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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009
4919
Guided Wave Analysis of Hollow Optical Fiber for
Mode-Coupling Device Applications
Sejin Lee, Jiyoung Park, Yoonseob Jeong, Hojoong Jung, and Kyunghwan Oh, Member, IEEE
Abstract—In this paper, optical mode characteristics of hollow
optical fibers are thoroughly analyzed using finite element method.
The guided modes along the ring core and cladding are identified
and their optical properties are investigated. For the core modes,
we investigated intensity distribution, higher order mode cutoff,
propagation constant, and chromatic dispersion. The mode coupling between the fundamental mode and the excited modes in both
core modes and cladding modes are discussed for applications in
mode-coupling devices.
Index Terms—Finite element methods, optical fibers, waveguides.
Fig. 1. (a) Waveguide structure and (b) cross section of a HOF.
I. INTRODUCTION
C
ONVENTIONAL optical fibers based on step index structure have led the revolution of telecommunications in the
last century. There have been intensive research and development efforts to modify the optical fiber waveguide structures
to flexibly control propagation properties such as the chromatic
dispersion and its slope, polarization of guided modes, bending
mode cutoff within optical communication bands
loss, and
[1]. These optical fibers extended their applications from optical
communications to sensor [2], endoscope [3], and high-power
laser [4]. Recently, air–silica structures have been embedded in
optical fibers to result in varieties of novel fibers: microstructured hole fibers [5], [6], omniguide fibers [7], [8], and hollow
IR transmitting fibers [9]. Another alternative structure of optical fibers called hollow optical fiber (HOF) has been recently
proposed by the authors for various photonic device applications
[10]. There have been intensive research reports on hollow photonic bandgap fibers, where the light is guided along the central
air hole region [11]–[13]. In contrast to those photonic bandgap
fibers, the HOF in this study guides the light by the index difference between the ring core and other boundaries in the central
air hole and the silica cladding. Therefore, the majority of the
light is guided along the ring core region along with the evanescent wave in the air hole region. The guided modes in the HOF,
therefore, can be flexibly controlled along with excellent longitudinal uniformity in optical characteristics.
Schematic structure of HOF is shown in Fig. 1, HOF consists
of three-layered structure. The central air hole of radius is
Manuscript received March 03, 2009; revised June 01, 2009, July 09, 2009.
First published July 21, 2009; current version published September 10, 2009.
This work was supported in part by the KOSEF under Grant ROA-2008-00020054-0 and Grant R15-2004-024-00000-0, in part by the KICOS under Grant
2008-8-1893(GOSPEL project), in part by the ITEP under Grant 2008-8-1901
and Grant 2009-8-0809, and in part by the Brain Korea 21 Project of the KRF.
The authors are with the Institute of Physics and Applied Physics, Yonsei
University, Seoul 120-749, Korea (e-mail: [email protected]).
Digital Object Identifier 10.1109/JLT.2009.2028162
surrounded by the germanosilicate glass ring core that carries
most of the guided core modes, and the cladding of radius is
made of fused silica. The ring core has the inner and outer radii
.
of and along with the thickness
HOF has drawn attention due to its unique advantages in the
following notions: 1) fiber devices can be fabricated by after
filling novel materials such as gas, liquid, and solid colloid in the
hollow region [14], 2) high form-birefringence could be induced
by elliptical ring core [15], and 3) HOF provides inherently high
compatibility with a standard single-mode fiber (SMF) by adiabatic mode transformation [16].
In comparison to various experimental investigations, theoretical analysis on HOF has been rather scarce and limited. It
is only recent that the guided core modes have been identified
in highly multimode HOFs [17], [18]. However, these multimode HOFs are not practical in device applications due to severe bending losses and modal interferences, which is detrimental to mode-coupling device applications. In mode-coupling
devices such as long-period grating (LPG) [19] or acousto-optic
tunable filters (AOTF) [20], the coupling between the fundamental core mode and a few excited modes either in the core or
the cladding has been of primary concern. In the case of HOF,
unique mode-coupling properties have not been fully exploited
thus far. Especially, the phase matching conditions for “the fundamental core modes to excited core modes coupling” and “the
fundamental core mode to the cladding modes” in HOF have
not been investigated at all.
In this paper, we systematically analyzed the guided modes
both along the ring core and the cladding along with the phase
matching conditions among them, for the first time to the best
knowledge of the authors. Utilizing a finite element method
(FEM) package, we report both the core and the cladding
modes and their modal distributions along with the normalized
propagation constants and chromatic dispersion. We also calcubetween
lated the difference in the propagation constant
the fundamental mode and excited modes both in the core and
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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009
cladding to understand unique phase matching conditions in
HOF for mode-coupling device applications.
