## Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles

Optics Express, Vol. 10, Issue 1, pp. 24-34 (2002)

http://dx.doi.org/10.1364/OE.10.000024

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### Abstract

Formulation of the Fourier modal method for multilevel structures with spatially adaptive resolution is presented for TE and TM polarizations using a slightly reformulated representation for the spatial coordinates. Projections to Fourier base in boundary value problem are used allowing extensions to multilayer profiles with differently placed transitions. We evade the eigenvalue problem in homogeneous regions demanded in the original formulation of the Fourier modal method with adaptive spatial resolution.

© Optical Society of America

## 1 Introduction

1. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978). [CrossRef]

2. 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]

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

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

5. 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]

5. 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]

*x*space will be used enabling different parametric representations in every modulated layer and the extra eigenvalue problems will be evaded. Projections of the field onto similar basis in every grating layer allows also arbitrary spacing of the transitions in each layer.

## 2 Theory

*y*-invariant geometry illustrated in fig. 1. The media in the regions 1 and 3 are assumed to be homogeneous with refractive indices

*n*

_{1}and

*n*

_{3}, respectively. In region 2 the relative permittivity

*∊*(

*x*) =

*n*(

*x*)

^{2}is assumed periodic in

*x*-direction, with period

*d*. Region 2 is also divided into

*J*layers: in each layer the relative permittivity can be arbitrarily modulated in

*x*-direction but is assumed

*z*-invariant. The structure is illuminated by an infinite plane wave from the negative

*z*-direction propagating in the

*xz*-plane with an angle

*θ*between the

*z*-axis and the wave vector

*, where |*

**k***| =*

**k***k*= 2

*π*/λ, and λ is the wavelength in vacuum. Since the problem is completely

*y*-invariant, all the partial

*y*-derivatives vanish in Maxwell’s equations and one has the so-called TE/TM-polarization decomposition [6

6. R. Petit, ed., *Electromagnetic theory of gratings* (Springer-Verlag, Berlin, 1980). [CrossRef]

### 2.1 Field in unmodulated regions

6. R. Petit, ed., *Electromagnetic theory of gratings* (Springer-Verlag, Berlin, 1980). [CrossRef]

*α*=

_{m}*α*

_{0}+ 2

*πm*/

*d*, and

*α*

_{0}=

*kn*

_{1}sin

*θ*. Corresponding representations for the magnetic field in TM polarization can be achieved by replacing

*E*by

*H*.

*R*and

_{m}*T*denote the complex amplitudes of the

_{m}*m*’th reflected or transmitted diffraction order and the diffraction efficiencies are given by

*ℜ*denotes the real part. In TE polarization the parameter

*C*= 1 and in TM polarization

*C*= (

*n*

_{1}/

*n*

_{3})

^{2}.

### 2.2 Eigenvalue problem

*y*invariant structures [1

1. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978). [CrossRef]

*xz*plane are

1. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. **68**, 1206–1210 (1978). [CrossRef]

*x*-direction the Gibbs phenomenon appears and slow convergence of the Fourier series is observed. Thus better convergence rate can be achieved by choosing a new Fourier space, where the variable

*x*is replaced by

*u*. The dependence between the variables

*x*and

*u*is presented by parametric equations that lead to new eigenvalue equations.

*x*→

*x*(

*u*):

5. 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]

*u*space the fields

*E*and

_{y}*H*are pseudoperiodic with period

_{y}*d*. Hence each mode can presented in the form

*E*is an eigenvector containing the Fourier coefficients of the mode, and

_{m}*γ*is an eigenvalue that defines the propagation in

*z*direction. Similar representation is valid also for the modes of the magnetic field

*H*.

