1852 J. Opt. Soc. Am. B/Vol. 26, No. 10 /October 2009 Grujic et al.
Modelling of photonic crystal fiberbased on layered inclusions
Thomas Grujic,1,* Boris T. Kuhlmey,1 C. Martijn de Sterke,1 and Chris G. Poulton2
1Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), School of Physics A28,University of Sydney, Sydney, NSW 2006, Australia
2Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), Department of Mathematical Sciences,University of Technology, Sydney, NSW 2007, Australia
. INTRODUCTIONhotonic crystal fibers (PCFs) are optical fibers composedf a lattice of inclusions in a background matrix surround-ng a central core [1]. The majority of PCFs studied con-ist of a lattice of simple high-index dielectric rods or cy-indrical holes around a central core. More recently,owever, fibers based on inclusions with a more compli-ated structure have begun to be investigated involvingayered, cylindrically symmetric inclusions as shownchematically in Fig. 1(a). Examples include PCFs basedn an array of high-index rings around a central core [2,3]nd fibers involving graded index inclusions [4,5]. A me-allic coating on the insides of the holes of a PCF opens uphe possibility of exploiting very strong surface plasmonesonances with potential applications in sensing andery strong light confinement [3,6,7]. Accurate modellings required to realize the potential of such fibers.
Existing mode-finding algorithms capable of dealingith fiber structures of arbitrary design include finite-lement methods (FEMs) [8] and plane-wave expansionethods (PWEMs) [9]. FEM methods approximate Max-ell’s equations by a system of ordinary differential equa-
ions and require the PCF profile to be discretized spa-ially. PWEM methods expand the modal fields in a plane-ave basis to reduce the numerical mode-finding problem
o find the eigenvalues of a matrix. This generally re-uires a large number of plane waves to describe theodes accurately, especially for structures involving high
efractive index contrasts. Both methods have the advan-age of being applicable to arbitrarily shaped fiber pro-les.There exists also a class of somewhat less general nu-erical methods for microstructured optical fibers, the so-
alled integral equation methods [10–14]. These methods
re generally more efficient than FEM and PWEM be-ause they assume that the geometry is piecewise homo-eneous, but the boundaries between regions are arbi-rary.
However, methods tailored toward a specific fiber ge-metry are more accurate and are usually much fasterhan generalized methods. In cases with complicated in-lusions, the advantages are even more pronounced.
In this paper, we present a set of numerical tools toharacterize the properties of PCFs constructed from cy-indrically symmetric inclusions with arbitrary radialrofiles. A useful approach in studying PCFs is to investi-ate the properties of their claddings, which in them-elves share properties with infinite, periodic lattices ashown in Fig. 1(b) [15]. Light is guided in the core of a fi-er if the cladding acts like a mirror—that is, if no modesf the infinite lattice exist. A plot showing for which theombinations of wavelength � and effective index neffodes of the cladding exist (propagation diagram) en-
bles us to predict the spectral transmission of PCFs [16].e are therefore chiefly concerned with constructing
ropagation diagrams for infinite lattices of layered inclu-ions. Here, we are interested in two regimes—one wherescalar treatment of the electromagnetic fields is suffi-
ient, and another that accounts for the fully vectorial na-ure of both the electric and magnetic fields. We find thatn both cases a transfer matrix approach is a streamlineday to track field quantities across boundaries within the
nclusion.Our scalar approach builds on the ideas of Birks et al.
17], which allows an investigation of the properties of annfinite periodic system of homogenous dielectric rods oflightly higher refractive index than the background ma-erial. We generalize their method to treat a lattice of lay-
009 Optical Society of America
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Grujic et al. Vol. 26, No. 10 /October 2009 /J. Opt. Soc. Am. B 1853
red inclusions of an arbitrary number of layers with aeak index contrast between all component materials
typically �n�1%).To treat the fully vectorial case, we turn to the Rayleighultipole methods. This class of methods has been par-
icularly useful in modelling PCFs involving arrays of cy-indrically symmetric homogeneous rods or holes [18].
ultipole methods enable the calculation of quantitiesuch as the band structure (in the infinite case), projectedand diagrams, propagation diagrams, and the photonicensity of states [19,20].These semi-analytic methods expand field quantities in
Fourier–Bessel basis, the natural solutions to the Helm-oltz equation in cylindrical coordinates, making themuited to problems with cylindrical symmetries. A farmaller number of terms is then required in the expan-ion to capture the details of the field relative to a moreeneral algorithm such as a plane-wave method [21]. Theeneral idea behind the method has been adapted totudy numerous PCF geometries [19], but not to inclu-ions with an arbitrary number of layers.
Multipole methods require the scattering matrix of thenclusions, which relates the incoming fields in the vicin-ty of the inclusion to those scattered from it. The scatter-ng matrix for a homogeneous cylindrical rod or hole isell known and available in analytic closed form [22]. The
cattering matrix for an inclusion of uniform refractive in-ex but arbitrary cross section can also be calculated nu-erically, but at the cost of decreased accuracy and speed
23]. To treat the cases of graded index and coated inclu-ions using the multipole method, we must calculate thecattering matrix for rotationally symmetric but layerednclusions.
Common to both the derivation of the scattering matrixfor use in the multipole method) and the approximatecalar approach is the need to know the fields in everyayer of the inclusion. We use a transfer matrix [24] toink the fields in adjacent layers of the inclusion, an ap-roach that allows us to deal with an arbitrary number ofayers.
