Abstract

Generalized Snell–Descartes and Fresnel laws are derived for harmonic inhomogeneous plane waves that are incident upon a static interface between two continuous absorbing dielectric media that are macroscopically characterized by their electric and magnetic permittivities and their conductivities. A coordinate-free formalism based on complex vector algebra is used to carry out all discussions. Surprisingly, the usual complex Snell–Descartes laws for reflection and refraction and Fresnel laws for polarization are recovered only in the special case in which the vector characterizing the direction of inhomogeneity is in the plane of incidence. In the more general case a new deflection angle between planes of incidence and refraction has to be introduced. An experiment is proposed to test this prediction. A generalized form of the TE and TM modes (with respect to the interface), which are elliptically polarized and which are called parallel electric and parallel magnetic modes, also emerges.

© 1994 Optical Society of America

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References

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  1. J. M. Saca, “On the velocity of light in uniformly moving dielectrics,” J. Mod. Opt. 37, 227–235 (1990) and references therein.
    [Crossref]
  2. M. v. Laue, “Die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. 32, 1085–1094 (1910).
    [Crossref]
  3. M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).
  4. W. König, “Elektromagnetische Lichttheorie,” in Licht als Wellenbewegung, H. Konen, ed., Vol. XX of Handbuch der Physik, H. Geiger, K. Scheel, eds. (Springer-Verlag, Berlin, 1928), pp. 141–262.
    [Crossref]
  5. A. Sommerfeld, “Das Reziprozitäts Theorem der drahlosen Telegraphie,” Jahrb. Drahtseil Telegr. 37–38, 167–169 (1931).
  6. I. S∼antavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
    [Crossref]
  7. Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
    [Crossref]
  8. V. A. Kizel, “Modern status of the theory of light reflection,” Sov. Phys. Usp. 10, 485–508 (1968).
    [Crossref]
  9. C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Braunschw. Wiss. Ges. 34, 25–29 (1987).
  10. L. Pincherle, “Refraction of plane non-uniform electromagnetic waves between absorbing media,” Phys. Rev. 72, 232–235 (1947).
    [Crossref]
  11. A. I. Mahan, “Reflection and refraction at oblique incidence on a dielectric–metallic interface as a boundary value problem in electromagnetic theory,” J. Opt. Soc. Am. 46, 913–926 (1956).
    [Crossref]
  12. C. K. Carniglia, L. Mandel, K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
    [Crossref]
  13. E. Lalor, E. Wolf, “Exact solution of the equations of molecular optics for refraction and reflection of an electromagnetic wave on a semi-infinite dielectric,” J. Opt. Soc. Am. 62, 1165–1174 (1972).
    [Crossref]
  14. A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983).
    [Crossref]
  15. P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
    [Crossref]
  16. M. Nieto-Vesperinas, E. Wolf, “Generalized Stokes reciprocity relations for scattering from dielectric objects of arbitrary shape,” J. Opt. Soc. Am. A 3, 2038–2046 (1986).
    [Crossref]
  17. Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
    [Crossref]
  18. W. N. Hugrass, “Angular momentum balance on light reflection,” J. Mod. Opt. 37, 339–351 (1990).
    [Crossref]
  19. A. I. Mahan, C. V. Bitterli, “Total internal reflection: a deeper look,” Appl. Opt. 17, 509–519 (1978).
    [Crossref] [PubMed]
  20. J. J. Regan, D. R. Andersen, “Reflection and refraction of optical beams at dielectric interfaces,” Computers Phys. 5(1), 49–61 (1991).
    [Crossref]
  21. B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
    [Crossref]
  22. Z. Knittl, Optics of Thin Films (An Optical Multilayer Theory) (Wiley, New York, 1976).
  23. F. R. Kessler, “Optics with gradients of free carrier concentration,” in Festkörperprobleme, Vol. 26 of Advances in Solid State PhysicsP. Grosse, ed. (Vieweg, Braunschweig, Germany, 1986), pp. 277–308.
  24. M. A. Dupertuis, M. Proctor, “Generalization of complex Snell-Descartes and Fresnel laws,” Opt. Photon. News 1(9), p. A-126 (1990).
  25. B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
    [Crossref]
  26. A. Sommerfeld, Optics, Vol. VI of Lectures on Theoretical Physics (Academic, San Diego, Calif., 1954), Chap. 38 and references therein.
  27. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).
  28. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  29. M. A Dupertuis, B. Acklin, M. Proctor, “Generalized energy balance and reciprocity relations for thin-film optics,” J. Opt. Soc. Am. A 11, 1167–1174 (1994).
    [Crossref]
  30. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).
  31. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  32. J. A. Stratton, Théorie del’ électromagnétisme (Dunod, Paris, 1961).
  33. M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, New York, 1966).
  34. C. K. Carniglia, “Reflection and transmission at a boundary between two absorbing media: a partial bibliography,” presented at the Thin Films Technical Group meeting, Optical Society of America Annual Meeting, November 8, 1990, Boston, Mass.

