Abstract

Microscopic thin films have shown wavelength selectivity in the context of radiative heat transfer. We propose a methodology to shift the wavelength selectivity in the desired location. This work deals with the far-field and near-field radiation from thin films embedded with nanoparticles. The calculations of emission spectra are performed using the Fresnel equations in the far-field limit, and using the dyadic Green’s function formalism for transmissivity between the closely spaced objects in the near-field regime. For the media doped with nanoparticles, an effective dielectric function using the Maxwell-Garnett-Mie theory is used to calculate emissivity and radiative heat transfer. It has been shown that the wavelength selectivity in the emission spectra can be influenced by varying the size and/or the volume fraction of nanoparticles. We characterize the wavelength selectivity using real and imaginary parts of the effective refractive index. We show that the influence of nanoparticles on wavelength selectivity is different depending on whether the particles are of polar materials or are metallic.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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  15. H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
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    [Crossref]
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2015 (1)

Y. Zheng and A. Ghanekar, “Radiative energy and momentum transfer for various spherical shapes: a single sphere, a bubble, a spherical shell and a coated sphere,” J. Appl. Phys. 117, 064314 (2015).
[Crossref]

2014 (6)

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
[Crossref]

A. Narayanaswamy, J. Mayo, and C. Canetta, “Infrared selective emitters with thin films of polar materials,” Appl. Phys. Lett. 104, 183107 (2014).
[Crossref]

Y. Zheng and A. Narayanaswamy, “Patch contribution to near-field radiative energy transfer and van der Waals pressure between two half-spaces,” Phys. Rev. A 89, 022512 (2014).
[Crossref]

A. Narayanaswamy and Y. Zheng, “A Green’s function formalism of energy and momentum transfer in fluctuational electrodynamics,” J. Quant. Spectrosc. Radiat. Transf. 132, 12–21 (2014).
[Crossref]

X. Liu, T. Bright, and Z. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transf. 136, 092703 (2014).
[Crossref]

2013 (2)

B. Liu and S. Shen, “Broadband near-field radiative thermal emitter/absorber based on hyperbolic metamaterials: Direct numerical simulation by the wiener chaos expansion method,” Phys. Rev. B. 87, 115403 (2013).
[Crossref]

S. J. Petersen, S. Basu, B. Raeymaekers, and M. Francoeur, “Tuning near-field thermal radiative properties by quantifying sensitivity of mie resonance-based metamaterial design parameters,” J. Quant. Spectrosco. Radiat. Transf. 129, 277–286 (2013).
[Crossref]

2012 (1)

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
[Crossref] [PubMed]

2011 (2)

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

M. Francoeur, S. Basu, and S. J. Petersen, “Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles,” Opt. Express. 19, 18774–18788 (2011).
[Crossref] [PubMed]

2010 (3)

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[Crossref]

Y. He and T. Zeng, “First-principles study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C. 114, 18023–18030 (2010).
[Crossref]

2008 (3)

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

H. Wei and H. Eilers, “Electrical conductivity of thin-film composites containing silver nanoparticles embedded in a dielectric fluoropolymer matrix,” Thin Solid Films 517, 575–581 (2008).
[Crossref]

C. McDonagh, C. S. Burke, and B. D. MacCraith, “Optical chemical sensors,” Chem. Rev. 108, 400–422 (2008).
[Crossref] [PubMed]

2007 (2)

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

C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
[Crossref]

2004 (1)

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70, 125101 (2004).
[Crossref]

2003 (1)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B. 107, 668–677 (2003).
[Crossref]

2002 (2)

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

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

2001 (1)

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

1995 (1)

M. Moharam, D. A. Pommet, E. B. Grann, and T. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A. 12, 1077–1086 (1995).
[Crossref]

1993 (1)

H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B. 48, 18178 (1993).
[Crossref]

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B. 39, 9852 (1989).
[Crossref]

1972 (1)

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B. 6, 4370 (1972).
[Crossref]

1906 (1)

J. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. ii,” Phil. Trans. Royal Soc. London Ser. A. 203, 237–288 (1906).
[Crossref]

Akiyama, Y.

