Abstract

Metallodielectric photonic crystals having hyperbolic dispersions are called indefinite materials because of their ability to guide modes with extremely large lateral wavevectors. While this is useful for enhancing near-field radiative heat transfer, it could also give rise to large lateral displacements of the energy pathways. The energy streamlines can be used to depict the flow of electromagnetic energy through a structure when wave propagation does not follow ray optics. We obtain the energy streamlines through two semi-infinite uniaxial anisotropic effective medium structures, separated by a small vacuum gap, using the Green functions and fluctuation-dissipation theorem. The lateral shifts are determined from the streamlines within two penetration depths. For hyperbolic modes, the predicted lateral shift can be several thousand times of the vacuum gap width.

© 2014 Optical Society of America

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    [CrossRef]

2014 (2)

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field radiative heat transfer with doped-silicon nanostructured metamaterials,” Int. J. Heat Mass Transfer 73, 389–398 (2014).
[CrossRef]

S. Lang, M. Tschikin, S.-A. Biehs, A. Yu. Petrov, and M. Eich, “Large penetration depth of near-field heat flux in hyperbolic media,” Appl. Phys. Lett. 104(12), 121903 (2014).
[CrossRef]

2013 (9)

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field thermal radiation between hyperbolic metamaterials: graphite and carbon nanotubes,” Appl. Phys. Lett. 103(21), 213102 (2013).
[CrossRef]

A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express 21(2), 1593–1598 (2013).
[CrossRef] [PubMed]

Y. Guo and Z. Jacob, “Thermal hyperbolic metamaterials,” Opt. Express 21(12), 15014–15019 (2013).
[CrossRef] [PubMed]

K. Park and Z. M. Zhang, “Fundamentals and applications of near-field radiative energy transfer,” Frontiers Heat Mass Transfer 4(1), 013001 (2013).
[CrossRef]

S. Shen, “Experimental studies of radiative heat transfer between bodies at small separations,” Annu. Rev. Heat Transfer 16(1), 327–343 (2013).
[CrossRef]

X. L. Liu and Z. M. Zhang, “Metal-free low-loss negative refraction in the mid-infrared region,” Appl. Phys. Lett. 103(10), 103101 (2013).
[CrossRef]

M. Tschikin, S.-A. Biehs, R. Messina, and P. Ben-Abdallah, “On the limits of the effective description of hyperbolic materials in the presence of surface waves,” J. Opt. 15(10), 105101 (2013).
[CrossRef]

X. L. Liu, L. P. Wang, and Z. M. Zhang, “Wideband tunable omnidirectional infrared absorbers based on doped-silicon nanowire arrays,” J. Heat Transfer 135(6), 061602 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, R. Messina, and P. Ben-Abdallah, “Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials,” Appl. Phys. Lett. 102(13), 131106 (2013).
[CrossRef]

2012 (3)

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[CrossRef]

S.-A. Biehs, M. Tschikin, and P. Ben-Abdallah, “Hyperbolic metamaterials as an analog of a blackbody in the near field,” Phys. Rev. Lett. 109(10), 104301 (2012).
[CrossRef] [PubMed]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

2011 (5)

S. Basu, L. P. Wang, and Z. M. Zhang, “Direct calculation of energy streamlines in near-field thermal radiation,” J. Quant. Spectrosc. Radiat. Transf. 112(7), 1149–1155 (2011).
[CrossRef]

S. Basu and M. Francoeur, “Penetration depth in near-field radiative heat transfer between metamaterials,” Appl. Phys. Lett. 99(14), 143107 (2011).
[CrossRef]

A. Eroglu, Y. H. Lee, and J. K. Lee, “Dyadic Green’s functions for multi-layered uniaxially anisotropic media with arbitrarily oriented optic axes,” IET Microwaves Antennas Propag. 5(15), 1779–1788 (2011).
[CrossRef]

