1756 OPTICS LETTERS / Vol. 25, No. 24 / December 15, 2000
Asymptotic analysis of Bragg fibers
Yong Xu, Reginald K. Lee, and Amnon Yariv
Department of Applied Physics, California Institute of Technology, MS 128-95, Pasadena, California 91125
Received July 31, 2000
Using an asymptotic analysis, we obtain an eigenvalue equation for the general mode dispersion in Bragg
fibers. The asymptotic analysis is applied to calculate the dispersion relation and the field distribution
of TE modes in a Bragg fiber. We compare the asymptotic results with exact solutions and find excellent
agreement between them. This asymptotic approach greatly simplifies the analysis and design of Bragg
fibers. © 2000 Optical Society of America
OCIS codes: 060.2280, 230.7370, 260.2110, 060.2310.
Bragg fibers, in which light confinement is due to
cylindrical Bragg ref lection instead of total internal
ref lection, were proposed by Yeh
et al.
1
An especially
appealing feature of these fibers is the possibility
of guiding light in an air core, which has attracted
considerable recent interest.
2–5
Potential advantages
of air-core fibers are lower absorption loss and higher
threshold power for nonlinear effects. These f ibers
also offer other possibilities such as atom guiding by
optical waves.
3
In the original proposal and analysis of Bragg
fibers,
1
it was shown that confined modes exist in a
Bragg fiber that consists of a low-index core, including
air, surrounded by a suitably designed alternating
cladding of high- and low-refractive-index media (see
Fig. 1). An experimental demonstration of wave-
guiding in Bragg fibers was recently reported.
4
The
exact theoretical analysis of Bragg f ibers in Ref. 1
is considerably more complicated than that of planar
Bragg waveguides.
6
One main reason is that the
Bloch theorem no longer applies to cylindrically sym-
metric geometries such as Bragg fibers. Therefore
it is difficult to find an equation that determines
the confined modes. Instead, we must treat the
confined modes as quasi modes and find the guided
modes by minimizing the radiation loss, which can
be quite complicated.
1
In this Letter we develop a
formalism to analyze guided modes in Bragg fibers
in the asymptotic limit. We find that the asymptotic
approach greatly simplifies the problem and provides
an excellent approximation to the exact solution, even
when the radius of the low-index core is relatively
small. Therefore we expect that the results in this
Letter will greatly facilitate the design and analysis
of Bragg fibers.
We consider a Bragg fiber composed of a low-
index core (refractive index
n
c
) surrounded by pairs of
high- and low-refractive-index layers with n
1
and n
2
,
respectively. Other parameters of the Bragg fiber are
defined in Fig. 1. For guided modes we can factor out
the temporal and the z dependence as exp关i共bz 2vt兲兴
and the azimuthal dependence as cos共lu兲 or sin共lu兲.
1
If
we use the asymptotic expressions
7
for the Bessel func-
tions at kr ! ` to describe the cladding fields, the
radial dependence of E
z
and H
z
becomes
E
z
苷 a
c
J
l
共k
c
r兲 0 , r ,r
1
,
E
z
苷
a
n
exp关ik
1
共r 2r
n
兲兴 1 b
n
exp关2ik
1
共r 2r
n
兲兴
p
k
1
r
,
r
n
, r ,r
n
1 l
1
,
E
z
苷
a
n
0
exp关ik
2
共r 2r
n
0
兲兴 1 b
n
0
exp关2ik
2
共r 2r
n
0
兲兴
p
k
2
r
,
r
n
0
, r ,r
n
0
1 l
2
, (1a)
H
z
苷 c
c
J
l
共k
c
r兲 ,0, r ,r
1
,
H
z
苷
c
n
exp关ik
1
共r 2r
n
兲兴 1 d
n
exp关2ik
1
共r 2r
n
兲兴
p
k
1
r
,
r
n
, r ,r
n
1 l
1
,
H
z
苷
c
n
0
exp关ik
2
共r 2r
n
0
兲兴 1 d
n
0
exp关2ik
2
共r 2r
n
0
兲兴
p
k
2
r
,
r
n
0
, r ,r
n
0
1 l
2
, (1b)
where k
c
苷 关n
c
2
共v兾c兲
2
2b
2
兴
1兾2
, k
1
苷 关n
1
2
共v兾c兲
2
2
b
2
兴
1兾2
, and k
2
苷 关n
2
2
共v兾c兲
2
2b
2
兴
1兾2
.
Fig. 1. Schematic of the r z cross section of a Bragg fiber.
The f iber core has refractive index n
c
and radius r
1
.The
fiber cladding is composed of pairs of alternating layers of
high- and low-index material. The high-index layer has
refractive index n
1
and thickness l
1
. The low-index layer
has refractive index n
2
and thickness l
2
.
0146-9592/00/241756-03$15.00/0 © 2000 Optical Society of America
December 15, 2000 / Vol. 25, No. 24 / OPTICS LETTERS 1757
With E
z
and H
z
known, other field components
can be easily found from the derivatives of E
z
and
H
z
.
