Abstract

An overview of the problems involved in the study of electromagnetic power transmission between lossy media is presented. Starting from the well-known problem of the transmission at a dielectric–conductor interface, the different representations of the complex propagation vector of the plane waves are introduced. Analytical expressions to convert from one formulation to the other are obtained. Moreover, the transmission of a plane wave at the interface between two lossy media is taken into account. An explanation of the strange behavior of the transmitted wave is developed by means of power considerations. Finally, the interesting effect of the parallel-attenuated transmitted wave is presented, and its properties as a function of the incident phase vector amplitude are deduced.

© 2012 Optical Society of America

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References

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  5. R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, 1964).
  6. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).
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  8. F. X. Canning, “Corrected Fresnel coefficients for lossy materials,” in Proceedings of IEEE International Symposium on Antennas and Propagation (IEEE, 2011), pp. 2123–2126.
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    [CrossRef]
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    [CrossRef]
  11. J. W. Gibbs and E. B. Wilson, Vector Analysis (Scribner, 1901).
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    [CrossRef]
  14. R. De Roo and C.-T. Tai, “Plane wave reflection and refraction involving a finitely conducting medium,” IEEE Antennas Propag. Mag. 45(5), 54–61 (2003).
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  19. W. König, Handbuch der Physik, Vol. XX (Springer-Verlag, 1928).
  20. P. E. Ciddor, “Refraction into an absorbing medium,” Am. J. Phys. 44, 786–787 (1976).
    [CrossRef]
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  22. T. Wriedt and A. Doicu, “Light scattering from particles on or near a surface,” Opt. Commun. 152, 376–384 (1998).
    [CrossRef]
  23. F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
    [CrossRef]
  24. R. D. Radcliff and C. A. Balanis, “Modified propagation constants for nonuniform plane wave transmission through conducting media,” IEEE Trans. Geosci. Remote Sens. GE-20, 408–411 (1982).
    [CrossRef]
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2012 (1)

2011 (2)

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

I. M. Besieris, “Comment on the ‘Corrected Fresnel coefficients for lossy materials’,” IEEE Antennas Propag. Mag. 53(4), 161–164 (2011).
[CrossRef]

2003 (2)

J. E. Roy, “New results for the effective propagation constants of nonuniform plane waves at the planar interface of two lossy media,” IEEE Trans. Antennas Propag. 51, 1206–1215 (2003).
[CrossRef]

R. De Roo and C.-T. Tai, “Plane wave reflection and refraction involving a finitely conducting medium,” IEEE Antennas Propag. Mag. 45(5), 54–61 (2003).
[CrossRef]

1998 (1)

T. Wriedt and A. Doicu, “Light scattering from particles on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

1997 (1)

J. R. Wait, “A note on the pseudo-Brewster angle,” IEEE Antennas Propag. Mag. 39(4), 68 (1997).
[CrossRef]

1992 (1)

E. I. Ivlev, “Scattering of inhomogeneous electromagnetic waves by a cylinder,” J. Mod. Opt. 39, 499–507 (1992).
[CrossRef]

1987 (1)

E. I. Ivlev, “Structure and properties of inhomogeneous waves,” J. Mod. Opt. 34, 1559–1569 (1987).
[CrossRef]

1984 (1)

1982 (1)

R. D. Radcliff and C. A. Balanis, “Modified propagation constants for nonuniform plane wave transmission through conducting media,” IEEE Trans. Geosci. Remote Sens. GE-20, 408–411 (1982).
[CrossRef]

1976 (1)

P. E. Ciddor, “Refraction into an absorbing medium,” Am. J. Phys. 44, 786–787 (1976).
[CrossRef]

1973 (1)

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

1967 (1)

1964 (1)

1963 (1)

T. Tamir and A. A. Oliner, “Complex guided waves,” Proc. IEE 110, 310–334 (1963).
[CrossRef]

Adler, R. B.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Transmission and Radiation (Wiley, 1972).

Balanis, C. A.

R. D. Radcliff and C. A. Balanis, “Modified propagation constants for nonuniform plane wave transmission through conducting media,” IEEE Trans. Geosci. Remote Sens. GE-20, 408–411 (1982).
[CrossRef]

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

Besieris, I. M.

I. M. Besieris, “Comment on the ‘Corrected Fresnel coefficients for lossy materials’,” IEEE Antennas Propag. Mag. 53(4), 161–164 (2011).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959).

Canning, F. X.

F. X. Canning, “Corrected Fresnel coefficients for lossy materials,” in Proceedings of IEEE International Symposium on Antennas and Propagation (IEEE, 2011), pp. 2123–2126.

Chu, L. J.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Transmission and Radiation (Wiley, 1972).

Ciddor, P. E.

P. E. Ciddor, “Refraction into an absorbing medium,” Am. J. Phys. 44, 786–787 (1976).
[CrossRef]

De Roo, R.

R. De Roo and C.-T. Tai, “Plane wave reflection and refraction involving a finitely conducting medium,” IEEE Antennas Propag. Mag. 45(5), 54–61 (2003).
[CrossRef]

Doicu, A.

