## Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces |

Optics Express, Vol. 20, Issue 9, pp. 9978-9990 (2012)

http://dx.doi.org/10.1364/OE.20.009978

Acrobat PDF (1393 KB)

### Abstract

This paper considers the electromagnetic scattering problem of periodically corrugated surface with local imperfection of structural periodicity, and presents a formulation based on the coordinate transformation method (C-method). The C-method is originally developed to analyze the plane-wave scattering from perfectly periodic structures, and uses the pseudo-periodic property of the fields. The fields in imperfectly periodic structures are not pseudo-periodic and the C-method cannot be directly applied. This paper introduces the pseudo-periodic Fourier transform to convert the fields in imperfectly periodic structures to pseudo-periodic ones, and the C-method becomes then applicable.

© 2012 OSA

## 1. Introduction

1. T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. **46**, 5435–5440 (2007). [CrossRef]

3. C. Yang, K. Shi, P. Edwards, and Z. Liu, “Demonstration of a PDMS based hybrid grating and Fresnel lens (G-Fresnel) device,” Opt. Express **18**, 23529–23534 (2010). [CrossRef] [PubMed]

## 2. Settings of the problem

*i*

*ω*

*t*) from imperfectly periodic surfaces, in which structural periodicity is locally collapsed. Figure 1 shows an example of the structures under consideration. The structure is uniform in the

*z*-direction and the

*y*-axis is perpendicular to the periodicity direction though the periodicity is not perfect. The equation of the corrugated surface is given by

*y*=

*g*(

*x*), where

*g*(

*x*) is supposed to be a continuous function. The minimum and the maximum values of

*g*(

*x*) are denoted by

*h*

_{1}and

*h*

_{2}, respectively. The surface profile function

*g*(

*x*) is supposed to be decomposed into periodic and aperiodic parts: where

*g*

^{(p)}(

*x*) is a periodic function with the period

*d*and

*g*

^{(a)}(

*x*) has nonzero value only at

*a*

_{1}≤

*x*≤

*a*

_{2}where the structural periodicity is locally collapsed. The surrounding region

*y*>

*g*(

*x*) is filled with a homogeneous and isotropic medium with the permittivity

*ε*and the permeability

_{s}*μ*, and the substrate region

_{s}*y*<

*g*(

*x*) is also filled with a homogeneous and isotropic medium described by the permittivity

*ε*and the permeability

_{c}*μ*. For lossy media, we use complex values for the permittivity and/or the permeability, and the terms concerning to the current densities are eliminated in the following formulation. The regions

_{c}*y*>

*g*(

*x*) and

*y*<

*g*(

*x*) are specified by

*s*and

*c*, respectively, and the wavenumber in each region is denoted by

*k*=

_{r}*ω*(

*ε*

_{r}*μ*)

_{r}^{1/2}for

*r*=

*s*,

*c*. The electromagnetic fields are uniform in the

*z*-direction and two-dimensional scattering problem is considered here. Two fundamental polarizations are expressed by TE and TM, in which the electric and the magnetic fields are respectively parallel to the

*z*-axis. We sometimes denote the

*z*-component of electric field for the TE-polarization and the

*z*-component of magnetic field for the TM-polarization by

*ψ*(

*x,y*) to express both polarizations simultaneously. The incident field is supposed to illuminate the corrugated surface from the upper or lower regions and there exists no source inside the inhomogeneous region

*h*

_{1}≤

*y*≤

*h*

_{2}.

