Abstract

Absorption and emission of light due to the resonant excitation of surface waves on a grating is a well-known phenomenon. We report the first complete study of the influence of the role of angle and polarization on thermal emission by lamellar gratings. We derive the emitted Stokes vectors in any direction. We find that a source can be quasi isotropic from the point of view of the intensity but strongly anisotropic for polarized light. It follows that the degree of polarization can vary between 0 and 1, depending on directions.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396 (1902).
  4. D. Maystre and M. C. Hutley, "The total absorption of light by a diffraction grating," Opt. Commun. 19, 431 (1976).
    [CrossRef]
  5. J. J. Greffet and M. Nieto-Vesperinas, "Field theory for the generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law," J. Opt. Soc. Am. A. 10, 2735 (1998).
    [CrossRef]
  6. M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, "Highly directional radiation generated by a tungsten thermal source," Opt. Lett. 30, 2623-2625 (2005).
    [CrossRef] [PubMed]
  7. A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, and J. Luther, "Radiation filters and emitters for the NIR based on periodically structured metal surfaces," J. Mod. Opt. 47, 2399-2419 (2000).
  8. M. Laroche, R. Carminati, and J. J. Greffet, "Coherent thermal antenna using a photonic crystal slab," Phys. Rev. Lett. 96, 4 (2006).
    [CrossRef]
  9. J. LeGall, M. Olivier, and J. J. Greffet, "Experimental and theoretical study of reflection and coherent thermal emissionby a SiC grating supporting a surface-phonon polariton," Phys. Rev. B 55, 10105-10114 (1997).
    [CrossRef]
  10. F. Marquier, K. Joulain, J. P. Mulet, R. Carminati, J. J. Greffet, and Y. Chen, "Coherent spontaneous emission of light by thermal sources," Phys. Rev. B 69, 11 (2004).
    [CrossRef]
  11. J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. P. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature 416, 61-64 (2002).
    [CrossRef] [PubMed]
  12. I. Celanovic, D. Perreault, and J. Kassakian, "Resonant-cavity enhanced thermal emission," Phys. Rev. B 72, 075127 (2005).
    [CrossRef]
  13. A. Battula and S. C. Chen, "Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance," Phys. Rev. B 74, 245407 (2006).
    [CrossRef]
  14. M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, "Tuning the optical response of nanocylinder arrays: An analytical study," Phys. Rev. B 74, 245422 (2006).
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  15. B. J. Lee and Z. M. Zhang, "Design and fabrication of planar multilayer structures with coherent thermal emission characteristics," J. Appl. Phys. 100, 063529 (2006).
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  16. B. J. Lee and Z. M. Zhang, "Coherent Thermal Emission From Modified Periodic Multilayer Structures, " ASME J. Heat Transf. 129, 17-26 (2007).
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  18. S. Enoch, J. J. Simon, L. Escoubas, Z. Elalmy, F. Lemarquis, P. Torchio, and G. Albrand, "Simple layer-by-layer photonic crystal for the control of thermal emission," Appl. Phys. Lett. 86, 261101 (2005).
    [CrossRef]
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    [CrossRef]
  22. L. Hu, A. Schmidt, A. Narayanaswamy, and G. Chen, "Effects of Periodic Structures on the Coherence Properties of Blackbody Radiation," ASME J. Heat Transf. 126, 786-792 (2004).
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    [CrossRef]
  25. S. Maruyama, T. Kashiwa, H. Yugami, and M. Esashi, "Thermal radiation from two-dimensionally confined modes in microcavities," Appl. Phys. Lett. 79, 1393-1395 (2001).
    [CrossRef]
  26. D. L. C. Chan, M. Soljacic, and J. D. Joannopoulos, "Thermal emission and design in 2D-periodic metallic photonic crystal slabs," Opt. Express 14, 8785-8796 (2006).
    [CrossRef] [PubMed]
  27. D. L. C. Chan, M. Soljacic, and J. D. Joannopoulos, "Direct Calculation of thermal emission for threedimensionally periodic photonic crystal slabs," Phys. Rev. E 74, 036615 (2006).
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  28. V. Yannopapas, "Thermal emission from three-dimensional arrays of gold nanoparticles," Phys. Rev. B 73, 113108 (2006).
    [CrossRef]
  29. M. Florescu, H. Lee, A. J. Stimpson, and J. Dowling, "Thermal emission and absorption of radiation in finite inverted opal photonic crystals," Phys. Rev. A 72, 033821 (2005).
    [CrossRef]
  30. R. Biswas, C. G. Ding, I. Puscasu, M. Pralle, M. McNeal, J. Daly, A. Greenwald, and E. Johnson, "Theory of subwavelength hole arrays coupled with photonic crystals for extraordinary thermal emission," Phys. Rev. B 74, 045107 (2006).
    [CrossRef]
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    [CrossRef]
  32. J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature (London) 417, 52-55 (2002).
    [CrossRef] [PubMed]
  33. N. Dahan, A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Space-variant polarization manipulation of a thermal emission by a SiO2 subwavelength grating supporting surface phonon-polaritons," Appl. Phys. Lett. 86, 191102 (2005).
    [CrossRef]
  34. N. Dahan, A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Thermal image encryption obtained with a SiO2 space-variant subwavelength grating supporting surface phonon-polaritons," Opt. Lett. 30, 3195-3197 (2005).
    [CrossRef] [PubMed]
  35. F. Marquier, M. Laroche, R. Carminati, and J. J. Greffet, "Anisotropic polarized emission of a doped silicon lamellar grating," ASME J. Heat Transf. 129, 11-16 (2007).
    [CrossRef]
  36. T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, "Plasma resonance absorption in conical diffraction: effects of groove depth," J. Opt. Soc. Am. B 3, 992-995 (1986).
    [CrossRef]
  37. S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, "Polarization conversion from diffraction gratings," Phys. Rev. B 44, 6393-6400 (1991).
    [CrossRef]
  38. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
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  40. T. Setala, J. Tervo, and A. T. Friberg, "Stokes parameters and polarization contrasts in Young’s interference experiment," Opt. Lett. 31, 2208-2210 (2006).
    [CrossRef] [PubMed]

