## Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing

Optics Express, Vol. 15, Issue 16, pp. 10231-10246 (2007)

http://dx.doi.org/10.1364/OE.15.010231

Acrobat PDF (1612 KB)

### Abstract

A boundary integral method [1] for calculating leaky and guided modes of microstructured optical fibers is presented. The method is rapidly converging and can handle a large number of inclusions (hundreds) of arbitrary geometries. Both, solid and hollow core photonic crystal fibers can be treated efficiently. We demonstrate that for large systems featuring closely spaced inclusions the computational intensity of the boundary integral method is significantly smaller than that of the multipole method. This is of particular importance in the case of hollow core band gap guiding fibers. We demonstrate versatility of the method by applying it to several challenging problems.

© 2007 Optical Society of America

## 1. Introduction

2. P. Russell“Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

4. T.A. Burks, J.C. Knight, and P.S.J. Russell “Endlessly single-mode photonic crystal fibers,” Opt. Lett. **22**, 961–963 (1997). [CrossRef]

6. A. Ferrando, E. Silvestre, J.J. Miret, P. Andres, and M.V. Andres “Full vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–278 (1999). [CrossRef]

7. T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. **17**, 1093–1102 (1999). [CrossRef]

8. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method,” Opt. Fiber Technol. **6**, 181–191 (2000). [CrossRef]

9. K. Saitoh and M. Koshiba, “Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers,” IEEE J. Quantum Electron. **38**, 297 (2002). [CrossRef]

10. A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**, 1530–1532 (2002). [CrossRef]

11. X. Wang, J. Lou, C. Lu, C. L. Zhao, and W. T Ang, “Modeling of PCF with multiple reciprocity boundary element method,” Opt. Express **12**, 961–966 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-961 [CrossRef] [PubMed]

12. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. **21**, 1787–1792 (2003). [CrossRef]

13. T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightwave Technol. **21**, 1793–1807 (2003). [CrossRef]

14. H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express **12**, 3791–3805 (2004),http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-16-3791 [CrossRef] [PubMed]

15. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

16. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B **19**, 2331–2340 (2002). [CrossRef]

17. S. Campbell, R. C. McPhedran, and C. Martijn de Sterke “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B **21**, 1919–1928 (2004). [CrossRef]

18. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Coupling between two collinear air-core Bragg fibers,” J. Opt. Soc. Am. B **21**, 2095–2101 (2004). [CrossRef]

19. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10851–10864 (2006). [CrossRef] [PubMed]

17. S. Campbell, R. C. McPhedran, and C. Martijn de Sterke “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B **21**, 1919–1928 (2004). [CrossRef]

18. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Coupling between two collinear air-core Bragg fibers,” J. Opt. Soc. Am. B **21**, 2095–2101 (2004). [CrossRef]

19. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10851–10864 (2006). [CrossRef] [PubMed]

20. S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. **8**, 1225–1231 (2002). [CrossRef]

## 2. Mathematical formulation

*E*and

_{z}*H*of the modal electromagnetic fields, all the other field components can be deduced from Maxwells equations. In each of the homogeneous dielectric regions, individual longitudinal components satisfy the Helmholtz equations:

_{z}*γ*. Moreover, on the boundaries between various dielectric regions the longitudinal and tangential components of the fields are continuous:

^{2}_{c,g}=n^{2}_{c,g}-n^{2}_{e}*L*,

*ε*is the dielectric constant of either the inclusion or the cladding. Throughout the paper

_{(c,g)}*H*represents the true magnetic field scaled by the free space impedance

_{z}*H*=

_{z}*μ*

_{0}

*cH*.