T. Wriedt and A. Doicu, “Light scattering from particles on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Eleftheriades, G. V.

Fano, R. M.

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Transmission and Radiation (Wiley, 1972).

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, 1964).

Frezza, F.

F. Frezza and N. Tedeschi, “Deeply penetrating waves in lossy media,” Opt. Lett. 37, 2616–2618 (2012).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
[CrossRef]

Gibbs, J. W.

J. W. Gibbs and E. B. Wilson, Vector Analysis (Scribner, 1901).

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002).

Helmy, A. S.

Holl, H. B.

Ivlev, E. I.

E. I. Ivlev, “Scattering of inhomogeneous electromagnetic waves by a cylinder,” J. Mod. Opt. 39, 499–507 (1992).
[CrossRef]

E. I. Ivlev, “Structure and properties of inhomogeneous waves,” J. Mod. Opt. 34, 1559–1569 (1987).
[CrossRef]

Jackson, D. R.

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

König, W.

W. König, Handbuch der Physik, Vol. XX (Springer-Verlag, 1928).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, 1964).

Leupacher, W.

Lindell, I. V.

I. V. Lindell, Methods for Electromagnetic Field Analysis (Wiley, 2002).

Oliner, A. A.

T. Tamir and A. A. Oliner, “Complex guided waves,” Proc. IEE 110, 310–334 (1963).
[CrossRef]

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

Pajewski, L.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
[CrossRef]

Penzkofer, A.

Ponti, C.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
[CrossRef]

Potter, R. F.

Radcliff, R. D.

R. D. Radcliff and C. A. Balanis, “Modified propagation constants for nonuniform plane wave transmission through conducting media,” IEEE Trans. Geosci. Remote Sens. GE-20, 408–411 (1982).
[CrossRef]

Roy, J. E.

J. E. Roy, “New results for the effective propagation constants of nonuniform plane waves at the planar interface of two lossy media,” IEEE Trans. Antennas Propag. 51, 1206–1215 (2003).
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, 1964).

Schettini, G.

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tai, C.-T.

R. De Roo and C.-T. Tai, “Plane wave reflection and refraction involving a finitely conducting medium,” IEEE Antennas Propag. Mag. 45(5), 54–61 (2003).
[CrossRef]

Tamir, T.

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

T. Tamir and A. A. Oliner, “Complex guided waves,” Proc. IEE 110, 310–334 (1963).
[CrossRef]

Tedeschi, N.

F. Frezza and N. Tedeschi, “Deeply penetrating waves in lossy media,” Opt. Lett. 37, 2616–2618 (2012).
[CrossRef]

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
[CrossRef]

Wait, J. R.

J. R. Wait, “A note on the pseudo-Brewster angle,” IEEE Antennas Propag. Mag. 39(4), 68 (1997).
[CrossRef]

Wang, Y.

Wilson, E. B.

J. W. Gibbs and E. B. Wilson, Vector Analysis (Scribner, 1901).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959).

Wriedt, T.

T. Wriedt and A. Doicu, “Light scattering from particles on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Am. J. Phys. (1)

P. E. Ciddor, “Refraction into an absorbing medium,” Am. J. Phys. 44, 786–787 (1976).
[CrossRef]

Appl. Opt. (1)

IEEE Antennas Propag. Mag. (3)

R. De Roo and C.-T. Tai, “Plane wave reflection and refraction involving a finitely conducting medium,” IEEE Antennas Propag. Mag. 45(5), 54–61 (2003).
[CrossRef]

J. R. Wait, “A note on the pseudo-Brewster angle,” IEEE Antennas Propag. Mag. 39(4), 68 (1997).
[CrossRef]

I. M. Besieris, “Comment on the ‘Corrected Fresnel coefficients for lossy materials’,” IEEE Antennas Propag. Mag. 53(4), 161–164 (2011).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. E. Roy, “New results for the effective propagation constants of nonuniform plane waves at the planar interface of two lossy media,” IEEE Trans. Antennas Propag. 51, 1206–1215 (2003).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

R. D. Radcliff and C. A. Balanis, “Modified propagation constants for nonuniform plane wave transmission through conducting media,” IEEE Trans. Geosci. Remote Sens. GE-20, 408–411 (1982).
[CrossRef]

J. Mod. Opt. (2)

E. I. Ivlev, “Scattering of inhomogeneous electromagnetic waves by a cylinder,” J. Mod. Opt. 39, 499–507 (1992).
[CrossRef]

E. I. Ivlev, “Structure and properties of inhomogeneous waves,” J. Mod. Opt. 34, 1559–1569 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

T. Wriedt and A. Doicu, “Light scattering from particles on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

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

Proc. IEE (1)

T. Tamir and A. A. Oliner, “Complex guided waves,” Proc. IEE 110, 310–334 (1963).
[CrossRef]

Other (12)

W. König, Handbuch der Physik, Vol. XX (Springer-Verlag, 1928).

F. Frezza, L. Pajewski, C. Ponti, G. Schettini, and N. Tedeschi, “Electromagnetic scattering by a metallic cylinder buried in a lossy medium with the cylindrical wave approach,” IEEE Geosci. Remote Sens. Lett.10, 179–183 (2013).
[CrossRef]

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1959).