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

Baneshi, M.

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
[Crossref]

Basu, S.

S. J. Petersen, S. Basu, B. Raeymaekers, and M. Francoeur, “Tuning near-field thermal radiative properties by quantifying sensitivity of mie resonance-based metamaterial design parameters,” J. Quant. Spectrosco. Radiat. Transf. 129, 277–286 (2013).
[Crossref]

M. Francoeur, S. Basu, and S. J. Petersen, “Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles,” Opt. Express. 19, 18774–18788 (2011).
[Crossref] [PubMed]

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

Bermel, P.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

Biswas, R.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Bright, T.

X. Liu, T. Bright, and Z. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transf. 136, 092703 (2014).
[Crossref]

Burke, C. S.

C. McDonagh, C. S. Burke, and B. D. MacCraith, “Optical chemical sensors,” Chem. Rev. 108, 400–422 (2008).
[Crossref] [PubMed]

Canetta, C.

A. Narayanaswamy, J. Mayo, and C. Canetta, “Infrared selective emitters with thin films of polar materials,” Appl. Phys. Lett. 104, 183107 (2014).
[Crossref]

Carminati, R.

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

Carole, D.

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Celanovic, I.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

Chen, G.

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70, 125101 (2004).
[Crossref]

Chen, L.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[Crossref]

Chen, Y.-B.

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

Chew, W. C.

W. C. Chew, Waves and fFelds in Inhomogeneous Media (IEEE, 1995).

Chiu, H.-T.

C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
[Crossref]

Choi, D.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Christy, R.-W.

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B. 6, 4370 (1972).
[Crossref]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B. 107, 668–677 (2003).
[Crossref]

Daly, J.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

de Abajo, F. J. G.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B. 39, 9852 (1989).
[Crossref]

Eilers, H.

H. Wei and H. Eilers, “Electrical conductivity of thin-film composites containing silver nanoparticles embedded in a dielectric fluoropolymer matrix,” Thin Solid Films 517, 575–581 (2008).
[Crossref]

El-Kady, I.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Fan, S.

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
[Crossref] [PubMed]

Ferro, G.

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Fietz, C.

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Francoeur, M.

S. J. Petersen, S. Basu, B. Raeymaekers, and M. Francoeur, “Tuning near-field thermal radiative properties by quantifying sensitivity of mie resonance-based metamaterial design parameters,” J. Quant. Spectrosco. Radiat. Transf. 129, 277–286 (2013).
[Crossref]

M. Francoeur, S. Basu, and S. J. Petersen, “Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles,” Opt. Express. 19, 18774–18788 (2011).
[Crossref] [PubMed]

Fritz, S.

H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B. 48, 18178 (1993).
[Crossref]

Funston, A. M.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Garnett, J. M.

J. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions. ii,” Phil. Trans. Royal Soc. London Ser. A. 203, 237–288 (1906).
[Crossref]

Gaylord, T.

M. Moharam, D. A. Pommet, E. B. Grann, and T. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A. 12, 1077–1086 (1995).
[Crossref]

George, T.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Ghanekar, A.

Y. Zheng and A. Ghanekar, “Radiative energy and momentum transfer for various spherical shapes: a single sphere, a bubble, a spherical shell and a coated sphere,” J. Appl. Phys. 117, 064314 (2015).
[Crossref]

Ghebrebrhan, M.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

Gonome, H.

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
[Crossref]

Grann, E. B.

M. Moharam, D. A. Pommet, E. B. Grann, and T. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A. 12, 1077–1086 (1995).
[Crossref]

Greenwald, A.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Greffet, J.-J.

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

Guha, B.

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
[Crossref] [PubMed]

Halterman, K.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[Crossref]

Hane, K.

