Abstract

We investigate the thermal transfer between finite-thickness planar slabs which support surface phonon polariton modes (SPhPs). The thickness-dependent dispersion of SPhPs in such layered materials provides a unique opportunity to manipulate and enhance the near field thermal transfer. The key accomplishment of this paper is the development of an ab-initio coupled mode theory that accurately describes all of its thermal transfer properties. We illustrate how the coupled mode parameters can be obtained in a direct fashion from the dispersion relation of the relevant modes of the system. This is illustrated for the specific case of a semi-infinite SiC substrate placed in close proximity to a thin slab of SiC. This is a system that exhibits rich physics in terms of its thermal transfer properties, despite the seemingly simple geometry. This includes a universal scaling behavior of the thermal conductance with the slab thickness and spacing. The work highlights and further increases the value of coupled mode theories in rapidly calculating and intuitively understanding near-field transfer.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
  3. S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices-a review,” International J. Energy Res. 31(6), 689–716 (2007).
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  4. S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
    [Crossref]
  5. J. A. Schuller, T. Taubner, and M. L. Brongersma, “Optical antenna thermal emitters,” Nature Photonics 3(11), 658–661 (2009).
    [Crossref]
  6. N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
    [Crossref]
  7. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4(10), 3303–3314 (1971).
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  8. A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
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    [Crossref] [PubMed]
  12. N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
    [Crossref]
  13. M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
    [Crossref]
  14. P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
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  15. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys.D: Appl. Phys. 43(7), 075501 (2010).
  16. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Coexistence of multiple regimes for near-field thermal radiation between two layers supporting surface phonon polaritons in the infrared,” Phys. Rev. B 84(7), 075436 (2011).
    [Crossref]
  17. M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
    [Crossref] [PubMed]
  18. C. Otey and S. Fan, “Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate,” Phys. Rev. B 84(24), 245431 (2011).
    [Crossref]
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    [Crossref]
  20. M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, “Fluctuation-induced phenomena in nanoscale systems: harnessing the power of noise,” Proceedings of the IEEE 101(2), 531–545 (2013).
    [Crossref]
  21. K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
    [Crossref]
  22. C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
    [Crossref]
  23. H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE 79(10), 1505–1518 (1991).
    [Crossref]
  24. C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
    [Crossref] [PubMed]
  25. J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32(3), 461 (1961).
    [Crossref]
  26. E. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
    [Crossref]
  27. W. Eckhardt, “First and second fluctuation-dissipation-theorem in electromagnetic fluctuation theory,” Opt. Commun. 41(5), 305–309 (1982).
    [Crossref]
  28. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
    [Crossref]
  29. S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
    [Crossref]
  30. P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82(12), 121419 (2010).
    [Crossref]
  31. W. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Physical Review 113(1), 127–132 (1959).
    [Crossref]
  32. W. Spitzer, D. Kleinman, and C. Frosch, “Infrared properties of cubic silicon carbide films,” Phys. Rev. 113(1), 133–136 (1959).
    [Crossref]
  33. R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17(26), 24112–29 (2009).
    [Crossref]

2014 (1)

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
[Crossref]

2013 (2)

M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, “Fluctuation-induced phenomena in nanoscale systems: harnessing the power of noise,” Proceedings of the IEEE 101(2), 531–545 (2013).
[Crossref]

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

2012 (2)

M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
[Crossref]

A. P. McCauley, M. T. H. Reid, M. Krüger, and S. G. Johnson, “Modeling near-field radiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique,” Phys. Rev. B 85(16), 165104 (2012).
[Crossref]

2011 (4)

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Coexistence of multiple regimes for near-field thermal radiation between two layers supporting surface phonon polaritons in the infrared,” Phys. Rev. B 84(7), 075436 (2011).
[Crossref]

M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
[Crossref] [PubMed]

C. Otey and S. Fan, “Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate,” Phys. Rev. B 84(24), 245431 (2011).
[Crossref]

K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
[Crossref]

2010 (5)

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref]

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82(12), 121419 (2010).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys.D: Appl. Phys. 43(7), 075501 (2010).