L. P. Wang, S. Basu, and Z. M. Zhang, “Direct and indirect methods for calculating thermal emission from layered structures with nonuniform temperatures,” J. Heat Transfer 133(7), 072701 (2011).
[CrossRef]

S.-A. Biehs, P. Ben-Abdallah, F. S. S. Rosa, K. Joulain, and J.-J. Greffet, “Nanoscale heat flux between nanoporous materials,” Opt. Express 19(S5Suppl 5), A1088–A1103 (2011).
[CrossRef] [PubMed]

2010 (1)

S. Basu, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of heavily doped silicon at room temperature,” J. Heat Transfer 132(2), 023301 (2010).
[CrossRef]

2009 (5)

O. A. Godin, “Wave refraction at an interface: Snell’s law versus Chapman’s law,” J. Acoust. Soc. Am. 125(4), EL117–EL122 (2009).
[CrossRef] [PubMed]

M. Francoeur, M. Pinar Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s functions and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[CrossRef]

S. Basu and Z. M. Zhang, “Ultrasmall penetration depth in nanoscale thermal radiation,” Appl. Phys. Lett. 95(13), 133104 (2009).
[CrossRef]

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79(24), 245127 (2009).
[CrossRef]

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[CrossRef]

2008 (5)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

B. J. Lee and Z. M. Zhang, “Lateral shifts in near-field thermal radiation with surface phonon polaritons,” Nanoscale Microscale Thermophys. Eng. 12(3), 238–250 (2008).
[CrossRef]

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[CrossRef]

D. M. F. Chapman, “Using streamlines to visualize acoustic energy flow across boundaries,” J. Acoust. Soc. Am. 124(1), 48–56 (2008).
[CrossRef] [PubMed]

2007 (2)

B. J. Lee, K. Park, and Z. M. Zhang, “Energy pathways in nanoscale thermal radiation,” Appl. Phys. Lett. 91(15), 153101 (2007).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

2006 (4)

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046618 (2006).
[CrossRef] [PubMed]

H. Shin and S. Fan, “All-angle negative refraction and evanescent wave amplification using onedimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89(15), 151102 (2006).
[CrossRef]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23(3), 498–505 (2006).
[CrossRef]

Z. M. Zhang and B. J. Lee, “Lateral shift in photon tunneling studied by the energy streamline method,” Opt. Express 14(21), 9963–9970 (2006).
[CrossRef] [PubMed]

2005 (3)

M. V. Bashevoy, V. A. Fedotov, and N. I. Zheludev, “Optical whirlpool on an absorbing metallic nanoparticle,” Opt. Express 13(21), 8372–8379 (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(3-4), 59–112 (2005).
[CrossRef]

A. Narayanaswamy and G. Chen, “Direct computation of thermal emission from nanostructures,” Annu. Rev. Heat Transfer 14(14), 169–195 (2005).
[CrossRef]

2004 (2)

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B Quantum Semiclassical Opt. 6(5), S404–S409 (2004).
[CrossRef]

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

2003 (1)

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50(9), 1419–1430 (2003).
[CrossRef]

1992 (1)

S. Barkeshli, “On the electromagnetic dyadic Green’s functions for planar multi-layered anistropic uniaxial material media,” Int. J. Infrared Millim. Waves 13(4), 507–527 (1992).
[CrossRef]

1987 (1)

1983 (1)

J. Lee and J. Kong, “Dyadic Green’s functions for layered anisotropic medium,” Electromagnetics 3(2), 111–130 (1983).
[CrossRef]

Akozbek, N.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Barkeshli, S.

S. Barkeshli, “On the electromagnetic dyadic Green’s functions for planar multi-layered anistropic uniaxial material media,” Int. J. Infrared Millim. Waves 13(4), 507–527 (1992).
[CrossRef]

Barnakov, Y. A.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[CrossRef]

Bartal, G.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Bashevoy, M. V.

Basu, S.