1994 (1)

1993 (1)

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[Crossref]

1992 (1)

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

1991 (1)

J. J. Regan, D. R. Andersen, “Reflection and refraction of optical beams at dielectric interfaces,” Computers Phys. 5(1), 49–61 (1991).
[Crossref]

1990 (3)

M. A. Dupertuis, M. Proctor, “Generalization of complex Snell-Descartes and Fresnel laws,” Opt. Photon. News 1(9), p. A-126 (1990).

W. N. Hugrass, “Angular momentum balance on light reflection,” J. Mod. Opt. 37, 339–351 (1990).
[Crossref]

J. M. Saca, “On the velocity of light in uniformly moving dielectrics,” J. Mod. Opt. 37, 227–235 (1990) and references therein.
[Crossref]

1989 (1)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[Crossref]

1987 (1)

C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Braunschw. Wiss. Ges. 34, 25–29 (1987).

1986 (1)

1983 (2)

A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983).
[Crossref]

P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[Crossref]

1978 (1)

1972 (2)

1968 (1)

V. A. Kizel, “Modern status of the theory of light reflection,” Sov. Phys. Usp. 10, 485–508 (1968).
[Crossref]

1962 (1)

Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
[Crossref]

1961 (1)

I. S∼antavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
[Crossref]

1956 (1)

1947 (1)

L. Pincherle, “Refraction of plane non-uniform electromagnetic waves between absorbing media,” Phys. Rev. 72, 232–235 (1947).
[Crossref]

1931 (1)

A. Sommerfeld, “Das Reziprozitäts Theorem der drahlosen Telegraphie,” Jahrb. Drahtseil Telegr. 37–38, 167–169 (1931).

1911 (1)

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

1910 (1)

M. v. Laue, “Die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. 32, 1085–1094 (1910).
[Crossref]

Acklin, B.

M. A Dupertuis, B. Acklin, M. Proctor, “Generalized energy balance and reciprocity relations for thin-film optics,” J. Opt. Soc. Am. A 11, 1167–1174 (1994).
[Crossref]

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

Andersen, D. R.

J. J. Regan, D. R. Andersen, “Reflection and refraction of optical beams at dielectric interfaces,” Computers Phys. 5(1), 49–61 (1991).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bagnoud, C.

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bitterli, C. V.

Born, M.

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, New York, 1966).

Carniglia, C. K.

C. K. Carniglia, L. Mandel, K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[Crossref]

C. K. Carniglia, “Reflection and transmission at a boundary between two absorbing media: a partial bibliography,” presented at the Thin Films Technical Group meeting, Optical Society of America Annual Meeting, November 8, 1990, Boston, Mass.

Chen, B.

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[Crossref]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

Drexhage, K. H.

Drummond, P. D.

A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983).
[Crossref]

P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[Crossref]

Dupertuis, M. A

Dupertuis, M. A.

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

M. A. Dupertuis, M. Proctor, “Generalization of complex Snell-Descartes and Fresnel laws,” Opt. Photon. News 1(9), p. A-126 (1990).

