Abstract

By combining stochastic electrodynamics and the Maxwell-Garnett description for effective media we study the radiative heat transfer between two nanoporous materials. We show that the heat flux can be significantly enhanced by air inclusions, which we explain by:(a) the presence of additional surface waves that give rise to supplementary channels for heat transfer throughout the gap, (b) an increase in the contribution given by the ordinary surface waves at resonance, (c) and the appearance of frustrated modes over a broad spectral range. We generalize the known expression for the nanoscale heat flux for anisotropic metamaterials.

© 2011 OSA

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    [CrossRef]
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    [CrossRef]
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  7. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
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  8. C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer 49, 1703 (2006).
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  9. S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “A mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010).
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    [CrossRef]
  11. A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2010 (5)

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

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96, 041112 (2010).
[CrossRef]

M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010).
[CrossRef]

P. Ben-Abdallah and K. Joulain, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 82, 121419 (2010).
[CrossRef]

2009 (4)

G. Bimonte, “Scattering approach to Casimir forces and radiative heat transfer for nanostructured surfaces out of thermal equilibrium,” Phys. Rev. A 80, 042102 (2009).
[CrossRef]

E. A. Vinogradov and I. A. Dorofeyev, “Thermally stimulated electromagnetic fields of solids,” Phys. Usp. 52, 425 (2009).
[CrossRef]

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

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

2008 (5)

U. F. Wischnath, J. Welker, M. Munzel, and A. Kittel, “Near-field scanning thermal microscope,” Rev. Sci. Instrum. 79, 073708 (2008).
[CrossRef] [PubMed]

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008).
[CrossRef]

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[CrossRef]

J. J. Saarinen, S. M. Weiss, P. M. Fauchet, and J. E. Sipe, “Reflectance analysis of a multilayer one-dimensional porous silicon structure: Theory and experiment,” J. Appl. Phys. 104, 013103 (2008).
[CrossRef]

T. G. Philbin and U. Leonhardt, “Alternative calculation of the Casimir forces between birefringent plates,” Phys. Rev. A 78, 042107 (2008).
[CrossRef]

2007 (2)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007).
[CrossRef]

A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291 (2007).
[CrossRef]

2006 (2)

C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer 49, 1703 (2006).
[CrossRef]

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

2005 (3)

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhanget, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005).
[CrossRef] [PubMed]

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, 59 (2005).
[CrossRef]

2004 (1)

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191 (2004).
[CrossRef]

2002 (1)

A. A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

2001 (2)

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
[CrossRef]

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[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, 1548 (2000).
[CrossRef] [PubMed]

1999 (1)

P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystal optics and homogenization of 2D periodic composites,” Phys. Rev. Lett. 82, 719 (1999).
[CrossRef]

1989 (1)

S.-F. Chuang, S. D. Collins, and R. L. Smith, “Porous silicon microstructure as studied by transmission electron microscopy,” Appl. Phys. Lett. 55, 1540 (1989).
[CrossRef]

1986 (1)

1980 (1)

M. L. Levin, V. G. Polevoi, and S. M. Rytov, “Theory of heat-transfer due to a fluctuation electromagnetic field,” Sov. Phys. JETP 50, 1054 (1980).

1971 (1)

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

1954 (1)

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Quantum theory of spontaneous and stimulated emission of surface plasmons,” Phys. Rev. B 79, 195414 (2009).

1851 (1)

J. J. Loomis and H. J. Maris, “Theory of heat transfer by evanescent electromagnetic waves,” Phys. Rev. B 50, 18517 (1994).

Archambault, A.

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Quantum theory of spontaneous and stimulated emission of surface plasmons,” Phys. Rev. B 79, 195414 (2009).

Arriaga, J.

A. A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystal optics and homogenization of 2D periodic composites,” Phys. Rev. Lett. 82, 719 (1999).
[CrossRef]

Ben-Abdallah, P.

P. Ben-Abdallah and K. Joulain, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 82, 121419 (2010).
[CrossRef]

Biehs, S.A.

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

Biehs, S.-A.

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

Bimonte, G.

