## Accurate analysis of electromagnetic scattering from periodic circular cylinder array with defects |

Optics Express, Vol. 20, Issue 10, pp. 10646-10657 (2012)

http://dx.doi.org/10.1364/OE.20.010646

Acrobat PDF (1817 KB)

### Abstract

This paper considers the two-dimensional electromagnetic scattering from periodic array of circular cylinders in which some cylinders are removed, and presents a formulation based on the recursive transition-matrix algorithm (RTMA). The RTMA was originally developed as an accurate approach to the scattering problem of a finite number of cylinders, and an approach to the problem of periodic cylinder array was then developed with the help of the lattice sums technique. This paper introduces the concept of the pseudo-periodic Fourier transform to the RTMA with the lattice sums technique, and proposes a spectral-domain approach to the problem of periodic cylinder array with defects.

© 2012 OSA

## 1. Introduction

1. S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics **2**, 52–56 (2007). [CrossRef]

3. Ch. Kang and S. M. Weiss, “Photonic crystal with multi-hole defect for sensor applications,” Opt. Express **16**, 18188–18193 (2008). [CrossRef] [PubMed]

4. A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron . **17**, 1693–1694 (2011). [CrossRef]

6. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A **11**, 2526–2538 (1994). [CrossRef]

7. H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl . **10**, 109–127 (1996). [CrossRef]

8. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res . **PIER 74**, 241–271 (2007). [CrossRef]

10. 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]

11. K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published). [PubMed]

## 2. Settings of the problem

*iωt*) from a periodic circular cylinder array with defects. Figure 1 shows an example of the structures under consideration. The structure consists of identical circular cylinders, which are infinitely long in the

*z*-direction and situated parallel to each other. The cylinders are made with a homogeneous and isotropic medium described by the permittivity

*ɛ*and the permeability

_{c}*μ*, and their radius is denoted by

_{c}*a*. For a lossy medium, 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 surrounding region is filled by a lossless, homogeneous, and isotropic medium with the permittivity

*ɛ*and the permeability

_{s}*μ*. The wavenumber and the characteristic impedance in each medium are respectively denoted by

_{s}*k*=

_{j}*ω*(

*ɛ*)

_{j}μ_{j}^{1/2}and

*ζ*= (

_{j}*μ*/

_{j}*ɛ*)

_{j}^{1/2}for

*j*=

*c,s*. The axis of the

*l*th-cylinder is located at (

*x,y*) = (

*l d,*0) for integer

*l*, though some cylinders are removed from the periodic array. To indicate the removed cylinders, we introduce a notation

*𝒟*, which is a finite subset of the integer set . If an integer

*l*′ is an element of

*𝒟*, the cylinder whose center is at (

*x,y*) = (

*l*′

*d,*0) is removed. Also, the complement of

*𝒟*in is denoted by

*𝒟*. The electromagnetic fields are supposed to be uniform in the

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

*z*-axis. We denote the

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

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

*ψ*(

*x,y*), and show the formulation for both polarizations simultaneously. The incident field

*ψ*

^{(i)}(

*x,y*) is supposed to illuminate the cylinders from outside and there exists no source inside the cylinders.

## 3. Formulation

*g*^{(Z)}(

*x,y*), in which the

*n*th-component is given as with where

*Z*specifies the cylindrical functions associating with the cylindrical-wave bases in such a way that

*Z*=

*J*denotes the Bessel function and

*Z*=

*H*

^{(1)}denotes the Hankel function of the first kind. Let (

*x*,

_{q}*y*) and (

_{q}*x*,

_{r}*y*) be the reference points of the bases. Then, when (

_{r}*x, y*) is inside a circle with center (

*x*,

_{r}*y*) and radius

_{r}*ρ*(

*x*), the translation relation for the cylindrical-wave bases is given by where

_{q}− x_{r}, y_{q}− y_{r}

*G*^{(Z)}(

*x,y*) denotes the Toeplitz matrix whose (

*n,m*)-entries are given by

*a*≤

*y*≤

*a*Therefore, if we choose the reference point at (

*x,y*) = (

*qd*,0) for an integer

*q*, the incident field in the surrounding medium can be expanded in a series of the cylindrical-waves associated with the Bessel functions as where the superscript