L. P. Wang, S. Basu, and Z. M. Zhang, “Direct and indirect methods for calculating thermal emission from layered structures with nonuniform temperatures,” J. Heat Transfer 133(7), 072701 (2011).
[CrossRef]

S. Basu, L. P. Wang, and Z. M. Zhang, “Direct calculation of energy streamlines in near-field thermal radiation,” J. Quant. Spectrosc. Radiat. Transf. 112(7), 1149–1155 (2011).
[CrossRef]

S. Basu and M. Francoeur, “Penetration depth in near-field radiative heat transfer between metamaterials,” Appl. Phys. Lett. 99(14), 143107 (2011).
[CrossRef]

S. Basu, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of heavily doped silicon at room temperature,” J. Heat Transfer 132(2), 023301 (2010).
[CrossRef]

S. Basu and Z. M. Zhang, “Ultrasmall penetration depth in nanoscale thermal radiation,” Appl. Phys. Lett. 95(13), 133104 (2009).
[CrossRef]

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[CrossRef]

Belov, P.

Ben-Abdallah, P.

M. Tschikin, S.-A. Biehs, R. Messina, and P. Ben-Abdallah, “On the limits of the effective description of hyperbolic materials in the presence of surface waves,” J. Opt. 15(10), 105101 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, R. Messina, and P. Ben-Abdallah, “Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials,” Appl. Phys. Lett. 102(13), 131106 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, and P. Ben-Abdallah, “Hyperbolic metamaterials as an analog of a blackbody in the near field,” Phys. Rev. Lett. 109(10), 104301 (2012).
[CrossRef] [PubMed]

S.-A. Biehs, P. Ben-Abdallah, F. S. S. Rosa, K. Joulain, and J.-J. Greffet, “Nanoscale heat flux between nanoporous materials,” Opt. Express 19(S5Suppl 5), A1088–A1103 (2011).
[CrossRef] [PubMed]

Biehs, S.-A.

S. Lang, M. Tschikin, S.-A. Biehs, A. Yu. Petrov, and M. Eich, “Large penetration depth of near-field heat flux in hyperbolic media,” Appl. Phys. Lett. 104(12), 121903 (2014).
[CrossRef]

M. Tschikin, S.-A. Biehs, R. Messina, and P. Ben-Abdallah, “On the limits of the effective description of hyperbolic materials in the presence of surface waves,” J. Opt. 15(10), 105101 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, R. Messina, and P. Ben-Abdallah, “Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials,” Appl. Phys. Lett. 102(13), 131106 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, and P. Ben-Abdallah, “Hyperbolic metamaterials as an analog of a blackbody in the near field,” Phys. Rev. Lett. 109(10), 104301 (2012).
[CrossRef] [PubMed]

S.-A. Biehs, P. Ben-Abdallah, F. S. S. Rosa, K. Joulain, and J.-J. Greffet, “Nanoscale heat flux between nanoporous materials,” Opt. Express 19(S5Suppl 5), A1088–A1103 (2011).
[CrossRef] [PubMed]

Bloemer, M. J.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Bright, T. J.

X. L. Liu, T. J. Bright, and Z. M. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transfer. in press.

Cappeddu, M. G.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[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(3-4), 59–112 (2005).
[CrossRef]

Centini, M.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Chapman, D. M. F.

D. M. F. Chapman, “Using streamlines to visualize acoustic energy flow across boundaries,” J. Acoust. Soc. Am. 124(1), 48–56 (2008).
[CrossRef] [PubMed]

Chen, G.

A. Narayanaswamy and G. Chen, “Direct computation of thermal emission from nanostructures,” Annu. Rev. Heat Transfer 14(14), 169–195 (2005).
[CrossRef]

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

Cortes, C. L.

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[CrossRef]

D’Orazio, A.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

de Ceglia, D.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Eich, M.

S. Lang, M. Tschikin, S.-A. Biehs, A. Yu. Petrov, and M. Eich, “Large penetration depth of near-field heat flux in hyperbolic media,” Appl. Phys. Lett. 104(12), 121903 (2014).
[CrossRef]

Elser, J.

