Abstract

In this review paper, we summarize the fundamental properties of inhomogeneous waves at the planar interface between two media. We point out the main differences between the wave types: lateral waves, surface waves, and leaky waves. We analyze each kind of inhomogeneous wave, giving a quasi-optical description and explaining the physical origin of some of their properties.

© 2015 Optical Society of America

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References

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  3. R. M. Whitmer, “Fields in nonmetallic waveguides,” Proc. IRE 36, 1105–1109 (1948).
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    [Crossref]
  33. T. Tamir, “The lateral wave,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 13.
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    [Crossref]
  35. F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
    [Crossref]
  36. R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, 1992).
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    [Crossref]
  38. T. Tamir and A. A. Oliner, “Role of the lateral wave in total reflection of light,” J. Opt. Soc. Am. 59, 942–949 (1969).
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    [Crossref]
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    [Crossref]
  42. K. Artman, “Berechung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
    [Crossref]
  43. A. W. Snyder and J. D. Love, “Goos–Hänchen shift,” Appl. Opt. 15, 236–238 (1976).
    [Crossref]
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    [Crossref]
  47. B. Friedman, “Surface waves over a lossy conductor,” IRE Trans. Antennas Propag. 7, 227–230 (1959).
    [Crossref]
  48. E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons–propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1119 (1974).
    [Crossref]
  49. C. A. Ward, R. J. Bell, R. W. Alexander, G. S. Kovener, and I. Tyler, “Surface electromagnetic waves on metals and polar insulators: some comments,” Appl. Opt. 13, 2378–2381 (1974).
    [Crossref]
  50. P. Halevi, “Polariton modes at the interface between two conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
    [Crossref]
  51. M. Fox, Optical Properties of Solids, 2nd ed. (Oxford University, 2010), Chap. 7.
  52. H. Nassenstein, “Superresolution by diffraction of subwaves,” Opt. Commun. 2, 231–234 (1970).
    [Crossref]
  53. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
  54. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
    [Crossref]
  55. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22, 475–477 (1997).
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    [Crossref]
  57. O. Madelung, Semiconductors: Data Handbook, 3rd ed. (Springer, 2004).
  58. R. Ruppin and R. Englman, “Optical phonon of small crystals,” Rep. Prog. Phys. 33, 149–196 (1970).
    [Crossref]
  59. D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
    [Crossref]
  60. P. Halevi, “Polaritons at the interface between two dielectric media,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 7.
  61. V. H. Rumsey, “Traveling wave slot antennas,” J. Appl. Phys. 24, 1358–1365 (1953).
    [Crossref]
  62. R. F. Harrington, “Propagation along a slotted cylinder,” J. Appl. Phys. 24, 1366–1371 (1953).
    [Crossref]
  63. L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas I: rectangular waveguides,” IRE Trans. Antennas Propag. 7, 307–319 (1959).
    [Crossref]
  64. P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectric-loaded open guiding structure,” IEEE Trans. Antennas Propag. 38, 1831–1836 (1990).

2012 (2)

2011 (3)

Y. Wang, S. H. Helmy, and G. V. Eleftheriades, “Ultra-wideband optical leaky-wave slot antennas,” Opt. Express 19, 12392–12401 (2011).
[Crossref]

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

S. Maci, “Metasurfing: addressing waves on impenetrable metasurfaces,” IEEE Antennas Wirel. Propag. Lett. 10, 1499–1502 (2011).
[Crossref]

2010 (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[Crossref]

2006 (1)

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A 8, S87–S93 (2006).
[Crossref]

2005 (1)

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).

1997 (2)

1993 (1)

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

1992 (1)

M. Wabia, “Lateral waves in anisotropic optical waveguides,” Acta Phys. Pol. A 81, 503–516 (1992).

1990 (1)

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectric-loaded open guiding structure,” IEEE Trans. Antennas Propag. 38, 1831–1836 (1990).

1987 (1)

J. H. Richmond, L. Peters, and R. A. Hill, “Surface waves on a lossy planar ferrite slab,” IEEE Trans. Antennas Propag. 35, 802–808 (1987).
[Crossref]

1980 (1)

1979 (1)

F. Bardati and P. Lampariello, “The modal spectrum of a lossy ferrimagnetic slab,” IEEE Trans. Microwave Theory Tech. 27, 679–688 (1979).

