Abstract

The optical modes in finite partially ordered arrays of dielectric particles are studied within the coupled dipole approach. It is shown that high-quality modes can be attained under conditions of small enough interparticle distance when the light-cone constraint is satisfied. We performed analytical and numerical investigations of these modes to determine their dependence on system size, dimensionality, and extent of disordering. The opportunity to use these modes to make high-performance random lasers is discussed.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2003

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

2002

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

S. Fan and J. D. Joannopolous, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

A. L. Burin, M. A. Ratner, H. Cao, and S. H. Chang, “Random laser in one dimension,” Phys. Rev. Lett. 88, 093904 (2002).
[CrossRef] [PubMed]

A. K. Sarychev and V. M. Shalaev, “Theory of nonlinear optical responses in metal–dielectric composites,” Top. Appl. Phys. 82, 169–184 (2002).
[CrossRef]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[CrossRef] [PubMed]

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

M. Bahoura, K. J. Morris, and M. A. Noginov, “Threshold and slope efficiency of Nd0.5La0.5Al3(BO3)4 ceramic random laser: effect of the pumped spot size,” Opt. Commun. 201, 405–411 (2002).
[CrossRef]

2001

S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett. 78, 16–18 (2001).
[CrossRef]

V. I. Kopp, A. Z. Genack, and Z. Q. Zhang, “Large coherence area thin-film photonic stop-band lasers,” Phys. Rev. Lett. 86, 1753–1756 (2001).
[CrossRef] [PubMed]

G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,” J. Mol. Struct.: THEOCHEM 573, 73–80 (2001).
[CrossRef]

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133–1135 (2001).
[CrossRef] [PubMed]

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

A. L. Burin, M. A. Ratner, H. Cao, and R. P. H. Chang, “Model for a random laser,” Phys. Rev. Lett. 87, 215503 (2001).
[CrossRef] [PubMed]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

2000

M. L. Brongersma, J. M. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[CrossRef] [PubMed]

D. Wiersma, “Laser physics—the smallest random laser,” Nature 406, 132–132 (2000).
[CrossRef] [PubMed]

1999

S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Laser-like emission in opal photonic crystals,” Opt. Commun. 162, 241–246 (1999).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

1998

V. M. Shalaev and A. K. Sarychev, “Nonlinear optics of random metal–dielectric films,” Phys. Rev. B 57, 13, 265–13, 288 (1998).
[CrossRef]

T. S. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

1997

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

1996

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

1994

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

1993

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281–2291 (1993).
[CrossRef]

1991

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulous, “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), 32–40 (1991).
[CrossRef]

1987

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

1968

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

1966

A. M. Afanas’ev and Yu. Kagan, “Change of resonance nuclear parameters during scattering by regular systems,” Sov. Phys. JETP 23, 178–184 (1966).

Afanas’ev, A. M.

A. M. Afanas’ev and Yu. Kagan, “Change of resonance nuclear parameters during scattering by regular systems,” Sov. Phys. JETP 23, 178–184 (1966).

Apalkov, V. M.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

Atwater, H. A.

S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett. 78, 16–18 (2001).
[CrossRef]

M. L. Brongersma, J. M. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Bahoura, M.

M. Bahoura, K. J. Morris, and M. A. Noginov, “Threshold and slope efficiency of Nd0.5La0.5Al3(BO3)4 ceramic random laser: effect of the pumped spot size,” Opt. Commun. 201, 405–411 (2002).
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Baughman, R. H.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Laser-like emission in opal photonic crystals,” Opt. Commun. 162, 241–246 (1999).
[CrossRef]

Bayram, S. B.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

Beenakker, C. W. J.

T. S. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

Bendickson, J. M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[CrossRef] [PubMed]

Bloemer, M. J.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Bowden, C. M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Brommer, K. D.

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

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett. 78, 16–18 (2001).
[CrossRef]

M. L. Brongersma, J. M. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Burin, A. L.

A. L. Burin, M. A. Ratner, H. Cao, and S. H. Chang, “Random laser in one dimension,” Phys. Rev. Lett. 88, 093904 (2002).
[CrossRef] [PubMed]

A. L. Burin, M. A. Ratner, H. Cao, and R. P. H. Chang, “Model for a random laser,” Phys. Rev. Lett. 87, 215503 (2001).
[CrossRef] [PubMed]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

Cao, H.

