Transverse Lightwave Circuits In Microstructured Optical Fibers ...
Transverse lightwave circuits in
microstructured optical fibers:
waveguides
Maksim Skorobogatiy
´
Ecole Polytechnique de Montr´eal, G´enie Physique, C.P. 6079, succ. Centre-Ville Montreal,
Qu´ebec H3C3A7, Canada
maksim.skorobogatiy@polymtl.ca
http://www.photonics.phys.polymtl.ca
Kunimasa Saitoh and Masanori Koshiba
Division of Media and Network Technologies, Hokkaido University, Sapporo 060-0814, Japan
Abstract:
Novel class of microstructured optical fiber couplers is
introduced that operates by resonant, rather than proximity, energy transfer
via transverse lightguides built into a fiber cross-section. Such a design
allows unlimited spatial separation between interacting fibers which, in
turn, eliminates inter-core crosstalk via proximity coupling. Controllable
energy transfer between fiber cores is then achieved by localized and highly
directional transmission through a transverse lightguide. Main advantage
of this coupling scheme is its inherent scalability as additional fiber cores
could be integrated into the existing fiber cross-section simply by placing
them far enough from the existing circuitry to avoid proximity crosstalk,
and then making the necessary inter-core connections with transverse light
“wires” - in a direct analogy with an on chip electronics integration.
© 2005 Optical Society of America
OCIS codes: 060.1810,130.3120
References and links
1. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Muller, J.A. West, N.F. Borrelli, D.C. Allan, K.W. Koch,
“Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657-659 ( 2003).
2. P. Russell, ”Photonic crystal fibers,” Science 299, 358-362 (2003).
3. M.A. van Eijkelenborg, A. Argyros, G. Barton, I.M. Bassett, M. Fellew, G. Henry, N.A. Issa, M.C.J. Large, S.
Manos, W. Padden, L. Poladian, J. Zagari, “Recent progress in microstructured polymer optical fibre fabrication
and characterisation,’ Opt. Fiber Techn. 9, 199-209 (2003).
4. B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres
with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650-653 (2002).
5. T. Katagiri, Y. Matsuura, M. Miyagi, “Photonic bandgap fiber with a silica core and multilayer dielectric
cladding,” Opt. Lett. 29, 557-559 (2004).
6. B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, and A.H. Greenaway, “Experimental study of dual-core
photonic crystal fibre,” Electron. Lett. 36, 1358-1359 (2000).
7. B.H. Lee, J.B. Eom, J. Kim, D.S. Moon, U.-C. Paek, and G.-H. Yang, “Photonic crystal fiber coupler,” Opt. Lett.
27, 812-814 (2002).
8. W.E.P. Padden, M.A. van Eijkelenborg, A. Argyros, N. A. Issa, “Coupling in a twin-core microstructured polymer
optical fiber,” Appl. Phys. Lett. 84, 1689-1691 (2004).
9. H. Kim, J. Kim, U.-C. Paek, B.H. Lee, and K. T. Kim, “Tunable photonic crystal fiber coupler based on a side-
polishing technique,” Opt. Lett. 29, 1194-1196 (2004).
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19 September 2005 / Vol. 13, No. 19 / OPTICS EXPRESS 7506
10. J. Laegsgaard, O. Bang, and A. Bjarklev, “Photonic crystal fiber design for broadband directional coupling,” Opt.
Lett. 29, 2473-2475 (2004).
11. K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers ,”
Opt. Express 11, 3100 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100
12. K. Saitoh, M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element
scheme: application to photonic vrystal fibers,” J. Quantum Electron. 38, 927-933 (2002).
13. K. Saitoh, M. Koshiba, “Full-vectorial imaginary-distance beam propagation method with perfectly matched
layers for anysotropic optical waveguides,” J. Ligthwave Techn. 19, 405-413 (2001).
14. M. Skorobogatiy, “Modeling the impact of imperfections in high-index-contrast photonic waveguides,” Phys.
Rev. E 70, 46609 (2004).
15. M. Skorobogatiy, “Hollow Bragg fiber bundles: when coupling helps and when it hurts,” Opt. Lett. 29, 1479-1481
(2004).
1.
