Note: Descriptions are shown in the official language in which they were submitted.
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METHOD FOR CONFIGURING AIR-CORE PHOTONIC-BANDGAP
FIBERS FREE OF SURFACE MODES
Background of the Invention
Field of the Invention
[0002] The present application is in the field of optical fibers for
propagating light, and more
particularly is in the field of photonic-bandgap fibers having a hollow core,
or a core with a
refractive index lower than the cladding materials.
Description of the Related Art
[0003] Air-core photonic-bandgap fibers (PBFs) have attracted great interest
in recent years
due to their unique advantages over conventional fibers. In particular, the
propagation loss in
an air-core PBF is not limited by the core material, and it is expected that
the propagation loss
= can be exceedingly low. The nonlinear effects in an air-core PBF are very
small, and the core
can be filled with liquids or gases to generate the desired light-matter
interaction. Numerous
new applications enabled by these advantages have been demonstrated recently.
Such
applications are described, for example, in Burak Temelkuran et al.,
Wavelength-scalable
hollow optical fibres with large photonic bandgaps for CO2 laser transmission,
Nature, Vol.
420, 12 December 2002, pages 650-653; Dimitri G. Ouzounov et al., Dispersion
and
nonlinear propagation in air-core photonic band-gap fibers, Proceedings of
Conference on
Laser and Electro-Optics (CLEO) 2003, Baltimore, USA, 1-6 June 2003, paper
CThV5, 2
pages; M.J. Renn et al., Laser-Guided Atoms in Hollow-Core Optical Fibers,
Physical
Review Letters, Vol. 75, No. 18, 30 October 1995, pages 3253-3256; F. Benabid
et al.,
Particle levitation and guidance in hollow-core photonic crystal fiber, Optics
Express,
Vol. 1 0,
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CA 02764731 2012-01-11
No. 21, 21 October 2002, pages 1195-1203; and Kazunori Suzuki et al.,
Ultrabroad band
white light generation from a multimode photonic bandgap fiber with an air
core,
Proceedings of Conference on Laser and Electro-Optics (CLEO) 2001, paper WIPD1-
11,
pages 24-25.
[0004] Calculations of selected properties of the fundamental mode of the PBFs
have also
been reported in, for example, R.F. Cregan et al., Single-Mode Photonic Band
Gap Guidance
of Light in Air, Science, Vol. 285, 3 September 1999, pages 1537-1539; Jes
Broeng et al.,
Analysis of air guiding photonic bandgap fibers, Optics Letters, Vol. 25, No.
2, January 15,
2000, pages 96-98; and Jes Broeng et al., Photonic Crystal Fibers: A New Class
of Optical
Waveguides, Optical Fiber Technology, Vol. 5, 1999, pages 305-330.
[0005] Surface modes, which do not exist in conventional fibers, are defect
modes that form
at the boundary between the air core and the photonic-crystal cladding.
Surface modes can
occur when an infinite photonic crystal is abruptly terminated, as happens for
example at the
edges of a crystal of finite dimensions. Terminations introduce a new set of
boundary
conditions, which result in the creation of surface modes that satisfy these
conditions and are
localized at the termination. See, for example, F. Ramos-Mendieta et al.,
Surface
electromagnetic waves in two-dimensional photonic crystals: effect of the
position of the
surface plane, Physical Review B, Vol. 59, No. 23, 15 June 1999, pages 15112-
15120. In a
photonic crystal, the existence of surface modes depends strongly on the
location of the
termination. See, for example, A. Yariv et al., Optical Waves in Crystals:
Propagation and
Control of Laser Radiation, John Wiley & Sons, New York, 1984, pages 209-214,
particularly at page 210; and J.D. Joannopoulos et al., Photonic Crystals:
Molding the flow of
light, Princeton University Press, Princeton, New Jersey, 1995, pages 54-77,
particularly at
page 73; and also see, for example, F. Ramos-Mendieta et al., Surface
electromagnetic waves
in two-dimensional photonic crystals: effect of the position of the surface
plane, cited above.
For example, in photonic crystals made of dielectric rods in air, surface
modes are induced
only when the termination cuts through rods. A termination that cuts through
air is too weak
to induce surface modes. See, for example, J.D. Joannopoulos et al., Photonic
Crystals:
Molding the flow of light, cited above.
[0006] Recent demonstrations have shown that surface modes play a particularly
important
role in air-core PBFs, and mounting evidence indicates that surface modes
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impose serious limitations in air-core photonic-bandgap fibers. See, for
example, Douglas C.
Allan et al., Surface modes and loss in air-core photonic band-gap fibers, in
Photonic
Crystals Materials and Devices, A. Adibi et al. (eds.), Proceedings of SPIE,
Vol. 5000, 2003,
pages 161-174; Wah Tung Lau et al., Creating large bandwidth line defects by
embedding
dielectric waveguides into photonic crystal slabs, Applied Physics Letters,
Vol. 81, No. 21,
18 November 2002, pages 3915-3917; and Dirk Muller et al., Measurement of
Photonic
Band-gap Fiber Transmission from 1.0 to 3.0 ,um and Impact of Surface Mode
Coupling,
Proceedings of Conference on Laser and Electro-Optics (CLEO) 2003, Baltimore,
USA, 1-6
June 2003, paper QTuL2, 2 pages, and also see, for example, J.D. Joannopoulos
et al.,
Photonic Crystals: Molding the flow of light, cited above; A. Yariv et al.,
Optical Waves in
Crystals: Propagation and Control of Laser Radiation, cited above; and F.
Ramos-Mendieta
et al., Surface electromagnetic waves in two-dimensional photonic crystals:
effect of the
position of the surface plane, cited above. Unless suitably designed, a fiber
will support
many surface modes.
[0007] In contrast to surface modes, a core mode (e.g., a fundamental core
mode) is one in
which the peak of the mode intensity is located in the core. In most cases,
most of the energy
will also be contained within the air core. The propagation constants of
surface modes often
fall close to or can even be equal to the propagation constant of the
fundamental core mode.
See, for example, Dirk Muller et al., Measurement of Photonic Band-gap Fiber
Transmission
from 1.0 to 3.0 ,um and Impact of Surface Mode Coupling, cited above. Strong
experimental
and analytical evidence indicates that the fundamental core mode couples to
one or more of
these surface modes. Such coupling may be caused, for example, by random
perturbations in
the fiber cross-section. Since surface modes are inherently lossy due to their
high energy
density of the Fiber, such coupling is a source of propagation loss.
Furthermore, since
surface modes occur across the entire bandgap, no portion of the available
spectrum is
immune to this loss mechanism. Recent findings have demonstrated that surface
modes are a
cause of the reduced transmission bandwidth in a 13-dB/km air-core PBF
manufactured by
Corning. See, for example, N. Venkataraman et al., Low loss (13dB/km) air core
photonic
band-gap fibre, Proceedings of European Conference on Optical Communication,
ECOC
2002, Copenhagen, Denmark, PostDeadline Session 1, PostDeadline Paper PD1.1,
September
12, 2002; and C.M. Smith, et al., Low-loss hollow-core silica/air photonic
bandgap fibre,
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Nature, Vol. 424, No. 6949, 7 August 2003, pages 657-659. This effect is
believed to be the
source of the remaining loss (approximately 13dB/lcm) in this air-core
photonic-bandgap
fiber. See, for example, Douglas C. Allan et al., Photonic Crystals Materials
and Devices,
cited above. Understanding the physical origin of surface modes and
identifying fiber
configurations that are free of such modes across the entire bandgap is
therefore of
importance in the ongoing search for low-loss PBFs.
