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As can be seen from Figs. 2 and 3, both approximate solutions
are in good agreement with the exact solution.
Conclusion: Poisson equation is solved analytically in a
AlGaAs/AlGaAs/GaAs structure under reverse bias subject to
ohmic contact boundary conditions. It is shown that the
position-dependent dielectric constant in the graded region
can be approximated by an average constant dielectric constant with good accuracy. This should facilitate the numerical
modelling of such device structures.
Acknowledgments: The author would like to thank E. Gardner
and F. Barnes for their encouragement.
S. C. KWOK
Department of Electrical and Computer Engineering
University of Colorado
Boulder, C O803094425. U S A
4th April 1990
References
CAPASSO,F.: ‘New multilayer and graded gap optoelectronic and
high speed devices by graded gap engineering’, Surface Sci., 1984,
142, pp. 513-528
2 CASEY, H. c., and PANISH,
M. B.: ‘Heterostructure lasers, part A:
fundamentalprinciples’ (Academic Press, New York, 1978)
3 ANDREWS, M.H.,MARSHAK, A.H.,and SHRIVASTAVA, R.: ‘The effect of
position-dependent dielectric constant on the electric field and
charge density in a p-n junction’, J . Appl. Phys., 1981,52, (ll),pp.
1
67834787
ANALYSIS OF COUPLING COEFFICIENT
BETWEEN TWO VERTICAL CAVITY
SURFACE E M l l T l N G LASERS FOR
TWO-DIMENSIONAL PHASE-LOCKED
ARRAY
In order to address this issue, we report the results of the
analysis of the coupling coellicient of two evanescently
coupled VCSE lasers and discuss the design of twodimensional phase-locked array of vertical cavity surface emitting lasers. If the radius of the laser is comparable to the
carrier diffusion length of a few microns, and the laser is
pumped high enough for the electron density within the active
region to be uniform, the laser can be approximated as a step
index profile cylindrical waveguide. A similar problem was
dealt with in optical fibre coupler analysis, however their
interest was focused on the optical fibre which can be approximated by the weakly guiding or degenerate fibre condition.
For a large refractive index difference between the laser and
the surrounding medium (or cladding), like the VCSE laser
array, a more rigorous analysis is necessary. In this analysis,
the InGaAs strained layer quantum well active layer and the
GaAs cavity are taken into account.2 The emission wavelength of the laser is l p m and the refractive index of the
cavity is 3.5.
A computer program was written to obtain the single transverse conditions for various values of the laser radius with the
refractive index of cladding as a parameter. If the surrounding
medium is air, the laser radius has to be <0.2pm for single
mode operation which is consistent with the observation of J.
L. Jewell.’ For an AIo.,Ga,.,As cladding, the cut-off radius is
increased to 0.55pm. It can be seen that the use of an
AI, ,Ga, ,As cladding region can alleviate the design constraint so that the proportion of the peripheral length to area
of the laser cross-section can be reduced. It also acts as a
passivation layer that is capable of reducing the surface
recombination currentof the laser which reduces the carrier
~
an increase of the refractive
lifetime to about ~ O O P S .With
index of the cladding and a decrease of radius of the laser the
optical field spreads into the cladding region.
If a second laser is brought into close proximity with the
first, the optical field will have a finite amplitude in the second
laser. Since the field has an overlap in the second laser it can
couple power into the second laser if the modal propagation
constants of the two lasers are identically matched. The eigenmode can be analysed by the coupled mode equations. The
coupling coefficient, c, between two layers of radius a and
centre to centre distanced is given byio
I ndexing terms: Lasers and laser applications, Semiconductor
lasers
The coupling coefficient between two adjacent circular vertical cavity surface emitting lasers having an InGaAs strained
quantum well active region (Ag = 1pm) and a GaAs cavity
has been analysed. The analysis was performed for
AI, ,Ga, ,As cladding ( n = 3.325)and air (n = 1). In order lo
obtain a coupling coefinent of IO-’, which is known for the
stable phase locking of array, the laser separation must be
less than 0.25pm for the lasers of radius 0.4prn with air
cladding and be less than 0.72pm for the lasers of radius
1.2pm with AI,,Ga,,As
cladding. Various structures of
two-dimensionalphase-locked arrays, such as periodic, circular and concentriccircular arrays, can be realised by bringing
lasers into close proximity of which value can be obtained
from the analysis.