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

2009 (5)

P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
[Crossref]

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

J. A. Schuller, T. Taubner, and M. L. Brongersma, “Optical antenna thermal emitters,” Nature Photonics 3(11), 658–661 (2009).
[Crossref]

R. D. Kekatpure, A. C. Hryciw, E. S. Barnard, and M. L. Brongersma, “Solving dielectric and plasmonic waveguide dispersion relations on a pocket calculator,” Opt. Express 17(26), 24112–29 (2009).
[Crossref]

2007 (2)

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices-a review,” International J. Energy Res. 31(6), 689–716 (2007).
[Crossref]

2005 (1)

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

2003 (1)

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[Crossref]

2002 (1)

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6(3), 209–222 (2002).
[Crossref]

2000 (1)

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
[Crossref] [PubMed]

1991 (1)

H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE 79(10), 1505–1518 (1991).
[Crossref]

1982 (1)

W. Eckhardt, “First and second fluctuation-dissipation-theorem in electromagnetic fluctuation theory,” Opt. Commun. 41(5), 305–309 (1982).
[Crossref]

1971 (1)

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4(10), 3303–3314 (1971).
[Crossref]

1969 (1)

E. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

1961 (1)

J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32(3), 461 (1961).
[Crossref]

1959 (2)

W. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Physical Review 113(1), 127–132 (1959).
[Crossref]

W. Spitzer, D. Kleinman, and C. Frosch, “Infrared properties of cubic silicon carbide films,” Phys. Rev. 113(1), 133–136 (1959).
[Crossref]

Barnard, E. S.

Basu, S.

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices-a review,” International J. Energy Res. 31(6), 689–716 (2007).
[Crossref]

Ben-Abdallah, P.

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82(12), 121419 (2010).
[Crossref]

P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
[Crossref]

Biehs, S.

S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref]

Biener, G.

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

Bimonte, G.

M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
[Crossref]

Brongersma, M. L.

Carminati, R.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6(3), 209–222 (2002).
[Crossref]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
[Crossref] [PubMed]

Chen, G.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Chen, Y.-B.

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices-a review,” International J. Energy Res. 31(6), 689–716 (2007).
[Crossref]

Chevrier, J.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

Comin, F.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

Dahan, N.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

Domingues, G.

P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
[Crossref]

Drevillon, J.

P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
[Crossref]

Eckhardt, W.

W. Eckhardt, “First and second fluctuation-dissipation-theorem in electromagnetic fluctuation theory,” Opt. Commun. 41(5), 305–309 (1982).
[Crossref]

Economou, E.

E. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

Emig, T.

M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
[Crossref]

M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
[Crossref] [PubMed]

Fan, S.

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
[Crossref]

C. Otey and S. Fan, “Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate,” Phys. Rev. B 84(24), 245431 (2011).
[Crossref]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Fleming, J. G.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[Crossref]

Francoeur, M.

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Coexistence of multiple regimes for near-field thermal radiation between two layers supporting surface phonon polaritons in the infrared,” Phys. Rev. B 84(7), 075436 (2011).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys.D: Appl. Phys. 43(7), 075501 (2010).

Frischwasser, K.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

Frosch, C.

W. Spitzer, D. Kleinman, and C. Frosch, “Infrared properties of cubic silicon carbide films,” Phys. Rev. 113(1), 133–136 (1959).
[Crossref]

Gorodetski, Y.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

Greffet, J.

S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref]

Greffet, J.-J.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6(3), 209–222 (2002).
[Crossref]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
[Crossref] [PubMed]

Hasman, E.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

Haus, H.

H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE 79(10), 1505–1518 (1991).
[Crossref]

Hryciw, A. C.

Huang, W.

H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE 79(10), 1505–1518 (1991).
[Crossref]

Johnson, S. G.

M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, “Fluctuation-induced phenomena in nanoscale systems: harnessing the power of noise,” Proceedings of the IEEE 101(2), 531–545 (2013).
[Crossref]

A. P. McCauley, M. T. H. Reid, M. Krüger, and S. G. Johnson, “Modeling near-field radiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique,” Phys. Rev. B 85(16), 165104 (2012).
[Crossref]

Joulain, K.

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82(12), 121419 (2010).
[Crossref]

P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6(3), 209–222 (2002).
[Crossref]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
[Crossref] [PubMed]

Jourdan, G.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

Kardar, M.