Summary of the Invention
[0008] The embodiments disclosed herein are based on information obtained in
an
investigation of the properties of the core modes and the surface modes of
PBFs using
numerical simulations. The investigation focused on the most common PBF
geometry,
namely fibers with a periodic, triangular pattern of cylindrical air-holes in
the cladding and a
circular core obtained by introducing an air defect. Such fibers are
described, for example, in
R.F. Cregan et al., Single-Mode Photonic Band Gap Guidance of Light in Air,
cited above;
Jes Broeng et al., Analysis of air-guiding photonic bandgap fibers, cited
above; and Jes
Broeng et al., Photonic Crystal Fibers: A new class of optical waveguides,
Optical Fiber
Technology, cited above. The results are also applicable to a broad range of
air-hole patterns
(e.g., hexagonal patterns, square patterns, etc.), hole shapes, and core
shapes. The results are
also applicable to other photonic-bandgap fibers, namely, fiber with similar
geometries that
operate on the same photonic-bandgap principle but with a core not necessarily
filled with air
(e.g., a core filled with another gas, a vacuum, a liquid, or a solid), with
cladding holes not
necessarily filled with air (e.g., cladding holes filled with another gas, a
vacuum, a liquid, or a
solid), and with solid portions of the cladding not necessarily made of silica
(e.g., the
cladding may comprise another solid or a multiplicity of solids). As used
herein, hole or a
core that is not filled with a solid or a liquid is referred to herein as
being hollow. It is
understood here that the respective refractive indices of the materials that
make up the core,
the cladding holes, and the solid portion of the cladding should be selected
such that the fiber
structure supports a guided mode via the photonic-bandgap effect. This implies
that the
refractive index of the core and the refractive index of the holes should be
lower than that of
the refractive index of the solid portions of the cladding, and that the
difference between
these indices should be large enough.
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[0009] New geometries are proposed herein for air-core fibers or fibers with a
core that
has a lower refractive index than the solid portions of the cladding. These
geometries
have ranges of core characteristic dimensions (e.g., core radii when the core
is circular)
for which the fiber core supports no surface modes over the entire wavelength
range of
the bandgap, and only core modes are present. In particular, for a circular
core with a
radius between about 0.7A and about 1.05A, where A is the hole-to-hole spacing
of the
triangular pattern, the core in a proposed new geometry supports a single mode
and does
not support any surface modes. The absence of surface modes suggests that
fibers within
this range of configuration should exhibit substantially lower losses than
current fibers.
As further shown below, the existence of surface modes in the defect structure
can be
readily predicted either from a study of the bulk modes alone or even more
simply by a
straightforward geometric argument. Because the structure is truly periodic,
prediction of
the existence of surface modes in accordance with the methods described below
is
quicker and less complicated than a full analysis of the defect modes.
[0010] The methods disclosed herein can be used to predict whether a
particular fiber
geometry will support surface modes so that fibers can be designed and
manufactured that
do not support surface modes. In particular, as illustrated below in the
detailed
description, the presence of surface modes can be avoided by selecting the
core radius or
other characteristic dimension such that the edge of the core does not cut
through any of
the circles inscribed in the veins (e.g., the solid intersection regions) of
the PBF lattice.
The technique works for broad ranges of geometries and hole sizes.
[0011] In order to avoid surface modes, the techniques described herein are
used to
design the core shape such that the core does not intersect any of the veins
of the PBF
lattice (e.g., the core intersects only the segments that join the veins of
the PBF lattice).
By following this general criterion, PBFs can be designed to be free of
surface modes.
[0012] An aspect in accordance with embodiments of the invention is a method
of
making a photonic-bandgap fiber. The method uses a photonic-bandgap fiber that
comprises a material with a pattern of regions formed therein to form a
photonic crystal
lattice. The material has a first refractive index. The pattern of regions has
a second
refractive index lower than the first refractive index. The method comprises
determining
intensity profiles of a highest frequency bulk mode proximate to the regions.
The method
forms a central core in the photonic crystal lattice. The core has an edge
that intersects
the pattern of regions at locations where the intensities of the highest
frequency bulk
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mode are low enough that the fiber supports no surface modes. Preferably, the
regions in
the material are circular; and the pattern of regions is periodic and arranged
such that each
group of three adjacent regions forms a triangle with a respective first
portion of the
material between each pair of regions and with a respective second portion of
the material
forming a central area within each group of three adjacent regions. The
central core is
formed in the photonic crystal lattice such that the edge of the central core
passes only
through the first portions of the material. In particular embodiments, the
regions in the
material are holes that have walls defined by the surrounding material.
Advantageously,
the holes in the material are hollow. The holes in the material may be filled
with air, gas
or liquid having the second refractive index. Alternatively, the circular
regions comprise
a solid having the second refractive index. In certain embodiments, the
material is a
dielectric, such as, for example, silica.
[0013] Another aspect in accordance with an embodiment of the invention is a
photonic-
bandgap fiber that comprises a photonic crystal lattice. The lattice comprises
a first
material having a first refractive index. The first material has a pattern of
a second
material formed therein. The second material has a second refractive index
lower than
the first refractive index. The photonic crystal lattice has a plurality of
first regions that
support intensity lobes of the highest frequency bulk mode and has a plurality
of second
regions that do not support intensity lobes of the highest frequency bulk
mode. A central
core is formed in the photonic crystal lattice. The central core has an edge
that passes
only through the second regions of the dielectric lattice. Preferably, the
pattern of the
second material is periodic and comprises a plurality of geometric regions.
Each
geometric region has a respective center, and adjacent geometric regions are
spaced apart
by a center-to-center distance A. Each geometric region of the second material
is circular
and has a radius p, wherein the radius p is less than 0.5A. Preferably, the
pattern is
triangular, and the first regions comprise circles inscribed between three
adjacent
geometric regions. In certain embodiments, each inscribed circle has a radius
a equal to
(A143)-p. Also, preferably, the radius p of each geometric region is
approximately 0.47A.
In certain embodiments, the core is generally circular, and the edge of the
core has a
radius within one of a plurality of ranges of radii. A first of the plurality
of ranges of core
radii extends from a radius of approximately 0.68A to a radius of
approximately 1.05A.
A second of the plurality of ranges of core radii extends from a radius of
approximately
1.26A to a radius of approximately 1.43A. A third of the plurality of ranges
of core radii
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extends from a radius of approximately 1.64A to a radius of approximately
1.97A. In
particularly preferred embodiments, the first of the plurality of ranges of
core radii
extends from a radius of approximately 0.685A to a radius of approximately
1.047A, the
second of the plurality of ranges of core radii extends from a radius of
approximately
1.262A to a radius of approximately 1.420A, and the third of the plurality of
ranges of
core radii extends from a radius of approximately 1.635A to a radius of
approximately
1.974A.
[0014] Another aspect in accordance with embodiments of the invention is a
geometric
method for selecting the dimensions of a core for producing a photonic-bandgap
fiber free
of surface modes. The photonic-bandgap fiber has a photonic crystal lattice
comprising a
first material having a first refractive index. The material encompasses a
periodic pattern
of regions of a second material. The second material has a second refractive
index lower
than the first refractive index. Each region of the second material is spaced
apart from an
adjacent region of the second material by a membrane of the first material and
each
intersection of membranes forms a vein of the first material. The method
comprises
defining an inscribed central area within each vein of the second material
such that an
outer periphery of the inscribed central area is tangential to the outer
peripheries of the
adjacent regions around the vein. The method further comprises defining a
plurality of
ranges of core characteristic dimensions wherein any core having a dimension
within one
of the plurality of ranges has an edge that does not pass through any of the
inscribed
central areas. The method further comprises selecting a core having a
dimension within
one of the plurality of ranges of core characteristic dimensions. Preferably,
each region
has a respective center, and adjacent regions are spaced apart by a center-to-
center
distance A. Also preferably, each region of the second material is circular
and has a
radius p, and the radius p is less than 0.5A. In certain embodiments, the
pattern is
triangular, and the inscribed central area circular. Preferably, the circle
has a radius a
equal to (A/43)-p. In particular embodiments, the radius p of each region is
approximately 0.47A. In such embodiments, the characteristic dimension of the
core is
the radius of a circle, and a first of the plurality of ranges of core
characteristic
dimensions extends from a radius of approximately 0.68A to a radius of
approximately
1.05A, a second of the plurality of ranges of core characteristic dimensions
extends from
a radius of approximately 1.26A to a radius of approximately 1.43A, and a
third of the
plurality of ranges of core characteristic dimensions extends from a radius of
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approximately 1.64A to a radius of approximately 1.97A. In particularly
preferred
embodiments, the first of the plurality of ranges of core characteristic
dimensions extends
from a radius of approximately 0.685A to a radius of approximately 1.047A, the
second
of the plurality of ranges of core characteristic dimensions extends from a
radius of
approximately 1.262A to a radius of approximately 1.420A, and the third of the
plurality
of ranges of core characteristic dimensions extends from a radius of
approximately
1.635A to a radius of approximately 1.974A.
