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The Theory for the Dielectric Slab Waveguide with Complex Refractive Index Applied toGaAs Lasers

Buus, Jens

Published in:7th European Microwave Conference

Link to article, DOI:10.1109/EUMA.1977.332400

Publication date:1977

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Buus, J. (1977). The Theory for the Dielectric Slab Waveguide with Complex Refractive Index Applied to GaAsLasers. In 7th European Microwave Conference (pp. 29-33). IEEE. https://doi.org/10.1109/EUMA.1977.332400

THE THEORY FOR THE DIELECTRIC SLAB WAVEGUIDE WITH COMPLEX REFRACTIVE INDEX

APPLIED TO GaAs LASERS

Jens Buus*

ABSTRACT

In this paper we investigate the hcmogeneous dielectric slab waveguidein the case of complex dielectric constants. A method for calculatingthe field distribution in a dielectric waveguide with an arbitrarysymmetrical variation in the refractive index is developed, and some ofthe results are presented. The results are applied to the GaAs laser.It is found that the guiding mechanism is a combination of gain guidingand index antiguiding.

Based on the calculations an explanation of the kinks on the light currentcharacteristics is suggested.

INTRODUCTION

The theory of the homogeneous dielectric slab waveguide in the case ofpure real dielectric constants is well-known. This theory is extendedto complex dielectric constants. Although the abrupt change inI £ is acrude approximation to the transverse structure of the active layer inGaAs lasers the theory gives qualitative information about the guidingmechanism and the mode structure.

The transverse field variition can be found explicitly in the case ofparabolic variation in s, but c is parabolic in a narrow region only, andthe results are of limited validity. Since the transverse intensityvariation is fundamental when coupling to optical fibers it is importantto obtain a belter description.

THEORY FOR THE HOMOGENEOUS SLAB

The geometry considered is shown in fig. 1. The field, E, is gaiven by

cos(u t) or sin(u x) (even or odd modes) for lxI<t and exp(-w It) fort t tlxI>t. Only TE modes are considered. The

normalized frequency, v, and normalizedpropagation constant, b, are given by

v2=U2+W2= (s1-Ec2) k2t2 and (1) E2E2 £ - El= E = £232 (2)__ ( *1jK)2 n2 j%2)2

b k 2 (2)£ - E

-t t

k and 3 are the propagation constants

*Electromagnetics Fig. 1. Geometry of the slabectromag_tir,cs- TInstitute waveguide.

Technical University of Denmark, Building 348DK-2800 Lyngby, Denmark

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in free space and the slab, respectively. The values of u and w arefound from the characteristic equation obtained from the condition thatdE should be continuous at x = ±t. b, v, u and w are related by u=vV]-band w=vv.

In order to get physically acceptable solutions we must require Im{e1-s2}> 0, and if furthermore Re{F-s2} > 0, the lowest order (o'th) modealways exists [1]. If, however, Re{£1-82} <0, a cutoff frequency forthe lowest mode exists. The guiding mechanism depends on the v-valueas shown schematically in fig. 2. The cutoff condition is Re{w}=Re{vA}=0, where b is found from the characteristic equation. Fig. 3 shows thecutoff values of v for the 5 lowest modes.

lvl

I gain guidingI index cantiguiding

n

I/I no guided /I_1I

( pure gain guiding

ndex and gainuiding

pure index guiding

6

5

4

i5s

Fig. 2. Guiding mechanisms forcomplex v-values.

The filling factor P is definedgating in the region lxl.t

Fig. 3. Cutoff conditions forthe lowest modes.

as the fraction of the intensity propa-

t 00

r = JE|E|2 dxJ E| 2 X (3)-t -00

For a pure real guide (K=K2=0) we have

r = + b + (1lb) vVb-even I + v/Bodd

For the complex guide we have derived the general expression

(4)

r = Re{w} Im{w}

eddn Re{w}Im{w} + Re{u}Im{u}= Re{b} + Re{v2} Im{b}

Im{v2 }

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(5)

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As an example P is calculated for the 2nd mode (fig. 4). It is found,that r decreases with increasingmode number, hence the effectivegain lvi