Sellmeier coefficients for SiO [28] and GeO in [29] were
used for the calculation. The refractive index of the SiO
cladding and the GeO -doped SiO ring core was derived from
II. MODE ANALYSIS METHOD AND MODAL
CHARACTERISTICS OF HOF
(3)
A. Mode Analysis Methodology
The electric fields of guided modes in HOF could be obtained
by solving the Maxwell’s equations for the given boundary conditions. For the propagating modes along -direction in HOF,
the transverse components are derived from the longitudinal
.
component
For a given waveguide structure as in Fig. 1(a), the longitudinal component of the electric field in a core mode is analytically given as
(1)
Here
, and
are Bessel functions of integer order
. Characteristic equation can be derived for the given boundary
conditions to satisfy continuity requirements for the fields and
their derivatives. The propagation constant and subsequent
propagation properties can be obtained from the characteristic
equation.
Although analytical solutions exist for this simple structure,
we have chosen to use the FEM because it can easily be adapted
to include the distortions in fabrication unavoidable in realistic
structures.
FEM has been one of the most accurate methods to investigate the propagation properties of guided modes providing
highly reliable estimates on modal characteristics such as effective refractive index, field distribution, and cutoff wavelength
[22]–[25]. In FEM analysis of an optical waveguide starts from
the scalar wave equation where the mode field function satisfies
equation
(2)
Subdividing the domain of interest into triangular meshes,
a set of matrix equations is obtained applying the variational
principle, and the eigenvalues of the equation could be solved
for the given boundary conditions using FEM [26].
In this study, we adopted FEM [27] to obtain modal characteristics for HOF such as effective refractive indices, electric field
distribution, and modal intensity for both the core modes and
the cladding modes.
where SA, Sl, GA, and Gl are the Sellmeier coefficients for the
SiO and GeO glasses, and is the mole fraction of GeO .
C. Electric Field Distribution and Intensity Profiles of Guided
Modes in HOF
Using FEM analysis, the fundamental mode and the first five
excited modes guided along the ring core were identified as in
Fig. 2 at the wavelength of 800 nm. HOF parameters were the air
m, thickness of the GeO -doped SiO
hole radius of
ring cores
m, and relative refractive index difference
between core and cladding
%.
The direction distribution of the electric fields in the core
modes was similar to that of conventional SMFs. It is notemode is linearly polarworthy to find that the fundamental
ized across the ring core. It is also highly notable that the excited
, and
are nondegenerate, as described
modes
in the following section, in contrast to the fourfold degenerate
mode in conventional SMFs. Therefore, the guided modes
in HOF were assigned following the hybrid mode nomenclature
rather than weakly guiding LP mode assignment in conventional
SMFs.
, and
have the
Three symmetric modes
twofold degeneracy for two orthogonal polarization directions.
and
in Fig. 2(d) and (e)
The antisymmetric modes
mode has the twofold degenare nondegenerate and the
eracy. The intensity distributions of these core modes were very
similar to one another in contrast to conventional SMFs, which
can make the overlap integral between the modes in HOF significantly larger than conventional SMFs.
We further analyzed the mode intensity profile and the results
are shown in contour plots in Fig. 3. It is clearly observed that
two high intensity lobes exist within the ring core for the funmode as in Fig. 3(a). This unique nonuniform
damental
mode in HOF is rather
intensity profile of the fundamental
unexpected and has not been reported yet, to the best knowledge
of the authors.
mode, uniform intensity within the
In contrast to the
and
modes, where
ring core was observed in the
the electric field directions conform well to the ring geometry.
In air–silica optical fibers, the amount of light in the air hole
is one of the important parameters to estimate the evanescent
wave interaction. In this HOF, the power fraction at the hollow
mode was calculated as a
channel of the fundamental
function of wavelength, and the results are plotted in Fig. 4.
The power fraction of light at the hollow channel for a certain guided mode is defined as
B. Material Dispersion
In order to include material dispersion effects in precise calculation of the effective indexes of the guided modes, we used
Sellmeier equations of GeO doped SiO and SiO for the ring
core and cladding, respectively.