A. L. Burin, M. A. Ratner, H. Cao, and S. H. Chang, “Random laser in one dimension,” Phys. Rev. Lett. 88, 093904 (2002).
[CrossRef] [PubMed]

A. L. Burin, M. A. Ratner, H. Cao, and R. P. H. Chang, “Model for a random laser,” Phys. Rev. Lett. 87, 215503 (2001).
[CrossRef] [PubMed]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Chabanov, A. A.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[CrossRef] [PubMed]

Chang, R. P. H.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

A. L. Burin, M. A. Ratner, H. Cao, and R. P. H. Chang, “Model for a random laser,” Phys. Rev. Lett. 87, 215503 (2001).
[CrossRef] [PubMed]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Chang, S. H.

A. L. Burin, M. A. Ratner, H. Cao, and S. H. Chang, “Random laser in one dimension,” Phys. Rev. Lett. 88, 093904 (2002).
[CrossRef] [PubMed]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

DeLong, M. C.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

DeLong, N. C.

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

Dowling, J. P.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Faleev, S. V.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[CrossRef] [PubMed]

Fan, S.

S. Fan and J. D. Joannopolous, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Frolov, S. V.

S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Laser-like emission in opal photonic crystals,” Opt. Commun. 162, 241–246 (1999).
[CrossRef]

Genack, A. Z.

V. I. Kopp, A. Z. Genack, and Z. Q. Zhang, “Large coherence area thin-film photonic stop-band lasers,” Phys. Rev. Lett. 86, 1753–1756 (2001).
[CrossRef] [PubMed]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[CrossRef] [PubMed]

Gmitter, T. J.

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

Hartman, J. M.

M. L. Brongersma, J. M. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Hinklin, T.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

Ho, S. T.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Joannopolous, J. D.

S. Fan and J. D. Joannopolous, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Joannopoulous, J. D.

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

John, S.

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133–1135 (2001).
[CrossRef] [PubMed]

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Johnson, S. G.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Kagan, Yu.

A. M. Afanas’ev and Yu. Kagan, “Change of resonance nuclear parameters during scattering by regular systems,” Sov. Phys. JETP 23, 178–184 (1966).

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Kopp, V. I.

V. I. Kopp, A. Z. Genack, and Z. Q. Zhang, “Large coherence area thin-film photonic stop-band lasers,” Phys. Rev. Lett. 86, 1753–1756 (2001).
[CrossRef] [PubMed]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Laine, R. M.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

Letokhov, V. S.

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

Ling, Y.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

Liu, X.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

Maier, S. A.

S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett. 78, 16–18 (2001).
[CrossRef]

Markel, V. A.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281–2291 (1993).
[CrossRef]

Meade, R. D.

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

Mekis, A.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

Misirpashaev, T. S.

T. S. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

Morris, K. J.

M. Bahoura, K. J. Morris, and M. A. Noginov, “Threshold and slope efficiency of Nd0.5La0.5Al3(BO3)4 ceramic random laser: effect of the pumped spot size,” Opt. Commun. 201, 405–411 (2002).
[CrossRef]

Noginov, M. A.

M. Bahoura, K. J. Morris, and M. A. Noginov, “Threshold and slope efficiency of Nd0.5La0.5Al3(BO3)4 ceramic random laser: effect of the pumped spot size,” Opt. Commun. 201, 405–411 (2002).
[CrossRef]

Polson, R. C.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

Raikh, M. E.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601 (2002).
[CrossRef] [PubMed]

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

Rand, S. C.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

Rappe, A. M.

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

Ratner, M. A.

A. L. Burin, M. A. Ratner, H. Cao, and S. H. Chang, “Random laser in one dimension,” Phys. Rev. Lett. 88, 093904 (2002).
[CrossRef] [PubMed]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

A. L. Burin, M. A. Ratner, H. Cao, and R. P. H. Chang, “Model for a random laser,” Phys. Rev. Lett. 87, 215503 (2001).
[CrossRef] [PubMed]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Sarychev, A. K.