[0015] Another aspect in accordance with embodiments of the invention is a
photonic-
bandgap fiber that supports no surface modes. The photonic-bandgap fiber
comprises a
photonic crystal lattice region comprising a material having a first
refractive index. The
material has a periodic pattern of regions formed therein. Each region has a
second
refractive index lower than the first refractive index. Each region is spaced
apart from an
adjacent region by a membrane of the material. Each group of adjacent regions
is formed
around a central area of the material. The central area within each group of
adjacent
regions is defined by an inscribed circle having a circumference tangential to
the
circumferences of the adjacent regions. A core is formed in the photonic-
bandgap fiber.
The core has a characteristic dimension selected such that the edge of the
core passes only
through portions of the material that are not within any of the inscribed
circles in the
central areas. Preferably, the material is a dielectric material, such as, for
example, silica.
Also, preferably, the pattern is triangular and each group of adjacent regions
comprises
three regions. In particularly preferred embodiments, the core is generally
circular, and
the characteristic dimension is the radius of the core.
[0016] Another aspect in accordance with embodiments of the invention is a
method for
producing a photonic-bandgap fiber that does not support surface modes. The
method
comprises selecting a photonic-bandgap fiber having a photonic crystal lattice
that
comprises a material having a first refractive index. The material has a
periodic
triangular pattern of regions formed therein. Each region has a refractive
index lower
than the first refractive index. The material comprises first areas between
adjacent holes
and comprises second areas between only two adjacent holes. The second areas
interconnect the first areas. The method further comprises forming a core in
the photonic
crystal lattice. The core has a characteristic dimension selected such that
the edge of the
core intersects only the second areas of the photonic crystal region.
Preferably, the
material is a dielectric material, such as, for example, silica. Also
preferably, the pattern
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is triangular, and each group of adjacent regions comprises three regions. In
particularly
preferred embodiments, the core is generally circular, and the characteristic
dimension is the
radius of the core.
[0016a] In accordance with an aspect of the present invention there is
provided a method of
making a photonic-bandgap fiber comprising a material with a pattern of
regions therein that
form a crystal lattice, the material having a first refractive index and the
pattern of regions
having a second refractive index lower than the first refractive index, the
method comprising:
determining intensity profiles of a highest frequency bulk mode proximate to
the
regions; and
forming a central core in the crystal lattice, the core having an edge that
intersects the
pattern of regions at locations where the intensities of the highest frequency
bulk mode are
low enough that the fiber supports no surface modes.
10016b1 In accordance with a further aspect of the present invention there is
provided a
photonic-bandgap fiber that supports no surface modes comprising:
a photonic-bandgap fiber comprising a crystal lattice region comprising a
material
having a first refractive index, the material having a periodic pattern of
regions formed
therein, each region having a second refractive index lower than the first
refractive index,
each region spaced apart from an adjacent region by a membrane of the
material, each group
of adjacent regions formed around a central area of the material, the central
area within each
group of adjacent regions defined by an inscribed circle having a
circumference tangential to
the circumferences of the adjacent regions; and
a core having a characteristic dimension selected such that an edge of the
core passes only
through portions of the material that are not within any of the inscribed
circles in the central
areas.
100160 In accordance with a further aspect of the present invention there is
provided a
method for producing a photonic-bandgap fiber that does not support surface
modes, the
method comprising:
selecting a photonic-bandgap fiber having a crystal lattice comprising a
material
having a first refractive index, the material having a periodic triangular
pattern of regions
formed therein, each region having a refractive index lower than the first
refractive index, the
material comprising first areas between adjacent holes and comprising second
areas between
only two adjacent holes, the second areas interconnecting the first areas; and
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forming a core in the photornc crystal lattice, the core having a
characteristic
dimension selected such that an edge of the core intersects only the second
areas of the
photonic crystal region.
[0016d] In accordance with a further aspect of the present invention there is
provided
a method of making a photonic-bandgap fiber comprising a material with a
pattern of regions
therein that form a crystal lattice, the material having a first refractive
index and the pattern
of regions having a second refractive index lower than the first refractive
index, the method
comprising:
obtaining information regarding intensity profile locations of a highest
frequency bulk
mode proximate to the regions; and
forming a hollow central core in the crystal lattice using the information to
select a
dimension of the central core in the crystal lattice such that the central
core has an edge that
intersects the pattern of regions at locations where the intensities of the
highest frequency
bulk mode are low enough that the fiber supports no surface modes, the core
having a
refractive index lower than the first refractive index.
[0016e] In accordance with a further aspect of the present invention there is
provided
a method for producing a photonic-bandgap fiber that does not support surface
modes, the
method comprising:
selecting a crystal lattice comprising a material having a first refractive
index, the
material having a periodic triangular pattern of regions formed therein, each
region having a
refractive index lower than the first refractive index, the material
comprising first areas
within a triangle formed by three adjacent regions and comprising second areas
between only
two adjacent regions, the second areas interconnecting the first areas; and
selecting a radius of a hollow circular core; and
forming the circular core at a position in the crystal lattice such that an
edge of the
core intersects only the second areas of the material, the core having a
refractive index lower
than the first refractive index.
Brief Description of the Drawings
[0017] Embodiments in accordance with the present invention are described
below in
connection with the accompanying drawing figures in which:
100181 Figure 1 illustrates a partial cross section of an exemplary triangular-
pattern air-core
photonic-bandgap fiber (PBF) for a core radius of 1.15A and a hole radius p of
approximately
. 0.47A.
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[00191 Figure 2 illustrates an enlarged view of the partial cross section of
Figure 1 to provide
additional detail on the spatial relationship between the air holes, the
segments (membranes)
between adjacent air holes and the veins (corners) at the intersections of the
segments;
[0020] Figure 3 illustrates contour lines that represent equal intensity lines
of a typical
surface mode for the air-core PBF of Figure 1;
[0021] Figure 4 illustrates contour lines that represent equal intensity lines
of the
fundamental core mode for the air-core PBF of Figure 1;
[0022] Figure 5 illustrates contour lines that represent equal intensity lines
of a typical bulk
mode for the triangular-pattern air-core PBF of Figure 1, but without the
removal of the
central structure to form the air core 106;
[0023] Figure 6 illustrates dispersion curves of the defect modes for the air-
core photonic-
bandgap fiber (PBF) of Figure 1 having a triangular-pattern of holes with a
photonic-crystal
structure of period (i.e., hole-to-hole spacing) A and a hole radius p of
approximately 0.47A,
surrounding an air-core having a radius R of approximately 1.15A, wherein the
shaded (cross
hatched) area represents the photonic bandgap of the crystal;
[0024] Figure 7 illustrates dispersion curves of the defect modes for an air-
core PBF having a
core radius R of approximately 1.8A;
[0025] Figure 8 illustrates a partial cross section showing the hole pattern
and air-core shape
of a PBF from which the dispersion curves of Figure 7 are obtained;
[0026] Figure 9 illustrates a graph of the number of core modes (diamonds) and
surface
modes (triangles) versus the air-core radius at the normalized frequency
roA/21tc=1.7;
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[0027] Figures 10A, 10B and 10C illustrate the core shapes for core radii of
0.9A, 1.2A,
and 2.1A, respectively, from which the information in Figure 9 was derived;
[0028] Figure 11 illustrates a graphical representation of the air-core radius
ranges that
support core modes only (unshaded rings) and both core and surface modes
(shaded rings;
[0029] Figure 12 illustrates the partial cross section of the triangular-
pattern air-core PBF
of Figure 1 with a core of radius R1 formed in the photonic crystal lattice,
wherein the
surface of the core intersects the corners of the crystal lattice and wherein
surface modes
are supported;
[0030] Figure 13 illustrates the partial cross section of the triangular-
pattern air-core PBF
of Figure 1 with a core of radius R2 formed in the photonic crystal lattice,
wherein the
surface of the core does not intersect the comers of the crystal lattice and
wherein surface
modes are not supported;
[0031] Figure 14 illustrates a plot (dotted curve) of the maximum intensity of
the highest
frequency bulk mode on a circle of radius R as a function of R overlaid on the
plot (solid
curve) of the maximum number of surface modes as a function of R from Figure
9;
[0032] Figures 15A and 15B illustrate intensity contour maps of the two
highest
frequency doubly degenerate bulk modes below the bandgap at the I" point,
wherein R1 is
an example of a core radius that supports both core modes and surface modes,
and R2 is
an example of a core radius that supports only core modes;
[0033] Figure 16 illustrates a graphical representation of a partial cross
section of the
triangular pattern air core PBF, wherein black circles at each dielectric
corner represent
dielectric rods, and wherein unshaded rings represent bands of core radii for
which the
surface of the core does not intersect the dielectric rods;
[0034] Figure 17 illustrates a graph (dashed lines) of the results of the
numerical
simulations of the number of surface modes and illustrates a graph (solid
lines) of the
number of surface modes predicted using the geometric model of Figure 16 and
counting
the number of rods intersected by the surface of the core, wherein the number
of surface
modes in each graph is plotted with respect to the normalized core radius RIA;
[0035] Figure 18 illustrates a plot of the normalized core radius RIA versus
the
normalized hole radius p/A to show the effect of the fiber air-filling ratio
on the presence
of surface modes;
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[0036] Figure 19 schematically illustrates a cross section of an alternative
air-core
photonic bandgap fiber having a non-circular (e.g., hexagonal) core shape and
no surface
modes; and
[0037j Figures 20A and 20B illustrate for comparison the effective refractive
indices of
core modes and surface modes for two commercially available photonic bandgap
fibers.