The successful fabrication of low threshold current vertical
cavity surface emitting (VCSE) lasers has been reported.’-,
The strong advantage of the VCSE lasers over other surface
emitting laser structures, such as grating coupled or 45” beam
deflector, comes from the fact that a high density twodimensional array of microlasers can be achieved.’ In the case
of a compact two-dimensional array the coupling or crosstalk
between adjacent lasers limits the distance between lasers, i.e.,
the integration density if independent operation is desired.
Alternatively if the lasers are in close proximity, one can use
its coupling property to achieve a phase locked array for highpower, narrow beam divergence4 The use of a twodimensional phase-locked array leads to more freedom in the
design of the arrays suggesting that the shape of the output
beam can be better controlled to conform to the pattern
which a specific application requires.’ However, the coupling
between VCSE lasers has been neglected in spite of its importan~e.~.’
896
The quantities U, w and ki are given as a (k: - j3’)”’, Q 2
- k:)‘/’ and k,? = nA2n/A))’, respectively, where n , is the
refractive index of the laser and n2 the refractive index of
cladding. The coupling coellicients against d / a for various
values of a ; for n2 = 1 and 3.325 (refractive index of
AI, &a, ,AS) are shown in Fig. 1. When the radius, a, is small
the field amplitude is spread out far into the cladding and a
fairly insensitive coupling coefficient is obtained. The coupling
coefficient is very small for large a because the field is essentially confined to the laser region and there is little evanescent
field penetrating the cladding region. At the same time, the
coupling coefficient decreases steeply with increasing d / a since
the evanescent field, which penetrates the cladding, falls off
exponentially. In addition, for the same radius and distance,
the AIo.,Gao ,As cladding gives a higher coupling coefficients
and less critical dependence of the coupling coellicient with
distance. This knowledge can be applied to the design of twodimensional phase-locked arrays.
Phase-locked laser arrays use optical coupling between
lasers to bring a spatial coherency across the array. It has
been known“.” that the coupling coellicent between lasers
for stable phase-locked operation of the array is in the range
of 10-3-10-4. If we choose a coupling coellicient of IO-, for
the design criteria for the array, the separation between lasers
of radius 1 pm should be less than 0.15pm and lasers of radius
ELECTRONICS LETERS
21st June 1990
Vol. 26 No. 13
0.4pm be less than 0.25pm with air cladding. With
Al,.,Ga,.,As
cladding, the separation between lasers of
1.2pm radius should be less than 0 . 7 2 ~ These
.
values of
1
F i 1 Coupling coef?ient against dla Md al.4
a Air cladding
b AI,.,Ga,.,As cladding
separation can be easily obtained using a standard e-beam
lithography. An example is shown in Fig. 2 where vertical
cavity surface emitting lasers of 0.6pm radius are in close
proximity of less than 0.1p n separation. This array shows a
laser action and its far-field beam width is as narrow as 7’
which is evidence of phase l ~ k i n g In
. ~ addition, use of twodimensional array makes it possible to achieve various
geometry of the phase locked array, such as rectangular, triangular, circular and concentric circular (centred polygonal)
arrays (Fig. 3). The separation and cladding material for the
various twodimensional phase locked array should be chosen
based on the analysis of the coupling coefficient.
In conclusion, we have analysed the use of different
materials for the cladding layer of circular vertical surface
emitting laser diodes. Based on this analysis, the coupling
coefficient between two identical VCSE lasers was calculated
for various values of separation and radius for air and
Alo.,Ga,.,As cladding region. As the refractive index of the
cladding material increases, the distance between the lasers
and the size of the laser decreases, the optical field penetrates
further into the cladding material increasing the coupling coefficient. This increase of the coupling coefficient can Emit the
design of the close packed two dimensional micro laser array
because of increased coupling loss. In contrast, we can fabricate two-dimensional phase locked array using the coupling
properties. In order to obtain a coupling coefficient of lo-,,
which is known to give a stable phase locking of array, the
laser separation must be less than 0.25pm for the lasers of
radius 0.4pm with air cladding and be less than 0.72pm for
the lasers of radius 1.2pm with Al,.,Ga,.,As
cladding. Twodimensional phase-locked arrays lead to a great freedom in
the design of array structures. We have suggested three different two-dimensional phase-locked array structures, i.e.,
periodic (rectangular, triangular), circular and concentric circular arrays.