A. K. Sarychev and V. M. Shalaev, “Theory of nonlinear optical responses in metal–dielectric composites,” Top. Appl. Phys. 82, 169–184 (2002).
[CrossRef]

V. M. Shalaev and A. K. Sarychev, “Nonlinear optics of random metal–dielectric films,” Phys. Rev. B 57, 13, 265–13, 288 (1998).
[CrossRef]

Scalora, M.

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Schatz, G. C.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,” J. Mol. Struct.: THEOCHEM 573, 73–80 (2001).
[CrossRef]

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Seelig, E. W.

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Shalaev, V. M.

A. K. Sarychev and V. M. Shalaev, “Theory of nonlinear optical responses in metal–dielectric composites,” Top. Appl. Phys. 82, 169–184 (2002).
[CrossRef]

V. M. Shalaev and A. K. Sarychev, “Nonlinear optics of random metal–dielectric films,” Phys. Rev. B 57, 13, 265–13, 288 (1998).
[CrossRef]

Shapiro, B.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

Shkunov, M. N.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

Stockman, M. I.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[CrossRef] [PubMed]

Stoytchev, M.

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[CrossRef] [PubMed]

Toader, O.

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133–1135 (2001).
[CrossRef] [PubMed]

Vardeny, Z. V.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Laser-like emission in opal photonic crystals,” Opt. Commun. 162, 241–246 (1999).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

Wang, Q. H.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Wiersma, D.

D. Wiersma, “Laser physics—the smallest random laser,” Nature 406, 132–132 (2000).
[CrossRef] [PubMed]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Williams, G. R.

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

Yablonovitch, E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

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

Zakhidov, A. A.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Laser-like emission in opal photonic crystals,” Opt. Commun. 162, 241–246 (1999).
[CrossRef]

Zeman, E. J.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Zhang, Z. Q.

V. I. Kopp, A. Z. Genack, and Z. Q. Zhang, “Large coherence area thin-film photonic stop-band lasers,” Phys. Rev. Lett. 86, 1753–1756 (2001).
[CrossRef] [PubMed]

Zhao, L. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Zhao, Y. G.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Adv. Funct. Mater.

M. N. Shkunov, Z. V. Vardeny, M. C. DeLong, R. C. Polson, A. A. Zakhidov, and R. H. Baughman, “Tunable, gap-state lasing in switchable directions for opal photonic crystals,” Adv. Funct. Mater. 12, 21–26 (2002).
[CrossRef]

Appl. Phys. Lett.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett. 78, 16–18 (2001).
[CrossRef]

J. Appl. Phys.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

J. Mod. Opt.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281–2291 (1993).
[CrossRef]

J. Mol. Struct.: THEOCHEM

G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,” J. Mol. Struct.: THEOCHEM 573, 73–80 (2001).
[CrossRef]

J. Phys. Chem.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

J. Phys. Chem. B

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Nature

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

A. A. Chabanov, M. Stoytchev, and A. Z. Genack, “Statistical signatures of photon localization,” Nature 404, 850–853 (2000).
[CrossRef] [PubMed]

D. Wiersma, “Laser physics—the smallest random laser,” Nature 406, 132–132 (2000).
[CrossRef] [PubMed]

Opt. Commun.

S. V. Frolov, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Laser-like emission in opal photonic crystals,” Opt. Commun. 162, 241–246 (1999).
[CrossRef]

M. Bahoura, K. J. Morris, and M. A. Noginov, “Threshold and slope efficiency of Nd0.5La0.5Al3(BO3)4 ceramic random laser: effect of the pumped spot size,” Opt. Commun. 201, 405–411 (2002).
[CrossRef]

Phys. Rev. A

G. R. Williams, S. B. Bayram, S. C. Rand, T. Hinklin, and R. M. Laine, “Laser action in strongly scattering rare-earth-metal-doped dielectric nanophosphors,” Phys. Rev. A 65, 013807 (2002).
[CrossRef]

