Coupling between two collinear air-core Bragg fibers.

M. Skorobogatiya, Kunimasa Saitoh and Masanori Koshibab,

a Ecole Polytechnique de Montreal, Genie physique,C.P. 6079, succ. Centre-Ville Montreal, QC, Canada H3C 3A7;

bDivision of Media and Network Technologies, Hokkaido University,Sapporo 060-0814, Japan

ABSTRACT

We characterize coupling between two identicalcollinear hollow core Bragg fibers, assuming TE01

launching condition. Using multipole method and fi-nite element method we investigate dependence of thebeat length between supermodes of the coupled fibersand supermode radiation losses as a function of theinter-fiber separation, fiber core radius and index ofthe cladding. We established that coupling is maximalwhen fibers are touching each other decreasing dramat-ically during the first tens of nanometers of separation.However, residual coupling with the strength propor-tional to the fiber radiation loss is very long range de-creasing as an inverse square root of the inter-fiber sepa-ration, and exhibiting periodic variation with inter-fiberseparation. Finally, coupling between the TE01 modesis considered in a view of designing a directional cou-pler. We find that for fibers with large enough core radiione can identify broad frequency ranges where inter-modal coupling strength exceeds super-mode radiationlosses by an order of magnitude, thus opening a pos-sibility of building a directional coupler. We attributesuch unusually strong inter-mode coupling both to theresonant effects in the inter-mirror cavity as well as aproximity interaction between the leaky modes local-ized in the mirror.

1. INTRODUCTION

Recently, hollow core Photonic Band Gap (PBG) mi-crostructured and Bragg fibers have been experimen-tally demonstrated to exhibit guidance and low trans-mission loss at 1.55µm,1 3.0µm and 10.6µm2 promis-ing considerable impact in the long haul and high powerguidance applications almost anywhere in the IR. Hol-low PBG fibers are able to guide light through thehollow (gaseous) core featuring very low material lossand nonlinearity, and achieving radiation confinementvia reflection from the surrounding high quality dielec-tric multilayer mirror. Development of such fibers mo-tivated the interest in design of directional couplers

Send e-mail to: maksim.skorobogatiy@polymtl.ca

based on the PBG fibers to provide a uniform guid-ing/switching fabric, where the same type of fiber isused to guide and to manipulate light.

Because of the wider availability of microstructuredPBG fibers, most of the recent experimental and the-oretical work has concentrated on the design of direc-tional couples based on such fibers.3–7 Particularly, onthe preform stage, silica rods are arranged to form twoclosely spaced silica or air cores separated by severalair-silica layers, all surrounded by a hexagonal latticeof silica rods. When preform is drawn the resultantmicrostructured fiber exhibits two closely spaced iden-tical cores surrounded by a photonic crystal reflector.Coupling between such cores is characterized by finiteelement or finite difference methods with integrated ab-sorbing boundary conditions.8–11

In this work we for the first time, to our knowledge,consider the coupling between another type of PBGfibers - hollow Bragg fibers. Our main interest is tocharacterize coupling strength between collinear PBGBragg fibers, and propagation losses of the lowest loss“telecommunication quality”12 TE01-like supermodes.The issue of inter-fiber modal coupling can be of im-portance when several hollow photonic crystal fibersare placed in the proximity of each other, because ofthe established in this work a very long radiation driveninteraction range between such fibers. We also addressthe possibility of building Bragg fiber based directionalcouplers for TE01 modes. Unlike for PBG microstruc-tured fibers, the current process of Bragg fiber fabri-cation does not allow placing the cores of two Braggfibers arbitrarily close, while creating a common PBGreflector on the outside of both cores. We show thatenhanced resonant coupling is still possible even withstandard Bragg fibers by tuning the separation betweenthem to specific values with Bragg fiber mirrors of ad-jacent fibers creating an open resonant cavity.