Friberg, A. T.

P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[Crossref]

A. T. Friberg, P. D. Drummond, “Reflection of a linearly polarized plane wave from a lossless stratified mirror in the presence of a phase-conjugate mirror,” J. Opt. Soc. Am. 73, 1216–1219 (1983).
[Crossref]

Hugrass, W. N.

W. N. Hugrass, “Angular momentum balance on light reflection,” J. Mod. Opt. 37, 339–351 (1990).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kessler, F. R.

F. R. Kessler, “Optics with gradients of free carrier concentration,” in Festkörperprobleme, Vol. 26 of Advances in Solid State PhysicsP. Grosse, ed. (Vieweg, Braunschweig, Germany, 1986), pp. 277–308.

Kizel, V. A.

V. A. Kizel, “Modern status of the theory of light reflection,” Sov. Phys. Usp. 10, 485–508 (1968).
[Crossref]

Knittl, Z.

Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
[Crossref]

Z. Knittl, Optics of Thin Films (An Optical Multilayer Theory) (Wiley, New York, 1976).

König, W.

W. König, “Elektromagnetische Lichttheorie,” in Licht als Wellenbewegung, H. Konen, ed., Vol. XX of Handbuch der Physik, H. Geiger, K. Scheel, eds. (Springer-Verlag, Berlin, 1928), pp. 141–262.
[Crossref]

Ladenburg, R.

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

Lalor, E.

Laue, M. v.

M. v. Laue, “Die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. 32, 1085–1094 (1910).
[Crossref]

Mahan, A. I.

Mandel, L.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[Crossref]

C. K. Carniglia, L. Mandel, K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[Crossref]

Martin, D.

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

Morier-Genoud, F.

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

Nelson, D. F.

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[Crossref]

Nieto-Vesperinas, M.

Ou, Z. Y.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[Crossref]

Pincherle, L.

L. Pincherle, “Refraction of plane non-uniform electromagnetic waves between absorbing media,” Phys. Rev. 72, 232–235 (1947).
[Crossref]

Proctor, M.

M. A Dupertuis, B. Acklin, M. Proctor, “Generalized energy balance and reciprocity relations for thin-film optics,” J. Opt. Soc. Am. A 11, 1167–1174 (1994).
[Crossref]

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

M. A. Dupertuis, M. Proctor, “Generalization of complex Snell-Descartes and Fresnel laws,” Opt. Photon. News 1(9), p. A-126 (1990).

Regan, J. J.

J. J. Regan, D. R. Andersen, “Reflection and refraction of optical beams at dielectric interfaces,” Computers Phys. 5(1), 49–61 (1991).
[Crossref]

S~antavý, I.

I. S∼antavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
[Crossref]

Saca, J. M.

J. M. Saca, “On the velocity of light in uniformly moving dielectrics,” J. Mod. Opt. 37, 227–235 (1990) and references therein.
[Crossref]

Sommerfeld, A.

A. Sommerfeld, “Das Reziprozitäts Theorem der drahlosen Telegraphie,” Jahrb. Drahtseil Telegr. 37–38, 167–169 (1931).

A. Sommerfeld, Optics, Vol. VI of Lectures on Theoretical Physics (Academic, San Diego, Calif., 1954), Chap. 38 and references therein.

Stratton, J. A.

J. A. Stratton, Théorie del’ électromagnétisme (Dunod, Paris, 1961).

von Fragstein, C.

C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Braunschw. Wiss. Ges. 34, 25–29 (1987).

Wolf, E.