This paper presents methods to construct propagationiagrams in both the scalar and fully vectorial regimes.ection 2 defines the geometry and coordinate system de-cribing our infinite lattice and gives the general form ofhe fields in the lattice. Section 3 is a mathematical de-cription of the transfer matrix method we use to find theelds in all layers of the inclusion. Section 4 details ourxtension of the scalar method developed by Birks et al. to
ig. 1. (Color online) (a) Cross section of an example fiber whoseladding consists of an array of layered inclusions—in this case,ielectric rings representing materials with homogeneous refrac-ive indices. (b) Many of the spectral properties of the fiber followrom those of the associated infinite periodic lattice.
he case of layered inclusions. Section 5 uses the transferatrix formalism to derive the scattering matrix for a
ayered inclusion. Section 6 compares the propagationiagrams for a photonic crystal composed of graded indexnclusions generated by the approximate and fully rigor-us multipole methods. Section 7 presents simulations ofPCF consisting of an array of holes coated with a thinetallic layer.
. GEOMETRY AND FIELDEPRESENTATIONe consider the 3D propagation of waves through a peri-
dic array of layered parallel dielectric cylinders. In thisaper, we consider only hexagonal lattices, though our ap-roach is valid for other lattice types. The lattice geom-try and hexagonal unit cell are shown in Fig. 2. Theethods developed here do not apply to the case of inter-
enetrating inclusions—the outer boundary r=�N mustie entirely within the hexagonal unit cell boundary.
We orient a coordinate system with the z axis parallelo the cylinders and the lattice periodicity in the x-ylane. Within a particular unit cell, we look for solutionsf the form
E�r,�,z,t� = E�r,��exp�i��z − �t�� + c.c. , �1�
H�r,�,z,t� = H�r,��exp�i��z − �t�� + c.c., �2�
here � is the angular frequency related to the free-spaceavenumber k=2� /� via �=kc, and � is the propagation
onstant. We also define the effective index as neff=� /k.All field components satisfy the Helmholtz equation in
he homogenous material around the inclusion [25]. Here,e are concerned with two regimes—one that requires a
ully vectorial description of both the electric and mag-etic field and one where a scalar approach is valid. In theormer case, the translational invariance along the z di-ection means that all field components can be generatedrom the longitudinal components of the electric and mag-etic fields Ez and Hz [25]. In the latter, as described inore detail in Section 4, the magnitude of a single trans-
erse component (which we denote by ) is sufficient toescribe the fields. Using the fact that the fields are nec-
ig. 2. (Color online) (a) Infinite hexagonal lattice geometry,ith coordinate system indicated. (b) Unit cell of hexagonal lat-
ice of layered inclusions of pitch representing different com-onent materials. The index of the pth annulus is np, with the in-erface between the pth and �p+1�th layers at � .
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1854 J. Opt. Soc. Am. B/Vol. 26, No. 10 /October 2009 Grujic et al.
ssarily 2� periodic in the angular coordinate, it followshat an arbitrary field around the inclusion can be writtens a Fourier–Bessel series,
Vp�r,�� = �m=−�
�
�AmV,pJm�k�
p r� + BmV,pHm
�1��k�p r��exp�im��,
�3�
here V� �Ez ,Hz ,�. Jm denotes the Bessel functions ofhe first kind of order m, and the Hm
�1� are the Hankelunctions of the first kind of order m [26]. The index p la-els the particular layer of the inclusion for which theeld expansion is valid, with different expansion coeffi-ients Am
V,p, BmV,p in each layer. The quantity k�
p , the trans-erse wavenumber in each layer, is defined as
k�p = �k2np
2 − �2. �4�
e choose the positive root for the transverse wavenum-er outside the inclusion to ensure that fields decay awayxponentially in the background medium. The sign of k�
p
nside the layered inclusion is immaterial in the cases in-estigated in this paper, and we choose the positive root toe consistent. A full discussion on the appropriate choicef the sign of k�
p can be found in Appendix A of [27].Eq. (3) can be broken into two parts—the Bessel func-
ions of the first kind are regular for every argument andorrespond to fields with sources exterior to the inclusion,hile the Hankel functions diverge at zero and satisfy theutgoing wave equation. Physically, these two terms inhe expansion are linked via a scattering of incomingaves from the inclusion, which generates outgoingaves. The major aim of the following sections is to derivescattering matrix formalism linking the expansion coef-cients B associated with sources within the inclusion tohose coefficients A associated with fields sourced withinhe inclusion.
. TRANSFER MATRIXhis section introduces and motivates the transfer matrix
ormalism we use to relate field expansion coefficients indjacent layers.The field components parallel to each boundary be-
ween layers must be continuous, as required by Max-ell’s equations. In a fully vectorial approach there are
our such field components. In the scalar approximation25], which reduces the fully vectorial fields to a singlecalar field, we require the field and its normal derivativeo be continuous across each boundary. To solve for theeld everywhere in a layered inclusion of N layers for aiven azimuthal order m in Eq. (3) then requires solving aN�4N system in the fully vectorial case, or a 2N�2Nystem in the scalar approximation.