(4)
The power fraction has a maximum value of about 0.035%
near 1.1 m. This value is significantly lower than some of
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LEE et al.: MODE-COUPLING DEVICE APPLICATIONS
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Fig. 2. Electric field direction distributions in HOF cladding modes. Here (a)–(c) are symmetric and (d)–(f) are antisymmetric core modes of HOF. The wavelength
of the light is at 800 nm. The intensity distributions are also shown below the electric field direction distribution as a function of radial position. (a) HE , (b) EH ,
(c) HE , (d) TE , (e) TM , and (f) HE .
Fig. 3. Contour plot of intensity profile of the core modes. (a) HE , (b) EH , (c) HE , (d) TE , (e) TM , and (f) HE .
photonic crystal fibers, which showed maximum fraction up to
50% [30]. The low power fraction in HOF directly indicates
strong confinement of the guided modes along the ring core and
is attributed to the large refractive index at the boundary between the air hole and the ring core region. In contrast to the air
hole and ring core boundary, the evanescent wave significantly
extends into silica cladding, as shown in the intensity profiles in
Fig. 2.
In recent fiber optic devices, the coupling between the
fundamental core mode and cladding modes has been widely
adopted especially in LPG [19] and AOTF [20]. In LPGs,
the uniform intensity grating in the core provides phase
matching conditions between the fundamental core mode and
symmetric cladding modes. On the contrary AOTF provides
phase matching conditions for the antisymmetric cladding
mode coupling.
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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009
Fig. 5. Electric field direction distribution of cladding modes in HOF.
(a)–(f) are antisymmetric and (g)–(i) symmetric cladding modes. (a) TE ,
(b) HE , (c) TM , (d) TE , (e) HE , (f) TM , (g) HE , (h) EH ,
and (i) HE .
D. Propagation Properties of the Core Modes
Fig. 4. Power fraction at the hollow channel of the HE
of wavelength.
mode as a function
We analyzed first few cladding modes and categorized into
symmetric and antisymmetric modes as in Fig. 6. Antisymmetric cladding modes are (a)–(f), and (g)–(i) are symmetric
and
modes are nondegenerate,
modes. Note that
and HE and EH are twofold degenerate modes. In the calculation, we assumed the cladding diameter of 125 m, surrounded
by air of refractive index 1. We also included the material
dispersion in both cladding and ring core.
Due to presence of the central air hole, the cladding modes
of HOF also showed significantly different intensity distribution compared with conventional SMFs, where the symmetric
cladding modes have intensity peaks at the center while the
antisymmetric modes have nodes. HOF both symmetric and
antisymmetric modes have the annular structures and modal
overlap integrals between the fundamental core mode
would be significantly different from those in conventional
fibers, which could provide a new opportunity in LPG and
AOTF device design.
Among the modes in Fig. 5, it is expected that the overlap
symmetric cladding modes
integral of
core mode would be significantly larger than the
with the
mode
rest of modes. In the case of antisymmetric modes,
is expected to provide the largest overlap integral. Further investigation on the modal overlap and mode-coupling strength
in HOF is being pursued by the authors.
The propagation properties of the guided modes in HOF were
of the
analyzed by studying the effective refractive index
guided modes as a function of wavelength. The plots of
for
the core and cladding modes are shown in Fig. 6(a) and (b).
, and
, are nondeThe first excited modes,
generate with distinctive cutoff wavelength as in Fig. 6(a).
The HOF cladding modes in Fig. 6(b) also showed distinctive
nondegenerate effective index dispersion due to the presence of
the central air hole.
Chromatic dispersion is one of the key factors to determine
the transmission capacity of an optical communication link. The
chromatic dispersion that includes waveguide and material dispersion of each mode is obtained from the following relation.
(5)
and first excited
For the fundamental core mode,
modes, we calculated the chromatic dispersion, and the results
are shown in Fig. 7. The
mode showed the zero dispersion near 1600 nm, which is quite different from 1310 nm of
conventional SMFs. Anomalous dispersion range is significantly reduced and in the U band the dispersion is less than
10 ps/nm km. The dispersion values in 1310 and 1550 nm are
and
ps/nm km, respectively. The sign of dispersion
slope near 1550 nm is also opposite to conventional SMF. The
excited modes have relatively large negative dispersion near the
cutoff as expected.
We further investigated the impact of the air hole radius on
chromatic dispersion and compared with the conventional SMF,
where the hole radius is equal to zero.
The results are summarized in Fig. 8 for the range of hole
radius from 0 to 4 m. It is noted that as soon as the central
air hole is introduced, the chromatic dispersion is in general
downshifted to result in a lower dispersion value.
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LEE et al.: MODE-COUPLING DEVICE APPLICATIONS
4923
Fig. 6. Effective refractive index of (a) the core modes (b) the cladding modes.
