Abstract

It has been suggested that spontaneous emission can be inhibited if atomic transition frequencies fall inside photonic band gaps, that is, three-dimensional frequency stop bands of electromagnetic waves generated by three-dimensional periodic dielectric materials (photonic crystals). There has been a growing interest in how atomic emission spectra are changed quantitatively inside the photonic crystals. We develop a classical theory for the calculation of the emission power from the electric dipole located in three-dimensional photonic crystals by incorporating the plane-wave method, dyadic Green’s function, Poynting theorem, and tetrahedron k-space integration. With the method we perform numerical computations for the emission power of an electric dipole located in the photonic crystals of the fcc lattice structure with spherical atoms. The results show the total inhibition of emission in the photonic band gap as well as strong enhancement around the band edges. In addition, the data indicate the strong dependencies of the emission spectrum on the dipole position and the dipole moment in the photonic crystal.

© 1995 Optical Society of America

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References

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  1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  3. J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612 (1992).
    [Crossref] [PubMed]
  4. J. P. Dowling and C. M. Bowden, “Beat radiation from dipoles near a photonic band edge,” J. Opt. Soc. Am. B 10, 353 (1993).
    [Crossref]
  5. E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
    [Crossref] [PubMed]
  6. S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
    [Crossref] [PubMed]
  7. K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
    [Crossref]
  8. K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646 (1990).
    [Crossref] [PubMed]
  9. Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
    [Crossref] [PubMed]
  10. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990).
    [Crossref] [PubMed]
  11. H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
    [Crossref]
  12. H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” J. Opt. Soc. Am. B 10, 296 (1993).
    [Crossref]
  13. E. Yablonovithch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Phys. B 175, 81 (1991);E. Yablonovithch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
    [Crossref]
  14. G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
    [Crossref]
  15. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
    [Crossref]
  16. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
    [Crossref]
  17. G. Lefmanm and M. Taut, “On the numerical computation of the density of the states and the related properties,” Phys. Status Solidi B 54, 469 (1972).
    [Crossref]
  18. H. L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984).
    [Crossref]
  19. C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971);Generalized Vector and Dyadic Analysis (Institute of Electric and Electronics Engineers, New York, 1992);W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), Chap. 7.

1993 (2)

1992 (5)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[Crossref] [PubMed]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

1991 (1)

E. Yablonovithch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Phys. B 175, 81 (1991);E. Yablonovithch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

1990 (5)

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

1989 (1)

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

1972 (1)

G. Lefmanm and M. Taut, “On the numerical computation of the density of the states and the related properties,” Phys. Status Solidi B 54, 469 (1972).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Beat radiation from dipoles near a photonic band edge,” J. Opt. Soc. Am. B 10, 353 (1993).
[Crossref]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[Crossref] [PubMed]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Beat radiation from dipoles near a photonic band edge,” J. Opt. Soc. Am. B 10, 353 (1993).
[Crossref]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[Crossref] [PubMed]

Gmitter, T. J.

E. Yablonovithch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Phys. B 175, 81 (1991);E. Yablonovithch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

Haus, J. W.

H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” J. Opt. Soc. Am. B 10, 296 (1993).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Inguva, R.

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

Joannopoulos, J. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

Lefmanm, G.

G. Lefmanm and M. Taut, “On the numerical computation of the density of the states and the related properties,” Phys. Status Solidi B 54, 469 (1972).
[Crossref]

Leung, K. M.

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Qian, G.-X.

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

Salehpour, M. R.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Satpathy, S.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

Skriver, H. L.

H. L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984).
[Crossref]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Sozuer, H. S.

H. S. Sozuer and J. W. Haus, “Photonic bands: simple-cubic lattice,” J. Opt. Soc. Am. B 10, 296 (1993).
[Crossref]

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

Tai, C.-T.

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971);Generalized Vector and Dyadic Analysis (Institute of Electric and Electronics Engineers, New York, 1992);W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), Chap. 7.

Taut, M.

G. Lefmanm and M. Taut, “On the numerical computation of the density of the states and the related properties,” Phys. Status Solidi B 54, 469 (1972).
[Crossref]

Yablonovitch, E.

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Yablonovithch, E.

E. Yablonovithch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Phys. B 175, 81 (1991);E. Yablonovithch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

Zhang, Z.

