行政院國家科學委員會專題研究計畫 期中進度報告

適於光電研究的先進有限差分相關數值方法與模型之發展(2/3)

期中進度報告(精簡版)

計 畫 類 別 ：個別型

計 畫 編 號 ： NSC 95-2221-E-002-380-

執 行 期 間 ： 95年 08 月 01 日至 96年 07 月 31 日

執 行 單 位 ：國立臺灣大學光電工程學研究所

計 畫主持人：張宏鈞

報 告 附 件 ：出席國際會議研究心得報告及發表論文

處 理 方 式 ：期中報告不提供公開查詢

中 華 民 國 96年 05 月 31 日

行政院國家科學委員會專題研究計畫期中進度報告 適於光電研究的先進有限差分相關數值方法與模型之發展(2/3) Development of Advanced Finite Difference Based Numerical

Methods and Models for Photonics Research (2/3) 計畫編號：NSC 95-2221-E-002-380

執行期限：95年 8月 1日至 96年 7月 31日 主持人：張宏鈞 台灣大學電機系、光電所暨電信所教授

計畫參與人員：江柏叡(台大光電所) 陳明昀(台大光電所)

林彥宏(台大光電所) 張峰偉(台大電信所)

江舜凡(台大光電所)

摘要

本計畫發展數種有限差分相關的頻域與時域數值模型，以做為研究各種光導波元件的工具。第二年至目前的執行情形，依有限差分時域法模型之發展、以有限差分頻域法計

算三維光子晶體與各向異性光子晶體能隙結構、譜方法模型之發展與應用、包含光子晶

體光纖之特殊光波導之分析與研究等四項研究課題，分述於本報告。

關鍵詞：光波導、光子晶體、光子晶體波導、波導元件、有限差分頻域法、有限差分時

域法、時域譜方法、頻域譜方法

Abstract

This research concerns development of several finite difference (FD) based frequency-domain and time-domain numerical models as tools for studying various photonic guided-wave devices. In this report we describe the research results up to date in the second year of this three-year project, including the development of the finite-difference time-domain (FDTD) method, the analysis of 3-D photonic crystal and anisotropic photonic crystal band structures using the finite-difference frequency-domain (FDFD) method, the development and applications of pseudo-spectral method models, and the analysis and investigation of photonic crystal fibers and special-type optical waveguides.

Keywords: Optical waveguides, photonic crystals, photonic crystal waveguides, optical waveguide devices, finite-difference frequency-domain method, finite-difference time-domain method, pseudo-spectral time-domain method, pseudo-spectral frequency method

I. Development of the FDTD Method

We have established both two-dimensional (2-D) and 3-D FDTD numerical models for studying various photonic structures. For 3-D FDTD modeling, the simulation is quite CPU time consuming and memory demanding and going for parallel computing is a natural way. In the first year, we had set up a personal computer cluster in our laboratory for such simulation and had made our 3-D FDTD code a parallel computing model [1]. We have continuously extended the capacity of our models to handle more general and more complicated structures, for example, the inclusion of material dispersion models and the treatment of nonlinear optics problems. We briefly show two numerical examples in the following.

1

The first example relates to the subwavelength imaging using an array of silver nanorods, as reported in [2]. The left panel of Fig. 1 is Fig. 1 of [2], which shows the nanorod-array structure and the FDTD simulation results of the image profiles at different places due to point sources shaped as the letter “λ” as an object. The rod height is 50 nm, the rod diameter is 20 nm, and the side-to-side distance between the rods is 40 nm. The right panel of Fig. 1 shows our 3-D FDTD simulation results corresponding to (f) and (g) places in the left panel. Our FDTD is with the Drude dispersion model and good agreement with {2} is seen. Fig. 2(a) is from Fig. 2 of [2], showing the light intensity distribution along one rod, and the blue line in Fig. 2(b) is our calculation. Again, excellent agreement with [2] is observed.

(Left panel) (Right panel)

Fig. 1 Left panel: Fig. 1 of [2]. Right panel: simulation results using our 3-D FDTD model.

(a) (b) Fig. 2 (a) From Fig. 2 of [2]: light intensity distribution along one rod. (b) The blue line is from our

FDTD calculation. The second example relates to the FDTD simulation of nonlinear medium phenomena.

2

We examine our established model using a four-wave mixing problem in the same way as was recently done in [3]. The simulation arrangement is as shown in Fig. 3(a), where uniform plane pump and signal waves are incident from a linear medium upon a nonlinear medium with 3rd-order instantaneous Kerr nonlinearity represented by the nonlinear susceptibility χ0(3) = 1.0 x 10-18 m2/V2, The linear index of both media is 1.2. The pump wave is at 192 THz and the signal wave is at 195 THZ, with their electric field amplitudes being 5.1 x 107 V/m and 5.1 x 106 V/m, respectively. The output spectrum of the electric field obtained from our FDTD simulation is given in Fig. 3(b), showing converted waves at 189 and 198 THz, which is consistent with the results of [3] and the analytical four-wave mixing theory.

(a) (b) Fig. 3 (a) FDTD simulation arrangement for a four-wave mixing problem. (b) The output spectrum

of the electric field obtained from our FDTD simulation.

II. Analysis of 3-D Photonic Crystal and Anisotropic Photonic Crystal Band Structures Using the FDFD Method

We have extended our FDFD photonic crystal solver based on the Yee mesh [4] to the analysis of 3-D photonic crystals (PCs) as well as PCs involving anisotropic materials. We have demonstrated that the FDFD method can be efficiently utilized to calculate the band diagrams of 3-D PC structures [5], [6]. We have considered two structures: the dielectric spheres in air and the scaffold structure [5]. When the index averaging scheme is used, the obtained band diagrams agree well with those calculated by the plane-wave-based transfer matrix method and the plane-wave expansion method. The established FDFD method can treat PCs with diagonal permittivity tensor. Please refer to Appendix I [5] for the detail. We have also developed an Yee-mesh-based FDFD method for calculating the band structures of 2-D PCs containing anisotropic materials [7]. As a numerical example, we have analyzed a square-lattice PC composed of circular cylinders of nematic liquid-crystal (LC) material with silicon as background. The LC director is assumed to be perpendicular to the PC extension direction so that the E and H waves are decoupled. The tenability of the band-gap size has been studied and symmetry properties of the gap size versus the orientation of the LC director observed [7]. Please refer to Appendix II [7].

III. Development and applications of pseudo-spectral method models

We have been working on the development of electromagnetic numerical models based on pseudo-spectral methods both in frequency domain and in time domain. A novel analysis

3

method based on a multidomain pseudospectral method has been proposed for calculating the band diagrams of 2-D PCs and is shown to possess excellent numerical convergence behavior and accuracy [8]. The proposed method shows uniformly excellent convergence characteristics for both the transverse-electric and transverse-magnetic waves in the analysis of different structures. The analysis of a mini band gap is also shown to demonstrate the extremely high accuracy of the proposed method. The first page of the paper published in Physical Review E [8] is attached as Appendix III.

In the development of the pseudospectral time-domain (PSTD) simulation method, the Drude dispersion model has been incorporated into the code for treating problems involving metallic materials in the optical frequency domain. In related work, optical properties of a metallic sphere based on the classical Drude model have been studied theoretically. The comparison between the metallic sphere and the perfectly conducting one shows conspicuous distinction in the electric field amplitude on the surface of the sphere and in the radar cross section behavior [9]. Please refer to Appendix IV [9].

IV. Analysis and Investigation of Photonic Crystal Fibers and Special-type Optical Waveguides

The FDFD waveguide eigenmode solver based on the Yee mesh [4] is a versatile method for the analysis of different optical waveguides. We have used it to study the propagation characteristics and chromatic dispersion coefficients of photonic crystal fibers [10]–[12]. We have recently applied the FDFD solver to investigate modal properties of guided modes on air-core terahertz waveguides with the air core surrounded by Teflon rods. The antiresonant reflection optical waveguide (ARROW) model was successfully adopted to explain the guiding behavior [13]. Please refer to Appendix V [13]. Based on such FDFD analysis, we have proposed a very simple procedure for numerical calculation of chromatic dispersion coefficients of photonic crystal fibers, which involves Chebyshev-Lagrange interpolation polynomials [14]. No direct numerical differentiation of the effective refractive index is needed. Please refer to Appendix VI [14].

References

[1] M. F. Chen and H. C. Chang, “Finite-Difference Time-Domain Analysis of Three-Dimensional Photonic-Wire Bends via Parallel Computing,” in Proceedings of Optics and Photonics Taiwan ’05 (OPT ’05) (CD-ROM), paper B-SA-IV10-5, Tainan, Taiwan, R.O.C., December 9–10, 2005.

[2]

[3]

[4]

A. Ono, J. Kato, and S. Kawata, “Subwavelength optical imaging through a metallic nanorod array,” Phys. Rev. Lett., vol. 95, 267407, 2005.

M. Fujii, C. Koos, C. Poulton, I. Sakagami, J. Leuthold, and W. Freude, “A simple and rigorous verification technique for nonlinear FDTD algorithms by optical parametric four-wave mixing,” Microwave and Optical Technology Letters, vol. 48, pp. 88–91, 2006.