The outline of our paper is as follows. We startby justifying most of the coupling properties betweencollinear PBG Bragg fibers based on the analogy with1D Bragg gratings with defects. We then characterizethe inter-fiber coupling as a function of core separa-tion and index of the cladding. Finally, we discuss the

Photonic Crystal Materials and Devices III, edited by Ali Adibi, Shawn-Yu Lin, Axel Scherer,Proc. of SPIE Vol. 5733 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.589732

206

nc

nclad

nh

nl

d

0 Rc Ro

σv(yz)

σv(x

z)

x

y

z

Oθ

d(θ)

Figure 1. Schematics of the two identical collinear hollowBragg fibers separated by the inter-mirror distance d. Di-electric profile along the inter-fiber center line resembles a1D Bragg grating with a central defect corresponding to theinter-mirror cavity.

possibility of building directional coupler based on theresonant interaction between the fibers.

2. ANALOGY TO THE 1D BRAGGGRATING WITH DEFECTS

We consider two collinear hollow PBG Bragg fibers ofcore radius Rc, outer mirror radius Ro and inter-mirrorseparation d. On Fig. 1 we present schematics of thesystem together with a dielectric profile along the linepassing through the fiber centers. We assume that thePBG mirror is made of two dielectrics with refractiveindexes nh > nl > nc, where nc is a core index (forhollow fibers nc = 1), while corresponding mirror layerthicknesses dh, dl are chosen to form a quarter-wavestack for the grazing angles of incidence.12 Thus, de-noting λ to be the center wavelength of the primaryBand Gap, then dh

√n2

h − n2c = dl

√n2

l − n2c = λ/4.

Cladding index nclad can be chosen at will.

By inspection of Fig. 1, dielectric profile along thefiber center line resembles a 1D Bragg grating made ofthe fiber reflector mirrors and a central defect of size dcorresponding to the inter-mirror cavity. Quarter-wavethickness of each mirror layer ensures the largest BandGap (stop band) of the reflector Bragg grating, so thatradiation incoming from the hollow core onto the con-taining mirror will be maximally reflected. However,when optical length of the central defect in the Bragggrating is λν/2, ν ∈ (0, 1, ...) it is known that transmis-sion trough such a grating will exhibit a narrow max-imum at λ, although transmission everywhere else in

the Bragg grating stop band will be still strongly sup-pressed. In the case of two identical Bragg fibers, thefirst resonance happens when the fibers are touchingd = 0 as the two outside high index layers of the fiber-mirrors create a λ/2 defect (see Fig. 1). Introducinga free space wave number k = 2π/λ, modal propaga-tion constant β, and transverse modal wave number inthe defect layer of refractive index n and thickness d askt

n =√

(kn)2 − β2, we rewrite resonant condition fora half-wavelength defect as dkt

n = πν. Thus, anytimethe inter-mirror separation d approaches its resonantvalue we expect increase in the inter-fiber coupling dueto enhanced radiation leakage from one core to anothermediated by the resonant inter-mirrors cavity. Spec-tral width and the maximum of an enhanced couplingpeak will be a strong function of the inter-mirror cav-ity Q factor. Because of the cylindrical shape of thefiber Bragg reflectors, inter-mirror cavity Q factor isultimately limited by the fiber finite curvature ∼ R−1

o .Finally, to increase the spectral width of a couplingpeak one can adopt a standard solution from the the-ory of flattened passband filters15 where the structure ofthe Bragg reflector is modified to present a sequence ofseveral low quality λ/4 stacks coupled together by λ/2defects, exhibiting a designable spectral width step-liketransmission response.

We now address the impact of cladding index nclad onthe PBG Bragg fiber coupling. It was established in,12

that low loss modes in the PBG Bragg fibers have theirpropagation constants situated closely to the core mate-rial light line, particularly for TE01 mode, 1−β/(knc) ∼(λ/Rc)2. Typical values of the core radii for a long-haulPBG Bragg fiber12 being Rc ∼ 10 − 15λ. Thus, in thecore material kt

nc=

√(knc)2 − β2 ∼ R−1

c , while inthe material with n > nc, kt

n � k√

n2 − n2c . Thus, if

cladding index is the same as the core index nclad = nc,one expects resonant increase in the coupling betweenPBG Bragg fibers at d = πν/(knc

) ∼ νRc, while ifnclad > nc, then d � λν/(2

√n2

clad − n2c) where in both

cases ν ∈ (0, 1, ...).

