Annular Bragg defect mode resonators
Jacob Scheuer and Amnon Yariv
Department of Applied Physics, 128-95 California Institute of Technology, Pasadena, California 91125
Received April 10, 2003; revised manuscript received July 8, 2003; accepted July 11, 2003
We propose and analyze a new type of a resonator in an annular geometry that is based on a single defect
surrounded by radial Bragg reflectors on both sides. We show that the conditions for efficient mode confine-
ment are different from those of the conventional Bragg waveguiding in a rectangular geometry. A simple and
intuitive approach to the design of optimal radial Bragg reflectors is proposed and employed, yielding chirped
gratings. Small bending radii and strong control over the resonator dispersion are possible by the Bragg con-
finement. A design compromise between large free-spectral-range requirements and fabrication tolerances is
suggested. © 2003 Optical Society of America
OCIS codes: 130.0130, 130.2790, 230.5750.
1. INTRODUCTION
The past few years have witnessed a substantial increase
of activity in utilization of ring resonators for optical com-
munication devices. Various ring-resonator-based appli-
cations such as modulators,
1
channel drop filters,
2
and
dispersion compensators
3
have been suggested and dem-
onstrated.
The important characteristics of the modes of ring reso-
nators are the free spectral range (FSR) and the loss per
revolution, or, equivalently, the Q factor. One method of
realizing tight confinement and high Q is to utilize Bragg
reflection instead of total internal reflection (as in ‘‘con-
ventional’’ resonators). Disk resonators based on Bragg
reflection have been analyzed in the past, both for laser
and passive-resonator applications,
4–12
employing both
coupled-mode theory and field transfer matrices.
In this paper we propose and analyze a new type of ring
resonator: an annular-defect-mode resonator which con-
sists of a single annular defect located between radial
Bragg reflectors. Bragg-reflection-based disk resonators
(i.e., a disk surrounded by concentric Bragg layers) and,
recently, ring resonators have been studied theoretically
and demonstrated experimentally.
4–13
Also recently, a
hexagonal-waveguide ring resonator based on photonic-
bandgap-crystal confinement on both sides of the wave-
guide was demonstrated experimentally.
14
However, this
structure exploited the specific symmetry of the triangu-
lar lattice that permits low-loss, 60° abrupt turns in order
to realize a closed resonator.
The basic geometry we propose is illustrated in Fig. 1.
A circumferentially-guiding defect is located within a me-
dium which consists of annular Bragg layers. As a result
of the circular geometry, the layer widths, unlike in rect-
angular geometry, are not constant,
15
and our task is to
determine the widths that lead to maximum confinement
in the defect.
In Section 2 we develop a matrix formalism in order to
solve for the modal field, and in Section 3 we describe the
rules for designing an annular Bragg-defect-mode resona-
tor. In Section 4 we describe the dispersion relation and
the modal profile of the field. In Section 5 we analyze the
properties of resonators that are based on higher-Bragg-
order reflectors, and in Section 6 we discuss the results
and summarize.
2. BASIC THEORY
We consider an azimuthally symmetric structure as illus-
trated in Fig. 1. The guiding defect, which is composed of
a material with refractive index n
def
, is surrounded by
distributed Bragg reflectors on both sides, where the re-
flectors’ layers are of refractive indices n
1
and n
2
alter-
nating. All the electromagnetic field components can be
expressed by the z component of the electrical and mag-
netic field
15
that satisfy the Helmholtz equation, which in
cylindrical coordinates is given by
冋
1
冉
冊
⫹
1
2
2
2
⫹ k
0
2
n
2
共
兲
⫹
2
z
2
册
冉
E
z
H
z
冊
⫽ 0,
(1)
where
, z, and
are the radial, axial, and azimuthal co-
ordinates, respectively, and k
0
is the wave number in
vacuum. The refractive index n(
) equals n
def
, n
1
,orn
2
according to the radius
. Assuming that the
,
, and z
dependence of the field can be separated, the electrical
field z component of the electrical field can be written as
E
z
⫽ R
共
兲
exp
关
i
共
m
⫾

z
兲
兴
m is an integer, (2)
with a similar expression for the magnetic field z compo-
nent. Introducing Eq. (2) into Eq. (1) leads to
2
2
R
2
⫹
R
⫹
兵
关
k
2
共
兲
⫺