C. P. Yu and H. C. Chang, “Chapter 7: Applications of the Finite Difference Frequency Domain Mode Solution Method to Photonic Crystal Structures (50 pages),” in Electromagnetic Theory and Applications for Photonic Crystals (448 pages), edited by Kiyotoshi Yasumoto, pp. 351–400. Boca Raton, Florida: Marcel Dekker/CRC Press, Inc., 2006. (ISBN: 0849336775)

[5] Y. H. Lin, C. P. Yu, and H. C. Chang, “Calculation of Three-Dimensional Photonic-Crystal Band Diagrams Using the Finite-Difference Frequency-Domain Method,” in Optics and Photonics Taiwan ’06 (OPT ’06) (CD-ROM), paper AO-61, Hsinchu, Taiwan, R.O.C., December 15–16, 2006.

H. C. Chang, Y. H. Lin, C. P. Yu, and M. M. Chen, “Finite Difference Analysis of 3D Anisotropic Photonic Crystals, in Technical Digest, PECS-VII: International Symposium on Photonic and Electromagnetic Crystal Structures VII (CD ROM), paper A21, Monterey, California, April 8–11, 2007.

[6]

M. M. Chen, S. M. Hsu, and H. C. Chang, “Analysis of 2-D Photonic Crystals Involving Liquid Crystals Using the Finite-Difference Frequency-Domain Method,” in OSA 2006 Integrated Photonics and

[7]

4

Nanophotonics Research and Applications (IPNRA ’07) Technical Digest (CD ROM), paper IMB3 (3 pages), Salt Lake City, Utah, July 8–11, 2007.

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of Two-Dimensional Photonic Crystals Using a Multidomain Pseudospectral Method,” Physical Review E, Vol. 75, pp. 026703-1–026703-14, 20 February 2007.

[8]

B. Y. Lin, C. H. Teng, and H. C. Chang, “Near-Field and Far-Field Behavior of a Metallic Nanoscale Sphere at Optical Frequencies Based on the Classical Drude Model,” accepted for presentation at URSI Commission B – International Symposium on Electromagnetic Theory (EMTS 2007), Ottawa, ON, Canada, August 26–28, 2007.

[9]

[10] H. C. Chang, C. P. Yu, P. J. Chiang, and S. M. Hsu, “Development and Applications of High-Accuracy Optical Waveguide Eigenmode Solvers,” presented at International Workshop on Photonics and Display Technologies, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., September 22–23, 2006. (Invited paper)

[11] H. C. Chang, “High-Accuracy Analysis of New-Type Optical Waveguide Structures,” Optics and Photonics Taiwan ’06, Hsinchu, Taiwan, R.O.C., December 15–16, 2006. (Invited paper)

[12] H. C. Chang and C. P. Yu, “Finite-Difference Numerical Analysis of Photonic Crystals and Photonic Crystal Fibers,” presented at Physical Society R.O.C. 2007 Annual Meeting, ChungLi, Taiwan, R.O.C., January 23–25, 2007. (Invited paper)

[13] C. P. Yu and H. C. Chang, “Air-Core Waveguides for Terahertz Signals,” in OSA 2006 Integrated Photonics and Nanophotonics Research and Applications (IPNRA ’07) Technical Digest (CD ROM), paper ITuH4 (3 pages), Salt Lake City, Utah, July 8–11, 2007.

[14] P. J. Chiang, C. P. Yu, and H. C. Chang, “Robust Calculation of Chromatic dispersion Coefficients of Optical Fibers from Numerically Determined Effective Indices Using Chebyshev-Lagrange Interpolation Polynomials,” IEEE/OSA Journal of Lightwave Technology, Vol. 24, No. 11, pp. 4411–4416, November 2006.

5

CALCULATION OF THREE-DIMENSIONAL PHOTONIC-CRYSTAL BAND DIAGRAMS

USING THE FINITE-DIFFERENCE FREQUENCY-DOMAIN METHOD

Yen-Hung Lin (林彥宏), Chin-Ping Yu (于欽平),and Hung-Chun Chang* (張宏鈞) Graduate Institute of Electro-Optical Engineering, National Taiwan University,

Taipei, Taiwan 106-17, R.O.C. *also with Department of Electrical Engineering and

Graduate Institute of Communication Engineering, National Taiwan University Phone: +886-2-23635251-513, Fax: +886-2-23638247, E-mail: hcchang@cc.ee.ntu.edu.tw

(NSC94-2215-E-002-022 and NSC95-2221-E-002-380) Abstract --- We adopt the finite-difference frequency-domain method to determine the 3D photonic crystal

band structures. The band diagrams of two different kinds of structures are efficiently obtained with good agreement with results from other methods.

Keywords: Finite-difference frequency-domain method, photonic band gap, photonic crystals.

INTRODUCTION Photonic crystals (PCs) have many applications, including the routing and guiding of light in all three

dimensions, due to the photonic band gaps (PBGs) resulting from periodic structures [1]. A simple cubic (sc) lattice structure is symmetrical with respect to x, y, and z directions and therefore is the most natural platform for constructing photonic crystal devices [2]. Besides, theoretical investigation suggests that it is possible to create such an sc structure with a complete PBG [3]. In this paper, we develop a finite-difference frequency-domain (FDFD) method based on the Yee mesh to calculate the band diagrams of three-dimensional (3D) PCs. Two-dimensional version of the FDFD method has been used to calculate the band structures of 2D PCs [4]. We will consider two kinds of 3D structures. The first one is dielectric spheres in air, and the second is scaffold structures, as shown in Fig. 1(a) and (b), respectively. The band diagrams of these two structures are obtained and compared with other numerical methods and experimental results.

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Hz( i , j +1,k+0.5)

Hx( i +0.5,j +1,k)

Hx( i +0.5,j +1,k+1)

(a) (b) (c)

Figure 1. (a) The 3D sc photonic crystal structure composed of dielectric spheres in air. (b) The 3D sc photonic crystal structure composed of scaffold structures. (c) 3D Yee’s mesh for the FDFD method.

NUMERICAL METHOD Since 3D PCs are periodic along three orthogonal directions, the propagation in all directions must be

taken into account, i.e., the three phase constants kx, ky, and kz may all have non-zero values. Starting from Maxwell’s curl equations and using the central difference scheme with 3D Yee’s mesh shown in Fig. 1(c), we can obtain the following six equations with six electromagnetic components as

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εωε

εωε

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where ω is the angular frequency, ε0 and µ0 are the permittivity and permeability of free space, respectively, and εx, εy, and εz are the relative permittivities at the corresponding grid points. Here, we have assumed the material of the PC has permeability µ0, but can have a diagonal permittivity tensor. Therefore, more general anisotropic materials can be considered by equations (1)−(6). Those equations can then be written into the matrix form. After some mathematical work, we can obtain an eigenvalue equation as

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where k02 = ω2µ0ε0, 0 is the zero matrix, εxx, εyy, and εzz are diagonal matrices representing εx, εy, and εz at the corresponding grid points, and Ux, Uy, Uz, Vx, Vy, and Vz are square matrices determined by (1)−(6) and the periodic boundary conditions at the edge of the computing window. By finding out the eigen frequencies of (7), we can construct the band diagrams of the PC structures.

NUMERICAL PROCESS AND RESULTS The first 3D PC under study is the sc structure containing dielectric spheres in air. The sphere has a refractive index of n = 3.4 and a radius of r = 0.2a with a being the lattice constant of the PC. The unit cell of the structure is shown in Fig. 2(a). The reciprocal lattice of this PC and some high-symmetry points are schematically displayed as the inset of Fig. 2(b). The band diagram is plotted along the high-symmetry line Γ-X-M-R. For this simple cubic structure, we have carried out the FDFD calculation to get the eigen frequencies of (7). Numerical computation of the photonic band structure uses 16×16×16 grid points. In order to solve the discontinuity at dielectric interfaces, the relative permittivity ε(i, j, k) at each mesh point is determined by the index averaging scheme through ε(i, j, k) = fa εa + (1－fa) εd where fa is the filling fraction of air in the mesh, and εa and εd are the relative permittivities of air and the dielectric medium, respectively. The band diagram of this structure is shown in Fig. 2(b) with the dashed lines being our results from the FDFD method and the dots being from the plane-wave-based transfer-matrix method (TMM) [5]. It can be seen that our results agree quite well with those of [5]. However, no complete band gap can be found in this structure.

a

r

r=0.3an=3.4

0

0.1

0.2

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0.4

0.5

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mal

ized

freq

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y (ω

a/2π

c)

(a) (b)

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Plane-wave-based TMM [5]FDFD method

kx

ky

kz

kx

ky

kz

X

RM

Γ

We proceed further to consider a more complex 3D PC structure: the scaffold PC which consists of layers of interconnected square rods joined by vertical posts to form a sc lattice. The reciprocal lattice of the scaffold structure is the same as the inset of Fig 2(b). The unit cell of the scaffold structure is shown in Fig. 3(a), and the width of the square rod is w = 0.13950a. The dielectric filling fraction is 19% and the refractive index of dielectric materials is n = 3.6. By applying the FDFD method with 16×16×16 grid points in the unit cell and the index averaging scheme for the permittivity at each grid point, the band diagram of this structure can then be obtained as the dashed lines as shown in Fig. 3(b). The triangular dots in Fig. 3(b) are from the plane-wave-expansion (PWE) method [6], and the dots are from experimental results [2].

a

w=0.13950an=3.6

w

0

0.1

0.2

0.3

0.4

0.5

Γ X M R

Experimental results [2]

FDFD method (this work)

PWE method [6]Nor

mal

ized

freq

uenc

y (ω

a/2π

c)

(a) (b)

Figure 3. (a) The unit cell of the 3D sc PC containing the scaffold structure. (b) The band diagram of the PC of (a) with dashed lines from the FDFD method, triangular dots from the PWE method [6], and the dots from experimental results [2].

