## Generation of adaptive coordinates and their use in the Fourier Modal Method |

Optics Express, Vol. 18, Issue 22, pp. 23258-23274 (2010)

http://dx.doi.org/10.1364/OE.18.023258

Acrobat PDF (1320 KB)

### Abstract

We present an improvement of the standard Fourier Modal Method (FMM) for the analysis of lamellar gratings that is based on the use of *automatically generated* adaptive coordinates for arbitrarily shaped material profiles in the lateral plane of periodicity. This allows for an accurate resolution of small geometric features and/or large material contrasts within the unit. For dielectric gratings, we obtain considerable convergence accelerations. Similarly, for metallic gratings, our approach allows efficient and accurate computations of transmittance and reflectance coefficients into various Bragg orders, the spectral positions of Rayleigh anomalies, and field enhancement values within the grating structures.

© 2010 Optical Society of America

## 1. Introduction

1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. **444**, 101–202 (2007). [CrossRef]

2. F. J. Garcia de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**, 1267–1290 (2007). [CrossRef]

3. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. **108**, 494–521 (2008). [CrossRef] [PubMed]

4. M. G. Moharam and T. K. Gaylord, “Diffraction analysis fo dielectric surface-relief gratings,” J. Opt. Soc. Am. **72**, 1385–1392 (1982). [CrossRef]

8. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

9. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1034 (1996). [CrossRef]

6. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A **13**, 779–784 (1996). [CrossRef]

7. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A **13**, 1019–1023 (1996). [CrossRef]

8. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

12. T. Schuster, J. Ruoff, N. Kerwien, Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A24, 2880–2890 (2007). [CrossRef]

13. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A **16**, 2510–2516 (1999). [CrossRef]

14. J. Chandezon, M. T. Dupuis, G. Gornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. **72**, 839–846 (1982). [CrossRef]

15. T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express10, 24–34 (2002). [PubMed]

17. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express **17**, 8051–8061 (2009). [CrossRef] [PubMed]

18. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. **8**, 247 (2006). [CrossRef]

19. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking devices, electromagnetic wormholes, and transformation optics,” SIAM Review **51**, 3–33 (2009). [CrossRef]

*automatically generated*and can thus be applied to arbitrary material profiles. This has not yet been fully accomplished in the early pioneering works [15

15. T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express10, 24–34 (2002). [PubMed]

17. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express **17**, 8051–8061 (2009). [CrossRef] [PubMed]

15. T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express10, 24–34 (2002). [PubMed]

13. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A **16**, 2510–2516 (1999). [CrossRef]

17. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express **17**, 8051–8061 (2009). [CrossRef] [PubMed]

20. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B **71**, 195108 (2005). [CrossRef]

21. F. Gygi, “Electronic-structure calculations in adaptive coordinates,” Phys. Rev. B **48**, 11692–11700 (1993). [CrossRef]

22. P. Götz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express **16**, 17295–17301 (2008). [CrossRef] [PubMed]

20. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B **71**, 195108 (2005). [CrossRef]

21. F. Gygi, “Electronic-structure calculations in adaptive coordinates,” Phys. Rev. B **48**, 11692–11700 (1993). [CrossRef]

20. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B **71**, 195108 (2005). [CrossRef]

21. F. Gygi, “Electronic-structure calculations in adaptive coordinates,” Phys. Rev. B **48**, 11692–11700 (1993). [CrossRef]

22. P. Götz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express **16**, 17295–17301 (2008). [CrossRef] [PubMed]

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

## 2. Fourier Modal Method in general coordinates

*Ox̄*

^{1}

*x̄*

^{2}

*x̄*

^{3}that are periodic in the (lateral)

*x̄*

^{1}

*x̄*

^{2}-plane and finite in the

*x̄*

^{3}-direction, the propagation direction (see Fig. 1). A plane wave with wave vector

**k**

_{in}is incident on this structure and the direction of the wave vector defines the polar angle and azimuthal angles,

*φ*and

*θ*. For concreteness and since this is a rather often occuring setup [2

2. F. J. Garcia de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**, 1267–1290 (2007). [CrossRef]

3. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. **108**, 494–521 (2008). [CrossRef] [PubMed]

*h*, and (iii) a semi-infinite substrate region in which the radiation that is transmitted into certain Bragg orders has to be determined.