2
兴
2
⫺ m
2
其
R ⫽ 0, (3)
where k(
) ⫽ k
0
n(
) is constant in each layer. The gen-
eral solution of Eq. (3) can be expressed by a superposi-
tion of Bessel functions of the first and second kind:
R
m
共
兲
⫽ AJ
m
共
冑
k
j
2
⫺

2
兲
⫹ BY
m
共
冑
k
j
2
⫺

2
兲
, (4)
where k
j
is the material wave number in the jth layer.
By combining Eqs. (3) and (4), the electrical and magnetic
z components of the field become
J. Scheuer and A. Yariv Vol. 20, No. 11 / November 2003/ J. Opt. Soc. Am. B 2285
0740-3224/2003/112285-07$15.00 © 2003 Optical Society of America
E
z
⫽
关
AJ
m
共
冑
k
j
2
⫺

2
兲
⫹ BY
m
共
冑
k
j
2
⫺

2
兲
兴
⫻ cos
共

z ⫹
兲
exp
共
im
兲
,
H
z
⫽
关
CJ
m
共
冑
k
j
2
⫺

2
兲
⫹ DY
m
共
冑
k
j
2
⫺

2
兲
兴
⫻ sin
共

z ⫹
兲
exp
共
im
兲
. (5)
The other fields’ components are derived from E
z
and H
z
:
E
⫽
i
␥
j
2
冉
H
z
⫺
m
E
z
z
冊
,
E
⫽
1
␥
j
2
冉
2
E
z
z
⫺
m
H
z
冊
,
H
⫽
⫺ i
␥
j
2
冉
⑀
E
z
⫹
m
H
z
z
冊
,
H
⫽
1
␥
j
2
冉
2
H
z
z
⫹
m
⑀
E
z
冊
, (6)
where
␥
j
⫽
冑
k
j
2
⫺

2
, and
⑀
and
are the dielectric and
magnetic susceptibilities, respectively.
Introducing Eq. (5) into Eq. (6) yields all the fields’ com-
ponents in the jth layer. The parallel components of the
fields—E
z
, H
z
, E
, H
—must be continuous at the inter-
faces. This requirement can be written in the form of a
transfer matrix connecting the amplitude vector
关
ABCD
兴
in the jth and ( j ⫹ 1)th layers:
where
is the optical angular frequency, the prime indi-
cates a derivative with respect to the function argument,
and M
j
ញ
is the matrix to the right of the equal sign. The
continuity consideration of the tangential electric and
magnetic fields at the boundary
j
separating the layers j
and j ⫹ 1 leads to
冉
A
j⫹1
B
j⫹1
C
j⫹1
D
j⫹1
冊
⫽ M
ញ
j⫹1
⫺1
共
j
兲
• M
ញ
j
共
j
兲
•
冉
A
j
B
j
C
j
D
j
冊
. (8)
It is obvious from the structure of M
ញ
j
that the natural po-
larizations of the structure are not pure TE or TM. In
this paper we are interested primarily in ring-resonator
modes such that

⬇ 0. In this case Eqs. (6) and (7) ad-
mit of two independent types of solutions: a TE mode
with E
z
, H
, and H
, and a TM mode with H
z
, E
, and
E
.
We consider the TE component of the electromagnetic
field, which is characterized by E
z
, H
, and H
.We
designate this component as TE because the primary di-
rection of the propagation is
. The M
ញ
matrix for this
component is given by Eq. (7) with

⬇ 0:
M
j
⫽
冋
J
共
␥
j
兲
Y
共
␥
j
兲
n
j
2
␥
j
J
⬘
共
␥
j
兲
n
j
2
␥
j
Y
⬘
共
␥
j
兲
册
. (9)
Using relation (8), the field components A and B can be
‘‘propagated’’ from the inner layers to the external layers.
We use the finiteness of the field at
⫽ 0 so that B
1
⫽ 0. The second boundary condition is that beyond the
last layer there is no inward-propagating field so that
B
N⫹1
⫽⫺iA
N⫹1
(for the TE mode), where N is the num-
ber of layers.
The employment of the transfer matrices is important
here because, in contrast to coupled mode theory,
5,7
it per-
mits an exact analysis of high-contrast Bragg structures
that cannot be considered as small perturbations.
3. DESIGN RULES
The formalism of Section 2 enables us to find the modal
field distribution in the case of an arbitrary arrangement
of annular, concentric, dielectric rings. We are especially
interested in structures that can lead to a concentration
of the modal energy at a predetermined radial distance,
i.e., in a radial defect.
High-efficiency Bragg reflectors in Cartesian coordi-
nates require a constant grating period that determines
冉
E
z
H
H
z
E
冊
⫽
冤
J
共
␥
j
兲
Y
共
␥
j
兲
00
n
j
2
␥
j
J
⬘
共
␥
j
兲
n
j
2
␥
j
Y
⬘
共
␥
j
兲
m