Abh. Braunschw. Wiss. Ges. (1)

C. von Fragstein, “The history of the mixed Poynting vector,” Abh. Braunschw. Wiss. Ges. 34, 25–29 (1987).

Am. J. Phys. (1)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989).
[Crossref]

Ann. Phys. (1)

M. v. Laue, “Die Wärmestrahlung in absorbierenden Körpern,” Ann. Phys. 32, 1085–1094 (1910).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

B. Acklin, C. Bagnoud, M. A. Dupertuis, M. Proctor, F. Morier-Genoud, D. Martin, “Thermally stable operation of a bistable Fabry–Perot étalon with a bulk GaAs spacer,” Appl. Phys. Lett. 60, 3099–3101 (1992).
[Crossref]

Computers Phys. (1)

J. J. Regan, D. R. Andersen, “Reflection and refraction of optical beams at dielectric interfaces,” Computers Phys. 5(1), 49–61 (1991).
[Crossref]

J. Appl. Phys. (1)

P. D. Drummond, A. T. Friberg, “Specular reflection cancellation in an interferometer with a phase-conjugate mirror,” J. Appl. Phys. 54, 5618–5625 (1983).
[Crossref]

J. Mod. Opt. (2)

W. N. Hugrass, “Angular momentum balance on light reflection,” J. Mod. Opt. 37, 339–351 (1990).
[Crossref]

J. M. Saca, “On the velocity of light in uniformly moving dielectrics,” J. Mod. Opt. 37, 227–235 (1990) and references therein.
[Crossref]

J. Opt. Soc. Am. (4)

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

Jahrb. Drahtseil Telegr. (1)

A. Sommerfeld, “Das Reziprozitäts Theorem der drahlosen Telegraphie,” Jahrb. Drahtseil Telegr. 37–38, 167–169 (1931).

Opt. Acta (2)

I. S∼antavý, “On the reversibility of light beams in conducting media,” Opt. Acta 8, 301–307 (1961).
[Crossref]

Z. Knittl, “The principle of reversibility and thin film optics,” Opt. Acta 17, 33–45 (1962).
[Crossref]

Opt. Photon. News (1)

M. A. Dupertuis, M. Proctor, “Generalization of complex Snell-Descartes and Fresnel laws,” Opt. Photon. News 1(9), p. A-126 (1990).

Phys. Rev. (1)

L. Pincherle, “Refraction of plane non-uniform electromagnetic waves between absorbing media,” Phys. Rev. 72, 232–235 (1947).
[Crossref]

Phys. Z. (1)

M. Born, R. Ladenburg, “Über das Verhältnis von Emissions- und Absorptionsvermögen bei Stark absorbieren- den Körpern,” Phys. Z. 12, 198–202 (1911).

Solid State Commun. (1)

B. Chen, D. F. Nelson, “Wavevector space method and its application to the optics near an exciton resonance,” Solid State Commun. 86, 769–773 (1993).
[Crossref]

Sov. Phys. Usp. (1)

V. A. Kizel, “Modern status of the theory of light reflection,” Sov. Phys. Usp. 10, 485–508 (1968).
[Crossref]

Other (11)

W. König, “Elektromagnetische Lichttheorie,” in Licht als Wellenbewegung, H. Konen, ed., Vol. XX of Handbuch der Physik, H. Geiger, K. Scheel, eds. (Springer-Verlag, Berlin, 1928), pp. 141–262.
[Crossref]

Z. Knittl, Optics of Thin Films (An Optical Multilayer Theory) (Wiley, New York, 1976).

F. R. Kessler, “Optics with gradients of free carrier concentration,” in Festkörperprobleme, Vol. 26 of Advances in Solid State PhysicsP. Grosse, ed. (Vieweg, Braunschweig, Germany, 1986), pp. 277–308.

A. Sommerfeld, Optics, Vol. VI of Lectures on Theoretical Physics (Academic, San Diego, Calif., 1954), Chap. 38 and references therein.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

J. A. Stratton, Théorie del’ électromagnétisme (Dunod, Paris, 1961).

M. Born, E. Wolf, Principles of Optics, 1st ed. (Pergamon, New York, 1966).

C. K. Carniglia, “Reflection and transmission at a boundary between two absorbing media: a partial bibliography,” presented at the Thin Films Technical Group meeting, Optical Society of America Annual Meeting, November 8, 1990, Boston, Mass.

