Abstract

A laser threshold condition derived from the instantaneous form of Maxwell’s equations is presented. This derivation incorporates the passive quality factor of the resonator and is particularly amenable to semiconductor microcavity lasers. The optical confinement factor is derived and compared to previous reports. The threshold condition derived here is compared to the results of active cavity finite-difference time-domain calculations, and excellent agreement is found.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  50. M.-K. Seo, G. H. Song, H. In-Kag, and Y.-H. Lee, “Nonlinear dispersive three-dimensional finite-difference time-domain analysis for photonic-crystal lasers,” Opt. Express 13, 9645–9651 (2005).
    [CrossRef] [PubMed]
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2010 (1)

2009 (3)

A. Mock, L. Lu, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “Modal analysis of photonic crystal double-heterostructure laser cavities,” IEEE J. Sel. Top. Quantum Electron. 15, 892–900 (2009).
[CrossRef]

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 μW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

2008 (4)

2007 (8)

B.-S. Song, T. Asano, and S. Noda, “Heterostructures in two-dimensional photonic-crystal slabs and their application to nanocavities,” J. Phys. D 40, 2629–2634 (2007).
[CrossRef]

W. C. Stumpf, M. Fujita, M. Yamaguchi, T. Asano, and S. Noda, “Light-emission properties of quantum dots embedded in a photonic double-heterostructure nanocavity,” Appl. Phys. Lett. 90, 231101 (2007).
[CrossRef]

M.-K. Seo, K.-Y. Jeong, J.-K. Yang, Y.-H. Lee, H.-G. Park, and S.-B. Kim, “Low threshold current single-cell hexapole mode photonic crystal laser,” Appl. Phys. Lett. 90, 171122 (2007).
[CrossRef]

T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G. Deppe, “Edge-emitting photonic crystal double-heterostructure nanocavity lasers with InAs quantum dot active material,” Opt. Lett. 32, 1153–1155 (2007).
[CrossRef] [PubMed]

T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G. Deppe, “Lasing characteristics of InAs quantum dot microcavity lasers as a function of temperature and wavelength,” Opt. Express 15, 7281–7289 (2007).
[CrossRef] [PubMed]

K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15, 7506–7514 (2007).
[CrossRef] [PubMed]

S. Shi and D. W. Prather, “Lasing dynamics of a silicon photonic crystal micro-cavity,” Opt. Express 15, 10294–10302 (2007).
[CrossRef] [PubMed]

W. H. P. Pernice, F. P. Payne, and D. F. G. Gallagher, “A finite-difference time-domain method for the simulation of gain materials with carrier diffusion in photonic crystals,” J. Lightwave Technol. 25, 2306–2314 (2007).
[CrossRef]

2006 (3)

M. H. Shih, W. Kuang, T. Yang, M. Bagheri, Z.-J. Wei, S.-J. Choi, L. Lu, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the optical loss of sapphire-bonded photonic crystal laser cavities,” IEEE Photon. Technol. Lett. 18, 535–537 (2006).
[CrossRef]

Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Investigation of point-defect cavity formed in two-dimensional photonic crystal slab with one-sided dielectric cladding,” Appl. Phys. Lett. 88, 011112 (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]

2005 (6)

W. Kuang, J. R. Cao, T. Yang, S.-J. Choi, P.-T. Lee, J. D. O’Brien, and P. D. Dapkus, “Classification of modes in suspended-membrane, 19-missing-hole photonic-crystal microcavities,” J. Opt. Soc. Am. B 22, 1092–1099 (2005).
[CrossRef]

M.-K. Seo, G. H. Song, H. In-Kag, and Y.-H. Lee, “Nonlinear dispersive three-dimensional finite-difference time-domain analysis for photonic-crystal lasers,” Opt. Express 13, 9645–9651 (2005).
[CrossRef] [PubMed]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