*t*denotes the transpose and

*ψ*

^{(s)}(

*x,y*) outside the cylinders consists of the outward propagating waves from the cylinders, and it can be expressed as where

*q*th-cylinder. Using the translation relation Eq. (4), the total field near but outside the

*q*th-cylinder (

*q*∈

*𝒟*) can be expressed as where

^{c}*𝒟*\{

^{c}*q*} denotes the set in which an element

*q*is removed from

*𝒟*. The first and the second terms on the right-hand side of Eq. (8) are given by superpositions of cylindrical-waves associated with the Bessel function and the Hankel function of the first kind, respectively, and they represent the incident and the scattered fields for the

^{c}*q*th-cylinder. Since the relation between the coefficient matrices of the incident and the scattered fields are given by the transition-matrix (T-matrix), we have The (

*n,m*)-entries of the T-matirx

**are given by**

*T**δ*stands for Kronecker’s delta.

_{n,m}6. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A **11**, 2526–2538 (1994). [CrossRef]

*f*(

*x*) be a function of

*x*. Then the PPFT and its inverse transform are formally defined by where

*ξ*is the transform parameter and

*k*= 2

_{d}*π/d*denotes the inverse lattice constant. The transformed functions have a pseudo-periodic property in terms of

*x: f̄*(

*x − qd*;

*ξ*) =

*f̄*(

*x*;

*ξ*) exp(−

*iqdξ*) for any integer

*q*, and also have a periodic property in terms of

*ξ*:

*f̄*(

*x*;

*ξ*

*− qk*) =

_{d}*f̄*(

*x*;

*ξ*). Applying the PPFT to the incident and the scattered fields, the transformed fields are expressed as follows with The column matrices

*ā*^{(i)}(

*ξ*) and

*ā*^{(s)}(

*ξ*) give the coefficients of the cylindrical-wave expansions for the reference point at (

*x,y*) = (0,0), and they are periodic in terms of

*ξ*with the period

*k*. The original coefficient matrices are inversely obtained by integrating on the transform parameter

_{d}*ξ*as for

*f*=

*i,s*.

**(**

*L**ξ*) are called the lattice sums [12

12. N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E **50**, 3143–3160 (1994). [CrossRef]

13. K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag . **47**, 1050–1055 (1999). [CrossRef]

*c*

^{(d)}(

*ξ, ξ*′) is given by a sum of finite terms, and it expresses the effects of defects in the periodic array.

*ψ*

^{(s)}(

*x,y*) is here decomposed into the scattered field by the periodic cylinder array without defects

*ψ*

^{(s,p)}(

*x,y*) and the residual field

*ψ*

^{(s,d)}(

*x,y*) =

*ψ*

^{(s)}(

*x,y*) −

*ψ*

^{(s,p)}(

*x,y*). Since the transformed field

*ψ̄*

^{(s,p)}(

*x*;

*ξ*,

*y*) can be expressed in the same form with Eq. (14), the coefficient matrix

*ā*^{(s)}(

*ξ*) is also decomposed as where

*ā*^{(s,p)}(

*ξ*) and

*ā*^{(s,d)}(

*ξ*) denote the coefficient matrices of

*ψ̄*

^{(s,p)}(

*x*;

*ξ*,

*y*) and

*ψ̄*

^{(s,d)}(

*x*;

*ξ,y*), respectively. If there is no defect in the cylinder array (

*M*^{(d)}(

*ξ*,

*ξ*′) and

*c*

^{(d)}(

*ξ,ξ*′) vanish. Therefore, from Eq. (18), the coefficient matrix for the scattered field by the periodic cylinder array is given as Substituting Eqs. (23) and (24) into Eq. (18), we may obtain the following equation for

*ā*^{(s,d)}(

*ξ*):