Eroglu, A.

A. Eroglu, Y. H. Lee, and J. K. Lee, “Dyadic Green’s functions for multi-layered uniaxially anisotropic media with arbitrarily oriented optic axes,” IET Microwaves Antennas Propag. 5(15), 1779–1788 (2011).
[CrossRef]

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction and evanescent wave amplification using onedimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89(15), 151102 (2006).
[CrossRef]

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79(24), 245127 (2009).
[CrossRef]

Fedotov, V. A.

Francoeur, M.

S. Basu and M. Francoeur, “Penetration depth in near-field radiative heat transfer between metamaterials,” Appl. Phys. Lett. 99(14), 143107 (2011).
[CrossRef]

M. Francoeur, M. Pinar Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s functions and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[CrossRef]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Godin, O. A.

O. A. Godin, “Wave refraction at an interface: Snell’s law versus Chapman’s law,” J. Acoust. Soc. Am. 125(4), EL117–EL122 (2009).
[CrossRef] [PubMed]

Greffet, J.-J.

S.-A. Biehs, P. Ben-Abdallah, F. S. S. Rosa, K. Joulain, and J.-J. Greffet, “Nanoscale heat flux between nanoporous materials,” Opt. Express 19(S5Suppl 5), A1088–A1103 (2011).
[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(3-4), 59–112 (2005).
[CrossRef]

Guo, Y.

Y. Guo and Z. Jacob, “Thermal hyperbolic metamaterials,” Opt. Express 21(12), 15014–15019 (2013).
[CrossRef] [PubMed]

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[CrossRef]

Haus, J. W.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Iorsh, I.

Jacob, Z.

Y. Guo and Z. Jacob, “Thermal hyperbolic metamaterials,” Opt. Express 21(12), 15014–15019 (2013).
[CrossRef] [PubMed]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[CrossRef]

Joulain, K.

S.-A. Biehs, P. Ben-Abdallah, F. S. S. Rosa, K. Joulain, and J.-J. Greffet, “Nanoscale heat flux between nanoporous materials,” Opt. Express 19(S5Suppl 5), A1088–A1103 (2011).
[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(3-4), 59–112 (2005).
[CrossRef]

King, W. P.

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[CrossRef]

Kivshar, Y.

Kong, J.

J. Lee and J. Kong, “Dyadic Green’s functions for layered anisotropic medium,” Electromagnetics 3(2), 111–130 (1983).
[CrossRef]

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79(24), 245127 (2009).
[CrossRef]

Kretzschmar, I.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Krishnamoorthy, H. N. S.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Lang, S.

S. Lang, M. Tschikin, S.-A. Biehs, A. Yu. Petrov, and M. Eich, “Large penetration depth of near-field heat flux in hyperbolic media,” Appl. Phys. Lett. 104(12), 121903 (2014).
[CrossRef]

Lee, B. J.

S. Basu, B. J. Lee, and Z. M. Zhang, “Infrared radiative properties of heavily doped silicon at room temperature,” J. Heat Transfer 132(2), 023301 (2010).
[CrossRef]

B. J. Lee and Z. M. Zhang, “Lateral shifts in near-field thermal radiation with surface phonon polaritons,” Nanoscale Microscale Thermophys. Eng. 12(3), 238–250 (2008).
[CrossRef]

B. J. Lee, K. Park, and Z. M. Zhang, “Energy pathways in nanoscale thermal radiation,” Appl. Phys. Lett. 91(15), 153101 (2007).
[CrossRef]

Z. M. Zhang and B. J. Lee, “Lateral shift in photon tunneling studied by the energy streamline method,” Opt. Express 14(21), 9963–9970 (2006).
[CrossRef] [PubMed]

Lee, J.