Introduction
Currently, one of the major trends in the development of all-fiber devices is an increasing in-
tegration of functionalities in a single fiber. The ultimate goal is to be able to fabricate in a
single draw a complete all fiber component provisioned on a preform level. Some of the ad-
vantages of all-fiber devices are: simplified packaging, absence of sub-component splicing
losses, environmental stability due to the absence of a free space optics. While the benefits
of integrated all-fiber devices are significant enough to encourage development of increasingly
complex components, the major roadblock to their realization is an unavoidable complexity of
the required transverse refractive index profile. From an experimental point of view, complex
preform geometries such as in microstructured optical fibers (MOF) [1, 2, 3], and multicom-
ponent material combinations such as in Bragg fibers [4, 5] result in major challenges for the
preform fabrication and drawing. Assuming that these technological issues could be resolved
there is still a conceptual difficulty in the implementation of increasingly complex fiber profiles.
Namely, many of the interesting functionalities that fiber devices offer such as modal dispersion
profile design, directional power transfer between several fiber cores [6, 7, 8, 9, 10], inter-mode
conversion, etc. rely critically on proximity interaction between the modes localized in different
spatial regions. Requirement of a finite overlap between the interacting modes, typically, forces
multi-core systems to be designed to operate on a principle of proximity interaction, where
different cores are placed in an immediate proximity of each other. Such arrangement forms an
all-interacting system where all the fibers have to be considered simultaneously with a com-
plexity of a system design increasing dramatically with the number of fiber cores. While it can
be beneficial to have the individual sub-components designed on a proximity principle, scal-
able integration of several of them in a single fiber would rather require the individual spatially
separated “non-interacting” sub-components connected by a “wire”-like transverse lightwave
circuitry. In this paper we will demonstrate the principles of design of such transverse light-
guides enabling long range energy transfer between the two widely separated cores.
2.
Structure and definitions
In this paper, we review a “work-horse” of the fiber components - a directional coupler. In its
traditional implementation two fiber cores are placed in a close proximity of each other. Inter-
core coupling is due to a finite overlap between the corresponding core modes leading to a
complete energy transfer from one core into the other after a certain “coupling” length Lc. By
choosing the overall coupler length to be finite and equal to Lc directionality of energy transfer
is achieved. When two cores are spaced further apart an inter-core coupling reduces dramati-
cally resulting in an exponential increase of a coupling length. In what follows we demonstrate
designing of a resonant rather than a proximity coupler which enables energy transfer between
the two fiber cores regardless of the inter-core separation. To demonstrate robustness of our
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design and to investigate an importance of the bandgap confinement on the coupler radiation
losses we choose the most challenging case - designing of a long range coupling between the
two hollow core fibers guiding in the band gap of a surrounding 2D photonic crystal cladding.
Structure and modal properties of the individual hollow waveguides are detailed in [11]. Briefly,
hollow core is formed in a silica based MOF with a cladding refractive index n = 1.45 by re-
moving two rows of tubes and smoothing the resulting core edges. The pitch is Λ = 2µm, while
d/Λ = 0.9 with a total of six hole layers in the cladding. Fundamental band gap where the core
guided modes are found extends between 1.29µm < λ < 1.40µm. To form a coupler we place
two hollow cores N periods apart from each other as shown in Fig. 1. Transverse waveguide
is then introduced by slightly reducing (high index defect) or increasing (low index defect) the
diameters of the holes along the line joining the cores. As hollow core size is relatively small,
in order to minimize the impact on the guided modes of a single fiber we keep the sizes of the
holes closest to the cores unchanged. In a stand alone fiber the lowest loss mode is a doublet
f
e
c
t
d
z
x
de
x
dw
y
i
nde
N
Λ
w
o
L
e
f
ect
e
x d
d
i
n
i
gh
H
Fig. 1. Schematic of a two hollow core MOF coupler. Cores are separated by N lattice
periods. Transverse waveguide is formed by the holes of somewhat different diameters
placed along the inter-core line. Holes closest to the cores are not modified.
with an electric field vector having either x or y dominant component. We call such fields as x or
y polarized. In what follows we consider each polarization separately, making all the derivations
only for a y polarized mode. When second identical core is introduced, interaction between the
core modes of the same polarization leads to an appearance of the even and odd (with respect
to the reflection in OY axis) supermodes with the effective refractive indexes defined respec-
tively as ny+ , ny− closely spaced around n0y
of a single core fiber mode. An inter-fiber
e f f
e f f
e f f
coupling strength is then defined by difference in the real parts of the supermode effective in-
dexes ∆ny
= |Re(ny+ − ny− )|, while modal radiation losses are defined by the imaginary
e f f
e f f
e f f
parts of their effective indexes. Coupling length after which the power launched in one core
will be completely transferred into the other is Lc = λ /(2∆ny ). To find an upper bound on the
e f f
radiation losses of a coupler we define the power decay length as Ld = λ /(4πmax(Im(ny± ))).