G. Bimonte, “Scattering approach to Casimir forces and radiative heat transfer for nanostructured surfaces out of thermal equilibrium,” Phys. Rev. A 80, 042102 (2009).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).

Boltasseva, A.

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

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, 59 (2005).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
[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, 1548 (2000).
[CrossRef] [PubMed]

Chen, G.

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

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[CrossRef]

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008).
[CrossRef]

Chen, X.

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008).
[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,” Nat. Photonics 3, 514 (2009).
[CrossRef]

Choy, H. K. H.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

Chuang, S.-F.

S.-F. Chuang, S. D. Collins, and R. L. Smith, “Porous silicon microstructure as studied by transmission electron microscopy,” Appl. Phys. Lett. 55, 1540 (1989).
[CrossRef]

Collins, S. D.

S.-F. Chuang, S. D. Collins, and R. L. Smith, “Porous silicon microstructure as studied by transmission electron microscopy,” Appl. Phys. Lett. 55, 1540 (1989).
[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,” Nat. Photonics 3, 514 (2009).
[CrossRef]

Cooke, R.

R. Cooke,Classical Algebra (John-Wiley & Sons, 2008).

DiMatteo, R. S.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

Dorofeyev, I. A.

E. A. Vinogradov and I. A. Dorofeyev, “Thermally stimulated electromagnetic fields of solids,” Phys. Usp. 52, 425 (2009).
[CrossRef]

Elser, J.

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Fainman, Y.

L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96, 041112 (2010).
[CrossRef]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhanget, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005).
[CrossRef] [PubMed]

Fauchet, P. M.

J. J. Saarinen, S. M. Weiss, P. M. Fauchet, and J. E. Sipe, “Reflectance analysis of a multilayer one-dimensional porous silicon structure: Theory and experiment,” J. Appl. Phys. 104, 013103 (2008).
[CrossRef]

Feng, L.

L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96, 041112 (2010).
[CrossRef]

Finberg, S. L.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

Fonstad, C. G.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

Fu, C. J.

C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer 49, 1703 (2006).
[CrossRef]

Greffet, J.-J.

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “A mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 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,” Nat. Photonics 3, 514 (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, 59 (2005).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
[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, 1548 (2000).
[CrossRef] [PubMed]

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Quantum theory of spontaneous and stimulated emission of surface plasmons,” Phys. Rev. B 79, 195414 (2009).

Greiff, P.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

Halevi, P.

A. A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystal optics and homogenization of 2D periodic composites,” Phys. Rev. Lett. 82, 719 (1999).
[CrossRef]

Holthaus, M.

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

Hu, L.

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).

Jacob, Z.

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

Joulain, K.

P. Ben-Abdallah and K. Joulain, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 82, 121419 (2010).
[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, 59 (2005).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
[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, 1548 (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,” Nat. Photonics 3, 514 (2009).
[CrossRef]

Kim, J.-Y.

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

Kittel, A.

U. F. Wischnath, J. Welker, M. Munzel, and A. Kittel, “Near-field scanning thermal microscope,” Rev. Sci. Instrum. 79, 073708 (2008).
[CrossRef] [PubMed]

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

Krokhin, A. A.

A. A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystal optics and homogenization of 2D periodic composites,” Phys. Rev. Lett. 82, 719 (1999).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhanget, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005).
[CrossRef] [PubMed]

Leonhardt, U.

T. G. Philbin and U. Leonhardt, “Alternative calculation of the Casimir forces between birefringent plates,” Phys. Rev. A 78, 042107 (2008).
[CrossRef]

Levin, M. L.

M. L. Levin, V. G. Polevoi, and S. M. Rytov, “Theory of heat-transfer due to a fluctuation electromagnetic field,” Sov. Phys. JETP 50, 1054 (1980).

Liscidini, M.

M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010).
[CrossRef]

Liu, Z.

L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96, 041112 (2010).
[CrossRef]

Lomakin, V.

L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96, 041112 (2010).
[CrossRef]

Loomis, J. J.

J. J. Loomis and H. J. Maris, “Theory of heat transfer by evanescent electromagnetic waves,” Phys. Rev. B 50, 18517 (1994).

Maris, H. J.

