Laser cavity modelling

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Laser cavity modelling

I. Damakoa, J. Audounet, G. Bouyssou, G. Vassilieff

To cite this version:

I. Damakoa, J. Audounet, G. Bouyssou, G. Vassilieff. Laser cavity modelling. Journal de Physique III, EDP Sciences, 1993, 3 (9), pp.1729-1737. �10.1051/jp3:1993233�. �jpa-00249034�

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Classification Physic-s Abstracts

02.70 42.20C

Laser cavity modelling

1. Damakoa ('), J. Audounet (2), G. Bouyssou (') and G. Vassilieff (') (') CNRS LAAS, 7 _avenue du Colonel Roche, 31077 Toulouse Cedex, France (2) Laboratoire d'Analyse Numdrique, UPS, 31077 Toulouse Cedex, France

(Received lo November 1992, revised 4 June 1993, acc.epted 21 June 1993)

R4sumk. Deux approches de moddlisation des cavit6s laser inhomogdnes sont prdsentdes. ^Elles reposent sur la mdthode de propagation d'onde (B.P.M.) permettant l'utilisation de la transformde de Fourier rapide (FFT). Les procddures rdsultantes foumissent des solutions auto-consistantes des

dquations de Maxwell _et de diffusion de porteurs. Des _cas typiques de calculs _sont donnds pour illustrer (es deux m6thodes.

Abstract. Two approachs of modelling nonhomogeneous cavity laser _are presented. They are

based _on the beam propagation method which allows the _use of fast Fourier transform (FFT). The resulting procedures provide selfconsistent solutions _to the Maxwell and diffusion equations.

Results _are given to illustrate the two methods.

Introduction.

Modelling semiconductor laser cavity has been the subject of _a number of investigations [I].

Considerable attention has focused _on providing self-consistent solutions _to the waveguide and carrier distribution equations. Various workers make _use of separation of transverse and lateral modes [2].

The objective of this paper is _to give two numerical methods in order _to study nonhomogeneous laser cavity. The first method adopts the approach of [3]. The transverse modes and the effective index _are first computed. ^Then ^the ^lateral ^modes are obtained using

beam propagation method and fast Fourier transform technique. As this method leads _to multisolution of transverse field distribution, _we _propose an altemative method which does not use effective index method [4].

1. Theoretical model.

For definiteness _we consider _a semiconductor laser shown in figure I.

WAVEGUIDE EQUATION. The optical ^field Ê îs âssumed to be solution of the Maxwell's

wave equation

V~E ~_~'§

= 0 (1)

c

t

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1730 JOURNAL DE PHYSIQUE III N° 9

L

w ++

h

~

x

active layer

Fig. I. Schematic illustration of the geometry ^and ^notation ^of a semiconductor laser.

where _c is the velocity of light in _vacuum and I is the complex refractive index given by :

p(X), _X

~

~_~~~ _~~ _~

i~ + R~ min (g(n ), 0) I ~ ~~

,

ix _w ^~

~~~

2 ko 2

where _A~ and i~ are the active layer and the cladding layer indices respectively,

R~ is the antiguiding _parameter, ko ₌ ^I

,

d is the active layer thickness, _g(n is the optical gain

c

defined by

g(n) ₌ a~ ⁿ b~

where a~, ^and b~ ^are ^the optical gain coefficients and _n is the carrier density distribution.

For the first approach we seek the optical field in the following form E

= iW(x) qr,~ ~~ ~~~, ^~~_

~

~ "b°~ ~) e'~'+ '~z)

where the polarization vector f is the unit vector in the y-direction, W (x) is the transverse field distribution, qt~ and qt~ are the forward and backward lateral field distributions, respectively.

Inserting equation (3) into equation (I) and separating W lx) from qt _we obtain _:

~~+x [k(i~-p~(y,z)]W=0 ₍₄₎

±2ik~~'+~~+~~+ [p~~,z)-k~]qt=0 (5)

?z

z y

where the upper or lower sign is chosen for qt~ or qtr, respectively.

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For the second method _we _assume _a field distribution of the form _: E

=

I(«<(x, _y, z) et"'- ^'~. ₊ «~(x, _y, z) et ^"< ⁺i~.j (6)

Substituting equation (6) in equation (I) qt~ ^and qt, _are found _to satisfy

± 2 ik ~~'+

~~~ + ~~~ + ~~~ + (k(^i~ k~) qt

= 0 (7)

?z Jz Jy ax

The function p~y, z) is required to solve equation (5). Thus, we first compute p and

W solving equation (4) under these conditions _: (I) W and ~'~'

are continuous at the interface between the layers

ax

(it) W is assumed _to be _zero at infinity _; (iii) W(o)

= i.

Noting that I is graduated in the x-direction, equation (4) together ^with the conditions (I), (it), (iii) _are solved using the staircase method [5].

Equations (5) and (7) are solved using the beam propagation method. An initial profile

~/ is given and equation (5) or (7) can be solved repeatedly to accomplish _a round trip with the boundary conditions _at the facets _:

~b(X, _y~ o)

"

fi _~'(X,

y, o)

~<(x, _y, L

=

fi _~b(x,

y, L ).