Detailed Description of the Preferred Embodiment
[00381 The following description is based on a photonic band-gap fiber (PBF)
with a
cladding photonic crystal region comprising a triangular lattice comprising a
plurality of
circular holes filled with a gas (e.g., air) in silica or other solids, where
the holes are
spaced apart by a period A. Silica-based PBFs are described, for example, in
R.F. Cregan
et al., Single-Mode Photonic Band Gap Guidance of Light in Air, cited above;
Jes Broeng
et al., Analysis of air-guiding photonic bandgap fibers, cited above; and Jes
Broeng et al.,
Photonic Ctystal Fibers: A New Class of Optical Waveguides, cited above. For
simplicity, such fibers are referred to herein as air-hole fibers; however, as
discussed
above, the following discussions and results are also applicable to photonic-
bandgap
fibers with a core and/or all or some of the cladding holes filled with other
materials
besides air (e.g., another gas, a vacuum, a liquid, or a solid) and with solid
portions of the
cladding made of materials other than silica (e.g., a different solid or a
multiplicity of
solids). Furthermore, the results are also adaptable to other patterns of
holes (e.g.,
hexagonal patterns, square patterns, etc.).
[0039] A partial cross section of an exemplary triangular-pattern air-core PBF
100 is
illustrated in Figure 1. As illustrated, the fiber 100 comprises a solid
dielectric lattice 102
comprising a plurality of air holes 104 surrounding an air core 106. Three
exemplary
adjacent holes 104 are shown in more detail in Figure 2. The portion of the
solid lattice
102 between any three adjacent holes 104 is referred to as a vein (or a
corner) 110, and
the thinner regions connecting two adjacent veins (i.e., a region between any
two adjacent
holes) is referred to as a segment (or a membrane) 112. In the illustrated
embodiment,
each air hole 104 has a radius p. The center-to-center spacing of adjacent air
holes 104 is
referred to as the period A of the photonic crystal.
[0040] As will be discussed in more detail below, each vein 110 can be
approximated by
an inscribed circle 114 of radius a, wherein the circumference of the
inscribed circle 114
is tangential to the circumferences of three holes 104 surrounding the vein
110. Simple
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CA 02764731 2012-01-11
geometric calculations readily show that the radius a of the inscribed circle
114 is related to
the radius p and the period A of the air holes 104 as follows:
a = (111 ¨ p
[0041] As illustrated in Figure 1, the air-core 106 of the PBF 100 is
advantageously created
by introducing a larger cylindrical air hole of radius R at the center of the
fiber. The location
of this cylinder, reproduced in Figure 1 as a dashed circle is referred to
herein as the edge of
the core 106. The radius R is referred to herein as the characteristic
dimension of the air-core
106. In the example of the circular core illustrated in Figure 1, the radius R
is the radius of
the circular core. The following discussion is adaptable to cores having other
shapes and
characteristic dimensions (e.g., the shortest distance from the center to the
nearest boundary
of a polygonal-shaped core). In the PBF 100 of Figures 1 and 2, the radius R
is selected to be
1.15A, and the radius p of each air hole 104 is selected to be 0.47A. For
example, the air-core
106 radius 1.15A is advantageously selected because the core radius
corresponds to a core
formed in practice by removing seven cylinders from the center of the PBF
preform (e.g.,
effectively removing the glass structure between the seven cylinders). Such a
configuration
is commonly used and is described, for example, in J.A. West et al., Photonic
Crystal Fibers,
Proceedings of 27th European Conference on Optical Communications (ECOC'01-
Amsterdam),Amsterdam, The Netherlands, September 30-October 4, 2001 paper
ThA2.2,
pages 582-585.
[0042] As discussed above, surface modes are defect modes that form at the
boundary
between the core 106 and the photonic-crystal cladding 102. A typical surface
mode for the
triangular-pattern air-core PBF 100 of Figures 1 and 2 is illustrated in
Figure 3. A typical
fundamental core mode for the PDF 100 of Figures 1 and 2 is illustrated in
Figure 4. In
Figures 3 and 4, the contour lines represent equal intensity lines. The
outmost intensity line
in each group has a normalized intensity of 0.1 and the innermost intensity
line has a
normalized intensity of 0.9, and each intervening intensity line represents a
normalized step
increase of 0.1.
[0043] In the absence of a core, a PBF carries only bulk modes. An example of
bulk mode is
illustrated in Figure 5. The bulk mode of Figure 5 is calculated for the same
triangular-
pattern air-core PBF 100 illustrated in Figure 1, but without the removal of
the
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CA 02764731 2012-01-11
central structure to form the air core 106. As in Figures 3 and 4, the contour
lines in Figure 5
represent equal intensity lines.
100441 The particular bulk mode illustrated in Figure 5 comprises a series of
narrow intensity
lobes centered on each of the thicker dielectric corners 110 of the photonic
crystal 102. Other
bulk modes may have different lobe distributions (e.g., all the lobes may be
centered on
membranes 112 rather than on corners 110).
[0045] As discussed above, a fiber will support many surface modes unless the
fiber is
suitably designed to eliminate all surface modes. As further discussed above,
the propagation
constants of the surface modes are often close to or equal to the propagation
constant of the
fundamental core mode, and, as a result, the core mode can easily be coupled
to the surface
modes (e.g., by random perturbations in the fiber cross section), which
results in an increased
propagation loss for the fundamental core mode. This problem is also present
for other core
modes besides the fundamental mode when the fiber is not single mode.
[0046] By varying the radius R of the air core 106, the effect of the core
radius on the core
modes and the effect of surface truncation on the surface mode behavior can be
systematically studied. One such study is based on simulations performed on
the University
of Michigan AMD Linux cluster of parallel Athlon 2000MP processors using a
full-vectorial
plane-wave expansion method. An exemplary full-vectorial plane wave expansion
method is
described, for example, in Steven G. Johnson et al., Block-iterative frequency-
domain
methods for Maxwell's equations in a planewave basis, Optic Express, Vol.8,
No. 3, 29
January 2001, pages 173-190.
100471 The simulations use a grid resolution of A116 and a supercell size of
8A x 8A. The
solid portion of the cladding was assumed to be silica, and all holes were
assumed to be
circular and filled with air. When running the simulations with 16 parallel
processors,
complete modeling of the electro-field distributions and dispersion curves of
all the core
modes and surface modes of a given fiber typically takes between 7 hours and
10 hours.
[0048] The results of the simulation for a triangular pattern indicate that a
photonic bandgap
suitable for air guiding exists only for air-hole radii p larger than about
0.43A. The largest
circular air-hole radius that can be fabricated in practice (e.g., so that
sufficient silica remains
in the membranes 112 between adjacent air holes 104 to provide a supporting
structure) is
slightly higher than 0.49A. In the study, a structure is simulated
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CA 02764731 2012-01-11
that has an air-hole radius p between these two extreme values. In particular,
p is
selected to be approximately 0.5A. Although the simulations described herein
are carried
out for p= 0.47A, similar results have been obtained for any value of p
between 0.43A to
0.5A, and the qualitative conclusions described herein are valid for any air-
hole size in the
range of 0.43A to 0.5A.