J. J. Loomis and H. J. Maris, “Theory of heat transfer by evanescent electromagnetic waves,” Phys. Rev. B 50, 18517 (1994).

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, 59 (2005).
[CrossRef]

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Quantum theory of spontaneous and stimulated emission of surface plasmons,” Phys. Rev. B 79, 195414 (2009).

Masaki, M. M.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[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, 59 (2005).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
[CrossRef]

Müller-Hirsch, W.

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

Munzel, M.

U. F. Wischnath, J. Welker, M. Munzel, and A. Kittel, “Near-field scanning thermal microscope,” Rev. Sci. Instrum. 79, 073708 (2008).
[CrossRef] [PubMed]

Naik, G. V.

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

Narayanaswamy, A.

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

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[CrossRef]

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008).
[CrossRef]

Narimanov, E. E.

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Parisi, J.

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191 (2004).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

Persson, B. N. J.

A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291 (2007).
[CrossRef]

Philbin, T. G.

T. G. Philbin and U. Leonhardt, “Alternative calculation of the Casimir forces between birefringent plates,” Phys. Rev. A 78, 042107 (2008).
[CrossRef]

Podolskiy, V. A.

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Polder, D.

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

Polevoi, V. G.

M. L. Levin, V. G. Polevoi, and S. M. Rytov, “Theory of heat-transfer due to a fluctuation electromagnetic field,” Sov. Phys. JETP 50, 1054 (1980).

Raether, H.

H. RaetherSurface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Reddig, D.

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

Rousseau, E.

S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “A mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 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,” Nat. Photonics 3, 514 (2009).
[CrossRef]

Rytov, S. M.

M. L. Levin, V. G. Polevoi, and S. M. Rytov, “Theory of heat-transfer due to a fluctuation electromagnetic field,” Sov. Phys. JETP 50, 1054 (1980).

Saarinen, J. J.

J. J. Saarinen, S. M. Weiss, P. M. Fauchet, and J. E. Sipe, “Reflectance analysis of a multilayer one-dimensional porous silicon structure: Theory and experiment,” J. Appl. Phys. 104, 013103 (2008).
[CrossRef]

Shalaev, V. M.

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007).
[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, 1548 (2000).
[CrossRef] [PubMed]

Shen, S.

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

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[CrossRef]

Sipe, J. E.

M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010).
[CrossRef]

J. J. Saarinen, S. M. Weiss, P. M. Fauchet, and J. E. Sipe, “Reflectance analysis of a multilayer one-dimensional porous silicon structure: Theory and experiment,” J. Appl. Phys. 104, 013103 (2008).
[CrossRef]

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481 (1986).
[CrossRef]

Siria, A.

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

Smith, R. L.

S.-F. Chuang, S. D. Collins, and R. L. Smith, “Porous silicon microstructure as studied by transmission electron microscopy,” Appl. Phys. Lett. 55, 1540 (1989).
[CrossRef]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhanget, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005).
[CrossRef] [PubMed]

Teperik, T. V.

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Quantum theory of spontaneous and stimulated emission of surface plasmons,” Phys. Rev. B 79, 195414 (2009).

van Hove, M.

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

Vinogradov, E. A.

E. A. Vinogradov and I. A. Dorofeyev, “Thermally stimulated electromagnetic fields of solids,” Phys. Usp. 52, 425 (2009).
[CrossRef]

Volokitin, A. I.

A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291 (2007).
[CrossRef]

Volz, S.

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

Wangberg, R.

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Weiss, S. M.

J. J. Saarinen, S. M. Weiss, P. M. Fauchet, and J. E. Sipe, “Reflectance analysis of a multilayer one-dimensional porous silicon structure: Theory and experiment,” J. Appl. Phys. 104, 013103 (2008).
[CrossRef]

Welker, J.

U. F. Wischnath, J. Welker, M. Munzel, and A. Kittel, “Near-field scanning thermal microscope,” Rev. Sci. Instrum. 79, 073708 (2008).
[CrossRef] [PubMed]

Wischnath, U. F.

U. F. Wischnath, J. Welker, M. Munzel, and A. Kittel, “Near-field scanning thermal microscope,” Rev. Sci. Instrum. 79, 073708 (2008).
[CrossRef] [PubMed]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

Young-Waithe, K. A.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

Zhang, Z. M.