Where R~ is the mirror reflectivity and L is the cavity length. The advantage of this technique

is _to uncouple the waveguide equation and diffusion equation. It also allows the _use of fast Fourier transform algorithm ^for which there exists vectorised and parallelised routine. Sine this method has been discussed elsewhere [6-8], only _a brief description is given.

The laser cavity îs divided into _nz I segments figure (2). Êach segment îs partitioned in

nr I sub-segments of equal spacing 2 6. Let qt~ ^be ^the ^field distribution _at the beginning of the j-th segment. ^If au, ^{is the} ^field distribution at the beginning of the f-th sub-segment ^of the

j-th segment, ^its ^value at the next sub-segment is obtained using _:

qty~j ₌ (,f~~e'~~f)e~~~J(3~'e'~~f)qti~ (I wiwn;-1) (8)

where 3 and &~ _are the Fourier and the inverse Fourier transform operator respectively, the function 4 _is _defined _in _the _Fourier _space by :

» ~j~j~~ k~ _ip12 _((p( _wk)

~~~

~ k+1 ip12~k2 _((p( _~k)

&

~j j~_ -k+ _(p(~ _((p( _Sk)

~ -k+ifi _((p(~k). ~j~~

The sign ₊ or is chosen for qt~ _or qt,, respectively.

The _term

q~ represents ^the changes in index and gain

q~ =

P ~ k~ f°r ~qUati°n 13)

(11) k(^i~ ^k~ for equation (6)

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cavity facet 0

~

cavity facet I

zj = 0 zj_ _j _~ _~ /~

j _nZ

x~segment

_~ ^j

26 gain sheet

z

~l,j- _I ^" Zj-1 ^{~~~ "~} ^' ^~l,j- ^I ^~nr,J- ^' ⁹ sub-segment I-

Fig. 2. Illustration of the B.P.M, procedure.

Assuming slow variation of the field distribution in the z-direction, its second derivative with respect to z can be neglected in equations (5) and (7). In this _case the function

4 _is _defined by

&((p()=±- ~j2 (12)

where the upper ^or lower sign is chosen for qt~ _or qt,, respectively.

The field distribution _at the next segment ^is

qt~~j=qt~,~ (lwjwnz-1), (13)

At the beginning of the j-th segmpent~ we set

qt~ = qt~ (I _w j _w _nz ). (14)

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The numerical procedure is initiated by choosing _a Gaussian field distribution of

qt) _at

z = 0. For the first forward beam propagation, qt~ is assumed _to be _zero and

W, is carried _out using equations (8-14) from _z

= 0 _to _z

=

L. Its value is kept for the _next backward beam propagation. At _z =L,qt~ is derived from qtr applying the reflectivity

condition. We then propagate qt~ back to _z

=

0 using equations (8-14). Its value is stocked for the _next forward beam propagation. Another reflection _at _z

= 0 provides qtr for the next round

trip. The whole procedure is repeated until _a self-consistent field distribution emerges. The

self-consistency is achieved [2] when the mode power

P=('-Rm)~~ _i(~i(~dx

changes by a constant fraction and the mode width

'~~ ~~ ~~~

8 w~joP

~2

~~~~ ~~~~~

remains constant on successive round trips with _some specified _error tolerance _F. In _our numerical results _s _was taken _to be I fl. The parameter _Ho ^is ^the permeability of free space.

DIFFUSION EQUATION. As equation (11) shows, I is needed _to compute q~. ^From

equation (2), _n is required to determine I. Now _n is derived using the model of [3] ^which

assumes a small variation of carrier transport ⁱⁿ ^the z-direction _:

D~rr ~

= ~~~j~ ⁺ ^~ ⁺ ^Bn~ ⁺ ^Cn~ ⁺ ^~~~ ^S(E) ⁽¹⁵⁾

3y q Tnr W

where S is defined by

with the boundary conditions _:

n(0)

= n(y~~,) ₌ _n~. (17)

The function J is the current density and depends _on the _contact voltage Vo, the contact

resistivity _r~, the passive layer resistivity _p and the parameters H and h shown in figure 3. The parameters D~rr, _T~~, ^B and C _are the effective coefficient diffusion, the nonradiative

recombination time, the spontaneous recombination coefficient and the Auger coefficient, respectively.

Equation (15) together with the boundary conditions (17) _are solved using a variable order, variable step ^size ^finite ^difference method with deferred corrections [9].

2. Computational results.

The values of the material parameters ûsed ⁱⁿ ^the analysis are given in table I. The values for a~, b~, D~,r, ^B ând ^C âre ^those ûsed ⁱⁿ [10]. The structural parameters are reported in table II.

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Table I. Values of material parameters ^used ⁱⁿ ^the analysis.

Parameter Value Unit

Ao ^0.84 ~Lm

a~ ^1.5 ^x ^10~~° ^m~

b 1.5 x 10~ m-'

/ 1.4 _x 10~'6 m/s

C 0 _x 10-4~ m/s

R~ ^2.67 x 10-5

D

~

33.0 _x 10-4 m~/s

T(~ 50 _ns

n~ 3.641

N~ I-O _x 10~~ m-~

R~ ^0.32

Table II. Values of ^structural parameters ^used ⁱⁿ ^the analysis.

