Abstract

Development of accurate numerical methods for the analysis of photonic-bandgap-based devices is a relevant issue in optimizing existing devices and/or developing new design solutions. Within this framework, we present an innovative and general approach for the evaluation of the electromagnetic behavior of two-dimensional finite-extent photonic crystals made of a finite set of parallel rods. The proposed approach is a generalization of the scattering-matrix method introduced by Maystre and co-workers and of its improved version proposed by the present authors, which exploits a suitable aggregation into “macrocells” to achieve a reduction of the number of unknowns. As a matter of fact, both of these approaches can be exploited only in those cases in which particular modal expansions for the fields hold true. In order to overcome this limitation, we propose a suitable exploitation of the method of auxiliary sources to provide a general and reliable method for the numerical computation of the scattering matrix of an object of arbitrary shape. By taking advantage of this, we can then generalize our improved scattering matrix method to further increase its computational effectiveness. A numerical analysis of some square-lattice configurations is reported to confirm the accuracy of the proposed method and the remarkable computational benefit.

© 2007 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).
  2. M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, "Design and fabrication of silicon photonic crystal optical waveguides," J. Lightwave Technol. 18, 1402-1411 (2000).
    [CrossRef]
  3. 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]
  4. M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
    [CrossRef]
  5. L. Crocco, F. Cuomo, and T. Isernia, "An improved scattering matrix method for the analysis of two-dimensional PBG devices," Microwave Opt. Technol. Lett. 48, 2564-2570 (2006).
    [CrossRef]
  6. R. S. Zaridze, D. D. Karkashadze, G. M. Khatiashvili, and Z. S. Tsverikmazashvili, "The method of auxiliary sources in applied electrodynamics," in Proceedings of the URSI International Symposium on Electromagnetic Theory (Budapest, 1986), pp. 104-106.
  7. D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Trans. Antennas Propag. 44, 48-64 (2002).
  8. M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  9. G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 11, 3323-3332 (1997).
    [CrossRef]
  10. O. M. Bucci, L. Crocco, and T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploitation of close-proximity setups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
    [CrossRef]
  11. D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering, P.Sabatier and E.R.Pike, eds. (Academic, 2001).
  12. G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems," Eng. Analysis Boundary Elements 27, 759-769 (2003).
    [CrossRef]
  13. O. M. Bucci and G. Franceschetti, "On the degrees of freedom of scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1989).
    [CrossRef]
  14. O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
    [CrossRef]
  15. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
    [CrossRef]
  16. M. A. Aleksidze, Fundamental Functions in Approximate Solutions to Boundary Value Problems (Nauka, 1991).
  17. A. G. Kyurkchan and S. A. Minaev, "Using of the wavelet technique for the solution of the wave diffraction problem," J. Quant. Spectrosc. Radiat. Transf. 89, 219-236 (2004).
    [CrossRef]
  18. J. Yonekura, M. Ikeda, and T. Baba, "Analysis of finite 2D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightwave Technol. 17, 1500-1508 (1999).
    [CrossRef]
  19. R. Penrose, "The role of aesthetics in pure and applied mathematical research," Bull. Inst. Math. Appl. 10, 266-271 (1974).

2006

L. Crocco, F. Cuomo, and T. Isernia, "An improved scattering matrix method for the analysis of two-dimensional PBG devices," Microwave Opt. Technol. Lett. 48, 2564-2570 (2006).
[CrossRef]

2004

A. G. Kyurkchan and S. A. Minaev, "Using of the wavelet technique for the solution of the wave diffraction problem," J. Quant. Spectrosc. Radiat. Transf. 89, 219-236 (2004).
[CrossRef]

2003

G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems," Eng. Analysis Boundary Elements 27, 759-769 (2003).
[CrossRef]

2002

D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Trans. Antennas Propag. 44, 48-64 (2002).

2000

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, "Design and fabrication of silicon photonic crystal optical waveguides," J. Lightwave Technol. 18, 1402-1411 (2000).
[CrossRef]

1999

1998

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

1997

G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 11, 3323-3332 (1997).
[CrossRef]

1994

1989

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1989).
[CrossRef]

1974

R. Penrose, "The role of aesthetics in pure and applied mathematical research," Bull. Inst. Math. Appl. 10, 266-271 (1974).