The lowest six bands are presented in Fig. 3(b) with bands 1 and 2 being the photonic valence bands (VBs), and bands 3 to 6 being the photonic conduction bands (CBs). One can see a complete band gap in Fig. 3(b), which is defined by the VB maximum at the R point and the CB minimum at the X point. The lower bound of the band gap is ω1 = 0.367, and the upper bound is ω2 = 0.390. In our numerical calculation, there is a visible difference between our results and those of [6] if the discontinuity of dielectric interfaces is not taken into account. By adopting the index averaging scheme, our results match well with those obtained by the PWE method [6], as shown in Fig. 3(b). Besides, from Fig. 3(b), it is seen that both our FDFD results and those of the PWE method are close to the trend of the dots from experimental results [2] at the X-M region in the VB region.

CONCLUSION We have demonstrated that the FDFD method can be efficiently utilized to calculate the band diagrams of

3D PC structures. We have considered two structures: the dielectric spheres in air and the scaffold structure. When the index averaging scheme is used, the obtained band diagrams agree well with those calculated by the plane-wave-based TMM and the PWE method. The established FDFD method can treat PCs with diagonal permittivity tensor.

REFERENCES [1] E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett.

58, pp. 2059-2062 (1987). [2] S. Y. Lin, J. G. Fleming, and R. Lin, "Complete three-dimensional photonic bandgap in a simple cubic

structure," J. Opt. Soc. Am. B 18, pp. 32-35 (2001). [3] H. S. Sozuer and J. W. Haus, "Photonic bands: simple cubic lattice," J. Opt. Soc. Am. B 10, pp. 296-302

(1993). [4] C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of

two-dimensional photonic crystals," Optics Express 12, pp. 1397-1408 (2004) [5] Z. Y. Li and L. L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method,"

Phys. Rev. E 67, 046607 (2003) [6] J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, "Focusing of light in a three-dimensional cubic

photonic crystal," Phys. Rev. B 67, 075109 (2003)

ω2

ω1

Analysis of 2-D Photonic Crystals Involving LiquidCrystals Using the Finite-Difference

Frequency-Domain Method

Ming-mung Chen, Sen-ming Hsu, and Hung-chun Chang∗Graduate Institute of Electro-Optical Engineering

National Taiwan University, Taipei, Taiwan 106-17, R.O.C.∗also with Department of Electrical Engineering and Graduate Institute of

Communication Engineering, National Taiwan University

Phone: +886-2-23635251-513 Fax: +886-2-23683824 E-mail: hcchang@cc.ee.ntu.edu.tw

Abstract: The finite-difference frequency-domain method is formulated for calculatingband structures of 2-D photonic crystals involving anisotropic materials such as liquid crys-tals. The director of the liquid crystal is allowed to rotate in the plane of the unit cell.c©2007 Optical Society of AmericaOCIS codes: (160.3710) Liquid crystals; (260.2110) Electromagnetic theory; (999.9999) Pho-

tonic crystals.

1. Introduction

Research on photonic crystals (PCs) has been actively conducted for two decades since the pioneering worksby Yablonovitch [1] and John [2] regarding that a periodic dielectric structure exhibits a forbidden band foroptical energy. Many applications of PCs have been demonstrated, including novel waveguiding structures,waveguide devices, cavities, filters, multiplexers, etc. Several studies have been performed on PCs madeof anisotropic materials. Zabel and Stroud [3] first analyzed three-dimensional (3-D) anisotropic PCs andshowed the narrowing of the band gap. Li et al. [4] demonstrated that for 2-D PCs, the complete band gapcan be achieved by including uniaxial materials with the optical axis along the extension direction of thePC. In this paper, we particularly consider the more involved structure with the optical axis perpendicularto the PC extension direction.

The analysis of PCs and their photonic band gaps (PBGs) have been most often based on the plane-wave expansion (PWE) method, including the study of anisotropic PCs. In this research, we develop afinite-difference frequency-domain (FDFD) method based on the Yee mesh for calculation of band struc-tures of 2-D PCs involving anisotropic materials. We will investigate a structure involving liquid crystals(LCs). The LC is a uniaxial material and its optical properties depend on the direction of the propagationand polarization state of lightwaves relative to the orientation of the LC director. When the director isperpendicular to the PC extension, the E and H waves are decoupled. Here, the E (H) wave refers to themode with the electric (magnetic) field parallel to the PC extension direction. Therefore, the E wave is likethat in an isotropic PC. We will study the variation of band gaps for the H wave for different radii of theLC cylinders and different orientations of the LC director in square lattice, and investigate the symmetryproperties of the PBG with respect to the orientation of the LC director.

2. The FDFD formulation

The FDFD formulation starts from the six component equations of the two Maxwell curl equations for ageneral anisotropic dielectric medium. The z-axis is defined as along the extension direction of the 2-D PC.Consider the situation that one principal axis of the anisotropic medium is along the z direction such thatthe permittivity tensor elements ²xz, ²yz, ²zx, and ²zy are zeros. We use the Yee-mesh-based finite-differencescheme [5]. For the case of H wave, since the different components of the electric field intensity vectorE and the electric flux density vector D are assigned at different grid points in the Yee mesh, the tensorconstitutive relation between E and D must be carefully treated at a given point. We employ a scheme ofmaking spatial averaging of the suitable component of D, as given by the following expressions [6]:

Ex(i + 0.5, j) =²yyΛ

Dx(i + 0.5, j)− ²xy4Λ {Dy(i, j − 0.5) + Dy(i + 1, j − 0.5)+ Dy(i, j + 0.5) + Dy(i + 1, j + 0.5)} (1a)

adminAppendix II

Ey(i, j + 0.5) =− ²yx4Λ {Dx(i + 0.5, j) + Dx(i + 0.5, j + 1) + Dx(i− 0.5, j)

+ Dx(i− 0.5, j + 1)}+ ²xxΛ Dy(i, j + 0.5) (1b)

where Λ = ²0(²xx²yy−²xy²yx) and ²0 is the free-space permittivity. Finally, an eigenvalue equation in termsof Hz in the following form can be obtained:

k20Hz =− {UxB22Vx − UxB21Vy + VyA11Uy − UyA12Vx}Hz (2)

where Ux, Uy, Vx, Vy, B22, B21, A11, and A12 are all matrices. For the E wave, the eigenvalue equation isobtained as

k20Ez =− ²−1zz {VxUx + VyUy}Ez. (3)

In the analysis of PCs, the periodic boundary conditions (PBCs) need to be considered with the unit cell.For a square unit cell with width a, the PBCs for the field ψ to be solved can be expressed as

ψ(x, y + a) =e−jkyaψ(x, y) (4a)

ψ(x + a, y) =e−jkxaψ(x, y). (4b)

3. Numerical results

As a numerical example, we consider a 2D PC composed of circular cylinder LC-filled regions with silicon

Figure 1: (a) The cross-section of the square-lattice 2D PC considered. (b) The entire first Brillouin zone.

as the background medium. The refractive index of silicon is taken to be n = 3.4. Figure 1(a) shows thecross-section of this 2D PC, where the cylinder rod of radius r = 0.5a is the region with the nematic LC,where a is the period of the lattice. The nematic LC is a homogenous uniaxial LC having the ordinaryand extraordinary refractive indices no = 1.5292 and ne = 1.7072, respectively. The permittivity tensorelements in the LC region are expressed as

²xx =n2o + (n2e − n2o) sin2 θ cos2 α (5a)

²xy =²yx = (n2e − n2o) sin2 θ sin α cosα (5b)²yy =n2o + (n

2e − n2o) cos2 θ sin2 α (5c)

where θ is the angle between the LC director and the z axis, and α is the angle between the projectionof the LC director on the x-y plane. We assume θ = 90◦ in this example. The first Brillouin zone (BZ)of the square lattice is shown in Fig. 1(b). With the existence of anisotropic materials, the symmetryproperties of the band structure become more complicated. For the present PC, we need to consider halfof the first BZ. Figure 2 presents the band structure for the PC with α = 45◦ for the H wave. This resulthas been checked with the calculation conducted by the finite element method [7] and very good agreementis achieved. A bandgap exists between the fourth and fifth bands. The calculated gap-to-midgap ratioversus the cylinder radius for α = 45◦ is shown in Fig. 3(a). It is seen that when r = 0.5a, the maximum

Figure 2: The band structure for α = 45◦

radius (lattice constant)

( )a ( )b

a ( )degree

0.4 0.42 0.44 0.46 0.48 0.5

a=45

Figure 3: (a) Gap-to-midgap ratio as a function of the cylinder radius. (b) Gap-to-midgap ratio as afunction of α.

gap size occurs. The calculated gap-to-midgap ratio as a function of α is plotted in Fig. 3(b), which showssymmetry with respect to 45◦.

4. Conclusion

We have developed an Yee-mesh-based FDFD method for calculating the band structures of 2-D PCscontaining anisotropic materials. As a numerical example, we have analyzed a square-lattice PC composedof circular cylinders of nematic LC material with silicon as background. The LC director is assumed to beperpendicular to the PC extension direction so that the E and H waves are decoupled. The tunability ofthe band-gap size has been studied and symmetry properties of the gap size versus the orientation of theLC director observed. The FDFD calculation has been found to agree with the finite element analysis [7].