10_

(K(K K K P (6) 00geff (2 1K 2 9 r 0.8

will be highest for the lowest 095 /UT OFFmodes. For hlgh values of lv|,however, r is close to 1 for 7

several modes. 6

APPLICATION TO GaAs LASERS 5 /

The theory can be applied to GaAs 4/lasers of the oxide insulated orproton-bombarded types. The 3

active layer is modelled by a 2-slab structure where the electrondensity in the central region ishigh due to the pump current; in theouter regions the electron density 2A 0 I-L5 Tt(from carrier diffusion) is low.From a calculation of the gain [21 and Fig. 4L Filling factor for theuse of the Kramers-Kronig relations 2nd mode.[31 we have An = n(N-N ) - n(N=O)-001 where Nc cis the critical electron density. This order of magnitude is confirmedby experiment [4]; the measured value is An Z -0.007. From the gainvalues in [51 we deduce K1 3 10-4, K2 - ll0-4; this gives for

X = 8800 A0 and n(N=0) = 3.6: lvi z 8.0 -t and arg (v) = 1.47.-1 -1

The effective gain for t = 5pum for the 3 lowest modes is 3209 m , 758 mand -3502 m-1 , for t = 10pm; 4168 m-1, 3820 m-1 and 3247 m"1. Thisshows that the effective gains are approaching each other for widestripes. If the end loss is lower for high order modes, the laser canshift to higher modes.

GENERAL MODEL

When a laser is operating at a high intensity level spatial holeburning inthe electron density profile is likely to occur [6]. This will changethe complex index profile, and the homogeneous slab model will be inade-quate. In order to solve the field equation the variation in s is ex-panded in orthogonal functions and Galerkin's method is used.

EFFECT OF HOLEBURNING

In order to investigate the effect of holeburning we have used an electrondensity profile given by

N(x) ={(l.1 - X3 cos(t x) - X cos(K)}N |x|<'2 t,t = 5pm (7)

The dielectric constant was assumed to be

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s(x)-E(co) = (3,6+A'N(x) + j*B*N(x))2 - (356-'2 43 -~2 7 3 2 8 3with N=1.710 M-nf A=-5ol0 m and B=9.10-m3

G by 0

00

j 0,0011)2 (8)

We define the modegain

00

G =

AK(X) IE(x) 12 dx / J E(X) 12 dX

-00

(9)-0)

G is shown in fig. 5 for the various modes.the O'th transversal mode and the inten-sity is increased we can expect holeburn-ing to occur (corresponding to higher X).This will produce a local increase in thereal part of the refractive index, hencethe mode will be self-focused. Further in-crease of the intensity will give a deeperhole and a better confinement, but on theother hand the mode-gain will be reduced, andanother mode begins to dominate. Theshift from one mode to another will producea re-distribution of the electrons, andwe suggest, that this gives rise to a kink.In order to get a better understanding ofthis mechanism, however, it is necessaryto couple the field equation to an equa-tion for the electron density profile.

CONCLUSION

The g;uiding mechanism and mode structurefor GaAs lasers wec discussed by applyingmodels for dielectric waveguides. Anexplanation of the kinks on the light-current characteristic was suggested, andsupported by numerical calculations.

If a laser is operating in

G103mr1

3

2

0 0.1 0.2 0.3

Fig. 5. Mode gain as a

function of X.

REFERENCES

[1] W.O. Schlosser, "Gain induced modes invol. 52, pp. 886-9055 1973.

planar structures", BSTJ,

[2] F. Stern, "Calculated spectral dependence of gain in excited GaAs",J. Appl. Phys., vol. 47, pp. 5382-5386, 1976.

[3] F. Stern, "Dispersion of the index of refraction near the absorptionedge of semiconductors"', Phys. Rev., vol. 133, pp. A1653-A1664, 1964.

[4] M.R. Matthews, R.B. Dyott, W.P. Carling, "Filaments as optical wave-guides in gallium-arsenide lasers", Elect. Lett., vol 8, pp. 570-572,1972.

[5] B.W. Hakki, "Striped GaAs lasers: Mode size and efficiency", J.Appl. Phys., vol. 46, pp. 2723-2730, 1975.

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[61 J. Buus, M. Danielsen, "Carrier diffusion and higher order trans-versal modes in spectral dynamics of the semiconductor lasers",to appear in IEEE J. Quant. Elect., October 1977.

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