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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 25 — Dec. 12, 2005
  • pp: 10367–10381
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Contour-path effective permittivities for the two-dimensional finite-difference time-domain method

Ahmad Mohammadi, Hamid Nadgaran, and Mario Agio  »View Author Affiliations


Optics Express, Vol. 13, Issue 25, pp. 10367-10381 (2005)
http://dx.doi.org/10.1364/OPEX.13.010367


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Abstract

Effective permittivities for the two-dimensional Finite-Difference Time-Domain (FDTD) method are derived using a contour path approach that accounts for the boundary conditions of the electromagnetic field at dielectric interfaces. A phenomenological formula for the effective permittivities is also proposed as an effective and simpler alternative to the previous result. Our schemes are validated using Mie theory for the scattering of a dielectric cylinder and they are compared to the usual staircase and the widely used volume-average approximations. Significant improvements in terms of accuracy and error fluctuations are demonstrated, especially in the calculation of resonances.

© 2005 Optical Society of America

1. Introduction

Research fields like advanced optical imaging and integrated photonics explore and push forward the terrific possibilities offered by Maxwell’s equations. These developments rely more and more on complex geometries, like photonic crystals [1

1 . Proceedings of the Fifth International Symposium on Photonic and Electromagnetic Crystal Structures (PECS-V) ( Kyoto, Japan , March 7 – 11 , 2004 ); H. Benisty , S. Kawakami , D.J. Norris , and C.M. Soukoulis , eds, Phot. Nanostructures Fund. Appl . 2 , 57 – 159 ( 2004 ); [CrossRef]

1 . C. Jagadish , D.G. Deppe , S. Noda , T.F. Krauss , and O.J. Painter , eds, IEEE J. Sel. Top. Area Commun. 23 , 1305 – 1423 ( 2005 ).

] or metamaterials [2

2 . Special issue on nanostructured optical meta-materials: beyond photonic band gap effects , N. Zheludev and V. Shalaev , eds., J. Opt. A: Pure and Applied Optics , 7 , S1 – S254 ( 2005 ). [CrossRef]

], as well as on complex field configurations [3

3 . Proceedings of the EOS Topical Meeting on Advanced Optical Imaging Techniques , ( London, UK , June 29 - July 1, 2005 ).

], so that full-vector numerical methods play a key role in understanding and designing them. In the past, several techniques have been proposed [4

4 . M.V.K. Chari and S.J. Salon , Numerical methods in electromagnetism ( Academic Press, San Diego, CA , 2000 )

], like finite-difference and finite-element methods, integral equation methods and so forth. Among them, the Finite-Difference Time-Domain (FDTD) method, proposed by K.S. Yee [5

5 . K.S. Yee , “ Numerical Solution of Initial Boundary Value Problems involving Maxwell’s Equations in Isotropic Media ,” IEEE Trans. Antennas Propag. AP-14 , 302 – 307 ( 1966 ).

], has gained much popularity for several reasons: it is rather easy to implement, the algorithm is intuitive, it can solve Maxwell’s equation for systems of arbitrary shape, it works in space and time domain.

In the FDTD method, space and time are discretized in a way that the derivatives in Maxwell’s curl equations can be written as finite central differences. The structure of the curl equations suggests that the field components are defined on different discrete positions in space, as shown in Fig. 1 for the two-dimensional case. The same holds for the time mesh, where the so-called leapfrog scheme is used. After a few simple rearrangements, the curl equations are finally transformed into loops that simulate the propagation of the electromagnetic field in space and time [6

6 . A. Taflove and S.C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method ( Artech House, Norwood, MA , 2005 ).

]. There are, however, some intrinsic problems that can make the FDTD method inaccurate, such as staircasing, numerical velocity dispersion and absorbing boundary conditions. The two latters will not be directly addressed in this work; the reader can refer to a reference book on FDTD [6

6 . A. Taflove and S.C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method ( Artech House, Norwood, MA , 2005 ).

]. Staircasing can be easily understood by looking at Fig. 1(a). Because the mesh can assign only discrete values to position, any object embedded in the simulation domain, will be pixelized in a way that a smooth interface becomes like a staircase. Consequently, the scattering properties of the system will change going from the real shape to the meshed shape, especially if the mesh is coarse and the dielectric contrast is large. This issue has limited the application of the FDTD method when the exact shape and permittivity values are important in determining the electromagnetic response.

A possible solution to staircasing is to depart from the simple Cartesian mesh picture and use non-orthogonal grids or curvilinear coordinates that follow exactly the shape of the objects [7

7 . K.K. Mei , A. Cangellaris , and D.J. Angelakos , “ Conformal Time Domain Finite-Difference Method ,” Radio Sci. 19 , 1145 – 1147 ( 1984 ). [CrossRef]

, 8

8 . R. Holland , “ Finite-Difference Solution of Maxwell’s Equations in Generalized Nonorthogonal Coordinates ,” IEEE Trans. Nucl. Sci. NS-30 , 4589 – 4591 ( 1983 ). [CrossRef]

, 9

9 . M. Fusco , “ FDTD Algorithm in Curvilinear Coordinates ,” IEEE Trans. Antennas Propag. 38 , 76 – 89 ( 1990 ). [CrossRef]

, 10

10 . V. Shankar , A. Mohammadian , and W.F. Hall , “ A Time-Domain Finite-Volume Treatment for the Maxwell Equations ,” Electromagnetics 10 , 127 – 145 ( 1990 ). [CrossRef]

, 11

11 . N.K. Madsen and R.W. Ziolkowski , “ A Three-Dimensional Modified Finite Volume Technique for Maxwell’s Equations ,” Electromagnetics 10 , 147 – 161 ( 1990 ). [CrossRef]

, 12

12 . P.H. Harms , J.-F. Lee , and R. Mittra , “ A Study of the Nonorthogonal FDTD Method Versus the Conventional FDTD Technique for Computing Resonant Frequencies of Cylindrical Cavities ,” IEEE Trans. Microwave Theory Tech. 40 , 741 – 476 ( 1992 ). [CrossRef]

]. However, while improving the accuracy, such approach not only considerably increases the complexity of the algorithm, but can also cause numerical artifacts due to a highly irregular grid, like time instability, velocity dispersion and spurious wave reflection [15

15 . C.J. Railton , I.J. Craddock , and J.B. Schneider , “ Improved locally distorted CPFDTD algorithm with provable stability ,” Electron. Lett. 31 , 1585 – 1586 ( 1995 ). [CrossRef]

]. Better is to exploit a Cartesian mesh as much as possible and introduce distorted cells only when it is really necessary. For the special cells, a Contour Path (CP) FDTD algorithm can be obtained directly from Maxwell’s equations in integral form [13

13 . T.G. Jurgens , A. Taflove , K. Umashankar , and T.G. Moore , “ Finite-Difference Time-Domain Modeling of Curved Surfaces ,” IEEE Trans. Antennas Propag. 40 , 357 – 365 ( 1992 ). [CrossRef]

, 14

14 . T.G. Jurgens and A. Taflove , “ Three-Dimensional Contour FDTD Modeling of Scattering from Single and Multiple Bodies ,” IEEE Trans. Antennas Propag. 41 , 1703 – 1708 ( 1993 ). [CrossRef]

