Abstract

We develop a frequency-domain formulation in the form of generalized eigenvalue problems for reciprocal microlasers and nanolasers. While the goal is to explore the resonance properties of dispersive cavities, the starting point of our approach is the mode expansion of arbitrary current sources inside the active regions of lasers. Due to the Lorentz reciprocity, a mode orthogonality relation is present and serves as the basis to distinguish various cavity modes. This scheme can also incorporate the asymmetric Fano lineshape into the emission spectra of cavities. We show how to obtain the important parameters of laser cavities based on this formulation. The proposed approach could be an alternative to other computation schemes such as the finite-difference-time-domain method for reciprocal cavities.

© 2011 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
    [CrossRef]
  29. J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
    [CrossRef]
  30. A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).
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    [CrossRef] [PubMed]
  32. B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
    [CrossRef]
  33. E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
    [CrossRef]
  34. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
    [CrossRef]
  35. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
    [CrossRef]
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    [CrossRef]
  40. N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
    [CrossRef] [PubMed]
  41. Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
    [CrossRef]
  42. H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
    [CrossRef]
  43. G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
    [CrossRef]
  44. A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
    [CrossRef]
  45. G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
    [CrossRef]
  46. S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).
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    [CrossRef] [PubMed]

2011 (2)

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
[CrossRef] [PubMed]

2010 (5)

K. Yu, A. Lakhani, and M. C. Wu, “Subwavelength metal-optic semiconductor nanopatch lasers,” Opt. Express 18, 8790–8799 (2010).
[CrossRef] [PubMed]

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010).
[CrossRef] [PubMed]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

2009 (3)

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

2007 (5)

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

2006 (3)

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[CrossRef]

Y. Huang and S. T. Ho, “Computational model of solid-state, molecular, or atomic media for FDTD simulation based on a multi-level multi-electron system governed by Pauli exclusion and Fermi-Dirac thermalization with application to semiconductor photonics,” Opt. Express 14, 3569–3587 (2006).
[CrossRef] [PubMed]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

2005 (4)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

2004 (5)

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
[CrossRef]

S. H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express 12, 3827–3833 (2004).
[CrossRef] [PubMed]

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
[CrossRef] [PubMed]

2003 (2)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

1998 (2)

B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
[CrossRef] [PubMed]

B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

1986 (1)

H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
[CrossRef]

1982 (1)

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
[CrossRef]

1980 (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Electromagn. Compat. 22, 191–202 (1980).
[CrossRef]

1977 (1)

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1896 (1)

H. A. Lorentz, “The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd. 4, 176–187 (1896).

Amanti, M. I.

G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Aoki, H.

H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
[CrossRef]

Arakawa, Y.

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
[CrossRef]

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

Baba, T.

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 1st ed. (Wiley and Sons, 1989).

Beck, M.

G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Beere, H.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Benson, R. M.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

Benson, T. M.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Bimberg, D.

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
[CrossRef] [PubMed]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Bokor, J.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

Bondarenko, O.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Boriskina, S. V.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

Brodwin, M. E.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech.23, 623–630 (1975).
[CrossRef]

Busch, K.

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
[CrossRef]

Chang, S. H.

Chang, S. W.

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010).
[CrossRef] [PubMed]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Chigrin, D. N.

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
[CrossRef]

Chuang, S. L.

C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
[CrossRef] [PubMed]

S. W. Chang, T. R. Lin, and S. L. Chuang, “Theory of plasmonic Fabry-Perot nanolasers,” Opt. Express 18, 15039–15053 (2010).
[CrossRef] [PubMed]

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[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
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B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

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C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

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G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
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B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

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B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
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N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
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M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

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Hoyler, N.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

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A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

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Iwamoto, S.

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

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S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
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S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
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B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
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B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
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K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
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M. W. Kim and P. C. Ku, “Semiconductor nanoring lasers,” Appl. Phys. Lett. 98, 201105 (2011).

Kumagai, N.

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

Kumar, S.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

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M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
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S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
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N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

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M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
[CrossRef] [PubMed]

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Lee, Y. H.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Liddle, A.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Coninuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

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Lloyd-Hughes, J.

G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

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M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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Lowther, M. A.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
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C. Y. Lu, S. L. Chuang, A. Mutig, and D. Bimberg, “Metal-cavity surface-emitting microlaser with hybrid metal-dbr reflectors,” Opt. Lett. 36, 2447–2449 (2011).
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C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Marell, M.