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (2)

Phys. B (1)

E. Yablonovithch and T. J. Gmitter, “Photonic band structure: nonspherical atoms in the faced-centered-cubic case,” Phys. B 175, 81 (1991);E. Yablonovithch and K. M. Leung, “Photonic band structure: the faced-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295 (1991);E. Yablonovitch, T. J. Gmitter, K. M. Leung, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Three-dimensional photonic band structure,” Opt. Quantum Electron. 24, S273 (1992).
[Crossref]

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (1)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with application to photonic band structures,” Phys. Rev. A 46, 612 (1992).
[Crossref] [PubMed]

Phys. Rev. B (5)

H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic bands: convergence problems with the plane wave method,” Phys. Rev. B 45, 13962 (1992).
[Crossref]

K. M. Leung and Y. F. Liu, “The photon band structures: the plane wave method,” Phys. Rev. B 41, 10188 (1990).
[Crossref]

G.-X. Qian and K. M. Leung, “Photonic band structure: the case of oval holes,” Phys. Rev. B 44, 11482 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434 (1992).
[Crossref]

R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13772 (1992).
[Crossref]

Phys. Rev. Lett. (6)

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic propagation in periodic structure: Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

E. Yablonovitch and T. J. Gmitter, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950 (1989).
[Crossref] [PubMed]

S. Satpathy, Z. Zhang, and M. R. Salehpour, “Theory of photon bands in three-dimensional periodic dielectric structure,” Phys. Rev. Lett. 64, 1239 (1990).
[Crossref] [PubMed]

Phys. Status Solidi B (1)

G. Lefmanm and M. Taut, “On the numerical computation of the density of the states and the related properties,” Phys. Status Solidi B 54, 469 (1972).
[Crossref]

Other (2)

H. L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984).
[Crossref]

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (International Textbook, Scranton, Pa., 1971);Generalized Vector and Dyadic Analysis (Institute of Electric and Electronics Engineers, New York, 1992);W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), Chap. 7.

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Figures (8)

Fig. 1
Fig. 1

Electric dipole located in the photonic crystal. The unit cell of the fcc lattice is partially drawn, with the dipole positioned at (0.5, 0.5, 0.5).

Fig. 2
Fig. 2

(a) Photonic band structure for the fcc lattice structure. (b) Density of states (DOS). (c) Total emission power. (d) Emission power (REP)total relative to that in free space. Here a = 1.0, b = 12.25, βυ = 0.74 (close packing), NP = 125, NK = 89, NT = 256, and NB = 15; the dipole position is (0.25, 0.25, 0), and the dipole moment is (0, 0, 1).

Fig. 3
Fig. 3

(a) Electric dipole shifted in the (1, 1, 1) direction. (b) Relative emission power (REP)total versus the position of the electric dipole in the (1, 1, 1) direction. Parameters are as in Fig. 2, except that the electric dipole is shifted in the (1, 1, 1) direction with the dipole moment fixed to the (0, 0, 1) direction. The emission power in the first Brillouin zone is normalized relative to the emission in free space. The thick curves indicate that the dipole is located outside the air spheres, and the thin curves that it is inside the air spheres.

Fig. 4
Fig. 4

(a) Electric dipole shifted in the (1, 1, 0) direction. (b) Relative emission power (REP)total versus the position of the electric dipole in the (1, 1, 0) direction. Parameters are as in Fig. 2, except that the electric dipole is shifted in the (1, 1, 0) direction with the dipole moment fixed to the (0, 0, 1) direction. The emission power in the first Brillouin zone is normalized relative to the emission in free space. The thick curves indicate that the dipole is located outside the air spheres, and the thin curves that it is inside the air spheres.

Fig. 5
Fig. 5

(a) Electric dipole shifted in the (1, 0, 0) direction. (b) Relative emission power (REP)total versus the position of the electric dipole in the (1, 0, 0) direction. The parameters are as in Fig. 2, except that the electric dipole is shifted in the (1, 0, 0) direction with the dipole moment fixed to the (0, 0, 1) direction. The emission power in the first Brillouin zone is normalized relative to the emission in free space. The thick curves indicate that the dipole is located outside the air spheres, and the thin curves that it is inside the air spheres.