]. The CP-FDTD method can still contain cells that potentially generate instability because of non-reciprocal nearest neighbor borrowing steps [15

15 . C.J. Railton , I.J. Craddock , and J.B. Schneider , “ Improved locally distorted CPFDTD algorithm with provable stability ,” Electron. Lett. 31 , 1585 – 1586 ( 1995 ). [CrossRef]

]. Moreover, the introduction of auxiliary field components and update equations slightly increases memory and CPU time. There have been improvements to CP-FDTD that solve the instability issue [15

15 . C.J. Railton , I.J. Craddock , and J.B. Schneider , “ Improved locally distorted CPFDTD algorithm with provable stability ,” Electron. Lett. 31 , 1585 – 1586 ( 1995 ). [CrossRef]

, 16

16 . Y. Hao and C.J. Railton , “ Analyzing Electromagnetic Structures with Curved Boundaries on Cartesian FDTD Meshes ,” IEEE Trans. Microwave Theory Tech. 46 , 82 – 88 ( 1998 ). [CrossRef]

, 17

17 . T.I. Kosmanis and T.D. Tsiboukis , “ A Systematic and Topologically Stable Conformal Finite-Difference Time-Domain Algorithm for Modeling Curved Dielectric Interfaces in Three Dimensions ,” IEEE Trans. Microwave Theory Tech. 51 , 839 – 847 ( 2003 ). [CrossRef]

].

Another possible way for reducing the staircasing error is refining the Cartesian mesh in proximity of the interfaces, the so-called subgridding method [18

18 . I.S. Kim and W.J.R. Hoefer , “ A Local Mesh Refinement Algorithm for the Time Domain-Finite Difference Method Using Maxwell’s Curl Equations ,” IEEE Trans. Microwave Theory Tech. 38 , 812 – 815 ( 1990 ). [CrossRef]

, 19

19 . S.S. Zivanovic , K.S. Yee , and K.K. Mei , “ A Subgridding Method for the Time-Domain Finite-Difference Method to Solve Maxwell’s Equations ,” IEEE Trans. Microwave Theory Tech. 39 , 471 – 479 ( 1991 ). [CrossRef]

]. However, this scheme implies modifications at the fields-marching level, making the algorithm more complicated to implement, besides other numerical issues like spurious wave reflection.

A different approach, specific to dielectrics, is using Effective Permittivities (EPs) for the partially filled cells, without any distortion of the Cartesian grid. The question is, what value for the permittivity has to be chosen in order to get the best approximation of the dielectric interface? An early attempt in this direction has been made for modeling thin material sheets [20

20 . J.G. Maloney and G.S. Smith , “ The Efficient Modeling of Thin Material Sheets in the Finite-Difference Time-Domain (FDTD) Method ,” IEEE Trans. Antennas Propag. 40 , 323 – 330 ( 1992 ). [CrossRef]

], even though there is still usage of auxiliary terms for field components normal to the interface and the procedure is limited to rectangular objects aligned with the mesh. A few years later, Kaneda et al. [21

21 . N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD Analysis of Dielectric Resonators with Curved Surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ). [CrossRef]

] proposed a phenomenological formula for the EP applicable to any kind of interface geometry, including curved surfaces. Their expression matches the rigorous result that can be obtained when the field component is perpendicular or parallel to the interface [22

22 . T. Hirono , Y. Shibata , W.W. Lui , S. Seki , and Y. Yoshikuni , “ The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme ,” IEEE Microwave Guided Wave Lett. 10 , 359 – 361 ( 2000 ). [CrossRef]

, 23

23 . K.-P. Hwang and A.C. Cangellaris , “ Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 158 – 160 ( 2001 ). [CrossRef]

]. These EPs improve the accuracy of the FDTD method, while keeping the same stability and simple structure of the original algorithm. However, there is no guarantee that the formula fulfills the proper boundary conditions at a curved interface or simply at a flat interface tilted with respect to the mesh axes. There are several works presenting other kinds of EPs: a volume average [24

24 . S. Dey and R. Mittra , “ A Conformal Finite-Difference Time-Domain Technique for Modeling Cylindrical Dielectric Resonators ,” IEEE Trans. Microwave Theory Tech. 47 , 1737 – 1739 ( 1999 ). [CrossRef]

], a first-neighbor average [25

25 . W. Yu and R. Mittra , “ On the modeling of periodic structures using the finite-difference time-domain algorithm ,” Microw. Opt. Technol. Lett. 24 , 151 – 155 ( 2000 ). [CrossRef]

, 26

26 . P. Yang , G.W. Kattawar , K.-N. Liou , and J.Q. Lu , “ Comparison of Cartesian grid configurations for application of the finite-difference time-domain method to electromagnetic scattering by dielectric particles ,” Appl. Opt. 43 , 4611 – 4624 ( 2004 ). [CrossRef] [PubMed]

], Maxwell-Garnett, inverted Maxwell-Garnett and Bruggeman formulae [27

27 . P. Yang , K.N. Liou , M.I. Mishchenko , and B.-C. Gao , “ Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols , Appl. Opt. 39 , 3727 – 3737 ( 2000 ). [CrossRef]

], and other phenomenological derivations [28

28 . W. Yu and R. Mittra , “ A Conformal Finite Difference Time Domain Technique for Modeling Curved Dielectric Surfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 25 – 27 ( 2001 ). [CrossRef]

]. Some of them have been tested together for the purpose of comparison [29

29 . W. Sun and Q. Fu “ Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices ,” Appl. Opt. 39 , 5569 ( 2000 ). [CrossRef]

]. These proposals are not fundamentally more accurate than Kaneda’s approach.

The main problem with the formulation of EPs resides in the vectorial nature of the electromagnetic field. In fact, the same discontinuity can lead to quite different EP values depending on the orientation of the electric field with respect to the interface [22

22 . T. Hirono , Y. Shibata , W.W. Lui , S. Seki , and Y. Yoshikuni , “ The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme ,” IEEE Microwave Guided Wave Lett. 10 , 359 – 361 ( 2000 ). [CrossRef]

, 23

23 . K.-P. Hwang and A.C. Cangellaris , “ Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 158 – 160 ( 2001 ). [CrossRef]

]. Therefore, it is crucial that in the derivation of the EP, not only the geometry, but also the proper boundary conditions are taken into account. Along this line a non-diagonal EP-tensor can be obtained via the homogenization of a partially filled cell [30

30 . J.-Y. Lee and N.-H. Myung , “ Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces ,” Microw. Opt. Technol. Lett. 23 , 245 – 249 ( 1999 ). [CrossRef]

, 31

31 . J. Nadobny , D. Sullivan , W. Wlodarczyk , P. Deuflhard , and P. Wust , “ A 3-D Tensor FDTD-Formulation for Treatment of Slopes Interfaces in Electrically Inhomogeneous Media ,” IEEE Trans. Antennas Propag. 51 , 1760 – 1770 ( 2003 ). [CrossRef]