C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer 49, 1703 (2006).
[CrossRef]

Zhanget, X.

N. Fang, H. Lee, C. Sun, and X. Zhanget, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005).
[CrossRef] [PubMed]

Appl. Phys. B (1)

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215 (2010).
[CrossRef]

Appl. Phys. Lett. (6)

L. Feng, Z. Liu, V. Lomakin, and Y. Fainman, “Form birefringence metal and its plasmonic anisotropy,” Appl. Phys. Lett. 96, 041112 (2010).
[CrossRef]

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

S.-F. Chuang, S. D. Collins, and R. L. Smith, “Porous silicon microstructure as studied by transmission electron microscopy,” Appl. Phys. Lett. 55, 1540 (1989).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931 (2001).
[CrossRef]

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79, 1894 (2001).
[CrossRef]

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding Planck’s blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008).
[CrossRef]

Contemp. Phys. (1)

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191 (2004).
[CrossRef]

Int. J. Heat Mass Transfer (1)

C. J. Fu and Z. M. Zhang, “Nanoscale radiation heat transfer for silicon at different doping levels,” Int. J. Heat Mass Transfer 49, 1703 (2006).
[CrossRef]

J. Appl. Phys. (1)

J. J. Saarinen, S. M. Weiss, P. M. Fauchet, and J. E. Sipe, “Reflectance analysis of a multilayer one-dimensional porous silicon structure: Theory and experiment,” J. Appl. Phys. 104, 013103 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

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

Nat. Photonics (2)

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

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41 (2007).
[CrossRef]

Phys. Rev. A (2)

T. G. Philbin and U. Leonhardt, “Alternative calculation of the Casimir forces between birefringent plates,” Phys. Rev. A 78, 042107 (2008).
[CrossRef]

G. Bimonte, “Scattering approach to Casimir forces and radiative heat transfer for nanostructured surfaces out of thermal equilibrium,” Phys. Rev. A 80, 042102 (2009).
[CrossRef]

Phys. Rev. B (7)

A. A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

M. Liscidini and J. E. Sipe, “Quasiguided surface plasmon excitations in anisotropic materials,” Phys. Rev. B 81, 115335 (2010).
[CrossRef]

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Quantum theory of spontaneous and stimulated emission of surface plasmons,” Phys. Rev. B 79, 195414 (2009).

P. Ben-Abdallah and K. Joulain, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 82, 121419 (2010).
[CrossRef]

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008).
[CrossRef]

J. J. Loomis and H. J. Maris, “Theory of heat transfer by evanescent electromagnetic waves,” Phys. Rev. B 50, 18517 (1994).

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

Phys. Rev. Lett. (5)

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

A. Kittel, W. Müller-Hirsch, J. Parisi, S.A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95, 224301 (2005).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[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, 1548 (2000).
[CrossRef] [PubMed]

P. Halevi, A. A. Krokhin, and J. Arriaga, “Photonic crystal optics and homogenization of 2D periodic composites,” Phys. Rev. Lett. 82, 719 (1999).
[CrossRef]

Phys. Usp. (1)

E. A. Vinogradov and I. A. Dorofeyev, “Thermally stimulated electromagnetic fields of solids,” Phys. Usp. 52, 425 (2009).
[CrossRef]

Rev. Mod. Phys. (1)

A. I. Volokitin and B. N. J. Persson, “Near-field radiative heat transfer and noncontact friction,” Rev. Mod. Phys. 79, 1291 (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

U. F. Wischnath, J. Welker, M. Munzel, and A. Kittel, “Near-field scanning thermal microscope,” Rev. Sci. Instrum. 79, 073708 (2008).
[CrossRef] [PubMed]

Science (1)

N. Fang, H. Lee, C. Sun, and X. Zhanget, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308, 534 (2005).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

M. L. Levin, V. G. Polevoi, and S. M. Rytov, “Theory of heat-transfer due to a fluctuation electromagnetic field,” Sov. Phys. JETP 50, 1054 (1980).