^{true field}_{z}*r*⃗ in the fiber cross section, they can be represented by the following contour integrals (also known as the single layer potentials) [20

20. S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. **8**, 1225–1231 (2002). [CrossRef]

*e(r⃗*and

_{s})*h(r⃗*are called the potential densities [21] and their specification is sufficient to obtain the longitudinal field components. The function

_{s})*G*(

*r⃗,r⃗*) is the Green’s function of the Helmholtz equation in a uniform medium and it is given by:

_{s}*H*() is the zeroth-order Hankel function of the first kind. The value of

^{(1)}0*γ*and the contour of integration depend on the position

*r*⃗. If

*r*⃗ is located inside of the

*k*th inclusion then

*γ=γ*and integration is taken along the inclusion’s boundary

_{c}*L=L*. If

^{(k)}*r*⃑ is located in the fiber cladding then

*λ*=

*λ*and integration is taken along all the inclusion boundaries

_{g}*r*⃑ is located exactly at the inclusion boundary one can use any of the two definitions. We also note here that such a formulation satisfies the Reichardt condition [20

20. S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. **8**, 1225–1231 (2002). [CrossRef]

*λ*to be positive or negative.

_{g}*x=x*and

_{k}(s)*y=y*, s being a parameter such that 0≤

_{k}(s)*s*≤2

*π*, and

*k*being the inclusion number. Inserting Eqs. (3) into the continuity conditions (2), we obtain the following equations for any point⃑

*r*∈

_{s}*L*

^{(j)}:

*k*. This is a system of four coupled linear integral equations for the scalar contour functions

*e*=1…

^{(k)}_{c}, e^{(k)}_{g}, h^{(k)}_{c}, h^{(k)}_{g}, k*N*. The value of

_{c}*n*, for which a nontrivial solution of (5) exists, defines the effective refractive index of the fiber mode. Note, that the terms

_{e}*r*⃗

*→*

_{s}*L*[21]. Consider now the discretization of Eqs. (5). A direct discretization would run into difficulties as Hankel functions and their tangential derivatives become singular when

^{(j)}*s*′→

*s*. Therefore, such a singularity has to be first removed from the formulation.We start with the case of circular inclusions.

### 2.1. Circular inclusions

*J*=

^{(k)}(s)*a*, where

_{k}*a*is a radius of the

_{k}*k*th inclusion. Let

*ψ*represent any of the contour functions

^{(k)}(s′)*e*, or h

^{(k)}_{c}(s′), e^{(k)}_{g}(s′), h^{(k)}_{c}(s′)^{(k)}

_{g}(s′)and let st =

*t*=0,…,2

*n*

^{(k)}-1, be an equidistant grid. Since

*ψ*is a periodic function with a period 2

^{(k)}(s′)*π*, we can choose the trigonometric interpolation to approximate

*ψ*:

^{(k)}(s′)*n*. Expansion (6) is a key to our formulation of the boundary integral method. As it will be shown later, in comparison to the prior formulations that use Fourier expansion, using the trigonometric interpolation reduces considerably the numerical cost of resolving Eqs. (5). Consider first the integrals along the contour

^{(k)}*L*. These are the most problematic because of the presence of the singular point⃑

^{(j)}*r*. When interpolations of the form (6) are inserted into the integrals along the contour

_{s}′=⃑r_{s}*L*in (5), these integrals take the following form:

^{(j)}*(s,s′)*stands for

*G(s,s′)*,

*s,s*′) in (7) can be evaluated analytically according to the following formulas [23

23. S. V. Boriskina, P. Sewell, and T. M. Benson “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A **21**, 393–402 (2004). [CrossRef]

*J*() and

_{m}*H*() are the

^{(1)}_{m}*m*th order Bessel and the first kind Hankel functions, while a prime denotes the derivative with respect to the argument. The first two relations can be derived from the Graf’s addition theorem [24], while the third relation is simply the

*ψ(s′)*.

*ψ*defined on the corresponding equidistant grids

^{(k)}(st)*t*=0,…,

*2n*-1. Since ⃑

^{(k)}*r*is not on the contour

_{s}*L*, no singularities are present in such integrals. We now distinguish two cases.

