Multipole analysis of photonic crystal fibers with coated inclusions

Multipole analysis of photonic crystalfibers with coated inclusions

Boris T. Kuhlmey, Karrnan Pathmanandavel and Ross C. McPhedranCentre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS) and School of Physics,

University of Sydney, New South Wales 2006, Australia

[email protected]

http://www.physics.usyd.edu.au/cudos/

Abstract: Photonic crystal fibers (PCF) containing coated holes haverecently been demonstrated experimentally, but haven’t been studiedtheoretically and numerically thus far. We extend the multipole formalismto take into account coated cylinders, and demonstrate its accuracy evenwith metallic coatings. We provide numerical tables for calibration of othernumerical methods. Further, we study the guidance properties of severalPCF with coated holes: we demonstrate that the confinement mechanismsof PCFs with high index coated holes depend on wavelength, and exhibitplasmonic resonances in metal coated PCFs.

© 2006 Optical Society of America

OCIS codes:(060.2310) Fiber optics; (240.6680) Surface plasmons; (999.9999) Photonic crys-tals

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#74753 - $15.00 USD Received 5 September 2006; revised 19 October 2006; accepted 21 October 2006

(C) 2006 OSA 30 October 2006 / Vol. 14, No. 22 / OPTICS EXPRESS 10851

mailto:[email protected]

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1. Introduction

Since their first experimental demonstration [1], photoniccrystal fibers (PCFs), optical fiberswith holes running along their length, have become a major topic of research. Not only have



http://setis.library.usyd.edu.au/adt/public

PCFs allowed technological breakthroughs such as hollow-core guidance or endlessly sin-gle mode fibers, with many applications in metrology, sensing, dispersion management, non-linear optics or particle guidance [2], they have also enabled researchers to unveil new as-pects of waveguidance and discover new physical phenomena.PCFs with high-index fluid filledholes [3] or with solid inclusions having a refractive indexhigher than that of the backgroundmaterial [4] can exhibit guidance which is not due to Bragg reflection from the surroundingphotonic crystal. Rather, as for planar anti-resonant reflecting optical waveguides (ARROWs),guidance in these fibers relies on enhanced back-scatteringat the anti-resonances of single in-clusions. Many of the properties of these fibers, also calledARROW fibers, can be deducedfrom the properties of single inclusions [5, 6]. From this model and other studies, it appearsthat many properties of PCFs can be designed by adjusting theresonances of single holes.Resonances of single cylinders, however, are well known andoffer only limited flexibility. Anatural way of gaining further control over the number, location and nature of resonances is forexample by adding a coating layer of dielectric, metal or composite material to the inclusions.

Coated spheres [7] and coated cylinders [8–10] have been shown to be able to widenbandgaps when used instead of their un-coated equivalent intwo- or three-dimensional pho-tonic crystals. Very recently, Sazioet al have demonstrated that holes of PCFs can be coatedwith a whole range of materials through high-pressure chemical vapor deposition over lengthsof up to 70 cm , and that they could also be coated locally, on micrometric length-scales, usinglaser assisted chemical vapor deposition [11]. Sazioet al’s technique promises the developmentof novel complex all-in-fiber passive as well as active photonic devices. For example, addinga metallic coating to all or some of the holes of a PCF allows exploitation of surface plasmonpolariton resonances to modify PCF properties. Although this is expected to increase losses be-cause of the metal’s material absorption, localized metallic coatings over micrometric lengthsin the fiber could enable the exploitation of very strong plasmonic resonances with minimalloss. However, the theory of PCFs with coated cylinders and their modal properties have so farto our knowledge not been studied.