Mayer, M. A.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
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M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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Nezhad, M. P.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
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K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
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M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
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Nozaki, K.

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

Oei, Y. S.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

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M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Coninuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

Pohl, U. W.

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

Qureshi, N.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

Register, L. F.

B. Klein, L. F. Register, M. Grupen, and K. Hess, “Numerical simulation of vertical cavity surface emitting lasers,” Opt. Express 2, 163–168 (1998).
[CrossRef] [PubMed]

B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

Reno, J. L.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

Rim, S.

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758–2760 (2004).
[CrossRef] [PubMed]

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Ritchie, D.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Sadowski, M. L.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Sakaki, H.

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
[CrossRef]

Sarmiento, T.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Scalari, G.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Schmidt, H.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

Sewell, P.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Shambat, G.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Simic, A.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Sirigu, L.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Skliarov, O. P.

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

Slutsky, B.

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

Smalbrugge, B.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Smirnov, D.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

Smit, M. K.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Smotrova, E. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Sun, M.

Taflove, A.

S. H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express 12, 3827–3833 (2004).
[CrossRef] [PubMed]

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Electromagn. Compat. 22, 191–202 (1980).
[CrossRef]

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech.23, 623–630 (1975).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Turcinková, D.

G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

Turkiewicz, J. P.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

van Otten, F. W. M.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

van Veldhoven, P. J.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Vlasov, A. G.

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

Vuckovic, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Wade, A.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

Walther, C.

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Wang, S.

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

Wiersig, J.

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[CrossRef]

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

Wilcut, E.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

Williams, B. S.

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

Wu, M. C.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, 1996).

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Yu, K.

Zhu, Y.

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y. S. Oei, R. Nötzel, C. Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

Zhukovsky, S. V.

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
[CrossRef]

Appl. Phys. Lett. (8)

S. Kita, S. Hachuda, K. Nozaki, and T. Baba, “Nanoslot laser,” Appl. Phys. Lett. 97, 161108 (2010).
[CrossRef]

C. Y. Lu, S. W. Chang, S. L. Chuang, T. D. Germann, and D. Bimberg, “Metal-cavity surface-emitting microlaser at room temperature,” Appl. Phys. Lett. 96, 251101 (2010).
[CrossRef]

M. Nomura, Y. Ota, N. Kumagai, S. Iwamoto, and Y. Arakawa, “Zero-cell photonic crystal nanocavity laser with quantum dot gain,” Appl. Phys. Lett. 97, 191108 (2010).

M. W. Kim and P. C. Ku, “Semiconductor nanoring lasers,” Appl. Phys. Lett. 98, 201105 (2011).

B. Klein, L. F. Register, K. Hess, D. G. Deppe, and Q. Deng, “Self-consistent Green’s function approach to the analysis of dielectrically apertured vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 73, 3324–3326 (1998).
[CrossRef]

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85, 431–433 (2004).
[CrossRef]

Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982).
[CrossRef]

H. Aoki, “Novel Landau level laser in the quantum Hall regime,” Appl. Phys. Lett. 48, 559–560 (1986).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and nonuniform gain: quasi-3-D modeling with accurate 2-D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady state electromagnetic penetration problems,” IEEE Trans. Electromagn. Compat. 22, 191–202 (1980).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

Nano Lett. (1)

N. Qureshi, S. Wang, M. A. Lowther, A. R. Hawkins, S. Kwon, A. Liddle, J. Bokor, and H. Schmidt, “Cavity-enhanced magnetooptical observation of magnetization reversal in individual single-domain nanomagnets,” Nano Lett. 5, 1413–1417 (2005).
[CrossRef] [PubMed]

Nat. Photonics (4)

A. Wade, G. Fedorov, D. Smirnov, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Magnetic-field-assisted terahertz quantum cascade laser operating up to 225 K,” Nat. Photonics 3, 41–45 (2009).
[CrossRef]

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, and J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5, 297–300 (2011).
[CrossRef]

Opt. Express (6)

Opt. Lett. (2)

Opt. Quantum Electron. (2)

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, R. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Mathematical study of the two-dimensional lasing problem for the whispering-gallery modes in a circular dielectric microcavity,” Opt. Quantum Electron. 36, 213–221 (2004).
[CrossRef]

Phys. Rev. A (5)

