It is known that electromagnetic waves propagating in a magnetized cold plasma are not necessarily purely transverse waves. The existence of a longitudinal component implies that the normal to the polarization plane is not, in general, parallel to the propagation vector. The offset angle between these two directions is computed for each characteristic wave. Our investigation shows that the offset angle can be large under certain conditions. For a wave field that is a superposition of two characteristic waves, its polarization plane wobbles in space as the field propagates while its normal precesses. Several examples are considered, computed, and given to illustrate the 3D nature of wave polarizations.
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 The discovery of Faraday rotation was made over 150 years ago [Faraday, 1846]. A theoretical explanation of this phenomenon was offered 51 years after its discovery [Bacquerel, 1897]. The issue of wave propagation in a magnetized cold plasma has been attacked for many decades by many researchers. Mature as it seems to be, attention is still attracted in recent years on Faraday rotation phenomenon since the complexity of its 3D nature does not yet close the question. While theoretical foundation has been soundly laid, the interpretation of the solutions and laconic and intuitive approach to lay out the theoretical results are still worthwhile. Recently, Yeh et al. [1999a, 1999b] introduce a 2D polarization diagram which makes the nature of polarization transformation clearer and simpler. They take advantage of the bilinear mapping in complex function theory to map propagation factor in the complex plane R = Ey(z)/Ex(z), called polarization ratio plane, which determines geometric features of the transverse polarization. It plays an equivalent role as Smith chart does in transmission line theory.
 Polarization of the wave field in a magnetized cold plasma, however, is 3D in nature because the electric field is not purely transverse in general. The work in this paper is intended to extend the earlier 2D results to 3D case. Under some circumstances the longitudinal component plays an important role as discussed later in this paper.
 Research interest on the topic is partly stimulated by the practical objective of NASA's IMAGE project, which has been on board a radio plasma imager (RPI) used for magnetospheric sounding. In a paper authored by Reinisch et al. , the working principle of the active Doppler radar sounding in IMAGE has been outlined. One of the RPI experiments determines the normal vector of the polarization plane of the returned echo. If the returned electric field is purely transverse, this normal vector gives the arrival direction of the echo. However, if the returned echo has a longitudinal component, as is generally true in a magnetized cold plasma, care must be exercised in its interpretation.
 In this paper we will discuss this important issue in detail to see under what conditions neglect of longitudinal component is justified and, if not, what is the angle difference between the polarization normal vector of a characteristic wave and the actual arrival angle.
 The calculation given in section 4 shows that a general elliptically polarized field, constructed as a superposition of ordinary and extraordinary waves, has the polarization plane spanned by two frame vectors which are turning their directions in two different planes, making the polarization plane either nutating or rolling along propagation path.
 This paper adopts a Hermitian coordinate system, which is constituted by two oppositely rotating circular polarization base vectors and the unit propagation vector. In this system every component of the wave field is found to be real valued. This turns out to be convenient in expressing the solutions and especially in determining the normal vector of a wave polarization. With these explicit vectorial expressions the problem of a superposition of two characteristic wave modes can be discussed further in detail, especially those pertaining to 3D geometric features.
2. Wave Dispersion Equation in a Circular Polarization Basis
 We start with Maxwell's two curl equations. For plane waves, all field variables have the time–space dependence of the form exp(ωt − k · r) for which the differential operators are simplified to algebraic ones. After eliminating the magnetic field, the electric field is found to satisfy
 In a magnetized cold plasma, the relative permittivity takes the dyadic form,
In (3), the coefficients α, β, and γ are expressed in two forms using the angular plasma frequency ωp and angular gyrofrequency ωb or the normalized frequencies X and Y popularly used in the ionospheric literature. These quantities are defined as follows:
where N, m, and e are the density, mass and electric charge of an electron respectively.
 Substituting the relative dielectric permittivity expression (2) into (1), the resulting equation becomes
where, , n = k/k0 is the refractive index and k02 = ω2μ0ε0.
 Consider a coordinate system constructed with the following three base vectors:
, a unit vector in the wave number direction
, where is a unit vector in the steady magnetic field direction, the angle between B0 and k is ;
It is convenient to combine , , and into the following Hermitian basis:
The left-hand circular polarization vector and the right-hand circular polarization vector have their properties discussed by Lee . With these coordinate transformations, the dyadic equation (4) expressed in the basis system (6) takes the matrix form
where λ = n2 − 1. Notice that all the elements of Π appearing in (8) and (9) are real. As we shall see later, the reality of Π is an important property useful in later development.