Here, we extend the multipole formalism for PCFs [12] to include coated cylinders. Usingthis formalism, we calculate modes and modal properties forseveral PCFs with metal and di-electric coated holes. Because of intrinsic geometrical limitations [12], the multipole methodis not the most versatile PCF simulation tool; although it isgenerally considered to be compu-tationally very efficient, other very fast and more versatile mode finders exist [13]. However,because it is rigorous and analytical, the multipole methodis certainly the most accurate andbecause of this has been widely used in calibration of subsequent methods [13–24]. Our aimhere, besides demonstrating future interesting possibilities of PCFs with coated holes, is toprovide accurate data for several examples of coated hole PCFs which may be used as a refer-ence to verify simulations made using other methods. Indeed, especially for structures coatedwith metals having large complex dielectric constants, simulations using other methods canbe delicate [25, 26], and it is good practice to verify their convergence using a few referenceexamples. For this purpose, we provide convergence studieson selected examples, with tablesof numeric data for easy comparison in Section 3. Finally, westudy the guidance properties oftwo types of PCF containing coated inclusions: First a PCF with high index coated holes, forwhich we demonstrate that the dominant guidance mechanism is antiresonant scattering at shortwavelength and modified total internal reflection at long wavelengths, and second a PCF con-taining silver coated holes for which we exhibit coupling ofthe core more to surface plasmonpolaritons, associated with strong field localization and very large group velocity dispersion.

The remainder of the paper is organized as follows: In Section 2 we describe the mathemat-ical framework of the multipole method for PCFs including coated cylinders; Section 3 givesnumerical examples and tables of convergence; Finally we study guidance properties of two



examples in Section 4.

2. Multipole method for coated cylinders

A PCF typically consists of a narrow rod of high index material with holes running along itslength (Fig. 1). Holes are usually arranged periodically, following a triangular lattice, around acentral core. In the present study, the core will consist of asingle missing hole at the center ofthe fiber.

In the multipole method [12], the longitudinal components of the fields are expanded inFourier-Bessel series, containing Bessel and Hankel functions of the local coordinates, aroundeach hole. The coefficients for Bessel and Hankel functions are linked through the scatteringmatrix of each inclusion, whereas coefficients in the seriesfor different holes are linked throughGraf’s addition theorem. Graf’s theorem along with the scattering matrices is sufficient to en-tirely determine the system, and since both are known in closed form for simple circularlysymmetric holes, the multipole method is rigorous, semi-analytic, fast and accurate.

The extension of the multipole method to the case of coated inclusions is hence straightfor-ward since only the scattering matrices of the inclusions need to be modified. Accordingly, wewill not re-derive the multipole method in its entirety (forthis, see Ref. 12), but solely detailthe derivation of the scattering matrices for coated cylinders, which form the elements ofR inthe mode’s eigenvalue equation (Ref. 12, Eq. (30)).

For coated cylinders with circular symmetry we obtain closed form expressions for the scat-tering matrices, starting from the scattering matrices fora single interface and using recurrencerelations; although the choice of basis functions and notations differ, the approach we follow toextract the scattering matrices is mathematically equivalent to that used in previous studies ofscattering by coated cylinders in oblique incidence [27].

We consider two concentric circularly symmetric interfaces, such as depicted in Fig. 2. Thestructure being infinitely long, all components of the electric and magnetic fields can be ob-tained [28] fromEz andHz, and at a given frequency their longitudinal and time dependencecan be separated from the transverse dependence:

V(r,θ ,z, t) = Vt(r,θ)exp(i(βz−ωt)) , (1)

whereV is eitherEz or Hz, β is the propagation constant andω is the light’s angular frequency.The transverse fieldsVt then satisfy the Helmholtz equation with propagation constant β andcan be expanded in each region in Fourier Bessel series:

Vt(r,θ ,z) = ∑ν

[

AV,lν Jν(kl

⊥r)+BV,lν H(1)

ν (kl⊥r)

]

exp(iνθ) (2)

whereV is eitherEz or Hz, l is one of e (exterior), s (shell or coating) or i (interior),k⊥ =

(k0nl 2 − β 2)1/2 and r is in region l . We define the column vectors for each field and regionAV,l = [AV,l

ν ] andBV,l = [BV,lν ], and for each region, the vectors

A l =

[

AEz,l

AHz,l

]