⑀
0
␥
j
2
J
共
␥
j
兲
m

⑀
0
␥
j
2
Y
共
␥
j
兲
00J
共
␥
j
兲
Y
共
␥
j
兲
m

␥
j
2
J
共
␥
j
兲
m

␥
j
2
Y
共
␥
j
兲
1
␥
j
J
⬘
共
␥
j
兲
1
␥
j
Y
⬘
共
␥
j
兲
冥
冉
A
j
B
j
C
j
D
j
冊
⬅ M
j
ញ
共
兲
冉
A
j
B
j
C
j
D
j
冊
, (7)
Fig. 1. Illustration of the annular-defect-mode resonator struc-
ture. Dark rings are of refractive index n
1
, narrow light rings
n
2
, wide light ring n
def
, center n
core
.
2286 J. Opt. Soc. Am. B/Vol. 20, No. 11 / November 2003 J. Scheuer and A. Yariv
the angles at which an incident wave would be reflected.
Generally, the grating wave number (2
/⌳, where ⌳ is the
grating period) multiplied by the reflection order should
be approximately twice the transverse component of the
incident wave’s wave vector.
15
However, when the struc-
ture is annular, the conditions for efficient reflection are
different.
Several methods for determining the thickness, and
thus the position, of the Bragg layers’ interfaces have
been suggested in previous publications.
5–8
Compared
with Bragg fibers,
16
the incidence angle of the waves at
the interfaces (measured from the normal to the inter-
face) is smaller; therefore, the asymptotic approxima-
tion
17
is not valid and the ‘‘conventional’’ /4 layers would
not be appropriate. The principle underlying these
methods is to position the layers’ interfaces at the zeros
and extrema of the field transverse profile. This strategy
ensures the decrease of the field intensity for larger radii
and the reduction of radiating power from the resonator.
Here we present a more intuitive, although equivalent,
approach to determine the widths of the layers.
We use the conformal transformation
18,19
⫽ R exp
共
U/R
兲
,
⫽ V/R, (10)
and the inverse transformation
U ⫽ R ln
共
/R
兲
,
V ⫽
R,
n
共
兲
⫽ n
eq
共
兲
R/
,(11)
where R is an arbitrary parameter. The transformation
(10) maps a circle in the (
,
) plane with radius R
0
to a
straight line in the (U, V) plane located at U
0
⫽ R ln(R
0
/R). The structure in Fig. 1 is transformed
into a series of straight lines. The wave equation in the
(U, V) plane is obtained by transforming Eq. (1):
2
E
U
2
⫹
2
E
V
2
⫹ k
0
2
n
eq
2
共
U
兲
E ⫽ 0, (12)
where n
eq
(U) ⫽ n(U)exp(U/R) is the profile of the refrac-
tive index in the (U, V) plane. Figure 2(B) depicts the
equivalent index profile n
eq
(U) in the (U, V) plane corre-
sponding to the real index profile n(
) shown in Fig. 2(A).
The latter exemplifies a ‘‘conventional’’ Bragg waveguide
design comprising /4 layers of alternating refractive in-
dex and a /2 defect. As seen in Fig. 2(B), the equivalent
index increases exponentially with the radius, and the
equivalent grating period (which is constant in the real
plane) also increases with the radius. This index profile
does not necessarily support a guided-defect mode.
In the (U, V) plane, the radial gratings are trans-
formed into a series of parallel gratings normal to the V
axis but with an exponential index profile. For this
structure to act as a Bragg reflector, the partial reflections
from each interface must interfere constructively (see Fig.
3). For that to happen, the total phase shift that the
wave accumulates while propagating through the layer
should be
/2. This condition determines the layer width
as follows:
/2 ⫽
冕
k
⬜
dU ⫽
冕
冑
k
0
2
n
eq
2
⫺