T. S. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

Y. Ling, H. Cao, A. L. Burin, M. A. Ratner, X. Liu, E. W. Seelig, and R. P. H. Chang, “Investigation of random lasers with resonant feedback,” Phys. Rev. A 64, 063808 (2001).
[CrossRef]

Phys. Rev. B

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

S. Fan and J. D. Joannopolous, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

V. M. Shalaev and A. K. Sarychev, “Nonlinear optics of random metal–dielectric films,” Phys. Rev. B 57, 13, 265–13, 288 (1998).
[CrossRef]

M. L. Brongersma, J. M. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Phys. Rev. E

J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite one-dimensional, photonic band-gap structures,” Phys. Rev. E 53, 4107–4121 (1996).
[CrossRef]

Phys. Rev. Lett.

A. L. Burin, M. A. Ratner, H. Cao, and S. H. Chang, “Random laser in one dimension,” Phys. Rev. Lett. 88, 093904 (2002).
[CrossRef] [PubMed]

A. L. Burin, M. A. Ratner, H. Cao, and R. P. H. Chang, “Model for a random laser,” Phys. Rev. Lett. 87, 215503 (2001).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Crossover between universality classes in the statistics of rare events in disordered conductors,” Phys. Rev. Lett. 89, 126601 (2002).
[CrossRef] [PubMed]

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002).
[CrossRef] [PubMed]

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

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, “3D wire mesh photonic crystals,” Phys. Rev. Lett. 76, 2480–2483 (1996).
[CrossRef] [PubMed]

V. I. Kopp, A. Z. Genack, and Z. Q. Zhang, “Large coherence area thin-film photonic stop-band lasers,” Phys. Rev. Lett. 86, 1753–1756 (2001).
[CrossRef] [PubMed]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[CrossRef] [PubMed]

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999).
[CrossRef]

Phys. Today

S. John, “Localization of light,” Phys. Today 44(5), 32–40 (1991).
[CrossRef]

Science

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133–1135 (2001).
[CrossRef] [PubMed]

Sov. Phys. JETP

V. S. Letokhov, “Generation of light by a scattering medium with negative resonance absorption,” Sov. Phys. JETP 26, 835–840 (1968).

A. M. Afanas’ev and Yu. Kagan, “Change of resonance nuclear parameters during scattering by regular systems,” Sov. Phys. JETP 23, 178–184 (1966).

Synth. Met.

M. N. Shkunov, N. C. DeLong, M. E. Raikh, Z. V. Vardeny, A. A. Zakhidov, and R. H. Baughman, “Photonic versus random lasing in opal single crystals,” Synth. Met. 116, 485–491 (2001).
[CrossRef]

Top. Appl. Phys.

A. K. Sarychev and V. M. Shalaev, “Theory of nonlinear optical responses in metal–dielectric composites,” Top. Appl. Phys. 82, 169–184 (2002).
[CrossRef]

Other

C. L. Haynes, A. D. McFarland, L. L. Zhao, R. P. Van Duyne, G. C. Schatz, L. Gunnarsson, J. Prikuli, B. Kasemo, and M. Käll, “Nanoparticle optics: the importance of radiative dipole coupling in two-dimensional nanoparticle arrays,” J. Phys. Chem. B (to be published).

P. Sievert, Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208–3113 (personal communication, 2003).

A. L. Burin, M. A. Ratner, and H. Cao, “Understanding and control of random lasing,” Physica B (to be published).

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Figures (10)

Fig. 1
Fig. 1

Emission of light from the 1-D chain excitations along the cone surface defined by the condition q cos(θ)=k, where q is the momentum of the emitted photon and k is the momentum of the excitation (see text for details). When k>q one gets cos(θ)>1, and light emission becomes impossible because of the light-cone constraint.

Fig. 2
Fig. 2

1-D chain of resonant spherical particles (diameter d) with period a. Arrows, directions of photon emission leading to the radiative decay of optical excitations.