1978 (1)

P. Halevi, “Polariton modes at the interface between two conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
[Crossref]

1976 (1)

1974 (3)

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons–propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1119 (1974).
[Crossref]

C. A. Ward, R. J. Bell, R. W. Alexander, G. S. Kovener, and I. Tyler, “Surface electromagnetic waves on metals and polar insulators: some comments,” Appl. Opt. 13, 2378–2381 (1974).
[Crossref]

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

1973 (2)

T. Tamir, “Inhomogeneous waves types at planar structures: II. Surface waves,” Optik 37, 204–228 (1973).

T. Tamir, “Inhomogeneous waves types at planar structures: III. Leaky waves,” Optik 38, 269–297 (1973).

1972 (1)

T. Tamir, “Inhomogeneous waves types at planar structures: I. The lateral wave,” Optik 36, 209–232 (1972).

1971 (1)

1970 (2)

H. Nassenstein, “Superresolution by diffraction of subwaves,” Opt. Commun. 2, 231–234 (1970).
[Crossref]

R. Ruppin and R. Englman, “Optical phonon of small crystals,” Rep. Prog. Phys. 33, 149–196 (1970).
[Crossref]

1969 (1)

1965 (1)

T. Tamir and L. B. Felsen, “On the lateral waves in slab configurations and their relation to other wave types,” IEEE Trans. Antennas Propag. 13, 410–422 (1965).
[Crossref]

1963 (1)

T. Tamir and A. A. Oliner, “Guided complex waves Part 1. Fields at an interface,” Proc. IEE 110, 310–324 (1963).

1959 (4)

G. Goubau, “Waves on interfaces,” IRE Trans. Antennas Propag. 7, 140–146 (1959).
[Crossref]

S. A. Schelkunoff, “Anatomy of surface waves,” IRE Trans. Antennas Propag. 7, 133–139 (1959).
[Crossref]

B. Friedman, “Surface waves over a lossy conductor,” IRE Trans. Antennas Propag. 7, 227–230 (1959).
[Crossref]

L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas I: rectangular waveguides,” IRE Trans. Antennas Propag. 7, 307–319 (1959).
[Crossref]

1957 (1)

R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106, 874–881 (1957).
[Crossref]

1953 (3)

J. Brown, “The types of wave which may exist near a guiding surface,” Proc. IEE 100, 363–364 (1953).

V. H. Rumsey, “Traveling wave slot antennas,” J. Appl. Phys. 24, 1358–1365 (1953).
[Crossref]

R. F. Harrington, “Propagation along a slotted cylinder,” J. Appl. Phys. 24, 1366–1371 (1953).
[Crossref]

1952 (1)

F. J. Zucker, “Theory and applications of surface waves,” Nuovo Cimento 9, 450–473 (1952).
[Crossref]

1951 (1)

N. Marcuvitz, “Field representation in spherically stratified regions,” Commun. Pure Appl. Math. 4, 263–315, 1951.
[Crossref]

1950 (1)

G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21, 1119–1128 (1950).

1948 (2)

R. M. Whitmer, “Fields in nonmetallic waveguides,” Proc. IRE 36, 1105–1109 (1948).
[Crossref]

K. Artman, “Berechung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[Crossref]

1933 (1)

M. Muskat, “Potential distribution about an electrode on the surface of the earth,” Physics 4, 129–147 (1933).
[Crossref]

1909 (1)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Physik 333, 665–736 (1909).
[Crossref]

1907 (1)

J. Zenneck, “Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Physik 328, 846–866 (1907).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Chap. 19.

Adler, R. B.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmission and Radiation (MIT, 1969), Chap. 8.

Alexander, R. W.

Artman, K.

K. Artman, “Berechung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 437, 87–102 (1948).
[Crossref]

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), Chap. 8.

Baños, A.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, 1966).

Bardati, F.