Finally, as the distance L = 2Ro+d between the fibercenters increase, intensities of the radiated fields fromthe core of one fiber at the position of the second fiberwill decrease with distance as E ∼ √

Im(β)/L, whereIm(β) is proportional to the modal radiation loss (fromthe energy conservation argument). Classical consid-eration of inter-fiber coupling between similar modessuggests that coupling strength is proportional to theoverlap of the fields of one fiber in the mirror region ofanother fiber, leading to the Im(β)/

√L dependence of

the PBG Bragg fiber coupling strength with the modalloss and inter-fiber separation.

Proc. of SPIE Vol. 5733 207

3. COUPLING AS A FUNCTION OFINTER-FIBER SEPARATION

We analyze coupling between the the lowest loss“telecommunication quality” TE01 modes of the twocollinear PBG Bragg fibers employing a modified mul-tipole method.13, 14 In a stand along fiber, TE01 isa singlet, with zero electric field components along thedirection of propagation and radial direction12 (elec-tric field vector is circling parallel to the dielectric in-terfaces). When second identical fiber is introduced,rotational symmetry of a single fiber is broken and in-teraction between the TE01 modes of the Bragg fibersleads to appearance of the two supermodes with prop-agation constants β− and β+ close to β. Remain-ing symmetry of a system is described by C2v groupthat includes reflections in (XZ), (Y Z) planes andinversion with respect to the system symmetry cen-ter O (see Fig. 1). Symmetry considerations result inthe following symmetries of the transverse electric fieldcomponents of supermodes: supermode− Ex(x,−y) =−Ex(x, y), Ey(x,−y) = Ey(x, y), Ex(−x, y) =Ex(x, y), Ey(−x, y) = −Ey(x, y), and supermode+

Ex(x,−y) = −Ex(x, y), Ey(x,−y) = Ey(x, y),Ex(−x, y) = −Ex(x, y), Ey(−x, y) = Ey(x, y). Thus,at the system symmetry center O, supermode+ willhave a local maximum of the electric field, whilesupermode− will have a node.

We first quantify coupling strength between fibersand radiation losses of the supermodes as a function ofthe inter-mirror separation d for the case when nc =nclad. We characterize inter-fiber coupling strength bythe difference in the real parts of supermode propaga-tion constants δβ = |β+ − β−|, while modal radiationlosses are defined by the imaginary parts of their propa-gation constants. Bragg fiber under study has 7 mirrorlayers (starting and ending with a high index layer),nc = 1, nh = 2.8, nl = 1.5, nclad = nc = 1, Rc = 5µm,operating wavelength is 1.55µm. On Fig. 2 normal-ized coupling strength and radiation losses of super-modes are presented. Normalization factor is radiationloss of a TE01 mode, which for this fiber is 11.2dB/m.Such a normalization choice allows us to largely decou-ple the structure of the reflector from other geometricalvariables such as inter-fiber separation. One observesthat coupling strength exhibits periodical variation asseparation between fibers increase. Locations of themaxima in coupling strength match well with the pre-dicted half wave condition for the optical defect lengthd = πν/(knc

) ∼ νRc (marked by the vertical dottedlines). Locations of the maxima in supermode losses arealso close to the half-wave separation between fiber mir-rors, suggesting that loss increase is due to field leakage

0.2

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radia

tion l

oss

es

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π 2π 3π 4π

Im(β±)/Im(β )TE

|Re(δβ)|/Im(β )TE

n = nc clad

Figure 2. Normalized coupling strength |Re(δβ)|/Im(βTE)and supermode radiation losses Im(β+)/Im(βTE),Im(β−)/Im(βTE) in a system of two collinear hollowBragg fibers as a function of inter-mirror separation d.Cladding index is the same as core index nc = nclad. Allthe curves are normalized by the radiation losses of theTE01 mode of a stand alone hollow Bragg fiber.

out of the open inter-mirror cavity, where at resonancefield intensity is enhanced. From Fig. 2 one also ob-serves a very slow decrease in coupling with inter-fiberseparation. By analyzing the values of the couplingmaxima as a function of distance up to d = 100µm aclear |δβ| ∼ (2Ro + d)−0.5 dependence is observed.