^{(k)}*d*between ⃑

^{j,k}_{min}*r*and the contour

_{s}*L*is relatively small

^{(k)}*d*≲

^{j,k}_{min}*ak*, which results in a sharply peaked function Φ(

*s,s*′) with a width at a half maximum ~

*d*(see Fig. 2). As before, we approximate

^{j,k}_{min}/ak*ψ*with a trigonometric interpolation and obtain integrals of the form (7). In this case, however, no analytical formulas are available for the fourier transforms present in (7) and we evaluate them numerically by performing a Fast Fourier Transform (FFT). The accuracy of FFT should be higher than the accuracy of the trigonometric interpolation in order to achieve the smallest error. In fact, because of a cusp in the Green’s function the number of points used in FFT should exceed ~

^{(k)}(s′)*a*.

_{k}/^{dj},k_{min}*r*and the contour

_{s}*L*

^{(k)}is relatively large

*d*≳

^{j,k}_{min}*a*, and the functions Φ(

_{k}*s,s*′) may be considered as relatively smooth. When a large number of inclusions is present, this becomes the most common case. Here, for the integration of the function

*ψ*

^{(k)}(

*s*′)Φ(

*s,s*′), instead of (7) we apply a much simpler trapezoidal rule:

*n*is that it has to be large enough to resolve all the oscillations in the Green’s functions Φ(

^{(k)}*s,s*′) (see Fig. 2(b)). Particularly, from the functional form of the Green’s function (4) it is straightforward to demonstrate that the number of oscillations in the interval of

*s-s*′∈ (0,2

*π*) is ~

*k*(where

_{0}γ2a_{k}/(2π) ~ γa_{k}/λ*λ*is a wavelength of light). Therefore, the number of points in the boundary discretization has to be larger than the number of oscillations. In what follows, as an empirical rule, we use the trigonometric interpolation (6) when

*d*, while we use trapezoidal rule (9) when

^{j,k}_{min}<6ak*d*.

^{j,k}_{min}≥6a_{k}*X*has 4

*N*elements which are the values of

_{c}∑2n^{(k)}*e*defined on their proper discrete lattices

^{(k)}_{c}(s_{t}), e^{(k)}_{g}(s_{t}),h^{(k)}_{c}(s_{t}), h^{(k)}_{g}(s_{t})*t*=0,…,2

*n*-1, and

^{(k)}*k*=1,…,

*N*. The elements of matrix

_{c}*A*(

*n*) depend non-linearly on

_{e}*n*. Modal effective refractive indexes are defined by the values of

_{e}*n*for which the determinant of

_{e}*A(n*is zero. We note here that, as in [15

_{e})15. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

*X*and the corresponding matrix

*A(n*) could be cut in half by considering only the first two equations of (2) and using the other two to express

_{e}*e*and

_{c}(s_{t})*h*as a function of

_{c}(s_{t})*e*and

_{g}(s_{t})*h*at each inclusion. However, this results in more complex matrix elements.

_{g}(s_{t})### 2.2. Comparison of the code performance with that of a multipole method

**8**, 1225–1231 (2002). [CrossRef]

*e*and

_{g}(s′)*h*are specified and the Graf’s addition theorem is applied twice (once to derive the so-calledWijngaard expansion and once to transform the the origin of the cylindrical waves) the multipole formulation is obtained.

_{g}(s′)*H*) for some coefficients

_{z}*c*. Here

_{m}*M*is the multipole order and this field expansion has the same accuracy as the trigonometric interpolation (6) if

*M*≃

*n*is assumed. The Graf’s addition theorem involves an infinite series which for consistency must be truncated to the same order

*M*as the field expansion. This induces some error which is independent from the error associated with the field expansion. For well separated inclusions this last error is smaller than the error in the field expansion and the method works well. As the distance between inclusions decreases, the error associated with the truncated Graf’s series increases and at some point it becomes bigger than the error associated with the fields expansion. In such case, the multipole method starts to loose accuracy unless the order of the field expansion is increased accordingly.