2007 (4)

B. J. Lee and Z. M. Zhang, "Coherent Thermal Emission From Modified Periodic Multilayer Structures, " ASME J. Heat Transf. 129, 17-26 (2007).
[CrossRef]

K. Joulain and A. Loizeau, "Coherent thermal emission by microstructured waveguides," J. Quantum Spectrosc. Radiat. Transfer 104, 208-216 (2007).
[CrossRef]

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, V. I. Pipa, and E. F. Venger, "Circular polarized coherent thermal radiation from semiconductor layers in an external magnetic field," Opt. Commun. 276, 131-134 (2007).
[CrossRef]

F. Marquier, M. Laroche, R. Carminati, and J. J. Greffet, "Anisotropic polarized emission of a doped silicon lamellar grating," ASME J. Heat Transf. 129, 11-16 (2007).
[CrossRef]

2006 (9)

T. Setala, J. Tervo, and A. T. Friberg, "Stokes parameters and polarization contrasts in Young’s interference experiment," Opt. Lett. 31, 2208-2210 (2006).
[CrossRef] [PubMed]

D. L. C. Chan, M. Soljacic, and J. D. Joannopoulos, "Thermal emission and design in 2D-periodic metallic photonic crystal slabs," Opt. Express 14, 8785-8796 (2006).
[CrossRef] [PubMed]

D. L. C. Chan, M. Soljacic, and J. D. Joannopoulos, "Direct Calculation of thermal emission for threedimensionally periodic photonic crystal slabs," Phys. Rev. E 74, 036615 (2006).
[CrossRef]

V. Yannopapas, "Thermal emission from three-dimensional arrays of gold nanoparticles," Phys. Rev. B 73, 113108 (2006).
[CrossRef]

R. Biswas, C. G. Ding, I. Puscasu, M. Pralle, M. McNeal, J. Daly, A. Greenwald, and E. Johnson, "Theory of subwavelength hole arrays coupled with photonic crystals for extraordinary thermal emission," Phys. Rev. B 74, 045107 (2006).
[CrossRef]

M. Laroche, R. Carminati, and J. J. Greffet, "Coherent thermal antenna using a photonic crystal slab," Phys. Rev. Lett. 96, 4 (2006).
[CrossRef]

A. Battula and S. C. Chen, "Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance," Phys. Rev. B 74, 245407 (2006).
[CrossRef]

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, "Tuning the optical response of nanocylinder arrays: An analytical study," Phys. Rev. B 74, 245422 (2006).
[CrossRef]

B. J. Lee and Z. M. Zhang, "Design and fabrication of planar multilayer structures with coherent thermal emission characteristics," J. Appl. Phys. 100, 063529 (2006).
[CrossRef]

2005 (10)

I. Celanovic, D. Perreault, and J. Kassakian, "Resonant-cavity enhanced thermal emission," Phys. Rev. B 72, 075127 (2005).
[CrossRef]