Fig. 6
Fig. 6

(a) Electric dipole rotated on xy = 0. (b) Relative emission power (REP)in BZ in D versus the direction of the dipole moment (rotated angle θ). The parameters are as in Fig. 2, except that the dipole position is (0.5, 0.5, 0.5) and the dipole moment is (2−1/2 sin θ, 2−1/2 sin θ, cos θ). The emission power in the irreducible Brillouin zone D is normalized relative to the emission in free space. The dipole is rotated by 2π rad at position (0.5, 0.5, 0.5) on the plane xy = 0.

Fig. 7
Fig. 7

(a) Electric dipole rotated on y = 0.5. (b) Relative emission power (REP)in BZ in D versus the direction of the dipole moment (rotated angle θ). Parameters are as in Fig. 2, except that the dipole position is (0.5, 0.5, 0.5) and the dipole moment is (sin θ, 0.0, cos θ). The emission power in the irreducible Brillouin zone D is normalized relative to the emission in free space. The dipole is rotated by 2π rad at position (0.5, 0.5, 0.5) on the plane y = 0.5.

Fig. 8
Fig. 8

Relative emission power (REP)total or (REP)ir BZ in D. Parameters are as in Fig. 2, except that the dipole position is (x, y, z) in units of the lattice constant a and the dipole moment is (dx, dy, dz). Long-dashed lines indicate the level of the emission in the dielectric, = 12.25. Dotted–dashed lines show the level of the emission for the average dielectric constant, b(1 − βυ) + aβυ = 3.925. (a) (REP)ir BZ(x, y, z) = (0, 0, 0), (dx, y, xz) = (0, 0, 1). (b) (REP)ir BZ(x, y, z) = (0.1, 0.1, 0.1), (dx, dy, dz) = (0, 0, 1). (c) (REP)ir BZ(x, y, z) = (0.5, 0.5, 0.0), (dx, dy, dz) = (0, 0, 1). (d) (REP)ir BZ(x, y, z) = (0.45, 0.0, 0.0), (dx, dy, dz) = (0, 0, 1). (e) (REP)ir BZ(x, y, z) = (0.35, 0, 0), (dx, dy, dz) = (0, 0, 1). (f) (REP)ir BZ(x, y, z) = (0.5, 0.5, 0.5), (dx, dy, dz) = (0, 0, 1). (g) (REP)ir BZ(x, y, z) = (0.5, 0.5, 0.5), (dx, dy, dz) = [sin(0.3π), 0, cos(0.3π)]. (h) (REP)ir BZ(x, y, z) = (0.5, 0.5, 0.5), (dx, dy, dz) = (1, 0, 0). (i) (REP)ir BZ(x, y, z) = (0.5, 0.5, 0.5), ( d x , d y , d z ) = ( 1 / 2 , 1 / 2 , 0 ). (j) (REP)ir BZ(x, y, z) = (0.5, 0.5, 0.5), ( d x , d y , d z ) = [ ( 1 / 2 ) sin ( 0.3 π ) ], ( 1 / 2 ) sin ( 0.3 π ), cos(0.3π)]. (k) (REP)ir BZ(x, y, z) = (0.25, 0.25, 0), (dx, dy, xz) = (0, 0, 1). (l) (REP)ir BZ(x, y, z) = (0.25, 0.25, 0), (dx, dy, xz) = (0, 1, 0). (m) (REP)ir BZ(x, y, z) = (0.25, 0.25, 0), (dx, dy, xz) = (1, 0, 0). (n) (REP)total(x, y, z) = (0, 0, 0), (dx, dy, xz) = (0, 0, 1). (o) (REP)total(x, y, z) = (0.5, 0.5, 0), (dx, dy, xz) = (0, 0, 1). (p) (REP)total(x, y, z) = (0.5, 0.5, 0.5), (dx, dy, xz) = (0, 0, 1).

Tables (2)

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Table 2 Dipole Moment and Position

Equations (63)