As argued in the previous section, when nc = nclad

position of the maxima of modal coupling scales pro-portionally to the core radius Rc. On Fig. 3 we verifythis scaling by plotting normalized coupling and su-permode radiation losses as a function of the inter-fiber separation for three Bragg fibers having differ-ent core radii and the same dielectric profile as be-fore. Predicted scaling is clearly observable from theplot. When comparing the values of the modal cou-pling at the first maxima, one observes that the cou-pling slowly increases as the fiber core radius increases.When looking in the region of small inter-mirror sepa-rations 0.2µm < d < 1µm (left subplot of Fig. 3) oneobserves a substantial increase in the coupling strengththat considerably surpasses the supermode radiationlosses when the distance between mirrors is decreased.Moreover, for the same inter-mirror separation d, thecoupling strength increases with an increase in the fibercore radius, signifying that the quality of the inter-mirror cavity resonator increases as Rc increases.

Overall, from Figs. 2, 3 we observe that coupling be-

208 Proc. of SPIE Vol. 5733

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1.5

2

2.5

d (µm)

Rc=15µm Rc=10µm

Rc=20µm

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d (µm)

0

Norm

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mode

radia

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es

Im(β±)/Im(β )TE

|Re(δβ)|/Im(β )TE

n = nc clad

Im(β±)/Im(β )TE

|Re(δβ)|/Im(β )TE

Figure 3. Normalized coupling strength and supermode ra-diation losses of coupled hollow Bragg fibers as a function ofinter-mirror fiber separation d. Cladding index is the sameas core index nc = nclad. Different line types correspond toBragg fibers of different core radii, solid lines - Rc = 10µm,dashed lines - Rc = 15µm, and dotted lines - Rc = 20µm.Left plot is a blow-up of the region where fibers nearly touch0.2µm < d < 1µm showing dramatic increase in the fibercoupling compared to almost constant radiation losses ofthe supermodes.

tween Bragg fibers stays comparable to the supermoderadiation losses even at very large separations, whilein the region of almost touching fibers d < 1µm cou-pling strength considerably exceeds supermode radia-tion losses. Thus, supermode beat length π/|δβ| (char-acteristic length of the fiber link after which consid-erable amount of power in one fiber gets transferredinto another fiber) remains smaller or comparable tothe supermode decay length 1/Im(β±) even for largeinter-fiber separations d ∼ 100µm. Hence, extra careshould be taken to absorb the radiation fields of theleaky modes when dealing with closely spaced Braggfibers.

Next, we investigate coupling strength between fibersand supermode radiation losses as a function of inter-mirror separation d when nclad > nc. Bragg fibers un-der study have the same dielectric profile and operat-ing wavelength as before except for the value of thecladding index nclad = 1.3. On Fig. 4 we present thenormalized coupling strengthm and supermode radia-tion losses for three different Bragg fibers of core radiiRc = 5µm, Rc = 10µm and Rc = 20µm as a functionof the inter-mirror separation around their second max-imum. By inspection of Fig. 4 we observe that indepen-dently of the fiber core radius, locations of the secondmaxima ν = 1 in the coupling strength match well withthe predicted based on the half-wave condition for the

0.75 0.8 0.85 0.9 0.95 1 1.050

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1.2

1.4

d (µm)

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up

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es

Rc=5µm

Rc=20µm

Rc=10µm

πIm(β±)/Im(β )

TE

|Re(δβ)|/Im(β )TE

n < nc clad

Figure 4. Normalized coupling strength and supermoderadiation losses of coupled hollow Bragg fibers as a functionof inter-mirror fiber separation d around second resonance.Cladding index is larger than core index nclad = 1.3, nc = 1.Different line types correspond to different Bragg fiber coreradii, solid lines - Rc = 5µm, dashed lines - Rc = 10µm,and dotted lines - Rc = 20µm. Maximum of the couplingstrength slowly decreases as fiber core radius increases.

optical defect length d = λν/(2√

n2clad − n2

c) (markedby the dotted line). When comparing the values of thefiber coupling at the first maxima, one observes thatthe coupling slowly decreases as the fiber core radiusincreases.