e f f
Maximum power loss over a single coupling length is then P(Lc)/P(0) = exp(−Lc/Ld). In
what follows, we use the FEM mode solver [12] to perform modal analysis of a coupler as well
as the FEM BPM solver [13] to confirm our findings.
3.
Proximity coupling between two hollow cores
We first investigate proximity coupling between two hollow cores without any line defects in
the cladding. In Fig. 2(a) dispersion relations of the y even and odd supermodes of a two core
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coupler with N = 4 are presented with respect to a dispersion relation of a y polarized mode
of a stand alone fiber; modal splitting due to the proximity coupling is evident. In the inset,
|Ey| field distribution of one of the supermodes is presented. Due to exponentially small field
presence in the cladding, plots of absolute field values will look very much the same for both
supermodes. Coupling length for N = 4 varies between 3 − 8cm achieving its maximum value
well inside of a band gap where modal confinement in the core is strongest. Although inter-core
coupling increases near the band gap edges as core mode becomes less well confined by a PBG
cladding, coupler losses increase at a much faster rate, thus resulting in the dramatic increase
of a coupler loss per coupling length P(Lc)/P(0) when approaching the band gap edges.
In Fig. 2(a) note that effective index curves of the two supermodes are not centered around
the effective index curve of an unperturbed mode as predicted by a standard perturbation theory
(PT). We have observed the same failure of a standard PT even when inter-core separation is
increased beyond 11 periods and mode coupling becomes negligibly small. This failure is a
vivid example of a general observation that predictions of a standard PT are generally not
applicable to the high index-contrast systems with shifting material boundaries (see [14] and the
references of thereof). Another example of a high index-contrast coupler with a similar anomaly
in the behavior of its supermodes is a case of two coupled hollow Bragg fibers detailed in [15].
As the fields of the core modes guided by PBG decay exponentially fast into the cladding,
x 10
-6
y polarization, N=4
N=11
15
y polarization
1
10
y-even
7
N=10
N=9
10
N=8
6
0
10
N=7
eff 5
Lc (cm)
-n0
y-odd
N=6
5
Lc (m)
effn 0
λ
N=5
.3 µ
-1
Re( )
1
10
2
4
N=4
3
-5
4
N=3
5
3
-2
10
1.29
29 1.3
1 3 1.31
1 31 1.32
1 32 1.33
1 33 1.34
1 34 1.35 1.36 1.37 1.38 1.39 1.40
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
λ
λ
(µm)
(µm)
a)
b)
Fig. 2. a) Proximity coupling between the modes of two cores separated by 4 periods,
dispersion relations, coupling length. b) Proximity coupling length as a function of λ for
various inter-core separations.
coupling between the cores also decreases exponentially fast when separation between the cores
is increased. In Fig. 2(b) we clearly observe this trend by plotting on a log scale coupling length
across the band gap for the various inter-core separations N. Thus, for an inter-core separation
N = 4 coupling length is ∼ 10cm, for N = 7 it is ∼ 1m, while for N = 11 it is ∼ 20m. In what
follows we will demonstrate that introduction of the transverse waveguides enables reduction
of a coupling length to a sub centimeter scale even for large separations N = 11 making such a
resonant coupling practically independent of an inter-core separation.
4.
Resonant coupling between two hollow cores
We now introduce the high index line defect (transverse waveguide) into the N = 4 coupler of
Fig. 2(a) by reducing inner diameters of all the holes along the inter-core line to dw/Λ = 0.85.
In the upper plot of Fig. 3(a) dispersion relation of the supermodes relative to the dispersion
relation of a fundamental mode of an isolated core is presented. At λ ∼ 1.305µm, y − even
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supermode experiences avoiding crossing with a localized state of a line defect. In this regime
there are three modes interacting: y − odd mode (point 2 in Fig. 3(a)), and two y − even modes
(points 1,3 in Fig. 3(a)). Field distribution in a y − odd mode is similar to that of a Fig. 2(a).