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Figures (2)

Fig. 1
Fig. 1

Coordinate systems, vectors, and angles introduced in the text to describe the incident, reflected, and transmitted HIPW’s at the interface. A figurative exponential tail has been drawn along n ̂ to represent the HIPW Although not well represented, n ̂ j is always perpendicular to n ̂ j and makes an angle ψj with ê1j (j = i, r, t).

Fig. 2
Fig. 2

Proposed experiment to test the generalized Snell–Descartes and Fresnel laws in an ITO wedge on a rutile slab. A homogeneous plane wave is incident upon the rutile slab, coupled onto the rutile–ITO interface, where the HIPW is generated. The orientation of the ITO–air final interface is such that ψπ/2; therefore only generalized laws are valid for that interface. Note that the different angles and parameters are indicated in Table 1 and are not to scale in this figure.

Tables (1)

Tables Icon

Table 1 Angles and Parameters of the HIPW and of Each of the Layers in the Proposed Experimenta

Equations (73)

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

E = 1 2 { E 0 exp [ i ( k r ω t ) ] + c . c . } ,
k E 0 = k H 0 = 0 ,
k E 0 = ω μ 0 μ H 0 ,
k H 0 = ω 0 E 0 ,
k 2 ω 2 μ 0 μ 0 = 0 .
k = ñ k 0 n ,
n = n ̂ cosh β + i n ̂ sinh β .
k k * = | ñ | 2 k 0 2 ( 2 i sinh β cosh β ) ( ñ n ̂ ) = 0 ,
k * E 0 = 1 ω 0 k * ( k H 0 ) = 1 ω 0 ( k * k ) H 0 = 0 .
( k k * ) ( E 0 E 0 * ) = | k E 0 * | 2 .
E 0 = E 01 ê 1 + E 02 ê 2 + E 0 n ̂ ,
E 0 = ( i tanh β ) ( E 01 cos ψ + E 02 sin ψ ) ,
E = { Re ( E 0 ) cos [ Re ( k ) r ω t ] + Im ( E 0 ) sin [ Re ( k ) r ω t ] } exp [ Im ( k ) r ] ,
n ̂ E Re ( E 0 ) Im ( E 0 ) = 1 4 i E 0 E 0 * .
E 0 E 0 * = 0 .
E 0 2 k = E 0 ( E 0 k ) .
( E 0 2 E 0 * 2 ) ( k k * ) = ( ω μ 0 μ ) 2 ( E 0 H 0 ) ( E 0 * H 0 * ) = ( ω μ 0 μ ) 2 [ ( H 0 H 0 * ) E 0 ] E 0 * + [ ( E 0 E 0 * ) H 0 ] H 0 * .
E 0 2 = 0 .
H 0 2 = 0 E 0 H 0 = 0 ,
E H = 1 2 Re ( E 0 H 0 * ) ( ñ n ñ * n * ) ( E 0 E 0 * ) .
E 01 = A exp ( i δ ) ( cos φ cos ɛ i sin φ sin ɛ ) ,
E 02 = A exp ( i δ ) ( sin φ cos ɛ i cos φ sin ɛ ) ,
cos θ j = ŝ n ̂ j , sin θ j = ê i j ( ŝ n ̂ j ) ,
cos Δ = ê 1 i ê 1 t , sin Δ = ŝ ( ê 1 i ê 1 t ) .
k i r = k r r = k t r ,
( ŝ k i ) ( ŝ r ) = ( ŝ k r ) ( ŝ r ) = ( ŝ k t ) ( ŝ r ) .
ŝ k i = ŝ k r = ŝ k t .
ŝ k r = ŝ k i ,
cosh β r sin θ r = cosh β i sin θ i ,