V
2
dU, (13)
where the integration beginning and ending coordinates
correspond to the interfaces of the layer, n
eq
⫽ n
¯
/R and
n
¯
⫽ n
j
2
/
␥
j
according to Eq. (9). The propagation factor

V
appearing in Eq. (13) is determined by the azimuthal
wave number m:

V
⫽ m/R. (14)
Equation (13) was used to calculate the structure re-
quired for the high-reflection Bragg mirrors surrounding
the defect. Assuming the Bragg reflectors on both sides
have identical reflection phase, then the defect width
must be /2 in the sense of Eq. (13), i.e., the defect must
satisfy
l
⫽
冕
k
⬜
dU ⫽
冕
冑
k
0
2
n
eq
2
⫺

V
2
dU, l ⫽ 1, 2, 3...,
(15)
where the integer l indicates the number of the Bessel pe-
riods (or the radial modal number) of the field in the de-
fect. It follows that the widths of the defect and the
Bragg layers depend on their coordinate U (or
) because
the equivalent index n
eq
is a function of U.
Fig. 2. Radial refractive index profile (A) and the equivalent in-
dex profile (B) of an annular defect surrounded by Bragg reflec-
tors. The maximal and minimal refractive indices are 1.5 and 1,
respectively, and the grating period is ⬃1
m.
Fig. 3. Illustration of the design rule used to realize a highly
efficient, radial Bragg reflector.
J. Scheuer and A. Yariv Vol. 20, No. 11 / November 2003/ J. Opt. Soc. Am. B 2287
Figure 4 depicts the index (A) and the modal field (B)
profiles of an annular defect mode resonator. The high-
index layers have an effective refractive index (n
¯
)of2
while the low-index layers and the defect have an effec-
tive refractive index of 1. The internal and external
Bragg reflectors have 5 and 10 periods, respectively, the
wavelength is 1.55
m, and the azimuthal wave number
is 7. The defect is located approximately at
⫽ 5.6
m
and it is 0.85
m wide.
Figure 5 shows the width of the high-index (stars) and
low-index (circles) layers. At small radii the layers’
width is greater because the equivalent index is lower
there. The layers’ width decreases for larger radii and
approaches asymptotically the ‘‘conventional’’ quarter-
wavelength width /4n. The two exceptionally wide low-
index layers in Fig. 5 are the first low-index layer (
⫽ 0 –2
m) and the defect, which has a /2 width.
4. MODAL SOLUTION PROPERTIES
Because of the design method (/4 layers and /2 defect),
the resonator has a single radial mode whose peak is lo-
cated almost exactly in the middle of the defect (see also
Fig. 4). This is unlike the field profile of conventional
ring resonators in which the field peak tends to shift to-
ward the exterior radius of the waveguide as a result of
the increase of the equivalent n
eq
index at larger radii.
Nevertheless, the asymmetry of the field profile (with re-
spect to the intensity peak) which is due to the radial
structure is noticeable.
Figure 6 shows the dispersion curve of the annular de-
fect resonator presented in Fig. 5. The vertical and hori-
zontal axes indicate, respectively, the wavelength and the
azimithal wave number m. The circles indicate the reso-
nance wavelengths and the solid curve represents a qua-
dratic interpolation: ⫽ 1.6309 ⫺ 4.8 ⫻ 10
⫺3
(m)
⫺ 9.63 ⫻ 10
⫺5
(m
2
). The resonator FSR at ⬃1.55
mis
approximately 20 nm and it increases for shorter wave-
lengths. It is interesting to note that the quadratic term
is the most dominant term in the determination of the
resonator FSR.
Figure 7 depicts the transverse profile of the modal
fields corresponding to changing the azimuthal wave
number from m ⫽ 6 to 10. It is evident that the trans-
Fig. 4. Radial index profile (A) and electrical field distribution
(B) of an annular-defect-mode resonator.
Fig. 5. High-index (stars) and low-index (circles) layer widths of
the resonator shown in Fig. 4.
Fig. 6. Resonance wavelengths (circles) and quadratic fit (solid
curve) of the resonator shown in Fig. 4.
Fig. 7. Modal field profiles for m ⫽ 6 (dotted curve), 7 (solid
curve) and 10 (dashed-dotted curve) of the resonator shown in
Fig. 4.
2288 J. Opt. Soc. Am. B/Vol. 20, No. 11 / November 2003 J. Scheuer and A. Yariv
verse profile is almost identical although the resonance
wavelength changes over more than 100 nm. The reason
for this is that the transverse profile is primarily deter-
mined by the Bragg layers width (or spatial frequency),
which are independent of wavelength. This feature is an
important advantage compared with conventional ring
resonators because coupling between resonators of this
type and Bragg waveguides, which is determined prima-
rily by the modal profiles’ overlap, can be expected to be
almost wavelength independent. Figures 8 and 9 show
the dispersion and the transverse profiles of m ⫽ 6to12
for a Bragg defect resonator utilizing lower refractive-
index contrast. For this structure, n
¯
def
⫽ n
¯
2
⫽ 3.0 and
n
¯
1
⫽ 3.5, the internal and external Bragg layers both
have 40 periods, and they were designed for m ⫽ 10 at
1.55
m. The defect is located at
⫽ 10.85
m and its
width is approximately 0.27
m. Because of the lower
contrast, more Bragg layers are needed to realize good
mode confinement, and as a result the resonator is larger
and the FSR is smaller (about 96 GHz at 1.55
m).
As shown in Fig. 8, the dispersion curve of this resona-
tor is also quadratic; it is given by ⫽ 1.5541 ⫺ 1.2
⫻ 10
⫺5
(m) ⫺ 3.98 ⫻ 10
⫺5
(m
2
). Similar to the high-
contrast case, the modal transverse profile exhibits small
wavelength dependence, which can be primarily seen in
the small-radii regime.
5. HIGHER-ORDER BRAGG REFLECTORS
Although the chirped quarter-wavelength Bragg layers
form an optimal reflector, their implementation could
prove to be difficult. Because the layers’ spatial period
changes, some conventional photolithography methods
that are employed for uniform (not chirped) Bragg
gratings
20
cannot be used. A possible approach to over-
come this problem is to position the interfaces in nonse-
quential zeros–extrema, i.e., allow the Bessel function in
each layer to complete a full period before changing the
index. From the Bragg reflection point of view, such ap-
proach is equivalent to using (2l ⫹ 1)/4 layers or em-
ploying higher reflection orders of the Bragg stack. Prac-
tically, the layer width can be evaluated in a manner
similar to that of the quarter-wavelength structure, but
the layers have to satisfy the following condition:
共
2l ⫹ 1
兲
/2 ⫽
冕
k
⬜
dU ⫽
冕
冑
k
0
2
n
eq
2
⫺