*N*identical inclusions of diameter

_{c}*d*forming a periodic lattice of pitch

*λ*(see Fig. 1(a), for example). In this case the total number of unknowns is 2

*N*(2

_{c}*M*+1), and the number of matrix elements

*Size*

*(A(N*. One can show that to construct such a matrix one needs ~

_{e}))~N^{2}_{c}M^{2}*N*Bessel function evaluations. In the case when inclusions are well separated from each other, the truncated Graf’s series are rapidly convergent when

^{2}_{c}M*M*≳

*k*

_{0}

*λ*. However, when inclusions become too close to each other

_{g}d*d*→Λ convergence of the truncated Graf’s series is achieved only when

*M*≳

*d*/(Λ-

*d*).

*N*point discretization of every boundary, the number of unknowns in a system is 8

_{p}*N*. The number of matrix elements becomes

_{c}N_{p}*Size(A(N*. One can show that to construct such a matrix one needs ~ (

_{e}))~N^{2}_{c}N^{2}_{p}*const*) Bessel function evaluations. In the case when inclusions are well separated from each other, as established in the previous subsection, we only have to resolve all the oscillations in the Green’s function resulting in (

_{1}·N^{2}_{c}N_{p}+const2 ·N_{c}N_{FFT}*N*. When inclusions become too close to each other

_{FFT}~N_{p}) ≳ k_{0}γgd*d*→L the FFT order has to be high enough to resolve the cusp in the Green’s function

*N*≳

_{FFT}*d*/(L-

*d*). However, due to the conception of the method, the number of boundary discretization points will still remain small

*N*≳

_{p}*k0γ*resulting in a considerable performance improvement over the multipole method.

_{g}d#### 2.3. Arbitrary shaped inclusions

*:*

^{(k)}(s′)=y^{(k)}(s′)J^{(k)}(s′)*r*∈

_{s}*L*

^{(j)}, discretization of the integrals along the contours

*L*is done exactly as in the previous section, obtaining linear equations in terms of the ψ

^{(k)}_{k≠j}

^{(k)}*(s*by employing

_{t})*L*. When interpolations (11) are substituted into (5), the following expressions are obtained:

^{(j)}*s,s*′) is evaluated along the boundary of an arbitrary shaped inclusion. In this case no analytical formulas are available for the Fourier transforms of Φ

*(*

^{a}*s,s′*). Moreover, we can not use FFT efficiently because of the singularity at

*s=s*′. To circumvent this problem we introduce a regularization circle following [20

**8**, 1225–1231 (2002). [CrossRef]

*j*th inclusion, we consider a circle of a comparable diameter

*a*as shown schematically in Fig. 2(c). We further distinguish two cases. In the first case, Φ

_{j}*(*

^{a}*s,s*′) represents the Green’s function

*G*(

^{a}*s,s*′). Then, the integral in (12) can be expressed as:

*c*denotes the regularization circle. Second integral in the righthand side can be evaluated analytically. Function

*G*is not singular any more, and its fourier transform can be evaluated by using FFT. The value of

^{a}-G^{c}*G*when

^{a}-G^{c}*s*′→

*s*is given in the Appendix A. In the second case, Φ

*(*

^{a}*s,s*′) represents the normal

*denotes the corresponding normal or tangential derivative of the Green’s function on the circle, and*

^{c}*J*is the jacobian of the inclusion contour at

^{(j)}(s)*s*. Again, the second integral on the righthand side is evaluated analytically while for the non singular function

*s*′ →

*s*is given in the Appendix A.

#### 2.4. Finding the modes

*n*for which the determinant of

_{e}*A(n*in Eq.(10) is zero. This can be done in several ways. One approach is the integral root searching technique [25

_{e})25. R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. **14**, 73–83 (2004). [CrossRef]

15. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

*n*is expected to be much smaller

_{e}*n*is calculated numerically using first order finite difference scheme.