]. However, its implementation requires the usage of both E and D, implying more storage and CPU time. Moreover, because in the FDTD method the field components are not defined in the same position, a nearest-neighbor average is required for linking E with D. Such average, can wash out the fulfillment of the boundary conditions at the interface. Recently, there have been other original ideas that improve the accuracy of the FDTD method under a rigorous treatment of the electromagnetic field at the dielectric interface, even though they increase the complexity of the algorithm more than the derivation of EPs does [32

32 . K.H. Dridi , J.S. Hesthaven , and A. Ditkowski , “ Staircase-Free Finite-Difference Time-Domain Formulation for General Materials in Complex Geometries ,” IEEE Trans. Antennas Propag. 49 , 749 – 756 ( 2001 ). [CrossRef]

, 33

33 . A. Ditkowski , K. Dridi , and J.S. Hesthaven , “ Convergent Cartesian Grid Methods for Maxwell’s Equations in Complex Geometries ,” J. Comp. Phys. 170 , 39 – 80 ( 2001 ). [CrossRef]

, 34

34 . M. Fujii , D. Lukashevich , I. Sakagami , and P. Russer , “ Convergence of FDTD and Wavelet-Collocation Modeling of Curved Dielectric Interface with the Effective Dielectric Constant Technique ,” IEEE Microwave Wireless Comp. Lett. 13 , 469 – 471 ( 2003 ). [CrossRef]

, 35

35 . T. Xiao and Q.H. Liu , “ A Staggered Upwind Embedded Boundary (SUEB) Method to Eliminate the FDTD Staircasing Error ,” IEEE Trans. Antennas Propag. 52 , 730 – 740 ( 2004 ). [CrossRef]

].

Fig. 1. (a) FDTD mesh showing the staircasing effect for a curved interface (blue line); (b) location of the field components for H-modes and integration lines for Ampére law (blue segment) and Faraday law (red segment) for Ex|i,j-1/2. ∆x and ∆y are the cell dimensions (mesh’s pitch); (i, j) refers to the position of Hz, while Ex and Ey are at (i, j - 1/2) and (i -1/2, j), respectively. Notice that the cells associated to the different field components partially overlap.

2. The Effective Permittivities

In order to obtain EPs, we start from Ampére and Faraday laws in integral form:

tD·nds=H·dl,tB·nds=E·dl.
(1)

Moreover, we restrict the discussion to dielectric non-magnetic media, so that D = εE and B = H, where the electric permittivity and the magnetic permeability are set to one (ε0 = μ0 = 1) for simplicity and e is the relative permittivity. In the two-dimensional FDTD method [6

6 . A. Taflove and S.C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method ( Artech House, Norwood, MA , 2005 ).

], the xy plane is discretized using space increments ∆x and ∆y along x and y, respectively, and the notation (i, j) represents a rectangular cell, with area ∆xy, centered at the position (ix,jy) on the mesh. According to the Yee scheme [5

5 . K.S. Yee , “ Numerical Solution of Initial Boundary Value Problems involving Maxwell’s Equations in Isotropic Media ,” IEEE Trans. Antennas Propag. AP-14 , 302 – 307 ( 1966 ).

, 6

6 . A. Taflove and S.C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method ( Artech House, Norwood, MA , 2005 ).

], the non-zero field components for H-modes are positioned as follows: Ex|i,j-1/2,Ey|j-1/2,j,Dx|i,j-1/2,Dy|i-1/2,j,Hz|i,j,Bz|i,j, where the notation F|i,j is used to name the field component F located in the cell (i,j), see Fig. 1(b). Thus, the electromagnetic field is associated to two orthogonal dual meshes. Hz is at the nodes of one mesh, while Ex and Ey are at the edges of the dual mesh, as it can be noticed from Fig. 1(a) and (b). This captures the topological structure of Maxwell’s equations and provides a geometrical interpretation of the FDTD method [37

37 . A. Bossavit , “ Generalized finite differences in computational electromagnetics ,” Progress in Electromagnetic Research, PIER 32 , 45 – 64 ( 2001 ). [CrossRef]

].

When Eqs. (1) are applied on these meshes, we keep following the Yee scheme for choosing the surface and line integrals: the flux is always computed through a surface normal to the field component, whereas the circulation is computed along the same direction of the field component, as shown in Fig. 1(b) with blue and red lines, respectively. Because the fields are homogeneous along the z direction, Eqs. (1) become:

t(j1)ΔyjΔyDxi,ydy=Hzi,jHzi,j1,t(i1)ΔxiΔxDy|x,jdx=Hz|i1,jHz|i,j,
(2)

for Ampère law and

t(i1/2)Δx(i+1/2)Δx(j1/2)Δy(j+1/2)ΔyHzx,ydxdy=(i1/2)Δx(i+1/2)Δx(Exx,j1/2Exx,j+1/2)dx+
(j1/2)Δy(j+1/2)Δy(Eyi+1/2,yEyi1/2,y)dy,
(3)

for Faraday law. In Eqs. (2) and (3), the subscripts with x or y mean that the field component along the whole integration path is needed, while the subscripts with i, j, i+1/2 and so forth, represent the field component only at the corresponding position on the mesh. Within each integral, if the medium is homogeneous and the mesh is sufficiently fine, the field components can be assumed to be constant, i.e. Ex|x,j-1/2 = Ex|i,j-1/2. With little manipulation and with the time discretisation, Eqs. (2) and (3) become the usual Yee algorithm for H-modes in two-dimensions [6

6 . A. Taflove and S.C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method ( Artech House, Norwood, MA , 2005 ).

].

Fig. 2. Partially filled cells: (a) interface parallel to the field component, (b) interface orthogonal to the field component, (c) inclusion without crossing the integration lines, (d) inclusion crossing both integration lines, (e) inclusion crossing only the integration line of Ampére law, (f) inclusion crossing only the integration line of Faraday law. d,f represent the line filling factors, 1 and 2 mean media with ε1 and ε2 respectively.

In the presence of a dielectric interface that crosses the line integrals, the electric field components cannot be assumed constant any more. The magnetic field is not affected, so that the integral on the left side of Eq. (3) follows the same treatment of the Yee algorithm. Therefore, we focus only on the left sides of Eqs. (2) and on the right side of Eq. (3). The two simplest situations of partially filled cells are sketched in Fig. 2(a) and (b), where a dielectric planar interface is parallel or perpendicular to the field component.

In Fig. 2(a), the interface crosses only the line integral of Eq. (2), while the integral of Eq. (3) can be treated as discussed earlier. The line integral of Eq. (2), which corresponds to the electric flux across the vertical dashed line, can be performed using the continuous tangential electric field, namely E ∥1 = E ∥2 = E , where 1 and 2 refer to the electric field inside materials with electric permittivity equal to ε1 and ε2, respectively. E∥ is the value stored in the computer memory for that cell. Inside the cell, the field component in each medium is assumed to be constant. Applying this rule to Dx, for instance, gives:

(j1)ΔyjΔyDxi,ydy=(j1)Δy(j1)Δy+dDxi,ydy+(j1)Δy+djΔyDxi,yd=[dε2+(Δyd)ε1]Exi,j1/2,
(4)

where d measures the crossing of the dielectric interface in the cell. Therefore, Eq. (4) leads to the original Yee algorithm with the electric permittivity of that cell replaced by an EP ε = ε2(d/∆y)+ε1(1 - d/∆y), as already reported by several works [20

20 . J.G. Maloney and G.S. Smith , “ The Efficient Modeling of Thin Material Sheets in the Finite-Difference Time-Domain (FDTD) Method ,” IEEE Trans. Antennas Propag. 40 , 323 – 330 ( 1992 ). [CrossRef]

,21

21 . N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD Analysis of Dielectric Resonators with Curved Surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ). [CrossRef]

, 22

22 . T. Hirono , Y. Shibata , W.W. Lui , S. Seki , and Y. Yoshikuni , “ The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme ,” IEEE Microwave Guided Wave Lett. 10 , 359 – 361 ( 2000 ). [CrossRef]

, 23

23 . K.-P. Hwang and A.C. Cangellaris , “ Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 158 – 160 ( 2001 ). [CrossRef]

]. The same result can be obtained for the other field component Ey.