M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
[Crossref]

M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
[Crossref] [PubMed]

Kekatpure, R. D.

Kleiner, V.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

Kleinman, D.

W. Spitzer, D. Kleinman, and C. Frosch, “Infrared properties of cubic silicon carbide films,” Phys. Rev. 113(1), 133–136 (1959).
[Crossref]

W. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Physical Review 113(1), 127–132 (1959).
[Crossref]

Krüger, M.

M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
[Crossref]

A. P. McCauley, M. T. H. Reid, M. Krüger, and S. G. Johnson, “Modeling near-field radiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique,” Phys. Rev. B 85(16), 165104 (2012).
[Crossref]

M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
[Crossref] [PubMed]

Lau, W. T.

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Lin, S. Y.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[Crossref]

Maguid, E.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

Marquier, F.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

McCauley, A. P.

A. P. McCauley, M. T. H. Reid, M. Krüger, and S. G. Johnson, “Modeling near-field radiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique,” Phys. Rev. B 85(16), 165104 (2012).
[Crossref]

Mengüç, M. P.

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Coexistence of multiple regimes for near-field thermal radiation between two layers supporting surface phonon polaritons in the infrared,” Phys. Rev. B 84(7), 075436 (2011).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys.D: Appl. Phys. 43(7), 075501 (2010).

Moreno, J.

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[Crossref]

Mulet, J.-P.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6(3), 209–222 (2002).
[Crossref]

Narayanaswamy, A.

K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
[Crossref]

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Niv, A.

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

Otey, C.

C. Otey and S. Fan, “Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate,” Phys. Rev. B 84(24), 245431 (2011).
[Crossref]

Otey, C. R.

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
[Crossref]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Ozeri, D.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

Planck, M.

M. Planck, The Theory of Heat Radiation (Blakiston’s Son & Co, 1914).

Polder, D.

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4(10), 3303–3314 (1971).
[Crossref]

Reid, M. T. H.

M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, “Fluctuation-induced phenomena in nanoscale systems: harnessing the power of noise,” Proceedings of the IEEE 101(2), 531–545 (2013).
[Crossref]

A. P. McCauley, M. T. H. Reid, M. Krüger, and S. G. Johnson, “Modeling near-field radiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique,” Phys. Rev. B 85(16), 165104 (2012).
[Crossref]

Rodriguez, A. W.

M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, “Fluctuation-induced phenomena in nanoscale systems: harnessing the power of noise,” Proceedings of the IEEE 101(2), 531–545 (2013).
[Crossref]

Rousseau, E.

S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref]

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

Sandhu, S.

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
[Crossref]

Sasihithlu, K.

K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
[Crossref]

Schuller, J. A.

J. A. Schuller, T. Taubner, and M. L. Brongersma, “Optical antenna thermal emitters,” Nature Photonics 3(11), 658–661 (2009).
[Crossref]

Shchegrov, A. V.

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
[Crossref] [PubMed]

Shen, S.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Shitrit, N.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

Siria, A.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

Spitzer, W.

W. Spitzer, D. Kleinman, and C. Frosch, “Infrared properties of cubic silicon carbide films,” Phys. Rev. 113(1), 133–136 (1959).
[Crossref]

W. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Physical Review 113(1), 127–132 (1959).
[Crossref]

Swihart, J. C.

J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32(3), 461 (1961).
[Crossref]

Taubner, T.

J. A. Schuller, T. Taubner, and M. L. Brongersma, “Optical antenna thermal emitters,” Nature Photonics 3(11), 658–661 (2009).
[Crossref]

Vaillon, R.

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Coexistence of multiple regimes for near-field thermal radiation between two layers supporting surface phonon polaritons in the infrared,” Phys. Rev. B 84(7), 075436 (2011).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys.D: Appl. Phys. 43(7), 075501 (2010).

Van Hove, M.

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4(10), 3303–3314 (1971).
[Crossref]

Veksler, D.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

Volz, S.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

Walsh, D.

W. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Physical Review 113(1), 127–132 (1959).
[Crossref]

Yulevich, I.

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

Zhang, Z. M.

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices-a review,” International J. Energy Res. 31(6), 689–716 (2007).
[Crossref]

Zhu, L.