Abramovitz, M.

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Aleksidze, M. A.

M. A. Aleksidze, Fundamental Functions in Approximate Solutions to Boundary Value Problems (Nauka, 1991).

Anastassiu, H. T.

D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Trans. Antennas Propag. 44, 48-64 (2002).

Baba, T.

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Bucci, O. M.

O. M. Bucci, L. Crocco, and T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploitation of close-proximity setups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
[CrossRef]

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1989).
[CrossRef]

Crocco, L.

L. Crocco, F. Cuomo, and T. Isernia, "An improved scattering matrix method for the analysis of two-dimensional PBG devices," Microwave Opt. Technol. Lett. 48, 2564-2570 (2006).
[CrossRef]

O. M. Bucci, L. Crocco, and T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploitation of close-proximity setups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
[CrossRef]

Cuomo, F.

L. Crocco, F. Cuomo, and T. Isernia, "An improved scattering matrix method for the analysis of two-dimensional PBG devices," Microwave Opt. Technol. Lett. 48, 2564-2570 (2006).
[CrossRef]

Doll, T.

Fairweather, G.

G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems," Eng. Analysis Boundary Elements 27, 759-769 (2003).
[CrossRef]

Felbacq, D.

Franceschetti, G.

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1989).
[CrossRef]

Gennarelli, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

He, S.

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Ikeda, M.

Isernia, T.

L. Crocco, F. Cuomo, and T. Isernia, "An improved scattering matrix method for the analysis of two-dimensional PBG devices," Microwave Opt. Technol. Lett. 48, 2564-2570 (2006).
[CrossRef]

O. M. Bucci, L. Crocco, and T. Isernia, "Improving the reconstruction capabilities in inverse scattering problems by exploitation of close-proximity setups," J. Opt. Soc. Am. A 16, 1788-1798 (1999).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Kaklamani, D. I.

D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Trans. Antennas Propag. 44, 48-64 (2002).

Karageorghis, A.

G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems," Eng. Analysis Boundary Elements 27, 759-769 (2003).
[CrossRef]

Karkashadze, D. D.

R. S. Zaridze, D. D. Karkashadze, G. M. Khatiashvili, and Z. S. Tsverikmazashvili, "The method of auxiliary sources in applied electrodynamics," in Proceedings of the URSI International Symposium on Electromagnetic Theory (Budapest, 1986), pp. 104-106.

Khatiashvili, G. M.

R. S. Zaridze, D. D. Karkashadze, G. M. Khatiashvili, and Z. S. Tsverikmazashvili, "The method of auxiliary sources in applied electrodynamics," in Proceedings of the URSI International Symposium on Electromagnetic Theory (Budapest, 1986), pp. 104-106.

Kyurkchan, A. G.

A. G. Kyurkchan and S. A. Minaev, "Using of the wavelet technique for the solution of the wave diffraction problem," J. Quant. Spectrosc. Radiat. Transf. 89, 219-236 (2004).
[CrossRef]

Loncar, M.

Martin, P. A.

G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems," Eng. Analysis Boundary Elements 27, 759-769 (2003).
[CrossRef]

Maystre, D.

G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 11, 3323-3332 (1997).
[CrossRef]

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]

D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering, P.Sabatier and E.R.Pike, eds. (Academic, 2001).

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Minaev, S. A.

A. G. Kyurkchan and S. A. Minaev, "Using of the wavelet technique for the solution of the wave diffraction problem," J. Quant. Spectrosc. Radiat. Transf. 89, 219-236 (2004).
[CrossRef]

Penrose, R.

R. Penrose, "The role of aesthetics in pure and applied mathematical research," Bull. Inst. Math. Appl. 10, 266-271 (1974).

Qiu, M.

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Saillard, M.

D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering, P.Sabatier and E.R.Pike, eds. (Academic, 2001).

Savarese, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

Scherer, A.

Stegun, I.