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

S. V. Zhukovsky, D. N. Chigrin, and J. Kroha, “Bistability and mode interaction in microlasers,” Phys. Rev. A 79, 033803 (2009).
[CrossRef]

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005).
[CrossRef]

T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
[CrossRef]

Phys. Rev. B (1)

G. Scalari, C. Walther, L. Sirigu, M. L. Sadowski, H. Beere, D. Ritchie, N. Hoyler, M. Giovannini, and J. Faist, “Strong confinement in terahertz intersubband lasers by intense magnetic fields,” Phys. Rev. B 76, 115305 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. 99, 073902 (2007).
[CrossRef] [PubMed]

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901 (2006).
[CrossRef]

Phys. Stat. Solidi B (1)

S. V. Zhukovsky and D. N. Chigrin, “Numerical modelling of lasing in microstructures,” Phys. Stat. Solidi B 244, 3515–3527 (2007).
[CrossRef]

Radio. Eng. Electron. Phys. (1)

A. G. Vlasov and O. P. Skliarov, “An electromagnetic boundary value problem for a radiating dielectric cylinder with reflectors at both ends,” Radio. Eng. Electron. Phys. 22, 17–23 (1977).

Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd. (1)

H. A. Lorentz, “The theorem of Poynting concerning the energy in the electromagnetic field and two general propositions concerning the propagation of light,” Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd. 4, 176–187 (1896).

Other (10)

J. Jin, The Finite Element Method in Electromagnetics (Wiley and Sons, 2002).

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photonics Rev., (2011).
[CrossRef]

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Coninuous Media, 2nd ed. (Butterworth-Heinemann, 1984).

C. A. Balanis, Advanced Engineering Electromagnetics, 1st ed. (Wiley and Sons, 1989).

G. Scalari, D. Tur?inková, J. Lloyd-Hughes, M. I. Amanti, M. Fischer, M. Beck, and J. Faist, “Magnetically assisted quantum cascade laser emitting from 740 GHz to 1.4 THz,” Appl. Phys. Lett.97, 081110 (2010).
[CrossRef]

S. W. Chang, C. Y. Lu, S. L. Chuang, T. D. Germann, U. W. Pohl, and D. Bimberg, “Theory of metal-cavity surface-emitting microlasers and comparison with experiment,” IEEE. J. Sel. Top. Quantum. Electron. (to be published).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, 1996).

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, 1st ed. (Wiley and Sons, 1995).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech.23, 623–630 (1975).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The schematic diagram of the laser cavity. The active region is denoted as Ωa, and Ω is an arbitrary region which contains Ωa. Sa and S are the surfaces corresponding to Ωa and Ω, respectively.

Fig. 2
Fig. 2

The effect of Δɛr,n(ω) on the emission spectrum Pn(ω′). The real part Re[Δɛr,n(ω)] shifts the resonance frequency ωn to ω, while the imaginary part Im[Δɛr,n(ω)] compensates the loss and converts Pn(ω′) into a delta function centered at ω.

Fig. 3
Fig. 3

(a) The locus of η(ω) = ωΔɛr,l(ω) parameterized by ω on the complex η plane. At resonance frequency ωl, η(ωl) is closest to the origin of the complex plane. (b) The comparison between the Lorentzian and Fano lineshapes. The Fano lineshape is asymmetric with respect to ωl, and the peak is shifted from that of the Lorentzian.

Fig. 4
Fig. 4

(a) The layout of the 1D FP cavity. The active region has the same permittivity as that (ɛc) of the whole cavity. (b) The resonance lineshapes of various cavity modes with even or odd electric fields when ɛc = 12.25. The lineshape of each mode closely resembles a Lorentzian near its resonance frequency.

Fig. 5
Fig. 5

The spectral characteristics of the mode at h̄ω3 = 1.42 eV. (a) The square magnitude of the mode profile. The pattern in the active region shows the amplification due to Im[Δɛr,3(ω3)]. (b) The locus of h̄η(ω) = h̄ωΔɛr,3(ω) on the complex h̄η plane. The real part Re[Δɛr,3(ω3)] = 0.00256 indicates a slight blueshift from ω3.

Fig. 6
Fig. 6

(a) The white-noise lineshapes of two even modes with ɛc = 3.0625 and ɛc = 12.25, respectively. The mode corresponding to ɛc = 3.0625 has the more asymmetric lineshape. (b) The mode profiles (square magnitudes) of the two even modes with ɛc = 3.0625 and ɛc = 12.25, respectively. The more significant field amplification in the active region with ɛc = 3.0625 indicates the more leaky cavity in this case.