*d*

_{1}and

*d*

_{2}that contain a distribution of isotropic and nonmagnetic dielectric materials whose profile is described by

*ɛ*

*̄*(

*x̄*

^{1},

*x̄*

^{2}). The corresponding constituent materials are thus described via frequency-dependent complex dielectric constants and this includes many dispersive materials, notably metals. However, we would like to note that the subsequent developments can easily be extended to optically anisotropic materials such as liquid crystals and non-rectangular systems such as hexagonal arrays.

*Ox*

^{1}

*x*

^{2}

*x*

^{3}that is connected to the original Cartesian system (

*Ox̄*

^{1}

*x̄*

^{2}

*x̄*

^{3}≡

*Oxyz*) via Since our goal is to develop adaptive spatial resolution (ASR) within the lateral grating plane, we have restricted the coordinate transformation (1)–(3) accordingly and further require that the transformation retains the periodicity in the lateral plane with periodicities

*d*

_{1}and

*d*

_{2}in

*x̄*

^{1}- and

*x̄*

^{2}-direction, respectively. The contravariant metric tensor associated with these curvilinear coordinates is defined as As a consequence, we obtain the covariant form of Maxwell’s curl equations for our material distribution as Here,

*E*and

_{m}*H*denote, respectively, the covariant electric and magnetic field components and we have introduced the vacuum wave number

_{m}*k*

_{0}=

*ω*/

*c*. Furthermore,

*g*represents the reciprocal of the determinant of the metric tensor (4),

*ξ*is the (totally antisymmetric) Levi-Civita tensor, and

*∂*is an abbreviation for

_{l}*∂*/

*∂x*.

_{l}*x̄*

^{3}-direction for which the application of FMM (including the use of Fourier factorization rules) has been described by Li [23].

*x*

^{1}

*x*

^{2}-plane into a Floquet-Fourier series Here,

*F*(

_{σ}*F*=

*E,H*and

*σ*= 1, 2, 3) stands for any field component with corresponding Floquet-Fourier coefficients

*f*and we have introduced the abbreviation

_{σmn}*α*=

_{m}*k*

_{in}sin

*θ*cos

*φ*+

*m*2

*π*/

*d*

_{1}and

*β*=

_{n}*k*

_{in}sin

*θ*sin

*φ*+

*n*2

*π*/

*d*

_{2}. In an actual computation the sum over reciprocal lattice vectors in (9) has to be truncated

*m*,

*n*= 0, ±1, ±2,.. so that in total

*N*Floquet-Fourier coefficients have to be determined by solving Maxwell’s equations in Fourier space. In general, we select those

*N*reciprocal lattice vectors (

*m*2

*π*/

*d*

_{1},

*n*2

*π*/

*d*

_{2}) that lie closest to the origin in reciprocal space. This allows for the most flexible representation of an arbitrary material distribution inside the unit cell. Different choices would be possible for special geometries in the unit cell. However, our (rather extensive) experience indicates that this leads to only marginal improvements in the convergence rates.

*ɛ*and

^{kl}*μ*for the general case of general curvilinear coordinate transformations. Although we consider only isotropic materials, a general coordinate transformation will, in the transformed space, induce effective anisotropic material properties (transformation optics).

^{kl}*x*

^{3}-dependence As a result, we are able to set up an eigenvalue problem for the propagation constant

*q*in each region. The resulting eigenvectors are the eigenmodes in the respective region and are used as an expansion basis for the field components. The expansions in the different regions are connected in curvilinear space by a scattering matrix formalism [24

24. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A **20**, 655–660 (2003). [CrossRef]

10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

25. G. Granet and B. Guizal, “Analysis of strip gratings using a parametric modal method by Fourier expansions,” Opt. Commun. **255**, 1–11 (2005). [CrossRef]

## 3. Generation of adaptive coordinates

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

**71**, 195108 (2005). [CrossRef]

*M*plane waves, giving a total number of 2

*M*expansion coefficients, i.e.,

*M*coefficients

*M*coefficients

*m*,

*n*= 0, ±1, ±2,.... We aim at the most flexible, i.e., the most symmetric representation of the transformation since this can be adapted best to general material distributions within the unit cell. Therefore, we construct this plane-wave representation by selecting the

*M*reciprocal lattice vectors (

*m*2

*π*/

*d*

_{1}

*x*

^{1},

*n*2

*π*/

*d*

_{2}

*x*

^{2}) to be those that lie within a circle around the origin in reciprocal space. In general, the coefficients

*M*free parameters.