S. Enoch, J. J. Simon, L. Escoubas, Z. Elalmy, F. Lemarquis, P. Torchio, and G. Albrand, "Simple layer-by-layer photonic crystal for the control of thermal emission," Appl. Phys. Lett. 86, 261101 (2005).
[CrossRef]

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

C. H. Seager, M. B. Sinclair, and J. G. Fleming, "Accurate measurements of thermal radiation from a tungsten photonic lattice," Appl. Phys. Lett. 86, 244105 (2005).
[CrossRef]

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

M. Laroche, F. Marquier, R. Carminati, and J. J. Greffet, "Tailoring silicon radiative properties," Opt. Commun. 250, 316-320 (2005).
[CrossRef]

H. Sai, Y. Kanamori, K. Hane, and H. Yugami, "Numerical study on spectral properties of tungsten onedimensional surface-relief gratings for spectrally selective devices," J. Opt. Soc. Am. A 22, 1805-1813 (2005).
[CrossRef]

M. Laroche, C. Arnold, F. Marquier, R. Carminati, J. J. Greffet, S. Collin, N. Bardou, and J. L. Pelouard, "Highly directional radiation generated by a tungsten thermal source," Opt. Lett. 30, 2623-2625 (2005).
[CrossRef] [PubMed]

N. Dahan, A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Thermal image encryption obtained with a SiO2 space-variant subwavelength grating supporting surface phonon-polaritons," Opt. Lett. 30, 3195-3197 (2005).
[CrossRef] [PubMed]

N. Dahan, A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Space-variant polarization manipulation of a thermal emission by a SiO2 subwavelength grating supporting surface phonon-polaritons," Appl. Phys. Lett. 86, 191102 (2005).
[CrossRef]

2004 (2)

L. Hu, A. Schmidt, A. Narayanaswamy, and G. Chen, "Effects of Periodic Structures on the Coherence Properties of Blackbody Radiation," ASME J. Heat Transf. 126, 786-792 (2004).
[CrossRef]

F. Marquier, K. Joulain, J. P. Mulet, R. Carminati, J. J. Greffet, and Y. Chen, "Coherent spontaneous emission of light by thermal sources," Phys. Rev. B 69, 11 (2004).
[CrossRef]

2003 (1)

2002 (3)

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. P. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature 416, 61-64 (2002).
[CrossRef] [PubMed]

M. U. Pralle, N. Moelders, M. P. McNeal, I. Puscasu, A. C. Greenwald, J. T. Daly, E. A. Johnson, T. George, D. S. Cho, I. El-Kady, and R. Biswas, "Photonic crystal enhanced narrow-band infrared emitters," Appl. Phys. Lett. 81, 4685-4687 (2002).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature (London) 417, 52-55 (2002).
[CrossRef] [PubMed]

2001 (1)

S. Maruyama, T. Kashiwa, H. Yugami, and M. Esashi, "Thermal radiation from two-dimensionally confined modes in microcavities," Appl. Phys. Lett. 79, 1393-1395 (2001).
[CrossRef]

2000 (1)

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, and J. Luther, "Radiation filters and emitters for the NIR based on periodically structured metal surfaces," J. Mod. Opt. 47, 2399-2419 (2000).

1999 (1)

.M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mittler-Neher, andW. Knoll, "Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons," Opt. Commun. 168, 117-122 (1999).
[CrossRef]

1998 (1)

J. J. Greffet and M. Nieto-Vesperinas, "Field theory for the generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law," J. Opt. Soc. Am. A. 10, 2735 (1998).
[CrossRef]

1997 (1)

J. LeGall, M. Olivier, and J. J. Greffet, "Experimental and theoretical study of reflection and coherent thermal emissionby a SiC grating supporting a surface-phonon polariton," Phys. Rev. B 55, 10105-10114 (1997).
[CrossRef]

1991 (1)

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, "Polarization conversion from diffraction gratings," Phys. Rev. B 44, 6393-6400 (1991).
[CrossRef]

1986 (2)

T. Inagaki, J. P. Goudonnet, and E. T. Arakawa, "Plasma resonance absorption in conical diffraction: effects of groove depth," J. Opt. Soc. Am. B 3, 992-995 (1986).
[CrossRef]

P. J. Hesketh, J. N. Zemel, and B. Gebhart, "Organ pipe radiant modes of periodic micromachined silicon surfaces," Nature (London) 324, 549 (1986).
[CrossRef]

1976 (1)

D. Maystre and M. C. Hutley, "The total absorption of light by a diffraction grating," Opt. Commun. 19, 431 (1976).
[CrossRef]

1902 (1)

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396 (1902).

