Abstract

Spherical radially periodic dielectric structures inhibit electromagnetic propagation over a complete solid angle and a finite frequency range. The range of inhibited frequencies is linear in the amplitude of the radial modulation. Depending on the phase, the amplitude, and the inner boundary of the dielectric modulation, localized photon modes may occur within the gap. The internal field distributions and scattering characteristics of spherical dielectric resonators are analyzed numerically.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
    [CrossRef]
  2. K. H. Drexhage, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 165.
    [CrossRef]
  3. D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
    [CrossRef] [PubMed]
  4. E. Yablonovitch, T. J. Gmitter, and R. Bath, “Inhibited and enhanced spontaneous emission in optically thin AsGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
    [CrossRef] [PubMed]
  5. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), pp. 184–193.
  6. A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
    [CrossRef] [PubMed]
  7. H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
    [CrossRef]
  8. S. Arnold, C. T. Liu, W. B. Whitten, and J. M. Ramsey, “Room-temperature microparticle-based persistent spectral hole burning memory,” Opt. Lett. 16, 420–422 (1991).
    [CrossRef] [PubMed]
  9. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  10. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  11. E. Yablonovitch, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
    [CrossRef] [PubMed]
  12. K. M. Leung, “Full vector wave calculation of photonic band structures in FCC dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [CrossRef] [PubMed]
  13. Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
    [CrossRef] [PubMed]
  14. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic band gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  15. E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: the FCC case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
    [CrossRef] [PubMed]
  16. J. Martorell and N. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
    [CrossRef] [PubMed]
  17. R. D. Meade, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13,772–13,774 (1991).
    [CrossRef]
  18. E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
    [CrossRef] [PubMed]
  19. S. John, “Localization of light,” Phys. Today 44(5), 32–41 (1991).
    [CrossRef]
  20. H. Sami Sözüer, R. Inguva, and J. W. Haus, “Electron-photon analogy analyzed,” Phys. Today 45(4), 121 (1992).
    [CrossRef]
  21. R. M. Shimpe, U.S. patent4,743,083 (May10, 1988).
  22. M. Toda, “Single mode behavior of a circular grating for potential disk shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
    [CrossRef]
  23. T. Erdogan and D. Hall, “Circularly symmetric distributed feedback semiconductor lasers: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [CrossRef]
  24. C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
    [CrossRef]
  25. C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.
  26. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  27. M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), pp. 126–132.
  28. A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in 2-D disordered media,” Opt. Commun. 91, 175–179 (1992).
    [CrossRef]
  29. M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
    [CrossRef]
  30. R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
    [CrossRef]
  31. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
    [CrossRef] [PubMed]
  32. M. N. Jones, “Electromagnetic theory of anisotropic media in spherical geometries: Part 1, transverse isotropy,” Int. J. Electron. 66, 457–467 (1989).
    [CrossRef]
  33. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 746.
  34. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.
  35. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 5.
  36. N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, New York, 1951).
  37. H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
    [CrossRef]
  38. D. G. Hall, “A comment on the coupled-mode equations used in guided wave optics,” Opt. Commun. 82, 453–455 (1991).
    [CrossRef]
  39. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  40. K. Kim and K. Y. Jang, “Hollow silica spheres of controlled size and porosity by sol-gel processing,” J. Am. Ceram. Soc. 74, 1987–1992 (1991).
    [CrossRef]

1992 (6)

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[CrossRef]

H. Sami Sözüer, R. Inguva, and J. W. Haus, “Electron-photon analogy analyzed,” Phys. Today 45(4), 121 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in 2-D disordered media,” Opt. Commun. 91, 175–179 (1992).
[CrossRef]

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

1991 (10)

M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

D. G. Hall, “A comment on the coupled-mode equations used in guided wave optics,” Opt. Commun. 82, 453–455 (1991).
[CrossRef]

K. Kim and K. Y. Jang, “Hollow silica spheres of controlled size and porosity by sol-gel processing,” J. Am. Ceram. Soc. 74, 1987–1992 (1991).
[CrossRef]

R. D. Meade, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13,772–13,774 (1991).
[CrossRef]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

S. John, “Localization of light,” Phys. Today 44(5), 32–41 (1991).
[CrossRef]

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: the FCC case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

