Plane waves in dissipative media

Abstract

The most general exponential wave that satisfies Maxwell's equations in an infinite homogeneous isotropic medium is derived. Linear and elliptically polarized transverse electromagnetic (TEM) waves, transverse magnetic (TM) hybrid waves, and transverse electric (TE) hybrid waves are shown to be special cases of the general wave. This can be written as H = A [ Be <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iΞ</sup> n + C (coshƛp + i sinh ƛm] .exp{i(ωt - θ) - i Γ e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-iv/2</sup> [cosh ƛ m - isinh ƛ p](r - r <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> )} E = Aze <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iv/2</sup> [Cn - Be <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iΞ</sup> (cosh ƛ p + isinh ƛ m)] .exp{i(ωt - θ) - i Γ e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-iv/2</sup> [cosh ƛ m - isinh ƛ p](r - r <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> )} where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, p</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> are unit vectors which form a right-handed orthogonal coordinate system. Here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ƛ</tex> (dia) is the ellipticity of the wave. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Tanh ƛ</tex> is the axial ratio of the ellipse traced by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> vector of the transverse electric component <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(C)</tex> and by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> vector of the transverse magnetic component <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(B)</tex> , and by the ellipse defining attenuation and phase vectors as functions of the loss tangent of the medium. The three expressions in the square brackets are complex unit vectors which define three planes of interest: 1) the propagation plane, which contains the propagation and attenuation vectors, 2) the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> plane, which contains the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> vector, and 3) the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> plane which contains the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> vector. The vector product of the complex unit vector, defining the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> ellipse with that defining the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> ellipse, is equal to the complex unit vector, defining the Γ ellipse. [Cn - Be <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iΞ</sup> (cosh ƛ p + isinh ƛ m) × [Be <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iΞ</sup> n + C (cosh ƛ p + isinh ƛ m)] = cosh ƛm - isinh ƛ p].From this, it is evident that in the most general exponential wave, both the electric and the magnetic vectors trace ellipses in planes that may make arbitrary angles with the plane of propagation. The geometrical relationships are treated in detail. The projections of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> ellipse and the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> ellipse on the Γ plane differ only by a scale factor. They have the same axial ratio as the Γ ellipse, but the major and minor axes are interchanged. There is no right and left handedness to the hybrid waves as there is with elliptically-poiarized waves. The projection of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> on the plane of propagation traces ellipses in the same direction. This direction is opposite from that in which the attenuation and propagation vectors are displaced, as the conductivity of the medium is increased from zero. The right and left handedness of the general wave depends upon the phase relation between the hybrid waves of which it is composed. The decomposition of the general wave into a TE hybrid wave and a TM hybrid wave is unique, except in the degenerate case wherethe ellipticity of the wave is zero, and the attenuation vector, if any, is coincident with the phase vector. This is the case of the TEM wave. When the ellipticity is zero, each hybrid wave becomes a linearly-polarized wave, so that the expression of elliptically-polarized waves, as the sum of two linearly-poiarized waves of arbitrary magnitude or arbitrary phase, is a special case of the expression of the general wave as the sum of a TE hybrid wave and the TM hybrid wave.