(3)

and similarlyBl . The vectorsA l andBl of adjacent regions are linked through the boundaryconditions at the interface between those regions, and for asingle interface one can definereflection and transmission scattering matrices such that

A− = S+−A+ + S−−B− (4)

B+ = S++A+ + S−+B− , (5)



where the+ and− superscripts refer to the outside and inside of a single interface respectively.For circularly symmetric interfaces, the transmission andreflection matrices are known ex-

plicitly and given in Ref. [12], Appendix C, whereas for non-circularly symmetric interfaces,they can be computed using a number of numerical techniques,egdifferential methods usingthe fast Fourier factorization method [24,29].

The overall scattering matrixSe-eof the coated inclusion required for the multipole method’seigenvalue equation is defined byBe = Se-eAe. When there are no sources in the inclusionBi = 0, so that through elementary matrix manipulations Eqs. (4-5) at the e-s and s-i interfaceyield

Se-e= S++e-s + S−+

e-s (I − S++s-i S−−

e-s )−1S++s-i S+−

e-s , (6)

whereI is the identity matrix and the subscripts refer to the two interfaces. Solving the eigen-value equation yieldsBe for all inclusions, and hence the fields everywhere in the PCF’s back-ground material [12]. The fields inside the inclusions are obtained using Eq. (2) whereA l andBl (with l = i, s) are obtained fromBe using Eqs. (4-5).

ρe

Λ

Background material

Coating

ρi

Core

Inclusion

Fig. 1. Geometry of a PCF with coated cylinders.

3. Numerical examples and convergence

Our first example is a coated version of the PCF studied in Ref.12, to enable direct comparisons.The inclusions are coated by dielectric material of higher refractive index than the backgroundmaterial. The geometry is a single hexagon of inclusions in abackground material extendingto infinity, with parameters defined in the caption of Table 1.Table 1 shows a convergence testdone on the fundamental mode (confined mode with highest realpart of the effective index) ofthat structure, at a wavelength of 1.45 µm, along with the Wijngaard parametersWE andWH

defined in Ref. 30.WE andWH can be seen as a very loose upper bound of the relative error ofthe fieldsE andH respectively [31]. Fig. 3 shows the field distributions of the studied mode,for M = 8. Note that the fundamental mode for such a geometry is doubly degenerate, and weshow field plots for only one of the two degenerate modes. As can be seen from Table 1, the realand imaginary part of the mode’s effective index converge very quickly with M. AboveM = 8,the Wijngaard parameters increase, showing that accuracy is lost in fields, and eventually alsoin the effective index. This is due to higher order coefficients of the Fourier Bessel expansionbecoming so small that they are numerically negligible and ill-defined, and only add numericalnoise to the general formulation. Although not documented before in the context of PCFs, thisis a general feature of the multipole method, which could be somewhat improved by differentnormalization of the Fourier Bessel coefficients. However its effect becomes relevant only afterthe effective indices converge almost to machine precision; when accuracy is paramount, it is



y

O x

Aiν , Bi

ν

Aeν , Be

ν

Asν , Bs

ν

r

ρiρe

ns

ni

ne

θ

Fig. 2. Geometry and field expansion coefficients for a single coated inclusion.

nevertheless important to choose the best possible value ofM, which varies with geometry, bycarrying out a convergence test.

The fields of Fig. 3 show that the mode is predominantly confined in the core (the regionbetween the inclusions). However, a non negligible fraction of the field resides in the high-index coating. This is to be compared with the same mode without coating in Ref. 12, Fig. 4,where the fraction of the field inside the low index inclusions is negligible. Also, the effectiveindex of the mode is raised compared to that of the same mode without high-index coating, andthe propagation loss is higher by almost two orders of magnitude. We will see in Section 4 thatthis is linked to coupling to leaky modes of the coatings, andis highly wavelength dependent.