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
[Crossref]

Appl. Phys. Lett. (1)

S. Y. Lin, J. Moreno, and J. G. Fleming, “Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation,” Appl. Phys. Lett. 83(2), 380 (2003).
[Crossref]

International J. Energy Res. (1)

S. Basu, Y.-B. Chen, and Z. M. Zhang, “Microscale radiation in thermophotovoltaic devices-a review,” International J. Energy Res. 31(6), 689–716 (2007).
[Crossref]

J. Appl. Phys. (2)

P. Ben-Abdallah, K. Joulain, J. Drevillon, and G. Domingues, “Near-field heat transfer mediated by surface wave hybridization between two films,” J. Appl. Phys. 106(4), 044306 (2009).
[Crossref]

J. C. Swihart, “Field solution for a thin-film superconducting strip transmission line,” J. Appl. Phys. 32(3), 461 (1961).
[Crossref]

J. Phys.D: Appl. Phys. (1)

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two thin silicon carbide films,” J. Phys.D: Appl. Phys. 43(7), 075501 (2010).

J. Quantitative Spectroscopy and Radiative Transfer (1)

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: a brief overview,” J. Quantitative Spectroscopy and Radiative Transfer 132, 3–11 (2014).
[Crossref]

Microscale Thermophysical Engineering (1)

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophysical Engineering 6(3), 209–222 (2002).
[Crossref]

Nano Lett. (1)

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[Crossref] [PubMed]

Nature Photonics (2)

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nature Photonics 3(9), 514–517 (2009).
[Crossref]

J. A. Schuller, T. Taubner, and M. L. Brongersma, “Optical antenna thermal emitters,” Nature Photonics 3(11), 658–661 (2009).
[Crossref]

Opt. Commun. (1)

W. Eckhardt, “First and second fluctuation-dissipation-theorem in electromagnetic fluctuation theory,” Opt. Commun. 41(5), 305–309 (1982).
[Crossref]

Opt. Express (1)

Phys. Rev. (2)

W. Spitzer, D. Kleinman, and C. Frosch, “Infrared properties of cubic silicon carbide films,” Phys. Rev. 113(1), 133–136 (1959).
[Crossref]

E. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[Crossref]

Phys. Rev. B (8)

K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
[Crossref]

C. Otey and S. Fan, “Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate,” Phys. Rev. B 84(24), 245431 (2011).
[Crossref]

A. P. McCauley, M. T. H. Reid, M. Krüger, and S. G. Johnson, “Modeling near-field radiative heat transfer from sharp objects using a general three-dimensional numerical scattering technique,” Phys. Rev. B 85(16), 165104 (2012).
[Crossref]

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82(12), 121419 (2010).
[Crossref]

N. Dahan, A. Niv, G. Biener, Y. Gorodetski, V. Kleiner, and E. Hasman, “Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves,” Phys. Rev. B 76(4), 045427 (2007).
[Crossref]

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4(10), 3303–3314 (1971).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Coexistence of multiple regimes for near-field thermal radiation between two layers supporting surface phonon polaritons in the infrared,” Phys. Rev. B 84(7), 075436 (2011).
[Crossref]

M. Krüger, G. Bimonte, T. Emig, and M. Kardar, “Trace formulas for nonequilibrium casimir interactions, heat radiation, and heat transfer for arbitrary objects,” Phys. Rev. B 86(11), 115423 (2012).
[Crossref]

Phys. Rev. Lett. (5)

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
[Crossref] [PubMed]

A. V. Shchegrov, K. Joulain, R. Carminati, and J.-J. Greffet, “Near-field spectral effects due to electromagnetic surface excitations,” Phys. Rev. Lett. 85(7), 1548–1551 (2000).
[Crossref] [PubMed]

S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010).
[Crossref]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Physical Review (1)

W. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Physical Review 113(1), 127–132 (1959).
[Crossref]

Proceedings of the IEEE (2)

M. T. H. Reid, A. W. Rodriguez, and S. G. Johnson, “Fluctuation-induced phenomena in nanoscale systems: harnessing the power of noise,” Proceedings of the IEEE 101(2), 531–545 (2013).
[Crossref]

H. Haus and W. Huang, “Coupled-mode theory,” Proceedings of the IEEE 79(10), 1505–1518 (1991).
[Crossref]

Science (1)

