The Theory for the Dielectric Slab Waveguide with Complex

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THE THEORY FOR THE DIELECTRIC SLAB WAVEGUIDE WITH COMPLEX REFRACTIVE INDEX. APPLIED TO GaAs LASERS, In this paper we investigate the hcmogeneous dielectric slab waveguide. in the case of complex dielectric constants A method for calculating. the field distribution in a dielectric waveguide with an arbitrary. symmetrical variation in the refractive index is developed and some of. the results are presented The results are applied to the GaAs laser. It is found that the guiding mechanism is a combination of gain guiding. and index antiguiding, Based on the calculations an explanation of the kinks on the light current. characteristics is suggested,INTRODUCTION, The theory of the homogeneous dielectric slab waveguide in the case of. pure real dielectric constants is well known This theory is extended. to complex dielectric constants Although the abrupt change inI is a. crude approximation to the transverse structure of the active layer in. GaAs lasers the theory gives qualitative information about the guiding. mechanism and the mode structure, The transverse field variition can be found explicitly in the case of. parabolic variation in s but c is parabolic in a narrow region only and. the results are of limited validity Since the transverse intensity. variation is fundamental when coupling to optical fibers it is important. to 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,t even or odd modes for lxI t and exp w It for. lxI t Only TE modes are considered The,normalized frequency v and normalized. propagation constant b are given by,v2 U2 W2 s1 Ec2 k2t2 and 1 E2E2 El E 2. 32 2 1jK 2 n2 j 2 2,k and 3 are the propagation constants. Electromagnetics TInstitute,ectromag tir cs,Fig 1 Geometry.
of the slab,Technical University of Denmark Building 348. DK 2800 Lyngby Denmark, Authorized licensed use limited to Danmarks Tekniske Informationscenter Downloaded on June 30 2010 at 13 30 50 UTC from IEEE Xplore Restrictions apply. in free space and the slab respectively The values of u and w are. found from the characteristic equation obtained from the condition that. dE should be continuous at x t b v u and w are related by u vV b. 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 mode. always exists 1 If however Re 1 82 0 a cutoff frequency for. the lowest mode exists The guiding mechanism depends on the v value. as 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 the. cutoff values of v for the 5 lowest modes,pure gain guiding. I gain guiding,I index cantiguiding,I no guided,ndex and gain 4. pure index guiding, Fig 2 Guiding mechanisms for Fig 3 Cutoff conditions for.
complex v values the lowest modes, The filling factor P is defined as the fraction of the intensity propa. gating in the region lxl t,r JE E 2 dxJ E 2 X 3,For a pure real guide K K 2 0 we have. r even b 1lb vVb 4, For the complex guide we have derived the general expression. r Re w Im w Re b Re v2 Im b 5,eddn Re w Im w Re u Im u Im v2. Authorized licensed use limited to Danmarks Tekniske Informationscenter Downloaded on June 30 2010 at 13 30 50 UTC from IEEE Xplore Restrictions apply. As an example P is calculated for the 2nd mode fig 4 It is found. that r decreases with increasing,mode number hence the effective.
K K K P 6 00,geff 2 1K 2 9 r 0 8,will be highest for the lowest 095 UT OFF. modes 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 or,proton bombarded types The 3. active layer is modelled by a 2,slab structure where the electron.
density in the central region is,high due to the pump current in the. outer regions the electron density A 0 I L5 2Tt,from carrier diffusion is low. From a calculation of the gain 21 and Fig 4L Filling factor for the. use of the Kramers Kronig relations 2nd mode,31 we have An n N Nc n N O 001 where Nc. is the critical electron density This order of magnitude is confirmed. by experiment 4 the measured value is An Z 0 007 From the gain. values 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. The effective gain for t 5pum for the 3 lowest modes is 3209 m 758 m. and 3502 m 1 for t 10pm 4168 m 1 3820 m 1 and 3247 m 1 This. shows that the effective gains are approaching each other for wide. stripes If the end loss is lower for high order modes the laser can. shift to higher modes,GENERAL MODEL, When a laser is operating at a high intensity level spatial holeburning in. the electron density profile is likely to occur 6 This will change. the 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 electron.
density 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, Authorized licensed use limited to Danmarks Tekniske Informationscenter Downloaded on June 30 2010 at 13 30 50 UTC from IEEE Xplore Restrictions apply. s x E co 3 6 A N x j B N x 2 356 j 0 0011 2 8,with N 1 710 M nf 3 A 5ol0 2 7 m 3 and B 9 10 m3. We define the modegain,G K X IE x 12 dx J E X 12 dX 9. G is shown in fig 5 for the various modes If a laser is operating in. the O th transversal mode and the inten,sity is increased we can expect holeburn G.
ing to occur corresponding to higher X 103mr1,This will produce a local increase in the. real part of the refractive index hence,the mode will be self focused Further in. crease of the intensity will give a deeper,hole and a better confinement but on the. other hand the mode gain will be reduced and,another mode begins to dominate The. shift from one mode to another will produce,a re distribution of the electrons and.
we suggest that this gives rise to a kink,In order to get a better understanding of 3. this mechanism however it is necessary,to couple the field equation to an equa. tion for the electron density profile 2,CONCLUSION. The g uiding mechanism and mode structure,for GaAs lasers wec discussed by applying. models for dielectric waveguides An A, explanation of the kinks on the light 0 0 1 0 2 0 3.
current characteristic was suggested and,supported by numerical calculations. Fig 5 Mode gain as a,function of X,REFERENCES, 1 W O Schlosser Gain induced modes in planar structures BSTJ. vol 52 pp 886 9055 1973, 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 absorption. edge 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. 5 B W Hakki Striped GaAs lasers Mode size and efficiency J. Appl Phys vol 46 pp 2723 2730 1975, Authorized licensed use limited to Danmarks Tekniske Informationscenter Downloaded on June 30 2010 at 13 30 50 UTC from IEEE Xplore Restrictions apply.
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, Authorized licensed use limited to Danmarks Tekniske Informationscenter Downloaded on June 30 2010 at 13 30 50 UTC from IEEE Xplore Restrictions apply. 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 waveguide

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