_{y}4. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

**16**, 2510–2516 (1999). [CrossRef]

**f**,

**a**, and

**b**are toeplitz matrices formed from the Fourier coefficients of the respective functions, and

*α*is a diagonal matrix formed from

*α*. The sign rule for the eigenvalues is known as

_{m}*ℜ*{

*γ*} + ℑ{

*γ*} > 0, where ℑ denotes the imaginary part. The exact eigenvalues are independent of the chosen representation of the coordinates; the eigenvectors

*and*

**E***depend on the coordinate system and must be transformed to a more convenient one.*

**H**### 2.3 Parametric representation of the coordinate x

*x*as a function of

*u*and the transition points are denoted by

*x*in the

_{l}*x*space and by

*u*in the

_{l}*u*space. Between the transitions

*l*and

*l*- 1 we use the function

*x*(

_{l}*u*) for the mapping between different spaces:

*G*=

*f*(

*u*

_{l-1}) =

*f*(

*u*). The difference between the representation (16) and the original parametric function [5

_{l}**16**, 2510–2516 (1999). [CrossRef]

08. G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A **18**, 2102–2108 (2001). [CrossRef]

*a*

_{3}allowing different spacing of the transitions in

*x*and

*u*spaces without discontinuities of the resolution function, which was our motivation for the reformulation of the method. In the original formulation the parameter

*a*

_{3}was defined as

*a*

_{3}= (

*G*- 1)(

*x*-

_{l}*x*

_{l-1}).

### 2.4 Boundary value problem

*j*can be written as a superposition of the modes given by Eq. 13:

*E*by

*H*.

*h*’s are the height transitions in the

_{j}*z*-direction. The field representations in each layer are in different bases exp[i

*α*

_{m}*u*(

_{j}*x*)] which depend on the locations of the transitions in the respective layer. Thereby we expand the eigenfunction in each layer in terms of similar base functions in the

*x*space. The boundary conditions between the layers

*j*and

*j*+ 1 in TE polarization are

*x*space is an apparent choice for the basis function allowing the use of general algorithms for calculating the projections, which are easily obtained by calculating the integral

**K**:

*x*space are denoted by

**E**

_{j}and

**H**

_{j}in layer the

*j*and the matrix

**Q**

_{j}includes the Fourier coefficients of the function

*Q*(

_{y,j}*x*) =

*H*(

_{y,j}*x*)/

*∊*(

_{j}*x*) in the equation (24); the superscript

*u*refers to the eigenmatrices in the

*u*space. By substituting the matrices in the

*x*space to the boundary conditions (21)–(24) one obtains the following matrix equation in TM mode

**Γ**

_{j}and

**X**

_{j}are diagonal matrices with elements

*γ*and exp[i

_{v,j}*γ*(

_{v,j}*h*

_{j+i}-

*h*)], while the vectors

_{j}**A**

_{j}and

*B*_{j}are defined in equation (20). The matrix equations for TE modes are obtained by replacing

**H**and

**Q**with

**E**. For a numerically stable treatment of the evanescent waves at the boundaries we use the S matrix algorithm [9

9. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A **13**, 1024–1035 (1996). [CrossRef]

*Q*(

*x*) =

*H*(

*x*)/

*∊*(

*x*) must be calculated in the

*u*space and transformed to the

*x*space by Eq. (26), not by using the Laurent’s rule in the

*x*space.

## 3 Numerical examples

11. D. Nyyssonen and C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry,” J. Opt. Soc. Am. A **5**, 1270–1280 (1988). [CrossRef]

### 3.1 Convergence of eigenvalues

*d*= 1α containing a single transition at

*c*= 0.9

_{x}*d*. The parameter

*G*is set to

*G*= 0.001 for all the examples presented here. The refractive indices around the transition are

*n*

_{2a}= 1 and

*n*

_{2b}= 5+15i and normal incidence is assumed. We calculate the eigenvalues by using a growing number of the truncation order and compare the convergence of the real part of the first eigenvalue. In fig. 2a and 2b we see how the parametric representation of the

*x*axis leads to faster convergence of the eigenvalue. We have also compared the convergence of the eigenvalue with two different locations of the transition in the

*u*space. The dotted line illustrates the error when similar transitions in both

*x*and

*u*space have been used, whereas the solid line represents the convergence when the transition in the

*u*space is located at

*c*= 0.5

_{u}*d*The use of equal transitions in each space gives us the parametric representation used in the original formulation [5

**16**, 2510–2516 (1999). [CrossRef]

**16**, 2510–2516 (1999). [CrossRef]

*u*space with differentiable mapping.