Field distributions in the y −even supermodes reflect properly symmetrized linear combinations
of the individual core modes and a line defect mode (see Fig. 3(b)); due to a small field overlap
between the core modes and a line defect mode, plots of absolute field values appear similar
for both supermodes. When three modes interact, complete power transfer between the cores is
-4
x 10
4
3
eff
2
y-even
-n0
y-odd
0
2
effn -2
Re( )
1
-4
dB 0
0
10
-1
-2
-3
0
Lc/Ld
10
-4
-5
Lc (cm)
-6
-7
-1
-8
10
-9
-10
y polarization, N=4
-1
101.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
λ
a)
b
b)
Fig. 3. Resonant coupling between the modes of two cores separated by 4 periods mediated
by a line defect. a) Upper plot: dispersion relations of the supermodes exhibiting avoiding
crossing with the transverse waveguide modes. Lower plot: coupling length, losses. b) Field
distribution of a y − even supermode at λ = 1.3052µm resonance. Excitation of a localized
mode of a line defect is observed in an inter-core region.
still possible with the same definition of coupling length as in section 2, given that the distance
between points 1 and 2 is the same as the one between 2 and 3. In what follows we call this
a resonance. In the lower plot of Fig. 3(a) we present coupling length across the band gap
(solid curve), using as coupling strength ∆ny
= min(|Re(ny1+ − ny− )|,|Re(ny2+ − ny−odd)|),
e f f
e f f
e f f
e f f
e f f
which at the resonances gives coupling length for a complete energy transfer. At λ = 1.3052µm
coupling length becomes as small as Lc = 2mm, while far from the point of resonance, two
mode interaction picture becomes valid again and coupling length becomes similar to that of
a proximity interaction. Another important characteristic of a resonant coupler is the radiation
loss over a single coupling length. In section 2 we estimated an upper bound for such a loss
to be 1 − P(Lc)/P(0) = 1 − exp(−Lc/Ld), which in the limit of a small coupling loss becomes
simply Lc/Ld. In Fig. 3(a) Lc/Ld parameter is presented across the band gap. At resonance,
the coupling loss exhibits large increase, while outside a resonance in the region dominated by
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proximity coupling, radiation loss follows a general trend to be the smallest inside of a band gap
while increasing towards the band gap edges. In this particular example an upper bound estimate
gives a −2.4dB loss per coupling length. For a system with a line defect, while at resonance,
there is a substantial power transfer to a strongly localized state of a transverse waveguide (in
the perpendicular to a fiber direction). As dispersion relation of a supermode is close to that
of an air line, confinement in a higher refractive index line defect is only possible by the band
gap of a cladding. Thus, if the cladding were infinite, radiation losses of a defect state would be
zero. To understand better the origin of the radiation losses when coupling via a transverse line
defect it is useful to think about such a line defect in a limit of an infinite transverse waveguide
connecting two infinitely separated fiber cores. Such system will exhibit a discrete translational
symmetry along the direction of a transverse waveguide having well defined guided states. In
the case of a finite cladding, losses of the line defect modes will be ultimately determined by
their location in the band gap of a cladding - the deeper into the band gap the lower would be
the losses. When creating a transverse waveguide by reducing the sizes of all the holes along
a certain direction, one “pulls” into the band gap of a cladding a defect state associated with
a transverse waveguide. Thus, the larger (up to a certain extent) is a change in the hole sizes
the deeper into the band gap of a cladding will the guided modes of a line defect be located,
consequently leading to the lower losses of the transversely guide modes of a line defect.