(i1/2)Δx(i+1/2)Δx(Exx,j1/2Exx,j+1/2)dx=(i1/2)Δx(i1/2)Δx+f(Exx,j1/2Exx,j+1/2)dx+
(i1/2)Δx+f(i+1/2)Δxε2ε1(Exx,j1/2Exx,j+1/2)dx
=[f+ε2ε1(Δxf)](Exi,j1/2Exi,j+1/2),
(5)

where f measures the crossing of the dielectric interface in the cell. Eq. (5) can be further simplified if the following change of variable is done for the value that has to be stored in the computer memory: E′x|i,j±1/2 = [(f/∆x) + (1 - f/∆x)ε 2/ε 1]Ex|i,j±1/2. With this substitution, Eq. (5) leads to the original Yee algorithm, while Eq. (2) leads to the original Yee algorithm as well, if E′x|i,j-1/2 is used in place of Ex|i,j-1/2 and the electric permittivity of the corresponding cell is replaced by the EP, ε = [(f/∆x)/ε2 + (1 -f/∆x)/ε1]-1. The same permittivity is discussed in [21

21 . N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD Analysis of Dielectric Resonators with Curved Surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ). [CrossRef]

, 22

22 . T. Hirono , Y. Shibata , W.W. Lui , S. Seki , and Y. Yoshikuni , “ The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme ,” IEEE Microwave Guided Wave Lett. 10 , 359 – 361 ( 2000 ). [CrossRef]

, 23

23 . K.-P. Hwang and A.C. Cangellaris , “ Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 158 – 160 ( 2001 ). [CrossRef]

] and more recent works. Notice that the value stored in the computer memory corresponds neither to the electric field before nor to the one after the dielectric interface. A similar approach holds for the Ey component. The EPs derived so far have been shown to preserve second-order accuracy in the Yee algorithm [22

22 . T. Hirono , Y. Shibata , W.W. Lui , S. Seki , and Y. Yoshikuni , “ The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme ,” IEEE Microwave Guided Wave Lett. 10 , 359 – 361 ( 2000 ). [CrossRef]

, 23

23 . K.-P. Hwang and A.C. Cangellaris , “ Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 158 – 160 ( 2001 ). [CrossRef]

].

The case (d), however, is different from the situations discussed so far and requires some more attention. In fact, the dielectric inclusion crosses both integration lines, one with the field component parallel and the other one with the field component perpendicular to the interface. Assuming that we are working with the Ex field component, the integral of Eq. (2) will give an EP ε = ε 2 (dy) + ε 1 (1 - dy). Besides, the integral of Eq. (3) will give the field transformation Ex|i,j-1/2 = [(fx)+(1 - fx)ε 2/ ε 1]Ex|i,j-1/2. The combination of these effects results in a new permittivity for E′x|i,j-1/2, the value stored in the computer memory, given by

εExi,j1/2=εExi,j1/2f/Δx+(1f/Δx)ε2/ε1=εεε2Exi,j1/2.
(6)

Once again, this kind of partially filled cell can be modeled with the usual Yee algorithm using the EP ε eff = ε ε|ε 2.

When the dielectric interfaces are neither parallel nor perpendicular to the field component, but tilted by a certain angle, the previously derived permittivities need to be modified. Again, we consider first the situation where only one integration line is crossed, then both. Figure 3(a) shows a tilted interface that affects only the integrals in Eqs. (2). n is the normal to the interface and d is measured exactly at the crossing with the integration line. In order to fulfill the boundary conditions, the field component has to be projected into normal and parallel components to the interface, using n, and re-projected to the usual component because of the scalar products in Eqs. (1). Therefore, if n is the projection of n along the field component, the integrals in Eqs. (2) become, referring to the Dx component,

(j1)ΔyjΔyDxi,ydy={d[ε1n2+ε2(1n2)]+(Δyd)ε1}Exi,j1/2.
(7)
Fig. 3. Partially filled cells with tilted interfaces: (a) tilted interface crossing only the integration line of Ampère law, (b) tilted interface crossing only the integration line of Faraday law, (c) tilted interface crossing both integration lines, (d) curved interface crossing only the integration line of Ampère law, (e) curved interface crossing only the integration line of Faraday law, (f) curved interface crossing both integration lines. n,m represent unit vectors normal to the interface, d, f represent the line filling factors, 1 and 2 mean media with ε 1 and ε 2, respectively.

The EP ε ∥,n = [ε 1 n 2 + ε 2(1 - n 2)](dy)+ ε 1 (1 - dy) reduces to ε for n = 0. In a similar way, if d > Δy/2, the permittivity is ε ∥,n = ε 2(dy) + [ε 2 n 2 + ε 1(1 - n 2)](1 - dy). The treatment for the other field component is completely analogous.

The opposite situation is shown in Fig. 3(b), where only the integral in Eq. (3) crosses the dielectric interface. By application of the boundary conditions and considering for the moment only the Ex component the integral is computed as follows,

(i1/2)Δx(i+1/2)Δx(Exx,j1/2Exx,j+1/2)dx={(Δxf)+[ε1ε2n2+(1n2)]f}(Exi,j1/2).
(8)

After exploiting a field transformation similar to the one performed for Eq. (5), the result is once again an EP ε ⊥,n = {(fx)[n 2/ε 2 + (1 - n 2)/ε 1]+(1 - fx)/ε 1)-1, which becomes ε for n = 1. In a similar way, if f > Δx/2, the permittivity is ε ⊥,n = {(fx)/ε 2+(1 - fx)[n 2/ε 1 + (1-n 2)/ε 2]}-1.

It is worth to briefly discuss another situation that is quite common in FDTD simulations: partially filled cells with curved interfaces. Some examples of them are shown in Fig. 3(d), (e) and (f). In these cases, the angle between the normal to the interface and the field component is not constant over the FDTD cell, because the dielectric inclusion has a curvature. Following our approach, the normal that is needed for the calculation of the EP is only at the crossing between the integration line and the dielectric interface. Therefore, the result will be as for a tilted interface: ε ∥,n = [ε 1 n 2 + ε 2(1 - n 2)](dy)+ ε 1(1 - dy) for case (d), ε ⊥,n = {(f/∆x)[n 22 + (1 - n 2)/ε1] +(1 - f/∆x)/ε1}-1 for case (e) and εeff,m,n = ε∥,mε⊥,n2 for case (f). Notice that for the latter there are two normals, n and m, inside the formula for the EP. Other situations with curved dielectric inclusions can be treated in a similar manner.