**16**, 2510–2516 (1999). [CrossRef]

*u*space, like the reformulated representation does. Thus the Gibb’s phenomenon will ruin the shape of the small detail. We can observe that the new parametric representation converges faster to the reliable value without substantial fluctuations. The difference between the representations is illustrated figure 4 where both the new and the old formulations are presented. We stress that the reformulation of the parametric representation is not necessarily the optimal one since several other mappings than the aforementioned sine function might prove more suitable. This primarily serves as a new basis for research aimed at obtaining better convergence of the FMM.

10. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A **12**, 1043–1056 (1995). [CrossRef]

10. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A **12**, 1043–1056 (1995). [CrossRef]

*M*× 3

*M*; the number of the Fourier coefficients and the modes is

*M*, respectively.

### 3.2 Accuracy of the diffraction efficiency

*z*invariant layers as illustrated in fig. 5 to determine convergence properties of diffraction orders. Results are presented in fig. 6 for TE polarized light, where the diffraction efficiency of the zeroth order analysed with FMM in fig. 6a is smaller than the correct value when five modes have been included in the analysis, but the method with parametric representation in fig. 6b is reliable even with 5 modes. The accurate results calculated by the FMM with 240 modes (---) agree well with the values of the parametric formulation. Figure 7 illustrates the corresponding results in TM polarization and remarkable changes in the curves with different number of modes can be observed in fig. 7a with FMM, while the parametric formulation lead to accurate values with smaller number of modes in fig. 7b.

10. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A **12**, 1043–1056 (1995). [CrossRef]

*n*= 1 and

_{a}*n*= 5. Figure 11 illustrates the efficiency of the zeroth transmitted diffraction order analyzed with FMM in fig. 11a and with the present method in fig. 11b. Note the evident change of the resonance peaks with FMM which is not observable when the adaptive resolution has been applied.

_{b}## 4 Conclusions

*x*and

*u*spaces was slightly reformulated to enable different spacing of the transitions in each space. The eigenvalue problem was solved in the adaptive

*u*space and the eigenvectors were transformed into the

*x*space for solving the boundary conditions between different grating layers. Good convergence rates both in TE and in TM polarizations were achieved and the method proved reliable also for structures causing strong resonance peaks.

**12**, 1043–1056 (1995). [CrossRef]

## References and links

1. | K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. |

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

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

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

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

6. | R. Petit, ed., |

7. | J. Turunen, “Diffraction theory of microrelief gratings,” Chap 2 in Micro-Optics: Elements, Systems and Applications,H. P. Herzig, ed. (Taylor & Francis, Cornwall, 1997) |

08. | G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” 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. | R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A |

11. | D. Nyyssonen and C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1960) Diffraction and gratings : Diffraction theory

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 4, 2001

Published: January 14, 2002

**Citation**

Tuomas Vallius and M. Honkanen, "Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles," Opt. Express **10**, 24-34 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-24

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### References

- K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978). [CrossRef]
- 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]
- G. Granet and B. Guizal, "Efficient implementation for 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]
- R. Petit, ed., Electromagnetic theory of gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
- J. Turunen, "Diffraction theory of microrelief gratings," Chap 2 in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, Cornwall, 1997)
- G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001). [CrossRef]
- L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996). [CrossRef]
- R. H. Morf, "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings," J. Opt. Soc. Am. A 12, 1043-1056 (1995). [CrossRef]
- D. Nyyssonen and C. P. Kirk, "Optical microscope imaging of lines patterned in thick layers with variable edge geometry," J. Opt. Soc. Am. A 5, 1270-1280 (1988). [CrossRef]

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