Authors would like to acknowledge a helpful discussion
with Dr. Tae-Kyung Yo0 of Gold Star Central Research Institute, Dr. Jung Woong of Kukje Co., and Drs. A. Scherer and
E. Kapon of Bellcore.
H.-J. YOO‘
J. U. HAYES
Bellcore, Red Bank, N J 07701-7040, U S A
12th March 1990
Y . 3 . KWON
Department of Electrical Engineering
Korea Advanced Institute ofscience and Technology
Cheongryang PO Box 150, Seoul, Korea
Hoi-Jun Yo0 is on leave from the Department of Electrical Engineering, Kaist, Seoul, Korea
References
1
IGA, K., KOVOMA, F., and KINOSHITA, s.: ‘Surface emitting semiconductor lasers’, IEEE J . QuMtwn Electron., 1988, QE24,pp. 1 8 4 5
1855
2
JEWELL, I. L., SCHERER, A., MCCALL, S. L.,
HARBINSON, 1. P., and FLOREZ, L. I.: ‘Low
Y. H., WALKER, S. I.,
threshold electricallypumped vertical cavity surface emitting micro-lasers’, Electron.
Lett., 1989,25, pp. 1123-1124
3
HARBISON,
I. P., and
a
b
lj@
Fig. 2 SEM photograph
of two-dimensional phase-locked array of verti-
cal cavity surface emitting lasers
a Whole structure
b Detail
periodic array
0000
0000
0000
0000
0000
circular array
0
0
0
0
0
0
centered polygon array
1950191
Fig. 3 Two-dimensional phase locked arrays
ELECTRONICS LETTERS 21st June 1990 Vol. 26 No. 13
E. G., CHANG, 0. K.,
FLOREZ, L. T.: ‘Low series resistance vertical
VOO, H.-I., HAVE& I. R., ANDREAKIDAS, N., PAEK,
cavity frong surface emitting laser diode (FSELD)’,to be published
in Appl. Phys. Lett.
4 YOO,H.-J., SCHERwl A., HARBISON, I., FLOREZ,L., PAEK, E. G., VAN DER
GAAG, B. P., HAYFS, 1. R., VON LEHMEN, A., KAPON, E., and KWON, Y.
s.: ‘Fabricationof two-dimensional phased array of vertical cavity
surface emitting lasers’, Appl. Phys. Lett., 1990
5 YOO, H.-I., HAYES, 1. R., PAEK, E. G., XHERUI, A., and KWON, Y. s.:
‘Arraymode analysis of twodimensional phased arrays of vertical
cavity surface emitting lasers’, to be published in June issue of
IEEE J . Quantum. Electron., 1990
6 IEWELL, I. L., SCHERER, A., MCCALL, s. L., COSSARD, A. c., and
ENGLISH, J. H.: ‘GaAs-AIAs monolithic microresonator arrays’,
Appl. Phys. Lett., 1987,51, pp. 94-96
7 SCHERER, A., JEWELL, I. L., HARBINSON,J., and FLOREZ, L. T.: ‘Fabrication of surface emitting lasers and microresonators’, to be
published
8 W C U S E , D.: ‘The coupling of degenerate modes in two parallel
dielectric waveguides’, Bell Syst. Tech. J., 1971, So, pp. 1791-1816
9 IEWELL, 1. L., MCCALL, S. L., SCHERER, A., HOUH, H. H., WHITAKER, N.
A., GOSSARD, A. c., and ENGLISH,
I. H . : ‘Transverse modes, waveguide dispersion and 30ps recovery in submicron GaAs/AIAs
microresonators’, Appl. Phys. Lett., 1989,55, pp. 22-24
10 V A N C L OR.,
O and
S ~ PHARISFAU,
,
P.: The coupling of two parallel
dielectric fibers, Part 11’, Physica, 1970,47, pp. 501-514
1 1 BUTLER, 1. K., ACKLEY, D. E., and ETTENBERG, M. E.: ‘COUpkd-mode
analysis of gain and wavelength oscillation characteristics of diode
laser phased arrays’, IEEE J. Quantum Electron., 1985, QE21, pp.
458463
12 ACKLEY, D. E., BUTLFR, 1. K., and ETTENBFRG, M. E.: ‘Phase-locked
injection laser arrays with variable stripe spacing’, IEEE J.
Quantum Electron., 1986, QE2f pp. 2204-221 1
897
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