Abstract

One can predict the thermal emission spectrum of any material from the knowledge of its absorbance and its temperature: this is the Kirchhoff–Planck law. We show that if McCumber’s relation holds and if the spatial distribution of the excited state is uniform, the Kirchhoff–Planck law can be generalized by introducing the chemical potential difference between the metastable and ground manifolds involved in the transition. The proposed formalism makes it possible to determine the emission spectra of an optical structure driven out of equilibrium solely from its transmission and reflection spectra and the level of excitation, considerably simplifying computations compared to a direct approach. An example is shown for a multilayer with embedded luminescent ions. Experimental emission spectra from Yb3+-doped Y2O3 taken at and out of thermal equilibrium are found to be in qualitative agreement with the theory.

© 2011 Optical Society of America

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  1. J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
    [CrossRef]
  2. P. Würfel, “Light with nonzero chemical potential,” J. Phys. C 15, 3967–3985 (1982).
    [CrossRef]
  3. P. Pigeat, D. Rouxel, and B. Weber, “Calculation of thermal emissivity for thin films by a direct method,” Phys. Rev. B 57, 9293–9300 (1998).
    [CrossRef]
  4. C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos,” Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93, 213905 (2004).
    [CrossRef] [PubMed]
  5. S. Chandrasekhar, Radiative Transfer (Courier Dover, 1960).
  6. F. LeBlanc, An Introduction to Stellar Astrophysics (Wiley, 2010).
  7. M. A. Weinstein, “On the validity of Kirchhoff’s law for a freely radiating body,” Am. J. Phys. 28, 123–125 (1960).
    [CrossRef]
  8. D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957(1964).
    [CrossRef]
  9. D. Kouznetsov, J.-F. Bisson, K. Takaichi, and K. Ueda, “High-power single-mode solid-state laser with a short wide unstable cavity,” J. Opt. Soc. Am. B 22, 1605–1619 (2005).
    [CrossRef]
  10. D. E. McCumber, “Theory of phonon-terminated optical masers,” Phys. Rev. 134, A299–A306 (1964).
    [CrossRef]
  11. L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
    [CrossRef]
  12. B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections and radiative lifetimes of rare earth ions in solids: application to TM3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
    [CrossRef]
  13. R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
    [CrossRef]
  14. M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
    [CrossRef]
  15. R. Baierlein, “The elusive chemical potential,” Am. J. Phys. 69, 423–434 (2001).
    [CrossRef]
  16. J. P. Dowling and C. M. Bowden, “Atomic rates in homogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
    [CrossRef] [PubMed]
  17. M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
    [CrossRef] [PubMed]
  18. W. W. Chow, “Theory of emission from an active photonic lattice,” Phys. Rev. A 73, 013821 (2006).
    [CrossRef]
  19. N.-P. Harder and M. A. Green, “Thermophotonics,” Semicond. Sci. Technol. 18, S270–S278 (2003).
    [CrossRef]
  20. P. Ben-Abdallah, “Thermal antenna behavior for thin-film structures,” J. Opt. Soc. Am. A 21, 1368–1371 (2004).
    [CrossRef]
  21. P. Ben-Abdallah, “Single-defect Bragg stacks for high-power narrow-band thermal emission,” J. Appl. Phys. 97, 104910 (2005).
    [CrossRef]
  22. C. M. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59, 4736–4746 (1999).
    [CrossRef]
  23. B. J. Lee and Z. M. Zhang, “Coherent thermal emission from modified periodic multilayer structures,” J. Heat Transfer 129, 17–26 (2007).
    [CrossRef]
  24. T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
    [CrossRef]
  25. M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, “Thermal emission and absorption of radiation in finite inverted-opal photonic crystals,” Phys. Rev. A 72, 033821 (2005).
    [CrossRef]
  26. M. Laroche, R. Carminati, and J. J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
    [CrossRef] [PubMed]
  27. S. E. Han, “Theory of thermal emission from periodic structures,” Phys. Rev. B 80, 155108 (2009).
    [CrossRef]
  28. P. Yeh, Optical Waves in Layered Media (Wiley, 2005).
  29. L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
    [CrossRef]
  30. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

2009

T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
[CrossRef]

S. E. Han, “Theory of thermal emission from periodic structures,” Phys. Rev. B 80, 155108 (2009).
[CrossRef]

2008

L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
[CrossRef]