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

Surf. Sci. Rep. (1)

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: radiative heat transfer, coherence properties and casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3), 59–112 (2005).
[Crossref]

Other (1)

M. Planck, The Theory of Heat Radiation (Blakiston’s Son & Co, 1914).

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of the Swihart geometry which is composed of a thick slab of SiC at a temperature T 1 separated by a vacuum gap of width d from a slab of SiC of thickness t and at a temperature T 2. The profiles of the three supported SPhP modes are shown in this structure in the limit of negligible coupling (b) Schematic of the mode profiles at a frequency right below the SPhP resonance frequency for three special cases of (i) t = 0.5d (ii) t = d (iii) t = 2d. Curves show the normalized absolute value of the magnetic field.

Fig. 2
Fig. 2

(a) Plot of normalized thermal conductance relative to its maximum value as a function of slab thickness for four different vacuum gaps (b) Log-log plot of the maximum thermal conductance achievable at each gap width as a function of the gap width.

Fig. 3
Fig. 3

Surface plot of S (ω, β) in ωβ plane for the case of (a) t = 0.5d (b) t = d, and (c) t = 2d. These plots are overlaid by the ℜ(ω), β dispersion diagram of the surface modes (dashed white lines). Insets show the calculations of S (ω, β) using the coupled mode theory, which are in close quantitative agreement with the exact calculations.

Fig. 4
Fig. 4

Comparison of variation of S (ω, β) versus ω for a fixed value of β as calculated by exact method (Exact) and coupled mode theory (CMT). Figures (a) to (c) are corresponding to βd = 0.1, βd = 2, and βd = 5, respectively.

Equations (28)