Fig. 3
Fig. 3

Light-cone constraint in A, a linear 1-D chain of scattering particles and B, a 2-D closely packed lattice of scattering particles. The original lattice is shown at the right, and the inverse lattice is shown at the left. The maximum wave vector is π/a for one dimension and 2π/(31/2a) for two dimensions.

Fig. 4
Fig. 4

Relative decay rate for the transverse mode of highest quality for wavelengths a/λ.

Fig. 5
Fig. 5

Relative decay rate for the longitudinal mode of an infinite linear array of highest quality for wavelengths a/λ [γ(1) is the single-particle decay rate].

Fig. 6
Fig. 6

Relative lasing threshold for finite chain and planar layers of resonant dipoles [γ(1) is the single-particle decay rate].

Fig. 7
Fig. 7

Dependence of lasing threshold on length for finite chain and planar layers of resonant dipoles.

Fig. 8
Fig. 8

Dependence of lasing threshold on site disorder for the planar layer (long wavelength, a/λ=0.45).

Fig. 9
Fig. 9

Dependence of lasing threshold on site disorder for the 1-D chain (long wavelength, a/λ=0.4).

Fig. 10
Fig. 10

Dependence of lasing threshold on site disorder for the 1-D chain (short wavelength, a/λ=0.53).

Tables (1)

Tables Icon

Table 1 Summary of Theoretical Predictions of Lasing Threshold Behavior in Various Situations

Equations (32)

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q=ω(k)cω0c.
a<λ2=πcω0.
a<λ3.
1-D1>3.8,2-D1>3.1,
1-D2>7.5,2-D2>5.4,
piα=β χαβ(z)E0iβ+ji,γ Vijβγpjγ,
Vijαβ=exp(iqr)r q2(δαβ-nαnβ)-δαβ-3nαnβr2×(1-iqr),q=zc,n=rr.
χαβ(z)=δαβ2ω0μ2ω02-z2-2izγ,γ=23μ2q3.
-z2+ω02-i 43ω0μ2q3piα
=μ2E0iα+ji,γ Vijαγpjγ.
z2=ω02+i 43ω0μ2q3-2ω0μ2q2af1(k, q)+iqa2f2(k, q)-1a3f3(k, q),
fp(k, q)=n0 exp[i(|n|qa-nka)]np
z2=ω02+i 43ω0μ2q3+2ω0μ2iqa2f2(k, q)-1a3f3(k, q)
f1(k, q)=2 ln|cos(ka)-cos(qa)|-iqa+iπ n=0{θ(-ka+qa-2πn)+θ[ka+qa-2π(n+1)]}.
Im z2=π ω0μ2q2a n=0θ(-ka+qa-2πn)×1+(k+2πn/a)2q2+θ[ka+qa-2π(n+1)]×1+[-k+2π(n+1)/a]2q2
Im z2=π ω0μ2q2a n=0θ(-ka+qa-2πn)×1-(k+2πn/a)2q2+θ[ka+qa-2π(n+1)]×1-[-k+2π(n+1)/a]22q2
Im z=2π nμ2q2a 1+(2n-1)(2n+1)3 πaq2, 2πn<qa<(2n+1)π,
Im z=π μ2q2a×1+2n+2n(n+1)(2n+1)3 πaq2, (2n+1)πn<qa<2πn.
Im z=2π nμ2q2a 1-(2n-1)(2n+1)3 πaq2, (2n-1)π<qa<(2n+1)π.
zfi=-Vm[fi-1(1-δi1)+fi+1(1-δiN)]-iγmfi(δi1+δiN).
z=-2Vm cos(ka),
-2Vm cos(ka)+iγmVm=-cos[ka(N/2-1)]cos(kaN/2).
k1πL-πaL2 1+iγm/Vm1+(γm/Vm)2,
k2πa-πaL-πL2 1+iγm/Vm1+(γm/Vm)2.
Im z2π2VmaL3 iγmVmVm2+γm2.
Im zμ2/L3.
-d2dx2+Uθ(L-x)θ(x)ψ(x)=Eψ.
Im E4/UL3.
(z+ig)2P=A(z)P,
gv ltrL2,
gC/L3,
gminv ltrL2, Altr2.

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