2007

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

B. J. Lee and Z. M. Zhang, “Coherent thermal emission from modified periodic multilayer structures,” J. Heat Transfer 129, 17–26 (2007).
[CrossRef]

2006

M. Laroche, R. Carminati, and J. J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef] [PubMed]

W. W. Chow, “Theory of emission from an active photonic lattice,” Phys. Rev. A 73, 013821 (2006).
[CrossRef]

2005

P. Ben-Abdallah, “Single-defect Bragg stacks for high-power narrow-band thermal emission,” J. Appl. Phys. 97, 104910 (2005).
[CrossRef]

M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, “Thermal emission and absorption of radiation in finite inverted-opal photonic crystals,” Phys. Rev. A 72, 033821 (2005).
[CrossRef]

D. Kouznetsov, J.-F. Bisson, K. Takaichi, and K. Ueda, “High-power single-mode solid-state laser with a short wide unstable cavity,” J. Opt. Soc. Am. B 22, 1605–1619 (2005).
[CrossRef]

2004

P. Ben-Abdallah, “Thermal antenna behavior for thin-film structures,” J. Opt. Soc. Am. A 21, 1368–1371 (2004).
[CrossRef]

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos,” Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[CrossRef] [PubMed]

2003

N.-P. Harder and M. A. Green, “Thermophotonics,” Semicond. Sci. Technol. 18, S270–S278 (2003).
[CrossRef]

2002

R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
[CrossRef]

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[CrossRef]

2001

R. Baierlein, “The elusive chemical potential,” Am. J. Phys. 69, 423–434 (2001).
[CrossRef]

1999

C. M. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59, 4736–4746 (1999).
[CrossRef]

1998

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections and radiative lifetimes of rare earth ions in solids: application to TM3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

P. Pigeat, D. Rouxel, and B. Weber, “Calculation of thermal emissivity for thin films by a direct method,” Phys. Rev. B 57, 9293–9300 (1998).
[CrossRef]

1996

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

1993

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

1992

J. P. Dowling and C. M. Bowden, “Atomic rates in homogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

1982

P. Würfel, “Light with nonzero chemical potential,” J. Phys. C 15, 3967–3985 (1982).
[CrossRef]

1964

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957(1964).
[CrossRef]

D. E. McCumber, “Theory of phonon-terminated optical masers,” Phys. Rev. 134, A299–A306 (1964).
[CrossRef]

1960

M. A. Weinstein, “On the validity of Kirchhoff’s law for a freely radiating body,” Am. J. Phys. 28, 123–125 (1960).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Ashrit, P. V.

T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
[CrossRef]

Baierlein, R.

R. Baierlein, “The elusive chemical potential,” Am. J. Phys. 69, 423–434 (2001).
[CrossRef]

Barnes, N. P.

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections and radiative lifetimes of rare earth ions in solids: application to TM3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

Ben-Abdallah, P.

P. Ben-Abdallah, “Single-defect Bragg stacks for high-power narrow-band thermal emission,” J. Appl. Phys. 97, 104910 (2005).
[CrossRef]

P. Ben-Abdallah, “Thermal antenna behavior for thin-film structures,” J. Opt. Soc. Am. A 21, 1368–1371 (2004).
[CrossRef]

Ben-Messaoud, T.

T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
[CrossRef]

Bisson, J. F.

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

Bisson, J.-F.

Bloemer, M. J.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Bowden, C. M.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

J. P. Dowling and C. M. Bowden, “Atomic rates in homogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Carminati, R.

M. Laroche, R. Carminati, and J. J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Courier Dover, 1960).

Chase, L. L.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Chen, G.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos,” Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[CrossRef] [PubMed]

Chow, W. W.

W. W. Chow, “Theory of emission from an active photonic lattice,” Phys. Rev. A 73, 013821 (2006).
[CrossRef]

Cornelius, C. M.

C. M. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59, 4736–4746 (1999).
[CrossRef]

DeLoach, L. D.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Di Bartolo, B.

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections and radiative lifetimes of rare earth ions in solids: application to TM3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

Digonnet, M. J. F.

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[CrossRef]

Dowling, J.

M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, “Thermal emission and absorption of radiation in finite inverted-opal photonic crystals,” Phys. Rev. A 72, 033821 (2005).
[CrossRef]

Dowling, J. P.