**71**, 195108 (2005). [CrossRef]

**48**, 11692–11700 (1993). [CrossRef]

_{c}and ℰ

_{s}represent, respectively, compression and shear energy [21

**48**, 11692–11700 (1993). [CrossRef]

*η*

_{c}and

*η*

_{s}control the relative strength of these two energy terms within the functional. Similar to the work of Pearce et al. [20

**71**, 195108 (2005). [CrossRef]

*η*

_{s}= 0.

*E*

_{g}and

*E*

_{t}, are responsible for adapting the coordinates to the geometry of the material profile contained in the unit cell as described by the permittivity distribution

*ɛ*(

*x*

^{1}

*,x*

^{2}). For the subsequent discussion, we assume that we are dealing with a binary grating, i.e., that we have two constituent materials within the unit cell, a host material matrix in which certain patches of a guest material are embedded. Since our curvilinear coordinates should faithfully represent the structural details without reference to the actual permittivity values of the host and guest material, we characterize the structural information via the function

*S*(

*x*

^{1}

*,x*

^{2}). This functions takes the values 0 or 1 for points (

*x*

^{1}

*,x*

^{2}) that, respectively, lie in the host or guest material regions. Since the gradient of

*S*(

*x*

^{1}

*,x*

^{2}) is ill-defined at step-like material interfaces, we apply a Gaussian smoothing to the structure function

*S*(

*x*

^{1}

*,x*

^{2}). This smoothing is performed in Fourier space by multiplying the Fourier coefficients of the permittivity distribution with a Gaussian (filter) function. As a result, the smoothed structure function

*S*

_{sm}(

*x̄*

^{1},

*x̄*

^{2}) in real space is where we have introduced the smoothing width

*w*

_{s}and the smoothed Fourier coefficients that are obtained from the Fourier coefficients

*S*of the structure function

_{pq}*S*(

*x*

^{1},

*x*

^{2}). As a result, the gradient of the smoothed structure function yields a vector field whose vectors are locally normal to the material interfaces in a vicinity of the interface and accept their largest magnitude directly at the interface. In Fig. 2, we display this vector field for a single circular patch of a guest material that lies at the center of a quadratic unit cell.

_{g}that tends to distort coordinate lines in such a way that their density is increased in regions with large gradients of the structure function. This term has been discussed in Ref. [20

**71**, 195108 (2005). [CrossRef]

*η*

_{c}and

*η*

_{s}, as introduced above (see (15) and (16)) as well as the weighting factor

*η*

_{t}of the tangential contribution to be introduced below (see (20)) weight these contributions relative to the gradient contribution (19).

22. P. Götz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express **16**, 17295–17301 (2008). [CrossRef] [PubMed]

_{t}that lowers the fictitious energy (see Eq. 14) when one of the coordinate lines of the transformation (locally) runs parallel to the material interfaces. Explicitly, the tangent vectors along the coordinate lines are given by the covariant basis vectors

**71**, 195108 (2005). [CrossRef]

26. http://www.gnu.org.

*x̄*

^{1},

*x̄*

^{2})-space. Specifically, we choose the number of grid points within the unit cell such that (i) sufficient oversampling relative to the Fourier representation of the coordinate transformation is achieved and (ii) that the structural features are adequately resolved. For the results shown in this paper, we typically use a grid of 200 × 200 points and have checked that a finer discretization in transformed space does not change the numerical results. In order to reduce the computational complexity, we enforce the symmetry of the system onto the coordinate transformation Eq. 12 and Eq. 13 whenever possible.

## 4. Performance characteristics

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

*ɛ*= 2.25 and an air superstrate (

*ɛ*= 1) that both occupy a half space. On the substrate, we deposit a single layer of periodically placed cylinders with height

*h*and a certain cross-section which we call the motif. These patches are arranged into a square lattice such that the quadratic unit cell exhibits side lengths

*d*

_{1}=

*d*

_{2}= 1000 nm (see Fig. 1 (a)). The description of these patches of guest material is completed by specificying the material’s dielectric constant

*ɛ*. Further, we assume that host material is air so that we describe a typical monolayer of metamaterials [1

1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. **444**, 101–202 (2007). [CrossRef]

2. F. J. Garcia de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. **79**, 1267–1290 (2007). [CrossRef]

3. M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. **108**, 494–521 (2008). [CrossRef] [PubMed]

*ω*and polarization

*ê*. Oblique illumination is equally possible but the corresponding results do not provide any further insight to the workings of ASR within FMM.