S. Arnold, C. T. Liu, W. B. Whitten, and J. M. Ramsey, “Room-temperature microparticle-based persistent spectral hole burning memory,” Opt. Lett. 16, 420–422 (1991).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

1990 (6)

J. Martorell and N. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

K. M. Leung, “Full vector wave calculation of photonic band structures in FCC dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic band gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

M. Toda, “Single mode behavior of a circular grating for potential disk shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

T. Erdogan and D. Hall, “Circularly symmetric distributed feedback semiconductor lasers: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

1989 (2)

E. Yablonovitch, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
[CrossRef] [PubMed]

M. N. Jones, “Electromagnetic theory of anisotropic media in spherical geometries: Part 1, transverse isotropy,” Int. J. Electron. 66, 457–467 (1989).
[CrossRef]

1988 (1)

E. Yablonovitch, T. J. Gmitter, and R. Bath, “Inhibited and enhanced spontaneous emission in optically thin AsGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef] [PubMed]

1987 (3)

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1981 (1)

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

1976 (1)

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[CrossRef]

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Arnold, S.

Bath, R.

E. Yablonovitch, T. J. Gmitter, and R. Bath, “Inhibited and enhanced spontaneous emission in optically thin AsGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef] [PubMed]

Blaauw, C.

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Campillo, A. J.

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic band gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), pp. 184–193.

Childs, J. J.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

Drexhage, K. H.

K. H. Drexhage, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 165.
[CrossRef]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan and D. Hall, “Circularly symmetric distributed feedback semiconductor lasers: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Eversole, J. D.

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

Fallahi, M.

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 5.

Field, M. S.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

Ghatak, A. K.

M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), pp. 126–132.

Glinski, J.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Gmitter, T.

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: the FCC case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and R. Bath, “Inhibited and enhanced spontaneous emission in optically thin AsGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef] [PubMed]

Hall, D.

T. Erdogan and D. Hall, “Circularly symmetric distributed feedback semiconductor lasers: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Hall, D. G.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

D. G. Hall, “A comment on the coupled-mode equations used in guided wave optics,” Opt. Commun. 82, 453–455 (1991).
[CrossRef]

Haus, H. A.

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[CrossRef]

Haus, J. W.

H. Sami Sözüer, R. Inguva, and J. W. Haus, “Electron-photon analogy analyzed,” Phys. Today 45(4), 121 (1992).
[CrossRef]

Heinzen, D. J.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic band gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Inguva, R.

H. Sami Sözüer, R. Inguva, and J. W. Haus, “Electron-photon analogy analyzed,” Phys. Today 45(4), 121 (1992).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 746.

Jang, K. Y.

K. Kim and K. Y. Jang, “Hollow silica spheres of controlled size and porosity by sol-gel processing,” J. Am. Ceram. Soc. 74, 1987–1992 (1991).
[CrossRef]

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13,772–13,774 (1991).
[CrossRef]

John, S.

S. John, “Localization of light,” Phys. Today 44(5), 32–41 (1991).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Jones, M. N.

M. N. Jones, “Electromagnetic theory of anisotropic media in spherical geometries: Part 1, transverse isotropy,” Int. J. Electron. 66, 457–467 (1989).
[CrossRef]

Kim, K.

K. Kim and K. Y. Jang, “Hollow silica spheres of controlled size and porosity by sol-gel processing,” J. Am. Ceram. Soc. 74, 1987–1992 (1991).
[CrossRef]

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Kleppner, D.

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

Knight, D. G.

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Lawandy, N.

J. Martorell and N. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

Leung, K.

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: the FCC case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Leung, K. M.

K. M. Leung, “Full vector wave calculation of photonic band structures in FCC dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Lin, H.-B.

H.-B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[CrossRef]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

Liu, C. T.

Maciejko, R.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Makino, T.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

Maradudin, A. A.

A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in 2-D disordered media,” Opt. Commun. 91, 175–179 (1992).
[CrossRef]

M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[CrossRef]

Maritan, C.

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Martorell, J.

J. Martorell and N. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

McGurn, A. R.

A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in 2-D disordered media,” Opt. Commun. 91, 175–179 (1992).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, New York, 1951).

Mead, R. D.

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13,772–13,774 (1991).
[CrossRef]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 5.

Najafi, S. I.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Ping, S.