[0049] Figure 6 illustrates the theoretical ark, diagram of the fiber geometry
under study
generated for a core radius R = 1.15A (see, for example, Figure 1). In Figure
6, the
vertical axis is the optical angular frequency co = 27ucl A normalized to
27cc/A (i.e., Abl),
where A is the free-space wavelength of the liPht signal, c is the velocity of
light in
vacuum, and A is the photonic-crystal structure period. Thus, the vertical
axis represents
coA/2ac = A/2, which is a dimensionless quantity. The horizontal axis in
Figure 6 is the
propagation constant along the axis of the fiber (z direction) kz, normalized
to 2/t/A
(i.e., kA/27c).
[0050] The first photonic bandgap supported by the infinite structure of the
simulated
fiber 100 of Figure 1 is represented by the shaded (cross-hatched) region. The
size and
shape of the first photonic bandgap depends on the value of the radii p of the
air holes
104 (which are equal to 0.47A in the illustrated simulation), but the bandgap
is very
nearly independent of the dimension of the core 106. The dashed line in Figure
6
represents the light line, below which no core modes can exist, irrespective
of the core
size and the core shape. The portion of the shaded region above the dashed
line shows
that in the simulated fiber 100, the normalized frequencies for which light
can be guided
in the air core range from approximately 1.53 to approximately 1.9.
[0051] The solid curves in Figure 6 represent the dispersion relations of the
core mode
and the surface modes. The air core actually carries two fundamental modes.
Each mode
is nearly linearly polarized, and the polarization of each mode is orthogonal
to the
polarization of the other mode. These two modes are very nearly degenerate. In
other
words, the two modes have almost exactly the same dispersion curve within the
bandgap.
The topmost curve in Figure 6 actually comprises two dispersion curves, one
for each of
these two fundamental modes; however, the two curves are so nearly identical
that they
cannot be distinguished on this graph. The related intensity profiles of
selected modes at
= 1.7 are plotted in Figure 4 for one of the two fundamental core modes and in
Figure 3 for an exemplary surface mode. These profiles indicate that the
highest-
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CA 02764731 2012-01-11
frequency modes inside the bandgap are the two fundamental core modes. All
other
modes in the bandgap are surface modes, which have their intensities localized
at the
core¨cladding boundary, as shown in Figure 3. The strength of the spatial
overlap with
the silica portions of the fiber is different for core and surface modes. The
difference in
strength results in the core mode having a group velocity close to c and the
surface modes
having a lower group velocity, as illustrated in Figure 6.
[0052] Figure 6 also illustrates another distinguishing feature of the core
and surface
modes. In particular, the curves for the surface modes always cross the light
line within
the bandgap. In contrast, the curves for the core modes never cross the light
line within
the bandgap.
[0053] The behaviors of the core mode and the surface modes are investigated
as a
function of defect size by changing the core radius R from 0.6A to 2.2A in
0.1A steps.
Figure 7 illustrates the eu-lcz diagram generated for the same fiber geometry
as used to
generate the information in Figure 6, but for a larger core radius (R = 1.8A).
As
illustrated by the partial fiber cross section in Figure 8, the larger core
radius is formed,
for example, by removing additional lattice structure beyond the central seven
cylinders
of the preform so that the surface of the core 106 intersects the thinner
membranes 112
between the holes 104 rather than intersecting the thicker dielectric corners
110. As
expected, the number of core modes appearing in Figure 7 for the embodiment of
Figure
8 is greater than for the embodiment of Figure 1. In addition, all the modes
are core
modes for this larger radius. As the frequency is increased from the low-
frequency cutoff
of the bandgap, the highest order core modes appear first, in a group of four
or more
modes (e.g., four in Figure 7). This feature depends on the core size and mode
degeneracy. See, for example Jes Broeng et al., Analysis of air-guiding
photonic
bandgap fibers, cited above. As the frequency is further increased, the number
of modes
reaches some maximum number (14 in the example illustrated in Figure 7) at a
normalized frequency (coA/27cc) of approximately 1.7. Above a normalized
frequency of
approximately 1.7, the number of modes gradually decreases to two (the two
fundamental
modes) at the high-frequency cutoff of the bandgap. The maximum number of core
modes occurs at or in the vicinity of the frequency where the light line
intersects the
lower band edge. In the embodiment illustrated by the plot in Figure 7, the
light line
intersects the lower band edge at a normalized frequency (coA127rc) having a
value of
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CA 02764731 2012-01-11
around 1.67. Note that in Figure 7, many of the curves represent multiple
modes that are
degenerate and thus overlap in the diagram.
[0054] Figure 9 illustrates the dependence of this maximum number of core
modes (i.e.,
the number of modes is plotted at coAl2ac= 1.7) on R. The number of surface
modes is
also shown in Figure 9. In addition, core shapes for representative radii of R
= 0.9A,
R= 1.2A, and R=2.1A are illustrated in Figure 10A, Figure 108 and Figure 10C,
respectively. As stated above, the grid resolution used to generate the data
points in
Figure 9 was A/16. However, to generate additional points in the more
interesting range
of core radii between 1.1A and 1.3A, the grid size was reduced to A/32 in that
range. As
a result, the absolute number of surface modes predicted in this range does
not scale the
same way as in the rest of the graph. This is inconsequential since the
primary interest in
generating the data points is to determine the boundaries of the surface mode
regions.
[0055] The behaviors of the core modes in PBFs and in conventional fibers
based on total
internal reflection have striking similarities. The fundamental mode, like an
LPoi mode,
is doubly degenerate (see Figures 6 and 7), is very nearly linearly polarized,
and exhibits
a Gaussian-like intensity profile. See, for example, Jes Broeng et al.,
Analysis of
air-guiding photonic bandgap fibers, cited above. The next four modes are also
degenerate, and the electric field distributions of these four modes are very
similar to
those of the 11E21odd, HE2leven, Thoi, and TMoi modes of conventional fibers.
Many of the
core modes, especially the low-order modes, exhibit a two-fold degeneracy in
polarization over much of the bandgap. As the core radius is increased, the
number of
core modes increases in discrete steps (see Figure 9), from two (the two
fundamental
modes) to six (these two modes plus the four degenerate modes mentioned
above), then
14 (because the next eight modes happen to reach cutoff at almost the same
radius), etc.
N0561 Figure 9 also illustrates another aspect of the modes. In particular,
when R falls in
certain bounded ranges, all modes are found to be core modes. The first three
of the
bounded ranges are:
range 1 from approximately 0.7A to approximately 1.1A;
range 2 from approximately 1.3A to approximately 1.4A; and
range 3 from approximately 1.7A to approximately 2.0A.
[01357] Figure 7 illustrates the case where R is equal to 1.8A, which is one
particular
example of a surface-mode-free PBF in range 3. The surface-mode-free ranges
determined by the computer simulation are illustrated schematically in Figure
11. In
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CA 02764731 2012-01-11
Figure 11, the background pattern of circles represents the infinite photonic
crystal
structure, the four shaded (cross hatched) annular areas represent the ranges
of core radii
that support surface modes, and the three unshaded annular areas (labeled as
band 1, band
2 and band 3) represent the first three ranges of radii that are free of
surface modes. Note
that for radii less than 0.5A (e.g., the central unshaded portion of Figure
11), the core does
not support core modes that are guided by the photonic-bandgap effect.
[0058] Figure 11 is simply a different way to graph the regions of no surface
modes
shown in Figure 9. Thus, the three ranges of radii in Figure 9 that support no
surface
modes, as shown by the open triangles that fall along the bottom horizontal
axis, are
graphed as the three white annular (unshaded) regions in Figure 11 (bands 1, 2
and 3).
The complementary (shaded) bands between the white bands correspond to the
ranges of
radii in Figure 9 where the triangles are above the horizontal axis and thus
represent radii
that support surface modes.
[0059] In the first of the unshaded ranges in Figure 11 (e.g., band I from
approximately
0.7A to approximately 1.1A), the core supports a single core mode and does not
support
any surface modes at all across the entire wavelength range of the bandgap,
i.e., the PBF
is truly single mode. There do not appear to be any previous reports of a
single-mode
all-silica PBF design in the literature. Note that in band 2, band 3 and all
other bands
representing larger radii, the fiber is no longer single mode.
[0060] An example of a terminating surface shape that falls in this single-
mode range
(e.g., range 1) is shown in Figure 10A for R equal to 0.9A. These particular
configurations may be fabricated using small tips of glass protruding into the
core using
an extrusion method and other known fabrication techniques.