5. References

[1]E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062

(1987).

[2]S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489

(1987).

[3]I. H. H. Zabel and D. Stround, “Photonic band structures of optically anisotropic periodic arrays,” Phys. Rev. B 48,

5004–5012 (1993).

[4]Z.-Y. Li, J. Wang, and B.-Y. Gu, “Creation of partial bandgaps in anisotropic photonic-band-gap structures,” Phys. Rev.

B 12, 1397–1408 (2004).

[5]C. P. Yu and H. C. Chang, “Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic

crystals,” Opt. Express 58, 3721–3729 (1998).

[6]A. P. Zhao, J. Juntunen and A. V. Raisanen, “An efficient FDTD algorithm for the analysis of microstrip patch antennas

printed on a general anisotropic dielectric substrate,” IEEE Trans. Microwave Theory Tech. 47, 1142–1146 (1999).

[7] S. M. Hsu, M. M. Chen, and H. C. Chang, “Band-structure analysis of 2D non-diagonal anisotropic photonic crystals by

the finite element method,” submitted to IPNRA 2007.

Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method

Po-jui Chiang,1 Chin-ping Yu,1 and Hung-chun Chang1,2,3,*1Graduate Institute of Electro-Optical Engineering, National Taiwan University, Taipei, Taiwan 106-17, Republic of China2Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 106-17, Republic of China

3Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106-17, Republic of China�Received 18 April 2006; revised manuscript received 21 November 2006; published 20 February 2007�

An analysis method based on a multidomain pseudospectral method is proposed for calculating the banddiagrams of two-dimensional photonic crystals and is shown to possess excellent numerical convergencebehavior and accuracy. The proposed scheme utilizes the multidomain Chebyshev collocation method. Byapplying Chebyshev-Lagrange interpolating polynomials to the approximation of spatial derivatives at collo-cation points, the Helmholtz equation is converted into a matrix eigenvalue equation which is then solved forthe eigenfrequencies by the shift inverse power method. Suitable multidomain division of the computationaldomain is performed to deal with general curved interfaces of the permittivity profile, and field continuityconditions are carefully imposed across the dielectric interfaces. The proposed method shows uniformly ex-cellent convergence characteristics for both the transverse-electric and transverse-magnetic waves in the analy-sis of different structures. The analysis of a mini band gap is also shown to demonstrate the extremely highaccuracy of the proposed method.

DOI: 10.1103/PhysRevE.75.026703 PACS number�s�: 02.70.Hm, 03.50.De

I. INTRODUCTION

Band structures are essential characteristics of photoniccrystals �PCs�, from which possible photonic band gaps�PBGs� can be identified �1–3�. For frequencies within thePBGs, wave propagation is forbidden and many photonicdevices have been proposed and designed based on this phe-nomenon. In particular, two-dimensional �2D� PCs com-posed of either dielectric rods or air columns have beenwidely employed in many applications such as waveguiding,resonant cavity formation, and wavelength filtering. In thispaper, we propose an analysis scheme with excellent numeri-cal convergence behavior and accuracy for calculating theband structures of 2D PCs. The currently most used numeri-cal methods for such calculations have been the plane-waveexpansion �PWE� method �3–6� and the finite-differencetime-domain �FDTD� method �7,8�. The finite-difference ei-genvalue problem formulation has also been employed byYang �9� and Shen et al. �10�, and more recently by Yu andChang �11� based on the Yee mesh as often employed in theFDTD method �12�. The Yee-mesh-based formulation wasnamed the finite-difference frequency-domain �FDFD�method. Yu and Chang �11� used the FDFD method to ana-lyze the band structures of 2D PCs with either square ortriangular lattices and adopted a fourth-order accurate com-pact finite-difference scheme �13� to increase numerical effi-ciency and accuracy. Although the FDFD method offers re-sults with accuracy comparable to those obtained using theMIT photonic-bands �MPB� package �14� based on the PWEmethod, the numerical convergent speed was found not to beuniformly fast among different bands in the two methods.

The numerical formulation proposed in this paper is basedon the multidomain pseudospectral method using Chebyshevpolynomials. The pseudospectral method has recently at-

tracted raised attention as an alternative treatment for com-putational electromagnetics because of its high-order accu-racy and fast convergence behavior over traditionaltechniques while retaining formulation simplicity. It has along history of being applied to fluid dynamics �15� and hasrecently been extended to the analysis of electromagneticsboth in the time �16–18� and in the frequency domain �19�.However, while the theory of the pseudospectral method hasbeen well elaborated, the application in the frequency do-main has not received much focus compared with that in thetime domain in the electromagnetics community. In �19�, thepseudospectral frequency-domain method was proposed andapplied to solve the nonhomogeneous �nonzero-source�Helmholtz equation in a simple two-subdomain problemwith rectangular structure shape. Our proposed scheme inthis paper utilizes the multidomain Chebyshev collocationmethod supported by the curvilinear mapping technique �20�to facilitate and ameliorate the simulation of 2D PCs of ar-bitrary permittivity profile. The formulation is derived in theform of an eigenvalue problem so that we can readily obtainthe eigenmodes by available mathematical tools. Here, weadopt the shift inverse power method �SIPM� for its particu-larly fast convergence characteristic over other conventionalmethods applying matrix inversion. Both the multidomainpseudospectral algorithm and the SIPM furnish our pseu-dospectral mode solver �PSMS� as a quite powerful and flex-ible method. To obtain high-accuracy full-vectorial modalsolutions for dielectric structures, proper satisfaction of di-electric interface conditions is essential, whether it is basedon the finite-difference method �21,22�, the finite-elementmethod �23�, or others. Such proper treatment of interfaceconditions will be carefully considered in our formulation.

The rest of this paper is outlined as follows. The physicalproblem involving the Helmholtz equations is described inSec. II along with the required Dirichlet and Neumann typeboundary conditions across the dielectric interfaces. The for-mulation of the PSMS is presented in Sec. III. Numericalexamples including 2D PCs with either a square or a trian-*Electronic address: hcchang@cc.ee.ntu.edu.tw

PHYSICAL REVIEW E 75, 026703 �2007�

1539-3755/2007/75�2�/026703�14� ©2007 The American Physical Society026703-1

http://dx.doi.org/10.1103/PhysRevE.75.026703adminAppendix III

admin

NEAR-FIELD AND FAR-FIELD BEHAVIOR OF A METALLIC NANOSCALE SPHERE AT OPTICAL FREQUENCIES BASED ON THE

CLASSICAL DRUDE MODEL

Bang-Yan Lin1, Chun-Hao Teng2, and Hung-chun Chang1 1Dept. of Electrical Eng., Graduate Institute of Electro-Optical Eng., and Graduate Institute of Comm. Eng., National Taiwan Univ., Taipei, Taiwan 106-17, R.O.C. hcchang@cc.ee.ntu.edu.tw 2Math. Dept., National Cheng Kung Univ., Tainan, Taiwan 701, R.O.C. tengch@mail.ncku.edu.tw Abstract: Optical properties of a metallic sphere based on the classical Drude model are studied theoretically. The comparison between the metallic sphere and the perfectly conducting one shows conspicuous distinction in the electric field amplitude on the surface of the sphere and in the radar cross section behavior. INTRODUCTION Recently, the field of nano-optics has been rapidly developing due to mature fabrication technologies. One relevant important subject is the phenomenon of the surface plasmon that was induced by the interaction of metal with electromagnetic radiation. The phenomenon is especially evident in some novel metals, such as silver or gold. In the scattering measurement problem, one of the usual geometric structures is the particle that can be approximated by a sphere. Fortunately, there has existed a well-known analytic solution for the spherical-particle scattering, i.e., the Mie solution [1], which not only can be applied to a perfectly conducting sphere but also to a metallic one with a complex value of the permittivity. The theory provides an initial prediction of the behavior in the near and far regions of the sphere. Traditionally, the use of the perfectly conducting sphere in the scattering problem can provide good approximation in the microwave region. But when the frequency moves to the optical regime, such as the visible region, and the particle shrinks to the nanometer-scale, one should care the appropriateness of using the perfectly conducting sphere. In this paper, to conform the real situation, we consider the classical Drude model for a metallic sphere. We discuss the near field and far field behavior, respectively. The study of the near field is aimed at whether the size of the particle will lead to a significant difference in the amplitude of electric field on the surface of the sphere. As for the far field, the radar cross section (RCS) will be reconsidered under the new situation. Besides using the analytic solution, we also employ a high-accuracy 3-D time-domain numerical scheme to verify the RCS results. NEAR-FIELD BEHAVIOR Fig. 1 shows a plane wave with the x-direction polarization propagating along the z direction and incident on a metallic sphere, where the dielectric constants of the background space and the

Fig. 1. Uniform plane wave incident on a metallic sphere.

sphere are ( )Iε and ( )IIε , respectively, and the radius of the sphere is a . We assume ( ) 1.0Iε =

adminAppendix IV

and the dielectric constant of the sphere is dominated by the classical Drude model with its dispersion characteristic expressed as ( ) 2 2( ) /( / ).II p iε ω ε ω ω ω τ∞= − + Here ε∞ is the high-frequency limit of the dielectric constant, pω is the plasma frequency, and τ is the relaxation time. Fig. 2 shows the experimental data of the dielectric constant of silver referring to [2], and the curve-fitting results based on two different sets of parameters for the classical Drude model, the first one being 14 144.0, 133.6848 10 Hz, and 0.9019 10 secpε ω τ

−∞ = = × = × from [3]

and the second one being 14 143.7, 136.73 10 Hz, and 3.6563 10 secpε ω τ−

∞ = = × = × from [4], denoted as classical Drude models 1 and 2, respectively.