Table 1. Real and imaginary parts ofneff and Wijngaard parametersWE andWH as a func-tion of truncation parameter M of the Fourier Bessel series. Single hexagon with pitchΛ = 6.75 µm of inclusions havingρe = 2.5 µm, ρi = 1.5 µm, ni = 1, ns = 1.7, ne = 1.45,λ = 1.45 µm.

M Re(neff) Im(neff) WE WH

1 1.4471543714882 4.5467559634287×10−6 4.51×10−2 0.7142 1.4468691509821 1.5144137973724×10−5 2.60×10−2 0.2073 1.4468375162036 1.6972392192901×10−5 5.61×10−3 2.21×10−2

4 1.4468324114032 1.6654346274542×10−5 4.42×10−3 1.18×10−2

5 1.4468318076468 1.6658431542624×10−5 1.89×10−3 5.86×10−3

6 1.4468318005963 1.6659398783238×10−5 3.53×10−4 1.10×10−3

7 1.4468317905788 1.6658763190748×10−5 7.69×10−5 2.60×10−4

8 1.4468317968727 1.6658892471164×10−5 7.85×10−5 2.51×10−4

9 1.4468317971238 1.6658890915135×10−5 1.72×10−4 5.30×10−4

10 1.4468317971244 1.6658891108097×10−5 1.79×10−2 5.53×10−2

11 1.4468317971280 1.6658891300929×10−5 24.7 76.3

Table 2 shows a convergence test for the fundamental mode of astructure containing 50nmmetallic (silver) coatings; the geometry and parameters ofthe structure are described in the



(a) (b) (c)0

Max. value

(d)

Fig. 3. Selected field distributions of the mode studied in Table 1: (a)|Ez|, (b) |Hz|, (c)Re(Sz). (d) color scale. All fields are computed withM = 8. The color scale for this and allsubsequent figures is linear.

(a) (b) (c) (d)

Fig. 4. Selected field distributions of the mode studied in Table 2: (a)|Ez|, (b) |Hz|, (c) |Sz|,(d) detail of|Sz| around the rightmost cylinder, showing the strong field concentration onthe silver surface. All fields computed withM = 9. The color scale is the same as in Fig. 3.

caption. Again, convergence is very rapid with increasingM, although nowM = 9 is nowrequired whenM = 7 was sufficient in the dielectric case. Losses are significantly higher, dueto the high absorption of silver at the chosen wavelength. Fig. 4 shows field distributions of thefundamental mode. Note how these differ significantly from those of similar geometries withnon-coated or dielectric coated cylinders. Fig. 4(d) demonstrates that a significant fraction ofthe power carried along the fiber is concentrated at the dielectric/silver interface. We will showin Section 4 that plasmon resonances indeed strongly influence guidance properties of this typeof PCF.

4. Properties of coated PCFs

4.1. High-index dielectric coated inclusions

Resonances of coated cylinders can differ substantially from those of uncoated cylinders. Thishas been used by Stone et al [33], who have recently exploitedthe fact that higher order reso-nances of annular high-index regions are frequency shiftedcompared to solid high index rods todesign all-solid photonic bandgap fibers with reduced bend losses. Here, we illustrate the diver-sity of phenomena accessible with coated PCFs using the example of a solid core PCF similarto that in Fig. 1, but with three rings of identical inclusions. Each inclusion has a hollow core(air, ni = 1) and a high index coatingns = 1.6, with ρe = 1 µm andρi = 0.8 µm. The back-ground material has refractive indexne = 1.45. The average refractive index of the claddingis hence lower than that of the core, and one could naively expect the PCF to guide light in



Table 2. Real and imaginary parts ofneff and Wijngaard parametersWE and WH as afunction of truncation parameter M of the Fourier Bessel series. Single hexagon withpitch Λ = 1.5 µm of inclusions havingρe = 0.4 µm, ρi = 0.35 µm, ni = 1, ns =4.33480043837×10−1+ i8.70529497278,λ = 1.45µm andne = 1.45. Note that the valueused forns results from an interpolation of measured data for silver [32]. The valuegivenis the one used in our simulation, however it should be noted that the numberof significantdigits given here bears no relation to the accuracy with which the refractive index of silveris known at that wavelength.

