## Three-dimensional adaptive coordinate transformations for the Fourier modal method |

Optics Express, Vol. 22, Issue 2, pp. 1342-1349 (2014)

http://dx.doi.org/10.1364/OE.22.001342

Acrobat PDF (5102 KB)

### Abstract

The concepts of adaptive coordinates and adaptive spatial resolution have proved to be a valuable tool to improve the convergence characteristics of the Fourier Modal Method (FMM), especially for metallo-dielectric systems. Yet, only two-dimensional adaptive coordinates were used so far. This paper presents the first systematic construction of three-dimensional adaptive coordinate and adaptive spatial resolution transformations in the context of the FMM. For that, the construction of a three-dimensional mesh for a periodic system consisting of two layers of mutually rotated, metallic crosses is discussed. The main impact of this method is that it can be used with any classic FMM code that is able to solve the large FMM eigenproblem. Since the transformation starts and ends in a Cartesian mesh, only the transformed material tensors need to be computed and entered into an existing FMM code.

© 2014 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]

*xy*-plane and finite in

*z*-direction. The system is sliced into layers with constant permittivity in

*z*-direction and in each of these layers an eigenvalue problem is solved which stems from Maxwell’s curl equations. This allows expanding the fields into eigenmodes. The layers are then connected using a scattering matrix algorithm which ensures the fulfillment of the continuity conditions [2

2. 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]

3. 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]

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

6. 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]

8. 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). [CrossRef] [PubMed]

9. 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]

11. J. Küchenmeister, T. Zebrowski, and K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express **20**, 17319–17347 (2012). [CrossRef] [PubMed]

*xy*-plane. However, complex structures occurring in different layers pose a problem since different adaptive meshes would be necessary. How to connect these different meshes optimally remains a challenging task since each mesh represents a different basis. Also, the incident plane waves need to be transformed which induces additional errors. These problems can be tackled by designing a three-dimensional adaptive coordinate transformation. This method trades an increased amount of slices in the method for an accurate representation of the structure’s surface in all three dimensions. In this paper, such a three-dimensional transformation is designed for a system that has gathered an extensive amount of interest in recent years, two periodic layers of mutually rotated, metallic crosses [12

12. M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. **34**, 2501–2503 (2009). [CrossRef] [PubMed]

*z*-direction.

13. V. Liu and S. Fan, “S^{4}: A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. **183**, 2233–2244 (2012). [CrossRef]

## 2. Covariant formulation of the Fourier Modal Method with generalized coordinates

9. 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]

11. J. Küchenmeister, T. Zebrowski, and K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express **20**, 17319–17347 (2012). [CrossRef] [PubMed]

15. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A **5**, 345–355 (2003). [CrossRef]

*Ox*

^{1}

*x*

^{2}

*x*

^{3}and a Cartesian coordinate system

*Ox̄*

^{1}

*x̄*

^{2}

*x̄*

^{3}. The three-dimensional adaptive coordinate transformations that are investigated in this paper have the form Eventually, we want to solve Maxwell’s curl equations which read in covariant form, see [15

15. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A **5**, 345–355 (2003). [CrossRef]

*ξ*denotes the Levi-Civita symbol and

*E*and

_{σ}*H*are covariant components of the electric and magnetic field. Throughout the manuscript, Greek indices run from 1 to 3. Furthermore, we use the Einstein sum convention, meaning that repeated indices are implicitly summed over. The vacuum wave number is denoted

_{σ}*k*

_{0}=

*ω/c*with the frequency

*ω*and the speed of light

*c*. The metric tensor reads and

*g*(as used in Eqs. (4) and (5)) denotes the reciprocal of its determinant. As illustrated in detail in [11

11. J. Küchenmeister, T. Zebrowski, and K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express **20**, 17319–17347 (2012). [CrossRef] [PubMed]

*ε̄*is the permittivity tensor in the Cartesian system. The permeability transforms identically. It is noteworthy that the matrix that is Fourier transformed when using FMM with AC and/or ASR is not the permittivity itself given in Eq. (7) but rather

^{τχ}*effective permittivity*. By entering Eqs. (1)–(3) in Eq. (7) one can observe that both the effective permittivity and the effective permeability become fully anisotropic. Therefore, the full anisotropic FMM eigenvalue problem has to be solved.

## 3. Three-dimensional adaptive coordinates

### 3.1. Two-dimensional mesh for a rotated cross

**20**, 17319–17347 (2012). [CrossRef] [PubMed]

*P*,

*Q*,

*R*and

*S*define specific coordinate lines (blue and red). Figure 2(b) depicts how these coordinate lines are mapped. The mapping in between these specific coordinate lines is given by a linear interpolation. The resulting mesh is depicted in Fig. 2(c). One may notice that we could have chosen different specific coordinate lines which would also lead to a grid-aligned cross in the effective permittivity. The reason we choose the ones shown in Fig. 2 becomes clear in the next paragraph.

### 3.2. Constructing the three-dimensional transformation

*x*

^{3}. This value of

*x*

^{3}directly translates into a rotation angle which in turn translates to a mesh like in section 3.1. Explicitly, this means that we perform coordinate transformations in the space between the crosses, too. This explains why we constructed the mesh in Fig. 2 the way we did–for a given

*x*

^{3}value we only compute the rotation angle and easily obtain the planar mesh. In particular, we hereby make sure that the grid-aligned crosses are directly above each other in the transformed space.