(a) (b)

Fig. 2. (a) The real part and (b) the imaginary part of the dielectric constant of silver. Fig. 3 shows the maximum amplitude of the electric field on the surface of the sphere in the wavelength range 0.25 μm ~ 1 μmλ = , where “Experimental Data” mean calculated results using the corresponding ( )IIε shown in Fig. 2. Fig. 3 (a) and (b) is for 0.25 μma = and Fig. 3 (c) and (d) for 0.025 μma = , with Fig. (b) and (d) being the expanded versions for

0.25 μm ~ 0.45 μmλ = We can see that the values for the silver sphere are larger than those for the perfectly conducting one for most of the wavelengths. In particular, for 0.3 ~ 0.4 μmλ = , the differences are more significant. In both cases, the amplitude reaches the highest for 0.36 μmλ : , partly due to the minimum in the imaginary part shown in Fig. 2(a). The highest values represented by the experimental data are about 7 and 19, respectively. The results show that the size of the sphere will affect the amplitude of the electric field and that for the metallic sphere is larger than that of the perfectly conducting one.

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

Fig. 3. The maximum amplitude of the electric field on the surface of the sphere. The radius of the sphere is 0.25 μm in (a) and (b). The radius of the sphere is 0.025 μm in (c) and (d).

FAR-FIELD BEHAVIOR The measurement of electromagnetic scattering in the far-field zone is one of the important approaches for investigating the properties of the target, which is well known to be described by the RCS quantitatively. In our study, we use the classical Drude model 1 because it fits better the

experimental data than the type 2. The definition of RCS can be written as 2 22 s i

3-D ( , , ) lim[4 E / E ]r rσ λ θ φ π→∞= , where iE is the incident field and sE is the scattering

field. Here we discuss two kinds of RCS, bistatic RCS and monostatic RCS, with calculated results shown in Fig. 4 (a) and (b), respectively. The parameters for Fig. 4(a) are 0.364 μmλ = ,

[0, ]θ π= , and φ π= , and those for Fig. 4(b) are / 0 ~ 1.5a λ = , , and θ π φ π= = . Both results are normalized to 2aπ and a used in the calculation is 0.25 μm . Besides comparing the exact results of the silver sphere with those of the perfectly conducting one, we also conduct time-domain numerical simulations using a high-accuracy scheme similar to that of [5] along with the technique of near-to-far field transformation [6]. We can see that the differences between the metallic sphere and the perfectly conducting one are obvious in both cases except in the Rayleigh region ( 0.1a λ< ) for the monostatic RCS. The numerical results for the bistatic RCS agree very well with the exact results. But for the monostatic RCS, there exists deviation at high frequencies. This is attributed to lack of enough resolution. Using denser meshes in the numerical scheme can improve the results.

(a) (b)

Fig. 4. (a) Normalized bistatic RCS. (b) Normalized monstatic RCS. CONCLUSION At optical frequencies, we have discovered that the size of silver sphere will make a large distinction on the amplitude of the electric field in the near field, as predicted by the Mie theory. In the far field, the RCS properties of the metallic sphere are unlike those of the perfectly conducting one. These theoretical results could provide a useful reference for related experiments involving nanoscale metallic spheres. REFERENCES [1] Born, M., and Wolf E. “Principle of Optics”, Cambridge University Press, U.K., 1999. [2] Johnson, P. B., and Christy R. W., “Optical Constants of Noble Metals,” Phys. Rev. B., vol. 6,

no. 12, Dec. 1972, pp. 4370–4379. [3] Etchegoin, P. G., and Ru L. E. C., “Multipolar emission in the vicinity of metallic nanostructures,” J. Phys.: Condens. Matter, vol. 18, 2006, pp. 1175–1188. [4] Saj, W. M., “FDTD simulations of 2D plasmon waveguide on silver nanorods in hexagonal lattice,” Opt. Express, vol. 13, 2005, pp. 4818–4827. [5] Hesthaven, J. S., and Gottlieb D., “A stable penalty method for the compressible Navier-Stokes equations. III. Multidimensional domain decomposition schemes,” SIAM J. Sci. Comput., vol. 20, 1998, pp. 62–93. [6] Taflove, A., and Hagness A. C., “Computational Electrodynamics: The Finite-Difference Time-Domain Method”, Artech House Inc., Boston, 2000, Chapter 8.

Air-Core Waveguides for Terahertz Signals Chin-ping Yu and Hung-chun Chang*

Graduate Institute of Electro-Optical Engineering, National Taiwan University, Taipei, Taiwan 106-17, R.O.C. *also with Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University

E-mail: hcchang@cc.ee.ntu.edu.tw

Abstract: The finite-difference frequency-domain method is adopted for the analysis of air-core THz waveguides formed by Teflon rods. The guiding mechanism is found to be based on the ARROW model. The calculated propagation characteristics demonstrate that well confined THz guided field can be obtained. ©2007 Optical Society of America OCIS codes: (060.2310) Fiber optics; (230.7370) Waveguides; (260.2110) Electromagnetic theory; (999.9999) THz optics.

1. Introduction

In most terahertz (THz) systems, the propagation of far infrared electromagnetic waves relies on traditional metal or dielectric waveguides which suffer from high conductivity losses or dielectric losses and consequently have limitation for practical applications. Thus, it is a great issue to develop effective and low-loss waveguiding structures for THz signals for a compact, reliable, and flexible THz system. One possible way is to use metal-based Sommerfeld wires to confine THz signals with low fraction of power within lossy dielectric materials to reduce the attenuation caused by materials [1]. However, the transmission of THz signals on Sommerfeld wires suffers from a large field extension into air and large radiation loss at waveguide bending. Another approach utilizes total internal reflection (TIR) to guide THz signals in small air region formed by dielectric materials [2]. Good confinement of THz signals can be observed in the small air gap but with unignorable amount of field in the thick dielectric layers, which might result in high transmission loss in the materials in THz frequency range.

In this paper we consider air-core waveguides formed by polytetrafluorethylene (Teflon) rods for THz waveguiding. Figs. 1(a) and 1(b) show two examples of such waveguides with the air core surrounded by variant numbers of Teflon rods. The propagation characteristics of the waveguides with various rod-sizes and distributions are calculated by applying a full-vectorial finite-difference frequency-domain (FDFD) method [3]. The performance of these THz waveguides is discussed based on the calculated propagation characteristics.

i+1/2

j+1/2

Ex(i-1/2, j )

Ey( i , j-1/2)

Ez(i , j)

ε1

ε2

Hx( i , j-1/2)Hz(i-1/2, j-1/2)

Hy(i-1/2, j )

Hz(i +1/2, j+1/2)

i-1/2

j-1/2

εz

Hz(i +1/2, j-1/2)

Hz(i -1/2, j+1/2)

Ex(i+1/2, j )

Hy(i+1/2, j )

Ey( i , j+1/2)

Hx( i , j+1/2)

Fig. 1. (a) The cross-section of an air-core waveguide with 8 Teflon rods. (b) The cross-section of an air-

core waveguide with 12 Teflon rods. (c) Yee’s mesh for the FDFD method.

2. The numerical method

The FDFD method which is an efficient and accurate numerical model for optical waveguide analysis [3] is utilized to analyze air-core waveguides in Figs. 1(a) and 1(b). Consider the time-harmonic Maxwell's curl equations

HjE 0ωμ−=×∇ and EjH rεε 0ω=×∇ where ω is the angular frequency, μ0 and ε0 are the permittivity and the permeability of free space, respectively, and εr is the relative permittivity of the dielectric medium. By discretizing the transverse (x-y) plane using the Yee’s mesh as shown in Fig. 1(c) and applying the central difference scheme for the differential operators, the curl equations can be converted into the matrix form [3]

adminAppendix V

(1)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

z

y

x

xy

x

y

z

y

x

z

y

x

z

y

x

xy

x

y

z

y

x

HHH

0VVV0I

VI0

EEE

εε

ε

EEE

UUUI

UI

HHH

jβjβ

j

jj

j

000000

00

0

0

0

ωε

ββ

ωμ

where β is the modal propagation constant, H and E are the vectors composed of the electric and magnetic field components, respectively, at the grid points, I is a square identity matrix, and εi's (i = x, y, z) are diagonal matrices representing the relative permittivities at the corresponding grid points. Ux, Uy, Vx, and Vy are square matrices determined by the central difference scheme and the boundary conditions. After some mathematical work, an eigenvalue matrix equation in terms of the transverse magnetic fields can be obtained as

2 .β⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x xx xy x x

y yx yy y y

H Q Q H HQ

H Q Q H H (2)

Solving the eigenvalue problem of (2) by the shift inverse power method, the propagation constants and the field distributions of guided modes on the air-core waveguides can be obtained.