1
The boundary conditions require E
z
, E
u
, H
z
,
and H
u
to be continuous at the interface between two
adjacent dielectric layers. Keeping only the leading
terms for E
u
and H
u
in the asymptotic limit, we find
the following matrix relations:
µ
a
n11
b
n11
∂
苷
"
A
TM
B
TM
B
TM
ⴱ
A
TM
ⴱ
#
µ
a
n
b
n
∂
,
(2a)
µ
c
n11
d
n11
∂
苷
"
A
TE
B
TE
B
TE
ⴱ
A
TE
ⴱ
#
µ
c
n
d
n
∂
,
(2b)
where A
TE
, B
TE
, A
TM
, and B
TM
are defined as
A
TE
苷 exp共ik
1
l
1
兲
∑
i
k
1
2
1 k
2
2
2k
1
k
2
sin共k
2
l
2
兲 1 cos共k
2
l
2
兲
∏
,
(3a)
B
TE
苷 i exp共2ik
1
l
1
兲
k
1
2
2 k
2
2
2k
1
k
2
sin共k
2
l
2
兲 , (3b)
A
TM
苷 exp共ik
1
l
1
兲
∑
i
n
2
4
k
1
2
1 n
1
4
k
2
2
2n
1
2
n
2
2
k
1
k
2
sin共k
2
l
2
兲
1 cos共k
2
l
2
兲
∏
, (3c)
B
TM
苷 i exp共2ik
1
l
1
兲
n
2
4
k
1
2
2 n
1
4
k
2
2
2n
1
2
n
2
2
k
1
k
2
sin共k
2
l
2
兲 . (3d)
Since A
TE
, B
TE
, A
TM
, and B
TM
are the same for all
cladding layers, we can apply the Bloch theorem to the
cladding fields:
µ
a
n11
b
n11
∂
苷 exp共iK
TM
L兲
µ
a
n
b
n
∂
,
µ
c
n11
d
n11
∂
苷 exp共iK
TE
L兲
µ
c
n
d
n
∂
,
(4a)
exp共iK
TM
L兲 苷 Re共A
TM
兲 6 兵关Re共A
TM
兲兴
2
2 1其
1兾2
, (4b)
exp共iK
TE
L兲 苷 Re共A
TE
兲 6 兵关Re共A
TE
兲兴
2
2 1其
1兾2
. (4c)
These results clearly indicate that in the asymp-
totic limit the properties of Bragg f iber cladding re-
semble those of planar Bragg stacks.
6
Consequently,
many results of planar Bragg stacks can be directly
applied to Bragg fibers. For example, the condition
Re共A
TE
兲 . 1 indicates the TE Bragg ref lection gap,
in which exp共iK
TE
L兲 becomes a real number. Also,
to achieve optimal confinement in Bragg fibers, we
should choose l
1
and l
2
such that k
1
l
1
苷 k
2
l
2
苷 p兾2
(i.e., quarter-wave layers).
To obtain the solution for propagation constant b
we need to match the values of E
z
, E
u
, H
z
, and H
u
at
r 苷 r
1
, which gives us
v
2
c
2
n
c
2
"
J
l
0
共k
c
r
1
兲
J
l
共k
c
r
1
兲
1 i
k
c
n
1
2
exp共iK
TM
L兲2A
TM
2B
TM
k
1
n
c
2
exp共iK
TM
L兲2A
TM
1B
TM
#
3
"
J
l
0
共k
c
r
1
兲
J
l
共k
c
r
1
兲
1 i
k
c
k
1
exp共iK
TE
L兲 2 A
TE
2 B
TE
exp共iK
TE
L兲 2 A
TE
1 B
TE
#
苷
b
2
l
2
k
c
2
r
1
2
.
(5)
It becomes clear that for any l fi 0 the guided modes are
a mixture of TE and TM modes. Only for the special
case l 苷 0 can the guided modes be classified as either
TE or TM modes.
Equation (5) governs the general mode dispersion in
any Bragg fiber. A natural question arises: How ac-
curate is this asymptotic result? It may seem that
asymptotic approximations are only valid for a core ra-
dius of at least several wavelengths. However, such
is not the case, as will be shown by the following ex-
ample: To simplify our analysis we restrict ourselves
to the TE modes. The low-index core of the fiber is
air 共n
c
苷 1兲, and the high and low refractive indices
of the f iber cladding are n
1
苷 3 and n
2
苷 1.5, respec-
tively. We choose the radius of hollow f iber core to
be r
1
苷 1 mm, and the f iber cladding parameters are
l
1
苷 0.130 mm and l
2
苷 0.265 mm. These parameters
are chosen so that the Bragg fiber cladding forms a
quarter-wave stack for l 苷 1.55 mm light. For the TE
mode with l 苷 0, Eq. (5) is simplified into
J
0
0
共k
c
r
1
兲
J
0
共k
c
r
1
兲
1 i
k
c
k
1
exp共iK
TE
L兲 2 A
TE
2 B
TE
exp共iK
TE
L兲 2 A
TE
1 B
TE
苷 0. (6)
Within the spectral range of 1.4 mm ,l,1.6 mm,
the Bragg fiber supports a single TE mode, whose
propagation constant b is calculated with Eq. (6)
and plotted in Fig. 2 together with the exact results
obtained with the method reported in Ref. 1. The
agreement between the asymptotic results and exact
solutions is excellent, with the difference between the
Fig. 2. Dispersion of the fundamental TE mode in an
air-core Bragg fiber. The parameters of the Bragg fiber
are n
c
苷 1, r
1
苷 1 mm, n
1
苷 3.0, l
1
苷 0.130 mm, n
2
苷 1.5,
and l
2
苷 0.265 mm.