Parameter Value Unit

d 0. I _~Lm

h 0. 3 ~Lm

H 1.6 _~Lm

w> 1. 5 _~Lm

w' 1. 5 _~Lm

1 200 _~Lm

x ~_

0.lpm

Y

h Metal

w

A1203 H w'

GaAIAS

xAJ ₌ 40%

GaAlAs xAl ₌ 5%

o ~~ p~

~

xAJ ₌ 40%

GaAlAs

Fig. 3. Schematic illustration of structure layers.

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Results _are obtained for AlGaAs array shown in figure 3. Carrier density, nearfield and farfield intensity shown in figures 4 _are provided using the first method. Five round trips _are

needed to achieve convergence with _nz

= nr = I I and 6

= ~Lm. The results obtained by the

second method _are presented in figure 5. We have chosen 6

=

0.016 _~Lm, _nz

=

26 and

nr ₌ 251. The self-consistency is achieved about six round trips. We have reported a three dimensional plot figure 5b of intensity field and the corresponding intensity profiles of lateral

figure 5c and figure 5d _transverse modes. The resulting current value is about 65 mA.

('~

# _~ U-B

w ~

C >

3 ^Z

c £

.~ g

£ Ko

~i b

~ $

.( C

fl 3

$ .~

~ £

fi 5~

0 IO 20 30 40 0 2 3 4 5

Y (pm) _x (Mm)

a) b)

Q~

j/ 24

20

-

q ~

16 ~

~ ,

~

W c

C '

~ "

~

C 8

~ o

@ .

.£

4

C4

0

y _(vlll) e (~)

c) d)

Fig. 4.

intensity ^rofile the cavity facet.

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~

(12

~

~~

. .

' '

~ ,W

0 20

Y _JLm)

3500 3500

3000 3000

- -

I I

£ 2500 £ 2500

§ §

~- ~-

# b

W 7

f ₁₅₀₀ f

£ 1000 £

~ ~

5l 5l

soo

o

0 10 20 30 40 0 2 3 4 5

Y (Mm)

x (vm)

c) d)

Fig. 5. al carrier density, b) field intensity profile, cl lateral mode intensity profile, d) transverse mode intensity profile _at the cavity facet.

3. Conclusion.

For the given example the _two methods provide the _same results.

The first _one is computationally fast but in the _case of complex geometry nonhomogeneous

structures, the effective index method requires _to control the chosen solution. This is the main

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difficulty of the method. The system being multisolution in _transverse mode, it is not always possible _to maintain the procedure _on the fundamental mode.

On the other hand in the second method this problem does not exist since the _transverse and lateral modes _are automatically treated. Simply in the _case of semiconductor laser structures,

the thin active layer thickness (0.I ~m) imposes a propagation _step size of order

10~~ ~m in order _to _account for the real structure and _to eliminate the parasite diffraction

produced by the spatial discretisation of the B.P.M. This _means _an important computation time which _can be reduced using parallelised and vectorised algorithm.

This method is general and useful for the modelling of any given nonhomogeneous cavity

laser.

References

[ii Buus J., Principles of semiconductor laser modelling, TEE Proceedings 1 (1985) 42-51.

[2] Agrawal G. P., Lateral-mode analysis of gain-guided and index-guided semiconductor laser arrays, J. Appl. Phys. 8 (1985) 2922-2931.

[3] Agrawal G. P., Joyce W. B., Dixon R. W., Lax M., Beam-propagation analysis of stripe-geometry semiconductor lasers _: Threshold behavior, J. Appl. Phys. Lett. 43 (1983) II-13.

[4] Buus J., The effective index method and its application to semiconductor lasers, IEEE J. Qttant.

Electron. 7 (1982) 1083-1089.

j5] Delort M.. Thkse de Doctorat de I'UPS 703 (1990) 12-15.

[6] Bamberger A., Coron F., Guidaglia J. M., Analyse de la B-P-M-, m6thode de rdsolution approchde

de l'dquation de Helmholtz dans _une fibre optique, moddlisation, convergence ^et stabilitd, Rapport inteme CMAP lsl (1986).

[7] Bamberger A., Engquist B., Halpern L., Joly P., Parabolic _wave equation approximations in heterogeneous media, SIAM J. Appl. Math. 48 (1988) 99-128.

[8] Bamberger A., Engquist B., Halpern L., Joly P., Higher order paraxial equation wave equation approximations in heterogeneous media, SIAM J. Appl. Math. 48 (1988) 129-154.

[9] Pereyra V., PASVA3 _: _an adaptive finite-difference FORTRAN program for first order _non linear

boundary value problems, Lecture Notes in Computer Science 76 (1978) 67-88.

[10] Agrawal ^G. P., Fast-Fourier-transform based beam-propagation model for stripe- geometry semiconductor lasers _: Inclusion of axial effects, J. Appl. Phys. 56 (1984) 3100-3109.