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

*ɛ*= 12) and metallic objects. In the latter case, we use a Drude model with parameters that reasonably well describe gold within the relevant frequency range (plasma frequency

*ω*

_{p}= 1.3544 × 10

^{16}1/s and collision frequency

*ω*

_{col}= 1.1536 × 10

^{14}1/s; see Ref. [27

27. A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B **71**, 085416 (2005). [CrossRef]

### 4.1. Square disk

*h*= 50 nm, square motif with side length

*w*= 500 nm; see Fig. 3) so that the disks are perfectly aligned to the Cartesian coordinate lines. As a result, this system exhibits rather good convergence within standard FMM [10

10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A **14**, 2758–2767 (1997). [CrossRef]

*G*= 0.001.

*M*= 97 plane wave coefficients, a smoothing width

*w*

_{s}= 400, and relative weights of the energy terms

*η*

_{c}= 0.1,

*η*

_{s}= 0, and

*η*

_{t}= 0.1. The 𝒞

_{4}

*symmetry of the structure has been enforced in the minimization process. For further comparison, we have also generated a mesh with the tangential energy term switched off, i.e., with*

_{v}*η*

_{c}= 0.1,

*η*

_{s}= 0, and

*η*

_{t}= 0 (not shown). This structure is normally illuminated by a plane wave with vacuum wavelength

*λ*and linear polarization along the system’s

*y*-axis.

*ɛ*= 12. The total resolution of the structure in real space is given on a 1024 × 1024 point grid. Table 1 shows our results on the convergence of the largest real-valued propagation constant

*q*for standard FMM [10

**14**, 2758–2767 (1997). [CrossRef]

*N*of plane waves that we have considered in this example.

*N*= 317 Fourier coefficients and, at first sight, rather good agreement between the different approaches is obtained. Nevertheless, we have examined the convergence characteristics of the system more carefully at the particle plasmon resonance of the square disk near

*λ*= 1600 nm. In Fig. 4 (b), we display the convergence of the transmittance values into the zeroth diffraction order as a function of the number of plane waves

*N*. The best convergence is reached for the analytical ASR, closely followed by the numerical ASR with tangential energy term. Based on this, we can draw the conclusion that grid-aligned structures can be treated rather satisfactorily within standard FMM, i.e., without ASR. Even in this case, the use of ASR – be it analytical or numerical – within FMM provides a rather welcome convergence acceleration relative to standard FMM. Based on the inner workings of standard FMM (see section 1), we expect a significantly different behavior for the case of structures that are not grid aligned, so that in the next section, we turn to the analysis of such structures.

### 4.2. Circular disk

*h*= 50 nm and radius

*r*= 500 nm (see Fig. 5 (a)). In order to generate the numerical ASR, we employ a reduced smoothing width

*w*

_{s}= 200 that accounts for the fact that the circular motif within the lateral plane does, in contrast to the square motif discussed in section 4.1, not exhibit any sharp corners. The corresponding parameters for the energy functional are kept the same, i.e., we use

*η*

_{c}= 0.1,

*η*

_{s}= 0,

*η*

_{t}= 0.1, employ

*M*= 97 coefficients to describe the coordinate transformation, and enforce the 𝒞

_{4v}-symmetry of the structure. We depict the resulting ASR in Fig. 5 (b). This numerically generated ASR exhibits a certain degree of similarity with the analytical ASR that has been described in Ref. [17

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

**17**, 8051–8061 (2009). [CrossRef] [PubMed]

*ɛ*= −110.9 + 11.24i; corresponding to gold at

*λ*= 1530 nm) that have been obtained within standard FMM (

*N*= 1257 plane wave), analytical ASR within FMM (

*N*= 317 plane waves), and numerical ASR within FMM with tangential energy term (see Fig. 5 (b);

*N*= 317 plane waves). The spectra of both ASR approaches agree well and – as we will demonstrate below – are essentially converged to within 5 significant digits even for as few as

*N*= 317 plane waves. In contrast, standard FMM is far from convergence even if as many as

*N*= 1257 plane waves are employed.

*λ*= 1530nm. In Fig. 7 (a), we display the convergence behavior for the transmittance into the zeroth diffraction order as a function of the number

*N*of plane waves. Clearly, both the analytical and the numerical ASR converge to the same value and

*N*= 317 is sufficient to obtain convergence to within 5 significant digits. The results for standard FMM show a rather poor convergence. To further illustrate this, we have implemented the symmetry reduction technique for standard FMM described in Ref. [28] that – for given computational resources and normal incidence – allows us to push the number of plane waves significantly higher.

