Modal properties of two types of

surface-emitting circular Bragg micro-lasers (CBMLs) are

compared. Disk CBMLs are suitable for low-threshold,

high-efficiency, ultracompact laser design, while ring CBMLs

are an excellent candiadate for high-efficiency, large-area,

high-power applications

Sun, Xiankai

Yariv, Amnon

English

Caltech Authors - Main

A Comparative Study of Modal Properties of 
Surface-Emitting Circular Bragg Micro-Lasers 
Xiankai Sun and Amnon Yariv 
Department of Applied Physics, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
E-mail: xksun@caltech.edu
Abstract: Modal properties of two types of 
surface-emitting circular Bragg micro-lasers (CBMLs) are 
compared. Disk CBMLs are suitable for low-threshold, 
high-efficiency, ultracompact laser design, while ring CBMLs 
are an excellent candiadate for high-efficiency, large-area, 
high-power applications. 
Keywords: Surface-emitting lasers; Semiconductor lasers; 
Bragg lasers; Modal property; Integrated optoelectronics
1. INTRODUCTION
Surface-emitting lasers with a circular emission aperture 
are particularly useful in optical communications because 
they can produce circularly-symmetric, narrow-divergence 
laser beams [1-3]. With optical gain, circular Bragg micro- 
resonators can be engineered for such applications – the radial 
Bragg reflector not only provides feedback for the in-plane 
fields but also couples the laser radiation out of the plane 
vertically. The lasers based on such circular Bragg resonance 
are referred to as circular Bragg micro-lasers (CBMLs). 
Similar to the circular resonators, the CBMLs can be 
classified into two configurations: disk- and ring- CBMLs. As 
shown in Fig. 1, disk CBML has a center disk surrounded by 
a radial Bragg grating extending from x0 to xb. Ring CBML 
has an annular defect surrounded by inner and outer gratings 
which extends, respectively, from the center to xL and from xR
to xb. In this paper we compare their modal properties such as 
modal field pattern, threshold gain, detuning factor, and 
far-filed pattern. In order to demonstrate their advantages as 
surface-emitting lasers, circular DFB laser will be introduced 
as a reference in comparing the quality factor, emission 
efficiency, and modal area. 
2. COMPARISON OF MODAL PROPERTIES OF 
SURFACE-EMITTING CBMLS
The characteristic equations which govern the laser modes 
were derived in [4] by resonance condition theory: 
02( )
0
0 0
sinh[ ( )] 1  for disk CBML,
( )sinh[ ( )] cosh[ ( )]
Ag i x
b
b b
e v S x x
u i S x x S S x x
?
?
? ? ?
? ? ? ?
2( )( ) sinh[ ( )]and 
( )sinh[ ( )] cosh[ ( )]
( )sinh[ ] cosh[ ] 1  for ring CBML.
( )sinh[ ] cosh[ ]
A R Lg i x x
b R
b R b R
L L
L L
e v S x x
u i S x x S S x x
u v i Sx S Sx
u v i Sx S Sx
?
?
?
?
? ? ?
? ? ? ?
? ? ?? ?
? ? ? ?
In the left equations, gA is the gain coefficient. ? = (?design–?)/?
is the frequency detuning factor, where ? is the in-plane 
propagation constant. u and v are defined as u = gA – h1, v = h1
TABLE I
MODAL FIELD PATTERNS, THRESHOLD GAINS (gA, IN UNIT OF 10?3), AND 
FREQUENCY DETUNING FACTORS (?, IN UNIT OF 10?3) OF THE DISK CBML
AND RING CBML WITH EXTERIOR BOUNDARY RADIUS xb = 200
Laser type Disk CBML Ring CBML 
 Modal field gA ? Modal field gA ?
Mode 1 0.127 49.8 0.457 55.9
Mode 2 0.288 21.2 1.06 66.9
Mode 3 0.454 ?8.09 1.92 71.0
Mode 4 0.690 ?37.4 3.14 84.4
Mode 5 1.21 ?66.5 4.09 91.6
Fig. 1. Illustrations of surface-emitting CBMLs: (a) Disk CBML and (b) 
Ring CBML. Laser radiation is coupled out of the plane in vertical direction 
by the first-order scattering.
0 5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
? (degree)
(b)
Ring CBML
0 5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
? (degree)
Fa
r-
fie
ld
 in
te
ns
ity
 (a
.u
.) (a)
Disk CBML
Fig. 2. Far-field intensity patterns of the fundamental modes of disk CBML
and ring CBML (both with xb = 200). 
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+ ih2, where h1 and h2 are the radiation- and feedback- 
coupling coefficients which can be precalculated. S is defined 
as S = [(u – i?)2 – v2]1/2. It should be noted that these 
characteristic equations implicitly includes the effect of 
vertical radiation for nonzero h1.
Modal field patterns, threshold gains gA, and frequency 
detuning factors ? of the disk- and ring- CBMLs are listed in 
Table I for devices with xb = 200. For disk CBML, the inner 
disk radius x0 = xb/2. For ring CBML, the annular defect is 
assumed to be located at xb/2 with its width being a 
wavelength such that xL + xR = xb and xR – xL = 2?. As shown 
in Table I, all the displayed modes of disk CBML are 
confined to the center disk with negligible peripheral power 
leakage and thus possess very low thresholds and very small 
modal areas. For ring CBML, each mode needs a higher 
threshold gain than that of the disk CBML, but they also 
possess large modal areas which is advantageous for 
large-aperture, high-power surface emission. 
Far-field intensities are calculated based on Fourier 