T. Yang, S. Lipson, J. D. O’Brien, and D. G. Deppe, “InAs quantum dot photonic crystal lasers and their temperature dependence,” IEEE Photon. Technol. Lett. 17, 2244–2246 (2005).
[CrossRef]

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244–1252 (2005).
[CrossRef]

K. Nozaki and T. Baba, “Carrier and photon analyses of photonic microlasers by two-dimensional rate equations,” IEEE J. Sel. Areas Commun. 23, 1411–1417 (2005).
[CrossRef]

2004 (5)

M. Kretschmann and A. Maradudin, “Lasing action in waveguide systems and the influence of rough walls,” J. Opt. Soc. Am. B 21, 150–158 (2004).
[CrossRef]

M. Bahl, H. Rao, N. C. Panoiu, and R. M. Osgood, “Simulation of mode-locked surface-emitting lasers through a finite-difference time-domain algorithm,” Opt. Lett. 29, 1689–1691 (2004).
[CrossRef] [PubMed]

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]

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1052–1062 (2004).
[CrossRef]

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

2002 (2)

G. H. Song, S. Kim, and K. Hwang, “FDTD simulation of photonic-crystal lasers and their relaxation oscillation,” Journal of the Optical Society of Korea 6, 87–95 (2002). ISSN 1276-4776.
[CrossRef]

P. Royo, R. Koda, and L. A. Coldren, “Vertical cavity semiconductor optical amplifiers: Comparison of Fabry–Pérot and rate equation approaches,” IEEE J. Quantum Electron. 38, 279–284 (2002).
[CrossRef]

1999 (2)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

J. Vŭcković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “Finite-difference time-domain calculation of the spontaneous emission coupling factor in optical micro-cavities,” IEEE J. Quantum Electron. 35, 1168–1175 (1999).
[CrossRef]

1998 (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

1997 (1)

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

1996 (2)

Y.-Z. Huang, Z. Pan, and R.-H. Wu, “Analysis of the optical confinement factor in semiconductor lasers,” J. Appl. Phys. 79, 3827–3830 (1996).
[CrossRef]

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

1994 (1)

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50, 1675–1680 (1994).
[CrossRef] [PubMed]

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk laser,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

1991 (1)

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[CrossRef]

1965 (1)

W. W. Anderson, “Mode confinement and gain in junction lasers,” IEEE J. Quantum Electron. QE-1, 228–236 (1965).
[CrossRef]

1955 (1)

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser—new type of microwave amplifier, frequency standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[CrossRef]

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

Anderson, W. W.

W. W. Anderson, “Mode confinement and gain in junction lasers,” IEEE J. Quantum Electron. QE-1, 228–236 (1965).
[CrossRef]

Arnold, J. M.

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1052–1062 (2004).
[CrossRef]

Asano, T.

B.-S. Song, T. Asano, and S. Noda, “Heterostructures in two-dimensional photonic-crystal slabs and their application to nanocavities,” J. Phys. D 40, 2629–2634 (2007).
[CrossRef]

W. C. Stumpf, M. Fujita, M. Yamaguchi, T. Asano, and S. Noda, “Light-emission properties of quantum dots embedded in a photonic double-heterostructure nanocavity,” Appl. Phys. Lett. 90, 231101 (2007).
[CrossRef]

Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Investigation of point-defect cavity formed in two-dimensional photonic crystal slab with one-sided dielectric cladding,” Appl. Phys. Lett. 88, 011112 (2006).
[CrossRef]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

Baba, T.

K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15, 7506–7514 (2007).
[CrossRef] [PubMed]

K. Nozaki and T. Baba, “Carrier and photon analyses of photonic microlasers by two-dimensional rate equations,” IEEE J. Sel. Areas Commun. 23, 1411–1417 (2005).
[CrossRef]

Baek, J. -H.