_{e}*A(n*smallest eigenvalue. When the determinant goes to zero, so does the smallest eigenvalue. We find that the smallest eigen value possess a wider convergence zone compared to that of the determinant. Furthermore, by working with the smallest eigenvalue, one avoids very large values typical for the determinants. Thus, for complex structures, we find the effective refractive index of a mode by performing root searching for the smallest eigenvalue of

_{e})*A(n*.

_{e})## 3. Study of the code accuracy for the simple test structures

**19**, 2322–2330 (2002). [CrossRef]

*d*=5

*µm*and a pitch Λ = 6.75µm. The glass cladding is assumed to have refractive index of

*n*=1.45, while the air holes have

_{g}*n*=1. The wavelength is

_{c}*λ*=1.45

*µm*. In the Table 3 we present convergence study of the effective refractive index of one of the low order modes as calculated by the boundary integral method, as well as comparison with a multipole method [15

**19**, 2322–2330 (2002). [CrossRef]

*n*; this is to be compared with the results of [14

_{e}14. H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express **12**, 3791–3805 (2004),http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-16-3791 [CrossRef] [PubMed]

*a*=5

*µm*, while the minor one is taken as

*b*=3

*µm*. Convergence data for one of the low order modes of this fiber is presented in [17

17. S. Campbell, R. C. McPhedran, and C. Martijn de Sterke “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B **21**, 1919–1928 (2004). [CrossRef]

**21**, 1919–1928 (2004). [CrossRef]

*n*) and two significant digits in the imaginary part.

_{e}19. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10851–10864 (2006). [CrossRef] [PubMed]

*n*=0.433480043837 +

_{m}*i*8.70529497278 at λ=1.45µm. Geometry of the structure is shown in Fig. 3(c); it consists of six coated cylinders with outer diameters

*d*=0.8

_{o}*µm*, inner diameters

*d*=0.7

_{i}*µm*, and a pitch Λ=1.5

*µm*. The glass cladding is assumed to have refractive index of

*n*=1.45, while the air holes have

_{g}*n*=1. The convergence analysis is presented in Table 4, including comparison with [19

_{c}**14**, 10851–10864 (2006). [CrossRef] [PubMed]

## 4. Demonstration of the code potential for the study of complex structures

### 4.1. Loss of the hollow core PCF featuring a large number of reflector layers

*µ*m, the hole diameter is

*d*=0.95Λ, and the core diameter is

*d*=2.5

_{c}*d*. The glass cladding is assumed to have refractive index of

*n*=1.45, while the refractive index of the air holes is

_{g}*n*=1. Dispersion curve for the fundamental core guided mode of this fiber is shown in Fig. 4(a) (dashed line). The minimum imaginary part, corresponding to the center of the bandgap, is obtained at

_{c}*λ*=1.51mm where we find

*n*=0.98451599741954+3.434721

_{e}*E*-8

*i*. For these calculations, the number of discretization points (2

*n*) per hole is chosen according to the following distribution: for the central hole

*n*=32, for the five rings of holes starting from the inner one we have taken respectively

*n*={16,16,14,12,10}. When

*n*is increased by 2, that is

*n*={34,18,18,16,14,12} the change in the value of

*n*equals to 2.6

_{e}*E*-9+5.4

*E*-10

*i*, signifying convergence of the small imaginary part. Convergence analysis is also performed for the order of FFT, which must be at least 256 to guarantee the low value of the overall error. This is a relatively high number and it increases the computational cost of the matrix elements, however note that we are dealing with a difficult case where the spacing between the inclusions is small. We also present in Fig. 4(b) radiation loss of the hollow core PCF as a function of the number of rings in the reflector. All the loss calculations are performed at a single wavelength

*λ*=1.51

*µm*. At this wavelength for the case of five rings we also show the |

*E*|, |

_{z}*H*| and

_{z}*S*of the fundamental mode in the outset of Fig. 4.