The method of the Contour Path Effective Permittivity (CP-EP) is expected to be more accurate than the staircase approximation, because much more information on the actual geometry is passed to the FDTD algorithm. On the other hand, because this information can be represented using EPs, the algorithm remains as fast and memory efficient (except a little more memory to store more permittivity values) as the basic Yee algorithm. In fact, the calculation of the permittivities can be done before time-marching the fields, maybe with external software, then passed to existing FDTD code. This preprocessing time can be very small, especially if the geometry is defined by means of analytical functions, like in vector graphics. The method can be readily extended to three-dimensions, keeping in mind that the line integrals of Eqs. (2) become surface integrals.

Since we are also interested in finding a trade-off between accuracy and complexity of coding, we propose a simple phenomenological formula for EP that still achieves a great improvement with respect to staircasing. First, instead of considering the crossing between the dielectric interface and the integration lines (surfaces in three-dimensions), we compute the cell filling ratio s, i.e. the fraction of material 2 with respect to the cell area (volume in three-dimensions). Secondly, the normal n to the interface is approximated by defining a unit vector along the connecting line between the center of the cell and the center of curvature of the interface. The idea is to weight ε and ε using the following formula:

εeff=ε(1n2)+εn2,
(9)

where n is the projection of n along the field component of that cell and ε and ε are computed using the filling ratio s, in place of d and f. Eq. (9) represents a Volume-average Polarized Effective Permittivity (VP-EP) that accounts for the orientation of the field with respect to the interface. The formula in Eq. (9) becomes en or ε if n = 0 or n = 1, respectively. The calculation of the VP-EP is much simpler than the CP-EP, but, as a drawback, it is not able to clearly distinguish among all the possible partial fillings that occur and it is not a rigorous derivation from Eqs. (2) and (3). Notice that this formula is similar to the result obtained by Kaneda et al. [21

21 . N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD Analysis of Dielectric Resonators with Curved Surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ). [CrossRef]

], but it is different from the approach presented in [30

30 . J.-Y. Lee and N.-H. Myung , “ Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces ,” Microw. Opt. Technol. Lett. 23 , 245 – 249 ( 1999 ). [CrossRef]

, 31

31 . J. Nadobny , D. Sullivan , W. Wlodarczyk , P. Deuflhard , and P. Wust , “ A 3-D Tensor FDTD-Formulation for Treatment of Slopes Interfaces in Electrically Inhomogeneous Media ,” IEEE Trans. Antennas Propag. 51 , 1760 – 1770 ( 2003 ). [CrossRef]

] by the fact that the resulting dielectric tensor is diagonal and its elements are computed at the position of the relative field component; as a result, the averaging step during the time-marching introduced by [30

30 . J.-Y. Lee and N.-H. Myung , “ Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces ,” Microw. Opt. Technol. Lett. 23 , 245 – 249 ( 1999 ). [CrossRef]

] is not required.

3. Numerical Tests and Discussion

Fig. 4. Layout of the FDTD calculation: CPML layers (gray), cylindrical scatterer (orange), the H-polarized incident plane wave is excited using the total field / scattered field method, the integration line is for the calculation of the total SCS.

The first test considers a low-index dielectric cylinder, ε = 3, with radius r = 400nm. Figure 5(a) shows the total cross section computed using CP-EP, V-EP and Mie theory for Nλ = 25. Nλ is the number of divisions for the shortest wavelength inside the cylinder, λ = 400nm/√ε. For Nλ = 25, a fine FDTD mesh, all EPs agree quite well with Mie theory, so that Fig. 5(a) would look the same also for staircase and VP-EP In order to see the small differences, almost not visible by eye, in Fig. 5(b) we plot the relative error errλ = (SCSFDTD - SCSMie)/SCSMie as a function of wavelength for Nλ = 25. Notice that the largest error accumulates where the SCS exhibits minima and maxima. The fact that the error oscillates between negative and positive values indicates that the SCS computed with FDTD is crossing several times the one computed using Mie theory. While staircase, VP-EP and CP-EP have comparable oscillations, the V-EP clearly exhibit a larger error though still quite small. Since the dielectric contrast is small, we do not expect that the EP converges much better than staircase. However, V-EP is actually doing worse, suggesting that a wrong choice of EP can damage the convergence of the FDTD method. Moreover, in the wavelength region above 0.7 μm, the error associated to staircase, VP-EP and CP-EP is further reduced to a value close to 0.1%, while V-EP is around 0.3%.

Fig. 5. Accuracy on the total SCS: (a) total SCS for Nλ = 25, (b) relative error on the total SCS for Nλ = 25, (c) average relative error on the total SCS. Nλ is the number of divisions for the shortest wavelength in the cylinder; i. e. Nλ = (400nm/∆)/√ε. The phase-velocity error in (c) is computed for Nλ. Parameters: ε = 3, r = 400nm and ∆λ = 1nm.

We also show how this error depends on Nλ. To this purpose, for each Nλ we compute an average error errNλ = Σλ |errλ|/N, where N is the total number of wavelengths used in the sum (N = NDFT). As shown in Fig. 5(c), errNλ exhibits fluctuations for the staircase case. In other words, a finer mesh does not always imply an improved result. In fact, during staircasing it could happen that a particular Nλ is able to match better the geometry than the successive finer division. Such magic-number like behavior is not predictable, especially when the geometry is more complex than a cylinder. On the other hand, the effective permittivities are always improving the result as the mesh gets finer and finer. However, if the EP does not properly preserve the scattering properties of the object, the convergence can be slower, as shown for the V-EP. In our tests we have noticed that also the two-dimensional Maxwell-Garnett EP [40

40 . A. Kirchner , K. Busch , and C.M. Soukoulis , “ Transport properties of random arrays of dielectric cylinders ,” Phys. Rev. B 57 , 277 – 288 ( 1998 ). [CrossRef]

] has given very good convergence results, but not for high-index contrasts.

Fig. 6. Accuracy on the total SCS and the resonant wavelengths: (a) total SCS for Nλ =25, (b) average relative error on the total SCS, (c) relative error on the wavelength of the resonance λ 1 = 675.8nm, (d) relative error on the wavelength of the resonance λ 2 = 5323nm. Nλ is the number of divisions for λ 2 in the cylinder; i. e. Nλ = (λ 2/∆)/√ε. The phase-velocity error in (b)-(d) is computed for Nλ. Parameters: ε = 12, r = 150nm and ∆λ = 0.25nm.

We now consider the error associated to the resonant wavelength for two peaks, the first and the second one starting from the right of Fig. 6(a). The relative error, computed as errpeak = (λ FDTD - λ Mei)λ Mie, is shown in Fig. 6(c) and (d) for the first and second peaks, respectively. In both Fig. 6(c) and (d), V-EP is the worst result. VP-EP and CP-EP are nearly the same and perform better than staircase, both in term of error and in terms of error fluctuations. For the finer meshes, the error is as small as the error on the discrete Fourier transform, which is of the order of 0.1%. A similar behavior has been found for the third sharp peak of Fig. 6(a) at λ ≃ 440nm.

