Abstract

Photonic band structures have been calculated for various structures with the periodicity of the simple-cubic lattice. Band gaps have been found, and the conditions for the appearances of such gaps are discussed. The effective long-wavelength dielectric constant is calculated and compared with the predictions of effective-medium and Maxwell–Garnett theories.

© 1993 Optical Society of America

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  1. E. M. Purcell, Phys. Rev. 69, 681 (1946).
    [Crossref]
  2. D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
    [Crossref]
  3. R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 20, 2137 (1985).
    [Crossref]
  4. S. Haroche and D. Kleppner, Phys. Today 42, 24 (1989).
    [Crossref]
  5. P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
    [Crossref]
  6. P. Meystre, in Nonlinear Optics in Solids, Vol. 9 of Springer Series in Wave Phenomena, O. Keller, ed. (Springer-Verlag, Berlin, 1990).
    [Crossref]
  7. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  8. H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
    [Crossref]
  9. E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
    [Crossref] [PubMed]
  10. K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
    [Crossref] [PubMed]
  11. Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
    [Crossref] [PubMed]
  12. K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
    [Crossref] [PubMed]
  13. C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16(6), 563 (1991).
    [Crossref]
  14. E. Yablonovitch and K. M. Leung, Physica B 175, 81 (1991).
    [Crossref]
  15. J. W. Haus, H. S. Sözüer, and R. Inguva, J. Mod. Opt. 39, 1991 (1992).
    [Crossref]
  16. Unless otherwise made clear, all lengths are in units of a/2π and all wave vectors (k, G) are in units of 2π/a, where a is the length of the side of the conventional unit cell.
  17. We say that there is a direct transition gap between the n th and (n+ 1)th levels when ωn+1(k) ωn(k) for all k whereas a gap is said to exist when ωn+1(k) ωn(k′) for all k, k′. A pseudogap, on the other hand, is a depression in the photon density of states ρ(ω).
  18. H. S. Sözüer, “Photonic bands,” Ph.D. dissertation (University of Wyoming, Laramie, Wyo., 1992).
  19. Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962).
    [Crossref]
  20. W. Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. B 21, 2248 (1980).
    [Crossref]
  21. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [Crossref] [PubMed]
  22. T. Nakayama, M. Takano, K. Yakubo, and T. Yamanaka, Opt. Lett. 17, 326 (1992); J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992).
    [Crossref] [PubMed]

1992 (3)

H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[Crossref]

J. W. Haus, H. S. Sözüer, and R. Inguva, J. Mod. Opt. 39, 1991 (1992).
[Crossref]

T. Nakayama, M. Takano, K. Yakubo, and T. Yamanaka, Opt. Lett. 17, 326 (1992); J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992).
[Crossref] [PubMed]

1991 (2)

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16(6), 563 (1991).
[Crossref]

E. Yablonovitch and K. M. Leung, Physica B 175, 81 (1991).
[Crossref]

1990 (3)

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

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

1989 (2)

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

S. Haroche and D. Kleppner, Phys. Today 42, 24 (1989).
[Crossref]

1987 (2)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

1985 (1)

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 20, 2137 (1985).
[Crossref]

1981 (1)

D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
[Crossref]

1980 (1)

W. Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. B 21, 2248 (1980).
[Crossref]

1973 (1)

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

1962 (1)

Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962).
[Crossref]

1946 (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Ashcroft, N. W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. B 21, 2248 (1980).
[Crossref]

Chan, C. T.

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16(6), 563 (1991).
[Crossref]

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

Gmitter, T. J.

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

Haroche, S.

S. Haroche and D. Kleppner, Phys. Today 42, 24 (1989).
[Crossref]

Hashin, Z.

Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962).
[Crossref]

Haus, J. W.

J. W. Haus, H. S. Sözüer, and R. Inguva, J. Mod. Opt. 39, 1991 (1992).
[Crossref]

H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[Crossref]

Hilfer, E. S.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 20, 2137 (1985).
[Crossref]

Ho, K. M.