A more streamlined and versatile method, and one wese in this paper, is to link the fields on either side of an
nterface via a transfer matrix. More specifically, theelds are expanded in a Bessel function basis, as in Eq.3). The fields anywhere within a particular layer are con-rolled by the expansion coefficients of that layer. The cy-indrical symmetry of the layered inclusion means thatach order m in the expansion of Eq. (3) may be decoupled
nd treated one at a time. Vectors v containing the fourin the full vector case) or two (in the scalar case) expan-ion coefficients for a particular expansion order m in ad-acent layers are linked via a 4�4 or 2�2 transfer matrix. Symbolically,
vp±1m = Tp,p±1
m vpm. �5�
mportantly, a product of transfer matrices can be used toxpress the field coefficients in the center of an inclusionith N layers (region 1 in Fig. 2(b)) to those outside the
nclusion (region N+1):
v1m = �T2,1
m T3,2m . . . TN+1,N
m �vN+1m . �6�
he physicality of the fields requires the coefficients con-rolling the diverging part of the fields in the center of thenclusion [the Bm
V,1 in Eq. (3)] to be zero; otherwise, theeld would diverge at the center. In the symbolic notationf this section, specific elements of v1
m must be zero. Weive examples of the implementation of this condition inwo situations described below. Equations (5) and (6), to-ether with the condition that the fields remain finite inhe center, allow us to link the fields at any point withinhe inclusion with those outside it. Importantly for the ex-ension to the multipole method we present in Section 5,he finite field condition additionally implies a connectionetween the diverging field coefficients Bm and the con-erging field coefficients Am outside the inclusion. Ourransfer matrix approach is key to both the approximatecalar theory of the next section and the full multipoleethod of Section 5.
. APPROXIMATE MODEL TO CONSTRUCTROPAGATION DIAGRAMSere we present our extension to the method of Birks etl. [17] to generate the edges of photonic bands on aropagation diagram for a low-index contrast structure.he particular approximations and assumptions underly-
ng the method may be found in [17].A propagation diagram consists of bands of modes of
he infinite lattice separated by bandgaps. To construct aropagation diagram, it is only necessary to map out theop (the maximum value of neff in a band for a given wave-ength) and bottom of the band (minimum value of neff inhe band for a given wavelength). It is known that the topnd bottom of a band correspond to Bloch states with theost bonding and anti-bonding character, respectively.he corresponding boundary conditions at the edge of thenit cell are =0 and d /ds=0, respectively, where s is aoordinate normal to the unit cell boundary [17]. Thesewo conditions correspond to particular vectors vN+1
�AN+1,BN+1�T,
=−Hm
�1���k�N+1rUC�
Jm� �k�N+1rUC�
,1�T
, for d/ds = 0, �7�
=−Hm
�1��k�N+1rUC�
Jm�k�N+1rUC�
,1�T
, for = 0, �8�
here, following Birks et al., we approximate the hexago-al unit cell to a circle with radius r (given in Appendix
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Grujic et al. Vol. 26, No. 10 /October 2009 /J. Opt. Soc. Am. B 1855
). Comparison with rigorous calculations (which weresent in Section 6) validates the approximation in theeak-index contrast limit. This approximation makes it
traightforward to impose the boundary conditions on n the perimeter of the unit cell [17].
Propagation diagrams are constructed by imposing thewo boundary conditions corresponding to the top and bot-om of a band and enforcing continuity of the scalar field
and its derivative d /dr at each interface between ad-acent layers of the inclusion through the application ofn appropriate 2�2 transfer matrix Ti,i±1
2�2 , whose form isiven explicitly in Appendix A:
A1
B1� = T2,1
2�2T3,22�2 . . . TN+1,N
2�2 AN+1
BN+1� . �9�
quation (9) gives the explicit form of the vector v in Eq.6), in the scalar case. Requiring that the fields be finite athe center of the inclusion (in this case, B1=0), allows uso solve for the two effective indices neff=� /k lying at theop and bottom of a band for a particular wavelength.arying the wavelength and repeating the procedureaps out the edges of the bands.Birks et al. treated analytically the case of a single
olid rod in each circular unit cell. An analogous, purelynalytic approach is impossible for the case of an arbi-rary number of layers. A transfer matrix approach en-bles any number of layers to be treated, simply by mul-iplying the appropriate transfer matrices.
As demonstrated in Section 6, the approximate band-dge finding algorithm works well in the limit of small re-ractive index contrasts ��n�1% � but breaks down out-ide this regime. A more general method is required toonstruct propagation diagrams for periodic lattices ofayered inclusions, and to this end we introduce our ex-ension to the Rayleigh multipole method in the next sec-ion.
. EXTENSION OF THE RAYLEIGHULTIPOLE METHOD
n this section, we extend the multipole method to enableodelling of an infinite, perfectly periodic lattice com-
osed of cylindrically symmetric inclusions of arbitraryndex profile.
The multipole method involves two essential ingredi-nts [18]—a description of how electromagnetic fields arecattered from an individual inclusion and a descriptionf the structure of the lattice. Our extension to theethod then involves only a modification of the formalism
escribing scattering from a simple cylinder to the case ofcattering from a cylindrically symmetric layered inclu-ion. We therefore do not trace the entire derivation of theultipole method, but show only the derivation of the
cattering matrix for a cylindrical inclusion with an arbi-rary number of layers.
Armed with our scattering matrix, the multipoleethod allows us to formulate a condition to determinehether a mode of the infinite lattice exists for a particu-
ar combination of wavelength �, Bloch vector k0, and ef-ective index n .
eff
erivation of the Scattering Matrixo describe scattering from an inclusion, we must relatehe diverging field coefficients Bm
V of Eq. (3) outside the in-lusion (controlling the outgoing waves), to the converg-ng field coefficients Am
V , also outside the inclusion. In ma-rix notation, we need to find the scattering matrix Sm,efined by
BmE,N+1
BmH,N+1� = SmAm
E,N+1
AmH,N+1� . �10�
cattering from an inclusion depends on the geometry ofhe inclusion, its refractive index profile, the propagationonstant, and the operating wavelength. The derivation ofhe scattering matrix requires knowledge of the field inll layers of the inclusion, particularly in the center.As in Eq. (6), we obtain the coefficients in the center of
he inclusion in terms of those outside [for a particular or-er m in the expansion of Eq. (3)] as
A1
E
0
A1H
0� = T2,1
4�4T3,24�4 . . . TN+1,N
4�4 AN+1
E
BN+1E
AN+1H
BN+1H
� T4�4 AN+1
E
BN+1E
AN+1H
BN+1H
� .