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Tayeb, G.

G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 11, 3323-3332 (1997).
[CrossRef]

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]

D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering, P.Sabatier and E.R.Pike, eds. (Academic, 2001).

Tsverikmazashvili, Z. S.

R. S. Zaridze, D. D. Karkashadze, G. M. Khatiashvili, and Z. S. Tsverikmazashvili, "The method of auxiliary sources in applied electrodynamics," in Proceedings of the URSI International Symposium on Electromagnetic Theory (Budapest, 1986), pp. 104-106.

Vuckovic, J.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Yonekura, J.

Zaridze, R. S.

R. S. Zaridze, D. D. Karkashadze, G. M. Khatiashvili, and Z. S. Tsverikmazashvili, "The method of auxiliary sources in applied electrodynamics," in Proceedings of the URSI International Symposium on Electromagnetic Theory (Budapest, 1986), pp. 104-106.

Bull. Inst. Math. Appl.

R. Penrose, "The role of aesthetics in pure and applied mathematical research," Bull. Inst. Math. Appl. 10, 266-271 (1974).

Eng. Analysis Boundary Elements

G. Fairweather, A. Karageorghis, and P. A. Martin, "The method of fundamental solutions for scattering and radiation problems," Eng. Analysis Boundary Elements 27, 759-769 (2003).
[CrossRef]

IEEE Trans. Antennas Propag.

O. M. Bucci and G. Franceschetti, "On the degrees of freedom of scattered fields," IEEE Trans. Antennas Propag. 37, 918-926 (1989).
[CrossRef]

O. M. Bucci, C. Gennarelli, and C. Savarese, "Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples," IEEE Trans. Antennas Propag. 46, 351-359 (1998).
[CrossRef]

D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Trans. Antennas Propag. 44, 48-64 (2002).

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

A. G. Kyurkchan and S. A. Minaev, "Using of the wavelet technique for the solution of the wave diffraction problem," J. Quant. Spectrosc. Radiat. Transf. 89, 219-236 (2004).
[CrossRef]

Microwave Opt. Technol. Lett.

L. Crocco, F. Cuomo, and T. Isernia, "An improved scattering matrix method for the analysis of two-dimensional PBG devices," Microwave Opt. Technol. Lett. 48, 2564-2570 (2006).
[CrossRef]

Phys. Rev. B

M. Qiu and S. He, "Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions," Phys. Rev. B 61, 12871-12876 (2000).
[CrossRef]

Other

R. S. Zaridze, D. D. Karkashadze, G. M. Khatiashvili, and Z. S. Tsverikmazashvili, "The method of auxiliary sources in applied electrodynamics," in Proceedings of the URSI International Symposium on Electromagnetic Theory (Budapest, 1986), pp. 104-106.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

M. A. Aleksidze, Fundamental Functions in Approximate Solutions to Boundary Value Problems (Nauka, 1991).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1970).

D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering, P.Sabatier and E.R.Pike, eds. (Academic, 2001).

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Figures (12)

Fig. 1
Fig. 1

General description of the problem and notations.

Fig. 2
Fig. 2

Single rod and macrocells for a crystal lattice with hexagonal symmetry, where ρ is the radius of the single rod and d is the distance among the cylinders.

Fig. 3
Fig. 3

Description of MAS formulation (IAS and EAS).

Fig. 4
Fig. 4

Generic macrocell with MAS notations.

Fig. 5
Fig. 5

Square macrocell ( 2 × 2 single rods).

Fig. 6
Fig. 6

Square macrocells: (a) 2 × 2 (8 single rods), (b) 3 × 3 ( 12 single rods), (c) 4 × 4 ( 16 single rods). The cylinders have permittivity ϵ = 12.25 and σ = 0 .

Fig. 7
Fig. 7

Behavior of the normalized singular values of A e for 2 × 2 square aggregation.

Fig. 8
Fig. 8

Behavior of the weighted error (a) for the scattered field on S e x t , (b) for the incident field in Ω.

Fig. 9
Fig. 9

Error for the left-hand singular functions with 8 IAS and different values of l S i n t .