Fig. 7
Fig. 7

The generic computation domain for the numerical implementation of the formulation. PMLs are inserted in the inner sides of the computation domain while the outer boundaries of the computation domain are set to PECs or PMCs.

Tables (1)

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Table 1 The Comparison of h̄ωn, Qn, and gth,n between the Theoretical and FP Estimations

Equations (70)

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× × E cav ( r ) = ( ω cav c ) 2 ɛ ¯ ¯ r ( r , ω cav ) E cav ( r ) ,
[ ɛ r ( r , ω ) ] α α = [ ɛ r ( r , ω ) ] α α , α , α = x , y , z r Ω ,
Ω a d r [ E 1 ( r ) J s , 2 ( r ) E 2 ( r ) J s , 1 ( r ) ] = S d a [ E 1 ( r ) × H 2 ( r ) E 2 ( r ) × H 1 ( r ) ] ,
Ω a d r [ E 1 ( r ) J s , 2 ( r ) E 2 ( r ) J s , 1 ( r ) ] = 0 .
× E ( r ) = i ω μ 0 H ( r ) ,
× H ( r ) = i ω ɛ 0 ɛ ¯ ¯ r ( r , ω ) E ( r ) + J s ( r ) ,
× × E ( r ) ( ω c ) 2 ɛ ¯ ¯ r ( r , ω ) E ( r ) = i ω μ 0 J s ( r ) .
J s ( r ) = { A vector field , r Ω a , 0 otherwise .
j s , n ( r , ω ) = i ω ɛ 0 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) ,
U ( r ) = { 1 , r Ω a , 0 , otherwise ,
× × f n ( r , ω ) ( ω c ) 2 ɛ ¯ ¯ r ( r , ω ) f n ( r , ω ) = i ω μ 0 j s , n ( r , ω ) = ( ω c ) 2 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) ,
g n ( r , ω ) = 1 i ω μ 0 × f n ( r , ω ) .
× × f n ( r , ω ) ( ω c ) 2 [ ɛ ¯ ¯ r ( r , ω ) + Δ ɛ r , n ( ω ) U ( r ) I ¯ ¯ ] f n ( r , ω ) = 0 ,
[ ɛ r , a ( ω ) + Δ ɛ r , n ( ω ) ] f n ( r , ω ) = 0 , r Ω a .
j s , n ( r , ω ) = i ω ɛ 0 Δ ɛ r , n ( ω ) f n ( r , ω ) = 0 , r Ω a .
J s ( r ) = n c n j s , n ( r , ω ) + m d m i s , m ( r , ω ) ,
E ( r ) = n c n f n ( r , ω ) + m d m u m ( r , ω ) ,
H ( r ) = n c n g n ( r , ω ) + m d m w m ( r , ω ) .
[ J s , 1 ( r ) , E 1 ( r ) ] = [ j s , n ( r , ω ) , f n ( r , ω ) ] = [ i ω ɛ 0 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) , f n ( r , ω ) ] ,
[ J s , 2 ( r ) , E 2 ( r ) ] = [ j s , n ( r , ω ) , f n ( r , ω ) ] = [ i ω ɛ 0 Δ ɛ r , n ( ω ) U ( r ) f n ( r , ω ) , f n ( r , ω ) ] ,
i ω ɛ 0 [ Δ ɛ r , n ( ω ) Δ ɛ r , n ( ω ) ] Ω a d r f n ( r , ω ) f n ( r , ω ) = 0 .
Ω a d r f n ( r , ω ) f n ( r , ω ) = δ n n Λ n ( ω ) ,
Ω a d r j s , n ( r , ω ) j s , n ( r , ω ) = δ n n Θ n ( ω ) ,