�11�
ere, Ti,i+14�4 is a 4�4 transfer matrix, and Eq. (11) gives
he explicit structure of the symbolic vector v of Eq. (5) inhe fully vectorial case. The explicit form of Ti,i+1
4�4 is givenn AppendixB. It links a vector of multipole coefficients onne side of a cylindrical boundary to the corresponding co-fficients on the other side for a particular order m.
The physicality of the fields requires B1E, B1
H=0; other-ise, the field would diverge at the center. Equation (11)
hen yields two equations in only the four multipole coef-cients outside the cylinder. These can be solved to givehe form of the scattering matrix Sm defined in Eq. (10).he procedure is illustrated in Appendix B.Knowledge of the scattering matrix for layered inclu-
ions is then sufficient to use the existing multipole for-alism to find the modes of finite or infinite arrays of lay-
red inclusions. A change of basis operator K is used toxpress the fields scattered from all other inclusions inhe local coordinates of a particular inclusion in the lat-ice. Conservation of the total ingoing and outgoing fieldsn the vicinity of the inclusion then allows a self-onsistent matrix equation to be written for the field ex-ansion coefficients [18].Symbolically, a mode of the system exists when
�I − SK�B P��,k0,neff�B = 0. �12�
ere, S is a scattering matrix combining the Sm for all or-ers m. K contains lattice sums describing the structuref the lattice in the case of an infinite periodic lattice28,29]. B is a vector holding the diverging field coeffi-ients of all orders for a single unit cell (in the infinitease), or for all inclusions (in the finite PCF case). Aropagation diagram for an infinite lattice can be con-tructed by scanning the Bloch vector k0 over the irreduc-ble Brillouin zone perimeter for a given effective �� ,n �
eff
ct
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1856 J. Opt. Soc. Am. B/Vol. 26, No. 10 /October 2009 Grujic et al.
ombination [30]. The existence of a mode corresponds tohe determinant of P being zero.
. NUMERICAL SIMULATIONSn this section, we compare the performance of the ap-roximate band-finding theory of Section 4 with that ofhe fully vectorial multipole method of Section 5. Thetructure we consider is a hexagonal array of pitch 6.7 m made up of graded index cylinders in a silicaackground. The characteristics of the inclusions matchhose of a fiber whose spectral properties were investi-ated numerically and experimentally by Kuhlmey et al.5]. In that work, differences between theoretical and ex-erimental results were in part attributed to the inabilityo accommodate both the material dispersion and an ac-urate description of the index profile of the cylinders inumerical simulations. The methods developed in this pa-er allow us to evaluate this claim. Each cylinder has araded refractive index distribution above that of theackground silica, given by
n�r� = �nsilica�1 + �nGI�1 − �r/r0����, if r � 0
nsilica, if r � r0� , �13�
here r is the distance from the cylinder’s center, ��4.7,nGI�0.0203, r0=1.5927, and nsilica is the wavelength-ependent index of fused silica, obtained using a Sell-eier expansion. This refractive index profile is the black
ashed curve in Fig. 3.To generate propagation diagrams for this structure,
e discretize the graded index profile, as illustrated byhe step index function in Fig. 3. This is done by dividinghe inclusion into N concentric layers of equal refractivendex steps �n, whose radii r̄i are calculated by taking thequare root of the average of the square of the radial co-rdinate r weighted by the index distribution over eachndex step:
r̄i =� 1
�n�ni
ni+1
r2�n�dn. �14�
ere, ni and ni+1 are the refractive indices on the edges ofach index step. Such an area-weighted discretization en-
0 0.1 0.2 0.3 0.4 0.50.995
1
1.005
1.01
1.015
1.02
1.025
r/Λ
n(r)
/nsi
lica
ig. 3. (Color online) Graded index profile of each cylinder inhe hexagonal infinite lattice (black curve) with an example dis-retization into four layers.
ures there are more steps approximating the graded pro-le where it varies most rapidly with r, while also taking
nto account the cylindrical geometry of the inclusion.
. Approximate Band Edgeso obtain accurate band edges using the approximateheory of Section 4, we must use an adequately largeumber of approximating layers N. To establish criteriaor constructing accurate propagation diagrams, we calcu-ate band edges for a number of different approximatingayers, as in Fig. 4.
The wavelength range we use corresponds to that in-estigated by Kuhlmey et al. Each band in the propaga-ion diagram corresponds to a particular m in the field ex-ansion of Eq. (3) and therefore consists of modes with aarticular azimuthal field dependence.We see that simply approximating the graded index
rofile by a single rod �N=1� or even a double-step indexistribution �N=2� is not sufficient to capture accuratelyll details of the band structure. The band edges for N8 and N=10 overlap to within the thickness of the linessed to plot them. Subsequent graded index inclusionimulations therefore used N=8 approximating layers.he dotted curves in Fig. 5, to be discussed below, are the=8 band edges generated by the approximate method.
. Full Multipole Propagation Diagramse now use the fully vectorial multipole formalism of Sec-
ion 5 to construct rigorous band diagrams for our struc-ure of interest. The scattering matrix Sm in Eq. (10) isalculated using eight approximating layers, and fielduantities were expanded in eleven terms (correspondingo m ranging from −5 to 5).