[0061] The number of surface modes is also strongly dependent on the core
radius, albeit
in a highly non-monotonic fashion. For core radii in the vicinities of
approximately 0.6 A, approximately 1.2A, approximately 1.6A, and approximately
2.1A,
many surface modes are introduced, resulting in the peaks in the number of
surface
modes. The peaks are apparent in Figure 9. Moreover, in these vicinities, the
number of
surface modes varies rapidly with R. Typical experimental PBFs are fabricated
by
removing the central 7 cylinders (R approximately equal to 1.15A) or 19
cylinders
(R approximately equal to 2.IA) from the preform to form the core 106;
however, these
particular values of R, which happen to be more straightforward to
manufacture, also
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CA 02764731 2012-01-11
happen to lead to geometries that support surface modes, as shown, for
example, in
Figure 9.
[0062] Based on the foregoing results of the computer simulations, the basic
conditions at
which surface modes occur have been investigated and new structures are
proposed that
have no surface modes. The basic conditions lead to the observation that
surface modes
are created when the surface of the core 106 intersects one or more of the
dielectric
comers 110 of the photonic crystal lattice 102. From this observation, a fast
and simple
geometric criterion is obtained for evaluating whether a particular fiber
configuration
supports surface modes. As discussed below, when the geometric criterion is
applied to
triangular-pattern PBFs 100 with a circular air core 106, the approximate
geometric
model yields quantitative predictions in acceptable agreement with the results
of
computer simulations described above.
[0063] As discussed above, surface modes can occur when an infinite photonic
crystal is
abruptly terminated, as happens for example at the edges of a crystal of
finite dimensions.
For example, in photonic crystals made of dielectric rods in air, surface
modes are
induced only when the termination cuts through rods. A termination that cuts
only
through air is too weak to induce surface modes.
[0064] In an air-core PBF 100, the core 106 also acts as a defect that
perturbs the crystal
lattice 102 and may introduce surface modes at the edge of the core 106.
Whether surface
modes appear, and how many appear, depends on how the photonic crystal is
terminated,
which determines the magnitude of the perturbation introduced by the defect.
In the
absence of an air core, a PBF carries only bulk modes, as discussed above with
respect to
Figure 5.
[0065] When the air core 106 is introduced as shown in Figures 1, 3 and 4, the
core 106
locally replaces the dielectric material of the crystal lattice 102 with air.
The portions of
the surface of the core 106 that cut through the cladding air holes 104 in
Figure 1 replace
air by air. Thus, just as in the case of a planar photonic crystal (as
described, for example,
in J.D. Joannopoulos et al., Photonic Crystals: Molding the flow of light,
cited above),
those portions of the core surface do not induce significant perturbation.
Only the
portions of the core surface that cut through the dielectric corners 110 or
the dielectric
membrane 112 of the crystal lattice 102 in Figure 1 replace dielectric by air
and thereby
perturb the bulk modes of Figure 5. Whether the perturbation is sufficient to
potentially
induce surface modes, such as the surface modes shown in Figure 3, is
discussed below.
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CA 02764731 2012-01-11
[0066] Since a core 106 of any size and shape always cuts through some
dielectric
material, some perturbation is always introduced by the core 106. The sign of
the
perturbation is such that in the co-k diagram, the bulk modes are all shifted
up in
frequency from their frequencies in their respective unperturbed positions.
For a silica/air
PBF 100, the perturbation is comparatively weak, and the frequency shift is
small such
that almost all perturbed bulk modes remain in a bulk mode band. Exceptions to
the
foregoing are modes from the highest frequency bulk-mode band of the lower
band
(referred to hereinafter as "HFBM"). Because such modes are located just below
the
bandgap in the co-k diagram, the perturbation moves them into the bandgap as
surface
modes. See, for example, J.D. Joannopoulos et al., Photonic Oystals: Molding
the flow
of light, cited above.
[0067] Surface modes can always be written as an expansion of bulk modes. For
the
weak perturbation considered here, it can be shown that the main term in this
expansion is
the HFBM, as expected in view of the origin of these surface modes. The HFBM
is the
bulk mode illustrated in Figure 5. As illustrated in Figure 5, the lobes of
the mode are all
centered on corners 110 of the crystal 102, which results in two important
consequences.
First, because surface modes are induced by a perturbation of this bulk mode,
the lobes of
the surface modes are also centered on the corners 110, as shown, for example,
in
Figure 3. Second, for the HFBM to be perturbed and yield surface modes, the
perturbation must occur in dielectric regions of the photonic crystal lattice
102 that carry
a sizable HFBM intensity, e.g., in regions at the corners 110 of the photonic
crystal 102.
These observations show that surface modes are strongly correlated with the
magnitude of
the perturbation introduced by the air core 106 on the HFBM. If the surface of
the core
106 intersects lobes of the HFBM at the corners 110 of the dielectric lattice
102 (as
illustrated, for example, by a core of radius R1 in Figure 12), the
perturbation is large and
surface modes are induced. The number of surface modes then scales like the
highest
intensity intersected by the core 106 in the dielectric 102. Conversely, if
the surface of
the core 106 does not intersect any of the lobes of this bulk mode (as
illustrated, for
example, by a core of radius R2 in Figure 13), no surface modes are created.
[0068] The foregoing is illustrated in Figure 14, which reproduces the plot of
the number
(values on the left vertical axis) of surface modes at caA127rc = 1.7 on a
circle of radius R
as a function of R normalized to A (horizontal axis) as a solid curve. Figure
14 also
includes a plot (dotted curve) of the maximum intensity (values in arbitrary
units on the
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CA 02764731 2012-01-11
right vertical axis) of the highest frequency bulk mode. Figure 14 clearly
shows the
relationship between the maximum intensity and the number of surface modes.
The two
curves in Figure 14 are clearly strongly correlated, which confirms that
surface modes
occur for radii R such that the edge of the core cuts through high-intensity
lobes of the
highest frequency bulk mode. Based on this principle, a first approximate
dependence of
the number of surface modes on the core radius was developed. By comparison to
the
results of exact simulations, the foregoing shows that the results obtained
using this
HFBM criterion predicts the presence or absence of surface modes fairly
accurately. Of
course, many other kinds of perturbations can induce surface modes in the
photonic
crystal 102, so that the foregoing condition for the absence of surface modes
is a
necessary condition but it is not always a sufficient condition.
[0069] In one criterion for determining the presence of the surface modes, the
electromagnetic intensity of the highest frequency bulk modes is integrated
along the
edge of the core. It is sufficient to perform such integration for either one
of the two
doubly degenerate modes, since the integrations for both modes are equal, as
required by
symmetry.
[0070] The foregoing determination of the radius R of the air core can be
performed in
accordance with a method of numerically computing the intensity distribution
of the bulk
modes of the infinite fiber cladding. In accordance with the method, the
intensity
distribution of the highest frequency bulk mode of the fiber of interest
without the air core
is first determined. Thereafter, a circular air core of radius R is superposed
on that
intensity distribution. As illustrated in Figures 15A and 15B, changing the
core radius R
causes the edge of the core to pass through different areas of this field
distribution. In
accordance with the computing method, the fiber will support surface modes
when the
edge of the core intersects high lobe regions of this field distribution. In
Figures 15A and
15B, a core of radius R = R1 is one example of a core radius that passes
through several
(six in this example) high intensity lobes of the highest frequency bulk mode.
The
computing method predicts that a core with such a radius will support surface
modes. At
the other extreme, when the core has a radius R = R2, as illustrated in
Figures 15A and
15B, the core edge does not pass through any of the high-intensity lobes of
the bulk
mode, and such a core of radius R2 does not support surface modes.
[0071] Although, described in connection with a circular core, it should be
understood
that the foregoing method is not limited to circular cores, and the method is
applicable to
any core shape.
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CA 02764731 2012-01-11
[0072] As described above, the computing method is qualitative. In accordance
with the
method, if the edge of a core of a selected radius R intersects high intensity
lobes of the
bulk mode, the fiber having a core of that radius will support surface modes.
As
described thus far, the method does not stipulate how many surface modes are
supported.
Furthermore, the method does not specify how high an intensity must be
intersected by
the edge of the core or how many high intensity lobes the edge of the core
must intersect
before surface modes appear (i.e., are supported).