Difference in behavior of the inter-fiber couplingstrength with the change of fiber core radius Rc forthe cases nc = nclad and nclad > nc can be rationalizedas following. Resonant phenomena in the inter-mirrorcavity comes from a coherent addition of the multiplyreflected radial waves originally radiated from the fibercore. Phase difference that radial wave experiences bytraversing from the first mirror to the second dependsstrongly on the inter-mirror separation. In turn, intermirror separation depends on the angle θ (see Fig. 1)at which the wave escapes the core. For small anglesand d � Ro, d(θ) � d(0) + Roθ

2. Phase shift thatradial wave experience after one trip is φ(θ) = d(θ)kt

n,where kt

n is a transverse wave number. Thus, wavestravelling at different angles will have somewhat differ-ent phases, and at some critical θc the phase differenceφ(θc) − φ(0) = π will lead to destructive interferenceof waves in the inter-mirror cavity. The larger the θc

the higher the quality of the resonator will be. Whennc = nclad, φ(θ) − φ(0) = Roθ

2knc, and as knc

∼ R−1c ,

Rc ∼ Ro we arrive at θc ∼ 1 which is independentof the core radius. In contrast, when nclad > nc,

Proc. of SPIE Vol. 5733 209

φ(θ) − φ(0) = Roθ2kn, and as kn � ω

√n2 − n2

c , thenθc ∼ √

λ/Rc, thus slowly decreasing as the core radiusincreases. Hence, in the case nclad > nc the qualityof the resonator decreases when the core radius is in-creased.

Finally, we have also observed that the multipolemethod13 while performing very efficiently at the inter-mirror separations d/Ro > 0.1, exhibits slow conver-gence at smaller separations, and at d/Ro < 0.01 con-vergence becomes problematic. To further study touch-ing fibers we resort to a finite element mode solver withabsorbing boundary conditions.

4. COUPLING BETWEEN TOUCHINGBRAGG FIBERS

In this section we study feasibility of designing a di-rectional coupler based on two touching hollow Braggfibers studied by the finite element mode solver.8 Inthe following we assume the same structure of the twoBragg fibers as before, now touching along their length(d = 0 in Fig.1), designed for 1.55µm operation wave-length with index of the cladding matched with that ofa core nclad = nc = 1. We first characterize couplingstrength between fibers and radiation losses of the su-permodes as a function of the fiber core radii Rc at afixed frequency of λ = 1.45µm. Because of the cylin-drical shape of the fiber Bragg reflectors, inter-mirrorcavity Q factor is limited by the mirror finite curva-ture, with Q and, thus, inter-fiber coupling increasingfor larger core radii.

In Fig. 5 normalized coupling strength and radiationlosses of supermodes are presented. Normalization fac-tor for each curve is a corresponding radiation loss ofTE01 mode of a stand alone fiber. One observes thatwith increasing core radius coupling strength exhibitsa tendency of gradual increase relative to the radiationlosses of the supermodes. For the core radii larger than10µm the ratio of the coupling strength to supermoderadiation loss approaches a factor of 10 allowing, inprinciple, to build a directional coupler. Resonant fea-tures correspond to the points of accidental degeneracyof TE01 with higher order modes.

In Fig. 6 we plot normalized coupling strength andsupermode radiation losses as a function of wavelengthλ for Rc = 15µm and three different inter-fiber sep-arations d = 0µm, d = 0.2µm and d = 0.5µm.For d = 0.5µm the coupling strength is weak and onthe order of the supermode radiation losses, changingsmoothly as a function of frequency. As inter-fiber sepa-ration decreases, the coupling strength exhibits a rapidincrease across a broad frequency range together with

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λ=1.45µm

Im(β+)Im(β−)

Rc(µm)

|Re(δβ)|

Figure 5. Normalized coupling strength and supermoderadiation losses as a function of fiber core radii Rc atλ = 1.45µm. With increasing core radius one observes atendency of gradual increase of the coupling strength rela-tive to the highest radiation loss of the supermodes.