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

Harradon, M.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

He, Y.

Y. He and T. Zeng, “First-principles study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C. 114, 18023–18030 (2010).
[Crossref]

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H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B. 48, 18178 (1993).
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H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B. 48, 18178 (1993).
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Joannopoulos, J. D.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
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M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
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J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6, 209–222 (2002).
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Kanamori, Y.

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B. 107, 668–677 (2003).
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H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
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Korobkin, D.

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Kreibig, U.

H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B. 48, 18178 (1993).
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U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, vol. 25 (Springer, 1995).
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C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
[Crossref]

Lipson, M.

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
[Crossref] [PubMed]

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B. Liu and S. Shen, “Broadband near-field radiative thermal emitter/absorber based on hyperbolic metamaterials: Direct numerical simulation by the wiener chaos expansion method,” Phys. Rev. B. 87, 115403 (2013).
[Crossref]

Liu, X.

X. Liu, T. Bright, and Z. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transf. 136, 092703 (2014).
[Crossref]

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V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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C. McDonagh, C. S. Burke, and B. D. MacCraith, “Optical chemical sensors,” Chem. Rev. 108, 400–422 (2008).
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H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
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A. Narayanaswamy, J. Mayo, and C. Canetta, “Infrared selective emitters with thin films of polar materials,” Appl. Phys. Lett. 104, 183107 (2014).
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C. McDonagh, C. S. Burke, and B. D. MacCraith, “Optical chemical sensors,” Chem. Rev. 108, 400–422 (2008).
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M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
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M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

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M. Moharam, D. A. Pommet, E. B. Grann, and T. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A. 12, 1077–1086 (1995).
[Crossref]

Mulet, J.-P.

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

Mulvaney, P.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Narayanaswamy, A.

A. Narayanaswamy and Y. Zheng, “A Green’s function formalism of energy and momentum transfer in fluctuational electrodynamics,” J. Quant. Spectrosc. Radiat. Transf. 132, 12–21 (2014).
[Crossref]

A. Narayanaswamy, J. Mayo, and C. Canetta, “Infrared selective emitters with thin films of polar materials,” Appl. Phys. Lett. 104, 183107 (2014).
[Crossref]

Y. Zheng and A. Narayanaswamy, “Patch contribution to near-field radiative energy transfer and van der Waals pressure between two half-spaces,” Phys. Rev. A 89, 022512 (2014).
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A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70, 125101 (2004).
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B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Nguyen, V. C.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[Crossref]

Novo, C.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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Okajima, J.

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

Otey, C.

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
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Pastoriza-Santos, I.

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
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Petersen, S. J.

S. J. Petersen, S. Basu, B. Raeymaekers, and M. Francoeur, “Tuning near-field thermal radiative properties by quantifying sensitivity of mie resonance-based metamaterial design parameters,” J. Quant. Spectrosco. Radiat. Transf. 129, 277–286 (2013).
[Crossref]

M. Francoeur, S. Basu, and S. J. Petersen, “Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles,” Opt. Express. 19, 18774–18788 (2011).
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M. Planck, The Theory of Heat Radiation(Dover Publications, 2011).

Poitras, C. B.

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
[Crossref] [PubMed]

Pommet, D. A.

M. Moharam, D. A. Pommet, E. B. Grann, and T. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A. 12, 1077–1086 (1995).
[Crossref]

Pralle, M.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Puscasu, I.

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

Raeymaekers, B.

S. J. Petersen, S. Basu, B. Raeymaekers, and M. Francoeur, “Tuning near-field thermal radiative properties by quantifying sensitivity of mie resonance-based metamaterial design parameters,” J. Quant. Spectrosco. Radiat. Transf. 129, 277–286 (2013).
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V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

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H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

Schatz, G. C.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B. 107, 668–677 (2003).
[Crossref]

Shen, S.