Appl. Phys. Lett. (5)

S. Enoch, J. J. Simon, L. Escoubas, Z. Elalmy, F. Lemarquis, P. Torchio, and G. Albrand, "Simple layer-by-layer photonic crystal for the control of thermal emission," Appl. Phys. Lett. 86, 261101 (2005).
[CrossRef]

M. U. Pralle, N. Moelders, M. P. McNeal, I. Puscasu, A. C. Greenwald, J. T. Daly, E. A. Johnson, T. George, D. S. Cho, I. El-Kady, and R. Biswas, "Photonic crystal enhanced narrow-band infrared emitters," Appl. Phys. Lett. 81, 4685-4687 (2002).
[CrossRef]

S. Maruyama, T. Kashiwa, H. Yugami, and M. Esashi, "Thermal radiation from two-dimensionally confined modes in microcavities," Appl. Phys. Lett. 79, 1393-1395 (2001).
[CrossRef]

C. H. Seager, M. B. Sinclair, and J. G. Fleming, "Accurate measurements of thermal radiation from a tungsten photonic lattice," Appl. Phys. Lett. 86, 244105 (2005).
[CrossRef]

N. Dahan, A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Space-variant polarization manipulation of a thermal emission by a SiO2 subwavelength grating supporting surface phonon-polaritons," Appl. Phys. Lett. 86, 191102 (2005).
[CrossRef]

ASME J. Heat Transf. (3)

L. Hu, A. Schmidt, A. Narayanaswamy, and G. Chen, "Effects of Periodic Structures on the Coherence Properties of Blackbody Radiation," ASME J. Heat Transf. 126, 786-792 (2004).
[CrossRef]

B. J. Lee and Z. M. Zhang, "Coherent Thermal Emission From Modified Periodic Multilayer Structures, " ASME J. Heat Transf. 129, 17-26 (2007).
[CrossRef]

F. Marquier, M. Laroche, R. Carminati, and J. J. Greffet, "Anisotropic polarized emission of a doped silicon lamellar grating," ASME J. Heat Transf. 129, 11-16 (2007).
[CrossRef]

J. Appl. Phys. (2)

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

B. J. Lee and Z. M. Zhang, "Design and fabrication of planar multilayer structures with coherent thermal emission characteristics," J. Appl. Phys. 100, 063529 (2006).
[CrossRef]

J. Mod. Opt. (1)

A. Heinzel, V. Boerner, A. Gombert, B. Blasi, V. Wittwer, and J. Luther, "Radiation filters and emitters for the NIR based on periodically structured metal surfaces," J. Mod. Opt. 47, 2399-2419 (2000).

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

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

J. J. Greffet and M. Nieto-Vesperinas, "Field theory for the generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law," J. Opt. Soc. Am. A. 10, 2735 (1998).
[CrossRef]

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

J. Quantum Spectrosc. Radiat. Transfer (1)

K. Joulain and A. Loizeau, "Coherent thermal emission by microstructured waveguides," J. Quantum Spectrosc. Radiat. Transfer 104, 208-216 (2007).
[CrossRef]

Nature (1)

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. P. Mainguy, and Y. Chen, "Coherent emission of light by thermal sources," Nature 416, 61-64 (2002).
[CrossRef] [PubMed]

Nature (London) (2)

P. J. Hesketh, J. N. Zemel, and B. Gebhart, "Organ pipe radiant modes of periodic micromachined silicon surfaces," Nature (London) 324, 549 (1986).
[CrossRef]

J. G. Fleming, S. Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, "All-metallic three-dimensional photonic crystals with a large infrared bandgap," Nature (London) 417, 52-55 (2002).
[CrossRef] [PubMed]

Opt. Commun. (4)

M. Laroche, F. Marquier, R. Carminati, and J. J. Greffet, "Tailoring silicon radiative properties," Opt. Commun. 250, 316-320 (2005).
[CrossRef]

.M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mittler-Neher, andW. Knoll, "Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons," Opt. Commun. 168, 117-122 (1999).
[CrossRef]

D. Maystre and M. C. Hutley, "The total absorption of light by a diffraction grating," Opt. Commun. 19, 431 (1976).
[CrossRef]

O. G. Kollyukh, A. I. Liptuga, V. Morozhenko, V. I. Pipa, and E. F. Venger, "Circular polarized coherent thermal radiation from semiconductor layers in an external magnetic field," Opt. Commun. 276, 131-134 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Philos. Mag. (1)

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396 (1902).