appearance of many sharp resonances (d = 0.2µm inFig. 6). When fibers are touching d = 0µm the couplingstrength strongly dominates supermode radiation lossesin a broad frequency range. For 1.4µm < λ < 1.55µm,for example, the radio of coupling strength to the su-permode radiation losses reaches a factor of 10. For acorresponding planar system of grating-defect-gratingwith a dielectric profile of Fig. 1 the resonance peak is,however, only several nanometers wide which is in con-tradiction with the Fig. 6 very broad resonant features.Thus, a simple picture of enhanced inter-fiber couplinghas to modified. As spectral width of the enhanced cou-pling peak depends strongly on the inter-mirror cavityQ factor, we believe that broad resonant features inFig. 6 can be explained by low Q factor of the reso-nant inter-mirror cavity due to a finite curvature of thefiber. As our mode solver was limited to the core radiiless than 20µm we were not able to investigate furtherthe narrowing of the resonance for larger radii. Anotherprominent feature of Fig. 6 is a presence of sharp andbroad regions of increase in the supermode losses. Be-cause of a multimoded nature of hollow Bragg fibers,lowest loss TE01 mode exhibits multiple points of ac-cidental degeneracies with higher loss modes. At suchdegeneracy points TE01-like supermode exhibits sharploss increase by “picking-up” some of the higher ordermode loss. In general, we find that broad frequencyregions of increase in the supermode losses are due tointeraction with low angular momenta modes. For ex-ample, by inspecting band diagram of a stand alone

210 Proc. of SPIE Vol. 5733

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10-1

d=0.5 µm

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d=0.2 µm

d=0 µm

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Im(β+)

|Re(δβ)|

Im(β−)

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es (

dB

/m)

Rc=15µm

Im(β )TE01

Im(β+)

Im(β−)

|Re(δβ)|

Im(β+)

Im(β−)

|Re(δβ)|

Im(β )TE01

Im(β )TE01

Figure 6. Normalized coupling strength and supermode ra-diation losses as a function of wavelength λ for Rc = 15µmand inter-fiber separations d = 0, 0.2, 0.5µm. For 1.4µm <λ < 1.55µm one observes a factor of 10 ratio of couplingstrength to the highest radiation loss of the supermodes.Sharp resonances in the coupling strength correspond tothe accidental mode crossing of TE01 with high angularmomenta modes of the reflector. In the insert d = 0µm,the |Ey| fields are presented at the frequencies close anddirectly at one of the sharp resonances. The region of1.55µm < λ < 1.65µm is dominated by a prolong cou-pling between TE01 and an m = 2 mode leading to a broadresonance and an increase in the supermode losses.

A

B

C

DE

F

G

H

Figure 7. Sharp resonances around the points of degen-eracies of a TE01 mode with the high angular momentummirror modes. A,B,C,D,E,F,G,H are the points around oneof such resonances at 1.5504µm.

fiber we find that in the region 1.55µm < λ < 1.65µman m = 2 mode crosses TE01 mode twice staying al-most degenerate with it in the whole interval. In Fig. 6,d = 0µm this broad modal interaction region is char-acterized by an increase in a supermode loss. We havefurther verified our assumption by modifying the loca-tion of the modal degeneracy region by adding morelayers to the reflector, and observed a consistent shiftof a broad resonance.

On the other hand, by inspecting the modal fieldsaround the sharp resonances (Figs. 7, 8) we concludethat such resonances correspond to the points of de-generacies of a TE01 mode with the high angular mo-mentum mirror modes. Note that Fig. 7 is the sameas Fig. 6, d = 0 but calculated with higher resolutionaround just a few sharp resonances. At resonance, a hy-brid mode has intensity maximum in the inter-mirrorcavity defect while the fields in the hollow fiber coresare reduced. We have verified that in a stand alonefiber there is a large number of high angular momentam > 6 leaky modes with propagation constants closeto the air line and fields concentrated mostly in thefiber reflector. In a region just outside of the fiber suchmodes exhibit a fast decay in the cladding. Thus, sharpresonances due to interaction with such modes disap-pear quickly with increase in the inter-fiber separationas clearly observable in Fig. 6 d = 0, 0.2, 0.5µm.