[0073] The HFBM criterion is advantageously simplified by recognizing that the
intensity
lobes of the HFBM are nearly azimuthally symmetric, as shown in Figure 5.
Thus, the
portion of each lobe confined in a dielectric corner 110 can be approximated
by the circle
114 inscribed in the corner 110, as illustrated in Figure 2. As discussed
above, the radius
a of the inscribed circle 114 is related to the period A and radius p of the
holes 104 of the
triangular pattern by a = (A143)-p.
[0074] The portions of the HFBM confined to the dielectric are approximated by
a
two-dimensional array of circles 114 centered on all the photonic-crystal
comers 110, as
illustrated in Figure 16, which is plotted for a triangular pattern and p=
0.47A. This
approximation enables a new, simpler existence criterion to be formulated for
surface
modes: surface modes exist when and only when the surface of the core 106
intersects
one or more of the circles 114. Of course, many other kinds of perturbations
can induce
surface modes in the photonic crystal 102, so that the foregoing condition for
the absence
of surface modes is a necessary condition but it is not always a sufficient
condition.
[0075] The same geometric criterion can also be derived using coupled-mode
theory. In
view of the symmetry of the lower-band bulk modes, each corner 110 can be
approximated by a dielectric rod inscribed in the comer 110, wherein the rod
extends the
length of the PBF 100. Each isolated rod is surrounded by air and constitutes
a dielectric
waveguide. The dielectric waveguide carries a fundamental mode with strong
fields in
the rod that decay evanescently into the surrounding air, so the field looks
much like the
individual lobes of the HFBM illustrated in Figure 5. Thus, the periodic array
of rods has
the pattern of the circles 114 illustrated in Figure 16. The waveguide modes
of the
individual rods are weakly coupled to each other due to the proximity of
neighboring rods
and form the bulk modes.
[0076] The HFBM is just one particular superposition of individual waveguide
modes. If
an air core 106 that cuts into one or more rods is introduced, the removal of
dielectric
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CA 02764731 2012-01-11
perturbs the waveguide modes in the opposite direction to that forming bulk
modes. The
waveguide modes of the ring of perturbed rods intersected by the surface of
the core 106
are then coupled to each other and form a surface mode. This surface mode is
supported
by the ring of rods and has fields that decrease outside each rod, as
evidenced by the
exemplary surface mode of Figure 3. If the surface of the core 106 cuts only
through
membranes 112 instead of corners 110, the rods are unperturbed, and the modes
couple to
each other much as they did without the presence of the core 106. Thus, no
surface mode
is formed. In accordance with this description, surface modes exist if and
only if the
surface of the core 106 intersects rods. This is the same criterion that was
derived above
by approximating the BFBM lobes by the inscribed circles 114.
[0077] To verify the validity of this new geometric criterion, the criterion
is applied to the
most widely studied class of air-core PBFs, namely fibers with circular air
holes in a
triangular pattern, as illustrated in Figure 16. The core 106 is a larger
circular air hole of
radius R at the center of the fiber 100. Again, this analysis postulates that
when R is
selected so that the surface of the core 106 intersects one or more rods
(e.g., the circles
114 in Figure 16), then surface modes will exist, and the number of surface
modes will be
proportional to the number of rods intersected. This scaling law is expected
because as
the number of intersected rods increases the perturbation magnitude increases
and the
number of surface modes also increases. Conversely, when the surface of the
core 106
does not intersect any rods, no surface modes occur. A simple diagram of the
fiber cross
section, such as the diagram illustrated in Figure 16, makes the application
of this
criterion to any fiber geometry very easy.
[0078] The result of the foregoing geometric analysis is graphed in Figure 16
for a
, triangular pattern. The shaded (cross hatched) rings in Figure 16 represent
the ranges of
core radii that intersect rods and thus support surface modes. As discussed
above with
respect to Figure 11, the unshaded rings between the shaded rings (band 1¨band
6)
represent ranges of radii that intersect no rods and thus do not support
surface modes.
The dependence of the number of surface modes on the core radius is calculated
straightforwardly by applying elementary trigonometry to Figure 16 to
determine the
number of rods crossed by the surface of a core 106 of a given radius. The
numbers are
plotted as a solid curve in Figure 17, wherein the horizontal axis of the
graph is the core
radius normalized to the crystal period A (e.g., RI A), and wherein the left
vertical axis
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CA 02764731 2012-01-11
represents the number of rods intersected by the surface of the core, as
predicted by the
geometric criterion.
[0079] The simple postulate predicts the important result illustrated in
Figure 17 that
several bands of radii for this type of PBF 100 support no surface modes at
all across the
entire bandgap. Six such bands occur in the range covered in Figure 17 for
radii 1? up to
3.5A, where A is the crystal period as defined above. The range in Figure 17
does not
encompass the band below R= 0.47A, for which the radii are too small to
support a core
mode. Although not shown in Figure 17, another eight bands occur for radii
larger than
3.5A. The last band is at R approximately equal to 8.86A.
[0080] Table 1 lists the boundaries and the widths of the 14 bands. As shown
in Table 1,
the first band is the widest. The first band is also the most important for
most purposes
because the first band is the only band that falls in the single-mode range of
this PBF 100
(e.g., in the range where R is less than about 1.2 for an air-hole radius p
equal to 0.47A).
All other bands, except for the third one, are substantially narrower.
Generally, the bands
where no surface modes are supported become narrower as the radius of the core
106
increases. Note that by nature of the rod approximation, these values are
independent of
the refractive index of the crystal lattice dielectric 102.
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CA 02764731 2012-01-11
Table 1
Band Range from Range from Range from simulations Width of
No. geometric HFBM (in units of A) Band
criterion criterion (in units of A)
(in units of A) (in units of A)
1 0.685-1.047 0.68-1.05 0.65 0.05-1.05 0.05 0.363
2 1.262-1.420 1.26-1.43 1.27 0.01-1.45 0.05 0.158
3 1.635-1.974 1.64-1.97 1.65 0.05-2.05 0.05 0.339
4 2.189-2.202 0.013
2.624-2.779 0.155
6 3.322-3.405 0.083
7 3.619-3.679 0.059
8 3.893-3.934 0.071
9 4.271-4.402 0.131
5.239-5.400 0.161
11 6.218-6.244 0.026
12 6.914-6.916 0.0022
13 7.875-7.914 0.039
14 8.844-8.856 0.0113
Location of the 14 bands of core radii that support no
surface modes in triangular PBFs with p= 0.47A.
[00811 To evaluate the accuracy of the foregoing quantitative predictions,
numerical
simulations of the surface modes of this same class of PBFs were conducted on
a
supercomputer using a full-vectorial plane wave expansion method, as discussed
above
The dielectric was defined to be silica and the radius p of the air-holes 104
was defined to
be equal to 0.47A. The results of the simulations are plotted in Figure 17 as
open
triangles joined by dashes, wherein the right vertical axis represents the
number of surface
modes predicted by the numerical simulations. Note that this curve of
triangular points is
exactly the same as the curve of triangular points of Figure 9. The agreement
with the
predictions of the geometric criterion (plotted as a solid curve in Figure 17)
is excellent.
This agreement is further apparent by comparing the information in the second
column of
Table 1 for the boundary values of the first three surface-mode-free bands
generated by
the geometric criterion with the information in the fourth column of Table 1
for the
boundary values produced by the simulations. The geometric criterion produces
values
that are within 5% of the values produced by the simulations. Note that the
exact
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CA 02764731 2012-01-11
boundary radii produced by the simulations were computed in limited numbers
(e.g., for
the radii encompassing the first three surface-mode-free bands) and were
computed with a
limited number of digits because the simulations are very time consuming
(e.g., about six
hours per radius). In contrast, the geometric criterion provided far more
information in a
small amount of time. Also note that although the geometric criterion does not
accurately
predict the exact number of surface modes (see Figure 17), the geometric
criterion does
exhibit the correct trend. In particular, the geometric criterion predicts
that surface modes
generally become more numerous with increasing radius R of the core 106, which
is
consistent with the original hypothesis.