transform of the radiation fields at the emission aperture. 
Shown in Fig. 2, the disk CBML has most of the energy 
concentrated in the zeroth-order Fourier component thus its 
peak is located at the center, while the ring CBML has the 
first-order lobe dominating thus features a donut shape. 
In what follows we also include circular DFB laser as a 
reference to show the advantages of CBMLs as surface- 
emitting lasers. To compare quality factor, an unnormalized 
grating
22
0 0
2 22
0
( ) ( )
( , 0) ( ) ( )
bD x
D
b b
Z z dz E x xdx
Q
E x z xdx Z z dz E x x x?
?
? ?
? ? ? ? ?
? ?
? ?
is used whose nominator represents the store energy and the 
denominator the total power loss. D is the resonator thickness 
and Z is the vertical modal profile. E(x) denotes the in-plane 
modal field and ?E(x,z) the radiation field where the emission 
aperture is located at z = 0. Fig. 3(a) shows increases in the 
device size (xb) result in enhanced Q?  values. Additionally, 
the disk CBML exhibits a much higher Q?  than the other 
two laser structures of identical dimensions, which is 
consistent with the threshold behaviors shown in Table I. 
The emission efficiency ? is naturally defined as the 
fraction of the total power loss which is represented by the 
useful vertical radiation. Fig. 3(b) demonstrates a larger ? for 
devices with a larger size (xb). As seen from comparison, only 
the disk- and ring- CBMLs achieve high emission efficiencies. 
This is because their fundamental modes are bandgap modes 
while that of the circular DFB laser is an in-band mode. 
The effective modal area in a 2-D case is defined as 
eff 2 2
mode | | max{| | },A xdxd?? ?? ? ??? E E
where x and ? are radial and angular coordinates, respectively. 
It is a measure of how the modal field is distributed in the 
laser. Plotted in Fig. 3(c), the modal area of the disk CBML 
was found to be the smallest. This is not surprising and can be 
inferred from their unique modal profiles listed in Table I. 
Actually our prior experimental work has demonstrated 
single-mode lasing with ultrasmall modal volume in a nano- 
sized disk CBML fabricated in InGaAsP [3]. Clearly, a disk 
configuration is preferable in ultracompact laser design. 
Alternatively, ring CBMLs have a large modal area with high 
emission efficiency, rendering them suitable for high- 
efficiency, high-power, large-area laser applications. 
3. CONCLUSIONS
By comparing the modal properties of disk- and ring- 
CBMLs, we demonstrated their advantages as surface- 
emitting lasers and concluded that disk CBMLs are most 
useful in low-threshold, high-efficiency, ultracompact laser 
design, while ring CBMLs are excellent candidates for high- 
efficiency, high-power, large-area lasers. 
REFERENCES
[1] T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. 
J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, 
surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," 
Appl. Phys. Lett., vol. 60, pp. 1921-1923, Apr. 1992. 
[2] M. Fallahi, M. Dion, F. Chatenoud, I. M. Templeton, K. A. McGreer, G. 
Champion, and R. Barber, "Low divergence electrically pumped 
circular-grating surface-emitting DBR laser on an InGaAs/GaAs 
structure," Electron. Lett., vol. 29, pp. 1412-1414, Aug. 1993. 
[3] J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from 
a circular Bragg nanocavity with an ultrasmall modal volume," Appl. 
Phys. Lett., vol. 86, p. 251101, Jun. 2005. 
[4] X. K. Sun and A. Yariv, "A unified theory for surface emitting chirped 
circular grating lasers," in Proc. SPIE. vol. 7218, J. E. Broquin and C. 
M. Greiner, Eds. SPIE Photonics West 2009, San Jose, CA: SPIE, 2009, 
p. 72180H. 
50 100 150 200 250
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Exterior boundary radius xb
E
m
is
si
on
 e
ffi
ci
en
cy
 ? Disk CBML
Ring CBML
Circular DFB laser
(b)
50 100 150 200 250
101
102
103
104
105
106
M
od
al
 a
re
a 
A
ef
f
m
od
e
Exterior boundary radius xb
Disk CBML
Top surface area ?xb
2
Ring CBML
Circular DFB laser
(c)
50 100 150 200 250
0
50
100
150
200
250
Exterior boundary radius xb
U
nn
or
m
al
iz
ed
 q
ua
lit
y 
fa
ct
or
 Q
'
Disk CBML
Ring CBML
Circular DFB laser
(a)
Fig. 3. (a) Unnormalized quality factors, (b) emission efficiencies, and (c) modal areas of the fundamental modes of disk CBMLs (dashed), ring CBMLs (dotted), 
and circular DFB lasers (dash-dot). 
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A Comparative Study of Modal Properties of Surface-Emitting Circular Bragg Micro-Lasers

0.9 Exterior boundary radius x b E m i s s i o n e f f i c i e n c y K Disk CBML Ring CBML Circular DFB laser (b)

A unified theory for surface emitting chirped circular grating lasers,&quot; in

Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,&quot;

Low divergence electrically pumped circular-grating surface-emitting DBR laser on an InGaAs/GaAs structure,&quot;

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A Comparative Study of Modal Properties of Surface-Emitting Circular Bragg Micro-Lasers

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