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

S = 0 0 ε 0 ω ( ε ) 2 π 3 ( Θ ( ω , T 1 ) Θ ( ω , T 2 ) ) V d β d ω
V = ω μ 0 8 ( k z ) | k z | 2 { [ k z ( 1 R s ) ( 1 + R s * ) k z v | T s | 2 ] + β 2 + | k z | 2 ε k 0 2 [ k z ( 1 R p ) ( 1 + R p * ) k z v | T p | 2 ] }
S = 0 0 S ( ω , β ) 4 π 2 ( Θ ( ω , T 1 ) Θ ( ω , T 2 ) ) β d β d ω
S ( ω , β ) = 4 ( ζ ) 2 ( e 2 β d e 2 β ( t + d ) ) ( 1 + e 2 β t | ζ | 2 ) × | 1 ζ 2 ( e 2 β d + e 2 β t e 2 β d e 2 β t ) | 2
S = 1 d 2 f ( t d )
S ( ω SPhP , β ) ε 2 e 2 β d ( 1 e 2 β t ) ( 1 + 4 ε 2 e 2 β t ) | e 2 β d + e 2 β t e 2 β d e 2 β t + ε 2 4 | 2
d A d t = ( i ω SPhP γ ) A + i κ A + 2 γ N
κ = Γ ω 0 [ 0 e β d ( 1 e 2 β t ) 1 2 0 0 e β t 0 ]
n 1 * ( ω ) n 1 ( ω ) = 2 π δ ( ω ω ) Θ ( ω , T 1 ) n 2 * ( ω ) n 2 ( ω ) = n 3 * ( ω ) n 3 ( ω ) = 2 π δ ( ω ω ) Θ ( ω , T 2 ) n i * ( ω ) n j ( ω ) = 0 for i j
S ( ω , β ) = ε 2 e 2 β d ( 1 e 2 β t ) | e 2 β d + e 2 β t e 2 β ( t + d ) ( ω ω SPhP ω 0 Γ + i ε 2 ) 2 | 2 × ( 1 + 4 ω 0 2 Γ 2 e 2 β t ω 0 2 Γ 2 ε 2 + 4 ( ω ω SPhP ) 2 )
S = 0 0 ε 0 ω ( ε ) 2 π 3 ( Θ ( ω , T 1 ) Θ ( ω , T 2 ) ) V 12 d β d ω
V 1 = ω μ 0 k z 8 ( k z ) | k z | 2 { ( 1 R s ) ( 1 + R s * ) + β 2 + | k z | 2 ε k 0 2 ( 1 R p ) ( 1 + R p * ) }
V 2 = ω μ 0 k z v 8 ( k z ) | k z | 2 { | T s | 2 + β 2 + | k z | 2 ε k 0 2 | T p | 2 }
R p = α 2 ξ ( 1 e 2 j k z v d + e 2 j k z v d 2 j k z t ) ξ 3 e 2 j k z t α 3 + α ξ 2 ( e 2 j k z v d 2 j k z t e 2 j k z t e 2 j k z v d ) T p = 4 ( α ξ ) e j k z v d e j k z t α 3 + α ξ 2 ( e 2 j k z v d 2 j k z t e 2 j k z t e 2 j k z v d )
S = d 2 0 0 f 1 ( ω , r , b ) 4 π 2 ( Θ ( ω , T 1 ) Θ ( ω , T 2 ) ) b d b d ω = d 2 f 2 ( r )
β 2 ω 2 c 2 β 2 ε ω 2 c 2 ε = { tanh ( 1 2 β 2 ω 2 c 2 d ) sym coth ( 1 2 β 2 ω 2 c 2 d ) asym
ω = i δ 2 + ω 0 1 + ε s 1 + ε ( 1 ± e β d ( ε s ε ) ( 1 + ε s ) ( 1 + ε ) )
β 2 ω 2 c 2 β 2 ε ω 2 c 2 ε = { tanh ( 1 2 β 2 ε ω 2 c 2 d ) asym coth ( 1 2 β 2 ε ω 2 c 2 d ) sym
d A d t = i ω SPhP A + i κ A
| κ + ( ω SPhP ω ) I | = 0
ζ sub 1 ζ f 2 ζ sub 1 e 2 β t ζ f 1 e 2 β d ( 1 e 2 β t ) = 0
( ω ω 1 ) ( ω ω 2 ) 2 Γ f 2 ω 0 , f 2 e 2 β t ( ω ω 1 ) + Γ f Γ sub ω 0 , f ω 0 , sub e 2 β d e 2 β t ( ω ω 2 ) Γ f Γ sub ω 0 , f ω 0 , sub e 2 β d ( ω ω 2 ) = 0
| ω 1 ω κ 12 κ 13 κ 12 * ω 2 ω κ 23 κ 13 * κ 23 * ω 2 ω | = 0
( ω ω 1 ) ( ω ω 2 ) 2 | κ 23 | 2 ( ω ω 1 ) | κ 12 | 2 ( ω ω 2 ) κ 12 κ 13 * κ 23 | κ 13 | 2 ( ω ω 2 ) κ 12 * κ 13 κ 23 * = 0
κ 12 = κ 21 = Γ ω 0 e β d ( 1 e 2 β t ) 1 2 κ 23 = κ 32 = Γ ω 0 e β t κ 13 = κ 31 = 0
[ A 1 ( ω , β ) A 2 ( ω , β ) A 3 ( ω , β ) ] = K 1 [ 2 γ n 1 2 γ n 2 2 γ n 3 ]
K = [ i ( ω ω SPhP ) + γ i Γ ω 0 e β d ( 1 e 2 β t ) 1 2 0 i Γ ω 0 e β d ( 1 e 2 β t ) 1 2 i ( ω ω SPhP ) + γ i Γ ω 0 e β t 0 i Γ ω 0 e β t i ( ω ω SPhP ) + γ ]
t ( ω , β ) = 4 π γ e β d ( 1 e 2 β t ) 1 2 | γ + i ( ω ω SPhP ) | 2 × | ( γ + i ( ω ω SPhP ) ) 2 + ω 0 2 Γ 2 ( e 2 β d + e 2 β t e 2 β ( t + d ) ) | 2 × [ i ω 0 Γ ( ( γ i ( ω ω SPhP ) ) 2 + ω 0 2 Γ 2 e 2 β t ) ( γ + i ( ω ω SPhP ) ) Θ ( ω , T 1 ) i ω 0 Γ ( γ i ( ω ω SPhP ) ) ( γ + i ( ω ω SPhP ) ) 2 Θ ( ω , T 2 ) i ω 0 3 Γ 3 e 2 β t ( γ + i ( ω ω SPhP ) ) Θ ( ω , T 2 ) ]

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