 For nontrivial solutions to (7) we require det Π = 0, obtaining the following dispersion equation:
The two roots of (10) give two characteristic wave numbers, known as Appleton–Lassen formula:
It is interesting to see how simple (9) becomes for the special case of θ = 0,
In this special θ = 0 case, the dispersion relation simplifies to
The two cases, represented by the vanishing of the first factor or the second factor of (13), correspond to the purely left-handed or the purely right-handed circularly polarized wave modes respectively. The case represented by the vanishing of the third factor corresponds to longitudinal plasma oscillations.
3. Characteristic Waves
 Taking first two rows of the matrix equation (7) yields
where the root n(1) is assumed as n in Π11 and Π22 Solving (14) by taking Uk as known and then multiplying the three components of the 3D vector U with an appropriate common factor gives the following ordinary characteristic wave mode vector.
Similarly, the extraordinary characteristic wave mode associated with the root n(2) can be shown to be
In (15) and (16), the polarization ratio R for the transverse components is, for respective modes,
A cos θ factor in (16) is used to assure continuity of the components over θ. In (17), the upper sign applies to R1 and the lower sign applies to R2. The two characteristic wave vectors can be normalized as follows:
It is easy to see that R1R2 = −1, and therefore
As it has been noted before, all elements of the characteristic vectors (15) and (16) are real valued. The physical implication of (19) is that all components in the characteristic wave modes are either in phase or 90° out of phase to one another. But, since two of the three base vectors are complex valued (see (6)), the characteristic wave vectors are complex valued vectors when expressed in real physical space. Let and , then
These two expressions turn out to be very useful in determining 3D polarization features of the wave modes as will be discussed later.
 The variations of the characteristic wave vectors with θ are illustrated in Figures 1a and 1b. The parameters used are: X = 0.005, Y2 = 0.00011 and X = 0.005, Y2 = 1.0001, respectively. Even though the longitudinal component is very small for these two examples, it is not necessarily always the case, even when X is small. This is shown next.
Figure 2 shows surfaces describing absolute value of the longitudinal component of the normalized characteristic wave electric field vector as a function of X and Y for ordinary and extraordinary wave modes respectively for three propagation angles θ = 30°, 74°, and 89°. It is seen that the longitudinal component of the extraordinary mode may be large even when X ≪ 1, provided Y is close to 1.
 To help discuss the wave propagation represented by a complex vector phasor, we consider the following general case. Let A be a constant vector phasor with A = Ar + jAi. For a plane wave of phase ψ = ωt − k · r, the instantaneous vector field becomes
When ψ = 0, U(t, r) coincides with Ar; while when, ψ = π/2, U(t, r) coincides with −Ai. At a fixed position r the tip of the vector U traces out an ellipse as a function of time in the plane formed by Ar and Ai. This plane is known as the polarization plane. In the polarization plane, the vector U(t, r) rotates with time in the sense from Ai to Ar. Define a normal to this polarization plane as
For transverse waves, N must necessarily be either parallel to k as for right-handed polarization or antiparallel to k as for left-handed polarization. Because of the possible presence of a longitudinal component in a magnetized cold plasma, there exists an angle Φ between N and k in general. If the angle Φ is acute, the sense of rotation can be defined as right handed. On the other hand, if Φ is obtuse, the sense of rotation is left handed. Such a definition is in complete agreement with the usual sense determination using the transverse components only. This is so because the acuteness of angle Φ (i.e., the sign of cos Φ) is determined by the transverse component ATrans only and not by the longitudinal component ALong as shown below
The calculation of (23) follows by noting the following vector relation,
 We now apply the above result to the characteristic wave vectors (18). Using (20a) and (20b), the normals can be calculated to be
for each of the two characteristic modes. This means that the normal vector is in the plane spanned by k and B0 (see (5)). The angle Φ between N and k, known as the offset angle, can be computed from