_{z}### 4.2. Large birefringence of a hollow elliptical core PCF

*µm*, the hole diameter is

*d*=0.9Λ, and the elliptical core has axis

*a*=2.3

*µm*and

*b*=4.6

*µm*. The glass cladding is assumed to have refractive index of

*n*=1.45, while refractive index of the air holes is

_{g}*n*=1. This structure is similar to the structure of a highly birefringent fiber proposed in [26

_{c}26. M. S. Alam, K. Saitoh, and M. Koshiba “High group birefringence in air-core photonic bandgap fibers,” Opt. Lett. **30**, 824–826 (2005). [CrossRef] [PubMed]

*λ*=1.41

*µm*. At the outset of this figure we show the

*S*fluxes for the

_{z}*x*and

*y*polarizations of the fiber fundamental mode at

*λ*=1.42

*µm*. The values for the effective refractive indices at this wavelength are as follows:

*n*=0.93903355+6.7418

^{x}_{e}*E*-4

*i*and

*n*=0.93816250+2.2133

^{y}_{e}*E*-3

*i*.

### 4.3. Loss birefringence of a MOF containing metal coated elliptical inclusions

**14**, 10851–10864 (2006). [CrossRef] [PubMed]

*n*=1.45, while the air holes have

_{g}*n*=1. The hole to hole pitch is Λ=1.5

_{c}*µm*. Six coated elliptical inclusions are described by the outer major axis

*a*=0.8

_{o}*µm*+δ,

*b*=0.8

_{o}*µm*-

*δ*and the inner major axis

*a*=0.7

_{i}*µm*+

*δ, b*=0.7

_{i}*µm*+

*δ*. In our simulations we use

*δ*=0.04

*µm*, which defines the hole ellipticity of

*a*-

*b*|/(

*a*+

*b*)=10%. We now characterize losses of the two fundamental mode polarizations as a function of the wavelength.

*δ*=0 (circular inclusions), both polarizations are degenerate. Here, loss curve of the fundamental mode is presented as dashed in Fig. 6. The wavelength of maximal loss ~1.41 corresponds to the point of phase matching of a core guided mode with a plasmon propagating on the interface between silver and glass. When ellipticity is introduced, wavelengths of phase matching of a fundamental core guided mode with a plasmon become somewhat different for the two polarizations. For example, for

*δ*=0.04

*µm*, dispersion

*x*-polarization the maximum of losses is at

*λ*=1.419

_{m}*µm*, while for the

*y*-polarization it is at

*λ*=1.407

_{m}*µm*. The corresponding

*S*fluxes are shown in the outset of Fig. 6. From the flux distributions it is clear that at the wavelengths of phase matching with a plasmon, core guided mode is well mixed with a plasmonic wave propagating on the silver-glass interface.

_{z}*λ*for the two polarizations of a fundamental core guided mode, one can envision detection of the hole ellipticity

_{p}*δ*̄. This principle can be used in pressure sensors. Thus, by starting with a fiber containing circular metallized inclusions and by compressing the fiber uniaxially one will induce ellipticity in the hole structure. Such an ellipticity can then detected by measuring splitting in the plasmon excitation wavelengths. To characterize sensitivity of a pressure sensor we define sensitivity as Sλ[nm]=

*S*=120

*nm*. Assuming that 0.1

*nm*shift between two plasmonic peaks can be resolved, ellipticity detection limit is estimated at 8·10

^{-4}.

## 5. Conclusion

## Appendix A: Normal and tangential derivatives of the Green’s functions

*rs(x*on a contour

_{s},y_{s})*L*described by the parametric expressions:

*x=x*(

*s*) and

*y=y(s)*,

*s*being a parameter such that 0≤

*s*≤2p. Let →

*r*′ (

_{s}*x*′,

_{s}*y*′) be an arbitrary point. Omitting the factor

_{s}*H*(

^{(1)}_{0}*k*), where

_{0}γR*R*=|→

*r*-→

_{s}*r*′|. The normal derivative at →

_{s}*r*is given by:

_{s}*s*, we obtain:

*x*′

*′*

_{s},y*), we obtain:*

_{s}*L*is a circle with radius

*a*and →

*r*′ is a point on that contour, the equations (15) and (16) take the following forms:

_{s}*κ(s)*being the curvature of

*L*at

*s*.