J. Lee and J. Kong, “Dyadic Green’s functions for layered anisotropic medium,” Electromagnetics 3(2), 111–130 (1983).
[CrossRef]

Lee, J. K.

A. Eroglu, Y. H. Lee, and J. K. Lee, “Dyadic Green’s functions for multi-layered uniaxially anisotropic media with arbitrarily oriented optic axes,” IET Microwaves Antennas Propag. 5(15), 1779–1788 (2011).
[CrossRef]

Lee, Y. H.

A. Eroglu, Y. H. Lee, and J. K. Lee, “Dyadic Green’s functions for multi-layered uniaxially anisotropic media with arbitrarily oriented optic axes,” IET Microwaves Antennas Propag. 5(15), 1779–1788 (2011).
[CrossRef]

Lenstra, D.

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B Quantum Semiclassical Opt. 6(5), S404–S409 (2004).
[CrossRef]

Li, H.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[CrossRef]

Liu, X. L.

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field radiative heat transfer with doped-silicon nanostructured metamaterials,” Int. J. Heat Mass Transfer 73, 389–398 (2014).
[CrossRef]

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field thermal radiation between hyperbolic metamaterials: graphite and carbon nanotubes,” Appl. Phys. Lett. 103(21), 213102 (2013).
[CrossRef]

X. L. Liu, L. P. Wang, and Z. M. Zhang, “Wideband tunable omnidirectional infrared absorbers based on doped-silicon nanowire arrays,” J. Heat Transfer 135(6), 061602 (2013).
[CrossRef]

X. L. Liu and Z. M. Zhang, “Metal-free low-loss negative refraction in the mid-infrared region,” Appl. Phys. Lett. 103(10), 103101 (2013).
[CrossRef]

X. L. Liu, T. J. Bright, and Z. M. Zhang, “Application conditions of effective medium theory in near-field radiative heat transfer between multilayered metamaterials,” J. Heat Transfer. in press.

Liu, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[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(3-4), 59–112 (2005).
[CrossRef]

Menon, V. M.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Messina, R.

S.-A. Biehs, M. Tschikin, R. Messina, and P. Ben-Abdallah, “Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials,” Appl. Phys. Lett. 102(13), 131106 (2013).
[CrossRef]

M. Tschikin, S.-A. Biehs, R. Messina, and P. Ben-Abdallah, “On the limits of the effective description of hyperbolic materials in the presence of surface waves,” J. Opt. 15(10), 105101 (2013).
[CrossRef]

Molesky, S.

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[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(3-4), 59–112 (2005).
[CrossRef]

Narayanaswamy, A.

A. Narayanaswamy and G. Chen, “Direct computation of thermal emission from nanostructures,” Annu. Rev. Heat Transfer 14(14), 169–195 (2005).
[CrossRef]

A. Narayanaswamy and G. Chen, “Thermal emission control with one-dimensional metallodielectric photonic crystals,” Phys. Rev. B 70(12), 125101 (2004).
[CrossRef]

Narimanov, E.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[CrossRef] [PubMed]

Narimanov, E. E.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23(3), 498–505 (2006).
[CrossRef]

Noginov, M. A.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[CrossRef]

Orlov, A.

Park, K.

K. Park and Z. M. Zhang, “Fundamentals and applications of near-field radiative energy transfer,” Frontiers Heat Mass Transfer 4(1), 013001 (2013).
[CrossRef]

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[CrossRef]

B. J. Lee, K. Park, and Z. M. Zhang, “Energy pathways in nanoscale thermal radiation,” Appl. Phys. Lett. 91(15), 153101 (2007).
[CrossRef]

Pendry, J. B.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50(9), 1419–1430 (2003).
[CrossRef]

Petrov, A. Yu.

S. Lang, M. Tschikin, S.-A. Biehs, A. Yu. Petrov, and M. Eich, “Large penetration depth of near-field heat flux in hyperbolic media,” Appl. Phys. Lett. 104(12), 121903 (2014).
[CrossRef]

Pinar Mengüç, M.