For purpose of investigating how large this offset angle Φ can be, let us adopt some numbers applicable to the RPI [Reinisch et al., 1999]. RPI uses an active Doppler sweeping in frequency from 3 kHz to 3 MHz for sensing plasma structures in the magnetosphere. As a first example, take f = 75 kHz, fp = 25 kHz, fb = 1.5 kHz, and θ = 74°. Theses parameters yield X = 0.1111 and Y = 0.02. The computed offset angles are 0.1319° and −0.1438° for the ordinary and extraordinary waves, respectively. Since the offset angles are very small, the normal to the polarization plane can be considered as the propagation direction with negligible error. In the second example, we drop the frequency to the middle of RPI range, say 30 kHz. Keeping other parameters the same, i.e., fp = 25 kHz, fb = 1.5 kHz, and θ = 74°, produces X = 0.6944 and Y = 0.05. The computed offset angle in this second example comes out to be 4.7121° and −8.2969° for the ordinary and extraordinary waves respectively. The offset angles are now larger and begin to matter in some applications. Actually, even in the same plasma (i.e., X = 0.6944 and Y = 0.05) the offset angle has a strong dependence on the propagation angle θ, especially for the extraordinary wave. This is shown in Figure 3a. In this figure the magnitude of the offset angle for the extraordinary wave is seen to increase drastically as the propagation direction becomes perpendicular to the steady magnetic field. If the local plasma frequency and gyrofrequency at RPI can be determined from other experiments, (25) can be used to correct the apparent echo arrival angle determined by using the wave polarization data from the propagation direction.
 In passing we mention two special cases of interest, both corresponding to the cutoff condition, one for the ordinary mode, the other for the extraordinary mode. For ordinary mode at X = 1, R1 turns out to be zero. The corresponding characteristic vector (15) simplifies to
When this is substituted into (25), the offset angle is found simply as
For this special case the normal N to the polarization plane is always perpendicular to B0. In ionospheric literature, a vertically incident ordinary ray on a horizontally stratified ionosphere is known to bend toward the nearest geomagnetic pole [Forsgren, 1951]. At the point of reflection, i.e., X = 1, the ray is exactly perpendicular to B0, implying N and the ray are parallel to each other. Figure 3b shows a case in which X = 0.9999 ≈ 1, Y = 0.004. This figure shows clearly that (27) is approximately satisfied for a wide range of θ.
 A second special case of interest occurs when X = 1 − Y for the extraordinary mode. In this case R2 in (17) simplifies to 1/cosθ and its characteristic vector becomes
That is to say, the offset angle Φ is equal to −θ. Because of the way Φ is defined, the normal vector of the polarization plane is always parallel to B0. In this case, however, the refractive index n = 0, corresponding to the cutoff condition. But if Y tends to 1 − X from below, (29) still holds approximately. One such example is shown in Figure 3b where X = 0.5, Y = 0.49 for which X is not exactly equal to 1 − Y. It is seen that the offset angle Φ is very close to −θ, showing approximate perpendicularity of the polarization plane to B0 for whatever direction in k.
 As seen from Figures 3a–3c, N of the ordinary mode always deviates away from B0 because Φ is positive when θ < π/2 and is negative when θ > π/2. This phenomenon may be called the “divergence effect.”
 The instantaneous electric field value of a characteristic mode, with the unimportant amplitude set to unity, is
where La(1,2) and Lb(1,2) are defined in (20a) and (20b) respectively. It is obvious that La(1,2) · Lb(1,2) implying the polarization ellipse described by (30) has a major axis aligned along either La(1,2) or Lb(1,2). Note that La(1,2) is the plane Pkb, spanned by the propagation path and the steady magnetic field, while Lb(1,2) is perpendicular to Pkb. The determination of the major axis can be decided by calculating the difference in squared lengths between La and Lb.
Under the assumption of negligible uk as considered in Figure 1, the major axis of the ellipse is determined by the relative signs between uL and uR. If uR and uL are of the same sign as in mode 1 shown in Figure 1, La is the major axis. On the other hand, when uL and uR have opposite signs as for mode 2, Lb is the major axis.
4. Wobbling of the Polarization Plane and Nutation
 In this section, we wish to consider the polarization transformation of a wave field propagating in a given direction, say z, and at a given frequency. Such a wave field can be decomposed as a sum of two characteristic wave modes as follows:
where the complex w takes care of the possible existence in relative magnitude and phase shift between the two characteristic modes.