## Appendix B: Coated inclusions

*j*th inclusion which is coated with a material of the refractive index

*n*as shown schematically in Fig. 7. In this case an additional inner contour is present, and four additional contours functions must be specified:

_{m}*e*(→

*r*) and

_{s}*h*(→

*r*) at both contours on the coating side. When the boundary conditions are considered at the outer contour of this inclusion, Eqs. (5) should be modified. The first one, corresponding to the continuity of

_{s}*E*, becomes:

_{z}*L*denotes the outer contour,

^{(j)}_{o}*L*denotes the inner one,

^{(j)}_{i}*e*denotes the potential density at the outer contour and

^{(j)}_{o}*e*denotes the potential density at the inner one (both densities are defined on the coating side). Suppose that the contours are circular. Since →

^{(j)}^{i}*r*is on the outer boundary, the first integral in (20) is discretized by using the analytical formulas in (8) while the second integral is discretized by performing a FFT.

_{s}## References and links

1. | Matlab implementation of the code is available at http://www.photonics.phys.polymtl.ca/codes.html |

2. | P. Russell“Photonic crystal fibers,” Science |

3. | A. Bjarklev, J. Broeng, and A.S. Bjarklev “Photonic crystal fibers,” Kluwer Academic Publishers, Boston, (2003). |

4. | T.A. Burks, J.C. Knight, and P.S.J. Russell “Endlessly single-mode photonic crystal fibers,” Opt. Lett. |

5. | M.C.J. Large, L. Poladian, G.W. Barton, and M.A. van Eijkelenborg, “Microstructured Polymer Optical Fibres,” Springer, Sydney, (2007) |

6. | A. Ferrando, E. Silvestre, J.J. Miret, P. Andres, and M.V. Andres “Full vector analysis of a realistic photonic crystal fiber,” Opt. Lett. |

7. | T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. |

8. | F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method,” Opt. Fiber Technol. |

9. | K. Saitoh and M. Koshiba, “Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers,” IEEE J. Quantum Electron. |

10. | A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. |

11. | X. Wang, J. Lou, C. Lu, C. L. Zhao, and W. T Ang, “Modeling of PCF with multiple reciprocity boundary element method,” Opt. Express |

12. | N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. |

13. | T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightwave Technol. |

14. | H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express |

15. | T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

16. | B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B |

17. | S. Campbell, R. C. McPhedran, and C. Martijn de Sterke “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B |

18. | M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Coupling between two collinear air-core Bragg fibers,” J. Opt. Soc. Am. B |

19. | B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express |

20. | S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. |

21. | D. Colton and R. Kress “Integral equation methods in scattering theory,” John Wiley & Sons, New York, (1983). |

22. | R. Kress “Linear integral equations,” Springer-Verlag, New York, (1989). |

23. | S. V. Boriskina, P. Sewell, and T. M. Benson “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A |

24. | M. Abramowitz and I. A. Stegun “Handbook of mathematical functions,” Dover, New York, (1965). |

25. | R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. |

26. | M. S. Alam, K. Saitoh, and M. Koshiba “High group birefringence in air-core photonic bandgap fibers,” Opt. Lett. |

**OCIS Codes**

(230.4170) Optical devices : Multilayers

(230.7370) Optical devices : Waveguides

(290.4210) Scattering : Multiple scattering

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: May 25, 2007

Revised Manuscript: July 24, 2007

Manuscript Accepted: July 25, 2007

Published: July 30, 2007

**Citation**

Elio Pone, Alireza Hassani, Suzanne Lacroix, Andrei Kabashin, and Maksim Skorobogatiy, "Boundary integral method for the challenging problems in bandgap
guiding, plasmonics and sensing," Opt. Express **15**, 10231-10246 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10231