M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[CrossRef]

Plihal, M.

M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[CrossRef]

Ramsey, J. M.

Rappe, A. M.

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13,772–13,774 (1991).
[CrossRef]

Robertson, W. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Sami Sözüer, H.

H. Sami Sözüer, R. Inguva, and J. W. Haus, “Electron-photon analogy analyzed,” Phys. Today 45(4), 121 (1992).
[CrossRef]

Satpathy, S.

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Shambrook, A.

M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[CrossRef]

Shank, C. V.

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[CrossRef]

Sheng, P.

A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in 2-D disordered media,” Opt. Commun. 91, 175–179 (1992).
[CrossRef]

Shimpe, R. M.

R. M. Shimpe, U.S. patent4,743,083 (May10, 1988).

Sodha, M. S.

M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), pp. 126–132.

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic band gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Svilans, M.

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Templeton, I.

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Thomas, J. E.

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

Toda, M.

M. Toda, “Single mode behavior of a circular grating for potential disk shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.

Whitten, W. B.

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Wu, C.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

Yablonovitch, E.

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: the FCC case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

E. Yablonovitch, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, and R. Bath, “Inhibited and enhanced spontaneous emission in optically thin AsGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zhang, Z.

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, “Circularly symmetric operation of a concentric-circle grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

R. D. Mead, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a photonic band gap in 2-D,” Appl. Phys. Lett. 61, 495–497 (1992).
[CrossRef]

IEEE J. Lightwave Technol. (1)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave-theory for circular gratings on planar dielectric waveguides,” IEEE J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Toda, “Single mode behavior of a circular grating for potential disk shaped DFB lasers,” IEEE J. Quantum Electron. 26, 473–481 (1990).
[CrossRef]

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[CrossRef]

Int. J. Electron. (1)

M. N. Jones, “Electromagnetic theory of anisotropic media in spherical geometries: Part 1, transverse isotropy,” Int. J. Electron. 66, 457–467 (1989).
[CrossRef]

J. Am. Ceram. Soc. (1)

K. Kim and K. Y. Jang, “Hollow silica spheres of controlled size and porosity by sol-gel processing,” J. Am. Ceram. Soc. 74, 1987–1992 (1991).
[CrossRef]

J. Appl. Phys. (1)

T. Erdogan and D. Hall, “Circularly symmetric distributed feedback semiconductor lasers: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

D. G. Hall, “A comment on the coupled-mode equations used in guided wave optics,” Opt. Commun. 82, 453–455 (1991).
[CrossRef]

A. R. McGurn, P. Sheng, and A. A. Maradudin, “Strong localization of light in 2-D disordered media,” Opt. Commun. 91, 175–179 (1992).
[CrossRef]

M. Plihal, A. Shambrook, A. A. Maradudin, and S. Ping, “2-D photonic band structures,” Opt. Commun. 80, 199–204 (1991).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

R. D. Meade, A. M. Rappe, and J. D. Joannopoulos, “Photonic bound states in periodic dielectric materials,” Phys. Rev. B 44, 13,772–13,774 (1991).
[CrossRef]

Phys. Rev. Lett. (14)

E. Yablonovitch, T. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, “Donor and acceptor modes in photonic band structure,” Phys. Rev. Lett. 67, 3380–3383 (1991).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Photonic band structure: the face-centered-cubic case,” Phys. Rev. Lett. 63, 1950–1953 (1989).
[CrossRef] [PubMed]

K. M. Leung, “Full vector wave calculation of photonic band structures in FCC dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic band gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: the FCC case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

J. Martorell and N. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a 2-D periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981).
[CrossRef]

D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Field, “Enhanced and inhibited visible spontaneous emission by atoms in a confocal resonator,” Phys. Rev. Lett. 58, 1320–1323 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, and R. Bath, “Inhibited and enhanced spontaneous emission in optically thin AsGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef] [PubMed]

A. J. Campillo, J. D. Eversole, and H.-B. Lin, “Cavity quantum electrodynamic enhancement of spontaneous emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[CrossRef] [PubMed]

Phys. Today (2)

S. John, “Localization of light,” Phys. Today 44(5), 32–41 (1991).
[CrossRef]

H. Sami Sözüer, R. Inguva, and J. W. Haus, “Electron-photon analogy analyzed,” Phys. Today 45(4), 121 (1992).
[CrossRef]

Other (10)

R. M. Shimpe, U.S. patent4,743,083 (May10, 1988).

C. Wu, M. Svilans, J. Glinski, C. Blaauw, C. Maritan, D. G. Knight, M. Fallahi, I. Templeton, R. Maciejko, and S. I. Najafi, “Circular grating surface emitting laser,” in Integrated Photonics Research, Vol. 4 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), post-deadline paper PD03, pp. 9–12.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990), pp. 184–193.