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Bagheri, M.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 μW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[CrossRef]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[CrossRef] [PubMed]

M. H. Shih, W. Kuang, T. Yang, M. Bagheri, Z.-J. Wei, S.-J. Choi, L. Lu, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the optical loss of sapphire-bonded photonic crystal laser cavities,” IEEE Photon. Technol. Lett. 18, 535–537 (2006).
[CrossRef]

Bahl, M.

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244–1252 (2005).
[CrossRef]

M. Bahl, H. Rao, N. C. Panoiu, and R. M. Osgood, “Simulation of mode-locked surface-emitting lasers through a finite-difference time-domain algorithm,” Opt. Lett. 29, 1689–1691 (2004).
[CrossRef] [PubMed]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Björk, G.

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50, 1675–1680 (1994).
[CrossRef] [PubMed]

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[CrossRef]

Blok, H.

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

Cao, J. R.

Chang, S. -H.

Cheng, D. K.

D. K. Cheng, Field and Wave Electromagnetics (Wesley, 1992).

Choi, S. -J.

M. H. Shih, W. Kuang, T. Yang, M. Bagheri, Z.-J. Wei, S.-J. Choi, L. Lu, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the optical loss of sapphire-bonded photonic crystal laser cavities,” IEEE Photon. Technol. Lett. 18, 535–537 (2006).
[CrossRef]

W. Kuang, J. R. Cao, T. Yang, S.-J. Choi, P.-T. Lee, J. D. O’Brien, and P. D. Dapkus, “Classification of modes in suspended-membrane, 19-missing-hole photonic-crystal microcavities,” J. Opt. Soc. Am. B 22, 1092–1099 (2005).
[CrossRef]

Coldren, L. A.

P. Royo, R. Koda, and L. A. Coldren, “Vertical cavity semiconductor optical amplifiers: Comparison of Fabry–Pérot and rate equation approaches,” IEEE J. Quantum Electron. 38, 279–284 (2002).
[CrossRef]

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

Corzine, S. W.

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

Dapkus, P. D.

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A. Mock and J. D. O’Brien, “Strategies for reducing the out-of-plane radiation in photonic crystal heterostructure microcavities for continuous wave laser applications,” J. Lightwave Technol. 28, 1042–1050 (2010).
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L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 μW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
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A. Mock, L. Lu, and J. D. O’Brien, “Spectral properties of photonic crystal double heterostructure resonant cavities,” Opt. Express 16, 9391–9397 (2008).
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A. Mock and J. D. O’Brien, “Direct extraction of large quality factors and resonant frequencies from Padé interpolated resonance spectra,” Opt. Quantum Electron. 40, 1187–1192 (2008).
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T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G. Deppe, “Edge-emitting photonic crystal double-heterostructure nanocavity lasers with InAs quantum dot active material,” Opt. Lett. 32, 1153–1155 (2007).
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M.-K. Seo, K.-Y. Jeong, J.-K. Yang, Y.-H. Lee, H.-G. Park, and S.-B. Kim, “Low threshold current single-cell hexapole mode photonic crystal laser,” Appl. Phys. Lett. 90, 171122 (2007).
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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
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B.-S. Song, T. Asano, and S. Noda, “Heterostructures in two-dimensional photonic-crystal slabs and their application to nanocavities,” J. Phys. D 40, 2629–2634 (2007).
[CrossRef]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

Song, G. H.

M.-K. Seo, G. H. Song, H. In-Kag, and Y.-H. Lee, “Nonlinear dispersive three-dimensional finite-difference time-domain analysis for photonic-crystal lasers,” Opt. Express 13, 9645–9651 (2005).
[CrossRef] [PubMed]

G. H. Song, S. Kim, and K. Hwang, “FDTD simulation of photonic-crystal lasers and their relaxation oscillation,” Journal of the Optical Society of Korea 6, 87–95 (2002). ISSN 1276-4776.
[CrossRef]

Stapleton, A.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 μW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[CrossRef]

Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

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W. C. Stumpf, M. Fujita, M. Yamaguchi, T. Asano, and S. Noda, “Light-emission properties of quantum dots embedded in a photonic double-heterostructure nanocavity,” Appl. Phys. Lett. 90, 231101 (2007).
[CrossRef]

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]

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

Tanaka, Y.

Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Investigation of point-defect cavity formed in two-dimensional photonic crystal slab with one-sided dielectric cladding,” Appl. Phys. Lett. 88, 011112 (2006).
[CrossRef]

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J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser—new type of microwave amplifier, frequency standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[CrossRef]

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T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

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J. Vŭcković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “Finite-difference time-domain calculation of the spontaneous emission coupling factor in optical micro-cavities,” IEEE J. Quantum Electron. 35, 1168–1175 (1999).
[CrossRef]

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M. H. Shih, W. Kuang, T. Yang, M. Bagheri, Z.-J. Wei, S.-J. Choi, L. Lu, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the optical loss of sapphire-bonded photonic crystal laser cavities,” IEEE Photon. Technol. Lett. 18, 535–537 (2006).
[CrossRef]

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Y.-Z. Huang, Z. Pan, and R.-H. Wu, “Analysis of the optical confinement factor in semiconductor lasers,” J. Appl. Phys. 79, 3827–3830 (1996).
[CrossRef]

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J. Vŭcković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “Finite-difference time-domain calculation of the spontaneous emission coupling factor in optical micro-cavities,” IEEE J. Quantum Electron. 35, 1168–1175 (1999).
[CrossRef]

Yamaguchi, M.

W. C. Stumpf, M. Fujita, M. Yamaguchi, T. Asano, and S. Noda, “Light-emission properties of quantum dots embedded in a photonic double-heterostructure nanocavity,” Appl. Phys. Lett. 90, 231101 (2007).
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H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Yang, J. -K.

M.-K. Seo, K.-Y. Jeong, J.-K. Yang, Y.-H. Lee, H.-G. Park, and S.-B. Kim, “Low threshold current single-cell hexapole mode photonic crystal laser,” Appl. Phys. Lett. 90, 171122 (2007).
[CrossRef]

Yang, T.

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 μW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[CrossRef]

T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G. Deppe, “Edge-emitting photonic crystal double-heterostructure nanocavity lasers with InAs quantum dot active material,” Opt. Lett. 32, 1153–1155 (2007).
[CrossRef] [PubMed]

T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G. Deppe, “Lasing characteristics of InAs quantum dot microcavity lasers as a function of temperature and wavelength,” Opt. Express 15, 7281–7289 (2007).
[CrossRef] [PubMed]

M. H. Shih, W. Kuang, T. Yang, M. Bagheri, Z.-J. Wei, S.-J. Choi, L. Lu, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the optical loss of sapphire-bonded photonic crystal laser cavities,” IEEE Photon. Technol. Lett. 18, 535–537 (2006).
[CrossRef]

T. Yang, S. Lipson, J. D. O’Brien, and D. G. Deppe, “InAs quantum dot photonic crystal lasers and their temperature dependence,” IEEE Photon. Technol. Lett. 17, 2244–2246 (2005).
[CrossRef]

W. Kuang, J. R. Cao, T. Yang, S.-J. Choi, P.-T. Lee, J. D. O’Brien, and P. D. Dapkus, “Classification of modes in suspended-membrane, 19-missing-hole photonic-crystal microcavities,” J. Opt. Soc. Am. B 22, 1092–1099 (2005).
[CrossRef]

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J. Vŭcković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “Finite-difference time-domain calculation of the spontaneous emission coupling factor in optical micro-cavities,” IEEE J. Quantum Electron. 35, 1168–1175 (1999).
[CrossRef]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
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A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

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J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser—new type of microwave amplifier, frequency standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[CrossRef]

Zhu, G.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

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G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1052–1062 (2004).
[CrossRef]

Appl. Phys. Lett. (5)