M. Francoeur, M. Pinar Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s functions and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[CrossRef]

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

R. Wangberg, J. Elser, E. E. Narimanov, and V. A. Podolskiy, “Nonmagnetic nanocomposites for optical and infrared negative-refractive-index media,” J. Opt. Soc. Am. B 23(3), 498–505 (2006).
[CrossRef]

Ramakrishna, S. A.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50(9), 1419–1430 (2003).
[CrossRef]

Rosa, F. S. S.

Scalora, M.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Schilling, J.

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046618 (2006).
[CrossRef] [PubMed]

Schouten, H. F.

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B Quantum Semiclassical Opt. 6(5), S404–S409 (2004).
[CrossRef]

Shen, S.

S. Shen, “Experimental studies of radiative heat transfer between bodies at small separations,” Annu. Rev. Heat Transfer 16(1), 327–343 (2013).
[CrossRef]

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction and evanescent wave amplification using onedimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89(15), 151102 (2006).
[CrossRef]

Sipe, J. E.

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Soukoulis, C. M.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79(24), 245127 (2009).
[CrossRef]

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Stewart, W. J.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50(9), 1419–1430 (2003).
[CrossRef]

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Tschikin, M.

S. Lang, M. Tschikin, S.-A. Biehs, A. Yu. Petrov, and M. Eich, “Large penetration depth of near-field heat flux in hyperbolic media,” Appl. Phys. Lett. 104(12), 121903 (2014).
[CrossRef]

M. Tschikin, S.-A. Biehs, R. Messina, and P. Ben-Abdallah, “On the limits of the effective description of hyperbolic materials in the presence of surface waves,” J. Opt. 15(10), 105101 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, R. Messina, and P. Ben-Abdallah, “Super-Planckian near-field thermal emission with phonon-polaritonic hyperbolic metamaterials,” Appl. Phys. Lett. 102(13), 131106 (2013).
[CrossRef]

S.-A. Biehs, M. Tschikin, and P. Ben-Abdallah, “Hyperbolic metamaterials as an analog of a blackbody in the near field,” Phys. Rev. Lett. 109(10), 104301 (2012).
[CrossRef] [PubMed]

Tumkur, T.

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[CrossRef]

Vaillon, R.

M. Francoeur, M. Pinar Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s functions and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[CrossRef]

Vincenti, M. A.

D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-IR ranges,” Phys. Rev. A 77(3), 033848 (2008).
[CrossRef]

Visser, T. D.

H. F. Schouten, T. D. Visser, and D. Lenstra, “Optical vortices near sub-wavelength structures,” J. Opt. B Quantum Semiclassical Opt. 6(5), S404–S409 (2004).
[CrossRef]

Wang, L. P.

X. L. Liu, L. P. Wang, and Z. M. Zhang, “Wideband tunable omnidirectional infrared absorbers based on doped-silicon nanowire arrays,” J. Heat Transfer 135(6), 061602 (2013).
[CrossRef]

S. Basu, L. P. Wang, and Z. M. Zhang, “Direct calculation of energy streamlines in near-field thermal radiation,” J. Quant. Spectrosc. Radiat. Transf. 112(7), 1149–1155 (2011).
[CrossRef]

L. P. Wang, S. Basu, and Z. M. Zhang, “Direct and indirect methods for calculating thermal emission from layered structures with nonuniform temperatures,” J. Heat Transfer 133(7), 072701 (2011).
[CrossRef]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Wangberg, R.

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50(9), 1419–1430 (2003).
[CrossRef]

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Zhang, R. Z.

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field radiative heat transfer with doped-silicon nanostructured metamaterials,” Int. J. Heat Mass Transfer 73, 389–398 (2014).
[CrossRef]

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field thermal radiation between hyperbolic metamaterials: graphite and carbon nanotubes,” Appl. Phys. Lett. 103(21), 213102 (2013).
[CrossRef]

Zhang, X.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[CrossRef] [PubMed]

Zhang, Z. M.