5. DISCUSSION

To summarize, we found that unlike in silica fibers,where the modal tail decays exponentially into the

Proc. of SPIE Vol. 5733 211

A

D

B

C

E

F

G

H

|Ey|

Figure 8. Amplitude of the modal field component |Ey|at the points A,B,C,D,E,F,G,H of Fig. 7 around one of thesharp resonances (black is zero). Points D and E correspondto the excited mirror-cavity modes.

cladding, the radiation field from a hollow PBG Braggfiber decays in the cladding very slowly as an inverseof the square root of the inter-fiber separation exhibit-ing periodic oscillations. Moreover, the beat lengthbetween supermodes π/Re(β+ − β−) stays on the or-der of the supermode decay length 1/Im(β±) even forvery large inter-fiber separations ∼ 100µm. When twostraight pieces of PBG Bragg fiber are spaced less than1µm from each other we observed a dramatic increasein the modal coupling without a substantial increase inthe supermode losses. We demonstrated that for twotouching PBG Bragg fibers of substantially large coreradius the frequency regions can be identified where alarge increase in the modal coupling is observed with-out a substantial increase in the supermode losses. Inthis regime, the supermode beat length becomes muchsmaller than the supermode decay length, opening apossibility of building a directional coupler exhibit-ing only a fraction of modal losses along the couplerlength. Because of the multimoded nature of hollowPBG Bragg fibers, special care should be taken to avoidaccidental modal degeneracies between the mode of op-eration and higher order modes at the frequency of in-terest.

REFERENCES

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 photonicbandgap fibre,” Nature 424, 657-659 ( 2003).

2. B. Temelkuran B, S.D. Hart, G. Benoit, J.D.Joannopoulos, Y. Fink, “Wavelength-scalable hollowoptical fibres with large photonic bandgaps for CO2laser transmission,” Nature 420, 650-653 ( 2002).

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. Za-gari, “Recent progress in microstructured polymeroptical fibre fabrication and characterisation,” Opt.Fiber Techn. 9, 199-209 (2003).

4. 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).

5. G. Kakarantzas, B.J. Mangan, T.A. Birks, J.C.Knight, P.S.J. Russell, “Directional coupling in atwin core photonic crystal fiber using heat treat-ment,” Technical Digest. Summaries, Conference onLasers and Electro-Optics, 599-600 (2001).

6. M. Kristensen, “Mode-coupling in photonic crystalfibers with multiple cores,” Conf. Digest. CLEO Eu-rope, 1 (2000).

7. B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Rus-sell, and A.H. Greenaway, “Experimental study ofdualcore photonic crystal fibre,” Electron. Lett. 36,1358-1359 (2000).

8. K. Saitoh, Y. Sato, and M. Koshiba, “Cou-pling characteristics of dual-core photonic crystalfiber couplers,” Opt. Express 11, 3188-3195 (2003),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-24-3188.

9. L. Zhang and C. Yang, “Polarizationsplitter based on photonic crystal fibers,”Opt. Express 11, 1015-1020 (2003),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1015

10. K. Saitoh and M. Koshiba, “Full-vectorialimaginary-distance beam propagation method basedon a finite element scheme: application to photoniccrystal fibers,” IEEE J. Quantum Electron. 38, 927-933 (2002).

11. F. Fogli, L. Saccomandi, P. Bassi, G. Bel-lanca, and S. Trillo, “Full vectorial BPM mod-eling of indexguiding photonic crystal fibersand couplers,” Opt. Express 10, 54-59 (2002),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54 12.

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12. Steven G. Johnson, Mihai Ibanescu, M. Sko-robogatiy, Ori Weisberg, Torkel D. Engeness,Marin Soljacic, Steven A. Jacobs, J. D. Joannopou-los, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,”Optics Express 9, 748 (2001).

13. T. P. White, B. T. Kuhlmey, R. C. McPhedran,D. Maystre, G.C. Martijn de Sterke, and L. C. Bot-ten, “Multipole method for microstructured opticalfibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322(2002).

14. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Cou-pling between two collinear air-core Bragg fibers,” J.Opt. Soc. Am. B 21, 2095 (2004).

15. R.A. Minasian, K.E. Alameh, E.H.W. Chan,“Photonics-based interference mitigation filters,”IEEE Transactions on Microwave Theory and Tech-niques 49, 1894 (2001)

16. P. Yeh, A. Yariv, and E. Marom, “Theory of Braggfiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).

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