L. Lu, A. Mock, T. Yang, M. H. Shih, E. H. Hwang, M. Bagheri, A. Stapleton, S. Farrell, J. D. O’Brien, and P. D. Dapkus, “120 μW peak output power from edge-emitting photonic crystal double heterostructure nanocavity lasers,” Appl. Phys. Lett. 94, 111101 (2009).
[CrossRef]

Y. Tanaka, T. Asano, R. Hatsuta, and S. Noda, “Investigation of point-defect cavity formed in two-dimensional photonic crystal slab with one-sided dielectric cladding,” Appl. Phys. Lett. 88, 011112 (2006).
[CrossRef]

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[CrossRef]

W. C. Stumpf, M. Fujita, M. Yamaguchi, T. Asano, and S. Noda, “Light-emission properties of quantum dots embedded in a photonic double-heterostructure nanocavity,” Appl. Phys. Lett. 90, 231101 (2007).
[CrossRef]

M.-K. Seo, K.-Y. Jeong, J.-K. Yang, Y.-H. Lee, H.-G. Park, and S.-B. Kim, “Low threshold current single-cell hexapole mode photonic crystal laser,” Appl. Phys. Lett. 90, 171122 (2007).
[CrossRef]

IEEE J. Quantum Electron. (6)

M. Bahl, N. C. Panoiu, and R. M. Osgood, “Modeling ultrashort field dynamics in surface emitting lasers by using finite-difference time-domain method,” IEEE J. Quantum Electron. 41, 1244–1252 (2005).
[CrossRef]

J. Vŭcković, O. Painter, Y. Xu, A. Yariv, and A. Scherer, “Finite-difference time-domain calculation of the spontaneous emission coupling factor in optical micro-cavities,” IEEE J. Quantum Electron. 35, 1168–1175 (1999).
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W. W. Anderson, “Mode confinement and gain in junction lasers,” IEEE J. Quantum Electron. QE-1, 228–236 (1965).
[CrossRef]

T. D. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

P. Royo, R. Koda, and L. A. Coldren, “Vertical cavity semiconductor optical amplifiers: Comparison of Fabry–Pérot and rate equation approaches,” IEEE J. Quantum Electron. 38, 279–284 (2002).
[CrossRef]

G. Björk and Y. Yamamoto, “Analysis of semiconductor microcavity lasers using rate equations,” IEEE J. Quantum Electron. 27, 2386–2396 (1991).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

K. Nozaki and T. Baba, “Carrier and photon analyses of photonic microlasers by two-dimensional rate equations,” IEEE J. Sel. Areas Commun. 23, 1411–1417 (2005).
[CrossRef]

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

A. Mock, L. Lu, E. H. Hwang, J. O’Brien, and P. D. Dapkus, “Modal analysis of photonic crystal double-heterostructure laser cavities,” IEEE J. Sel. Top. Quantum Electron. 15, 892–900 (2009).
[CrossRef]

G. M. Slavcheva, J. M. Arnold, and R. W. Ziolkowski, “FDTD simulation of the nonlinear gain dynamics in active optical waveguides and semiconductor microcavities,” IEEE J. Sel. Top. Quantum Electron. 10, 1052–1062 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

M. H. Shih, W. Kuang, T. Yang, M. Bagheri, Z.-J. Wei, S.-J. Choi, L. Lu, J. D. O’Brien, and P. D. Dapkus, “Experimental characterization of the optical loss of sapphire-bonded photonic crystal laser cavities,” IEEE Photon. Technol. Lett. 18, 535–537 (2006).
[CrossRef]

T. Yang, S. Lipson, J. D. O’Brien, and D. G. Deppe, “InAs quantum dot photonic crystal lasers and their temperature dependence,” IEEE Photon. Technol. Lett. 17, 2244–2246 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