## 3. Formulation

*f*(

*u*) be a function of

*u*and

*d*be a positive real constant. Then the PPFT and its inverse transform are formally defined by where

*ξ*is the transform parameter and

*k*= 2

_{d}*π*

*/d*. The transformed functions have pseudo-periodic property in terms of

*u: f̄*(

*u*−

*md*;

*ξ*) =

*f̄*(

*u*;

*ξ*) exp(−

*imd*

*ξ*) for any integer

*m*, and also have periodic property in terms of

*ξ*:

*f̄*(

*u*;

*ξ*−

*mk*) =

_{d}*f̄*(

*u*;

*ξ*). We use the period of

*g*

^{(p)}(

*x*) for the positive real constant

*d*for the PPFT. Then,

*k*becomes the inverse lattice constant and the periodicity cell of the transformed function gives the Brillouin zone. Applying the PPFT to Eqs. (5)–(7), we obtain the following relations:

_{d}*u*, they can be approximately expanded in the truncated generalized Fourier series. For example, the

*z*-component of electric field is written as with where

*N*denotes the truncation order and

*Ē*(

_{z,n}*ξ*

*,v*) are the

*n*th-order coefficients. To treat the coefficients systematically, we introduce (2

*N*+ 1) × 1 column matrices; for example, the coefficients of

*Ē*(

_{z}*u*;

*ξ*

*,v*) are expressed by a column matrix

*ē**(*

_{z}*ξ*

*,v*) in such a way that its

*n*th-entries are given by

*Ē*(

_{z,n}*ξ*

*,v*). Then Eqs. (10)–(12) yield the following relations:

**(**

*Ū**ξ*), 〚

*ġ*

^{(p)}〛, and 〚

*ġ̄*

^{(a)}〛(

*ξ*) are (2

*N*+ 1) × (2

*N*+ 1) square matrices whose (

*n,m*)-entries are given by The symbol

*δ*stands for Kronecker’s delta. The integrands appeared in Eqs. (10) and (12) are products of pseudo-periodic functions with no concurrent jump discontinuities, and the Laurent rule [12

_{n,m}12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

*ξ*, Eqs. (15)–(17) have to be satisfied for arbitrary

*ξ*∈ (−

*k*/2,

_{d}*k*/2]. However, we take

_{d}*L*sample points inside the Brillouin zone and assume that Eqs. (15)–(17) are satisfied at these sample points. Also the integrations in Eqs. (15) and (17) are approximated by an appropriate numerical integration scheme using the same sample points. Let

*h**̃*(

_{x}*v*) and

*h**̃*(

_{y}*v*) are similar to

*ẽ**(*

_{z}*v*). After a simple calculation, Eqs. (21)–(23) yield the following coupled differential-equation set: with where

**denotes the identity matrix and the superscript “−1” stands for the matrix inverse. The general solution to the coupled differential equation set (28) can be obtained by solving the eigenvalue-eigenvector problems because the matrix of coefficients**

*I*

*M**is constant. The 2*

_{r}*L*(2

*N*+ 1) eigenvalues can be divided into two sets, each containing

*L*(2

*N*+ 1) eigenvalues. The first set contains the negative real eigenvalues and the complex eigenvalues with positive imaginary parts, and the second set contains those with the opposite signs. We denote the reciprocals of the eigenvalues of

*M**by*

_{r}

*p**denote the eigenvector of*

_{r,n}

*M**associating with the eigenvalue 1/*

_{r}*η*. Then the matrix for diagonalization is constructed as and the general solution to the coupled differential-equation set (28) is written in the following form: where the column matrices

_{r,n}*v*-directions, respectively. The relation between the modal amplitudes at

*v*=

*v*′ and

*v*=

*v*″ is given by with The covariant component of the magnetic field in terms of

*u*is given by

*H*(

_{t}*u, v*) =

*H*(

_{x}*u, v*) +

*ġ*(

*u*)

*H*(

_{y}*u, v*), which gives the tangential component of the magnetic field on the grating surface

*v*= 0. From Eqs. (22), (23), and (31), the generalized Fourier coefficients of

*Ē*(

_{z}*u*;

*ξ*

_{l}*, v*) and

*H̄*(

_{t}*u*;

*ξ*

_{l}*, v*) are expressed in the following form: with for

*q*= 1,2.