The solid black regions of Fig. 5 show where modes ofhe infinite lattice exist, as found by the multipoleethod. The white regions between the black photonic
ands show where the bandgaps lie in neff−� space. Theurves show the results of a commercial plane-wave nu-erical package. The density of these lines gives some
imited information on the density of photonic states. Weote that our formulation can also calculate detailed den-ity of states data, though we are here only interested in
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−2
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ig. 4. (Color online) Band edges for an infinite lattice of gradednclusions, generated by the scalar method of Section 4, for dif-erent numbers of approximating discretized layers N. Here nbgs the wavelength dependent background index of silica, obtainedsing a Sellmeier expansion [31].
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Grujic et al. Vol. 26, No. 10 /October 2009 /J. Opt. Soc. Am. B 1857
he combinations of effective index and wavelength forhich a bandgap exists (i.e., when the structure supportso modes).The two photonic bandgaps beneath the light line neff
nbg between bands can be used to guide a core mode in anite photonic crystal fiber. The gray dashed curves inig. 5 are core modes calculated for a fiber consisting ofhree rings of graded inclusions around a central solidore that were formed by removing an inclusion. Thehite dots are the cladding modes of the same fiber and
oincide almost exactly with the photonic bands of the in-nite unperturbed lattice.While these results, which take into account both the
ctual nature of the graded index inclusion profile and theispersion of silica, differ appreciably from the numericalesults previously obtained by Kuhlmey et al. [5], thegreement with experimental results is not improved.he discrepancy may be due to other factors, such as an
naccurate characterization of the experimental fiber’sroperties, in particular the exact index profile of each in-lusion, and the pitch of the acoustic grating used.
Figure 5 shows that the approximate theory gives veryood agreement with the full multipole simulations, asxpected in this limit of a weak index contrast betweenhe maximum index of the inclusion and the backgroundndex of silica. In weak contrast situations where only theositions of the bands and bandgaps are required, it isuch faster to use the approximate approach than a fully
ectorial method.However, the assumption that the electromagnetic
elds can be treated as scalar quantities breaks down inystems with a higher contrast between the permittivitiesf the component materials, rendering the approximateethod inaccurate. The multipole formalism deals wellith inclusions consisting of concentric layers with high
efractive index contrasts between them, as it does notuffer from convergence issues inherent in other ap-roaches such as the plane-wave class of methods.
ig. 5. (Color online) Comparison of approximate band-edgending algorithm (curves) with fully vectorial multipole method
black solid regions). Also shown are the results of a commerciallane-wave package (lines). White dots are the cladding modes of
finite fiber consisting of three layers of graded inclusionsround a central missing inclusion. Gray dashed curves are coreodes for this finite fiber.
. ANALYSIS OF THIN METAL COATINGSn example of a situation involving a very high contrast
n the permittivity of component materials is that of a lat-ice of air-filled thin metallic rings in a silica background.hotonic crystal fibers involving metals open up the pos-ibility of exploiting surface plasmon polariton reso-ances, with potential applications in sensing and opto-lectronic components [32].
We simulate structures similar to those studied bothxperimentally by Schmidt et al. [33] and numerically byoulton et al. [6] , who studied the properties of an infi-ite hexagonal lattice of solid metallic nanowires. We in-tead model lattices of air-filled thin metal rings of vary-ng thicknesses in a silica background. Following Poultont al., we take the permittivity of the rings �m=−125.3 toe that of silver at the vacuum wavelength 1.55 m andonstruct propagation diagrams by varying the scale ofhe structure rather than the wavelength. Additionally,e neglect material absorption �Im��m�=0�. The exten-
ion of our method to the case of a structure involving ma-erials with a finite imaginary component to their permit-ivities is straightforward but beyond the scope of thisaper, as discussed in the conclusion. The permittivity ofhe background silica at this wavelength is �s=2.085. Wex the outer ring radius at b=0.15 and vary the inneradius a.
The photonic bands and bandgaps for a number of dif-erent ring thicknesses are shown in Fig. 6 in terms oformalized frequency and the quantity �neff−nbg�k0��−nbgk0�. The white regions indicate where modes of
he infinite lattice exist, and the gray areas in betweenre the photonic bandgaps. We remark that propagationiagrams are not appreciably different from those for theorresponding lattice of solid metal rods until the ringhickness becomes comparable to the skin depth of theetal. All three propagation diagrams in Fig. 6 corre-
pond to such a regime in which the ringlike nature of thenclusions is important.
Bound modes of single thin metallic rings in a silicaackground are also plotted as dots in Fig. 6 and are cutff at neff=nbackground. For an infinite periodic lattice, thelasmonic modes of the individual rings couple together,roaden into the white bands, and extend below the cutoff34].
Figure 6(a) indicates the azimuthal order m of theodes of the isolated rings that broaden into the three
ands visible for neff�nbackground. Between ring radius ra-ios a /b=0.95 and 0.98, we see a transition in the behav-or of the m=1 band of modes. For a /b=0.95, the m=1
ode of the isolated metal ring broadens into a photonicand near k0=0, while for a /b=0.98 a bandgap haspened up at low frequencies. Figure 6(b) shows the inter-ediate behavior.We can explain this transition by realizing that in the
ow-frequency limit, the wavelength is much greater thanhe overall size of individual inclusions. The details of theings cannot be resolved by the fields, and we may treathe inclusions as a uniform material with a homogenizedermittivity. Below a critical ring thickness, the homog-nized permittivity of the inclusions passes from negativeo positive. Plasmonic modes can only exist when there isdifference in sign between the permittivities of adjacent
mf
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poip
tblgo
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cidlsms
mocpimtmoem
Fnnai
1858 J. Opt. Soc. Am. B/Vol. 26, No. 10 /October 2009 Grujic et al.
aterials, and so the bandgap apparent in Fig. 6(c) at lowrequencies opens up below this critical ring thickness.