*N*= 23993 plane waves.

29. J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, “Fabrication of Crescent-Shaped Optical Antennas,” Adv. Mater. **17**, 2131–2134 (2005). [CrossRef]

31. Y. Choi, S. Hong, and L. P. Lee, “Shadow Overlap Ion-beam Lithography for Nanoarchitectures,” Nano Lett. **9**, 3726–3731 (2009). [CrossRef] [PubMed]

## 5. Realistic system

29. J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, “Fabrication of Crescent-Shaped Optical Antennas,” Adv. Mater. **17**, 2131–2134 (2005). [CrossRef]

31. Y. Choi, S. Hong, and L. P. Lee, “Shadow Overlap Ion-beam Lithography for Nanoarchitectures,” Nano Lett. **9**, 3726–3731 (2009). [CrossRef] [PubMed]

*w*= 500 nm and lateral thickness of

*t*= 200 nm. In order to faithfully reproduce the actual experimental realizations we have added corner roundings, each with radius of 10nm, at the two tips. As for the previous structures, the height of the crescents is

*h*= 50 nm and they are situated on a glass substrate.

*w*

_{s}= 200 and the same number

*M*= 97 of Fourier coefficients as before but have used different relative weights of the energy terms according to

*η*

_{c}= 0.2,

*η*

_{s}= 0, and

*η*

_{t}= 0.2 and enforce the mirror symmetry of the structure. Here, we choose different weights as in section 4 since the crescent shape has smaller features and thus the weights have to be larger as otherwise the gradient contribution would become too large at these points. In turn, this would lead to an absurdly large point density at that same grid position.

*w*

_{s}= 200,

*η*

_{c}= 0.2,

*η*

_{s}= 0, and

*η*

_{t}= 0 (not shown).

*x*- and

*y*-polarized illumination by normally incident plane waves. Similar to the case of the circular disk array, standard FMM with

*N*= 1257 reciprocal lattice vectors is not converged while the computations that utilize numerical ASR within FMM require only

*N*= 317 reciprocal lattice vectors to achieve well converged results (see also Fig. 10). A comparison of these spectra with the corresponding spectra of U-shape nanoparticles (see Ref. [1

1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. **444**, 101–202 (2007). [CrossRef]

*x*-polarized excitation is located at

*λ*= 1100 nm (see also Fig. 11 (b)) while the so-called magnetic resonance under

*y*-polarized excitation is located at

*λ*= 1900 nm (see also Fig. 11 (a)). These resonances as well as Rayleigh anomalies due to the periodicity of the array are clearly observed in the numerical ASR within FMM results.

*N*= 1257 reciprocal lattice vectors. A comparison with results for different numbers of plane waves yields that these field values are within 5% of the converged results. The corresponding enhancement of the electric field in a plane 25 nm above the glas substrate, i.e., halfway through the metal film that is 50 nm high, are shown in Fig. 11 (a) and (b), respectively. The field enhancement at the tips is clearly visible and the field distributions of these modes are rather similar those of their counterparts at the electric and magnetic resonances of U-shaped nanoparticles [1

**444**, 101–202 (2007). [CrossRef]

## 6. Conclusion

*automatically generated*through the minimization of an appropriate fictitious energy functional in such a way that (i) the point density near material interfaces is increased and (ii) one of the coordinate lines is (locally) aligned parallel relative to the material boundary that is provided by the local shape of the object. While an energy functional that incorporates the former aspect has been used in the context of bandstructure computations before [20

**71**, 195108 (2005). [CrossRef]

29. J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, “Fabrication of Crescent-Shaped Optical Antennas,” Adv. Mater. **17**, 2131–2134 (2005). [CrossRef]

30. H. Rochholz, N. Bocchio, and M. Kreiter, “Tuning resonances on crescent-shaped noble-metal nanoparticles,” N. J. Phys. **9**, 53 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. |

2. | F. J. Garcia de Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. |

3. | M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured plasmonic sensors,” Chem. Rev. |

4. | M. G. Moharam and T. K. Gaylord, “Diffraction analysis fo dielectric surface-relief gratings,” J. Opt. Soc. Am. |

5. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation of stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

6. | P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A |

7. | G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A |

8. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

9. | L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

10. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

11. | M. Nevière and E. Popov, |

12. | T. Schuster, J. Ruoff, N. Kerwien, Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A24, 2880–2890 (2007). [CrossRef] |