TABLE I
CUTOFF WAVELENGTH OF GUIDED CORE MODES IN HOF
Fig. 7. Chromatic dispersion of core modes of the HOF.
index air hole would result in a lower net material dispersion. In
fact, the chromatic dispersion is addition of material dispersion
and waveguide dispersion, and material dispersion has positive
value contribution while the waveguide dispersion has negative
value in the range of 1.2–1.6 m. Inclusion of air in the core
region will effectively lower the material dispersion to reduce
the chromatic dispersion. Further increase of the air hole radius
beyond 3 m did not change the dispersion value itself but the
slope was affected significantly, which is directly related to the
mode cutoff at longer wavelength.
One of unique propagation characteristics of HOF is the existence of the fundamental mode cutoff. In Fig. 7(a), it is noted
of
mode cross zero near
. The cutoff
that the
wavelength of the core modes in a HOF with the given waveguide parameters are listed in Table I. Single mode window is
and
mode cutoff.
bounded by the
E. Mode-Coupling Characteristics
Fig. 8. Impacts of the central air hole radius (m) on chromatic dispersion.
Fiber mode-coupling devices have been developed for various applications such as tunable filter, bandpass filter, band
rejection filter, and frequency shifter. In Fig. 9, schematic diagrams for the mode-coupling mechanism are shown for the
cases of the core-to-core mode and the core-to-cladding mode
coupling.
The fundamental mode of HOF is converted to higher order
core modes or the cladding modes by satisfying the phase
matching conditions imposed by the periodic perturbations.
The primary principle behind the mode-coupling devices lies
in the phase matching condition.
This lower chromatic dispersion is attributed to a speculation
that replacing the germanosilicate glass by the lower refractive
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(6)
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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 22, NOVEMBER 15, 2009
Fig. 9. Schematic diagram for the principle of mode-coupling devices.
Fig. 10. Propagation constant difference between the HE
core mode
and (a) higher order core modes, (b) cladding modes. (c) Magnified image
1.2–1.65 m of (b).
Fig. 11. Phase matching pitches for the mode coupling between (a) HE
mode and higher order core modes (b) HE core mode and cladding.
where
is the propagation constant of the fundamental mode
and is that of an excited mode where the fundamental mode
is coupled. is the pitch of the periodic perturbation caused by
either acoustic wave modulations or permanent index gratings.
In this study, we investigated the phase matching conditions
for both core-to-core mode and core-to-cladding mode coupling
and their behavior as a function of wavelength in HOF, for the
first time.
is the fundamental basis for mode-couInformation on
in HOF:
pling device design. We calculated two types of
core mode to the higher order core mode coupling and
1)
2)
core mode to cladding mode coupling, which can be
readily applicable for fiber grating and AOTF devices. The results are summarized in Fig. 10.
It is well known that the bandwidth of the mode coupling
or equivalently
would directly depend on the slope of
such that if it is close to zero at a certain wavelength,
, then the coupling results in a broadband spectra near . If
increase or decrease monotonically with a nonzero
the
slope, then the coupling results in a narrowband spectra [31].
core mode to antisymmetric core mode coupling
In the
plot in Fig. 10(a) shows a plateau with a near-zero
in HOF,
slope in the range from 0.9 to 1.0 m. Therefore, broadband
spectra in the above wavelength range can be provided in a HOF.
to
However, the symmetric coupling of the
modes shows a monotonic increase with a positive slope, and
they are expected to result in narrow line spectra.
core mode to cladding mode coupling, both symIn the
metric and antisymmetric coupling showed a similar pattern:
1) monotonic decrease with a negative slope up to 1.4 m and
2) a plateau extending from 1.4 to 1.6 m, which corresponds
to narrowband and broadband spectra, respectively. Broadband
coupling in HOF has been observed in the 1.5 m region with
bandwidth exceeding 100 nm [32], [33], which agrees well with
theoretical prediction in Fig. 11(b) and (c).
We obtained the modulation pitches for the phase matching
core mode and
conditions between the fundamental
higher order core modes or the cladding modes. The results are
summarized in Fig. 11. As in Fig. 11(a), the phase matching
mode to higher order core modes in HOF
pitch for the
shows unique parabolic shapes in contrast to monotonically
increasing behaviors in conventional SMFs [32]. These parabolic shapes in the phase matching condition will provide very
wide coupling bandwidth. For example, a wide band coupling
is expected between 0.9 to 1.1 m with the bandwidth over
to
coupling for the periodic pitch
200 nm for the
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core
LEE et al.: MODE-COUPLING DEVICE APPLICATIONS
of 1.2 mm. Broadband coupling in HOF has been reported
in AOTF by the authors [32], and this tendency is in good
agreement with the experimental results in the reference.