In the high-frequency limit, the dispersion curves of theings approach those of equivalent thin metallic slabshose widths are equal to the relevant ring thickness,ounded by air on one side and silica on the other.
ig. 6. (Color online) Propagation diagrams for infinite hexago-al lattices, pitch =1 m, of metallic rings of varying thick-esses. Dots indicate the bound modes of single metallic rings insilica background, which exist for neff�nbg. Metal permittivity
s �m=−125.4 and background permittivity is 2.085.
. DISCUSSION AND CONCLUSIONhe methods presented in this paper allow fast, accuratealculation of the properties of photonic crystal fiberased on cylindrically symmetric layered inclusions. Thepproximate method of Section 4 allows us to map out thehotonic band edges on a propagation diagram for infi-ite, periodic structures with weak refractive index con-rasts. Within the domain of applicability, the advantagesn using this method over the computationally intensive
ultipole method to generate band edge information rap-dly are marked, requiring minutes (rather than hours) toocate the positions of the bands. It is then possible to rap-dly predict the wavelength ranges where core modes of anite layered PCF exist.The ability to discretize an inclusion with a radially de-
endent refractive index profile into an arbitrary numberf layers, as in the case of the smoothly varying gradedndex cylinders of Section 6, allows us to treat PCFs com-osed of such inclusions accurately.For fibers with a higher permittivity contrast, the vec-
orial nature of the electromagnetic fields can no longere ignored. The scattering matrix derived in Section 5 al-ows efficient, accurate generation of propagation dia-rams and calculation of core mode dispersion in the casef finite PCF.
In common with existing multipole algorithms thatolve simple circular inclusion problems, the circularymmetry of the inclusions is built in, making a multipolepproach to the simulation of layered inclusion PCF veryfficient and accurate when compared with algorithms ca-able of simulating more general index profiles. The effi-iency becomes further pronounced as more layers aredded, as the element of the multipole method that re-uires the most computational resources is the evaluationf lattice sums, with the scattering matrix calculation re-uiring far less computing power. Coupled with the facthat high permittivity contrasts between layers presento additional numerical convergence issues, this makesur multipole formulation particularly well suited to mod-lling PCFs composed of cylindrically symmetric inclu-ions.
The numerical analysis of PCF involving thin metallicoatings in Section 7 required the same number of termsn the field expansions as did the modelling of the all-ielectric graded index fiber of Section 6. The extremelyarge permittivity contrasts present in metal–dielectrictructures present appreciable convergence difficulties inore general approaches, such as the plane-wave expan-
ion method [22].Though we have here studied only the case of losslessetals (i.e., metals with purely real permittivities) at one
perating wavelength, the multipole formalism easily ac-ommodates more general dispersive materials with com-lex permittivities. The effects of material dispersion arencluded easily as the multipole algorithm searches for
odes at a fixed frequency, which is then varied acrosshe simulation window. To include the effects of loss, theode-finding algorithm, which involves finding the roots
f the determinant of the matrix P in Eq. (12), must bextended in this case to search over complex values of theodal effective index rather than just purely real ones.We also note that while the multipole results presented
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Grujic et al. Vol. 26, No. 10 /October 2009 /J. Opt. Soc. Am. B 1859
n this paper give only a binary indication of whether orot one or more modes exist at a given point in a propa-ation diagram, density of states information can be gen-rated by counting the number of times Eq. (12) is satis-ed while tracing around the perimeter of the irreduciblerillouin zone.
PPENDIX Ahis appendix presents a transfer matrix method to maput the band edges on a propagation diagram for an infi-ite lattice of layered dielectric inclusions in the limit of aeak refractive index contrast. The assumption of a cir-
m m
ei
S
Sce
F==
ular unit cell with radius rUC= ��3/2��1/2 (this choicereserves the filling fraction) is identical to that of Birkst al. [17].
The scalar field has the form given in Eq. (3). We iso-ate one particular order m= l in the expansion for theeld in region p (different m yield modes with differentzimuthal dependence):
�p = �AlpJl�k�
p r� + BlpHl�k�
p r��exp�il��. �A1�
equiring that both � and d� /dr be continuous across annterface at radial coordinate r=�p, we can link the vectorf field coefficients �ApBp�T on one side of the interface tohose on the other �Ap+1Bp+1�T via a 2�2 transfer matrix2�2 :
i,i+1
Ti,i+12�2 =
1
k�i+1�Jl�y�Hl
�1���y� − Jl��y�Hl�1��y��
�k�i+1Hl
�1���y�Jl�x� − k�i Hl
�1��y�Jl��x� k�i+1Hl
�1���y�Hl�1��x� − k�
i Hl�1��y�Hl
�1���x�
k�i Jl�y�Jl��x� − k�
i+1Jl�x�Jl��y� k�i Jl�y�Hl
�1���x� − k�i+1Hl
�1�Jl��y�� ,
�A2�
here x=k�p �p and y=k�
p+1�p. To find the photonic banddges, we implement the boundary conditions discussedn Section 4 on the edge of our circular unit cell. For in-tance, the top of the band corresponds to a zero deriva-ive d /dr at r=rUC, which corresponds to the particular
hoice �AN+1BN+1�T= �−Hl�1���k�
N+1� /Jl��k�N+1�1�T. The trans-
er matrix is then used to repeatedly step across dielectricnterfaces, as in Eq. (9).
PPENDIX Be derive here the explicit form of the 4�4 transfer ma-
rix Ti,i+14�4 defined in Eq. (11). Our approach is similar to
hat of Yeh and Yariv [24], though we use a basis of Besselunctions of the first kind and Hankel functions of therst kind, while Yeh and Yariv used a basis with Besselunctions of the second kind in place of Hankel functions.