3. Results and discussion

In the following computation, the perfectly matched layers (PMLs) are applied as the boundary conditions around the computational window and the material loss is taken into account. The real part of the refractive index of Teflon is set to be 1.443 independent of frequency, and the value of its imaginary part is obtained by fitting the experimental results [4] with a second order polynomial as a function of frequency. We first consider an air-core waveguide with the air-core surrounded by 8 Teflon rods as shown in Fig. 1(a). The core size R is set to be 4.0 mm and the diameter of Teflon rods is d = 0.3R = 1.2 mm. By applying the FDFD method within the frequency range from 750 GHz to 1100 GHz, the modal refractive indices and the corresponding propagation losses of guided modes on such waveguide are shown in Figs. 2(a) and 2(b), respectively. The propagation loss is obtained from the modal propagation constant by the relation: Loss (dB/m) = 20·103·Im(β)/ln(10). In Fig. 2(a) it can be observed that over the entire frequency range, there always exist guided modes except at some particular frequencies. Fig. 2(c) illustrates the guided magnetic field profiles at f = 850 and 980 GHz. At f = 850 GHz, one can see that most of the field is well confined in the air core with relatively small field in the Teflon rods, thus reducing the possible attenuation loss caused by materials. As the frequency approaches the discontinuities in the modal dispersion curve, relatively large field appears in the Teflon rods and penetrates into the outer air region, as shown in Fig. 2(c) for f = 980 GHz, resulting in relatively large propagation losses at such frequencies shown in Fig. 2(b). The guiding mechanism of the air-core waveguide is just like the antiresonant reflecting optical waveguide (ARROW) model discussed in [5]. The discontinuities along the modal dispersion curve correspond to resonant frequencies of the Teflon rods at which the mode field leaks into the cladding region as in Fig. 2(c). At other frequencies, the field can be reflected by the Teflon rods to support centrally localized guided modes with relatively low propagation losses.

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.00.9988

0.9990

0.9992

0.9994

0.9996

0.9998

Mod

al in

dex

Frequency (Hz)

x1011

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.00.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

Loss

(dB

/m)

Frequency (Hz)

x1011

Fig. 2. (a) The modal index and (b) the propagation loss of guided modes on the air-core waveguide of Fig.

1(a). (c) Field distributions at f = 850 and 980 GHz.

If we keep the size of each Teflon rod the same and increase the number of rods surrounding the air core, similar propagation characteristics of guided modes can be obtained. Fig. 3(a) shows the comparison of modal dispersion curves for 8 and 12 Teflon rods. Slightly smaller modal indices are observed for 12 Teflon rods, and the

positions of discontinuities are almost the same for both arrangements. Besides, the propagation-loss results given in Fig. 3(b) also show similar distributions for 8 and 12 rods with the peaks appearing at similar locations.

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.00.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

8 rods 12 rods

Loss

(dB

/m)

Frequency (Hz)

x1011

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.00.9988

0.9990

0.9992

0.9994

0.9996

0.9998

8 rods 12 rods

Mod

al in

dex

Frequency (Hz)

x1011

Fig. 3. (a) The modal index and (b) the propagation loss of guided modes on air-core waveguides with

variant numbers of rods.

Now we keep the size of the air core the same and change the diameter of the Teflon rods. Figs. 4(a) and 4(b) show the modal indices and propagation losses, respectively, for 8 teflon rods with d = 1.2 mm and 1.6 mm. One can see that the discontinuities in modal dispersion curves and the positions of the propagation-loss peaks have obvious differences between the two sizes. This is because the resonant frequencies highly dependent on the size of the surrounding rods and so do the propagation characteristics. If the sizes of the Teflon rods are kept the same, the number of rods will not affect the resonant frequencies, making the air-core waveguides possess similar propagation characteristics as shown in Fig. 3. Practically, these rods need an inner or outer shell to be connected to realize a workable structure, which might influence the propagation properties and is worth further discussions.

7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0 R = 4.0 mm d = 1.2 mm R = 4.0 mm d = 1.6 mm

Loss

(dB

/m)

Frequency (Hz)

x10117.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0

0.9988

0.9990

0.9992

0.9994

0.9996

0.9998

R = 4.0 mm d = 1.2 mm R = 4.0 mm d = 1.6 mm

Mod

al in

dex

Frequency (Hz)

x1011

Fig. 4. (a) The modal index and (b) the propagation loss of guided modes on 8-rod air-core waveguides with

variant size of rods.

4. Conclusions

We have investigated modal properties of guided modes on air-core THz waveguides with the air core surrounded by Teflon rods. The guided modes can be found over almost the frequency range concerned and the ARROW model is successfully adopted to explain the discontinuities of the modal index curves where the propagation losses become relatively huge. It is also observed that the propagation characteristics of the air-core waveguides are highly dependent on the size of surrounding rods and the number of rod has little influence on the performance of these waveguides.

5. References [1] K. Wang and M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432, 376–379 (2004).

[2] M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14, 9944–9954 (2006).

[3] C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express 12, 6165–6177 (2004).

[4] J. R. Birch, J. D. Dromey, and J. Lesurf, “The optical constants of some common low-loss polymers between 4 and 40 cm-1,” Infrared Phys. 21, 225–228 (1981).

[5] T. P. White, R. C. McPhedran, C. M. Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002).

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006 4411

Robust Calculation of ChromaticDispersion Coefficients of Optical Fibers From

Numerically Determined Effective Indices UsingChebyshev–Lagrange Interpolation Polynomials

Po-Jui Chiang, Chin-Ping Yu, and Hung-Chun Chang, Senior Member, IEEE, Member, OSA

Abstract—Numerical calculation of chromatic dispersion co-efficients of optical fibers is conducted using a procedureinvolving Chebyshev–Lagrange interpolation polynomials. Onlynumerically determined effective indices at several wavelengthsare needed for obtaining the dispersion curve, and no direct nu-merical differentiation of the effective refractive index is involved.A silica-filled metallic rectangular waveguide having analyticalsolutions for the effective refractive index and the chromaticdispersion is used as an example for confirming the accuracy andefficiency of the proposed method. The method is then also appliedto the analysis of holey fibers.

Index Terms—Dispersion, fiber properties, optical waveguides,photonic-crystal fibers (PCFs), spectral method.

I. INTRODUCTION

THE CHROMATIC dispersion coefficient of an opticalfiber in units of picoseconds per nanometer kilometer isa key quantity in various analysis and design issues in fibertransmission systems, including recent intensive investigationof photonic-crystal fibers (PCFs) or holey fibers [1]–[3]. It isdefined as D = −(λ/c)d2neff/dλ2, where neff is the effectiverefractive index of the optical fiber mode, λ is the wavelength,and c is the speed of light in a vacuum. For a given fiber orwaveguide structure, neff can be solved at different wavelengthsusing various numerical methods. Then, in principle, one sim-ple way to obtain D at a particular wavelength is to performdirect numerical differentiation using the central differencescheme based on three neff values at three nearby wavelengths.Therefore, to obtain D values at N wavelengths, one needsto calculate neff at 3N wavelengths. From the basic theory of

Manuscript received May 27, 2006; revised August 14, 2006. This work wassupported in part by the National Science Council of the Republic of Chinaunder Grant NSC94-2215-E-002-022.

P.-J. Chiang and C.-P. Yu are with the Graduate Institute of Electro-OpticalEngineering, National Taiwan University, Taipei 106-17, Taiwan, R.O.C.(e-mail: d90941006@ntu.edu.tw; chinpingyu@ntu.edu.tw).

H.-C. Chang is with the Department of Electrical Engineering, Gradu-ate Institute of Electro-Optical Engineering, and the Graduate Institute ofCommunication Engineering, National Taiwan University, Taipei 106-17,Taiwan, R.O.C. (e-mail: hcchang@cc.ee.ntu.edu.tw).

Color versions of Figs. 1, 3, 4, and 6 are available online athttp://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2006.883646

the finite difference method, the separation between adjacentwavelengths should be small enough to maintain the requiredaccuracy but should not be too small such that the error wouldexplode [4]. Thus, some suitable ∆λ has to be determined.Furthermore, one might worry about the effect of the numericalaccuracy of the calculated neff on the degree of accuracy ofits second derivative. Kuhlmey et al. [5] have reported intheir analysis of holey fibers that they generally considered∼1000 points per unit λ/Λ, with Λ being the pitch of the air-hole lattice of the holey fiber, to obtain satisfactory results ofdispersion curves.

To ensure accurate determination of D, over the past twodecades, some more elaborate approaches have been proposedincluding, for example, that based on the matrix perturba-tion method [6] and those through formulating two associatedproblems for evaluating the first and second derivatives ofthe propagation constant [7]–[10]. In this paper, we proposea simple approach utilizing Chebyshev–Lagrange interpola-tion polynomials as in the Chebyshev spectral method or theChebyshev collocation method (CCM) [11] and demonstrateits efficiency and robustness in calculating D. For instance,over the wavelength range from 0.6 to 1.5 µm, calculation ofneff at only 13 wavelength points would guarantee accurateprediction of the D curves. Most importantly, no numericaldifferentiations of neff are required, and thus, computationalburden can be greatly reduced.

The rest of this paper is organized as follows: The CCMis described in Section II. The proposed procedure for chro-matic dispersion coefficient calculation is given in Section III,along with the analysis of a silica-filled metallic rectangularwaveguide. The application to holey fibers and comparison withpublished results are presented in Section IV. Conclusions aredrawn in Section V.