sinh β r sin ψ r cos θ r = sinh β i sin ψ i cos θ i ,
sinh β r cos ψ r = sinh β i cos ψ i .
| ñ t | cosh β t sin θ t cos Δ = | ñ i | [ cos ( η i η t ) cosh β i sin θ i + sin ( η i η t ) sinh β i sin ψ i cos θ i ] ,
| ñ t | cosh β t sin θ t sin Δ = | ñ i | sin ( η i η t ) sinh β i cos ψ i ,
| ñ t | ( sinh β t ) [ cos ψ t sin Δ + sin ψ t cos θ t cos Δ ] = | ñ i | [ sin ( η i η t ) cosh β i sin θ i + cos ( η i η t ) sinh β i sin ψ i cos θ i ] ,
| ñ t | ( sinh β t ) [ cos ψ t cos Δ sin ψ t cos θ t sin Δ ] = | ñ i | cos ( η i η t ) sinh β i cos ψ i .
sin θ r = sin θ i ,
ñ t sin θ t = ñ i sin θ i ,
θ j = θ j i β j , j = i , r , t .
ψ i = π 2 ψ t = ψ r = π 2 .
ŝ ( n n * ) = 0 or ŝ ( n ̂ n ̂ ) = 0 .
θ r = π θ i ,
ψ r = ψ i ,
β r = β i .
sinh 2 β r = sinh 2 β i sin 2 θ i cos 2 ψ i cos 2 θ i .
| ñ t | sinh β t sin ψ t cos θ t = | ñ i | sin ( η i η t ) ( sin θ i cos Δ ) .
ŝ E i + ŝ E r ŝ E t = 0 ,
μ i ŝ ( H i + H r ) μ t ŝ ( H t ) = 0 ,
ŝ H i + ŝ H r ŝ H t = 0 ,
i ŝ ( E i + E r ) t ŝ ( E t ) = Σ 2 .
i σ t t σ i 0 .
q 1 = ŝ n [ ( ŝ n ) 2 ] 1 / 2 , q 2 = n q 1 ,
E PE = E PE q 1 , H PE = H PE q 2 ,
E PM = E PM q 2 , H PM = H PM q 1 ,
E PE = q 1 E 0 , H PE = q 2 H 0 ,
E PM = q 2 E 0 , H PM = q 1 H 0 .
E PE = A PE exp ( i δ PE ) [ e PE ( i tanh β ) ( n ̂ e PE ) n ̂ ] ,
e PE = ê 1 + χ PE ê 2 ( 1 + | χ PE | 2 ) 1 / 2 ,
χ PE = i tanh β cos θ cos ψ sin θ i tanh β cos θ sin ψ .
E PE r = r PE E PE i with r PE = ( ŝ k i / μ i ŝ k t / μ t ŝ k i / μ i + ŝ k t / μ t ) ,
E PE t = t PE E PE i with t PE = ( 2 ŝ k i / μ i ŝ k i / μ i + ŝ k t / μ t ) .
H PM r = r PM H PM i with r PM = ( ŝ k i / i ŝ k t / t ŝ k i / i + ŝ k t / t ) ,
H PM t = t PM H PM i with t PM = ( 2 ŝ k i / i ŝ k i / i + ŝ k t / t ) ,
E PM t = [ 2 ñ t ( ŝ k i ) ñ i ( t / i ) ( ŝ k i ) + ñ i ( ŝ k t ) ] E PM i .
a b = i = 1 3 a i b i = b a ,
a b = i = 1 3 i j k a j b k = b a ,
a ( b c ) = c ( a b ) = b ( c a ) ,
a ( b c ) = ( a c ) b ( a b ) c ,
a ( b c ) + b ( c a ) + c ( a b ) = 0 ,
( a b ) ( c d ) = ( c a ) ( d b ) ( d a ) ( c b ) ,
( a b ) ( c d ) = [ ( c d ) a ] b [ ( c d ) b ] a = [ ( b d ) a ] c [ ( b c ) a ] d .
[ ( b c ) a ] x = [ ( b c ) x ] a + [ ( c a ) x ] b + [ ( a b ) x ] c .
a b = i = 1 3 a i * b i = a * b
a b = i = 1 3 ijk a j * b k * = a * b * = b a .

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