The field components tangential to a cylindrical inter-ace between layers in an inclusion (the z and � compo-ents) must be continuous. The multipole expansion coef-cients Am
V , BmV generate the longitudinal components Vz
f the electromagnetic fields, and we can generate the azi-uthal components from the longitudinal components us-
ng Maxwell’s equations as
E� =i�
k2np2 − �2� �
r��Ez −
�
�
�
�rHz� , �B1�
H� =i�
k2np2 − �2� �
r��Hz +
�
�
�
�rEz� . �B2�
his allows us to write continuity relations for the fourangential field components across an interface in termsf the multipole expansion coeffcients AV , BV .
The continuity of the longitudinal component of thelectric field Ez across a boundary between regions i and+1 at radial coordinate r=� is then written:
ApEJm�k�,i�� + Bp
EHm�k�,i�� = Ai+1E Jm�k�,i+1��
+ Bi+1E Hm�k�,i+1��. �B3�
imilarly, continuity of E� gives the following relation:
1
�k�p �2� im
��Ap
EJm�k�p �� + Bp
EHm�k�p ���
−�k�
p
��Ap
HJm� �k�p �� + Bp
HHm� �k�p ���� , �B4�
=1
�k�i+1�2� im
��A2
EJm�k�i+1�� + B2
EHm�k�i+1���
−�k�
i+1
��A2
HJm� �k�i+1�� + B2
HHm� �k�i+1���� . �B5�
imilar relations hold for the longitudinal and azimuthalomponents of the magnetic field. The four resultingquations can then be written in the matrix form
M�i,�� Ap
E
BpE
ApH
BpH� = M�i + 1,��
Ai+1E
Bi+1E
Ai+1H
Bi+1H� . �B6�
inally then, the transfer matrix can be found as Ti,i+14�4
M−1�i+1,��M�i ,��. Explicitly, defining x=k�p � and y
ki+1� and writing T4�4 as
� i,i+1
w
Wfe
Tofi
w
ATsEE
R
1
1
1
1
1
1
1
1
1
1860 J. Opt. Soc. Am. B/Vol. 26, No. 10 /October 2009 Grujic et al.
Ti,i+14�4 =
�y
2 �t11 t12 t13 t14
t21 t22 t23 t24
t31 t32 t33 t34
t41 t42 t43 t44
� , �B7�
e obtain the elements as
t11 = Jl�x�Hl�1���y� − �k�
i+1�p/k�p �i+1�Jl��x�Hl
�1��y�,
t12 = Hl�1��x�Hl
�1���y� − �k�i+1�p/k�
p �i+1�Hl�1���x�H�1��y�,
t13 = �i�l/��i+1��1/y − k�i+1/�xk�
p ��Jl�x�Hl�1��y�,
t14 = �i�l/��i+1��1/y − k�i+1/�xk�
p ��Hl�1��x�Hl
�1��y�,
t21 = �k�i+1�p/k�
p �i+1�Jl��x�Jl�y� − Jl�x�Jl��y�,
t22 = Hl�1��x�Hl
�1���y� − �k�i+1�p/k�
p �i+1�Hl�1���x�H�1��y�,
t23 = �i�l/��i+1��1/y − k�i+1/�xk�
p ��Jl�x�Hl�1��y�,
t24 = �i�l/��i+1��1/y − k�i+1/�xk�
p ��Hl�1��x�Hl
�1��y�,
t31 = �i�l/���k�i+1/�xk�
p � − 1/y�Jl�x�Hl�1��y�,
t32 = �i�l/���k�i+1/�xk�
p � − 1/y�Hl�1��x�Hl
�1��y�,
t33 = Jl�x�Hl�1���y� − �k�
i+1/k�p �Jl��x�Hl
�1��y�,
t34 = H�1��x�Hl�1���y� − �k�
i+1/k�p �H�1���x�Hl
�1��y�,
t41 = �i�l/���1/y − k�i+1/�xk�
p ��Jl�x�Jl�y�,
t42 = �i�l/���1/y − k�i+1/�xk�
p ��Hl�1��x�Jl�y�,
t43 = �k�i+1/k�
p �Jl��x�Jl�y� − Jl�x�Jl��y�,
t44 = �k�i+1/k�
p �Hl�1���x�Jl�y� − Hl
�1��x�Jl��y�.
e now show how the scattering matrix Sm is obtainedrom Eq. (11). Writing the elements of T4�4 as Tij, we canxpand the matrix equation of Eq. (11) to obtain
T21AN+1E + T22BN+1
E + T23AN+1H + T24BN+1
H = 0, �B8�
T41AN+1E + T42BN+1
E + T43AN+1H + T44BN+1
H = 0. �B9�
hese equations only involve field expansion coefficientsutside the inclusion and can be solved for the divergingeld coefficients in terms of the converging coefficients as
BE
BH� =1
�T24T41 − T44T21 T24T43 − T44T23
T42T21 − T22T41 T42T23 − T22T43�AE
AH� SmAE
AH� , �B10�
here �=T22T44−T42T24.
CKNOWLEDGMENTShis research was supported under the Australian Re-earch Council’s (ARC) Discovery Project and Centre ofxcellence funding schemes. CUDOS is an ARC Centre ofxcellence.
EFERENCES1. J. C. Knight, “Photonic crystal fibers,” Nature 424, 847–851
(2003).2. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight,
A. K. George, and D. M. Bird, “An improved photonicbandgap fiber based on an array of rings,” Opt. Express 14,6291–6296 (2006).
3. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran,“Multipole analysis of photonic crystal fibers with coatedinclusions,” Opt. Express 14, 10851–10864 (2006).
4. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight,and D. M. Bird, “Bend loss in all-solid bandgap fibers,” Opt.Express 14, 5688–5698 (2006).
5. B. T. Kuhlmey, F. Luan, L. Fu, D. I. Yeom, B. J. Eggleton,A. Wang, and J. Knight, “Experimental reconstruction ofbands in solid core photonic bandgap fibers using acousticgratings,” Opt. Express 16, 13845–13856 (2008).
6. C. G. Poulton, M. A. Schmidt, G. J. Pearce, G.Kakarantzas, and P. S. J. Russell, “Numerical study ofguided modes in arrays of metallic nanowires,” Opt. Lett.32, 1647–1649 (2007).
7. J. Hou, D. Bird, A. George, S. Maier, B. Kuhlmey, and J. C.Knight, “Metallic mode confinement in microstructuredfibers,” Opt. Express 16, 5983–5990 (2008).
8. C. T. Chan, Q. L.Yu, and K. M. Ho, “Order-N spectralmethod for electromagnetic waves,” Phys. Rev. B 51,16635–16642 (1995).
9. K. M. Leung and Y. F. Liu, “Full vector wave calculation ofphotonic band structures in face-centered-cubic dielectricmedia,” Phys. Rev. Lett. 65, 2646–2649 (1990).
0. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada,“Boundary element method for analysis of holey opticalfibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
1. T. L. Wu and C. H. Chao, “Photonic crystal fiber analysisthrough the vector boundary-element method: effect ofelliptical air hole,” IEEE Photon. Technol. Lett. 16,126–128 (2004).
2. H. Cheng, W. Crutchfield, M. Doery, and L. Greengard,“Fast, accurate integral equation methods for the analysisof photonic crystal fibers. I: Theory,” Opt. Express 12,3791–3805 (2004).
3. E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M.Skorobogatiy, “Boundary integral method for thechallenging problems in bandgap guiding, plasmonics, andsensing,” Opt. Express 15, 10231–10246 (2007).
4. A. Hochman and Y. Leviatan, “Efficient and spurious-freeintegral-equation-based optical waveguide mode solver,”Opt. Express 15, 14431–14453 (2007).
5. T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlesslysingle-mode photonic crystal fiber,” Opt. Lett. 22, 961–963(1997).
6. P. S. J. Russell, T. A. Birks, J. C. Knight, and B. J. Mangan,“Photonic crystal fibers,” US Patent 6,990,282 (2006).
7. T. A. Birks, G. J. Pearce, and D. D. M. Bird, “Approximateband structure calculation for photonic bandgap fibers,”Opt. Express 14, 9483–9490 (2006).
8. L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A.Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P.White, D. P. Fussell, and B. T. Kuhlmey, From MultipoleMethods to Photonic Crystal Device Modelling (CRC Press,
2005).
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
Grujic et al. Vol. 26, No. 10 /October 2009 /J. Opt. Soc. Am. B 1861
9. L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A.Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H.Smith, and T. N. Langtry, “Rayleigh multipole methods forphotonic crystal calculations,” PIER 41, 21–60 (2003),doi:10.2528/PIER02010802.
0. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan,C. M. de Sterke, and N. A. Nicorovici, “Density of statesfunctions for photonic crystals,” Phys. Rev. E 69, 016609(2004).
1. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C.Botten, C. M. de Sterke, and R. C. McPhedran, “Multipolemethod for microstructured optical fibers. II.Implementation and results,” J. Opt. Soc. Am. B 19,2331–2340 (2002).
2. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre,G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipolemethod for microstructured optical fibers. I. Formulation,”J. Opt. Soc. Am. B 19, 2322–2330 (2002).
3. S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C.Botten, “Differential multipole method for microstructuredoptical fibers,” J. Opt. Soc. Am. B 21, 1919–1928 (2004).
4. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J.Opt. Soc. Am. 68, 1196–1201 (1978).
5. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Academic, 1983).
6. M. Abramowitz and I. A. Stegun, Handbook ofMathematical Functions with Formulas, Graphs, andMathematical Tables (Dover, 1964), ninth Dover printing,tenth GPO printing ed.
7. B. Kuhlmey, “Theoretical and numerical investigation of
the physics of microstructured optical fibers,” Ph.D. thesis,University of Sydney and Université Aix-Marseille III(2006). http://setis.library.usyd.edu.au/adt/public html/adt-NU/public/adt-NU20040715.171105/.
8. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, and K. A.Grubits, “Lattice sums for gratings and arrays,” J. Math.Phys. 41, 7808–7816 (2000).
9. S. K. Chin, N. A. Nicorovici, and R. C. McPhedran, “Green’sfunction and lattice sums for electromagnetic scattering bya square array of cylinders,” Phys. Rev. E 49, 4590–4602(1994).
0. T. P. White, R. C. McPhedran, L. C. Botten, G. Smith, andC. M. de Sterke, “Calculations of air-guided modes inphotonic crystal fibers using the multipole method,” Opt.Express 9, 721–732 (2001).
1. G. P. Agrawal and R. W. Boyd, Nonlinear Fiber Optics(Springer, 2001).
2. P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R.Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D. J.Won, F. Zhang, and E. R. Margine, “Microstructured opticalfibers as high-pressure microfluidic reactors,” Science 311,1583–1586 (2006).
3. M. A. Schmidt, L. N. P. Sempere, H. K. Tyagi, C. G.Poulton, and P. S. J. Russell, “Waveguiding and plasmonresonances in two-dimensional photonic lattices of gold andsilver nanowires,” Phys Rev B 77, 033417 (2007).
4. J. Laegsgaard, “Gap formation and guided modes inphotonic bandgap fibers with high-index rods,” J. Opt. A 6,798–804 (2004).