C. M. Cornelius and J. P. Dowling, “Modification of Planck blackbody radiation by photonic band-gap structures,” Phys. Rev. A 59, 4736–4746 (1999).
[CrossRef]

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

J. P. Dowling and C. M. Bowden, “Atomic rates in homogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Dubinskii, M.

L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
[CrossRef]

Falquier, D. G.

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[CrossRef]

Florescu, M.

M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, “Thermal emission and absorption of radiation in finite inverted-opal photonic crystals,” Phys. Rev. A 72, 033821 (2005).
[CrossRef]

Fredrich-Thornton, S.

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

Green, M. A.

N.-P. Harder and M. A. Green, “Thermophotonics,” Semicond. Sci. Technol. 18, S270–S278 (2003).
[CrossRef]

Greffet, J. J.

M. Laroche, R. Carminati, and J. J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef] [PubMed]

Haché, A.

T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
[CrossRef]

Han, S. E.

S. E. Han, “Theory of thermal emission from periodic structures,” Phys. Rev. B 80, 155108 (2009).
[CrossRef]

Harder, N.-P.

N.-P. Harder and M. A. Green, “Thermophotonics,” Semicond. Sci. Technol. 18, S270–S278 (2003).
[CrossRef]

Huber, G.

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

Joannopoulos, J. D.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos,” Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[CrossRef] [PubMed]

Kouznetsov, D.

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

D. Kouznetsov, J.-F. Bisson, K. Takaichi, and K. Ueda, “High-power single-mode solid-state laser with a short wide unstable cavity,” J. Opt. Soc. Am. B 22, 1605–1619 (2005).
[CrossRef]

Krupke, W.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Kway, W. L.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Laroche, M.

M. Laroche, R. Carminati, and J. J. Greffet, “Coherent thermal antenna using a photonic crystal slab,” Phys. Rev. Lett. 96, 123903 (2006).
[CrossRef] [PubMed]

LeBlanc, F.

F. LeBlanc, An Introduction to Stellar Astrophysics (Wiley, 2010).

Lee, B. J.

B. J. Lee and Z. M. Zhang, “Coherent thermal emission from modified periodic multilayer structures,” J. Heat Transfer 129, 17–26 (2007).
[CrossRef]

Lee, H.

M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, “Thermal emission and absorption of radiation in finite inverted-opal photonic crystals,” Phys. Rev. A 72, 033821 (2005).
[CrossRef]

Luo, C.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos,” Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[CrossRef] [PubMed]

McCumber, D. E.

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. 136, A954–A957(1964).
[CrossRef]

D. E. McCumber, “Theory of phonon-terminated optical masers,” Phys. Rev. 134, A299–A306 (1964).
[CrossRef]

Melanson, A.

T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
[CrossRef]

Merkle, L. D.

L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, 1976).

Michael, A.

L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
[CrossRef]

Murphy-Chutorian, E.

M. J. F. Digonnet, E. Murphy-Chutorian, and D. G. Falquier, “Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers,” IEEE J. Quantum Electron. 38, 1629–1637 (2002).
[CrossRef]

Narayanaswamy, A.

C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos,” Thermal radiation from photonic crystals: a direct calculation,” Phys. Rev. Lett. 93, 213905 (2004).
[CrossRef] [PubMed]

Newburgh, G. A.

L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
[CrossRef]

Payne, S. A.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Petermann, K.

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

Pigeat, P.

P. Pigeat, D. Rouxel, and B. Weber, “Calculation of thermal emissivity for thin films by a direct method,” Phys. Rev. B 57, 9293–9300 (1998).
[CrossRef]

Quimby, R. S.

R. S. Quimby, “Range of validity of McCumber theory in relating absorption and emission cross sections,” J. Appl. Phys. 92, 180–187 (2002).
[CrossRef]

Riordon, J.

T. Ben-Messaoud, J. Riordon, A. Melanson, P. V. Ashrit, and A. Haché, “Photoactive periodic media,” Appl. Phys. Lett. 94, 111904 (2009).
[CrossRef]

Rouxel, D.

P. Pigeat, D. Rouxel, and B. Weber, “Calculation of thermal emissivity for thin films by a direct method,” Phys. Rev. B 57, 9293–9300 (1998).
[CrossRef]

Scalora, M.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Smith, L. K.

L. D. DeLoach, S. A. Payne, L. L. Chase, L. K. Smith, W. L. Kway, and W. Krupke, “Evaluation of absorption and emission properties of Yb3+ doped crystals for laser applications,” IEEE J. Quantum Electron. 29, 1179–1191 (1993).
[CrossRef]

Stimpson, A. J.

