Abstract

A two-dimensional scattering of a plane wave from a periodic array of dielectric cylinders with arbitrary shape using the multigrid-moment method is examined. The scattered field is expressed in terms of the integral form by an infinite summation of the surface integral over the cross section of the reference cylinder. The integral form is converted into the matrix equation by using the moment method. The integration in the elements of the matrix equation is evaluated by the lattice-sums technique to obtain a precise solution. The multigrid method is applied to the matrix equation to improve the CPU time. The CPU time and the residual norm are examined numerically for a given number of iterations and cycle indices. Then the effects of shape and material of the periodic structure on the power reflection coefficient of the fundamental Floquet mode are shown. In addition, the results indicate the effect of changing the relative permittivity of the dielectric coated body and the reflection coefficient.

© 2008 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  3. K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystals (Taylor & Francis, 2006).
  4. G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid fem-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973-980 (2000).
    [CrossRef]
  5. E. Popov and B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926-4933 (2000).
    [CrossRef]
  6. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Low Temp. Phys. 18, 102-110 (2000).
  7. H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401-444.
  8. R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
    [CrossRef]
  9. P. Wesseling, An Introduction to Multigrid Methods (Wiley, 1992).
  10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).
  11. U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid (Academic, 2001).
  12. A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333-390 (1977).
    [CrossRef]
  13. M. Yokota and K. Aoyama, “Scattering of a Gaussian beam by dielectric cylinders with arbitrary shape using multigrid-moment method,” Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E90-C, 258-264 (2007).
  14. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).
  15. 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]
  16. Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).
    [CrossRef]
  17. K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” PIER 29, 69-85 (2000).
    [CrossRef]

2007

M. Yokota and K. Aoyama, “Scattering of a Gaussian beam by dielectric cylinders with arbitrary shape using multigrid-moment method,” Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E90-C, 258-264 (2007).

2006

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401-444.

K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystals (Taylor & Francis, 2006).

2001

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid (Academic, 2001).

2000

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Low Temp. Phys. 18, 102-110 (2000).

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid fem-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

E. Popov and B. Bozhkov, “Differential method applied for photonic crystals,” Appl. Opt. 39, 4926-4933 (2000).
[CrossRef]

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” PIER 29, 69-85 (2000).
[CrossRef]

1999

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]

1993

R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
[CrossRef]

1992

P. Wesseling, An Introduction to Multigrid Methods (Wiley, 1992).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).

1991

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).

1986

Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).
[CrossRef]

1980

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

1977

A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333-390 (1977).
[CrossRef]

Aoyama, K.

M. Yokota and K. Aoyama, “Scattering of a Gaussian beam by dielectric cylinders with arbitrary shape using multigrid-moment method,” Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E90-C, 258-264 (2007).

Bozhkov, B.

Brandt, A.

A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333-390 (1977).
[CrossRef]

Cocchi, A.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid fem-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).

Harrington, R.

R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
[CrossRef]

Hikari, M.

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Low Temp. Phys. 18, 102-110 (2000).

Ikuno, H.

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401-444.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).

Koshiba, M.

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Low Temp. Phys. 18, 102-110 (2000).

Kushta, K.

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” PIER 29, 69-85 (2000).
[CrossRef]

Monorchio, A.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid fem-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

Naka, Y.

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401-444.

Oosterlee, C.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid (Academic, 2001).

Pelosi, G.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid fem-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

Petit, R.

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

Popov, E.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).

Saad, Y.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Schüller, A.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid (Academic, 2001).

Schultz, M. H.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).

Trottenberg, U.

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid (Academic, 2001).

Tsuji, Y.

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Low Temp. Phys. 18, 102-110 (2000).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).

Wesseling, P.

P. Wesseling, An Introduction to Multigrid Methods (Wiley, 1992).

Yasumoto, K.

K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystals (Taylor & Francis, 2006).

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” PIER 29, 69-85 (2000).
[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]

Yokota, M.

M. Yokota and K. Aoyama, “Scattering of a Gaussian beam by dielectric cylinders with arbitrary shape using multigrid-moment method,” Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E90-C, 258-264 (2007).

Yoshitomi, K.

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]

Appl. Opt.

IEEE Trans. Antennas Propag.

G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid fem-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[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]

J. Low Temp. Phys.

M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Low Temp. Phys. 18, 102-110 (2000).

Math. Comput.

A. Brandt, “Multi-level adaptive solutions to boundary-value problems,” Math. Comput. 31, 333-390 (1977).
[CrossRef]

PIER

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” PIER 29, 69-85 (2000).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 7, 856-869 (1986).
[CrossRef]

Trans. Inst. Electron., Inf. Commun. Eng., Sect. E

M. Yokota and K. Aoyama, “Scattering of a Gaussian beam by dielectric cylinders with arbitrary shape using multigrid-moment method,” Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E90-C, 258-264 (2007).

Other

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1991).

H. Ikuno and Y. Naka, “Finite-difference time-domain method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K.Yasumoto, ed. (Taylor & Francis, 2006), pp. 401-444.

R. Harrington, Field Computation by Moment Methods (IEEE, 1993).
[CrossRef]

P. Wesseling, An Introduction to Multigrid Methods (Wiley, 1992).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN (2nd ed.) (Cambridge U. Press, 1992).