F. Bardati and P. Lampariello, “The modal spectrum of a lossy ferrimagnetic slab,” IEEE Trans. Microwave Theory Tech. 27, 679–688 (1979).

Barlow, H. M.

H. M. Barlow and J. Brown, Radio Surface Waves (Clarendon, 1962).

Barnes, W. L.

W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A 8, S87–S93 (2006).
[Crossref]

Bell, R. J.

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[Crossref]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).

Brown, J.

J. Brown, “The types of wave which may exist near a guiding surface,” Proc. IEE 100, 363–364 (1953).

H. M. Barlow and J. Brown, Radio Surface Waves (Clarendon, 1962).

Burstein, E.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons–propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1119 (1974).
[Crossref]

Chen, W. P.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons–propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1119 (1974).
[Crossref]

Chen, Y. J.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons–propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1119 (1974).
[Crossref]

Chu, L. J.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmission and Radiation (MIT, 1969), Chap. 8.

Cincotti, G.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Eleftheriades, G. V.

Englman, R.

R. Ruppin and R. Englman, “Optical phonon of small crystals,” Rep. Prog. Phys. 33, 149–196 (1970).
[Crossref]

Fano, R. M.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmission and Radiation (MIT, 1969), Chap. 8.

Felsen, L. B.

T. Tamir and L. B. Felsen, “On the lateral waves in slab configurations and their relation to other wave types,” IEEE Trans. Antennas Propag. 13, 410–422 (1965).
[Crossref]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 5.

L. B. Felsen, “Lateral waves,” (Polytechnic Institute of Brooklyn, New York, Microwave Research Institute, 1965).

Fox, M.

M. Fox, Optical Properties of Solids, 2nd ed. (Oxford University, 2010), Chap. 7.

Frezza, F.

F. Frezza and N. Tedeschi, “On the electromagnetic power transmission between two lossy media: discussion,” J. Opt. Soc. Am. A 29, 2281–2288 (2012).
[Crossref]

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectric-loaded open guiding structure,” IEEE Trans. Antennas Propag. 38, 1831–1836 (1990).

Friedman, B.

B. Friedman, “Surface waves over a lossy conductor,” IRE Trans. Antennas Propag. 7, 227–230 (1959).
[Crossref]

Furnò, F.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Goldstone, L. O.

L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas I: rectangular waveguides,” IRE Trans. Antennas Propag. 7, 307–319 (1959).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[Crossref]

Gori, F.

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

Goubau, G.

G. Goubau, “Waves on interfaces,” IRE Trans. Antennas Propag. 7, 140–146 (1959).
[Crossref]

G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21, 1119–1128 (1950).

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[Crossref]

Halevi, P.

P. Halevi, “Polariton modes at the interface between two conducting or dielectric media,” Surf. Sci. 76, 64–90 (1978).
[Crossref]

P. Halevi, “Polaritons at the interface between two dielectric media,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 7.

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[Crossref]

Harrington, R. F.

R. F. Harrington, “Propagation along a slotted cylinder,” J. Appl. Phys. 24, 1366–1371 (1953).
[Crossref]

Hartstein, A.

E. Burstein, W. P. Chen, Y. J. Chen, and A. Hartstein, “Surface polaritons–propagating electromagnetic modes at interfaces,” J. Vac. Sci. Technol. 11, 1004–1119 (1974).
[Crossref]

Helmy, S. H.

Hill, R. A.

J. H. Richmond, L. Peters, and R. A. Hill, “Surface waves on a lossy planar ferrite slab,” IEEE Trans. Antennas Propag. 35, 802–808 (1987).
[Crossref]

Horowitz, B. R.

Jackson, D. R.

D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, ed. (Wiley, 2008), Chap. 7.

King, R. W. P.

R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, 1992).

Kobayashi, T.

Kovener, G. S.

Lampariello, P.

P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectric-loaded open guiding structure,” IEEE Trans. Antennas Propag. 38, 1831–1836 (1990).

F. Bardati and P. Lampariello, “The modal spectrum of a lossy ferrimagnetic slab,” IEEE Trans. Microwave Theory Tech. 27, 679–688 (1979).

Love, J. D.

Maci, S.