K. H. Drexhage, Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, p. 165.
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), pp. 126–132.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 746.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Chap. 5.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. Press, New York, 1951).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Rays propagating in a radially periodic dielectric.

Fig. 2
Fig. 2

Imaginary part of q for TE and TM modes in the first-order stop band of V(r) = Vg cos(Kr). Here Vg = 0.3. For Vg ≤ 0.1, the value of q is the same for the TE and TM modes.

Fig. 3
Fig. 3

Normalized first-order stop band edges for the TE and TM modes as functions of Vg. k is the frequency at which the characteristic exponent of the Bloch functions switches from real to complex.

Fig. 4
Fig. 4

(a) S+(r) for the TE and TM modes corresponding to V(r) = 0.05 cos(Kr) and k = K/2. (b) S(r) for the TE and TM modes corresponding to V(r) = 0.05 cos(Kr) and k = K/2.

Fig. 5
Fig. 5

(a) clpm as a function of kp for lp = 1, Vg = 0.05, and Kr0 = 90π. The zero crossing of the curve corresponds to a localized mode. (b) Same as (a), except that Vg = 0.2.

Fig. 6
Fig. 6

(a) Values of kp where clpm = 0 as a function of r0 for lp = 1, Vg = 0.05. These values of kp are where localized modes occur. The dashed line on the curve corresponds to Fig. 5(a). Note that the second value of kp that produces a localized mode is just cut off in Fig. 5(a). (b) Same as (a), except that Vg = 0.2. The dashed line corresponds to Fig. 5(b).

Fig. 7
Fig. 7

Same as Fig. 6(a) with S±(r) → S±(rπ/2). This phase shift in the Bloch functions corresponds to a phase shift in the original modulation. The values where local modes occur are significantly different from those in Fig. 6(a) because of the sensitivity of the local modes on the phase of the modulation.

Fig. 8
Fig. 8

clm versus Kr/2 for lp = 1, Vg = 0.05, Kr0 = 40.5π, and 2k = K.

Fig. 9
Fig. 9

cl at Kr = 400π for Kr0 = 40.5π as a function of k for the same parameters as in Fig. 8. The value of kp where cl = 0 is evidence of a local mode for the periodic structure. The value where 2kp = K corresponds to the asymptotic value in Fig. 8. Note that the dielectric modulation is continuous at the inner boundary.

Fig. 10
Fig. 10

cl at Kr = 1000 for the Kr0 = π as a function of k. The dielectric modulation is 0.05 cos(Kr + φ). (a) φ = 0, (b) φ = π/2, (c) φ = π, (d) φ = 3π/2.

Fig. 11
Fig. 11

cl at Kr = 2000 for Kr0 = 40π as a function of k. The dielectric modulation is 0.05 cos(Kr). (a) l = 1, (b) l = 2, (c) l = 3, (d) l = 4, (e) l = 11, (f) l = 14, (g) l = 22, (h) l = 29.

Fig. 12
Fig. 12

Qe as a function of the outer boundary, rf of a finite resonator. Here Vg = 0.02 and Kr0 = 100π. We consider the lp = 1 mode the resonance frequency 2k/K = 0.997008.

Fig. 13
Fig. 13

|α1m/a1m|2 versus (2k/K − 1)/Vg for Vg = 0.02, Kr0 = 100π, and krf = 400π for (a) l = 1, (b) l = 30, (c) = 60, (d) l = 100, (e) l = 200.

Fig. 14
Fig. 14

|α1m/a1m|2 as a function of (2k/K − 1)/Vg for Krf/2 for l = 1 and Vg = 0.02.

Fig. 15
Fig. 15

Quarter-wave stack stop band (2kn1/Kωc)nn versus Δn/n, where ωc = 2n/(2n + Δn) is the center of the gap. For small Δn/n, ωc ≈ 1, n1n, and the size of the gap is ≈2/π.