Fig. 10
Fig. 10

Comparison between the scattered fields evaluated with the SMM and the G-SMM aproaches: (a) amplitude, (b) phase.

Fig. 11
Fig. 11

Square-lattice PBG T-junction device: (a) 2 × 2 macrocell covering, (b) amplitude of the total electric field evaluated with G-SMM, (c) amplitude of the total electric field evaluated with SMM,(d) 4 × 4 macrocell covering, (e) amplitude of the total electric field evaluated with G-SMM.

Fig. 12
Fig. 12

Square-lattice PBG waveguide with 90 ° bends device: (a) macrocell covering, (b) amplitude of the total electric field evaluated with G-SMM, (c) amplitude of the total electric field evaluated with SMM.

Tables (1)

Tables Icon

Table 1 Number of Parameters for Different Macrocells and Methods

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

E t o t i ( P ) = m = + a i , m J m [ k 0 r i ( P ) ] e j m θ i ( P ) + m = + b i , m H m ( 2 ) [ k 0 r i ( P ) ] e j m θ i ( P ) ,
a i , m = e j k 0 r i sin ( θ i n c θ i ) e j m θ i n c + k i , 1 N q = + b k , q H m q ( 2 ) ( k 0 r i k ) e j ( q m ) θ i k .
[ I S 1 T 1 , 2 S 1 T 1 , N S 2 T 2 , 1 I S 2 T 2 , N S N T N , 1 S N T N , 2 I ] [ b 1 b 2 b N ] = [ S 1 Q 1 S 2 Q 2 S N Q N ] ,
Field scattered by the i th inclusion = = Field scattered due to the primary incident field + Fields scattered due to the secondary incident fields coming from all other inclusions.
E s c i ( ρ ) = k = 1 N I β k i H 0 ( 2 ) ( k 0 ρ r k I ) ,
E i n c i ( ρ ) = j = 1 N E α j i H 0 ( 2 ) ( k 0 ρ r j E ) ,
E s c i ( ρ ) = j = 1 N E α j i k = 1 N I σ k j H 0 ( 2 ) ( k 0 ρ r k I ) ,
β = S α ,
E s c j i ( r p ) = k = 1 N I σ k j H 0 ( 2 ) ( k 0 r p r k I ) , p = 1 , , P ,
β i = S i [ q i + h = 1 , h i M ψ i h ] ,
E i n c i ( r p ) = j = 1 N E i α j i H 0 ( 2 ) ( k 0 r p r j i E ) , p = 1 , , P ,
q i = [ P i E ] 1 E i n c i ,
E s c h ( r p ) = j = 1 N E i ψ j i h H 0 ( 2 ) ( k 0 r p r j i E ) , p = 1 , , P ,
E s c h ( r p ) = k = 1 N I h β k h H 0 ( 2 ) ( k 0 r p r k h I ) ,
ψ i h = [ P i E ] 1 Π i , h β h = T i , h β h ,
β i h i M S i [ P i E ] 1 Π i , h β h = S i [ P i E ] 1 E i n c i ,
β i S i h i M T i , h β h = S i q i , i = 1 , , M ,
[ I S 1 T 1 , 2 S 1 T 1 , M S 2 T 2 , 1 I S 2 T 2 , M S M T M , 1 S M T M , 2 I ] [ β 1 β 2 β M ] = [ S 1 q 1 S 2 q 2 S M q M ] .
A e : J L 2 ( Ω i n t ) Ω i n t G ( r , r ) J ( r ) d r = E s c ( r ) L 2 ( S e x t ) ,
E s c ( r ) = A e [ J ] = n = 0 + σ n J ( r ) , u n ( r ) v n ( r ) ,
v n ( r ) k = 1 N I β k H 0 ( 2 ) ( k 0 r r k I ) 2 ,
A p : J L 2 ( S e x t ) S e x t G ( r , r ) J ( r ) d r = E i n c ( r ) L 2 ( Ω ) ,
u n * ( r ) j = 1 N E α j H 0 ( 2 ) ( k 0 r r j E ) 2 ,
E s c S M M Γ E s c G S M M Γ E s c S M M Γ

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