1758 OPTICS LETTERS / Vol. 25, No. 24 / December 15, 2000
Fig. 3. H
z
and E
u
field of the guided TE mode at l 苷
1.55 mm in the same Bragg fiber as before. The dotted
lines indicate cladding interfaces.
two being less than 2%. Considering the small
air-core radius, it is somewhat surprising that the
asymptotic approximation performs so well. In our
example, k
c
r
1
is approximately 3.8. Owing to the
high refractive index of the first layer, k
1
r
1
is much
larger, typically ⬃12. In the exact analysis, the fields
in the first cladding layer are described by J
0
共k
1
r兲 and
Y
0
共k
1
r兲.
1
At x 艐 12, the asymptotic approximations
for J
0
共x兲 and Y
0
共x兲 are already quite accurate. This
explains why the asymptotic limit provides such a
good approximation even for relatively small fiber
cores. We point out that cladding layers with a large
refractive index have been experimentally fabricated,
where cladding indices n
1
苷 4.6 and n
2
苷 1.59 are
used.
4
Notice that in Fig. 2 the effective index b兾k
0
is al-
ways less than unity, which is a unique feature of guid-
ing light in hollow-core Bragg fibers. The dispersion
of the TE mode, as can be seen from Fig. 2, is quite
strong, which is a direct results of the small air core.
A larger air core reduces the dispersion of the Bragg
fiber. At the same time, however, the Bragg fiber may
support higher-order modes. Therefore, in the design
of Bragg fibers, we need to carefully balance the re-
quirements of single-mode operation and small disper-
sion. The asymptotic results in this Letter provide an
accurate and fast way of analyzing this problem.
It is also of interest to calculate the field distribu-
tion of the guided modes. For a confined TE mode,
E
z
苷 0 and exp共iK
TE
L兲, as given by Eq. (4), should be a
real number with an absolute value less than 1. With
these observations, from Eqs. (2) and (4) we find
µ
c
n
d
n
∂
苷 C exp关i共n 2 1兲K
TE
L兴
"
B
TE
exp共iK
TE
L兲 2 A
TE
#
,
(7a)
µ
c
n
0
d
n
0
∂
苷
1
2
s
k
2
k
1
2
6
6
6
4
k
1
1 k
2
k
1
exp共ik
1
l
1
兲
k
1
2 k
2
k
1
exp共2ik
1
l
1
兲
k
1
2 k
2
k
1
exp共ik
1
l
1
兲
k
1
1 k
2
k
1
exp共2ik
1
l
1
兲
3
7
7
7
5
3
µ
c
n
d
n
∂
,
(7b)
where the overall constant C is determined by the con-
dition of c
c
苷 1 in Eq. (1), which gives
C 苷
J
0
共k
c
r
1
兲
p
k
1
r
1
exp共iK
TE
L兲 2 A
TE
1 B
TE
.
(8)
The E
u
component can be easily found from the deriva-
tive of H
z
.
1
We calculate both H
z
and the E
u
field for
the guided TE mode at l 苷 1.55 in the Bragg fiber
studied in Fig. 2. In Fig. 3 we show the asymptotic
solutions, together with the exact results obtained with
the method presented in Ref. 1. The two results agree
well with each other, which again confirms the accu-
racy of the asymptotic analysis. A particularly inter-
esting feature of Fig. 3 is that the field decays to almost
zero within only a few pairs of cladding layers, a di-
rect result of large index contrast between the cladding
materials. In fact, a short calculation of exp共iK
TE
L兲
shows that it is possible to achieve 0.2 dB兾km loss with
less than 20 pairs of cladding layers.
In conclusion, we have developed an asymptotic for-
malism that greatly simplif ies the analysis of Bragg
fibers. Even for an example with relatively small cen-
tral air core, the asymptotic results agrees well with
the exact solutions. If more-accurate results are de-
sired, we can treat the first several cladding layers ex-
actly and use this asymptotic formalism for the rest of
the cladding structure.
This research was sponsored by the U.S. Office of
Naval Research, whose support is gratefully acknowl-
edged. Y. Xu’s e-mail address is yong@its.caltech.edu.
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