Fig. 16
Fig. 16

Qe as a function of the number of periods in the layered structure. Here ni = n2 = 1.5, n1 = 1.47 (Δn/n ≈ 0.02), nf = 1, and Kr0 = 100. We consider the lp = 1 mode with the resonance frequency 2kn1/K ≈ 0.98804, which is near the center of the gap.

Equations (61)

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

( r ) = a [ 1 + V ( r ) ] ,
× × E = k 2 [ 1 + V ( r ) ] E ,
E = E l m ( 1 ) P l m + E l m ( 2 ) B l m + E l m ( 2 ) C l m ,
× × E l m ( 1 ) P l m = L 2 r 2 E l m ( 1 ) P l m + L r E l m ( 1 ) B l m , × × E l m ( 2 ) B l m = - ( E l m ( 2 ) + 2 E l m ( 2 ) r ) B l m - ( E l m ( 2 ) + E l m ( 2 ) r ) L r P l m , × × E l m ( 3 ) C l m = - ( E l m ( 3 ) + 2 E l m ( 3 ) r - L 2 r 2 E l m ( 3 ) ) C l m ,
d 2 u l m d r 2 + k 2 u l m + ( k 2 V ( r ) - L 2 r 2 ) u l m = 0 ,
d 2 g l m d r 2 + k 2 g l m + ( k 2 Q ( r ) - L 2 r 2 ) g l m = 0 ,
E l m ( 3 ) = u l m ( r ) k r , E l m ( 1 ) = L g l m ( r ) k 2 r 2 [ 1 + V ( r ) ] 1 / 2 , E l m ( 2 ) = 1 k 2 r [ 1 + V ( r ) ] d d r { [ 1 + V ( r ) ] 1 / 2 g l m ( r ) } ,
Q ( r ) = V ( r ) - 3 ( V ) 2 4 k 2 ( 1 + V ) 2 + V 2 k 2 ( 1 + V ) .
E ( r ) = l = 1 m = - 1 l u l m ( r ) k r C l m + 1 [ 1 + V ( r ) ] k × [ 1 + V ( r ) ] 1 / 2 g l m ( r ) k r C l m .
H ( r ) = - i k 0 × E = - i l = 1 m = - 1 l × u l m ( r ) k 0 k r C l m + [ 1 + V ( r ) ] 1 / 2 g l m ( r ) k 0 r C l m ,
× f ( r ) C l m = 1 r d d r r f ( r ) B l m + L r f ( r ) P l m .
ψ l ( k r ) = k r j l ( k r ) , χ l ( k r ) = - k r n l ( k r ) , ζ l ( k r ) = k r h l ( 1 ) ( k r ) ,
S ( r ) = exp ( i q r ) n = - s n exp ( i n K r ) ,
k = K m 2 ( 1 - V g ) 1 / 2 ,
( 1 - V g 4 ) < 2 k K < ( 1 + V g 4 )
lim r V ( r ) < - 1.
V ( r ) > 1 a - 1 > - 1.
u l m ( k r ) = n = 0 μ n k n r n .
V ( r ) = n = 0 ν n k n r n
[ ( n + 2 ) ( n + 1 ) - l ( l + 1 ) ] ν n + 2 + μ n + m = 0 ν m μ m - n = 0.
V ( r ) = { 0 for             r < r 0 V g cos ( K r ) + l p ( l p + 1 ) k p 2 r 2 for             r > r 0 ,
E = { m = - l p l p α l p m j l p ( k p r ) C l p m for             r < r 0 m = - l p l p 1 k p r [ c l p m S + ( r ) + d l p m S - ( r ) ] C l p m for             r > r 0 ,
H = { - i m = - l p l p α l p m ( 1 k 0 r d d r r j l p ( k p r ) B l p m + L p j l p ( k p r ) k 0 r P l p m ) for             r < r 0 - i m = - l p l p ( 1 k 0 k p r B l p m d d r + L p k 0 k p r 2 P l p m ) [ c l p m S + ( r ) + d l p m S - ( r ) ] for             r > r 0 ,