J. Appl. Phys. (1)

Y.-Z. Huang, Z. Pan, and R.-H. Wu, “Analysis of the optical confinement factor in semiconductor lasers,” J. Appl. Phys. 79, 3827–3830 (1996).
[CrossRef]

J. Lightwave Technol. (2)

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

J. Phys. D (1)

B.-S. Song, T. Asano, and S. Noda, “Heterostructures in two-dimensional photonic-crystal slabs and their application to nanocavities,” J. Phys. D 40, 2629–2634 (2007).
[CrossRef]

Journal of the Optical Society of Korea (1)

G. H. Song, S. Kim, and K. Hwang, “FDTD simulation of photonic-crystal lasers and their relaxation oscillation,” Journal of the Optical Society of Korea 6, 87–95 (2002). ISSN 1276-4776.
[CrossRef]

Nature (1)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Na-rimonov, S. Stout, and E. Herz, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Nature Mater. (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[CrossRef]

Opt. Express (9)

M.-K. Seo, G. H. Song, H. In-Kag, and Y.-H. Lee, “Nonlinear dispersive three-dimensional finite-difference time-domain analysis for photonic-crystal lasers,” Opt. Express 13, 9645–9651 (2005).
[CrossRef] [PubMed]

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]

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]

T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G. Deppe, “Lasing characteristics of InAs quantum dot microcavity lasers as a function of temperature and wavelength,” Opt. Express 15, 7281–7289 (2007).
[CrossRef] [PubMed]

K. Nozaki, S. Kita, and T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15, 7506–7514 (2007).
[CrossRef] [PubMed]

S. Shi and D. W. Prather, “Lasing dynamics of a silicon photonic crystal micro-cavity,” Opt. Express 15, 10294–10302 (2007).
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A. Mock, L. Lu, and J. D. O’Brien, “Spectral properties of photonic crystal double heterostructure resonant cavities,” Opt. Express 16, 9391–9397 (2008).
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J. T. Robinson, K. Preston, O. Painter, and M. Lipson, “First-principle derivation of gain in high-index-contrast waveguides,” Opt. Express 16, 16659–16669 (2008).
[CrossRef] [PubMed]

L. Lu, A. Mock, M. Bagheri, E. H. Hwang, J. D. O’Brien, and P. D. Dapkus, “Double-heterostructure photonic crystal lasers with reduced threshold pump power and increased slope efficiency obtained by quantum well intermixing,” Opt. Express 16, 17342–17347 (2008).
[CrossRef] [PubMed]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Mock and J. D. O’Brien, “Direct extraction of large quality factors and resonant frequencies from Padé interpolated resonance spectra,” Opt. Quantum Electron. 40, 1187–1192 (2008).
[CrossRef]

Phys. Rev. (1)

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser—new type of microwave amplifier, frequency standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[CrossRef]

Phys. Rev. A (1)

G. Björk, A. Karlsson, and Y. Yamamoto, “Definition of a laser threshold,” Phys. Rev. A 50, 1675–1680 (1994).
[CrossRef] [PubMed]

Radio Sci. (1)

S. C. Hagness, R. M. Joseph, and A. Taflove, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations,” Radio Sci. 31, 931–941 (1996).
[CrossRef]

Science (2)

H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju, H.-K. Yang, J.-H. Baek, S.-B. Kim, and Y.-H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

Other (10)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

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

A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).

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A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth Heinemann, 2004).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic diagram of a photonic crystal double heterostructure resonant cavity. A photonic well diagram is shown below the cavity to illustrate the confinement mechanism. (b) H z ( x , y ) at the midplane of the semiconductor slab.

Fig. 2
Fig. 2

1 / Q tot versus peak material gain where the spatial gain distribution is varied. The gain is uniform in the plane of the slab but absent in the air holes. The three curves correspond to different fractions of the vertical slab providing gain.

Fig. 3
Fig. 3

Illustration of field evolution versus time for (a) 1 / Q tot > 0 , (b) 1 / Q tot = 0 , and (c) 1 / Q tot < 0 .