II. CCM

We first explain the approximation of a function f(x) inthe domain −1 ≤ x ≤ 1 under the CCM [11]. The Chebyshevpolynomial TN (x) of degree N is defined as TN (x) =cos(N cos−1 x), where |x| ≤ 1, and the collocation points tobe used are given by the Chebyshev–Gauss–Lobatto pointsdefined as the roots of the polynomial (1 − x2)T ′N , where

0733-8724/$20.00 © 2006 IEEE

adminAppendix VI

4412 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006

Fig. 1. Chromatic dispersion coefficient curve for the TE10 mode of a silica-filled metallic rectangular waveguide obtained from the exact solution and bythe CCM of different degrees.

the prime denotes the derivative. One merit to employ theChebyshev polynomial is the available analytical formula forits collocation points, which is given by xi = cos(iπ/N),i = 0, 1, 2, . . . , N . The CCM provides an N th-order approx-imation of f(x) as (LNf)(x) =

∑Ni=0 f(xi)gi(x), where the

Chebyshev–Lagrange interpolation polynomials are given bygi(x) = [(1 − x2)T ′N (x)(−1)i+1]/[ciN2(x − xi)] with c0 =cN = 2, and ci = 1 for 1 ≤ i ≤ N − 1. The polynomials gi(x)have the properties that gi(xj) = 1 for i = j and gi(xj) = 0for i �= j. Therefore, (LNf)(xi) = f(xi) and the derivativesof f(x) at a collocation point xi can be computed by a ma-trix operator, with the matrix entries Dij = g′j(xi), throughdf(xi)/dx =

∑Nj=0 g

′j(xi)f(xj) =

∑Nj=0 Dijf(xj). The ex-

plicit expressions for Dij , as given in [11], are Dij =(ci/cj)[(−1)i+j/(xi − xj)] for i �= j, i, j = 0, 1, 2, . . . , N ,Dii =−xi/[2(1−x2i )] for 1 ≤ i ≤ N−1, D00 = (2N2+ 1)/6,and DNN = −(2N2 + 1)/6. The matrix with elements Dijis termed the differential matrix and will be denoted as ¯̄D inthe following. In summary, in the CCM, the function f(x)is approximated by a summation of N + 1 terms involvingN + 1 known continuous interpolating functions gi(x), and thenumerical derivative of f(x) is easily evaluated in terms of theknown derivative values of gi(x).

III. IDEALISTIC WAVEGUIDE HAVING ANALYTICALCHROMATIC DISPERSION CHARACTERISTICS

We describe in this section our proposed method and ex-amine its accuracy in calculating chromatic dispersion coef-ficient curves by referring to some analytical results of anidealistic waveguide, as shown in the inset of Fig. 1. It is asilica-material-filled metallic rectangular waveguide with sidewidths a = 5.28 µm and b = 4.13 µm. The waveguide isassumed to be enclosed by perfect electric conductor (PEC)walls. From the theory of metallic waveguide [12], the prop-agation constant of the fundamental TE10 mode has the ex-pression β = [k20ε − (π/a)2]1/2 = neffk0, or we have neff =[ε − (λ/2a)2]1/2, where k0 is the wavenumber in free space,and ε is the relative permittivity of silica. Even when takinginto account the material dispersion of silica described by thefour-term Sellmeier formulas [13], [14] for ε as a function ofλ, the derivatives of neff with respect to λ can be analytically

TABLE ISELLMEIER COEFFICIENTS [13], [14]

derived, and thus, D versus λ has an analytical expression. Therelated formulas are given as follows:

dneffdλ

=(

12

dε

dλ− λ

4a2

)[ε −

(λ

2a

)2]−1/2(1)

d2neffdλ2

=(

12

d2ε

dλ2− 1

4a2

) [ε −

(λ

2a

)2]

−(

12

dε

dλ− λ

4a2

)2 [ε −

(λ

2a

)2]−3/2(2)

ε = 1 +4∑

i=1

Aiλ2

λ2 − B2i(3)

dε

dλ=

4∑i=1

Ai

[2λ

(λ2 − B2i

)−1 − 2λ3 (λ2 − B2i )−2]

=4∑

i=1

Ai−2λB2i

(λ2 − B2i )2(4)

d2ε

dλ2=

4∑i=1

Ai

[2(λ2 − B2i

)−1 − 10λ2 (λ2 − B2i )−2+ 8λ4

(λ2 − B2i

)−3]

=4∑

i=1

Ai6λ2B2i + 2B

4i

(λ2 − B2i )3(5)

with the Sellmeier coefficients [13], [14] given in Table I. Theexact analytical result is shown in Fig. 1 as the solid line forwavelength varying from 0.6 to 1.5 µm.

To apply the CCM, we first do a linear coordinate transforma-tion by mapping the domain λ1 ≤ λ ≤ λ2 to the domain −1 ≤x ≤ 1 of the x-axis. By employing Chebyshev polynomial ofdegree N = 12, we have 13 Chebyshev–Gauss–Lobatto pointsin the domain −1 ≤ x ≤ 1 and the corresponding ones in thewavelength domain λ1 ≤ λ ≤ λ2. We calculate neff of thefundamental core mode of the rectangular waveguide of Fig. 1at these 13 wavelengths and use these 13 numbers to constructa 13 × 1 column vector [neff ]. The mode solver we use isthat based on the finite-difference frequency-domain (FDFD)method [15]. Then, the D values to be determined at these 13wavelengths are taken to be the entries of the 13 × 1 vector [D].It is easy to obtain the following relationship using the simplechain rule given as follows:

[D] = − (4λ/c(λ2 − λ1)2) ¯̄D ¯̄D[neff ]. (6)Therefore, once the vector [neff ] is known, the vector [D]

can be obtained readily, and no numerical differentiation isneeded. The 13 circles in Fig. 1 show such results, and itis seen that they agree with the analytical solution. In fact,

CHIANG et al.: CALCULATION OF DISPERSION COEFFICIENTS USING CHEBYSHEV–LAGRANGE INTERPOLATION 4413

TABLE IICOMPARISON OF FDFD CALCULATIONS OF THE EFFECTIVE INDICES OF

THE TE10 MODE OF THE SILICA-FILLED METALLIC RECTANGULARWAVEGUIDE USING DIFFERENT GRID SIZES

TABLE IIICOMPARISON OF CHROMATIC DISPERSION COEFFICIENTS OBTAINED

BASED ON THE CCM USING THE CORRESPONDINGneff(FDFD) VALUES OF TABLE II

we can obtain a continuous curve for D since d2f(x)/dx2 =∑Nj=0(d

2gj(x)/dx2)f(xj). In addition, if we plot the obtainedcontinuous curve for D in Fig. 1, it would be indistinguishablefrom the solid line. To examine what the enough value forN would be, we have considered N = 8 and 4, and the twocontinuous curves are plotted in Fig. 1. For this waveguide,we found that the N = 8 curve is almost consistent with theN = 12 one, but the N = 4 curve is far from enough. Wehave found that N = 12 would provide a very good resultfor waveguide structures generally encountered, including theexamples to be discussed in the next section.

In the FDFD method [15], the waveguide cross section ismade up of uniform finite-difference grids (Yee mesh). It is wellknown that numerical accuracy in the calculation of neff wouldincrease with decreasing grid size. Assuming equal numericalgrid sizes in the horizontal and vertical directions (∆x =∆y = ∆), we have examined the possible dependence ofthe accuracy of D on the numerical accuracy of the calcu-lated neff . Table II lists the calculated neff values for ∆ =0.02, 0.1, and 0.66 µm, respectively, along with the exactvalues at the 13 collocation wavelength points, and Table IIIlists the corresponding D values obtained using the CCM withN = 13 along with the exact ones. It is observed in Table IIthat for ∆ = 0.02 µm, the FDFD-calculated neff has seven

Fig. 2. (a) Cross section of the three-ring holey fiber. (b) Cross section of thesix-ring holey.

digits of accuracy after the decimal point at shorter wavelengthsand six digits of accuracy at longer wavelengths and that for∆ = 0.66 µm, it decreases to four digits and three digits,respectively. It is found in Table III that with ∆ = 0.02 µm,the error in the prediction of D is less than 0.09% over thewavelength range considered and that even with quite coarsegrids of ∆ = 0.66 µm, the error is at most 3.82% near the zero-dispersion wavelength. The proposed method for calculating Dcurves is, thus, quite robust as far as good numerical accuracyis concerned. That is, with only three to four digits of accuracyafter the decimal point for neff , D curves of reasonable accu-racy can be obtained for this case. Such accuracy in neff canbe easily achieved using typical waveguide eigenmode solversbased on different numerical methods.

IV. DISPERSION IN HOLEY FIBERS

One of the attractive features of holey fibers is the betterflexibility in designing the D curves through varying the geo-metrical arrangement of the air holes. Here, we first compareour calculation with that of Kuhlmey et al. [5] by consideringthe lowest core mode of one three-air-hole-ring structure intheir Fig. 3. As mentioned in Section I, Kuhlmey et al. [5]generally considered ∼1000 points per unit λ/Λ in their cal-culation to obtain satisfactory results of the D curves. The

4414 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 11, NOVEMBER 2006

Fig. 3. Chromatic dispersion coefficient curve for the lowest core mode ofthe three-ring holey fiber of Fig. 2(a), with Λ = 2.3 µm and d/Λ = 0.6obtained by the CCM of different degrees. Also shown is the result calculatedby Kuhlmey et al. [5].

cross section of the three-ring holey fiber is shown in Fig. 2(a).The parameters considered are Λ = 2.3 µm and d/Λ = 0.6,where d is the diameter of each air hole. Again, we employour Yee-mesh-based FDFD waveguide eigenmode solver [15]using the grid size ∆x = ∆y = ∆ = 0.1 µm to calculate neffat 13 wavelengths from λ = 0.6 to 1.7 µm, corresponding to theChebyshev–Gauss–Lobatto points for N = 12. The materialdispersion of silica has been taken into account by using thefour-term Sellmeier formulas [13], [14]. Then, the D curve isdetermined using the CCM as in Section III. The D values at the13 collocation points are presented as the circles in Fig. 3 withthe solid line being that shown in [5, Fig. 3]. The agreement isobviously very good. If we plot our calculated continuous curvefor D in Fig. 3, it would, again, be indistinguishable from thesolid line. We have also considered N = 8, 6, and 4, and thethree continuous curves are plotted in Fig. 3. Again, N = 4 isnot enough for obtaining correct curve. As N is increased, theresults converge with the maximum errors occurring near thetwo ends. This is the characteristic of the Chebyshev spectralmethod. It can be seen that there exist some noticeable differ-ences at the two ends between the N = 8 and N = 12 lines.This example demonstrates a general feature that a suitablevalue of degree N can be determined by examining the numeri-cal convergence at the ends of the wavelength domain. As longas the difference between the results for degrees N − 1 and N ,respectively, at the ends is small enough for specific application,the whole curve of degree N will surely be acceptable.