Fig. 7. Accuracy on resonant wavelengths: (a) resonant peak for Nλ = 11 and (b) Nλ = 19, (c) relative error on the wavelength of the resonance λo = 679 4nm. Nλ is the number of divisions for λo in the cylinder; i. e. Nλ = (λo/∆)/√ε. The phase-velocity error in (c) is computed for Nλ. Parameters: ε = 20, r = 120nm and ∆λ = 0.2nm.

As the last example, we present another calculation of resonant wavelengths for a larger dielectric constant, ε = 20, and slightly smaller radius r = 120nm. The peak in the total SCS is shown for two values of Nλ in Fig. 7(a) and (b). Notice the anomalous behavior of staircasing: the calculation seems to be more accurate for the coarser (see Fig. 7(a)) than for the finer mesh (see Fig. 7(b)). This effect is even more evident in Fig. 7(c), where the relative error exhibits significant jumps even for very fine meshes. Once again, V-EP is much less accurate than VP-EP and CP-EP. For the finest mesh, the error is comparable to the wavelength discretization due to the Fourier transform.

Fig. 8. Color maps of the EPs: (a) staircase, (b) V-EP, (c) VP-EP, (d) CP-EP.

Regarding resonance wavelengths, the staircase result jumps around the correct value, while the V-EP approaches to it without fluctuations, but with a larger error. That is why, especially in the absence of analytical solutions or other references, one is led to think that V-EP is better than staircasing. On the other hand, VP-EP and CP-EP are shown to be better than staircasing in terms of accuracy and stability of the error. Moreover, they give very much the same error. To gain more insight on these facts, we look at the permittivity values given by staircase, V-EP, VP-EP and CP-EP. A color map for them is shown in Fig. 8 for the Ex component. Obviously, for staircase (a) only two values are possible, corresponding to black or white. Notice that while V-EP, VP-EP and CP-EP have similar values for the cells close to the top and bottom of the cylinder, V-EP departs from VP-EP and CP-EP for cells close to the left and right sides of the cylinder. Indeed, the interface at the top and bottom is almost parallel to the field component, so that V-EP is close to the correct result. On the contrary, at the right and left sides, the interface is nearly perpendicular to the field component and V-EP is larger than the proper EP. This points out the importance of taking into account the polarization of the field for the derivation of EPs. The fact that VP-EP and CP-EP have nearly the same color maps and yield similar convergence properties suggests that one could use VP-EP, which is very easy to implement, rather than CP-EP.

4. Conclusion

We have proposed two schemes for the calculation of effective permittivities to improve the accuracy of the two-dimensional FDTD method for dielectric media. The CP-EP results from the rigorous application of the field boundary conditions at the crossings between the integration lines and the interface and can be applied to any kind of partially filled cell. The VP-EP is a phenomenological formula that tries to find a compromise between accuracy and complexity of coding. It has been shown that VP-EP is as good as CP-EP over a wide range of tests, so that it can be used in place of CP-EP, being much simpler to implement.

Our results were compared with Mie theory and two other FDTD schemes: staircase and V-EP. All methods approach the Mie-theory result for very fine meshes, but the CP-EP and VP-EP exhibit faster convergence. The major problem of staircasing is the instability of the error, so that one does not know if the mesh is fine enough to fulfill a determined benchmark. On the contrary, V-EP has the advantage of exhibiting a stable error, allowing benchmarking, but it introduces a significant bias that worsens the performances reachable with properly chosen EPs. Even though these tests do not represent a mathematical proof of the convergence of FDTD with EPs, as shown for instance in Ref. [33

33 . A. Ditkowski , K. Dridi , and J.S. Hesthaven , “ Convergent Cartesian Grid Methods for Maxwell’s Equations in Complex Geometries ,” J. Comp. Phys. 170 , 39 – 80 ( 2001 ). [CrossRef]

], we consider them quite convincing and of practical importance.

It would be interesting to extend these ideas to dispersive media and, more importantly, to metals at optical wavelengths, where accurate modeling of complex structures is still a challenging problem with FDTD, especially when surface plasmon resonances are involved.

Acknowledgments

We thank Vahid Sandoghdar for continuous support and encouragement. A.M. acknowledges financial support from the Iranian Ministry of Science and ETH Zurich. This work was performed within the Innovation Initiative project Composite Doped Metamaterials (CDM) of ETH Zurich.

References and links

1 .

Proceedings of the Fifth International Symposium on Photonic and Electromagnetic Crystal Structures (PECS-V) ( Kyoto, Japan , March 7 – 11 , 2004 ); H. Benisty , S. Kawakami , D.J. Norris , and C.M. Soukoulis , eds, Phot. Nanostructures Fund. Appl . 2 , 57 – 159 ( 2004 ); [CrossRef]

C. Jagadish , D.G. Deppe , S. Noda , T.F. Krauss , and O.J. Painter , eds, IEEE J. Sel. Top. Area Commun. 23 , 1305 – 1423 ( 2005 ).

2 .

Special issue on nanostructured optical meta-materials: beyond photonic band gap effects , N. Zheludev and V. Shalaev , eds., J. Opt. A: Pure and Applied Optics , 7 , S1 – S254 ( 2005 ). [CrossRef]

3 .

Proceedings of the EOS Topical Meeting on Advanced Optical Imaging Techniques , ( London, UK , June 29 - July 1, 2005 ).

4 .

M.V.K. Chari and S.J. Salon , Numerical methods in electromagnetism ( Academic Press, San Diego, CA , 2000 )

5 .

K.S. Yee , “ Numerical Solution of Initial Boundary Value Problems involving Maxwell’s Equations in Isotropic Media ,” IEEE Trans. Antennas Propag. AP-14 , 302 – 307 ( 1966 ).

6 .

A. Taflove and S.C. Hagness , Computational Electrodynamics: The Finite-Difference Time-Domain Method ( Artech House, Norwood, MA , 2005 ).

7 .

K.K. Mei , A. Cangellaris , and D.J. Angelakos , “ Conformal Time Domain Finite-Difference Method ,” Radio Sci. 19 , 1145 – 1147 ( 1984 ). [CrossRef]

8 .

R. Holland , “ Finite-Difference Solution of Maxwell’s Equations in Generalized Nonorthogonal Coordinates ,” IEEE Trans. Nucl. Sci. NS-30 , 4589 – 4591 ( 1983 ). [CrossRef]

9 .

M. Fusco , “ FDTD Algorithm in Curvilinear Coordinates ,” IEEE Trans. Antennas Propag. 38 , 76 – 89 ( 1990 ). [CrossRef]

10 .

V. Shankar , A. Mohammadian , and W.F. Hall , “ A Time-Domain Finite-Volume Treatment for the Maxwell Equations ,” Electromagnetics 10 , 127 – 145 ( 1990 ). [CrossRef]

11 .

N.K. Madsen and R.W. Ziolkowski , “ A Three-Dimensional Modified Finite Volume Technique for Maxwell’s Equations ,” Electromagnetics 10 , 147 – 161 ( 1990 ). [CrossRef]

12 .