B. Liu and S. Shen, “Broadband near-field radiative thermal emitter/absorber based on hyperbolic metamaterials: Direct numerical simulation by the wiener chaos expansion method,” Phys. Rev. B. 87, 115403 (2013).
[Crossref]

Shen, W.-Y.

C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
[Crossref]

Sheng, P.-S.

C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
[Crossref]

Shvets, G.

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Soljacic, M.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

Vollmer, M.

H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, “Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping,” Phys. Rev. B. 48, 18178 (1993).
[Crossref]

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, vol. 25 (Springer, 1995).
[Crossref]

Wang, C.-H.

C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
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H. Wei and H. Eilers, “Electrical conductivity of thin-film composites containing silver nanoparticles embedded in a dielectric fluoropolymer matrix,” Thin Solid Films 517, 575–581 (2008).
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M. S. Wheeler, “A scattering-based approach to the design, analysis, and experimental verification of magnetic metamaterials made from dielectrics,” Ph.D. thesis (2010).

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H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

Yeng, Y. X.

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

Yugami, H.

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

Zeng, T.

Y. He and T. Zeng, “First-principles study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C. 114, 18023–18030 (2010).
[Crossref]

Zhang, Z.

X. Liu, T. Bright, and Z. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transf. 136, 092703 (2014).
[Crossref]

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

Zhao, L. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B. 107, 668–677 (2003).
[Crossref]

Zheng, Y.

Y. Zheng and A. Ghanekar, “Radiative energy and momentum transfer for various spherical shapes: a single sphere, a bubble, a spherical shell and a coated sphere,” J. Appl. Phys. 117, 064314 (2015).
[Crossref]

Y. Zheng and A. Narayanaswamy, “Patch contribution to near-field radiative energy transfer and van der Waals pressure between two half-spaces,” Phys. Rev. A 89, 022512 (2014).
[Crossref]

A. Narayanaswamy and Y. Zheng, “A Green’s function formalism of energy and momentum transfer in fluctuational electrodynamics,” J. Quant. Spectrosc. Radiat. Transf. 132, 12–21 (2014).
[Crossref]

Appl. Phys. Lett. (2)

M. Pralle, N. Moelders, M. McNeal, I. Puscasu, A. Greenwald, J. Daly, E. Johnson, T. George, D. Choi, I. El-Kady, and R. Biswas, “Photonic crystal enhanced narrow-band infrared emitters,” Appl. Phys. Lett. 81, 4685–4687 (2002).
[Crossref]

A. Narayanaswamy, J. Mayo, and C. Canetta, “Infrared selective emitters with thin films of polar materials,” Appl. Phys. Lett. 104, 183107 (2014).
[Crossref]

Chem. Mater. (1)

C.-H. Wang, W.-Y. Shen, P.-S. Sheng, C.-Y. Lee, and H.-T. Chiu, “Deposition of mesoporous silicon carbide thin films from (me3si) 4sn: tin nanoparticles as in situ generated templates,” Chem. Mater. 19, 5250–5255 (2007).
[Crossref]

Chem. Rev. (1)

C. McDonagh, C. S. Burke, and B. D. MacCraith, “Optical chemical sensors,” Chem. Rev. 108, 400–422 (2008).
[Crossref] [PubMed]

Chem. Soc. Rev. (1)

V. Myroshnychenko, J. Rodríguez-Fernández, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzán, and F. J. G. de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[Crossref] [PubMed]

Int. J. Energy Res. (1)

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

J. Appl. Phys. (1)

Y. Zheng and A. Ghanekar, “Radiative energy and momentum transfer for various spherical shapes: a single sphere, a bubble, a spherical shell and a coated sphere,” J. Appl. Phys. 117, 064314 (2015).
[Crossref]

J. Heat Transf. (1)

X. Liu, T. Bright, and Z. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transf. 136, 092703 (2014).
[Crossref]

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

M. Moharam, D. A. Pommet, E. B. Grann, and T. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A. 12, 1077–1086 (1995).
[Crossref]

