Lecture 9

Waves in Gyrotropic Media, Polarization

9.1 Gyrotropic Media

This section presents deriving the permittivity tensor of a gyrotropic medium in the ionsphere.

Our ionosphere is always biased by a static magnetic field due to the Earth’s magnetic field [67]. But in this derivation, one assumes that the ionosphere has a static magnetic field polarized in the z direction, namely that B = ẑB0. Now, the equation of motion from the Lorentz force law for an electron with q = −e, in accordance with Newton’s law, becomes

me dv/dt = −e(E + v × B) (9.1.1)

Next, let us assume that the electric field is polarized in the xy plane. The derivative of v is the acceleration of the electron, and also, v = dr/dt. And in the frequency domain, the above equation becomes

meω²x = e(Ex + jωB0y) (9.1.2)

meω²y = e(Ey − jωB0x) (9.1.3)

The above equations cannot be solved easily for x and y in terms of the electric field because they correspond to a two-by-two matrix system with cross coupling between the unknowns x and y. But they can be simplified as follows: We can multiply (9.1.3) by ±j and add it to (9.1.2) to get two decoupled equations [68]:

meω²(x + jy) = e[(Ex + jEy) + ωB0(x + jy)] (9.1.4)

meω²(x − jy) = e[(Ex − jEy) − ωB0(x − jy)] (9.1.5)

Defining new variables such that

s± = x ± jy (9.1.6)

E± = Ex ± jEy (9.1.7)

84 Electromagnetic Field Theory

then (9.1.4) and (9.1.5) become

meω²s± = e(E± ± ωB0s±) (9.1.8)

Thus, solving the above yields

s± = e / (meω² ∓ eB0ω) E± = C±E± (9.1.9)

where

C± = e / (meω² ∓ eB0ω) (9.1.10)

By this manipulation, the above equations (9.1.2) and (9.1.3) transform to new equations where there is no cross coupling between s± and E±. The mathematical parlance for this is the diagnolization of a matrix equation [69]. Thus, the new equation can be solved easily.

Next, one can define Px = −Nex, Py = −Ney, and that P± = Px ± jPy = −Nes±. Then it can be shown that

P± = ε0χ±E± (9.1.11)

The expression for χ± can be derived, and they are given as

χ± = −NeC±/ε0 = −Ne/ε0 · e/(meω² ∓ eBoω) = −ωp²/(ω² ∓ Ωω) (9.1.12)

where Ω and ωp are the cyclotron frequency1 and plasma frequency, respectively.

Ω = eB0/me, ωp² = Ne²/(meε0) (9.1.13)

At the cyclotron frequency, a solution exists to the equation of motion (9.1.1) without a forcing term, which in this case is the electric field E = 0. Thus, at this frequency, the solution blows up if the forcing term, E± is not zero. This is like what happens to an LC tank circuit at resonance whose current or voltage tends to infinity when the forcing term, like the voltage or current is nonzero.

Now, one can rewrite (9.1.11) in terms of the original variables Px, Py, Ex, Ey, or

Px = (P+ + P−)/2 = ε0/2 (χ+E+ + χ−E−) = ε0/2 [χ+(Ex + jEy) + χ−(Ex − jEy)]

= ε0/2 [(χ+ + χ−)Ex + j(χ+ − χ−)Ey] (9.1.14)

Py = (P+ − P−)/(2j) = ε0/(2j) (χ+E+ − χ−E−) = ε0/(2j) [χ+(Ex + jEy) − χ−(Ex − jEy)]

= ε0/(2j) [(χ+ − χ−)Ex + j(χ+ + χ−)Ey] (9.1.15)

1 This is also called the gyrofrequency.

The above relationship can be expressed using a tensor where

P = ε0χ · E (9.1.16)

where P = [Px, Py], and E = [Ex, Ey]. From the above, χ is of the form

χ = 1/2

((χ+ + χ−) j(χ+ − χ−)

−j(χ+ − χ−) (χ+ + χ−))

=

(−ωp²/(ω²−Ω²) −j ωp²Ω/(ω(ω²−Ω²))

j ωp²Ω/(ω(ω²−Ω²)) −ωp²/(ω²−Ω²))

(9.1.17)

Notice that in the above, when the B field is turned off or Ω = 0, the above resembles the solution of a collisionless, cold plasma.

For the electric field in the z direction, it will drive a motion of the electron to be in the z direction. In this case, v × B term is zero, and the electron motion is unaffected by the magnetic field as can be seen from the Lorentz force law or (9.1.1). Hence, it behaves like a simple collisionless plasma without a biasing magnetic field. Consequently, the above can be generalized to 3D to give

χ =

[ χ0 jχ1 0

−jχ1 χ0 0

0 0 χp ] (9.1.18)

where χp = −ωp²/ω².

Using the fact that D = ε0E + P = ε0(I + χ) · E = ε · E, the above implies that

ε = ε0

[ 1 + χ0 jχ1 0

−jχ1 1 + χ0 0

0 0 1 + χp ] (9.1.19)

Please notice that the above tensor is a hermitian tensor. We shall learn later that this is the hallmark of a lossless medium.

Another characteristic of a gyrotropic medium is that a linearly polarized wave will rotate when passing through it. This is the Faraday rotation effect [68], which we shall learn later. This phenomenon poses a severe problem to Earth-to-satellite communication, using linearly polarized wave as it requires the alignment of the Earth-to-satellite antennas. This can be avoided using a rotatingly polarized wave, called a circularly polarized wave that we shall learn in the next section. Also, the ionosphere of the Earth is highly dependent on temperature, and the effect of the Sun. The fluctuation of particles in the ionosphere gives rise to scintillation effects that affect radio wave communication systems [70].