M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, “Thermal emission and absorption of radiation in finite inverted-opal photonic crystals,” Phys. Rev. A 72, 033821 (2005).
[CrossRef]

Takaichi, K.

Ter-Gabrielyan, N.

L. D. Merkle, G. A. Newburgh, N. Ter-Gabrielyan, A. Michael, M. Dubinskii, “Temperature-dependent lasing and spectroscopy of Yb:Y2O3 and Yb:Sc2O3,” Opt. Commun. 281, 5855–5861(2008).
[CrossRef]

Tocci, M. D.

M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef] [PubMed]

Ueda, K.

J. F. Bisson, D. Kouznetsov, K. Ueda, S. Fredrich-Thornton, K. Petermann, and G. Huber, “Switching of emissivity and photo-conductivity in highly doped Yb3+:Y2O3 and Lu2O3 ceramics,” Appl. Phys. Lett. 90, 201901 (2007).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Effective absorption (solid curve) and emission (dashed curve) cross sections used in the simulations as a function of v / v ref . The emission cross section is calculated from the absorption cross section using McCumber’s formula.

Fig. 2
Fig. 2

Bulk absorption coefficient of material 1 as a function of v / v ref in thermal equilibrium (solid curve) and when the population of the upper manifold is 100 times larger than in thermal equilibrium (dashed curve).

Fig. 3
Fig. 3

Calculated reflectance spectrum of a 50-period quarter-wave multilayer at normal incidence for f 2 f 1 = f 2 f 1 | e q . × 100 .

Fig. 4
Fig. 4

Calculated normal absorbance spectra of the multilayer as a function of v / v ref in (black thick solid curve) and out of (red thin solid curve) thermal equilibrium. The absorbance spectrum of the homogeneous material 1 of the same thickness as the total thickness of material 1 in the multilayer is also shown (green dotted curve). The absorbance is reduced out of thermal equilibrium compared to thermal equilibrium but remains positive (no gain). A zoom of the region with the peak absorbance is shown in the inset.

Fig. 5
Fig. 5

Calculated normal emissivity spectra of the multilayer in (black thick solid curve) and out of (red thin solid curve) thermal equilibrium. The emissivity of a homogeneous slab in thermal equilibrium with the same total thickness as high-index material 1 is also shown (green dotted curve). The periodic structure provides a 20-fold enhancement of emissivity compared to the homogeneous slab, while placing the structure out of equilibrium provides further enhancement.

Fig. 6
Fig. 6

Similarity of the out-of-equilibrium and thermal luminescence spectra of Yb 3 + . The purely thermal luminescence spectrum (black thick solid curve), CO 2 laser 5.5 W , was multiplied by a constant factor (black dotted curve) in order to show the similarity with the nonthermal spectrum obtained by illuminating with 100 mW of LD (red thin solid curve). Scattering of the pump at around λ = 910 nm was removed from the figure.

Tables (1)

Tables Icon

Table 1 Parameters Used for the Simulations of the PBG

Equations (54)