A. Polemi and S. Maci, “A leaky-wave groove antenna at optical frequency,” J. Appl. Phys. 112, 074320 (2012).
[Crossref]

S. Maci, “Metasurfing: addressing waves on impenetrable metasurfaces,” IEEE Antennas Wirel. Propag. Lett. 10, 1499–1502 (2011).
[Crossref]

Madelung, O.

O. Madelung, Semiconductors: Data Handbook, 3rd ed. (Springer, 2004).

Maier, S. A.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).

Marcuvitz, N.

N. Marcuvitz, “Field representation in spherically stratified regions,” Commun. Pure Appl. Math. 4, 263–315, 1951.
[Crossref]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 5.

Mills, D. L.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817–926 (1974).
[Crossref]

Morimoto, A.

Moskalova, M. A.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, and V. A. Yakovlev, “Surface electromagnetic wave propagation on metal surfaces,” in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, 1982), Chap. 3

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[Crossref]

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[Crossref]

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[Crossref]

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P. Lampariello, F. Frezza, and A. A. Oliner, “The transition region between bound-wave and leaky-wave ranges for a partially dielectric-loaded open guiding structure,” IEEE Trans. Antennas Propag. 38, 1831–1836 (1990).

T. Tamir and A. A. Oliner, “Role of the lateral wave in total reflection of light,” J. Opt. Soc. Am. 59, 942–949 (1969).

T. Tamir and A. A. Oliner, “Guided complex waves Part 1. Fields at an interface,” Proc. IEE 110, 310–324 (1963).

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F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
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M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Chap. 19.

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T. Tamir and A. A. Oliner, “Role of the lateral wave in total reflection of light,” J. Opt. Soc. Am. 59, 942–949 (1969).

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T. Tamir and A. A. Oliner, “Guided complex waves Part 1. Fields at an interface,” Proc. IEE 110, 310–324 (1963).

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F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
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S. Stein and M. Wysession, An Introduction to Seismology, Earthquakes, and Earth Structure (Blackwell, 2003), Chap. 3.

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G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, and V. A. Yakovlev, “Surface electromagnetic wave propagation on metal surfaces,” in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, 1982), Chap. 3

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G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, and V. A. Yakovlev, “Surface electromagnetic wave propagation on metal surfaces,” in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, 1982), Chap. 3

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J. Zenneck, “Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Ann. Physik 328, 846–866 (1907).
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F. J. Zucker, “Theory and applications of surface waves,” Nuovo Cimento 9, 450–473 (1952).
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Opt. Commun. (3)

G. Cincotti, F. Gori, M. Santarsiero, F. Frezza, F. Furnò, and G. Schettini, “Plane wave expansion of cylindrical functions,” Opt. Commun. 95, 192–198 (1993).
[Crossref]

F. Frezza, G. Schettini, and N. Tedeschi, “Generalized plane-wave expansion of cylindrical functions in lossy media convergent in the whole complex plane,” Opt. Commun. 284, 3867–3871 (2011).
[Crossref]

H. Nassenstein, “Superresolution by diffraction of subwaves,” Opt. Commun. 2, 231–234 (1970).
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Opt. Lett. (2)

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T. Tamir, “Inhomogeneous waves types at planar structures: I. The lateral wave,” Optik 36, 209–232 (1972).

T. Tamir, “Inhomogeneous waves types at planar structures: II. Surface waves,” Optik 37, 204–228 (1973).

T. Tamir, “Inhomogeneous waves types at planar structures: III. Leaky waves,” Optik 38, 269–297 (1973).

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Kh. V. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett. A 237, 103–105, 1997.
[Crossref]

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R. W. P. King, M. Owens, and T. T. Wu, Lateral Electromagnetic Waves (Springer-Verlag, 1992).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972), Chap. 19.

G. N. Zhizhin, M. A. Moskalova, E. V. Shomina, and V. A. Yakovlev, “Surface electromagnetic wave propagation on metal surfaces,” in Surface Polaritons, V. M. Agranovich and D. L. Mills, eds. (North-Holland, 1982), Chap. 3

P. Halevi, “Polaritons at the interface between two dielectric media,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 7.