Another notable point in the HOF mode coupling is the
fact that the phase matching pitches are several times longer
core mode
than conventional SMF. For example, the
to cladding mode coupling in conventional SMFs requires
MHz for the case of the
pitch length of about 500 m or
index grating and AOTF, respectively. In contrast to SMF, the
modulation pitches in HOF are in the order of a few millimeter,
which is longer by several factors. From this HOF phase
matching conditions, we could expect that HOF-AOTF requires
RF frequency in tens of kHz range, which is also lower than
the conventional SMF by several factors.
These changes in the modulation pitch would be of significant benefit to reduce tolerance in the fabrication and facilitate mass production. Furthermore, longer modulation pitch
would also indicate robust optical characteristics against environmental perturbation. Fabrication and characterization of
mode-coupling devices based on HOF are being pursued by the
authors for sensor applications.
III. CONCLUSION
Guided modes in HOF have been theoretically investigated
using FEM for three-layered boundary conditions: central air
hole, germanosilicate glass ring core, and silica cladding. Even
with core–cladding relative index difference of 0.0045, which
is similar to weakly guiding conventional solid core fibers, the
, and
were found to be nondeexcited modes
mode was
generate. Intensity profile of the fundamental
not uniform along the ring core, yet showed two peaks at oppomode showed cutoff
site sides. In HOF, the fundamental
at longer wavelength and single-mode window is bounded by
cutoff. By inclusion of low refracthe short wavelength
tive index air within the core resulted in reduction of chromatic
dispersion in 1.2–1.7 m range with zero dispersion wavelength
near 1.6 m. For the antisymmetric core mode to core mode
coupling, a broadband resonance in 0.9–1.0 m was predicted.
In the case of core mode to cladding mode coupling, broadband
resonance in 1.4 to 1.6 m was predicted irrespective of the
symmetry of the coupled modes. Unique mode intensity profile, cutoff, chromatic dispersion, and mode beating dispersion
of HOF can endow new degree of freedom in all-fiber photonic
device design.
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Sejin Lee was born in 1983, Gyung-ki prefecture, Korea. He received the B.S.
degree in physics (summa cum laude) from Yonsei University, Seoul, Korea, in
2007, where currently he is working toward the Ph.D. degree in applied physics
since 2007.
His recent research interests include specialty optical fiber and waveguide for
photonic devices and components.
Jiyoung Park was born in 1985. She received the B.S. degree in physics from
Yonsei University, Seoul, Korea, in 2009, where she is currently working toward
the Ph.D. degree in physics and applied physics.
Her research interests include photonic device and fiber sensors.
Yoonseob Jeong was born in 1983. He received the B.S. degree in physics from
Yonsei University, Seoul, Korea, in 2008, where he is currently working toward
the M.S. degree in physics and applied physics.
His major is optical tapers and coupling devices and will be working toward
optical trapping and optical communication.
Hojoong Jung was born in 1982. He received the B.S. degree in physics from
Yonsei University, Seoul, Korea, in 2008, where he is currently working toward
the M.S. degree in physics and applied physics.
His research interests include photonic device and fiber optics.
Kyunghwan Oh (M’96) was born in 1963, Seoul,
Korea. He received the B.S. degree in physics from
Seoul National University, Seoul, Korea, and the
M.S. degree in engineering and the Ph.D. degree in
physics from Brown University, Providence, RI, in
1991 and 1994, respectively.
He was with Gwangju institute of Science and
Technology (GIST) from 1996 to 2006. He was a
Visiting Scientist at Murray Bell laboratory, optical
fiber research division, from September 2000 to
February 2002, and a Visiting Scientist at the Institute of Physical High Technology (IPHT), Jena, Germany, in the summer
and winter of 2004. Currently, he is a Professor at the Department of Physics,
Yonsei University, Seoul. His research interests include specialty optical fiber
and waveguide for photonic devices and components.
Dr. Oh is a Topical Editor of Optics Letters (OSA), Optical Fiber Technology
(Elsevier Science) and Editorial Board Member of Optics Communications
(Elsevier Science). He was awarded with Alexander von Humboldt Research
Fellow (Germany), Chevening Scholar, (UK) and JSPS Invitation Fellow
(Japan).
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