P.H. Harms , J.-F. Lee , and R. Mittra , “ A Study of the Nonorthogonal FDTD Method Versus the Conventional FDTD Technique for Computing Resonant Frequencies of Cylindrical Cavities ,” IEEE Trans. Microwave Theory Tech. 40 , 741 – 476 ( 1992 ). [CrossRef]

13 .

T.G. Jurgens , A. Taflove , K. Umashankar , and T.G. Moore , “ Finite-Difference Time-Domain Modeling of Curved Surfaces ,” IEEE Trans. Antennas Propag. 40 , 357 – 365 ( 1992 ). [CrossRef]

14 .

T.G. Jurgens and A. Taflove , “ Three-Dimensional Contour FDTD Modeling of Scattering from Single and Multiple Bodies ,” IEEE Trans. Antennas Propag. 41 , 1703 – 1708 ( 1993 ). [CrossRef]

15 .

C.J. Railton , I.J. Craddock , and J.B. Schneider , “ Improved locally distorted CPFDTD algorithm with provable stability ,” Electron. Lett. 31 , 1585 – 1586 ( 1995 ). [CrossRef]

16 .

Y. Hao and C.J. Railton , “ Analyzing Electromagnetic Structures with Curved Boundaries on Cartesian FDTD Meshes ,” IEEE Trans. Microwave Theory Tech. 46 , 82 – 88 ( 1998 ). [CrossRef]

17 .

T.I. Kosmanis and T.D. Tsiboukis , “ A Systematic and Topologically Stable Conformal Finite-Difference Time-Domain Algorithm for Modeling Curved Dielectric Interfaces in Three Dimensions ,” IEEE Trans. Microwave Theory Tech. 51 , 839 – 847 ( 2003 ). [CrossRef]

18 .

I.S. Kim and W.J.R. Hoefer , “ A Local Mesh Refinement Algorithm for the Time Domain-Finite Difference Method Using Maxwell’s Curl Equations ,” IEEE Trans. Microwave Theory Tech. 38 , 812 – 815 ( 1990 ). [CrossRef]

19 .

S.S. Zivanovic , K.S. Yee , and K.K. Mei , “ A Subgridding Method for the Time-Domain Finite-Difference Method to Solve Maxwell’s Equations ,” IEEE Trans. Microwave Theory Tech. 39 , 471 – 479 ( 1991 ). [CrossRef]

20 .

J.G. Maloney and G.S. Smith , “ The Efficient Modeling of Thin Material Sheets in the Finite-Difference Time-Domain (FDTD) Method ,” IEEE Trans. Antennas Propag. 40 , 323 – 330 ( 1992 ). [CrossRef]

21 .

N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD Analysis of Dielectric Resonators with Curved Surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ). [CrossRef]

22 .

T. Hirono , Y. Shibata , W.W. Lui , S. Seki , and Y. Yoshikuni , “ The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme ,” IEEE Microwave Guided Wave Lett. 10 , 359 – 361 ( 2000 ). [CrossRef]

23 .

K.-P. Hwang and A.C. Cangellaris , “ Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 158 – 160 ( 2001 ). [CrossRef]

24 .

S. Dey and R. Mittra , “ A Conformal Finite-Difference Time-Domain Technique for Modeling Cylindrical Dielectric Resonators ,” IEEE Trans. Microwave Theory Tech. 47 , 1737 – 1739 ( 1999 ). [CrossRef]

25 .

W. Yu and R. Mittra , “ On the modeling of periodic structures using the finite-difference time-domain algorithm ,” Microw. Opt. Technol. Lett. 24 , 151 – 155 ( 2000 ). [CrossRef]

26 .

P. Yang , G.W. Kattawar , K.-N. Liou , and J.Q. Lu , “ Comparison of Cartesian grid configurations for application of the finite-difference time-domain method to electromagnetic scattering by dielectric particles ,” Appl. Opt. 43 , 4611 – 4624 ( 2004 ). [CrossRef] [PubMed]

27 .

P. Yang , K.N. Liou , M.I. Mishchenko , and B.-C. Gao , “ Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols , Appl. Opt. 39 , 3727 – 3737 ( 2000 ). [CrossRef]

28 .

W. Yu and R. Mittra , “ A Conformal Finite Difference Time Domain Technique for Modeling Curved Dielectric Surfaces ,” IEEE Microwave Wireless Comp. Lett. 11 , 25 – 27 ( 2001 ). [CrossRef]

29 .

W. Sun and Q. Fu “ Finite-difference time-domain solution of light scattering by dielectric particles with large complex refractive indices ,” Appl. Opt. 39 , 5569 ( 2000 ). [CrossRef]

30 .

J.-Y. Lee and N.-H. Myung , “ Locally tensor conformal FDTD method for modeling arbitrary dielectric surfaces ,” Microw. Opt. Technol. Lett. 23 , 245 – 249 ( 1999 ). [CrossRef]

31 .

J. Nadobny , D. Sullivan , W. Wlodarczyk , P. Deuflhard , and P. Wust , “ A 3-D Tensor FDTD-Formulation for Treatment of Slopes Interfaces in Electrically Inhomogeneous Media ,” IEEE Trans. Antennas Propag. 51 , 1760 – 1770 ( 2003 ). [CrossRef]

32 .

K.H. Dridi , J.S. Hesthaven , and A. Ditkowski , “ Staircase-Free Finite-Difference Time-Domain Formulation for General Materials in Complex Geometries ,” IEEE Trans. Antennas Propag. 49 , 749 – 756 ( 2001 ). [CrossRef]

33 .

A. Ditkowski , K. Dridi , and J.S. Hesthaven , “ Convergent Cartesian Grid Methods for Maxwell’s Equations in Complex Geometries ,” J. Comp. Phys. 170 , 39 – 80 ( 2001 ). [CrossRef]

34 .

M. Fujii , D. Lukashevich , I. Sakagami , and P. Russer , “ Convergence of FDTD and Wavelet-Collocation Modeling of Curved Dielectric Interface with the Effective Dielectric Constant Technique ,” IEEE Microwave Wireless Comp. Lett. 13 , 469 – 471 ( 2003 ). [CrossRef]

35 .

T. Xiao and Q.H. Liu , “ A Staggered Upwind Embedded Boundary (SUEB) Method to Eliminate the FDTD Staircasing Error ,” IEEE Trans. Antennas Propag. 52 , 730 – 740 ( 2004 ). [CrossRef]

36 .

C.F. Bohren and D.R. Huffman , Absorption and Scattering of Light by Small Particles ( Wiley Interscience, New York , 1983 ).

37 .

A. Bossavit , “ Generalized finite differences in computational electromagnetics ,” Progress in Electromagnetic Research, PIER 32 , 45 – 64 ( 2001 ). [CrossRef]

38 .

K.L. Shlager and J.B. Schneider , “ Comparison of the Dispersion Properties of Several Low-Dispersion Finite-Difference Time-Domain Algorithms ,” IEEE Trans. Antennas Propag. 51 , 642 – 652 ( 2003 ). [CrossRef]

39 .

J.A. Roden and S.D. Gedney , “ Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media ,” Microw. Opt. Technol. Lett. 27 , 334 – 339 ( 2000 ). [CrossRef]

40 .