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).

A. Otto, “Experimental investigation of surface polaritons on plane interfaces,” in Advances in Solid State Physics, H. J. Queisser, ed. (Pergamon, 1974).

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, 1966).

C. A. Balanis, Advanced Engineering Electromagnetics, 2nd ed. (Wiley, 2012), Chap. 8.

D. R. Jackson and A. A. Oliner, “Leaky-wave antennas,” in Modern Antenna Handbook, C. A. Balanis, ed. (Wiley, 2008), Chap. 7.

H. M. Barlow and J. Brown, Radio Surface Waves (Clarendon, 1962).

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Energy Transmission and Radiation (MIT, 1969), Chap. 8.

L. B. Felsen, “Lateral waves,” (Polytechnic Institute of Brooklyn, New York, Microwave Research Institute, 1965).

S. Stein and M. Wysession, An Introduction to Seismology, Earthquakes, and Earth Structure (Blackwell, 2003), Chap. 3.

T. Tamir, “The lateral wave,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, 1982), Chap. 13.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), Chap. 5.

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Figures (15)

Fig. 1.
Fig. 1. Geometry of a slab with thickness d between two half-spaces.
Fig. 2.
Fig. 2. Complex plane of the variable k x = k x j k x of the integral in Eq. (1), in the case of a line current above a dielectric slab. The black points represent the branch points of the spectrum, and the crosses represent the poles. The four dashed lines represent the branch cuts of the spectrum.
Fig. 3.
Fig. 3. (a) Sketch of a plane wave V i incident at the angle of total reflection, with the reflected and transmitted waves. (b) Sketch of a plane wave traveling along the boundary and transmitted in medium 1 at the angle of total reflection. (c) Quasi-optical representation of a lateral wave.
Fig. 4.
Fig. 4. Geometrical rays representing the incidence of a cylindrical wave generated by a line current centered on C. Three paths are described: CP ¯ is the path of the direct ray from the source to the observation point, CT ¯ + TP ¯ is the path of the reflected ray, and CA ¯ + AB ¯ + BP ¯ is the path of the lateral ray.
Fig. 5.
Fig. 5. (a) Representation of the cylindrical wavefronts of the incident, reflected, and transmitted waves, V i , V r , and V t , respectively, due to a line source centered on a point on the z axis and placed at a distance h from the interface. The segment ST ¯ represents the plane front of the lateral wave that makes the field at the interface continuous. (b) Representation of the elementary plane wave, relevant to the spectrum of the field excited by the line source, at the incident angle θ L .
Fig. 6.
Fig. 6. Representation of a ray with a finite plane front, of width 2 w , incident on a prism with refraction index n 1 immersed in a medium with index n 2 < n 1 . The incident angle of the beam on the horizontal interface of the prism is θ L . The beam is totally reflected. The dashed arrows represent the geometrical reflected beam, while the solid arrows represent the actual reflected beam, shifted a distance D . The gray arrows represent the lateral wave because of the total reflection.
Fig. 7.
Fig. 7. (a) Magnitude, in a logarithmic scale, normalized with respect to its maximum value and (b) phase, in degrees, of the field V 0 in Eq. (22) (dashed line). Also, (a) magnitude, in a logarithmic scale, normalized with respect to the maximum value of V 0 and (b) phase, in degrees, of the field V 1 in Eq. (23) (solid line). The interface is considered between the medium with refractive index n 1 = 1.94 and air, n 2 = 1 , while the incident wave has a half-width of w = 1000 λ 2 . The plots are considered in the case z r = 0 .
Fig. 8.
Fig. 8. Dispersion behavior (solid line) of the component of the propagation vector parallel to the interface, k x , of a surface wave, Eq. (38), for the interface between air and silver, where the permittivity of silver is obtained by the Drude model in Eq. (42). The variables are normalized with respect to the plasma angular frequency ω p , Eq. (43). The light speed line is represented by the dashed line. The two horizontal dotted lines correspond to the angular frequencies ω sp and ω p .
Fig. 9.
Fig. 9. Dispersion behavior (solid line) of the component of the propagation vector parallel to the interface, k x , of a surface wave, Eq. (38), for the interface between air and gallium arsenide, where the permittivity of the gallium arsenide is obtained by Eq. (47). The variables are normalized with respect to the TO angular frequency ω TO . The light speed line is represented by the dotted line. The two horizontal dashed lines represent the angular frequencies ω TO and ω LO .
Fig. 10.
Fig. 10. (a) Sketch of a grounded dielectric slab with thickness t and relative permittivity ε 2 . (b) The equivalent transmission line of the grounded slab in the direction transverse to the propagation.
Fig. 11.
Fig. 11. Dispersion curves of a grounded dielectric slab, with n = 1.5 , as a function of normalized quantities. The curves correspond to the real solutions of Eqs. (55) and (56), indicated with (dashed line) TE n and (solid line) TM n , respectively, with n = 0,1 , 2 .
Fig. 12.
Fig. 12. Sketch of the (a) forward and (b) backward leaky waves. The dashed lines represent the constant-amplitude planes. The direction of the components of the phase and attenuation vectors are specified.
Fig. 13.
Fig. 13. Dispersion curve in the transition region from a leaky wave to a surface wave.
Fig. 14.
Fig. 14. Complex propagation vector of (a) a leaky wave, (b) a nonphysical wave in the transition region, (c) a surface wave at the cutoff, and (d) a surface wave above the cutoff.
Fig. 15.
Fig. 15. Geometrical-optics interpretation of a leaky wave: an optic ray is generated in A, refracted inside a dielectric slab, partially reflected at the points O, C, E, and G, and totally reflected on the ground plane at the points B, D, and F. The wave that emerges in the upper space ( z > 0 ) occupies a wedge-shaped region, is attenuated in the x direction, and is amplified in the z direction.

