Abstract

I present a treatment of highly localized defect states in photonic band-gap crystals. Vector Wannier functions, which are expected to be highly localized about the atomic sites, are introduced as an orthonormal basis to expand the fields within the crystal. A Green’s function formalism is employed to take advantage of the fact that the defect occupies only a small fraction of the volume within a primitive cell and the localized nature of the vector Wannier functions. This suggests that the size of the matrices involved in determining the defect mode frequencies may be relatively modest.

© 1993 Optical Society of America

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  1. E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
    [Crossref]
  2. K. M. Ho, C. Soukoulis, and C. T. Chan, Phys. Rev. Lett. 65, 3152 (1990).
    [Crossref] [PubMed]
  3. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
    [Crossref] [PubMed]
  4. E. Yablonovitch and K. M. Leung, Nature (London) 351, 278 (1991).
    [Crossref]
  5. G.-X. Qian and K. M. Leung, Phys. Rev. B 44, 11482 (1991).
    [Crossref]
  6. C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
    [Crossref]
  7. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
    [Crossref] [PubMed]
  8. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
    [Crossref] [PubMed]
  9. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
    [Crossref]
  10. R. D. Meade, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Mass. 02139 (personal communication).
  11. J. Callaway, Quantum Theory of the Solid State (Academic, New York, 1976), Chap. 5.

1992 (1)

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

1991 (7)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch and K. M. Leung, Nature (London) 351, 278 (1991).
[Crossref]

G.-X. Qian and K. M. Leung, Phys. Rev. B 44, 11482 (1991).
[Crossref]

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[Crossref]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[Crossref]

1990 (1)

K. M. Ho, C. Soukoulis, and C. T. Chan, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Brommer, K. D.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[Crossref]

Callaway, J.

J. Callaway, Quantum Theory of the Solid State (Academic, New York, 1976), Chap. 5.

Chan, C. T.

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[Crossref]

K. M. Ho, C. Soukoulis, and C. T. Chan, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Dalichaouch, R.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

Ho, K. M.

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[Crossref]

K. M. Ho, C. Soukoulis, and C. T. Chan, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Joannopoulos, J. D.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[Crossref]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch and K. M. Leung, Nature (London) 351, 278 (1991).
[Crossref]

G.-X. Qian and K. M. Leung, Phys. Rev. B 44, 11482 (1991).
[Crossref]

McCall, S. L.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

Meade, R. D.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[Crossref]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

R. D. Meade, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Mass. 02139 (personal communication).

Platzman, P. M.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

Qian, G.-X.

G.-X. Qian and K. M. Leung, Phys. Rev. B 44, 11482 (1991).
[Crossref]

Rappe, A. M.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[Crossref]

Schultz, S.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

Soukoulis, C.

K. M. Ho, C. Soukoulis, and C. T. Chan, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Soukoulis, C. M.

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

E. Yablonovitch and K. M. Leung, Nature (London) 351, 278 (1991).
[Crossref]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

Europhys. Lett. (1)

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16, 563 (1991).
[Crossref]

Nature (London) (1)

E. Yablonovitch and K. M. Leung, Nature (London) 351, 278 (1991).
[Crossref]

Opt. Quantum Electron. (1)

E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

Phys. Rev. B (2)

G.-X. Qian and K. M. Leung, Phys. Rev. B 44, 11482 (1991).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. B 44, 13772 (1991).
[Crossref]

Phys. Rev. Lett. (4)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991); R. Dalichaouch, J. Armstrong, S. Schultz, S. L. McCall, and P. M. Platzman, Nature (London) 354, 53 (1991).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67, 3380 (1991).
[Crossref] [PubMed]

K. M. Ho, C. Soukoulis, and C. T. Chan, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

Other (2)

R. D. Meade, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Mass. 02139 (personal communication).

J. Callaway, Quantum Theory of the Solid State (Academic, New York, 1976), Chap. 5.

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Equations (35)

