg/dN(26)

0.78 0.79 0.80 0.81 0.82 0.83energy,eV

0.84

Fig. 15 Linewidth enhancement factor <x against photon energy (twodevices)

a is usually known as the linewidth enhancement factor, asthe linewidth of a solitary laser mode is broadened, due tophase noise, by a factor (1 + a2). We note that the strongdependence of a on photon energy is consistent with thecalculations of Vahala et al. [16] with a increasingdramatically for photon energies near to the band edge.

5 Conclusions

The dispersion in the gain constant dg/dN and the changein refractive index with injected carriers dn/dN have beenmeasured for the first time in 1.5 jim InGaAsP lasers.Measurements were performed on ridge-waveguide lasers,measuring the carrier lifetime directly, and the values ofdg/dN and dn/dN at the lasing wavelength were found tobe 2.7 x 10~16 cm"2 and -1 .8 x 10~20 cm"3, respec-tively, which are approximately twice as large as thosereported previously by Stubkjaer for 1.6 (im buried hetero-structure lasers.

6 Acknowledgments

The author has pleasure in thanking Dr. D. Cooper for theridge-waveguide laser diodes and MJ. Adams and I.D.Henning for suggesting the investigation. Acknowl-edgement is made to the Director of Research of BritishTelecom for permission to publish this paper. Finally, ananonymous referee is thanked for useful comments.

7 References

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3 STUBKJAER, K., ASADA, M., ARA1, S., and SUEMATSU, Y.:'Spontaneous recombination, gain and refractive index variation for1.6 ^m wavelength InGaAsP/InP lasers', Jap. J. Appl. Phvs., 1981, 20,pp. 1499-1505

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4 BOULEY, J., CHARIL, J , SOREL, Y., and CHAMINANT, G.:'Injected carrier effects on modal properties of 1.55 f.im InGaAsPlasers', IEEE J. Quantum Electron., 1983, QE-19, pp. 969-973

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8 GOLDBERG, L., TAYLOR, H.F., and WELLER, J.F.: 'Intermodalinjection locking and gain profile measurement of GaAIAs lasers',IEEE J. Quantum Electron., 1984, QE-20, pp. 1226-1229

9 PRINCE, F.C., MATTOS, T.J.S., and PATEL, N.B.: 'Optical gainmeasurements of 1.3 ;/m InGaAsP as a function of injected currentdensity', Electron. Lett., 1982, 18, pp. 1054-1055

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12 HAKKI, B.W., and PAOLI, T.L.: 'CW degradation at 300 K of GaAsdouble-heterostructure junction lasers. II Electronic gain', J. Appl.Phys., 1973, 44, pp. 4113-4119

13 STERN, F.: 'Calculated spectral dependence of gain in excited GaAs',ibid., 1976,47, pp. 5382-5386

14 OSINSKI, M., and ADAMS, M.J.: 'Gain spectra of quaternary semi-conductors', IEE Proc. I, Solid-State & Electron. Dev., 1982, 129,pp. 229-238

15 VAN DER ZIEL, J.P., and LOGAN, R.A.: 'Dispersion of the groupvelocity refractive index in GaAs double-heterostructure lasers', IEEEJ. Quantum Electron., 1983, QE-19, pp. 164-169

16 VAHALA, K., CHIU, L.C., MARGALIT, S., and YARIV, A.: 'On thelinewidth enhancement factor a in semiconductor injection lasers',Appl. Phys. Lett., 1983, 42, pp. 631-633

8 Appendix

8.1 Calculation of the wave confinement factor l~The vertical refractive-index profile of our ridge-waveguidelasers forms a symmetrical five-layer dielectric slab wave-guide (see Fig. 16). n^, n2 and n3 are the refractive indicesof the active layer, symmetric buffer layers and binarycladding layers, respectively. The thickness 2h( = d) is thatof the active layer, whereas t is the thickness of each of thebuffer layers. The transverse-electric (TE) polarised fieldsolutions fall into two catagories: those where the guided-mode propagation constant /? is greater than n2k0 (k0

being the propagation constant of free space) and thosewhere /? is less than n2 k0.

8.1.1 Case where 0 > n2ko: The electric field componentEx\s

A cos (Ky) | y \ < h

A cos (fch)(cosh (<5(| y \ — h)

+ B sinh (<5(| y \ — h))) h < \ y \ < h + t (27)

A cos (Kh)(cosh (dt)

+ B sinh (dt)) exp ( - y(| y \ - h - t)) h + t < \ y \

where

(28)

Matching the magnetic field at the waveguide boundariesresults in an eigenvalue equation for K

B = tan (ich) = —(y + d tanh (5t))

(S + y tanh (dt))(29)

n2

Fig. 16 Five-layer dielectric slab

The confinement factor may be calculated from

r = M dx

\S E2, dx

hence

r =

ra + rb + rcsin (2KH) h

AK

(30)

(31)

(32)

+ 2B(cosh (2<5r) - 1) + 2(1 - B2)dt) (33)

r = cos (cosh((5t) + flsinh (dt)): (34)

8.1.2 Case where /3 <n2ko'.

i A cos (/cy) | y \ < h

\ A COS Ocfc)(cos (<5(l y I -h))

Bsin(S(\y\-h))) h<\y\<h + t (35)

A cos (K/I)(COS (dt)

+ B s i n (dt)) e x p ( - y(\ y | - h - t)) h + t < \ y \

where K and y are as previously defined, but d is nowdefined by

d2 = n2k20-p

2 (36)

Again, matching the fields at the waveguide boundariesgives

-K (y-d tan (dt))B = —— tan (KH) = -— ——

d (d + y tan (dt))

and the confinement factor is

T—

rfl + r; + r;, cos2 (KH)

4(5

(37)

(38)

- 2B(cos (2(5f) ~ 1) + 2(1 + B2) dt) (39)

((1 - B2) sin (2dt)

and

cos (K,II\Vc = — - ^ — i (cos (dt) + B sin (dt))2 (40)

142 IEE PROCEEDINGS, Vol. 133, Pt. J, No. 2, APRIL 1986