M Re(neff) Im(neff) WE WH

1 1.3216268828523 1.0497034621751×10−2 0.115 0.1582 1.3185004345335 1.0142408514349×10−2 2.19×10−2 1.67×10−3

3 1.3184913660031 1.0225211849014×10−2 8.83×10−4 5.34×10−3

4 1.3185271981923 1.0238906035349×10−2 7.58×10−4 8.75×10−4

5 1.3185289649829 1.0238740992047×10−2 1.87×10−4 3.65×10−4

6 1.3185290956223 1.0238773184090×10−2 1.19×10−5 1.23×10−4

7 1.3185291032524 1.0238771574621×10−2 6.15×10−6 1.73×10−5

8 1.3185291033515 1.0238771546462×10−2 1.48×10−6 7.14×10−6

9 1.3185291034001 1.0238771555248×10−2 3.63×10−7 2.70×10−6

10 1.3185291034042 1.0238771553832×10−2 5.69×10−5 7.83×10−5

11 1.3185291019762 1.0238770394985×10−2 1.66×10−2 2.30×10−2

the core through “index guidance” or “modified total internal reflection.” [34] Figures 5 and 6respectively show the real and imaginary part of the effective index of the fundamental mode.For comparison, Figs. 5 and 6 also show the real and imaginarypart of the effective index ofa PCF with equivalent homogenized holes,ie with three rings of homogeneous inclusions ofradius 1µm and refractive index( f n2

i +(1− f )n2s)

1/2 ≃ 1.24964. This value is derived fromthe mean of dielectric constants (f = ρ2

i /ρ2e is the filling fraction of air in the cylinder), and

corresponds to the homogenized refractive index parallel to the fiber axis [35]; using this ho-mogenization for a single inclusion does not rely on any rigorous approach, and other choices(eg Maxwell-Garnett type formulas) could also be argued for [35]. Note that the scale on thehorizontal axis for both figures is reciprocal, so that frequencies are evenly spaced: the x-axis ontop of the figures indicates the normalized frequencyV, which we define relative to the coatingparameters:

V =2πλ

ρe(

n2e−n2

s

)1/2. (7)

Figure 5 also shows the imaginary part of the effective indexof the fundamental mode of aPCF with homogeneous inclusions having refractive index 1.6 (ARROW fiber), which guidespurely by antiresonant effects [5]. In Fig. 5 the green vertical lines mark the cutoff wavelengthof modes of a single coated inclusion, while in Fig. 6 we also plot the real part of the effectiveindex of leaky and guided modes of single coated inclusions.

At long wavelengths, losses of the coated PCF asymptotically follow those of the equiva-lent fiber with homogenized inclusions, demonstrating all characteristics of index guidance. Onthe contrary, the ARROW fiber losses diverge rapidly with increasing wavelength, as expectedfrom the ARROW model [36]. At shorter wavelengths however, guidance in the coated PCFexhibits all characteristics of ARROW guidance: losses peak at cutoffs of the modes of indi-vidual inclusions, with low loss bands in between cutoffs. Note that while the definition of anormalized frequency is standard for the ARROW fiber, there is no unique way of defining thenormalized frequency for the coated holes, so that we can’t directly compare frequency values



1 2 3 4 51E-12

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

8 7 6 5 4 3 2 1

Coated PCF Homogenized holes ARROW fibre

Single inclusion cutoffs

Imag

(nef

f)

Wavelength ( m)0.5

V

Fig. 5. Imaginary part ofneff as a function of wavelength for a PCF with high index coatedholes, for an equivalent PCF with homogenized inclusions and for a PCFwith inclusionshaving the refractive index of the coating (ARROW fiber). The three structures are detailedin the text. Green vertical lines denotes cutoffs of the modes of a single coated inclusion.The imaginary part ofneff for the ARROW fiber forV & 5 are too small to be computedwith the multipole method.