We now increase the number of periods between the cores to N = 11. If there were no trans-
verse waveguide connecting the cores proximity coupling would be very small resulting in a
very large coupling length of Lc ∼ 20m. In the upper plot of Fig. 4(a) dispersion relation of
the supermodes relative to the dispersion relation of a fundamental mode of an isolated core is
presented for a line defect of dw/Λ = 0.85. As distance between the cores increases, several res-
onances appear inside of the band gap. These states are associated with the points of avoiding
crossing of the dispersion relation of a fundamental mode of a hollow core with the dispersion
relations of the transversely guided modes of a line defect (which in the limit of infinite inter-
core separation becomes a true transverse waveguide). Excitation of the transversely guided
modes of a transverse waveguide is possible as hollow core guided mode has a considerable
transverse wavevector component in a PBG cladding. Coupling lengths at resonances become
very small and on the order of the few millimeters (see Fig. 4(a) lower plot). Long range cou-
pling and complete power transfer at resonances have been confirmed with beam propagation
method (BPM) simulations. In the lower plot of Fig. 4 results of BPM simulations for the cou-
pling length, as well as the loss over a single coupling length are presented as solid curves with
circles, exhibiting excellent agreement with the predictions of a modal analysis. At the reso-
nances complete power transfer over a single coupling length is observed between the cores,
while out of the resonances only a partial power transfer is observed. In the media file, results
of a BPM simulation of a power transfer between the two hollow cores at the λ = 1.30639µm
resonance is demonstrated over the propagation distance of ten coupling lengths confirming our
modal analysis predictions. At λ ∼ 1.306µm, a y − even supermode exhibits avoiding crossing
with a localized line defect state. In this regime there are three modes interacting: y − odd
mode (point 2 in Fig. 4(a)), and two y − even modes (points 1,3 in Fig. 4(a)). Field distribution
in a y − odd mode is similar to that of Fig. 2(a), while field distribution in the y − even modes
exhibits a strong mixing with a line defect mode Fig. 4(b).
In Fig. 5 at four consecutive resonances we present the amplitude distributions of the y
components of the electric fields in the supermodes strongly mixed with the transverse guided
modes of a line defect. Standing wave pattern in the inter-core region arises due to the excita-
tions of both forward and backward propagation modes in a transverse waveguide. Also, toward
shorter wavelengths the number of the spatial oscillations in the defect region increases. In this
respect, the modes of a line defect are analogous to the Fabry-Perot resonances excited in a
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-4
x 10
2
3
y-odd
1
y-even
eff
-n0 0
2
effn
-1
Re( )
0
1
0
-2
10
1
2
2
10
3
4
Lc
5
6
-1 /Ld
7
10
1
9
10
Lc (cm)
10
0
10
y polarization, N=11
-2
10
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
λ
a)
b
b)
Fig. 4. Resonant coupling between the modes of two cores separated by 11 periods medi-
ated by a line defect. a) Upper plot: dispersion relations of the supermodes exhibit avoid-
ing crossing with waveguide modes. Lower plot: coupling length from the modal analysis
(solid) and BPM (solid with circles), losses from the modal analysis (dashed) and BPM
(dashed with circles). b) Field distribution of a y − even supermode at the λ = 1.3064µm
resonance. Excitation of a localized line defect mode is observed in the inter-core region.
short transverse waveguide terminated by the hollow cores. Origin of the localized Fabry-Perot
states in the inter-core region can be understood from a simplified interaction model. Partic-
ularly, we consider coupling in our system to be analogous to the coupling between the core
guided modes of the two collinear fibers via the in-plane guided modes of a slab waveguide;
fiber center lines are assumed to be in the plane of a slab. Modes of a slab waveguide are de-
generate with respect to a travelling direction along the waveguide plane. At a given frequency
ω we define a propagation constant of a guided slab state as k(ω) with a direction of propaga-
tion anywhere in the plane of a slab. Now assume that a guided fiber mode is characterized by
an in-plane propagation vector (β ,0), where OX axis is along the fiber propagation direction,
and OY axis is perpendicular to the fiber and is in the plane of a slab waveguide. For a given
frequency ω guided fiber mode will be phase matched with two modes of a slab waveguide
with the propagation vectors (β ,±kt), where the transverse propagation constant is defined as
kt =
k(ω)2 − β 2. If kt is real, guided slab modes will form a standing wave (Fabry-Perot
resonance) whenever constructive interference condition is satisfied kt L ∼ π p, where L is a
distance between the fibers and p is an integer. As inter-core distance is increased more res-
onances will appear in the inter-core region with spectral separation between the resonances
being inversely proportional to an inter-core separation. In a hollow core MOF coupler β ∼ ω,
k(ω) ∼ n
ω
clad
, thus for an inter-core separation of NΛ spectral separation between the reso-
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λ=1.294µm ; Loss(Lc)~21%
λ=1.331µm ; Loss(Lc)~80%
y-od
dd
y-even
λ=1.306µm ; Loss(Lc)~17%
λ=1.320µm ; Loss(Lc)~29%
y-eveen
y-odd
Fig. 5. Field distribution of the supermodes at the consecutive resonances for N = 11. The
number of spatial oscillations in the inter-core region increases towards shorter wavelengths
signifying Fabry-Perot nature of the localized states in a transverse waveguide.
nances will be ∆λ ∼ λ 2/(2NΛ n2
− 1). In Fig. 6 (right plot) coupling length as a function
clad
of the inter-core separation is presented across the band gap for a line defect dw/Λ = 0.85.