Fig. 4
Fig. 4

(a) Two-dimensional Gaussian gain distribution superimposed on a photonic crystal heterostructure cavity. Pump spot diameter is 1 a , where a is the photonic crystal lattice constant. (b) 1 / Q tot versus peak material gain for two different pump spot sizes.

Tables (3)

Tables Icon

Table 1 Expressions for the Confinement Factor Γ Reported in the Literature a

Tables Icon

Table 2 Comparison between Threshold Gains Estimated from Fig. 2 and Calculated Using Eq. (34) a

Tables Icon

Table 3 Comparison between Threshold Gains Estimated from Fig. 4 and Calculated Using Eq. (34) a

Equations (35)

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α mir + Γ i α i = Γ g g ,
A S d A = U ( t ) t P a + P s .
U ( t ) = V [ 1 2 ϵ E E + 1 2 μ H H ] d V .
Q p , k = ω k U k d U k d t ,
U ( t ) = 1 m T 0 m T U ( t ) d t ,
U k ( t ) = U k ( 0 ) exp ( ω k t / Q p , k ) .
F k ( t ) = F k ( 0 ) e ω k t / 2 Q p , k e j ω k t ,
F k ( t ) = F k ( 0 ) e ω k t / 2 Q p , k cos ( ω k t ) .
A S k d A = U k t .
A S k d A = U k t P a , k + P s , k .
A S k d A = ω k U k Q p , k .
ω k U k Q p , k = U k t P a , k + P s , k .
ω k U k Q p , k = ω k U k Q k P a , k .
ω k U k Q p , k = P a , k + P s , k .
ω k U k Q k = P s , k ,
P a = J E d V = σ E E d V .
( α i / 2 + j β ) 2 = j ω μ σ ω 2 ϵ μ ,
α i = σ 2 c ϵ 0 [ ϵ r + ϵ r 2 + ( σ ω ϵ 0 ) 2 ] 1 / 2 ,
β = ω 2 c [ ϵ r + ϵ r 2 + ( σ ω ϵ 0 ) 2 ] 1 / 2 .
σ = α i c ϵ 0 ϵ r + ( α i c 2 ω ) 2 ,
P a , k = σ E k E k d V = α i c ϵ 0 ϵ r + ( α i c 2 ω k ) 2 E k E k d V .
P s , k = σ E k E k d V = g c ϵ 0 ϵ r + ( g c 2 ω k ) 2 E k E k d V ,
ω k Q p , k + α i c ϵ 0 [ ϵ r + ( α i c 2 ω k ) 2 ] 1 / 2 E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V = g c ϵ 0 [ ϵ r + ( g c 2 ω k ) 2 ] 1 / 2 E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V .
α i σ 2 c ϵ 0 n ,
β ω c n ,
ω k Q p , k + ( c n ) α i ( x , y , z ) ϵ E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V = ( c n ) g ( x , y , z ) ϵ E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V .
α mir = ω k c Q p , k ,
Γ i , k α i = ( 1 n ) α i ( x , y , z ) α i , max ϵ E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V α i , max ,
Γ i , k = ( 1 n ) α i ( x , y , z ) α i , max ϵ E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V .
Γ g , k g = ( 1 n ) g ( x , y , z ) g max ϵ E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V g max ,
Γ g , k = ( 1 n ) g ( x , y , z ) g max ϵ E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V .
Γ g , k = n active active E k E k d V [ 1 2 ϵ E k E k + 1 2 μ H k H k ] d V .
ω 0 Q tot = ω 0 Q p ( c n ) g ( x , y , z ) ϵ E E d V [ 1 2 ϵ E E + 1 2 μ H H ] d V = ω 0 Q p , k c g max Γ g .
g max thr = ω k c Γ g Q p , k ,
Γ g = g ( x , y , z ) g max | E E | 2 d V | E E | 2 d V .

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