H. Sai, H. Yugami, Y. Akiyama, Y. Kanamori, and K. Hane, “Spectral control of thermal emission by periodic microstructured surfaces in the near-infrared region,” J. Opt. Soc. Am. A. 18, 1471–1476 (2001).
[Crossref]

J. Phys. Chem. B. (1)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B. 107, 668–677 (2003).
[Crossref]

J. Phys. Chem. C. (2)

B. Neuner, D. Korobkin, C. Fietz, D. Carole, G. Ferro, and G. Shvets, “Midinfrared Index Sensing of pL-Scale Analytes Based on Surface Phonon Polaritons in Silicon Carbide,” J. Phys. Chem. C. 114, 7489–7491 (2010).
[Crossref]

Y. He and T. Zeng, “First-principles study and model of dielectric functions of silver nanoparticles,” J. Phys. Chem. C. 114, 18023–18030 (2010).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

A. Narayanaswamy and Y. Zheng, “A Green’s function formalism of energy and momentum transfer in fluctuational electrodynamics,” J. Quant. Spectrosc. Radiat. Transf. 132, 12–21 (2014).
[Crossref]

J. Quant. Spectrosco. Radiat. Transf. (3)

S. J. Petersen, S. Basu, B. Raeymaekers, and M. Francoeur, “Tuning near-field thermal radiative properties by quantifying sensitivity of mie resonance-based metamaterial design parameters,” J. Quant. Spectrosco. Radiat. Transf. 129, 277–286 (2013).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, N. Yamada, and S. Maruyama, “Control of thermal barrier performance by optimized nanoparticle size and experimental evaluation using a solar simulator,” J. Quant. Spectrosco. Radiat. Transf. 149, 81–89 (2014).
[Crossref]

H. Gonome, M. Baneshi, J. Okajima, A. Komiya, and S. Maruyama, “Controlling the radiative properties of cool black-color coatings pigmented with cuo submicron particles,” J. Quant. Spectrosco. Radiat. Transf. 132, 90–98 (2014).
[Crossref]

Microscale Thermophys. Eng. (1)

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

Nano Lett. (1)

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-Field radiative cooling of nanostructures,” Nano Lett. 12, 4546–4550 (2012).
[Crossref] [PubMed]

Nanoscale Res. Lett. (1)

P. Bermel, M. Ghebrebrhan, M. Harradon, Y. X. Yeng, I. Celanovic, J. D. Joannopoulos, and M. Soljacic, “Tailoring photonic metamaterial resonances for thermal radiation,” Nanoscale Res. Lett. 6, 1–5 (2011).
[Crossref]

Opt. Express. (1)

M. Francoeur, S. Basu, and S. J. Petersen, “Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles,” Opt. Express. 19, 18774–18788 (2011).
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Phil. Trans. Royal Soc. London Ser. A. (1)

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Figures (6)