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× E ( r ) = i μ 0 ω H ( r ) ,
× H ( r ) = i 0 ( r ) ω E ( r ) ,
[ ( r ) E ( r ) ] = 0 ,
H ( r ) = 0 .
E ( r ) = K E ( K ) exp [ i ( k + K ) r ] ,
H ( r ) = K H ( K ) exp [ i ( k + K ) r ] ,
E ( K ) = E x ( K ) x ̂ + E y ( K ) ŷ + E z ( K ) ̂ ] ,
H ( K ) = H x ( K ) x + H y ( K ) ŷ + H z ( K ) ] .
( r ) V ( r ) = K V ( K ) exp ( i K r ) .
H p [ H p ( K 1 ) H p ( K 2 ) H p ( K 3 ) H p ( K NP ) ] , p = x , y , z ,
E p [ E p ( K 1 ) E p ( K 2 ) E p ( K 3 ) E p ( K NP ) ] , p = x , y , z ,
K p [ ( K + K 1 ) p 0 0 0 0 ( K + K 2 ) p 0 0 0 0 ( K + K 3 ) p 0 00 0 0 ( K + K NP ) p ] , p = x , y , z ,
U [ V 11 V 12 V 1 NP V 21 V 22 V 2 NP V NP 1 V NP 2 V NP NP ] ,
K y E z K z E y = μ 0 ω H x , K z E x K x E z = μ 0 ω H y , K x E y K y E x = μ 0 ω H z ,
K y H z K z H y = 0 ω UE x , K z H x K x H z = 0 ω UE y , K x H y K y H x = 0 ω UE z ,
K x H x + K y H y + K z H z = 0 ,
K x UE x + K y UE y + K z UE z = 0 .
E = [ E x E y E z ] ,
H = [ H x H y H z ] ,
T = [ 0 K z K y K z 0 K x K y K x 0 ] ,
U = [ U 0 0 0 U 0 0 0 U ] ,
K = [ K x K y K z ] .
TE = μ 0 ω H ,
TH = 0 ω UE ,
KH = 0 ,
KUE = 0 .
[ TU 1 T ] H = ω 2 c 2 H ,
[ T 2 ] E = ω 2 c 2 UE .
× × E ( r , t ) + 1 c 2 ( r ) 2 t 2 E ( r , t ) = μ 0 t J ( r , t ) .
× × ( r , t ; r , t ) + 1 c 2 ( r ) 2 t 2 ( r , t ; r , t ) = δ ( r r ) δ ( t t ) .
× × E 0 n ̂ ( r , t ) + 1 c 2 ( r ) 2 t 2 E 0 n ̂ ( r , t ) = 0 ,
E 0 n ̂ ( r , t ) = E 0 n ̂ ( r , k ) exp [ i ω n t ] = K [ E 0 n ̂ ( K ) ] exp [ i ( k + K ) r ] exp ( i ω n t ) ,
× × E 0 n ̂ ( r , k ) ω n 2 c 2 ( r ) E 0 n ̂ ( r , k ) = 0 .
1 ( 2 π ) 3 d 3 k ( r ) n = 1 3 NP E 0 n ̂ ( r , k ) E 0 n * ̂ ( r , k ) = η 2 ( NP ) δ ( r r ) ,
n = 1 3 NP H n ̂ H n a ̂ = I ,
η 2 [ n = 1 3 NP E n ̂ E n a ̂ ] U = I ,
n = 1 3 NP [ j = 1 NP E 0 p n ̂ ( K l ) E 0 q n ̂ ( K j ) * ] V j m = η 2 δ p , q δ l , m ( p , q = x , y , z ) ,
n = 1 3 NP l = 1 NP m = 1 NP [ j = 1 NP E 0 p n ̂ ( K l ) E 0 q n ̂ ( K j ) * ] V j m = η 2 ( NP ) δ p , q ( p , q = x , y , z ) ,
( r , t ; r , t ) = c 2 η 2 ( 2 π ) 3 d 3 k 1 ( NP ) n = 1 3 NP E 0 n ̂ ( r , k ) × E 0 n * ̂ ( r , k ) θ ( t t ) sin [ ω n ( t t ) ] ω n .
× × ( r , ω ) ω 2 c 2 ( r ) ( r , ω ) = i ω μ 0 J ( r , ω ) .
× × G * ¯ ( r , ω ; r , ω ) ω 2 c 2 ( r ) G * ¯ ( r , ω ; r , ω ) = δ ( r r ) δ ( ω ω ) 2 π ,