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· I ( k , ν , e ) = α ( k , ν , e ) I ( k , ν , e ) + j ( k , ν , e ) .
I ( x ) = ( j / α ) [ 1 exp ( 2 Im ( n 2 cos θ ) k 0 x ) ] for     cos θ 0 , I ( x ) = ( j / α ) [ 1 exp ( 2 Im ( n 2 cos θ ) k 0 ( d x ) ) ] for     cos θ < 0 ,
I ( x ) = ( j / α ) [ 1 exp ( α x / cos θ ) ] for     cos θ 0 , I ( x ) = ( j / α ) [ 1 exp ( α ( d x ) / cos θ ) ] for     cos θ < 0.
I ( d ) = ( j / α ) [ 1 exp ( α d ) ] ,
j ( k , ν , e ) = W ( k , ν , e ) h ν N f 2 ,
W ( k , ν , e ) = σ e ( k , ν , e ) ρ ( k , ν , e ) d ω d k ,
j = σ e ρ d ω d k h ν N f 2 .
ρ ( k , ν , e ) = ( n 2 ( k , ν , e ) ν c ) 2 d k d ω ( k , ν , e ) .
j = σ e ( n 2 ν c ) 2 h ν N f 2 .
α ( k , ν , e ) = N [ f 1 σ a ( k , ν , e ) f 2 σ e ( k , ν , e ) ] .
I ( d ) = f 2 / f 1 σ e σ a f 2 / f 1 σ e h ν 3 n 2 2 c 2 [ 1 exp ( α d ) ] .
T i = ( 1 R i ) n 2 2 ( 1 1 R i 2 exp ( 2 α d ) + R i exp ( α d ) 1 R i 2 exp ( 2 α d ) ) ,
T i = ( 1 R i ) n 2 2 ( 1 1 R i exp ( α d ) ) .
I = f 2 / f 1 σ e σ a f 2 / f 1 σ e h ν 3 c 2 A ,
A ( k , ν , e ) = ( 1 R i ) ( 1 exp ( α d ) 1 R i exp ( α d ) ) .
A = 1 R Tr ,
R ( k , ν , e ) = R i + R i ( 1 R i ) 2 exp ( 2 α d ) 1 R i 2 exp ( 2 α d )
Tr ( k , ν , e ) = ( 1 R i ) 2 exp ( α d ) 1 R i 2 exp ( 2 α d ) .
σ a = σ e exp [ ( h ν ε ) / k b T ] ,
exp ( ε / k b T ) f 2 / f 1 | e q . .
I = f 2 / f 1 f 2 / f 1 | e q exp [ h ν / k b T ] f 2 / f 1 h ν 3 c 2 A .
f 2 / f 1 = f 2 / f 1 | e q . .
I = 1 exp ( h ν / k b T ) 1 h ν 3 c 2 A ,
I B B ( k , ν , ε ) = h ν 3 / c 2 exp ( h ν / k b T ) 1 .
E I / I B B .
f 2 / f 1 = f 2 / f 1 | e q . exp ( Δ μ / k b T ) ,
I = h ν 3 / c 2 A exp ( ( h ν Δ μ ) / k b T ) 1 ,
α = N f 1 σ a [ 1 exp ( ( Δ μ h ν ) / k b T ) ] .
I exp ( Δ μ / k b T ) I w A ,
I w = h ν 3 / c 2 exp ( h ν / k b T )
σ a = σ a max exp [ ( ν ν 0 δ ν ) 2 ] .
1 τ s p = e ν ( 4 π W ( k , ν , e ) d Ω ) d ν .
1 τ s p = e ν ( 4 π W ( k , ν , e ) d Ω ) d ν = 8 π Re ( n 1 ) 2 c 2 ν σ e ( ν ) ν 2 d ν .
M = D 10 P 1 ( M 1 ) p D 01 ,
M 1 = D 21 P 2 D 12 P 1 .
D i j = 1 t i j ( 1 r i j r i j 1 ) ,
P i = ( exp ( j ϕ i ) 0 0 exp ( j ϕ i ) ) ,
ϕ i = 2 π n i d i λ ,
R = | M 21 M 11 | 2 and Tr = | 1 M 11 | 2 ,
r st = σ e ( k , ν , e ) ρ ( k , ν , e ) F B E ( ν ) d ω d k N f 2 ,
r abs = σ a ρ F B E d ω d k N f 1 .
r sp = W N f 2 .
r st + r sp = r abs ,
f 2 f 1 = f 2 f 1 | e q . = exp ( ε / k b T ) ,
σ a = σ e f 2 f 1 | e q . exp ( h ν / k b T ) .
W = σ e exp ( h ν / k b T ) ρ F B E d ω d k σ e ρ F B E d ω d k ,
W ( k , ν , e ) = σ e ( k , ν , e ) ρ ( k , ν , e ) d ω d k ,
E ( r + L i ^ ) = E ( r + L j ^ ) = E ( r + L k ^ ) = E ( r ) ,
E ( r ) = E 0 e exp ( i k · r ) ,
k x = 2 π L n x , k y = 2 π L n y , k z = 2 π L n z ,
N | k | k , e = k 3 / 3 ( 2 π L ) 3 = k 3 L 3 / 24 π 3 .
ρ ( k , ν , e ) = ρ k d k d ν = k 2 ( ν ) 8 π 3 d k d ν ,
k = n ( k , ν , e ) 2 π ν c ,
ρ ( k , ν , e ) = ( n ( k , ν , e ) 2 π ν c ) 2 1 8 π 3 d k d ν = n 2 ν 2 c 2 d k d ω ,

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