between the two cases. Propagation losses for the PCF with coated inclusions at their minima inhigher order bands are substantially lower than those of theequivalent PCF with homogenizedinclusions, and are also lower than those of a PCF with air holes of radiusρe or ρi (data notshown). From Fig. 6 it appears that loss peaks are due to avoided crossings of the core-guidedmode with leaky modes of the individual coated inclusions near cutoff, a characteristic featureof ARROW guidance [6,37,38].

The PCF with high-index coated holes hence exhibits two distinct guidance mechanisms,depending on wavelength. At long wavelengths, when the transverse wavelength becomes largeenough compared to the size of the inclusion for homogenization arguments to hold, the PCF isindex-guiding. As soon as the wavelength becomes small enough for resonances of the coatingto appear, the ARROW mechanism becomes the dominant waveguiding mechanism.

4.2. Metallic coated inclusions

By coating PCF holes with thin metallic layers, PCF guidancecan be combined with surfaceplasmon resonant effects, with the prospects of strong fieldlocalization and sharp resonances.Here we demonstrate excitation of a surface plasmon in a PCF with holes coated by a thinlayer of silver. The structure is that of Fig. 1, with only thesecond ring of holes being coated.Fig. 7 shows the real and imaginary parts of the effective index of the fundamental mode as afunction of wavelength. Aroundλ = 1.72µm an avoided crossing with another mode occurs.



8 7 6 5 4 3 2 1

0.5 1 1.5 2 2.5 3 4 51.4480

1.4482

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Homogenized holes Coated hole PCF single inclusion (leaky mode) single inclusion (guided mode)

Rea

l(nef

f)

Wavelength ( m)

Fig. 6. Real part ofneff as a function of wavelength for the fundamental mode of a PCFwith high index coated holes (red) and of an equivalent PCF with homogenized inclusions(blue). The real part ofneff of modes of a single coated inclusion are also shown; the latterare leaky for Re(neff) < ne = 1.45 and guided when Re(neff) > ne = 1.45. The structuresare detailed in the text.

Fig. 8 shows the field distributions around the avoided crossing. It appears clearly that the modecausing the avoided crossing is a surface plasmonic resonance of the ring of silver coated holes:the fields are strongly localized at the surface of the coatedinclusions. Fig. 9 shows a detail ofthe electric field across a coated inclusion. The fields are evanescent in the dielectric matrix aswell as in the air hole, and strongly decaying in the silver region, characteristic of a surfaceplasmon resonance.

The configuration studied here is somewhat reminiscent of the one used in dispersion com-pensating PCFs [39]: both use PCFs with an annular defect surrounding the core. In the case ofdispersion compensating PCFs, the second, third or fourth ring of holes around the core is mod-ified typically by reducing the hole-size, so that the ring can support guided modes. Avoidedcrossings of the fundamental core mode with modes of the ringdefect lead to strong groupvelocity dispersion values. The silver coated PCF studied here is very similar, except that thering mode is now a surface plasmon polariton. One could therefore expect a sharper resonance,and hence even larger values of the group velocity dispersion. Fig. 10 shows the chromatic dis-persion parameterD of the mode of the lower branch of Fig. 7; values ofD for the upper branchare similar, but of opposite sign. The unoptimized silver coated PCF achieves values ofD com-parable to the highest published values for non-coated designs [40], over a similar wavelengthrange. It is expected that optimization would lead to valuesof D up to an order of magnitudelarger, with comparable bandwidth. However, it must be noted that because of absorption in



1.4 1.5 1.6 1.7 1.81.30

1.32

1.34

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1.38

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1.42

1E-4

1E-3

0.01

0.1

1

Rea

l(nef

f)

Wavelength ( m)

ab

c d

Imag

(nef

f)