[ c l p m d l p m ] = - 1 w [ S - ( r 0 ) - S - ( r 0 ) - S + ( r 0 ) S + ( r 0 ) ] [ ψ l p ( k p r 0 ) α l m ψ l p ( k p r 0 ) α l m ] ,
ψ l p ( k p r 0 ) sin ( k p r 0 - l π 2 ) .
V ( r ) = { 0 for             r < r 0 V g cos ( K r ) for             r > r 0 .
G k ( l ) ( r , r ) = - ψ l ( k r < ) χ l ( k r > ) k ,
u l m = ϕ ( r ) + k χ l ( k r ) 0 r ψ l ( k r ) V ( r ) u l m ( r ) d r + k ψ l ( k r ) r χ l ( k r ) V ( r ) u l m ( r ) d r ,
u l m = a l m ( r ) ψ l ( k r ) + b l m ( r ) χ l ( k r ) ,
a l m ( r ) = C a + k 2 r χ l ( k r ) V ( r ) u l m r d r ,
b l m ( r ) = C b + k 2 0 r ψ l ( k r ) V ( r ) u l m ( r ) d r
d a l m d r = - k χ l ( k r ) V ( r ) [ a l m ( r ) ψ l ( k r ) + b l m ( r ) χ l ( k r ) ] , d b l m d r = k ψ l ( k r ) V ( r ) [ a l m ( r ) ψ l ( k r ) + b l m ( r ) χ l ( k r ) ] .
d 2 u l m d r 2 + k 2 u l m + k 2 V ( r ) u l m = 0.
G ( r , r ) = S + ( r < ) S - ( r > ) w ,
u l m = c l m ( r ) S + ( r ) + d l m ( r ) S - ( r ) ,
d c l m d r = L 2 w r 2 S - ( r ) [ c l m ( r ) S + ( r ) + d l m ( r ) S - ( r ) ] , d d l m d r = - L 2 w r 2 S + ( r ) [ c l m ( r ) S + ( r ) + d l m ( r ) S - ( r ) ] .
ψ l ( k r ) a l m = - χ l ( k r ) b l m , S + ( r ) c l m = - S - ( r ) d l m .
[ c l m ( r ) d l m ( r ) ] = - 1 w [ S - - S - - S + S + ] [ ψ l ( k r ) χ l ( k r ) ψ l ( k r ) χ l ( k r ) ] [ a l m ( r ) b l m ( r ) ]
S + ( r ) = exp ( ν r ) s + ( r ) , S - ( r ) = exp ( - ν r ) s - ( r ) ,
d c l m d r = L 2 w r 2 [ s - ( r ) s + ( r ) c l m ( r ) + s - 2 ( r ) d ¯ l m ( r ) ] , d d l m d r = - L 2 w r 2 [ s + 2 ( r ) c l m ( r ) + s - ( r ) s + ( r ) d ¯ l m ( r ) ] - 2 ν d ¯ l m ,
( r ) = { a for             r < r 0 a [ 1 + V g cos ( K r ) ] + l p ( l p + 1 ) k p 2 r 2 for             r 0 < r < r f a for             r > r f .
u l p m ( r ) = { ψ l p ( k p r ) for             r < r 0 d l p m S - ( r ) for             r 0 < r < r f α l p m ζ l p ( k p r ) + β l p m ζ l p * ( k p r ) for             r > r f .
Q e = ω U / P ,
U = 1 4 π k o 2 0 r f d r [ 1 + V ( r ) ] u l p m ( r ) 2 ,
P = c 8 k p π α l p m 2 r f 2 d Ω × R { i C l m ζ l p ( k p r ) k p r × ( × C l m * ζ l p * ( k p r ) k o r ) } · r ^ = ω α l p m 2 8 k o 2 k p π .
( r ) = { a for             r < r 0 a [ 1 + V g cos ( K r ) ] for             r 0 < r < r f a for             r > r f .
E = { E i + E sc for             r > r f E t for             r < r f , H = { H i + H sc for             r > r f H t for             r < r f .