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16(6), 563 (1991).
[Crossref]

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

Hulet, R. G.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 20, 2137 (1985).
[Crossref]

Inguva, R.

H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[Crossref]

J. W. Haus, H. S. Sözüer, and R. Inguva, J. Mod. Opt. 39, 1991 (1992).
[Crossref]

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

Kleppner, D.

S. Haroche and D. Kleppner, Phys. Today 42, 24 (1989).
[Crossref]

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 20, 2137 (1985).
[Crossref]

D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
[Crossref]

Knight, P. L.

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

Lamb, W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. B 21, 2248 (1980).
[Crossref]

Leung, K. M.

E. Yablonovitch and K. M. Leung, Physica B 175, 81 (1991).
[Crossref]

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Liu, Y. F.

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Meystre, P.

P. Meystre, in Nonlinear Optics in Solids, Vol. 9 of Springer Series in Wave Phenomena, O. Keller, ed. (Springer-Verlag, Berlin, 1990).
[Crossref]

Milonni, P. W.

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

Nakayama, T.

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Satpathy, S.

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

Shtrikman, S.

Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962).
[Crossref]

Soukoulis, C. M.

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16(6), 563 (1991).
[Crossref]

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

Sözüer, H. S.

J. W. Haus, H. S. Sözüer, and R. Inguva, J. Mod. Opt. 39, 1991 (1992).
[Crossref]

H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[Crossref]

H. S. Sözüer, “Photonic bands,” Ph.D. dissertation (University of Wyoming, Laramie, Wyo., 1992).

Takano, M.

Wood, D. M.

W. Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. B 21, 2248 (1980).
[Crossref]

Yablonovitch, E.

E. Yablonovitch and K. M. Leung, Physica B 175, 81 (1991).
[Crossref]

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Yakubo, K.

Yamanaka, T.

Zhang, Z.

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

Europhys. Lett. (1)

C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett. 16(6), 563 (1991).
[Crossref]

J. Appl. Phys. (1)

Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962).
[Crossref]

J. Mod. Opt. (1)

J. W. Haus, H. S. Sözüer, and R. Inguva, J. Mod. Opt. 39, 1991 (1992).
[Crossref]

Opt. Commun. (1)

P. W. Milonni and P. L. Knight, Opt. Commun. 9, 119 (1973).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Phys. Rev. B (2)

H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B 45, 13, 962 (1992).
[Crossref]

W. Lamb, D. M. Wood, and N. W. Ashcroft, Phys. Rev. B 21, 2248 (1980).
[Crossref]

Phys. Rev. Lett. (8)

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[Crossref] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

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

D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
[Crossref]

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 20, 2137 (1985).
[Crossref]

Phys. Today (1)

S. Haroche and D. Kleppner, Phys. Today 42, 24 (1989).
[Crossref]

Physica B (1)

E. Yablonovitch and K. M. Leung, Physica B 175, 81 (1991).
[Crossref]

Other (4)

P. Meystre, in Nonlinear Optics in Solids, Vol. 9 of Springer Series in Wave Phenomena, O. Keller, ed. (Springer-Verlag, Berlin, 1990).
[Crossref]

Unless otherwise made clear, all lengths are in units of a/2π and all wave vectors (k, G) are in units of 2π/a, where a is the length of the side of the conventional unit cell.

We say that there is a direct transition gap between the n th and (n+ 1)th levels when ωn+1(k) ωn(k) for all k whereas a gap is said to exist when ωn+1(k) ωn(k′) for all k, k′. A pseudogap, on the other hand, is a depression in the photon density of states ρ(ω).