Fig. 1
Fig. 1 Configurations of thin-film structures embedded with nanoparticles. (a) A thin film of SiC (or Polystyrene) on top of gold film of 1 µm placed on a substrate, and (b) two multi-layered half-spaces in near-field. The top film will be mixed with nanoparticles of radius r and volume fraction f.
Fig. 2
Fig. 2 Hemispherical emissivity spectra for SiC or polystyrene (PS) thin film of thickness 0.4 µm mixed with BN and Au nanoparticles of 25 nm radius and different volume fractions (a) SiC film mixed with BN nanoparticles of various volume fractions, (b) SiC film mixed with Au nanoparticles of various volume fractions, (c) Polystyrene film mixed with BN nanoparticles of various volume fraction, and (d) Polystyrene film mixed with Au nanoparticles of various volume fractions. All spectral ranges begin at 0.35 µm.
Fig. 3
Fig. 3 Refractive index characteristics of SiC doped with BN and Au nanoparticles: (a) Real part of refractive index n, and (b) imaginary part of refractive index κ, for SiC and SiC doped with 30% BN or Au nanoparticles.
Fig. 4
Fig. 4 Refractive index characteristics of Polystyrene doped with BN and Au nanoparticles: (a) Real part of refractive index n, and (b) imaginary part of refractive index κ, for PS and PS doped with 30% BN or Au nanoparticles.
Fig. 5
Fig. 5 Hemispherical emissivity spectra for SiC or polystyrene (PS) thin film of thickness 0.4 µm mixed with BN and Au nanoparticles of volume fraction 10% and different radii (a) SiC film doped with BN nanoparticles and (b) Polystyrene thin film embedded with Au nanoparticles.
Fig. 6
Fig. 6 Near-field radiative heat flux between multilayered structures at 300 K and 301 K mixed with nanoparticles of radius 25 nm for different volume fractions, and the normalized spectral heat flux is displayed in the inset for the same configuration at a distance of 100 nm- each half space has nanoparticle-embedded thin layer of 0.4 µm on the top deposited on Au layer of 1 µm placed on substrate. Volume fraction of nanoparticles is varied. (a) SiC film doped with BN nanoparticles (b) SiC film doped with Au nanoparticles (c) polystyrene (PS) film mixed with BN nanoparticles and (d) PS film mixed with Au nanoparticles.

Equations (11)

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R ˜ i , i + 1 ( μ ) = R i , i + 1 ( μ ) + R ˜ i + 1 , i + 2 ( μ ) e 2 j k i + 1 , z ( d i + 1 d i ) 1 + R i , i + 1 ( μ ) + R ˜ i + 1 , i + 2 ( μ ) e 2 j k i + 1 , z ( d i + 1 d i )
T ˜ 1 , N ( μ ) = i = 1 N 1 e j k i z ( d i d i 1 ) S i , i + 1 ( μ )
e ( ω ) = c 2 ω 2 0 ω / c d k ρ k ρ μ = s , p ( 1 | R ˜ h 1 ( μ ) | 2 | T ˜ h 1 ( μ ) | 2 )
q 1 2 ( T 1 , T 2 , L ) = 0 d ω 2 π [ Θ ( ω , T 1 ) Θ ( ω , T 2 ) ] T 1 2 ( ω )
T 1 2 ( ω ) = 0 ω / c k ρ d k ρ 2 π μ = s , p ( 1 | R ˜ h 1 ( μ ) | 2 ) ( 1 | R ˜ h 2 ( μ ) | 2 ) | 1 R ˜ h 1 ( μ ) R ˜ h 2 ( μ ) e 2 j k h z L | 2 + ω / c k ρ d k ρ 2 π μ = s , p 4 ( R ˜ h 1 ( μ ) ) ( R ˜ h 2 ( μ ) ) e 2 | k h z | L | 1 R ˜ h 1 ( μ ) R ˜ h 2 ( μ ) e 2 | k h z | L | 2
ε e f f = ε m ( r 3 + 2 α r f r 3 α r f )
α r = 3 j c 3 2 ω 2 ε m 3 / 2 a 1 , r
a 1 , r = ε n p Ψ 1 ( x n p ) Ψ 1 ( x m ) ε m Ψ 1 ( x m ) Ψ 1 ( x n p ) ε n p Ψ 1 ( x n p ) ξ 1 ( x m ) ε m ξ 1 ( x m ) Ψ 1 ( x n p )
ε ( ω ) = ε ( ω 2 ω L O 2 + j ω γ ) ( ω 2 ω T O 2 + j ω γ )
ε ( ω , r ) = ε b ( ω ) + ω p 2 ω 2 + j ω γ 0 ω p 2 ω 2 + j ω ( γ 0 + A ν f / r )
ε ( ω ) = 1 + i = 1 i = 4 f i ( w i 2 ω 2 j g i ω )

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