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### References

- Matlab implementation of the code is available at http://www.photonics.phys.polymtl.ca/codes.html
- P. Russell "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
- A. Bjarklev, J. Broeng and A.S. Bjarklev "Photonic crystal fibers," Kluwer Academic Publishers, Boston, (2003).
- T.A. Burks, J.C. Knight and P.S.J. Russell "Endlessly single-mode photonic crystal fibers," Opt. Lett. 22, 961-963 (1997). [CrossRef]
- M.C.J. Large, L. Poladian, G.W. Barton, M.A. van Eijkelenborg, "Microstructured Polymer Optical Fibres," Springer, Sydney, (2007)
- A. Ferrando, E. Silvestre, J.J. Miret, P. Andres and M.V. Andres "Full vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999). [CrossRef]
- T.M. Monro, D.J. Richardson, N.G.R. Broderick and P.J. Bennett "Holey optical fibers: an efficient modal model," J. Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
- F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, "Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method," Opt. Fiber Technol. 6, 181-191 (2000). [CrossRef]
- K. Saitoh,M. Koshiba, "Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," IEEE J. Quantum Electron. 38, 297 (2002). [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincent and M. Zoboli, "Holey fiber analysis through the finite element method," IEEE Photon. Technol. Lett. 14, 1530-1532 (2002). [CrossRef]
- X. Wang, J. Lou, C. Lu, C. L. Zhao andW. T Ang, "Modeling of PCF with multiple reciprocity boundary element method," Opt. Express 12, 961-966 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-961 [CrossRef] [PubMed]
- N. Guan, S. Habu, K. Takenaga, K. Himeno and A. Wada "Boundary element method for analysis of holey optical fibers," J. Lightwave Technol. 21, 1787-1792 (2003). [CrossRef]
- T. Lu and D. Yevick, "A vectorial boundary element method analysis of integrated optical waveguides," J. Lightwave Technol. 21, 1793-1807 (2003). [CrossRef]
- H. Cheng,W. Crutchfield,M. Doery, and L. Greengard, "Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory," Opt. Express 12, 3791-3805 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-16-3791 [CrossRef] [PubMed]
- T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke and L. C. Botten "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002). [CrossRef]
- B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran "Multipole method for microstructured optical fibers. II. Implementation and results," J. Opt. Soc. Am. B 19, 2331-2340 (2002). [CrossRef]
- S. Campbell, R. C. McPhedran, and C. Martijn de Sterke "Differential multipole method for microstructured optical fibers," J. Opt. Soc. Am. B 21, 1919-1928 (2004). [CrossRef]
- M. Skorobogatiy, K. Saitoh, and M. Koshiba, "Coupling between two collinear air-core Bragg fibers," J. Opt. Soc. Am. B 21, 2095-2101 (2004). [CrossRef]
- B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, "Multipole analysis of photonic crystal fibers with coated inclusions," Opt. Express 14, 10851-10864 (2006). [CrossRef] [PubMed]
- S. V. Boriskina, T.M. Benson. P. Sewell and A. I. Nosich "Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides," IEEE J. Sel. Top. Quantum Electron. 8, 1225-1231 (2002). [CrossRef]
- D. Colton and R. Kress "Integral equation methods in scattering theory," John Wiley & Sons, New York, (1983).
- R. Kress "Linear integral equations," Springer-Verlag, New York, (1989).
- S. V. Boriskina, P. Sewell and T. M. Benson "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004). [CrossRef]
- M. Abramowitz and I. A. Stegun "Handbook of mathematical functions," Dover, New York, (1965).
- R. Rodriguez-Berral, F. Mesa, and F. Medina, "Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides," Int. J. RF Microw. Comp. Eng. 14, 73-83 (2004). [CrossRef]
- M. S. Alam, K. Saitoh, and M. Koshiba "High group birefringence in air-core photonic bandgap fibers," Opt. Lett. 30, 824-826 (2005). [CrossRef] [PubMed]

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