Tables (1)

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Table 1. Summary of Surface Waves

Equations (69)

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V int ( x , z ) = 1 2 π + s ( k x ) e j ( k x x + k z z ) d k x ,
Z ( k z ) + Z 1 ( k z ) = 0 .
k x = k 2 sin θ i = k 2 k 1 θ i = θ L ,
V L = A 2 j m ε 2 ε 1 ε 2 exp { j [ k 1 ( L 1 + L 2 ) + k 2 L ] } ( k 2 L ) 3 / 2 ,
ϕ L = k 1 ( L 1 + L 2 ) + k 2 L .
ϕ i = k 1 x sin θ 1 ,
ϕ r = k 1 x sin θ 2 ,
ϕ L = k 1 ( L 1 + L 2 ) + k 1 L sin θ L = k 1 { h cos θ L + z cos θ L + [ x ( h + z ) tan θ L ] sin θ L } = k 1 x ( h + z x cos θ L + sin θ L ) .
h + z x = cot θ 2 ,
cos ( θ L θ 2 ) sin θ 2 = cos θ L cot θ 2 + sin θ L .
ϕ L = k 1 x sin θ 2 cos ( θ L θ 2 ) .
ϕ L = ϕ i sin θ 1 sin θ 2 cos ( θ L θ 2 ) .
x c = 2 h cos θ L 1 sin θ L = 2 h ( v 2 + v 1 v 2 v 1 ) 1 / 2 .
OT ¯ = OS ¯ + S T ¯ ,
OS ¯ = h tan θ L ,
S T ¯ = S S ¯ sin θ L = CS ¯ CS ¯ sin θ L = 1 sin θ L ( c t n 1 h cos θ L ) .
c ( t t L ) n 2 = h tan θ L + 1 sin θ L ( c t n 1 h cos θ L ) .
c t L n 2 = c t ( 1 n 2 1 n 1 sin θ L ) + h cot θ L .
1 n 2 1 n 1 sin θ L = 0 sin θ L = n 2 n 1 ,
c t L n 1 = h cos θ L .
V i = exp [ j k 1 z i ( x i / w ) 2 ] π w .
V 0 = R ( k x i ) exp [ j k 1 z r ( x r / w ) 2 ] π w ,
V 1 = A ( θ i ) V 0 { ( δ ) 1 / 2 [ 2 exp ( γ 2 j π ) ] 1 / 4 k 1 w D 1 / 2 ( γ ) } ,
A ( θ i ) = 4 m cos 2 θ L sin θ i [ cos 2 θ i + m 2 ( sin 2 θ i sin 2 θ L ) ] cos θ i ( sin θ i + sin θ L ) ,
γ = 2 ( i k 1 w δ 2 x r w ) ,
δ = ( sin θ i sin θ L ) sec θ θ i θ L ,
V 1 k 1 2 π R ( k x ) A ( θ i ) exp [ ( k 1 w δ / 2 ) 2 ] exp [ j ( k 2 x + π / 4 ) ] ( k 1 x cos θ i ) 3 / 2 .
E 1 y = A 1 exp [ j ( k x x + k 1 z z ) ] ,