Phys. Rev. A (1)

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

Phys. Rev. B (8)

R. Biswas, C. G. Ding, I. Puscasu, M. Pralle, M. McNeal, J. Daly, A. Greenwald, and E. Johnson, "Theory of subwavelength hole arrays coupled with photonic crystals for extraordinary thermal emission," Phys. Rev. B 74, 045107 (2006).
[CrossRef]

V. Yannopapas, "Thermal emission from three-dimensional arrays of gold nanoparticles," Phys. Rev. B 73, 113108 (2006).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, "Polarization conversion from diffraction gratings," Phys. Rev. B 44, 6393-6400 (1991).
[CrossRef]

I. Celanovic, D. Perreault, and J. Kassakian, "Resonant-cavity enhanced thermal emission," Phys. Rev. B 72, 075127 (2005).
[CrossRef]

A. Battula and S. C. Chen, "Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance," Phys. Rev. B 74, 245407 (2006).
[CrossRef]

M. Laroche, S. Albaladejo, R. Gomez-Medina, and J. J. Saenz, "Tuning the optical response of nanocylinder arrays: An analytical study," Phys. Rev. B 74, 245422 (2006).
[CrossRef]

J. LeGall, M. Olivier, and J. J. Greffet, "Experimental and theoretical study of reflection and coherent thermal emissionby a SiC grating supporting a surface-phonon polariton," Phys. Rev. B 55, 10105-10114 (1997).
[CrossRef]

F. Marquier, K. Joulain, J. P. Mulet, R. Carminati, J. J. Greffet, and Y. Chen, "Coherent spontaneous emission of light by thermal sources," Phys. Rev. B 69, 11 (2004).
[CrossRef]

Phys. Rev. E (1)

D. L. C. Chan, M. Soljacic, and J. D. Joannopoulos, "Direct Calculation of thermal emission for threedimensionally periodic photonic crystal slabs," Phys. Rev. E 74, 036615 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

M. Laroche, R. Carminati, and J. J. Greffet, "Coherent thermal antenna using a photonic crystal slab," Phys. Rev. Lett. 96, 4 (2006).
[CrossRef]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

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Figures (7)

Fig. 1.
Fig. 1.

Emission of a grating in the direction (θ,ϕ). The electric field E makes an angle Ψ with the plane (z,k). Ψ=0° corresponds to the p-polarization and Ψ=90° to the s-polarization.

Fig. 2.
Fig. 2.

Polar representation of the emissivity at λ=11.36 µm for the directional source in both p-polarization (a) and s-polarization (b) (numerical simulations).

Fig. 3.
Fig. 3.

Dispersion relation of surface plasmon-polaritons in a (kx ,ky ) plane for two gratings with different period at a fixed frequency. In both figures, the radius of the dashed circle is the wave vector modulus in free space. Inside this circle (in gray), the emitted waves have a real z-component of k, so that they are propagating waves. Outside the dashed circle, the waves are evanescent waves. In (a), the solid circle at the center has a radius equal to the modulus of the wave vector of the surface wave k SP. With a grating of period Λ, this solid circle is reproduced with a period 2π/Λ: it represents the surface waves diffraction at the orders ±1. One can see in the first figure (a), corresponding to the case of section ??, that a part of the solid line is lying now in the light cone, so that the corresponding surface polaritons are coupled to propagating waves. In the second figure (b), corresponding to the case of section ??, the period Λ is lower so that the linear part of the dispersion relation is always outside the light cone. Only the surface waves lying on the asymptotic part of the dispersion relation can be coupled to propagating waves. This is represented by the parallel red lines which cover the whole gray area. This means that propagating waves can be coupled to surface waves in every directions (kx ,ky ) at the asymptotic wavelength.

Fig. 4.
Fig. 4.

Polar representation of the emissivity at λ=10.88 µm for the quasi-isotropic source in both p-polarization (a) and s-polarization (b) (numerical simulations).

Fig. 5.
Fig. 5.

Polar representation of the average emissivity in both p- and s-polarization at λ=10.88 µm for the quasi-isotropic source (a) (numerical simulation). Average emissivity of the same source measured with three different azimuthal angles ϕ at θ=20° (b) (experimental measurements).

Fig. 6.
Fig. 6.

Polar representation of the emissivity in different polarization at λ=10.88 µm for the quasi-isotropic source: linear polarization Ψ=+45° (a), linear polarization Ψ=-45° (b), left-hand circular polarization (c) and right-hand circular polarization (d).

Fig. 7.
Fig. 7.

Polar representation of the degree of polarization P given by Stokes’s parameters.

Equations (1)

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S 0 = ε S + ε p S 1 = ε s + ε p S 2 = ε a + ε b S 3 = ε L + ε R

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