*H̄*(

_{z}*u*;

*ξ*

_{l}*, v*) and

*Ē*(

_{t}*u*;

*ξ*

_{l}*, v*) are expressed in the following form: with for

*q*= 1,2. It is worth noting that the coefficient matrix

*M**is independent of the polarization and, therefore,*

_{r}

*P̃**for*

_{r,pq}*p, q*= 1, 2 are also independent of the polarization.

*s*and

*c*can be matched at the grating surface

*v*= 0 by using the boundary conditions, which are given by the continuities of the tangential components of the fields. Then the scattering matrix that relates the amplitudes of the incoming and outgoing fields is derived as with for

*f*=

*e,h*. From Eq. (31), the amplitudes of the eigenmodes are written as where

**(**

*ψ**̃**v*) denotes

*ẽ**(*

_{z}*v*) for the TE-polarization and

*h̃**(*

_{z}*v*) for the TM-polarization. The column matrices

**(**

*ψ**̃**v*) and

*ψ*(

*x,y*). The amplitudes of the incoming fields

## 4. Numerical experiments

*g*

^{(p)}(

*x*) is given by and the fields are excited by a line-source situated parallel to the

*z*-axis at (

*x,y*) = (

*x*

_{0},

*y*

_{0}) for

*y*

_{0}>

*h*

_{2}. The grating parameters are chosen as follows:

*d*= 0.6

*λ*

_{0},

*h*

_{1}= −0.5

*d*,

*h*

_{2}= 0,

*ε*=

_{s}*ε*

_{0},

*ε*= (1.3 +

_{c}*i*7.6)

^{2}

*ε*

_{0}(a typical value of Aluminum at

*λ*

_{0}= 632.8nm), and

*μ*=

_{s}*μ*=

_{c}*μ*

_{0}.

### 4.1. Treatment of line-source excitation

*s*, the incident field

*ψ*

^{(i)}(

*x,y*) is expressed as for

*y*>

*g*(

*x*), where

*i*

*β*(

*h*

_{2}−

*g*

^{(p)}(

*u*))) in the Fourier series as Then, the first integrals on the right-hand sides of Eqs. (48) and (49) can be written as with near the corrugated surface. The periodic part of surface profile function is given by Eq. (46) and, then, the Fourier coefficients appeared in Eq. (51) are given by where

*J*denotes the

_{m}*m*th-order Bessel function. We truncate the infinite sums in Eqs. (52) and (53) by using the same truncation order

*N*with Eq. (13) and obtain approximate values of the first integrals on the right-hand sides of Eqs. (48) and (49). On the other hand, the second integrals are with a finite interval, and can be evaluated by standard numerical integration.

### 4.2. Grating with a defect

*g*

^{(a)}(

*x*) as and, then, one groove of the original grating is filled by the substrate medium. We set the position of line-source at (

*x*

_{0},

*y*

_{0}) = (

*d*, 2

*d*) and the observation point for convergence tests (

*x,y*) = (0

*,d*).

*L*. The truncation order for each generalized Fourier series expansion is set to

*N*= 4 to calculate these results. The dotted curves are the results of the trapezoidal scheme that uses equidistant sample points and equal weights. Since the transformed field

*ψ*

*̄*(

*x*;

*ξ*

*,y*) is not a smooth function of

*ξ*, they converge very slowly. On the other hand, for the solid curves, we split the Brillouin zone at the Wood-Rayleigh anomalies, which is the non-smooth points, and the double exponential scheme [14

14. H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. **9**, 721–741 (1974). [CrossRef]

7. K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express **19**, 25799–25811 (2011). [CrossRef]

*N*are shown in Fig. 2(b). The values are computed with

*L*= 80, and the sample points and the weights are determined by applying the double exponential scheme for the subintervals. The convergence with respect to the truncation order

*N*is very fast. Since the present formulation does not include the generalized Fourier series expansion of discontinuous function and the surface profile under consideration is smooth, the convergence speed is slightly faster than that shown in Fig. 2(b) of Ref. [7

7. K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express **19**, 25799–25811 (2011). [CrossRef]

*L*= 80 and

*N*= 4 by changing the observation point, and shown in Fig. 3. The positions of surfaces are indicated by the white dashed lines and the obtained results seem to be proper.