9.2 Wave Polarization

Studying wave polarization is very important for communication purposes [31]. A wave whose electric field is pointing in the x direction while propagating in the z direction is a linearly polarized (LP) wave. The same can be said of one with electric field polarized in the y direction. It turns out that a linearly polarized wave suffers from Faraday rotation when

86 Electromagnetic Field Theory

it propagates through the ionosphere. For instance, an x polarized wave can become a y polarized due to Faraday rotation. So its polarization becomes ambiguous: to overcome this, Earth to satellite communication is done with circularly polarized (CP) waves [71]. So even if the electric field vector is rotated by Faraday’s rotation, it remains to be a CP wave. We will study these polarized waves next.

We can write a general uniform plane wave propagating in the z direction as

E = x̂Ex(z, t) + ŷEy(z, t) (9.2.1)

Clearly, ∇ · E = 0, and Ex(z, t) and Ey(z, t) are solutions to the one-dimensional wave equation. For a time harmonic field, the two components may not be in phase, and we have in general

Ex(z, t) = E1 cos(ωt − βz) (9.2.2)

Ey(z, t) = E2 cos(ωt − βz + α) (9.2.3)

where α denotes the phase difference between these two wave components. We shall study how the linear superposition of these two components behaves for different α’s. First, we set z = 0 to observe this field. Then

E = x̂E1 cos(ωt) + ŷE2 cos(ωt + α) (9.2.4)

For α = π/2

Ex = E1 cos(ωt), Ey = Ez cos(ωt + π/2) (9.2.5)

Next, we evaluate the above for different ωt’s

ωt = 0, Ex = E1, Ey = 0 (9.2.6)

ωt = π/4, Ex = E1/√2, Ey = −E2/√2 (9.2.7)

ωt = π/2, Ex = 0, Ey = −E2 (9.2.8)

ωt = 3π/4, Ex = −E1/√2, Ey = −E2/√2 (9.2.9)

ωt = π, Ex = −E1, Ey = 0 (9.2.10)

The tip of the vector field E traces out an ellipse as show in Figure 9.1. With the thumb pointing in the z direction, and the wave rotating in the direction of the fingers, such a wave is called left-hand elliptically polarized (LHEP) wave.

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Figure 9.1: If one follows the tip of the electric field vector, it traces out an ellipse as a function of time t.

When E1 = E2, the ellipse becomes a circle, and we have a left-hand circularly polarized (LHCP) wave. When α = −π/2, the wave rotates in the counter-clockwise direction, and the wave is either right-hand elliptically polarized (RHEP), or right-hand circularly polarized (RHCP) wave depending on the ratio of E1/E2. Figure 9.2 shows the different polarizations of the wave wave for different phase differences and amplitude ratio.

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Figure 9.2: Due to different phase difference between the Ex and Ey components of the field, and their relative amplitudes E2/E1, different polarizations will ensure. The arrow indicates the direction of rotation of the field vector. In this figure, ψ = −α in our notes, and A = E2/E1 (Courtesy of J.A. Kong, Electromagnetic Wave Theory [31]).

Figure 9.3 shows a graphic picture of a CP wave propagating through space.

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Figure 9.3: The rotation of the field vector of a right-hand circular polarization wave as it propagates in the right direction [72] (Courtesy of Wikipedia).

9.2.1 Arbitrary Polarization Case and Axial Ratio

The axial ratio (AR) is an important figure of merit for designing CP antennas (antennas that will radiate CP or circularly polarized waves). The closer is this ratio to 1, the better the antenna design. We will discuss the general polarization and the axial ratio of a wave.

For the general case for arbitrary α, we let

Ex = E1 cos ωt, Ey = E2 cos(ωt + α) = E2(cos ωt cos α − sin ωt sin α) (9.2.11)

Then from the above, expressing Ey in terms of Ex, one gets

Ey = (E2/E1)Ex cos α − E2[1 − (Ex/E1)²]1/2 sin α (9.2.12)

Rearranging and squaring, we get

aEx² − bExEy + cEy² = 1 (9.2.13)

where

a = 1/(E1² sin² α), b = 2 cos α/(E1E2 sin α), c = 1/(E2² sin² α) (9.2.14)

After letting Ex → x, and Ey → y, equation (9.2.13) is of the form,

ax² − bxy + cy² = 1 (9.2.15)

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Figure 9.4: This figure shows the parameters used to derive the axial ratio (AR) of an elliptically polarized wave.

9.3 Polarization and Power Flow

For a linearly polarized wave,

E = x̂E0 cos(ωt − βz), H = ŷ E0/η cos(ωt − βz) (9.3.1)

Hence, the instantaneous power becomes

S(t) = E(t) × H(t) = ẑ E0²/η cos²(ωt − βz) (9.3.2)

indicating that for a linearly polarized wave, the instantaneous power is function of both time and space. It travels as lumps of energy through space. In the above E0 is the amplitude of the linearly polarized wave.

Next, we look at power flow for for elliptically and circularly polarized waves. It is to be noted that in the phasor world or frequency domain, (9.2.1) becomes

E(z, ω) = x̂E1e^(−jβz) + ŷE2e^(−jβz+jα) (9.3.3)

Page 11 onward contains the bibliography, which has been omitted here due to length limits.