V
2
dU,
l ⫽ 1, 2, 3... . (16)
The resulting structure would have wider layers and
would therefore be larger and exhibit smaller FSR. Fig-
ure 10 compares the field transverse profile of Fig. 4(A)
and the transverse profile of a resonator designed for
similar mode parameters (5 internal periods, 10 external
periods, and m ⫽ 7 for ⫽ 1.55
m) but utilizing wider
layers (B). The radius at which the field amplitude
peaks is more than twice as large as the original (11.35
m versus 5.6
m), and the radial decay of the field rate
is smaller. Figure 11 depicts the dispersion curve of the
higher-Bragg-order-based resonator. As expected, the
FSR of the resonator is significantly smaller than that of
the original (approximately 3 nm at 1.55
m). The main
reason for this decrease in the FSR is the increase in the
defect radius. However, since the low-index layers are
inherently wider (especially in the lower-radii regime),
Fig. 8. Resonance wavelengths (circles) and quadratic fit (solid
curve) of a resonator based on lower-contrast Bragg reflectors.
Fig. 9. Modal field profile of a resonator based on lower-contrast
Bragg reflectors.
Fig. 10. Comparison of the modal field profile shown in Fig. 4(A)
and the modal field of a resonator based on second-order Bragg
reflectors with similar parameters (B).
J. Scheuer and A. Yariv Vol. 20, No. 11 / November 2003/ J. Opt. Soc. Am. B 2289