X. L. Liu, R. Z. Zhang, and Z. M. Zhang, “Near-field radiative heat transfer with doped-silicon nanostructured metamaterials,” Int. J. Heat Mass Transfer 73, 389–398 (2014).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Illustration of near-field radiative transfer between two hyperbolic metamaterials whose optical axis is normal to the surface. A vacuum gap d separate the two semi-infinite media. Cylindrical coordinates are used in which ρ specifies the component parallel to the plane. The wavevector k is decomposed into a parallel component β and a perpendicular component γ. The source (medium 1) is at 300 K and the receiver (medium 2) is at 0 K.

Fig. 2
Fig. 2

Real part of the dielectric function predicted by EMT: (a) doped Si and Ge with f = 0.5; (b) SiC and Ge with f = 0.3. The multilayered structure is illustrated by the inset for each case. These filling ratios are used in the property calculations throughout this paper.

Fig. 3
Fig. 3

Power transmission factor and spectral heat flux with a gap spacing d = 10 nm. Contour plots of the transmission factor for p-polarization between two semi-infinite hyperbolic metamaterials made of (a) D-Si/Ge and (b) SiC/Ge multilayered structures; (c) Spectral heat flux between D-Si/Ge multilayered structures plotted together with that between blackbodies in the far field; (d) Spectral heat flux between SiC/Ge multilayered structures. The hyperbolic bands for the multilayered structures are highlighted. Note that the unit of Sω is heat flux [W/m2] per frequency [rad/s] interval.

Fig. 4
Fig. 4

Poynting vector and cumulative heat flux: (a) z-component of the Poynting vector versus lateral wavevector at select frequencies for (a) D-Si/Ge and (b) SiC/Ge multilayer metamaterials; Integration of the total heat flux over the wavevector space for (c) D-Si/Ge and (d) SiC/Ge. The horizontal line in (c) and (d) shows where 50% of the heat flux falls above and below the median wavevector for each frequency. Note that line styles in (c) and (d) correspond to the line styles and frequencies in (a) and (b), respectively. The unit of Sz is the heat flux [W/m2] per unit frequency [rad/s], wavevector [rad/m], and azimuthal angle [rad].

Fig. 5
Fig. 5

Penetration depth δ ( ω , β ) and spectral penetration depth δ ω ( ω ) for d = 10 nm: (a,b) δ / d for selected frequencies as a function of β/k0 for D-Si/Ge and SiC/Ge, respectively; (c,d) δ ω / d as a function of frequency for D-Si/Ge and SiC/Ge structures, respectively. Note that the hyperbolic bands are highlighted in (c,d).

Fig. 6
Fig. 6

Energy streamlines at select frequencies based on the median wavevector for (a,b) D-Si/Ge and (c,d) SiC/Ge multilayer structures at d = 10 nm. Here, β * = β median / k 0 that corresponds to 50% of the cumulative heat flux for each frequency as shown in Figs. 4(c) and 4(d).

Fig. 7
Fig. 7

Average lateral displacements in vacuum and emitter for (a) D-Si/Ge structure and (b) SiC/Ge structure, with a gap spacing d = 10 nm. The highlighted regions correspond to the hyperbolic band (type I or type II).

Equations (28)