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, 59 (2005).
[CrossRef]

Other (5)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

The other branch is connected with the so called Brewster modes [34], that are propagating waves for which rp,p vanishes.

R. Cooke,Classical Algebra (John-Wiley & Sons, 2008).

H. RaetherSurface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).

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Figures (10)

Fig. 1
Fig. 1

Sketch of two porous slabs with different temperatures separated by a vacuum gap.

Fig. 2
Fig. 2

Plot of the dispersion curves (white dashed lines) from Eq. (22) in the (ω, κ) plane for filling factors (a) f = 0.1, (b) f = 0.3, and (c) f = 0.5. The white dash-dotted line represents the light line in vacuum (ω = κ c). Furthermore the dark (blue) areas mark the region for which γ p is purely real, whereas the bright (red) areas are the regions for which γ p is purely imaginary.

Fig. 3
Fig. 3

Plot of ln(1/|r p,p|2) in (ω, κ) plane for (a) f = 0.1, (b) f = 0.3, and (c) f = 0.5.

Fig. 4
Fig. 4

Heat flux between two SiC plates over distance with T 1 = 300K and T 2 = 0K. The flux is normalized to the value for two black bodies S BB = 459.6Wm–2. The contribution of the s- and p-polarized part are shown as well.

Fig. 5
Fig. 5

Heat flux between two porous SiC plates over distance with T 1 = 300K and T 2 = 0K. The flux is normalized to the value for two SiC plates shown in Fig. 4.

Fig. 6
Fig. 6

As in Fig 5 but for the (a) s- and (b) p-polarized contribution only.

Fig. 7
Fig. 7

Transmission coefficient T p(ω, κ; d) in the (ω, κ)-plane for two porous SiC slabs with different filling factors (a) f = 0, (b) f = 0.1, (c) f = 0.3, and (d) f = 0.5. The distance is fixed at d = 100nm.

Fig. 8
Fig. 8

Spectral mean Poynting vector 〈Sω 〉 defined in Eq. (7) for two porous SiC slabs with different filling factors f = 0, 0.1, 0.3, 0.5 considering only the p-polarized contribution. The distance is fixed at d = 100nm.

Fig. 9
Fig. 9

Mean transmission coefficient defined in Eq. (24) for different filling factors normalized to the isotropic case (f = 0). The distance is fixed at d = 100nm and the temperature at T = 300K.

Fig. 10
Fig. 10

Plot of the normalized cutoff value κ uni/κ iso over filling factor f.

Equations (48)