Θ n ( ω ) = [ ω ɛ 0 Δ ɛ r , n ( ω ) ] 2 Λ n ( ω ) .
Ω a d r i s , m ( r , ω ) i s , m ( r , ω ) = δ m m Ξ m ( ω ) ,
Ω a d r i s , m ( r , ω ) j s , n ( r , ω ) = Ω a d r j s , n ( r , ω ) i s , m ( r , ω ) = 0 .
c n = 1 Θ n ( ω ) Ω a d r j s , n ( r , ω ) J s ( r ) .
d m = 1 Ξ m ( ω ) Ω a d r i s , m ( r , ω ) J s ( r ) .
E ( r ) = Ω a d r G ¯ ¯ ee ( r , r , ω ) [ i ω μ 0 J s ( r ) ] ,
G ¯ ¯ ee ( r , r , ω ) = n f n ( r , ω ) j s , n T ( r , ω ) i ω μ 0 Θ n ( ω ) + n u m ( r , ω ) i s , m T ( r , ω ) i ω μ 0 Ξ m ( ω ) = n f n ( r , ω ) f n T ( r , ω ) U ( r ) ( ω c ) 2 Δ ɛ r , n ( ω ) Λ n ( ω ) + m u m ( r , ω ) i s , m T ( r , ω ) i ω μ 0 Ξ m ( ω ) .
Ω a d r f l ( r , ω ) f l ( r , ω + Δ ω ) δ l l .
J s ( r ) = a ( ω ) j s , l ( r , ω ) = a ( ω ) ( i ω ) ɛ 0 Δ ɛ r , l ( ω ) U ( r ) f l ( r , ω ) ,
E ( r ) = a ( ω ) f l ( r , ω ) ,
Ω a d r | J s ( r ) | 2 = | a ( ω ) | 2 ( ɛ 0 ω ) 2 | Δ ɛ r , l ( ω ) | 2 Ω a d r | f l ( r , ω ) | 2 𝒥 2 V a ,
| a ( ω ) | 2 = 𝒥 2 V a ( ɛ 0 ω ) 2 | Δ ɛ r , l ( ω ) | 2 Ω a d r | f l ( r , ω ) | 2 ,
P l ( ω ) = Ω a d r 1 2 Re [ J s * ( r ) E ( r ) ] = | a ( ω ) | 2 ɛ 0 ω Im [ Δ ɛ r , l ( ω ) ] 2 Ω a d r | f l ( r , ω ) | 2 = 𝒥 2 V a 2 ɛ 0 Im [ 1 ω Δ ɛ r , l ( ω ) ] .
δ η η ( ω l ) = i δ ω ( Δ ω l / 2 ) ,
Δ ω l = 2 i η ( ω l ) η ( ω l ) = 2 i [ ω l Δ ɛ r , l ( ω l ) ] { [ ω Δ ɛ r , l ( ω ) ] ω | ω = ω l } 1 .
η ( ω ) = ω Δ ɛ r , l ( ω ) ω l Δ ɛ r , l ( ω l ) [ 1 i ( ω ω l ) ( Δ ω l / 2 ) ] ,
P l ( ω ) 𝒥 2 V a 2 ɛ 0 ( Δ ω l / 2 ) | ω l Δ ɛ r , l ( ω l ) | { Im [ ω l Δ ɛ r , l ( ω l ) ] | ω l Δ ɛ r , l ( ω l ) | ( Δ ω l / 2 ) ( ω ω l ) 2 + ( Δ ω l / 2 ) 2 } + Re [ ω l Δ ɛ r , l ( ω l ) ] | ω l Δ ɛ r , l ( ω l ) | ( ω ω l ) ( ω ω l ) 2 + ( Δ ω l / 2 ) 2 } .
Q l = ω l Δ ω l = i 2 Δ ɛ r , l ( ω l ) [ ω Δ ɛ r , l ( ω ) ] ω | ω = ω l .
g th , l = 2 ( ω l c ) Im [ ɛ a ( ω l ) + Δ ɛ r , l ( ω l ) ɛ a ( ω l ) ] ( ω l c ) Im [ Δ ɛ r , l ( ω l ) ] Re [ ɛ a ( ω l ) ] .
J sp , α * ( r , ω ) J sp , α ( r , ω ) = δ ( r r ) c , v D c v ( r , ω ) j sp , c v , α * ( ω ) j sp , c v , α ( ω ) ¯ ,
j sp , c v ( ω ) = 2 i ω q d c v ( ω ) ,
r sp ( ω ) = 1 h ¯ ω Ω a d r 1 2 Re [ J sp * ( r , ω ) E ( r , ω ) ] = 1 h ¯ ω Ω a Ω a d r d r 1 2 Re [ J sp * ( r , ω ) G ¯ ¯ ee ( r , r , ω ) i ω μ 0 J sp ( r , ω ) ] = μ 0 2 h ¯ ( α , α ) , ( c , v ) Ω a d r D c v ( r , ω ) Im { [ G ee ( r , r , ω ) ] α α j sp , c v , α * ( ω ) j sp , c v , α ( ω ) ¯ } ,