Fig. 7. Avoided crossing of the fundamental core mode and a plasmonicresonance in aPCF containing silver coated holes. The figure shows the real (upper curves) and imagi-nary (lower curves) of the two modes involved in the avoided crossing, as a function ofwavelength. The structure is that of Fig. 1, with the second ring of holes coated with 20nmof silver. Λ = 1.5µm, ρe = 262.5nm,ρi = 242.5nm,ne = 1.45, ni = 1.0 andns is wave-length dependent, with values resulting from interpolation of published data for silver [32].Letters (a-d) indicate the location of field plots in Fig 8.

the silver coating, losses are extremely high. The imaginary part shown in Fig. 7 correspondsto prohibitive losses of the core mode of 10dB/mm off-resonance (λ = 1.4µm, lower branch),of which 9dB/mm is due to absorption and the remaining 1dB/mmis confinement losses [41],and 136dB/mm at resonance (λ = 1.72µm, upper branch), of which 110dB/mm are due toabsorption. While confinement loss can be reduced by increasing the size of the inclusions oradding rings of holes, the plasmonic nature of the resonancerequires surface currents leading,in all but theoretical metals, to high absorption losses which can hardly be reduced.

Fig. 10 also shows the chromatic dispersion for the same PCF but with a coating thicknessof 30nm. Compared to the PCF with a coating thickness of 20nm,the resonant coupling tothe plasmonic mode has shifted considerably, and the strength of the resonance has also beenlargely modified. This proves how sensitive the plasmonic resonance is to the actual structure,a fact commonly usedeg in sensing to measure minute changes in refractive indices in theimmediate surroundings of a thin metallic film or particle [42]. This suggests that even with arange of propagation limited by absorption loss to a few tensof micrometers, a PCF with holeslocally coated at its tip could be used for all-optical sensing. Previous approaches to surfaceplasmon polariton optical fiber sensors required post processing of the fibers (such as tapering,etching or polishing) to be able to create a metal thin film near the core [42]. PCFs “naturally”provide holes near the core without any post-processing, limiting the required post-processingof the fiber to the deposition of a metallic thin film, potentially cutting costs and improvingrepeatability.



(a) (b)

(c) (d)

Fig. 8. Field distribution|E| of the mode on the upper and lower branch of Fig. 7 just beforeand just after the avoided crossing of the core mode with a plasmonic resonance: (a) lowerbranch,λ = 1.7µm; (b) upper branch,λ = 1.7µm; (c) lower branch,λ = 1.74µm; (d)upper branch,λ = 1.74µm. The color scale is the same as in Fig. 3. The green segment in(c) represents the coordinates along which the field is plotted in Fig. 9.



-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.60

2

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|E| (

arb.

uni

ts)

Radial position ( m)

Fig. 9. Norm of the electric field across the upper central coated cylinderin Fig. 8(c).

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D (p

s/nm

/km

)

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Coating Thickness 20nm 30nm

Fig. 10. Group velocity dispersion parameterD as a function of wavelength for the structureused in Fig. 7, with silver coatings in the second ring of thickness 20nm and 30nm.



5. Conclusion

The recently demonstrated ability of coating the holes of PCFs opens up a new dimension ac-cessible to PCF designers. We have given here a framework based on the multipole methodto explore the physics of these new PCFs further, along with tables for numerical compar-isons and calibration of other methods. Our preliminary study of some properties of PCFs withcoated inclusions in Section 4 already exhibited two novel phenomena not accessible to non-coated PCFs, namely the ability to have both index and ARROW guidance in a single PCF, andsurface plasmon polariton resonances. Although the latterhave very high absorptive losses,they may prove useful with short distances of coating at the tip of a fiber for localized sensingapplications.

Acknowledgements

This research was supported under the Australian Research Council’s (ARC) Discovery Project(Projects DP0665032 and DP0665923) and Centre of Excellence funding schemes. CUDOS isan ARC Centre of Excellence.



Multipole analysis of photonic crystal fibers with coated inclusions

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