### 4.3. Grating with period modulation

*d*+

*Δ*(

*x*) in the region −

*a*≤

*x*≤

*a*. The local period deviation

*Δ*(

*x*) is a raised-cosine function and the minimum value of the local period is denoted by

*d*

_{min}. The parameter

*a*is chosen to be

*a*=

*d*/[(

*d/d*

_{min})

^{1/2}− 1], and the surface profile function

*g*(

*x*) is smooth even at

*x*= ±

*a*. Here, the positions of the line-source and the observation point are respectively chosen at (

*x*

_{0},

*y*

_{0}) = (

*d*, 2

*d*) and (

*x*,

*y*) = (0,

*d*), and the minimum local period is

*d*

_{min}= 0.5

*d*.

*N*= 8. The convergence characteristics in terms of the sample point number

*L*are similar to those for the grating with a defect. In contrast, the convergence characteristics in terms of the truncation order

*N*are slower than that for the grating with a defect. Since the surface profile function

*g*(

*x*) under consideration has larger components in high spatial frequency range than that for the grating with a defect, the generalized Fourier series expression requires more terms to construct the original function accurately. The field intensity distributions computed with

*L*= 80 and

*N*= 8 and shown in Fig. 6 and the obtained results seem to be proper for both of the TE- and the TM-polarized cases. Figure 7 shows the results of the same reciprocity test with Fig. 2 though the values are computed with

*L*= 80 and

*N*= 8. The largest values are about 1.5 × 10

^{−3}for the TE-polarization and 1.2 × 10

^{−3}for the TM-polarization, and the reciprocity relation is well satisfied.

## 5. Concluding remarks

7. K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express **19**, 25799–25811 (2011). [CrossRef]

## References and links

1. | T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. |

2. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

3. | C. Yang, K. Shi, P. Edwards, and Z. Liu, “Demonstration of a PDMS based hybrid grating and Fresnel lens (G-Fresnel) device,” Opt. Express |

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

5. | R. Petit, ed., |

6. | K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res. |

7. | K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express |

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

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

10. | E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A |

11. | K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A |

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

13. | W. C. Chew, |

14. | H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. |

15. | P. J. Davis and P. Rabinowitz, |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(050.1950) Diffraction and gratings : Diffraction gratings

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: January 3, 2012

Revised Manuscript: March 8, 2012

Manuscript Accepted: March 8, 2012

Published: April 17, 2012

**Citation**

Koki Watanabe, Jaromír Pištora, and Yoshimasa Nakatake, "Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces," Opt. Express **20**, 9978-9990 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-9978

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### References

- T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys.46, 5435–5440 (2007). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).
- C. Yang, K. Shi, P. Edwards, and Z. Liu, “Demonstration of a PDMS based hybrid grating and Fresnel lens (G-Fresnel) device,” Opt. Express18, 23529–23534 (2010). [CrossRef] [PubMed]
- J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am.72, 839–846 (1982). [CrossRef]
- R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
- K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res.PIER 74, 241–271 (2007). [CrossRef]
- K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express19, 25799–25811 (2011). [CrossRef]
- K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am.68, 1206–1210 (1978). [CrossRef]
- M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am.72, 1385–1392 (1982). [CrossRef]
- E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A19, 33–42 (2002). [CrossRef]
- K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A19, 2245–2252 (2002). [CrossRef]
- L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A13, 1870–1876 (1996). [CrossRef]
- W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
- H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ.9, 721–741 (1974). [CrossRef]
- P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).

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