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× 1 ( r ) × H ( r ) = ω 2 c 2 H ( r ) ,
H ( r ) = H n ( k , r ) = 1 ( 2 π ) 3 / 2 G λ = 1 2 h n , λ ( k + G ) e ^ λ ( k + G ) × exp [ i ( k + G ) · r ] ,
( r ) = G G exp ( i G · r ) ,
G = 1 Ω cell d r ( r ) exp ( - i G · r ) ,
G k + G k + G G , G - 1 × [ e ^ 2 ( k + G ) · e ^ 2 ( k + G ) - e ^ 2 ( k + G ) · e ^ 1 ( k + G ) - e ^ 1 ( k + G ) · e ^ 2 ( k + G ) e ^ 1 ( k + G ) · e ^ 1 ( k + G ) ] × [ h n , 1 ( k + G ) h n , 2 ( k + G ) ] = w n 2 ( k ) c 2 [ h n , 1 ( k + G ) h n , 2 ( k + G ) ] .
G λ h n , λ * ( k + G ) h n , λ ( k + G ) = δ n n ,
d r H n * ( k , r ) · H n ( k , r ) = δ n n δ ( k - k ) .
a n ( R , r ) = Ω 1 / 2 ( 2 π ) 3 / 2 BZ d k exp ( - i k · R ) H n ( k , r ) .
H n ( k , r ) = Ω 1 / 2 ( 2 π ) 3 / 2 R exp ( i k · R ) a n ( R , r ) .
d r a n * ( R , r ) · a n ( R , r ) = δ n , n δ R , R .
× [ 1 ( r ) + U ( r ) ] × H ( r ) = ω 2 c 2 H ( r ) ,
U ( r ) = - ( r ) ( r ) [ ( r ) + ( r ) ] .
H ( r ) = n R c n ( R ) a n ( R , r ) .
× 1 ( r ) × H ( r ) = 1 c 2 n R R c n ( R ) ω n 2 ( R ) a n ( R + R , r ) ,
ω n 2 ( r ) = Ω 1 / 2 ( 2 π ) 3 BZ d k exp ( i k · r ) ω n 2 ( k ) .
d r a n * ( R , r ) · × 1 ( r ) × H ( r ) = 1 c 2 R ω n 2 ( R - R ) .
d r a n * ( R , r ) · × U ( r ) × H ( r ) = n R c n ( R ) d r b n * ( R , r ) · U ( r ) b n ( R , r ) ,
b n ( R , r ) × a n ( R , r ) ,
b n ( R , r ) = Ω 1 / 2 ( 2 π ) 3 BZ d k G k + G exp [ i ( k + G ) · ( r - R ) ] × [ h n , 1 ( k + G ) e ^ 2 ( k + G ) - h n , 2 ( k + G ) e ^ 1 ( k + G ) ] .
n R { δ n , n [ ω n 2 ( R - R ) - ω 2 δ R , R ] + M n , n ( R , R ) } c n ( R ) = 0 ,
M n , n ( R , R ) = d r b n * ( R , r ) · U ( r ) b n ( R , r ) ,
M = [ M a a 0 0 0 ] .
g n ( R - R ) = c 2 Ω ( 2 π ) 3 BZ d k exp [ i k · ( R - R ) ] ω 2 - ω n 2 ( k ) .
1 C 2 R [ ω n 2 ( R - R ) - ω 2 δ R , R ] g n ( R - R ) = - δ R , R .
( - G - 1 + M ) c = 0.
[ c a c b ] = [ G a a M a a c a G b a M a a c a ] .
( G a a - 1 - M a a ) c a = 0.
det ( G a a - 1 - M a a ) = 0.
c b = G b a M b b c a .
G k + G k + G - G + G - 1 [ e ^ 1 ( - k - G ) · e ^ 2 ( - k - G ) - e ^ 2 ( - k - G ) · e ^ 1 ( - k - G ) - e ^ 1 ( - k - G ) · e ^ 2 ( - k - G ) e ^ 1 ( - k - G ) · e ^ 1 ( - k - G ) ] [ h n , 1 ( - k - G ) h n , 2 ( - k - G ) ] = w n 2 ( k ) c 2 [ h n , 1 ( - k - G ) h n , 2 ( - k - G ) ] ,
e ^ 1 ( - k - G ) = e ^ 2 ( k + G ) , e ^ 2 ( - k - G ) = e ^ 1 ( k + G ) ,
h n , 1 ( - k - G ) = h n , 2 ( k + G ) , h n , 2 ( - k - G ) = h n , 1 ( k + G ) .
a n ( - r + R ) = a n ( r - R ) ,
b n ( - r + R ) = - b n ( r - R ) .
M n , n ( R , R ) = M n , n ( R , R ) .

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