U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid (Academic, 2001).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

K. Yasumoto, ed., Electromagnetic Theory and Applications for Photonic Crystals (Taylor & Francis, 2006).

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Figures (12)

Fig. 1
Fig. 1

Geometry of a periodic array of dielectric cylinders with arbitrary shape.

Fig. 2
Fig. 2

Structure of W cycle with cycle index γ = 3 and three levels.

Fig. 3
Fig. 3

Normalized residual norm and CPU time for several cycle indices. The number of unknowns N = 8100 , and the matrix equation on the finest grid is solved under the condition that the normalized residual norm is less than 10 5 .

Fig. 4
Fig. 4

Scheme for the improvement of the initial values for the multigrid.

Fig. 5
Fig. 5

Normalized residual norm and CPU time for several cycle indices using the scheme indicated in Fig. 4.

Fig. 6
Fig. 6

Power reflection coefficient R 0 2 of the fundamental space harmonics for several different relative permittivities as a function of the normalized frequency d λ ; radius of the circular cylinder is 0.3 d .

Fig. 7
Fig. 7

Power reflection coefficient R 0 2 of the fundamental space harmonics for several different inclination angles Φ g r as a function of the normalized frequency d λ ; ε r = 2.25 , ε 2 = 2.0 , r 1 = 0.15 d , r 2 = 0.3 d .

Fig. 8
Fig. 8

Geometry of a periodic array of composite elliptical cylinders.

Fig. 9
Fig. 9

Power reflection coefficient R 0 2 of the fundamental space harmonics for several different eccentricities as a function of the normalized frequency d λ ; a = 0.3 d and ε 2 = 2.0 .

Fig. 10
Fig. 10

Power reflection coefficient R 0 2 of the fundamental space harmonics for several different relative permittivities as a function of the normalized frequency d λ ; a = 0.3 d and eccentricity is 0.86.

Fig. 11
Fig. 11

Power reflection coefficient R 0 2 of the fundamental space harmonics for an elliptical cylinder containing two circular cylinders, as a function of the normalized frequency d λ ; a = 0.3 and eccentricity is 0.3.

Fig. 12
Fig. 12

Distribution of the total intensity contour distribution map; d = 0.902 λ , ε r 1 = 3.23 , R 1 = 0.1 d , and the eccentricity of the elliptical cylinder is 0.3 with a = 0.3 d .

Equations (16)

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E ( r ) = E i ( r ) j k 2 4 l = s 0 H 0 ( 2 ) ( k ρ l ) E ( r 0 ) exp ( j k x l d ) ( ε r 1 ) d s 0 ,
n = 1 N C m n E n = E m i , m = 1 , , N .
C m n = j π 2 [ ε r ( n ) 1 ] k a n J 1 ( k a n ) l = exp ( j k x l d ) H 0 ( 2 ) ( k ρ m n l ) ( m n ) ,
C n n = 1 + j π 2 [ ε r 0 ( n ) 1 ] { [ k a n H 1 ( 2 ) ( k a n ) 2 j π ] + k a n J 1 ( k a n ) l = , l 0 exp ( j k x l d ) H 0 ( 2 ) ( k ρ n n l ) } ( m = n ) ,
E s ( x , y ) = n = 1 N C m n E n ,
l = exp ( j k l d sin θ i ) H 0 ( 2 ) ( k ρ m n l ) = [ H 0 2 ( k ρ ) + S 0 ( k d , θ i ) J 0 ( k ρ ) + 2 p = 1 S p ( k d , θ i ) J p ( k ρ ) cos ( p ϕ ) ] ,
S p ( k d , θ i ) = l = 1 H p ( 2 ) ( l k d ) exp ( j l k d sin θ i ) + l = 1 ( 1 ) p H p ( 2 ) ( l k d ) exp ( j l k d sin θ i ) ,
l = 1 H p ( 2 ) ( l k d ) exp ( j l k d sin θ i ) = ( 1 ) p exp { j ( π 4 + k d sin θ i ) } 2 π 0 a [ G p ( τ ) + G p ( τ ) ] F ( τ ; k d , sin θ i ) d t ,
G p = ( t j 1 t 2 ) p ,
F ( τ ; k d , sin θ i ) = exp ( j k d 1 t 2 ) 1 t 2 { 1 exp [ j k d ( 1 t 2 sin θ i ) ] } ,
E s = j 2 n = 1 N E n [ ε r 0 ( n ) 1 ] k a n J 1 ( k a n ) × l = 1 κ ( k x , l ) exp [ + j k x , l x n 0 + j κ ( k x , l ) y n 0 ] exp [ j k x , l x j κ ( k x , l ) y ] ,
k x , l = k x + l 2 π d ,
κ ( k x , l ) = k 2 k x , l 2 , Im ( κ ( k x , l ) ) 0 .
l = exp [ j k x l d + j ξ l d ] = 2 π d l = δ ( ξ k x l 2 π d ) ,
R l = j 2 n = 1 N E n [ ε r 0 ( n ) 1 ] k a n J 1 ( k a n ) 1 κ ( k x , l ) exp [ + j k x , l x n 0 + j κ ( k x , l ) y n 0 ] .
b = a 1 e 2 .

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