H. S. Sözüer, “Photonic bands,” Ph.D. dissertation (University of Wyoming, Laramie, Wyo., 1992).

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Figures (14)

Fig. 1
Fig. 1

Photonic band structure for close-packed air spheres in the sc lattice. b = 13, a = 1, and β = π/6, with 750 plane waves (solid curves) by the E method. Also plotted are the lowest six bands, calculated with only 81 plane waves (overlapping dashed curves) for comparison. c ¯ c / ¯, where ¯ is the spatial average of (r). The inset shows the path in the sc BZ.

Fig. 2
Fig. 2

Photonic band structure for overlapping air spheres in the sc lattice. b = 13, a = 1, and β = 0.81. The E method was used with 750 plane waves.

Fig. 3
Fig. 3

Relative photonic band gap versus the radius Rs of air spheres in the sc lattice. b = 13 and a = 1. Rs = π corresponds to the close-packed case, and at R s = π 2 the background medium becomes disconnected.

Fig. 4
Fig. 4

Square-rod structure. A 3 × 3 × 3 section is shown.

Fig. 5
Fig. 5

Circular-rod structure. A 3 × 3 × 3 section is shown.

Fig. 6
Fig. 6

eff E (solid) and eff H (dashed) for the square-rod structure as a function of N−1/3. b = 13, a = 1, and β = 0.82.

Fig. 7
Fig. 7

eff calculated with the plane-wave method (●) for the square-rod structure in the sc lattice. Also plotted are the Hashin–Shtrikman bounds (Maxwell–Garnett results) (long dashed curves) and the effective-medium result (solid curve). β13 is the volume fraction of the = 13 material.

Fig. 8
Fig. 8

Photonic bands for the square-rod structure in the sc lattice. b = 13, a = 1, and β = 0.82.

Fig. 9
Fig. 9

Density of states computed from a random sample of 300 k points in the BZ for the same dielectric structure as in Fig. 8.

Fig. 10
Fig. 10

Photonic band gap versus N−1/3 for the same structure as in Fig. 8 with the E and the H methods.

Fig. 11
Fig. 11

Photonic bands for the same dielectric structure as in Fig. 8, calculated within the periodicity of the fcc lattice. The vertical scale is different from that of Fig. 8 because afcc = 2asc.

Fig. 12
Fig. 12

Photonic bands for the scaffold structure in the bcc lattice. b = 13, a = 1, and β = 0.84. 1481 plane waves were used with the E method. The H method yields the same degeneracies, but its convergence is slow.

Fig. 13
Fig. 13

Photonic band structure for the circular-dielectric-rod structure in the sc lattice. b = 1, a = 13, and β = 0.19. The radius of the dielectric rods is Rc = 1.

Fig. 14
Fig. 14

Photonic band structure for the circular-air-rod structure in the sc lattice. b = 13, a = 1, and β = 0.81. The radius of the air rods is Rc = 2.6.

Equations (27)