A. Kirchner , K. Busch , and C.M. Soukoulis , “ Transport properties of random arrays of dielectric cylinders ,” Phys. Rev. B 57 , 277 – 288 ( 1998 ). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(260.5740) Physical optics : Resonance
(290.0290) Scattering : Scattering

ToC Category:
Research Papers

Citation
Ahmad Mohammadi, Hamid Nadgaran, and Mario Agio, "Contour-path effective permittivities for the two-dimensional finite-difference time-domain method," Opt. Express 13, 10367-10381 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-25-10367


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References

  1. Proceedings of the Fifth International Symposium on Photonic and Electromagnetic Crystal Structures (PECSV) (Kyoto, Japan, March 7-11, 2004); H. Benisty, S. Kawakami, D.J. Norris, and C.M. Soukoulis, eds, Phot. Nanostructures Fund. Appl. 2, 57-159 (2004); C. Jagadish, D.G. Deppe, S. Noda, T.F. Krauss, and O.J. Painter, eds, IEEE J. Sel. Top. Area Commun. 23, 1305-1423 (2005). [CrossRef]
  2. Special issue on nanostructured optical meta-materials: beyond photonic band gap effects, N. Zheludev, and V. Shalaev, eds., J. Opt. A: Pure and Applied Optics, 7, S1-S254 (2005). [CrossRef]
  3. Proceedings of the EOS Topical Meeting on Advanced Optical Imaging Techniques, (London, UK, June 29 - July 1, 2005).
  4. M.V.K. Chari, and S.J. Salon, Numerical methods in electromagnetism (Academic Press, San Diego, CA, 2000)
  5. K.S. Yee, "Numerical Solution of Initial Boundary Value Problems involving Maxwell's Equations in Isotropic Media," IEEE Trans. Antennas Propag. AP-14, 302-307 (1966).
  6. A. Taflove, and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, MA, 2005).
  7. K.K. Mei, A. Cangellaris, and D.J. Angelakos, "Conformal Time Domain Finite-Difference Method," Radio Sci. 19, 1145-1147 (1984). [CrossRef]
  8. R. Holland, "Finite-Difference Solution of Maxwell's Equations in Generalized Nonorthogonal Coordinates," IEEE Trans. Nucl. Sci. NS-30, 4589-4591 (1983). [CrossRef]
  9. M. Fusco, "FDTD Algorithm in Curvilinear Coordinates," IEEE Trans. Antennas Propag. 38, 76-89 (1990). [CrossRef]
  10. V. Shankar, A. Mohammadian, andW.F. Hall, "A Time-Domain Finite-Volume Treatment for the Maxwell Equations," Electromagnetics 10, 127-145 (1990). [CrossRef]
  11. N.K. Madsen, and R.W. Ziolkowski, "A Three-Dimensional Modified Finite Volume Technique for Maxwell's Equations," Electromagnetics 10, 147-161 (1990). [CrossRef]
  12. P.H. Harms, J.-F. Lee, and R. Mittra, "A Study of the Nonorthogonal FDTD Method Versus the Conventional FDTD Technique for Computing Resonant Frequencies of Cylindrical Cavities," IEEE Trans. Microwave Theory Tech. 40, 741-476 (1992). [CrossRef]
  13. T.G. Jurgens, A. Taflove, K. Umashankar, and T.G. Moore, "Finite-Difference Time-Domain Modeling of Curved Surfaces," IEEE Trans. Antennas Propag. 40, 357-365 (1992). [CrossRef]
  14. T.G. Jurgens, and A. Taflove, "Three-Dimensional Contour FDTD Modeling of Scattering from Single and Multiple Bodies," IEEE Trans. Antennas Propag. 41, 1703-1708 (1993). [CrossRef]
  15. C.J. Railton, I.J. Craddock, and J.B. Schneider, "Improved locally distorted CPFDTD algorithm with provable stability," Electron. Lett. 31, 1585-1586 (1995). [CrossRef]
  16. Y. Hao, and C.J. Railton, "Analyzing Electromagnetic Structures with Curved Boundaries on Cartesian FDTD Meshes," IEEE Trans. Microwave Theory Tech. 46, 82-88 (1998). [CrossRef]
  17. T.I. Kosmanis, and T.D. Tsiboukis, "A Systematic and Topologically Stable Conformal Finite-Difference Time- Domain Algorithm for Modeling Curved Dielectric Interfaces in Three Dimensions," IEEE Trans. Microwave Theory Tech. 51, 839-847 (2003). [CrossRef]
  18. I.S. Kim, and W.J.R. Hoefer, "A Local Mesh Refinement Algorithm for the Time Domain-Finite Difference Method Using Maxwell's Curl Equations," IEEE Trans. Microwave Theory Tech. 38, 812-815 (1990). [CrossRef]
  19. S.S. Zivanovic, K.S. Yee, and K.K. Mei, "A Subgridding Method for the Time-Domain Finite-Difference Method to Solve Maxwell's Equations," IEEE Trans. Microwave Theory Tech. 39, 471-479 (1991). [CrossRef]
  20. J.G. Maloney, and G.S. Smith, "The Efficient Modeling of Thin Material Sheets in the Finite-Difference Time- Domain (FDTD) Method," IEEE Trans. Antennas Propag. 40, 323-330 (1992). [CrossRef]
  21. N. Kaneda, B. Houshmand, and T. Itoh, "FDTD Analysis of Dielectric Resonators with Curved Surfaces," IEEE Trans. Microwave Theory Tech. 45, 1645-1649 (1997). [CrossRef]
  22. T. Hirono, Y. Shibata, W.W. Lui, S. Seki, and Y. Yoshikuni, "The Second-Order Condition for the Dielectric Interface Orthogonal to the Yee-Lattice Axis in the FDTD Scheme," IEEE Microwave Guided Wave Lett. 10, 359-361 (2000). [CrossRef]
  23. K.-P. Hwang, and A.C. Cangellaris, "Effective Permittivities for Second-Order Accurate FDTD Equations at Dielectric Interfaces," IEEE Microwave Wireless Comp. Lett. 11, 158-160 (2001). [CrossRef]
  24. S. Dey, and R. Mittra, "A Conformal Finite-Difference Time-Domain Technique for Modeling Cylindrical Dielectric Resonators," IEEE Trans. Microwave Theory Tech. 47, 1737-1739 (1999). [CrossRef]
  25. W. Yu, and R. Mittra, "On the modeling of periodic structures using the finite-difference time-domain algorithm," Microw. Opt. Technol. Lett. 24, 151-155 (2000). [CrossRef]
  26. P. Yang, G.W. Kattawar, K.-N. Liou, and J.Q. Lu, "Comparison of Cartesian grid configurations for application of the finite-difference time-domain method to electromagnetic scattering by dielectric particles," Appl. Opt. 43, 4611-4624 (2004). [CrossRef] [PubMed]
  27. P. Yang, K.N. Liou, M.I. Mishchenko, and B.-C. Gao, "Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols," Appl. Opt. 39, 3727-3737 (2000). [CrossRef]
  28. W. Yu, and R. Mittra, "A Conformal Finite Difference Time Domain Technique for Modeling Curved Dielectric Surfaces," IEEE Microwave Wireless Comp. Lett. 11, 25-27 (2001). [CrossRef]
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