E 2 y = A 2 exp [ j ( k x x + k 2 z z ) ] ,
A 1 = A 2 ,
A 1 k 1 z = A 2 k 2 z .
α 1 z + α 2 z = 0 .
H 1 y = A 1 exp [ j ( k x x + k 1 z z ) ] ,
H 2 y = A 2 exp [ j ( k x x + k 2 z z ) ] ,
A 1 = A 2 ,
A 1 k 1 z ω ε 0 ε 1 = A 2 k 2 z ω ε 0 ε 2 .
ε 2 α 1 z + ε 1 α 2 z = 0 .
k x = k 0 ε 1 ε 2 ε 1 + ε 2 .
R TE = k 1 z k 2 z k 1 z k 2 z ,
R TM = ε 2 k 1 z ε 1 k 2 z ε 2 k 1 z ε 1 k 2 z .
s ( k x ) = R ( k x ) s ( k x ) ,
ε 2 = 1 ω p 2 ω 2 ,
ω p = N e 2 ε 0 m 0
ω < ω sp = ω p 1 + ε 1 ,
λ s = 2 π k x = λ 0 ε 1 + ε 2 ε 1 ε 2 .
v ϕ = ω k x = ω λ s 2 π = c ε 1 + ε 2 ε 1 ε 2 .
ε 2 = ε + ( ε st ε ) ω TO 2 ω TO 2 ω 2 ,
ω LO = ω TO ε st ε .
ω s = ω TO ε st + ε 1 ε + ε 1 .
Z 0 = ω μ 0 k 0 z , Z 2 = ω μ 0 k 2 z in TE polarization ;
Z 0 = k 0 z ω ε 0 , Z 2 = k 2 z ω ε 0 ε 2 in TM polarization ;
Z + Z = 0 .
Z = j Z 2 tan ( k 2 z t ) .
Z 0 cos ( k 2 z t ) + j Z 2 sin ( k 2 z t ) = 0 .
k 2 z cos ( k 2 z t ) + j k 0 z sin ( k 2 z t ) = 0 in TE polarization ,
k 0 z cos ( k 2 z t ) + j k 2 z ε 2 sin ( k 2 z t ) = 0 in TM polarization .
| β ̲ | 2 | α ̲ | 2 = β 2 α 2 = k 0 2 ,
β ̲ · α ̲ = β x α x + β z α z = 0 .
sin θ 0 β x k 0 .
L λ 0 0.183 k 0 α x ,
Δ θ α x k 0 .
E 0 = T e j k 1 ( x sin θ 0 + z cos θ 0 ) ,
E r = ( R ) r T e 2 r j k 2 d cos θ 1 e j k 1 ( x sin θ 0 + z cos θ 0 ) .
x z tan θ 0 = 2 r d tan θ 1 .
R = ( 1 δ ) e j Δ ,
R e δ e j Δ .
E r T exp { j [ x ( k 1 sin θ 0 + k 2 sin θ 1 + Δ j δ 2 d tan θ 1 ) + z ( k 1 cos θ 0 k 2 tan θ 0 sin θ 1 Δ j δ 2 d tan θ 1 tan θ 0 ) ] } .
k x = k 1 sin θ 0 + k 2 sin θ 1 + Δ j δ 2 d tan θ 1 ,
k z = k 1 cos θ 0 k 2 tan θ 0 sin θ 1 Δ j δ 2 d tan θ 1 tan θ 0 .

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