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× × E ( r , t ) + 1 c 2 2 t 2 ( r ) E ( r , t ) = 0 ,
× η ( r ) × H ( r , t ) + 1 c 2 2 t 2 H ( r , t ) = 0 ,
( r ) = ( r - R ) > 0.
E n k ( r , t ) = exp [ i ( k · r - ω n k t ) ] G E n k ( G ) exp [ i ( G · r ) ] ,
H n k ( r , t ) = exp [ i ( k · r - ω n k t ) ] G H n k ( G ) exp [ i ( G · r ) ] ,
( k + G ) × [ ( k + G ) × E G ] + ω 2 c 2 G GG E G = 0 ,
( k + G ) × [ G η GG ( k + G ) × H G ] + ω 2 c 2 H G = 0 ,
all r d r exp [ - i ( k - k ) · r ] ( r ) E n k * ( r ) · E n k ( r ) = C n k δ n n δ ( k - k ) ,
all r d r exp [ - i ( k - k ) · r ] H n k * ( r ) · H n k ( r ) = C n k δ n n δ ( k - k ) ,
× × η ( r ) D ( r , t ) + 1 c 2 2 t 2 D ( r , t ) = 0.
( r ) = b + R 0 ( r - R ) ,
( G ) = 1 V cell WS cell d r exp ( - i G · r ) ( r ) = b δ G 0 + 1 V cell all r d r exp ( - i G · r ) 0 ( r ) ,
0 ( r ) = ( a - b ) θ ( R s - r ) ,
( G ) = b δ G 0 + 3 β ( a - b ) sin x - x cos x x 3 ,
0 ( r ) = ( a - b ) [ θ ( π - x ) θ ( s 2 - y ) θ ( s 2 - z ) + θ ( s 2 - x ) θ ( π - y ) θ ( s 2 - z ) + θ ( s 2 - x ) θ ( s 2 - y ) θ ( π - z ) - 2 θ ( s 2 - x ) θ ( s 2 - y ) θ ( s 2 - z ) ] ,
( G ) = b δ G 0 + ( a - b ) ( 2 π s 2 V cell sin Y Y sin Z Z δ G x 0 + 2 π s 2 V cell sin Z Z sin X X δ G y 0 + 2 π s 2 V cell sin X X sin Y Y δ G z 0 - 2 s 3 V cell sin X X sin Y Y sin Z Z ) ,
N - 1 / 3 ~ 1 / G max
G fcc = ( n x , n y , n z ) 2 π a fcc ,             n x , n y , n z all even or all odd ,
G sc = ( m x , m y , m z ) 2 π a sc ,             m x , m y , m z any integers .
G sc = ( m x , m y , m z ) 2 π a fcc 2 ,             m x , m y , m z any integers ,
G sc = ( m x , m y , m z ) 2 π a fcc ,             m x , m y , m z even integers .
E n k even ( r , t ) = exp [ i ( k · r - ω n k t ) ] even G E n k ( G ) exp [ i ( G · r ) ] ,
E n k odd ( r , t ) = exp [ i ( k · r - ω n k t ) ] odd G E n k ( G ) exp [ i ( G · r ) ] .
0 ( r ) = ( a - b ) [ θ ( π - x ) θ ( R c - ρ y z ) - θ ( R c - x ) θ ( R c - ρ y z ) + θ ( π - y ) θ ( R c - ρ z x ) - θ ( R c - y ) θ ( R c - ρ z x ) + θ ( π - z ) θ ( R c - ρ x y ) - θ ( R c - z ) θ ( R c - ρ x y ) + θ ( R c - x ) θ ( R c - y ) θ ( R c - z ) ] - C 0 ( r ) ,
C 0 ( r ) = { a - b i f             x , y , z < R c , ρ x y , ρ y z , ρ z x > R c 0 otherwise .
( G ) = b δ G 0 + ( a - b ) [ ( 2 π 2 R c 2 V cell ) 2 J 1 ( G y z R c ) G y z R c δ G x 0 - ( 2 π R c 3 V cell ) sin ( G x R c ) G x R c 2 J 1 ( G y z R c ) G y z R c + ( 2 π 2 R c 2 V cell ) 2 J 1 ( G z x R c ) G z x R c δ G y 0 - ( 2 π R c 3 V cell ) sin ( G y R c ) G y R c 2 J 1 ( G z x R c ) G z x R c + ( 2 π 2 R c 2 V cell ) 2 J 1 ( G x y R c ) G x y R c δ G z 0 - ( 2 π R c 3 V cell ) sin ( G z R c ) G z R c 2 J 1 ( G x y R c ) G x y R c + ( R c 3 V cell ) sin ( G x R c ) G x R c sin ( G y R c ) G y R c sin ( G z R c ) G z R c ] - C 0 ( G ) ,
C 0 ( G ) = 1 V cell d r C 0 ( r ) cos ( G · r ) 1 V cell ( R c N grid ) 3 i , j , k = - N grid N grid - 1 C 0 ( x i , y j , z k ) × cos ( G x x i + G y y j + G z z k ) ,

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