r sp ( ω ) = n r sp , n ( ω ) + m r ˜ sp , m ( ω ) ,
r sp , n ( ω ) = 1 2 ɛ 0 ( c , v ) Ω a d r D c v ( r , ω ) h ¯ ω Im { [ j sp , c v * ( ω ) f n ( r , ω ) ] [ j sp , c v ( ω ) f n ( r , ω ) ] ¯ [ ω Δ ɛ r , n ( ω ) ] Λ n ( ω ) } ,
r ˜ sp , m ( ω ) = 1 2 ( c , v ) Ω a d r D c v ( r , ω ) h ¯ ω Im { [ j sp , c v * ( ω ) u m ( r , ω ) ] [ j sp , c v ( ω ) i s , m ( r , ω ) ] ¯ i Ξ m ( ω ) } ,
R sp = 0 d ( h ¯ ω ) r sp ( ω ) = n R sp , n + m R ˜ sp , m ,
R sp , n = 0 d ( h ¯ ω ) r sp , n ( ω ) ,
R ˜ sp , m = 0 d ( h ¯ ω ) r ˜ sp , m ( ω ) ,
β l = R sp , l R sp .
D c v ( r , ω ) = δ ( r r s ) δ ( h ¯ ω h ¯ ω c v ) , r s Ω a ,
R sp , l = 2 ω c v ɛ 0 h ¯ Im { [ q d c v * f l ( r s , ω c v ) ] [ q d c v f l ( r s , ω c v ) ] [ ω c v Δ ɛ r , l ( ω c v ) ] Λ l ( ω c v ) } .
W sp , l = 2 π h ¯ | q d c v E ^ l ( r s ) 2 | 2 ( Δ ω l / 2 ) π h ¯ 1 ( ω c v ω l ) 2 + ( Δ ω l / 2 ) 2 ,
E ^ l ( r s ) = 2 h ¯ ω l ɛ 0 f l ( r s , ω l ) Im [ Δ ɛ r , l ( ω l ) ] Q l Λ l ( ω l ) .
1 = ± e i k a , n L a [ r a , p + r p , fs e 2 i k p L p 1 + r a , p r p , fs e 2 i k p L p ] ,
k a , n = ( ω c ) ɛ c + Δ ɛ r , n ( ω ) , k p = ( ω c ) ɛ c ,
r a , p = ɛ c + Δ ɛ r , n ( ω ) ɛ c ɛ c + Δ ɛ r , n ( ω ) + ɛ c , r p , fs = ɛ c 1 ɛ c + 1 ,
ω n m n π c ɛ c L c ,
1 Q n v g ω n L c ln ( 1 | r p , fs | 2 ) = 2 m n π ln ( ɛ c + 1 ɛ c 1 ) ,
g th , n 1 Γ z , n L c ln ( 1 | r p , fs | 2 ) = 2 [ 1 ± sinc ( m n π L a / L c ) ] L a ln ( ɛ c + 1 ɛ c 1 ) ,
i s , m ( r , ω ) i ω ɛ 0 Ω a d r P ¯ ¯ ( c ) ( r , r , ω ) Φ m ( r , ω ) ,
P ¯ ¯ ( c ) ( r , r , ω ) = δ ( r r ) U ( r ) I ¯ ¯ n j s , n ( r , ω ) j s , n T ( r , ω ) Θ n ( ω ) .
i s , m ( r , ω ) = i ω ρ m ( r , ω ) .
ρ m ( r , ω ) = ɛ 0 ( ω c ) 2 Δ κ r , m ( ω ) U ( r ) Φ m ( r , ω ) ,
[ Ω a d r P ¯ ¯ ( c ) ( r , r , ω ) Φ m ( r , ω ) ] = ( ω c ) 2 Δ κ r , m ( ω ) U ( r ) Φ m ( r , ω ) ,
× × u m ( r , ω ) ( ω c ) 2 ɛ ¯ ¯ r ( r , ω ) u m ( r , ω ) = i ω μ 0 i s , m ( r , ω ) ,
w m ( r , ω ) = 1 i ω μ 0 × u m ( r , ω ) .
[ U ( r ) Φ m ( r , ω ) ] = ( ω c ) 2 Δ κ r , m ( ω ) U ( r ) Φ m ( r , ω ) , Φ m ( r , ω ) = 0 , r S a ,

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