One can clearly see an appearance of Fabry-Perot resonances with reducing spectral separation
between them as the inter-core distance is increased. At resonances coupling length is on the
order of several millimeters regardless of the inter-core separation Fig. 6 (left plot). Note that all
the resonances are located in the region λ
1.34µm suggesting that transverse guided modes
of a high index line defect are pulled down from the upper edge of a band gap. Towards longer
wavelengths, coupling becomes dominated by the proximity effects as there are no line defect
states available at the lower frequencies.
Finally, we would like to address polarization dependence of a resonance coupler described
in this work. In Figs. 7(a,b) coupling length for a high index defect dw/Λ = 0.85 is presented.
As inter fiber separation increases, y polarized resonant states of a transverse waveguide are
pulled down from the upper band gap edge (shorter wavelengths), while no resonant states
exist for x polarization. In Figs. 7(c,d) coupling length for a low index defect dw/Λ = 0.95 is
presented. As inter fiber separation increases, resonant states of a transverse waveguide exist
for both y and x polarization and they are pulled up from the lower band gap edge (longer
wavelengths). We believe that design of polarization independent long range coupler is possible
by a proper choice of a line defect geometry. Our simulations suggest that low index defects
are the prime candidates for polarization independent performance; work in this direction is
currently underway.
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1
10
N=11
N=10
N=9
0
1
10
N=8
10
N=7
0
10
N=6
-1
10
-1
N=5
10
-2
Lc (m) 10
Lc (m)
N=4
-3
10
N=3
11
10
-2
10
9
8
1.4
1.39
y-polarization
7
1.38
1.37
no defect
6
1.36
1.35
high ind. def.
5
1.34
-3
1.33
10
1.32
4
1.31
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
N
1.3
λ
3
1.29
λ
(µm)
(µm)
Fig. 6. Coupling length across the band gap as a function of the inter-core separation N.
Left: at resonances coupling length is on the order of several millimeters regardless of
the inter-core separation, at longer wavelength coupling is dominated by the proximity
interaction. Right: Fabry-Perot like resonances. Spectral separation between the resonances
decreases with an increasing inter-core distance.
5.
Conclusions
Novel class of microstructured optical fiber couplers is introduced that operates by resonant,
rather than proximity energy transfer via the transverse line defects introduced between the
fiber cores. Introduction of such transverse waveguides enables significant spatial separation
between the interacting fibers which, in turn, eliminates an inter-core crosstalk via the proximity
coupling and allows a scalable integration of the multiple fiber sub-components. Controllable
energy transfer between the cores is then achieved over a coupling length of several millimeters
regardless of the inter-core separation. Findings based on the modal analysis were confirmed
by the BPM simulations. Polarization independent coupler is believed to be possible based on
the low index line defect design.
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1
1
10
10
y-polarization
N=11
N=8
high
index defect
N=10
N=7
N=9
0
0
10
N=8
10
N=6
N=7
N=6
N=5
-1
N=5
-1
10
10
N=4
Lc (m)
N=4
Lc (m)
N=3
N=3
-2
-2
10
10
x-polarization
high index defect
-3
-3
10
10
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
a)
λ (µm)
b)
λ (µm)
1
1
10
10
N=7
N=9
N=8
N=6
0
0
10
N=7
10
N=6
N=5
N=5
N=4
-1
-1
10
10
N=4
Lc (m)
Lc (m)
N=3
-2
-2
10
10
y-polarization
x-polarization
low index defect
-3
-3
low index defect
10
10
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4
λ (µm)
c)
d)
λ (µm)
Fig. 7. Coupling length across the band gap as a function of the line defect length N, mode
polarization and defect type.
#8214 - $15.00 USD
Received 18 August 2005; revised 5 September 2005; accepted 7 September 2005
(C) 2005 OSA
19 September 2005 / Vol. 13, No. 19 / OPTICS EXPRESS 7515