G * ¯ ( r , ω ; r , t ) = 1 2 π + d t G * ¯ ( r , t ; r , t ) exp ( i ω t ) , G * ¯ ( r , ω ; r , ω ) = 1 2 π + d t G * ¯ ( r , ω ; r , t ) × exp ( i ω t ) , J ( r , ω ) = 1 2 π + d t J ( r , t ) exp ( i ω t ) .
× × ( r , ω ) G * ¯ ( r , ω ; r , ω ) ω 2 c 2 ( r ) ( r , ω ) G * ¯ ( r , ω ; r , ω ) = i ω μ 0 J ( r , ω ) G * ¯ ( r , ω ; r , ω ) .
( r , ω ) × × G * ¯ ( r , ω ; r , ω ) ω 2 c 2 ( r ) ( r , ω ) G * ¯ ( r , ω ; r , ω ) = ( r , ω ) δ J ( r r ) δ ( ω ω ) 2 π .
υ d V { ( r , ω ) [ × × G * ¯ ( r , ω ; r , ω ) ] [ × × ( r , ω ) ] G * ¯ ( r , ω ; r , ω ) } = i ω μ 0 υ d V J ( r , ω ) G * ¯ ( r , ω ; r , ω ) + υ d V ( r , ω ) δ ( r r ) δ ( ω ω ) 2 π .
E ( r , t ) = i μ 0 ( 2 π ) + d ω exp ( i ω t ) + d ω ω × V d υ G ¯ * ( r , ω ; r , ω ) J ( r , ω ) .
S d S n { ( r , ω ) × [ × * ¯ ( r , ω ; r , ω ) ] + [ + ( r , ω ) ] × G * ¯ ( r , ω ; r , ω ) } .
J ( r , t ) = J ( r ) J ( t ) = I 0 Ω d δ ( r r 0 ) θ ( t ) exp ( i Ω t ) ,
P ( t ) = Re [ V d V J * ( r , t ) E ( r , t ) ] .
U ( T ) = T d t P ( t ) .
P ( T ) = T U ( T ) .
lim T + P ( T ) = P lattice ,
lim T + sin [ { Ω + ω n ( k ) } T ] π [ Ω + ω n ( k ) ] = δ ( Ω + ω n ) ,
lim T + sin [ { Ω ω n ( k ) } T ] π [ Ω ω n ( k ) ] = δ ( Ω ω n ) ,
P lattice = ( I 0 Ω ) 2 μ 0 c 2 η 2 π 2 ( 2 π ) 3 1 NP n = 1 3 NP d 3 k | d E 0 n ̂ ( r 0 , k ) | 2 × { δ [ Ω + ω n ( k ) ] + δ [ Ω ω n ( k ) ] } .
P lattice = ( I 0 Ω ) 2 μ 0 c 2 η 2 π 2 ( 2 π ) 3 lim NP 1 NP × j = 1 NP ( the j th zone d 3 k n = 1 3 NP | d E 0 n ̂ ( r 0 , k ) | 2 × { δ [ Ω + ω n ( k ) ] + δ [ Ω ω n ( k ) ] } ) .
P lattice = ( I 0 Ω ) 2 μ 0 c 2 η 2 π 2 ( 2 π ) 3 n = 1 2 NP 1 st BZ d 3 k | d E 0 n ̂ ( r 0 , k ) | 2 × { δ [ Ω ω n ( k ) ] } .
P lattice = ( I 0 Ω ) 2 μ 0 c 2 η 2 π 2 ( 2 π ) 3 n = 1 2 NP Ω = ω n ( k ) , 1 st BZ d S | ω n ( k ) | × | d E 0 n ̂ ( r 0 , k ) | 2 .
P free = ( I 0 Ω ) 2 d 2 η π 2 ( 2 π ) 3 ( 8 π 3 Ω 2 c 2 ) , where = 1.0 for free space .
( REP ) total = P lattice P free = n = 1 2 NP ν 0 = ν n ( k ) , 1 st BZ d S | ω n ( k ) | | d ̂ E 0 n ̂ ( r 0 , k ) | 2 8 π 3 ν 0 2 ,
( REP ) ir zone = n = 1 2 NP ν 0 = ν n ( k ) , ir zone d S | ν n ( k ) | | d ̂ E 0 n ̂ ( r 0 , K ) | 2 8 π 3 ν 0 2 .
( REP ) total = 16 [ ( REP ) ir zone | d = ( 0.01 ) + ( REP ) ir zone | d = ( 0.1.0 ) + ( REP ) ir zone | d = ( 1.0.0 ) ] .
DOS = V unit ( 2 π ) 3 n = 1 2 N 1 st BZ d 3 k δ [ Ω ω n ( k ) ] .

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