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ε ¯ ¯ = ε t I ¯ ¯ ( ε t ε z ) z ^ z ^ = [ ε t 0 0 0 ε t 0 0 0 ε z ]
γ o 2 + β 2 = ε t k 0 2   ,  for ordinary waves
γ e 2 ε t + β 2 ε z = k 0 2 , for extraordinary waves
E( r,ω )=iω μ 0 V G ¯ ¯ ( r, r ,ω )· J r ( r ,ω ) d 3 r
H( r,ω )= V Γ ¯ ¯ ( r, r ,ω )· J r ( r ,ω ) d 3 r
J r, i ( r,ω ) J r,k * ( r ,ω ) = Θ( ω,T ) ε 0 ε ik ( ω )ω π δ( r r )
G ¯ ¯ ( r, r ,ω )= 1 2π 0 g ¯ ¯ ( β,z, z ,ω ) e iβ(ρ ρ ) βdβ
g ¯ ¯ 1 ± ( β,z, z ,ω )= i 2 γ o,1 ( e ±i γ o,1 ( z z ) + R s e i γ o,1 ( z z ) ) o ^ o ^ + i F(β,ω)( e ^ 1 ± e ^ 1 ± e ±i γ e,1 ( z z ) + R p e i γ e,1 ( z z ) e ^ 1 e ^ 1 + )
g ¯ ¯ 0 ( β,z, z ,ω )= i 2 γ o,1 T s e i γ o,1 z t 02,s ( e i γ 0 (zd) + r 02,s e i γ 0 z ) o ^ o ^ + i F(β,ω) Z 10 (β,ω) T p e i γ e,1 z t 02,p ( e i γ 0 (zd) e ^ 0 + + r 02,p e i γ 0 z e ^ 0 ) e ^ 1 +
g ¯ ¯ 2 ( β,z, z ,ω )= i 2 γ o,1 T s e i γ o,1 z e i γ o,2 ( zd ) o ^ o ^ + i F(β,ω) Z 12 ( β,ω ) T p e i γ e,1 z e i γ e,2 ( zd ) e ^ 2 + e ^ 1 +
F(β,ω)= k 0 2 ( ε t,1 + ε z,1 ) k e,1 2 2 γ e,1 k 0 2 ε z,1
Z 10 (β,ω)= k 0 / γ e,1 2 ε t,1 2 + β 2 ε z,1 2 ; Z 12 (β,ω)= γ e,2 2 ε t,2 2 + β 2 ε z,2 2 / γ e,1 2 ε t,1 2 + β 2 ε z,1 2
e ^ l ± = γ e,l ε z,l ρ ^ +β ε t,l z ^ ( γ e,l ε z,l ) 2 + ( β ε t,l ) 2
e ^ 0 ± = γ 0 ρ ^ +β z ^ k 0
S z ( z,ω,β ) = k 0 2 Θ( ω,T )β 2 π 3 Re[ i 0 ( ε t,1 g 11 h 21 * + ε z,1 g 13 h 23 * ε t,1 g 22 h 12 * )d z ]
S ρ ( z,ω,β ) = k 0 2 Θ( ω,T )β 2 π 3 Re[ i 0 ( ε t,1 g 22 h 32 * ε t,1 g 31 h 21 * ε z,1 g 33 h 23 * )d z ]
m= S ρ S z
q net = 1 4 π 2 0 [ Θ( ω, T 1 )Θ( ω, T 2 ) ] dω 0 β j=s,p ξ j ( ω,β ) dβ
ξ j ( ω,β )={ ( 1 | r 10,j | 2 ) 2 / | 1 r 10,j 2 e 2i γ 0 d | 2 , β< k 0 4 [ Im( r 10,j ) ] 2 e 2i γ 0 d / | 1 r 10,j 2 e 2i γ 0 d | 2 , β> k 0 }
ε t =f ε m +( 1f ) ε d
ε z = ε d ε m f ε d +( 1f ) ε m
δ(ω,β)= 1 ±Re( ε t / ε z ) 1 2β
δ ω (ω)= 0 β ξ p δ(ω,β)dβ 0 β ξ p dβ
ζ { z / δ for z < 0 z / d for 0 < z < d 1 + ( z d ) / δ for z > d
Δ(ω,β)=ρ( z 2 )ρ( z 1 )
Δ ω (ω)= 0 β ξ p Δ(ω,β))dβ 0 β ξ p dβ
e c = 1+ | ε z | ε t for type I hyperbolic dispersion
e c = 1+ | ε t | ε z for type II hyperbolic dispersion

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