In our FDFD eigenmode solver, as discussed in [15], we havetwo schemes in treating the dielectric interface problem, thatis, the interface between silica and air, namely 1) the simplestaircase approximation and 2) the more elaborative boundarycondition (BC) matching method. The latter has been shown tobe much more accurate than the former. We have analyzed thestructure of Fig. 3 using both schemes, and the results are givenin Table IV. It can be observed that although the neff valuesof the staircase scheme agree with those of the BC matchingscheme only up to three digits after the decimal point, thedifference in D results is at most 0.64%, which occurs nearλ = 1.7 µm. Again, this demonstrates the robustness of theproposed method for calculating D.

We then consider a six-ring holey fiber recently analyzedby Saitoh and Koshiba using the finite element method

TABLE IVCOMPARISON OF EFFECTIVE INDICES AND CHROMATIC DISPERSION

COEFFICIENTS OBTAINED BASED ON THE FDFD METHOD USINGTHE BOUNDARY MATCHING SCHEME AND THE STAIRCASEAPPROXIMATION, RESPECTIVELY, FOR THE CASE OF FIG. 3

Fig. 4. Chromatic dispersion coefficient curves for the lowest core modeof the six-ring holey fiber of Fig. 2(b), with Λ = 1.0 µm for different d/Λvalues obtained by the CCM of different degrees. Also shown are sample pointsextracted from the results calculated by Saitoh and Koshiba [16, Fig. 16(a)].

(FEM) [16], with the cross section as shown in Fig. 2(b). In[16, Fig. 16(a)], six D curves of the lowest core mode wereplotted for d/Λ = 0.5, 0.6, 0.7, 0.8, and 0.9, respectively,and Λ = 1.0 µm. We employ the FDFD eigenmode solver [15]using the grid size ∆x = ∆y = ∆ = 0.067 µm and the CCMwith N = 12 to obtain the corresponding results and plot themas the five lines in Fig. 4. These five lines agree well with thosein [16, Fig. 16(a)]. The circles are sample points extracted fromthe results of [16].

In the numerical examples presented so far, as in most sit-uations in the holey-fiber study, the dispersion variations wereseen to vary by over hundreds of picoseconds per nanometerkilometer within the interested wavelength range. We finallyconsider an extreme situation in which the holey fiber is de-signed to possess nearly zero dispersion-flattened characteris-tics. One such structure was demonstrated in [16], with thecross section as shown in Fig. 5 and the dispersion designedto remain between 0.1 and 0.3 ps/(nm · km) for the wavelengthrange from 1.41 to 1.68 µm. The structure parameters of thisten-air-hole-ring holey fiber were Λ = 1.6 µm, d1 = 0.47 µm,d2 = 0.71 µm, d3 = 0.74 µm, d4 = 0.62 µm, and d5 = · · · =d10 = 0.65 µm, where di is the air-hole diameter of theith air-hole ring. For this structure, we have found that thenumerical accuracy in the calculation of neff is quite criticalin determining the D curve. As shown in Fig. 6, the dots are

CHIANG et al.: CALCULATION OF DISPERSION COEFFICIENTS USING CHEBYSHEV–LAGRANGE INTERPOLATION 4415

Fig. 5. Cross section of the nearly zero dispersion-flattened holey fiber withten air-hole rings.

Fig. 6. Chromatic dispersion coefficient curve for the lowest core mode ofthe ten-ring holey fiber of Fig. 5, with Λ = 1.6 µm and a designed air-holediameter distribution obtained by the FDFD eigenmode solver using differentgrid sizes and the CCM of degree 12. Also shown are sample points extractedfrom the results calculated by Saitoh and Koshiba [16, Fig. 19].

sample points extracted from the D curve of the lowest coremode given in [16, Fig. 19], and the three continuous lines areobtained using our FDFD eigenmode solver with the grid sizes∆x = ∆y = ∆ = 0.1, 0.05, and 0.033 µm, respectively, andthe CCM with N = 12. The BC matching scheme has been em-ployed in the FDFD calculation of neff to ensure high accuracy.It is seen that the dispersion calculation for this case is quitesensitive to the grid size in the FDFD method. The grid size of0.1 µm is obviously not fine enough for this calculation. Itshould be emphasized here that the CCM still works as anefficient scheme in determining the D curve from the neffcalculation. The key point is that to obtain enough accurate Dcurve, we need to have neff values of enough high accuracy,and in the present case, the demand of the neff accuracy is quitehigh, as can be understood from the following discussion.

We approximate the D curve to be determined for the pre-sent holey fiber in the wavelength range of Fig. 6 by the expres-sion A + B sin[π(λ̄ − 1.55)/0.3], which can be considered asthe first two terms in a Taylor series expansion, where λ̄ = λ/(1 µm) is a dimensionless variable. According to the result of[16] in Fig. 6, A≈ 0.2 ps/(nm · km) and B≈ 0.1 ps/(nm·km).

The expression above should be equal to −(λ/c)d2neff/dλ2,which can be approximately written as −[(λ0/c)/(1 µm)2]d2neff/dλ̄

2 = −(1.55/3)×104d2neff/dλ̄2ps/(nm·km), with λ0being taken to be 1.55 µm. Thus, we have 0.2 + 0.1 sin[π(λ̄ −1.55)/0.3] ≈ −(1.55/3)×104d2neff/dλ̄2, which gives neff ≈−(0.6/1.55)×10−4(C2+ C1λ̄+λ̄2/2)−(0.3/1.55)(0.3/π)2×10−4 sin[π(λ̄−1.55)/0.3]= −3.87×10−5(C2+C1λ̄+λ̄2/2)−1.76 × 10−7 sin[π(λ̄ − 1.55)/0.3], where C1 and C2 areconstants. It can be seen that due to the small value ofB(≈0.1 ps/(nm · km)), the amplitude of the sine term inneff is as small as in the order of 10−7, which implies thatone needs to achieve very high accuracy in the numericalcalculation of neff in order to be able to resolve this sinevariation. This implication is also true even when A is notsmall (dispersion-flattened but not nearly zero). When B is inthe order of hundreds of picoseconds per nanometer kilometeras in the earlier examples, the amplitude of the sine termbecomes three orders larger, and the demand of the numericalaccuracy of neff is greatly reduced.

V. CONCLUSION

We have proposed a simple procedure for the numeri-cal calculation of chromatic dispersion coefficients of op-tical fibers. The procedure employs the CCM involvingChebyshev–Lagrange interpolation polynomials. No directnumerical differentiation of the propagation constant or of theeffective index is involved. From the application of the pro-posed method to an idealistic silica-filled metallic waveguide,which provides analytical solutions for the effective indices andchromatic dispersion coefficients of the waveguide modes, wehave demonstrated the robustness of this simple scheme andits efficiency and accuracy. Our analysis of a three- and a six-ring holey fiber also shows very good agreement with publishedresults based on different methods. We have found that calcu-lating the effective indices at 13 wavelengths, corresponding tothe Chebyshev–Gauss–Lobatto collocation points for N = 12,would generally provide very good results for the D curves. Theproposed method can be very easily applied to any structureby using (6) with the entries of the differential matrix ¯̄Dgiven at the end of Section II. It should be noticed that in thedetermination of a dispersion-flattened curve, as demonstratedin the final example of the ten-ring holey fiber, the CCM stillworks well as an efficient scheme, but the achievement of thecorrect result relies highly on the numerical accuracy of theeffective index calculation.

ACKNOWLEDGMENT

The authors would like to thank the National Center forHigh-Performance Computing, Hsinchu, Taiwan, R.O.C., forproviding useful computing resources.

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Po-Jui Chiang was born in Kaohsiung, Taiwan,R.O.C., on November 26, 1974. He received theB.S. degree in electronics engineering from the Na-tional Taiwan University of Science and Technology,Taipei, Taiwan, in 1998 and the M.S. degree inelectro-optical engineering from the National TaiwanUniversity, Taipei, in 2000. He is currently work-ing toward the Ph.D. degree at the Graduate Insti-tute of Electro-Optical Engineering, National TaiwanUniversity. His research interests include numericalmethods for solving optical waveguide problems.

He is also currently with the Graduate Institute of Electro-Optical Engi-neering, National Taiwan University. His research interests include numericalmethods for solving optical waveguide problems.

Chin-Ping Yu was born in Tainan, Taiwan, R.O.C.,on Januar