*ξ*. Considering the periodicity, we take

*L*sample points

*k*/2 <

_{d}*ξ*≤

*k*, and assume that Eq. (25) is satisfied at these points. Also, the integral on the left-hand side is approximated by an appropriate numerical integration scheme with the use of same sample points. We denote the respective weights of the sample points by

_{d}## 4. Numerical experiments

*d*= 0.8

*λ*

_{0},

*a*= 0.4

*d*,

*ɛ*= 4

_{c}*ɛ*

_{0},

*ɛ*=

_{s}*ɛ*

_{0},

*μ*=

_{c}*μ*=

_{s}*μ*

_{0}, and three cylinders with the center at (

*x, y*) = (−2

*d*, 0), (0, 0), (2

*d,*0) are removed from the perfectly periodic array (

*𝒟*= {−2, 0, 2}). When implementing a practical computation, the cylindrical-wave expansions must be truncated. We denote the truncation order for the expansions by

*K*that truncates the expansions from −

*K*th- to

*K*th-order.

### 4.1. Line-source excitation

*z*-axis at (

*x,y*) = (

*x*

_{0},

*y*

_{0}) for

*y*

_{0}>

*a*or

*y*

_{0}< −

*a*. The incident field is then expressed as Using the Fourier integral representation for the Hankel function of the first kind of order zero [5], the

*n*th-components of the column matrices

*ā*^{(i)}(

*ξ*) are given by

*K*+ 1 terms (from −

*K*th- to

*K*th-order terms) to obtain the following results. Also, we use the same sample points

*ā*^{(i)}(

*ξ*), and the column matrix

*b**given in Eq. (28) are then approximated as*

_{l}*x*

_{0},

*y*

_{0}) = (

*d*, 2

*d*), and the obtained field intensities at (

*x,y*) = (0, ±

*d*) are shown in Fig. 2. Figure 2(a) shows the convergence characteristics in terms of the number of sample points

*L*. The truncation order is set to

*K*= 4 for calculating these results. The dotted curves are the results of the trapezoidal scheme, which uses equidistant sample points and equal weights. They oscillate and converge very slowly. The solid curves are obtained by the discretization scheme used in Refs. [8

8. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res . **PIER 74**, 241–271 (2007). [CrossRef]

*K*. The values are computed with

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

*K*is very fast. From the physical point of view, the results of the present formulation are not expected to noticeably different in many cases from those of the conventional RTMA [5] for the scattering from a finite number of cylinders, if the cylinder number is large enough and the observation points are located near the line-source. Here, we consider 98 cylinders located at (

*x,y*) = (

*qd*,0) for

*q*= ±1, ±3, ±4, ±5,...,±50, and the field intensities at (

*x,y*) = (0,

*d*) are calculated by the conventional RTMA. The obtained values are 0.338 for the TM-polarization and 0.264 for the TE-polarization. These values are in good agreement with the results shown in Fig. 2. The total field intensities near the defects are computed with

*L*= 80 and

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

*ρ*

^{(o)},

*ϕ*

^{(o)}) and applying the saddle-point method for

*ρ*

^{(o)}→ ∞ [14], an expression of the scattered field in the far-zone is obtained, and Fig. 4 shows the absolute values of the scattering pattern function.

*ψ*(

*x*,

_{p}*y*;

_{p}*x*,

_{q}*y*) denotes the field observed at (

_{q}*x*,

_{p}*y*) for a line-source located at (

_{p}*x*,

_{q}*y*). The reciprocity theorem requires that this function is zero when both (

_{q}*x*,

_{p}*y*) and (

_{p}*x*,

_{q}*y*) are located in the surrounding medium. We fix one point at (

_{q}*x*,

_{p}*y*) = (0, 2

_{p}*d*) and the other point (

*x*,

_{q}*y*) is moved on the lines

_{q}*y*= ±

*d*. The reciprocity errors at 101 equidistant points in −8

*d*≤

*x*≤ 8

*d*are calculated with

*L*= 80 and

*N*= 4 in the standard double-precision arithmetic, and the obtained results are shown in Fig. 5. The obtained values are smaller than 3 × 10

^{−14}and, considering the computation precision, it can be said that the reciprocity relation is perfectly satisfied.