13. | G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A |

14. | J. Chandezon, M. T. Dupuis, G. Gornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. |

15. | T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express10, 24–34 (2002). [PubMed] |

16. | G. Granet and J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A |

17. | T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express |

18. | U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” N. J. Phys. |

19. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Cloaking devices, electromagnetic wormholes, and transformation optics,” SIAM Review |

20. | G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B |

21. | F. Gygi, “Electronic-structure calculations in adaptive coordinates,” Phys. Rev. B |

22. | P. Götz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express |

23. | L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A5, 345–355 (2003). |

24. | L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A |

25. | G. Granet and B. Guizal, “Analysis of strip gratings using a parametric modal method by Fourier expansions,” Opt. Commun. |

26. | |

27. | A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B |

28. | B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C |

29. | J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, “Fabrication of Crescent-Shaped Optical Antennas,” Adv. Mater. |

30. | H. Rochholz, N. Bocchio, and M. Kreiter, “Tuning resonances on crescent-shaped noble-metal nanoparticles,” N. J. Phys. |

31. | Y. Choi, S. Hong, and L. P. Lee, “Shadow Overlap Ion-beam Lithography for Nanoarchitectures,” Nano Lett. |

**OCIS Codes**

(090.1970) Holography : Diffractive optics

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 27, 2010

Revised Manuscript: September 10, 2010

Manuscript Accepted: September 13, 2010

Published: October 20, 2010

**Citation**

Sabine Essig and Kurt Busch, "Generation of adaptive coordinates and their use in the Fourier Modal Method," Opt. Express **18**, 23258-23274 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23258

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### References

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- L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996). [CrossRef]
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1034 (1996). [CrossRef]
- L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
- M. Nevière, and E. Popov, Light propagation in periodic media: Differential theory and design, Marcel Dekker (New York), 2003.
- T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, "Normal vector method for convergence improvement using the RCWA for crossed gratings," J. Opt. Soc. Am. A 24, 2880-2890 (2007). [CrossRef]
- G. Granet, "Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999). [CrossRef]
- J. Chandezon, M. T. Dupuis, G. Gornet, and D. Maystre, "Multicoated gratings: a differential formalism applicable in the entire optical region," J. Opt. Soc. Am. 72, 839-846 (1982). [CrossRef]
- T. Vallius, and M. Honkanen, "Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles," Opt. Express 10, 24-34 (2002). [PubMed]
- G. Granet, and J.-P. Plumey, "Parametric formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. A 4, S145-S149 (2002).
- T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, "Matched coordinates and adaptive spatial resolution in the Fourier modal method," Opt. Express 17, 8051-8061 (2009). [CrossRef] [PubMed]
- U. Leonhardt, and T. G. Philbin, "General relativity in electrical engineering," N. J. Phys. 8, 247 (2006). [CrossRef]
- A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, "Cloaking devices, electromagnetic wormholes, and transformation optics," SIAM Rev. 51, 3-33 (2009). [CrossRef]
- G. J. Pearce, T. D. Hedley, and D. M. Bird, "Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals," Phys. Rev. B 71, 195108 (2005). [CrossRef]
- F. Gygi, "Electronic-structure calculations in adaptive coordinates," Phys. Rev. B 48, 11692-11700 (1993). [CrossRef]
- P. G¨otz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Opt. Express 16, 17295-17301 (2008). [CrossRef] [PubMed]
- L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).
- L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003). [CrossRef]
- G. Granet, and B. Guizal, "Analysis of strip gratings using a parametric modal method by Fourier expansions," Opt. Commun. 255, 1-11 (2005). [CrossRef]
- http://www.gnu.org.
- A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. Lamy de la Chapelle, "Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method," Phys. Rev. B 71, 085416 (2005). [CrossRef]
- B. Bai, and L. Li, "Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with C4 symmetry," J. Opt. A 7, 783-789 (2005).
- J. S. Shumaker-Parry, H. Rochholz, and M. Kreiter, "Fabrication of Crescent-Shaped Optical Antennas," Adv. Mater. 17, 2131-2134 (2005). [CrossRef]
- H. Rochholz, N. Bocchio, and M. Kreiter, "Tuning resonances on crescent-shaped noble-metal nanoparticles," N. J. Phys. 9, 53 (2007). [CrossRef]
- Y. Choi, S. Hong, and L. P. Lee, "Shadow Overlap Ion-beam Lithography for Nanoarchitectures," Nano Lett. 9, 3726-3731 (2009). [CrossRef] [PubMed]

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