*h*and the distance between the crosses is denoted

*b*. Since we want the upper cross to be rotated by the angle

*φ*

_{0}, we obtain for the rotational dependence of the planar mesh on the

*x*

^{3}coordinate.

*x*

^{3}. The system is a square lattice with lattice constant 600 nm. The cross is 250 nm in diameter and the width of the arms of the crosses is 50 nm. The height of the crosses is

*h*= 25 nm and the spacing between them is

*b*= 50 nm. The angle by which the upper cross is rotated is

*φ*= 15°. We assume the crosses to consist of gold, described by a Drude model with the parameters

*ε*

_{∞}= 9.0685, a plasma frequency

*ω*= 1.3544 · 10

_{D}^{16}Hz and a damping coefficient

*γ*= 1.1536 · 10

^{14}Hz, see [16

16. 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]

*C*

_{4}symmetric. This results in

*x*

^{3}=

*h*+

*b*/2, i.e., between the crosses. As shown, the effective permittivity of this layer of air becomes fully anisotropic due to Eqs. (1), (2) and (7). In Fig. 3(b) we display the effective permittivity at

*x*

^{3}=

*h*+

*b*, i.e., in the layer with the rotated cross. Due to the form of the coordinate transformation in Fig. 2, the gold cross is grid-aligned in the transformed space. The origin of the discontinuities in the effective permittivity is the fact that the meshes above are not differentiable. This however does not affect the overall performance of the method as long as the discretization parameters are chosen wisely, see [11

**20**, 17319–17347 (2012). [CrossRef] [PubMed]

## 4. Three-dimensional adaptive spatial resolution

8. 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). [CrossRef] [PubMed]

**20**, 17319–17347 (2012). [CrossRef] [PubMed]

*x*

^{3}-coordinate. Thereby, the basis functions of the problem represent the real, physical space. Therefore, we can easily start an ordinary plane wave in the incoming, Cartesian, physical half-space. Then, we can introduce several intermediate layers to start the adaptive spatial resolution. The general procedure is sketched in Fig. 4(a). In this sketch, we start by a Cartesian layer at the bottom, cf. Fig. 4(b). In the next three layers, we gradually introduce the ASR as discussed in great detail in [11

**20**, 17319–17347 (2012). [CrossRef] [PubMed]

*φ*

_{0}, we can compute the layer of the rotated cross, again shaded in yellow in Fig. 4(a). The mesh that is used to compute the effective permittivity in this layer is depicted in Fig. 4(g). Like above, we then gradually reverse the mesh changes—first, the mesh is rotated back, then the ASR is decreased until we reach the outgoing layer with a Cartesian mesh.

*x*

^{3}∈ [0,

*c*], then the mapping has to obey Here, ASR denotes the compression function (see

*x̄*(

*x*) in Section 8 in [11

**20**, 17319–17347 (2012). [CrossRef] [PubMed]

*x̄*

^{2}mapping is constructed similarly.

*x*

^{3}we first compress the coordinate lines and then apply the adaptive coordinate transformation. The result is meshes like in Fig. 4. The great advantage of such a procedure is that it can be easily incorporated into any classical FMM code which can solve the large eigenproblem. Since the incoming and outgoing layer are Cartesian, this is perfectly compatible. So any classical FMM code that can solve the large eigenproblem can just be given the transformed permittivity and permeability and can, thereby, incorporate three-dimensional coordinate transformations. Moreover, the issues of two-dimensional transformations that were discussed above are avoided.

## 5. Conclusion

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

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

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

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

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

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

8. | T. Vallius and M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express |

9. | 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 |

10. | S. Essig and K. Busch, “Generation of adaptive coordinates and their use in the Fourier Modal Method,” Opt. Express |

11. | J. Küchenmeister, T. Zebrowski, and K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express |

12. | M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. |

13. | V. Liu and S. Fan, “S |

14. | H. Kim, J. Park, and B. Lee, |

15. | L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A |

16. | 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 |

**OCIS Codes**

(050.1970) Diffraction and gratings : 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: October 11, 2013

Revised Manuscript: December 9, 2013

Manuscript Accepted: December 13, 2013

Published: January 14, 2014

**Citation**

Jens Küchenmeister, "Three-dimensional adaptive coordinate transformations for the Fourier modal method," Opt. Express **22**, 1342-1349 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1342

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### References

- K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007). [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]
- P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996). [CrossRef]
- G. Granet, 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]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [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]
- G. Granet, J.-P. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A 4, S145–S149 (2002). [CrossRef]
- T. Vallius, M. Honkanen, “Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles,” Opt. Express 10, 24–34 (2002). [CrossRef] [PubMed]
- T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17, 8051–8061 (2009). [CrossRef] [PubMed]
- S. Essig, K. Busch, “Generation of adaptive coordinates and their use in the Fourier Modal Method,” Opt. Express 18, 23258–23274 (2010). [CrossRef] [PubMed]
- J. Küchenmeister, T. Zebrowski, K. Busch, “A construction guide to analytically generated meshes for the Fourier Modal Method,” Opt. Express 20, 17319–17347 (2012). [CrossRef] [PubMed]
- M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009). [CrossRef] [PubMed]
- V. Liu, S. Fan, “S4: A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012). [CrossRef]
- H. Kim, J. Park, B. Lee, Fourier Modal Method and its Applications in Computational Nanophotonics (CRC Press, 2012).
- L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003). [CrossRef]
- A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, 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]

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