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

P ( T 1 , T 2 , a ) = A 12 dA S ,
P ( T 1 , T 2 , a ) = A S z ,
S Z = 0 d ω 2 π [ Θ ( ω , T 1 ) Θ ( ω , T 2 ) ] S ω ,
Θ ( ω , T ) = h ¯ ω e h ¯ ω k B T 1 ,
S ω = 2 Re Tr [ A d r | | ( 𝔾 ( r , r ) z z 𝔾 ( r , r ) z 𝔾 ( r , r ) z 𝔾 ( r , r ) ) ] z = z = z 0 .
[ × × ω 2 c 2 ε ( r , ω ) ] 𝔾 ( r , r , ω ) = δ ( r r ) 𝕀 .
S ω = d 2 κ ( 2 π ) 2 T ( ω , κ ; d ) .
T ( ω , κ ; d ) = { Tr [ ( 𝟙 2 2 ) 𝔻 12 ( 𝟙 1 1 ) 𝔻 12 ] , κ < ω / c Tr [ ( 2 2 ) 𝔻 12 ( 1 1 ) 𝔻 12 ] e 2 | γ r | d , κ > ω / c
i = [ r i s , s ( ω , κ ) r i s , p ( ω , κ ) r i p , s ( ω , κ ) r i p , p ( ω , κ ) ] ,
r i s , s ( ω , κ ) = γ r ε i ( ω ) ω 2 / c 2 κ 2 γ r + ε i ( ω ) ω 2 / c 2 κ 2 , r i p , p ( ω , κ ) = ε i ( ω ) γ r ε i ( ω ) ω 2 / c 2 κ 2 ε i ( ω ) γ r + ε i ( ω ) ω 2 / c 2 κ 2 , r i s , p ( ω , κ ) = r i p , s ( ω , κ ) = 0 ,
𝔻 12 = ( 𝟙 1 2 e 2 i γ r d ) 1 ,
ε = ε | | [ e x e x + e y e y ] + ε e z e z
ε || = ε h ε i ( 1 + f ) + ε h ( 1 f ) ε i ( 1 f ) + ε h ( 1 + f ) ,
ε = ε h ( 1 f ) + ε i f ,
r s , s ( ω , κ ) = γ r γ s γ r + γ s ,
r p , p ( ω , κ ) = ε || γ r γ p ε || γ r + γ p ,
r s , p = r p , s = 0 ,
γ s = ε || ω 2 / c 2 κ 2 ,
γ p = ε || ω 2 / c 2 ε || ε κ 2 ,
( γ r + γ s ) = 0 ,
( ε || γ r + γ p ) = 0.
κ = ω c ε ( ε || 1 ) ε || ε 1 .
ε h = ε ω 2 ω L 2 i ω Γ ω 2 ω T 2 i ω Γ
T ¯ p ( κ ) = 3 π 2 0 d u f ( u ) T p ( u , κ ; d )
S z = π 2 3 k B 2 T h d κ 2 π κ T ¯ p ( κ ) Δ T .
κ iso > log ( 2 Im ( ε ) ) 1 2 d
κ uni > log ( 2 Im ( ε || ε ) ) 1 2 d
𝔾 ( r , r ) = d 2 κ ( 2 π ) 2 ie i κ ( x x ) 2 γ r e i γ r ( z z ) 𝟙
𝟙 = a ^ s + a ^ s + + a ^ p + a ^ p + .
a ^ s + = 1 κ ( k y k x 0 ) and a ^ p + = c κ ω ( k x γ r k y γ r κ 2 ) .
𝔾 ( ( r , r ) = d 2 κ ( 2 π ) 2 𝔾 ( κ ; z , z ) e i κ ( ( x x ) .
𝔾 A ( κ ; z , z ) = i 2 γ r [ 𝟙 e i γ r ( z z ) + e 2 i γ r d e i γ r ( z + z ) 2 ]
2 = i , j = { s , p } r i , j 2 a ^ i a ^ j +
1 = i , j = { s , p } r i , j 1 a ^ i + a ^ j
𝔾 A ( κ ; z , z ) = i 2 γ r [ 𝟙 e i γ r ( z , z ) + e 2 i γ r d e i γ r ( z + z ) 2 + e 2 i γ r d e i γ r ( z z ) 1 2 + e 4 i γ r d e i γ r ( z + z ) 2 1 2 + ] .
𝔾 A ( κ ; z , z ) = i 2 γ r [ 𝔻 12 e i γ r ( z z ) + 𝔻 21 2 e 2 i γ r d e i γ r ( z + z ) ]
𝔻 12 = ( 𝟙 1 2 e 2 i γ r d ) 1 ,
𝔻 21 = ( 𝟙 2 1 e 2 i γ r d ) 1 .
𝔾 B ( κ ; z , z ) = i 2 γ r [ 𝔻 12 1 e i γ r ( z + z ) + 𝔻 21 2 1 e 2 i γ r d e i γ r ( z z ) ]
𝔾 intra = i 2 γ r [ 𝔻 12 ( 𝟙 e i γ r ( z z ) + 1 e i γ r ( z + z ) ) + 𝔻 21 ( 2 1 e i γ r ( z z ) e 2 i γ r d + 2 e 2 i γ r d e i γ r ( z + z ) ) ]
ε || ε = 1.
a ε h 3 + b ε h 2 + c ε h + d = 0 ,
a = ( 1 f ) 2 ,
b = ε i ( 1 f ) ( 2 f + 1 ) ,
c = ( 1 + f ) ( ε i 2 f 1 ) ,
d = ε i ( f 1 ) .
ε h , n = 2 p 3 cos [ 1 3 arccos ( q 2 27 p 3 ) + 2 π n 3 ]
p = 3 a c b 2 3 a 2 and q = 27 a 2 d 9 a b c + 2 b 3 27 a 3

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