### 4.2. Plane-wave incidence

*θ*

^{(i)}is measured counter-clockwise from the

*x*-axis, and the incident field of unit amplitude is expressed as with

*k*= −

_{x}*k*cos(

_{s}*θ*

^{(i)}) and

*k*= −

_{y}*k*sin(

_{s}*θ*

^{(i)}). Then, the column matrices

*ā*^{(i)}(

*ξ*),

*ā*^{(s,p)}(

*ξ*), and

*b**are given by where*

_{l}**denotes the identity matrix and the**

*I**n*th-components of the column matrix

*θ*

^{(i)}= 70°. The convergence characteristics are similar to those for the line-source excitation, though the amplitudes of the oscillations appeared on the dotted curves are smaller. The results of the present formulation are compared with the those of the conventional RTMA for 98 cylinders located at (

*x,y*) = (

*qd,*0) for

*q*= ±1, ±3, ±4, ±5,...,±50. The values obtained by the conventional RTMA are 1.377 for the TM-polarization and 0.417 for the TE-polarization. These values are in good agreement with the results shown in Fig. 6. The total field intensities near the defects and the scattering pattern of the residual field

*ψ*

^{(s,d)}are computed with

*L*= 80 and

*K*= 4, and shown in Figs. 7 and 8, respectively. In this computation, the Brillouin zone is split at the Wood-Rayleigh anomalies, and the sample points and the weights are determined by applying the Gauss-Legendre scheme for the subintervals.

## 5. Concluding remarks

8. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res . **PIER 74**, 241–271 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics |

2. | J. Ouellette, “Seeing the future in photonic crystals,” Ind. Phys . |

3. | Ch. Kang and S. M. Weiss, “Photonic crystal with multi-hole defect for sensor applications,” Opt. Express |

4. | A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron . |

5. | W. C. Chew, |

6. | D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A |

7. | H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl . |

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

9. | K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B |

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

11. | K. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published). [PubMed] |

12. | N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E |

13. | K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag . |

14. | C. A. Balanis, |

**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 26, 2012

Revised Manuscript: March 13, 2012

Manuscript Accepted: March 23, 2012

Published: April 24, 2012

**Citation**

Koki Watanabe, Yoshimasa Nakatake, and Jaromír Pištora, "Accurate analysis of electromagnetic scattering from periodic circular cylinder array with defects," Opt. Express **20**, 10646-10657 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-10646

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### References

- S. A. Rinne, F. García-Santamaría, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics2, 52–56 (2007). [CrossRef]
- J. Ouellette, “Seeing the future in photonic crystals,” Ind. Phys. 7, 14–17 (2001).
- Ch. Kang and S. M. Weiss, “Photonic crystal with multi-hole defect for sensor applications,” Opt. Express16, 18188–18193 (2008). [CrossRef] [PubMed]
- A. V. Giannopoulos, J. D. Sulkin, Ch. M. Long, J. J. Coleman, and K. D. Choquette, “Decimated photonic crystal defect cavity lasers,” IEEE J. Sel. Top. Quantum Electron. 17, 1693–1694 (2011). [CrossRef]
- W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
- D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A11, 2526–2538 (1994). [CrossRef]
- H. Roussel, W. C. Chew, F. Jouvie, and W. Tabbara, “Electromagnetic scattering from dielectric and magnetic gratings of fibers — a T-matrix solution,” J. Electromagn. Waves Appl. 10, 109–127 (1996). [CrossRef]
- K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagn. Res. PIER 74, 241–271 (2007). [CrossRef]
- K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B31, 219–237 (2011).
- 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. Watanabe, J. Pištora, and Y. Nakatake, “Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces,” Opt. Express(to be published). [PubMed]
- N. A. Nicorovici and R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E50, 3143–3160 (1994). [CrossRef]
- K. Yasumoto and K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999). [CrossRef]
- C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989).

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