100821 The effect of the fiber air-filling ratio on the presence of surface
modes can also
be quickly evaluated with the above-described geometric criterion by simply
recalculating
the boundary radii for different values of the hole radius p. The results of
the calculations
are illustrated in Figure 18, which plots the normalized boundary core radius
RI A, from
RI A= 0.6 to RI A= 2.0, on the vertical axis versus the normalized hole radius
p/ A, from
p/ A= 0.43 to p/ A= 0.50, on the horizontal axis. The possible values for p
are
constrained between approximately 0.43A, below which the photonic crystal has
no
bandgap, and below approximately 0.50A, at which the thickness of the
membranes 112
becomes zero. The ranges of core radii versus hole radii that support surface
modes are
shaded (cross hatched) and the ranges of core radii that do not support
surface modes are
unshaded. Figure 18 shows that larger holes 104, which have greater air-
filling ratios,
yield wider surface-mode-free bands because increasing the radius p of the air-
holes 110
decreases the radius a of the rods (represented by the inscribed circles 114).
Because of
the smaller rod size, the ranges of core radii R that intersect the rods are
narrower, and the
bands of surface-mode-free radii become wider.
[00831 Other interesting observations can be obtained from the results of the
studies
described above. First, in experimental PBFs 100, the core 106 is typically
created by
removing the central seven tubes or the central nineteen tubes from the
preform. These
configurations correspond to core radii R of approximately 1.15A and
approximately
2.1A, respectively. The geometric criterion defined herein confirms the
predictions of
exact simulations that both of these configurations exhibit surface modes, as
shown, for
example, in Figure 17. The existence of the surface modes explains, at least
in part, the
high propagation loss of most photonic-bandgap fibers fabricated to date.
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CA 02764731 2012-01-11
[0084] Second, the simulated curve in Figure 17 shows that a small change in
core radius
is all it takes to go from a surface-mode-free PBF to a PBF that supports
surface modes.
The abruptness of the transitions is consistent with the perturbation process
that creates
surface modes, and supports the credibility of the rod approximation discussed
above.
[0085] Third, the trends in Table 1 discussed earlier can be explained with
simple
physical arguments. As the core radius increases, adjacent concentric layers
of rods
become closer to each other, as shown in Figure 16. For larger radii, it is
increasingly
more difficult to find room for a circular radius that avoids all rods. Also,
a larger radius
tends to intersect more rods, and thus the number of surface modes generally
increases.
A manifestation of this effect can readily be seen in the fifth and sixth
layers of rods,
which lie between band 4 and band 5 in Figure 16. The fifth and sixth layers
overlap
radially and thus merge into a single, wider zone of core radii that support
surface modes.
In other words, there is no surface-mode-free band between the fifth and sixth
layers of
rods. The same effect occurs with respect to the seventh, eighth and ninth
layers, which
lie between band 5 and band 6 in Figure 16 and cause the large numerical
difference
between the maximum radius of band 5 (R = 2.77911) and the minimum radius of
band 6
(R = 3.322A) in Table 1. Conversely, as the radius R of the core 106
increases, the
surface-mode-free bands become increasingly narrower, as can readily be seen
in the fifth
column of Table 1, which lists the width of each surface-mode-free band in
units of A.
[0086] It can be expected intuitively that cores 106 with radii larger than
some critical
value Rc will all support surface modes, and thus, only a fmite number of
surface-mode-
free bands are available. This intuitive expectation is consistent with the
results of Table
1. In particular, for the structure evaluated herein for a radius p of the
holes 104 of
0.47A, the number of surface-mode-free bands is limited (i.e., only 14 bands),
and a
critical radius /2, (i.e., approximately 8.86A) exists above which the surface
modes form a
continuum. As indicated by the values in Table 1, the last four surface-mode-
free bands
are so narrow (e.g., AR of a few percent of A) that the last four bands are
probably
unusable for most practical applications. A corollary of this observation is
that
multimode PBFs with the particular geometry illustrated herein and with a core
radius R
greater than 5.411 will likely be plagued with surface modes.
[0087] The average value of the 1/e2 radius of any of the lobes of the actual
bulk mode in
Figures 15A and 15B is approximately 0.22A. In comparison to the intensity
lobe, the
radius a of the inscribed (dashed) circle in Figure 8 is approximately 0.107A.
A more
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CA 02764731 2012-01-11
refined figure and a better quantitative agreement can be obtained by refining
the value of
the equivalent radius a of the silica rod, and by calculating the average
radius of the
fundamental mode of a solid rod suspended in air.
[00881 A final observation obtained from the study described herein is that
surface modes
can be avoided in principle for any core size by selecting a non-circular core
shape having
a surface that does not intersect any rods. A schematic of an example of a non-
circular
core having a characteristic dimension corresponding to the shortest distance
from the
center to the nearest boundary of the core is shown in Figure 19. With a
hexagon-shaped
core (as outlined by a dashed line in Figure 19 to assist in visualizing the
shape of the
core), the introduction of any surface mode is avoided even when the core
region is large.
Such a structure could represent an improvement over the above-described
circular core
structures in applications where multi-mode operation is desired.
[0089] The geometric criterion described herein is not limited to the
particular triangular
geometry with circular cladding holes and the circular cores. It is applicable
to other
shapes and geometries.
.[0090] In accordance with the foregoing description, a simple geometric
criterion quickly
evaluates whether an air-core PBF exhibits surface modes. Comparison of the
results of
the geometric criterion to the results of numerical simulations demonstrates
that when
applied to fibers with a triangular-pattern cladding and a circular core, the
geometric
criterion accurately predicts the presence of a finite number of bands of core
radii that
support no surface modes. For sufficiently large circular cores (i.e., for
radii above the
largest of these bands), the fiber supports surface modes for any core radius.
This
versatile criterion provides an expedient new tool to analyze the existence of
surface
modes in photonic-crystal fibers with an arbitrary crystal structure and an
arbitrary core
profile.
[0091] Figures 20A and 20B illustrate plots of the effective refractive
indices of the
modes as a function of wavelength. The plot in Figure 20A illustrates indices
of the fiber
manufactured by Crystal Fibre. The plot in Figure 20B illustrates the indices
of the fiber
manufactured by Corning. The plots were generated using numerical simulations.
The
fundamental core modes are shown in bold curves, and the less intense lines
are the
surface modes. The Crystal Fibre core mode (Figure 20A) has a measured minimum
loss
of the order of 100 dB/km while the Corning core mode (Figure 20B) has a
measured
minimum loss of 13 dB/km. The loss of the core mode is believed to be mainly
due to
coupling of the core mode to surface modes, which are inherently lossy due to
the
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CA 02764731 2012-08-10
concentration of energy near the surface of the core. Hence surface modes
suffer from
enhanced Rayleigh scattering. The total power coupled from core modes to
surface
modes will be enhanced, and thus the loss will be larger, if the core supports
a large
number of surface modes. In addition, it is well known from coupled mode
theory that
the coupling of two modes, in this case the core mode to a surface mode, will
be stronger
when the effective refractive indexes of the two modes are closer.
[0092] When considering the modes at a wavelength of 1.50 nm in Figures 20A
and 20B,
it can be seen that there are far more surface modes in the Crystal Fibre
structure (Figure
20A) than in the Coming structure (Figure 20B). Furthermore, the effective
refractive
indices of the Corning surface modes are less than 0.986, while the core mode
has an
effective refractive index of 0.994, a 0.8% difference. On the other hand, the
core mode
in the Crystal Fibre structure has an effective refractive index of 0.996,
while the nearest
surface mode has an effective refractive index of 0.994, only a 0.2%
difference.
Everything else being the same, in particular the level of geometrical
perturbation present
in the core of the two fibers, coupling of the core mode to surface modes is
expected to be
stronger in the fiber manufactured by Crystal Fibre. Thus, the Crystal Fibre
fiber
supports more surface modes, and the surface modes couple more strongly, which
is
consistent with the higher propagation loss of the Crystal Fibre fiber. From
the foregoing,
it can be concluded that to design air-guided PBFs with a low loss, the
preferred approach
is to completely eliminate surface modes, as described above. If it is not
possible to
completely eliminate the surface modes, a second approach is to reduce the
number of
surface modes (e.g., by assuring that the core does not cut through too many
comers of
the cladding lattice), to increase the effective index detuning between the
core modes and
the remaining surface modes, or both.
[0093] Although described above in connection with particular embodiments of
the
present invention, it should be understood that the descriptions of the
embodiments are
illustrative of the invention and are not intended to be limiting. Various
modifications
and applications inay occur to those skilled in the art without departing from
the scope
of the invention as defined in the appended claims.
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