E i = l , m [ a l m j l ( k r ) C l 1 + b l m × j l ( k r ) C l 1 k ] , H i = - i l , m [ b l m j l ( k r ) C l 1 + a l m × j l ( k r ) C l 1 k 0 ] .
E sc = l , m [ α l m h l ( 1 ) ( k r ) C l 1 + β l m × h k ( 1 ) ( k r ) C l 1 k ] , H sc = - i l , m [ β l m h l ( 1 ) ( k r ) C l 1 + α l m × h l ( 1 ) ( k r ) C l 1 k 0 ] .
E t = l , m [ γ l m u l ( r ) k r C l 1 + δ l m [ 1 + V ( r ) ] × [ 1 + V ( r ) ] 1 / 2 k 2 r g l ( r ) C l 1 ] , H t = - i l , m [ γ l m × u l ( r ) k o k r C l 1 + δ l m [ 1 + V ( r ) ] 1 / 2 g l ( r ) k r C l 1 ] ,
a l m ψ l ( k r f ) + α l m ζ l ( k r f ) = γ l m u l ( r f ) , a l m ψ l ( k r f ) + α l m ζ l ( k r f ) = γ l m u l ( r f ) , b l m ψ l ( k r f ) + β l m ζ l ( k r f ) = δ l m [ 1 + V ( r f ) ] 1 / 2 g l ( r f ) , b l m ψ l ( k r f ) + β l m ζ l ( k r f ) = δ l m [ 1 + V ( r ) ] d d r [ 1 + V ( r ) ] 1 / 2 g l ( r ) .
= { i for             r < r 0 1 for             r 1 n < r < r 2 n 2 for             r 2 n < r < r 1 ( n + 1 ) f for             r > r f ,
E ( r ) = { l , m α l ψ l ( k i r ) k i r C l m for             r < r 0 l , m ( β l ( n ) ψ l ( k 1 r ) k 1 r + γ l ( n ) χ l ( k 1 r ) k 1 r ) C l m for             r 1 n < r < r 2 n l , m ( δ l ( n ) ψ l ( k 2 r ) k 2 r + κ l ( n ) χ l ( k 2 r ) k 2 r ) C l m for             r 2 n < r < r 1 ( n + 1 ) l , m ( ν l ζ l ( k f r ) k f r + σ l ζ l * ( k f r ) k f r ) C l m for             r > r f ,
H ( r ) = - i k 0 × E .
[ β l ( n ) γ l ( n ) ] = α l M ¯ ¯ 1 ( n ) × [ m = 1 n P ¯ ¯ 2 ( m ) M ¯ ¯ 2 ( m - 1 ) P ¯ ¯ 1 ( m - 1 ) M ¯ ¯ 1 ( m - 1 ) ] × [ 1 k i ψ l ( k i r 0 ) 1 k o k i ψ l ( k i r 0 ) ] , [ δ l ( n ) κ l ( n ) ] = M ¯ ¯ 2 ( n ) P ¯ ¯ 1 ( n ) [ α l ( n ) β l ( n ) ] ,
[ ν 1 σ l ] = [ 1 k f ζ l ( k f r f ) 1 k f ζ l * ( k f r f ) 1 k f k 0 ζ l ( k f r f ) 1 k f k 0 ζ l * ( k f r f ) ] - 1 P ¯ ¯ 2 ( N + 1 ) × [ δ l ( N ) κ l ( N ) ] ,
M ¯ ¯ 1 ( n ) = [ ψ l ( k 1 r 1 n ) χ l ( k 1 r 1 n ) 1 k 1 ψ l ( k 1 r 1 n ) 1 k 1 χ l ( k 1 r 1 n ) ] - 1 ,
P ¯ ¯ 1 ( n ) = [ ψ l ( k 1 r 2 n ) χ l ( k 1 r 2 n ) 1 k 2 ψ l ( k 1 r 2 n ) 1 k 2 χ l ( k 1 r 2 n ) ] ,
M ¯ ¯ 2 ( n ) = [ ψ l ( k 2 r 2 n ) χ l ( k 2 r 2 n ) 1 k 2 ψ l ( k 2 r 2 n ) 1 k 2 χ l ( k 2 r 2 n ) ] - 1 ,
P ¯ ¯ 2 ( n ) = [ ψ l ( k 2 r 2 n ) χ l ( k 2 r 1 n ) 1 k 1 ψ l ( k 2 r 1 n ) 1 k 1 χ l ( k 2 r 1 n ) ] .
M ¯ ¯ = P ¯ ¯ 2 ( m ) M ¯ ¯ 2 ( m - 1 ) P ¯ ¯ 1 ( m - 1 ) M ¯ ¯ 1 ( m - 1 ) .

Metrics