R. P. Feynman, R. B. Leighton, and M. Sands, Lectures on Physics (Addison-Wesley, 1964).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

E. Hecht, Optics (Addison-Wesley, 2002).

F. X. Canning, “Corrected Fresnel coefficients for lossy materials,” in Proceedings of IEEE International Symposium on Antennas and Propagation (IEEE, 2011), pp. 2123–2126.

J. W. Gibbs and E. B. Wilson, Vector Analysis (Scribner, 1901).

I. V. Lindell, Methods for Electromagnetic Field Analysis (Wiley, 2002).

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

R. B. Adler, L. J. Chu, and R. M. Fano, Electromagnetic Transmission and Radiation (Wiley, 1972).

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Figures (8)

Fig. 1.
Fig. 1.

Geometry.

Fig. 2.
Fig. 2.

Behavior of the angle of the transmitted phase vector as a function of the angle of the incident phase vector. The properties of the media are [10] ϵ1=4, σ1=0.01S/m, ϵ2=10, and σ2=0.001S/m, at a frequency of f0=1MHz. Different values of η1 are considered. The two different behaviors are shown: the monotonic (dotted line) and the nonmonotonic (solid line).

Fig. 3.
Fig. 3.

Behavior of the angle of the transmitted attenuation vector as a function of the angle of the incident phase vector. The properties of the media are the same as in Fig. 2. Different values of η1 are considered. The two different behaviors are shown: the monotonic (dotted line) and the nonmonotonic (solid line).

Fig. 4.
Fig. 4.

Electromagnetic power |P| for unit length in a square cylinder with normalized side k0w=1, as a function of the angle of the incident phase vector. The cases of the nonmonotonic determination of ξ2 and ζ2, with η1=π/4 (solid curve), η1=0 (dotted line), and η1=π/4 (dashed line), are shown. Moreover, the monotonic determination with η1=0, ±π/4 (circles) is shown.

Fig. 5.
Fig. 5.

Real-power-only direction of the incident (dashed lines) and transmitted (solid lines) waves, as a function of the angle of the incident phase vector, in the same scenario of Figs. 2 and 3.

Fig. 6.
Fig. 6.

Real-power-only direction of the incident (dashed lines) and transmitted (solid lines) waves, as a function of the angle of the incident phase vector for all the cases in which ξ2>0 shown in [10]. The horizontal dashed-dotted line indicates the 90° angle: the pairs of lines that cross on it are related to same value of η1.

Fig. 7.
Fig. 7.

Behavior of the critical angle ξc and of the associated transmitted angle ξ2, as a function of the normalized incident phase vector amplitude, when the first medium is a vacuum, and the second medium has ϵ2=5+i0.03 at a wavelength λ=1550nm.

Fig. 8.
Fig. 8.

Behavior of the Fresnel transmission coefficient T and of the normalized incident attenuation vector amplitude as a function of the incident phase vector amplitude. The media involved are the same considered in Fig. 7.

Equations (28)

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

k1sinθ1=k2sinθ2,
k=k(cosθx0+sinθy0),
k=β+iα=β(cosξx0+sinξy0)+iα(cosζx0+sinζy0),
p=k×k*ik·k*,
q=|k·k|k·k*Re(k/k·k)|Re(k/k·k)|.
p=z0Im(k2)Re(k2)tanη=z0tanh(2θI),
q=|k2||k|2kRβ+kIαkR2β2+kI2α2+2(kRkI)2=|k2||k|2(cosθRx0+sinθRy0),
cosθR=kRβcosξ+kIαcosζkR2β2+kI2α2+2(kRkI)2,
sinθR=kRβsinξ+kIαsinζkR2β2+kI2α2+2(kRkI)2,
θI=12arctanh[Im(k2)Re(k2)tanη].
θI=12arctanh(2βαk2).
β1sinξ1=β2sinξ2,
α1sinζ1=α2sinζ2,
β22α22=Re(k22),
2β2α2cosη2=Im(k22).
β2=|kiτ|2+Re(k22)+|k22kiτ2|2,
α2=|kiτ|2Re(k22)+|k22kiτ2|2.
sinξ2=β1sinξ1β2,
sinζ2=α1sinζ1α2.
P=Pr+Ps+Pd=0,
ξ2={arcsin(β1sinξ1β2)ξ1<ξ1ξπarcsin(β1sinξ1β2)ξ1>ξ1ξ,
ζ2={arcsin(α1sinζ1α2)ξ1<ξ1ζπarcsin(α1sinζ1α2)ξ1>ξ1ζ,
SE=|E0|2ωμe2α·r(βiα),
SH=|H0|2ωμe2α·r(β+iαωϵ+iσ),
ξ1ξ,ζ=tanη1±tan2η14χ(χ1)2(χ1),
ξ1ξ,ζ=1tanη1.
ξ1ξ,ζ={ξ1ξifRe(kiτ2)Re(k22)ξ1ζifRe(kiτ2)<Re(k22).
ξ1ξ,ζ=12arcsin[Im(k22)β1α1].

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