 Let and ϕw = Arg(w), (32a) can be converted into the following form:
The instantaneous value of E, i.e., the electrical field polarization, is
For a fixed position, the tip of the electric field E still traces out an ellipse in time. However, because of the dependence of Fr and Fi on q and thus also on z, the polarization ellipse changes as the wave field propagates.
 From (32b), the normal vector (unnormalized) of the polarization plane can be expressed as
As discussed in connection with (23), the sense of rotation of the composite wave field is determined by the inner product
In above equations, , U(1) and U(2) are given in (15) and (16) respectively. Since C is a positive number, Ac is negative when X > 1 and positive when X < 1. Therefore, when X < 1, as in some practical cases, Nk is varying between two extreme values.
As mentioned above, the sense of rotation is determined by the sign of the Nk. Consequently, the behavior of the normal vector N(q) falls into the following three types:
Always right handed. This happens when Nkmin > 0, i.e.,
Always left handed. This happens when Nkmax < 0, i.e.,
Sense of rotation alternating between right handed and left handed. This happens when NkminNkmax < 0,
 To connect the results obtained here with existing results given by Yeh et al. [1999a, 1999b], we calculate the electrical field components along x and y axes, respectively,
From these expressions, the polarization ratio of the transversal components can be computed as
where the several quantities appeared in (51) are defined as
The bilinear transformation (51) maps circles in complex ξ-plane into circles in complex R-plane. In Figure 4, all three types of normal vector's behavior are illustrated by (a), (b), and (c), respectively. The parameters are marked in the figures.
 To describe the behavior of the polarization plane, the zenith angle θn(q) and azimuth angle φn(q) of its normal vector can be expressed as
In Figure 4, θn(q) is shown on the top left panel, φn(q) on the bottom left panel, the unit normal vector of the polarization plane on the bottom right panel and the polarization ratio plane, including the circular locus indicating the polarization transformation along the propagation path, on the top right panel. The types of polarization transformation represented by (a) and (b) are like nutation of a spin top where the normal vector precesses, while (c) can be called polarization plane rolling since the sense of rotation alternates along the propagation path. In this case, the tip of the electric wave vector at a fixed position still draws an ellipse in 3D space as a function of time on the polarization plane. But according to (39), this plane, spanned by two vectors Fr(q) and Fi(q) (called frame vectors of the ellipse), wobbles as a function of q and thus also as a function of the propagation distance z.
 When X ≪ 1 and Y ≪ 1, the longitudinal component of a wave field is much smaller than the transversal ones (Figure 1a describes a typical case of this). Therefore, the angle between the planes spanned by (Lr, Li) and (Mr, Mi), respectively, is very small. Even then, provided that the angle is not exactly zero, it is clear that the two frame vectors Fr and Fi can never be collinear with each other for the case of nonzero vectors (Lr, Li) and (Mr, Mi). Because if so, Fr and Fi must be on the intersection line of the planes spanned by (Lr, Li) and (Mr, Mi) respectively; and this line is along the direction of (since Li, Mi, and are collinear, see (36) and (38)). But from (33) and (34) this is impossible since sin(q) and cos(q) can never be zero simultaneously. Therefore, in three dimensions, the case of the linear polarization never occurs. Thus, on the wave front, the change of sense of rotation of the transverse polarization across a critical linear polarization case in the propagation path (as seen in the work of Yeh et al. [1999a, Figure 4] and Figure 4c in this paper when the loci are crossing real axis of complex R-plane) turns out to be a projection on wave front of a continuous rolling process. The 2D critical linear polarization occurs when the 3D elliptical polarization is projected on the wave front plane.
 From discussions given in previous sections, we can conclude as follows:
The polarization ellipses of the two characteristic electromagnetic wave modes in a magnetized cold plasma are in planes whose normals are in the plane Pkb spanned by the propagation wave number vector and the steady magnetic field. The normal vector to the polarization plane offsets from the propagation direction by an angle Φ. This offset angle can be very large, especially for extraordinary waves near their cutoff when the propagation direction tends to be orthogonal to the magnetic field. Besides, the offset angle can be large even in the case X ≪ 1, provided Y is not much less than 1. When the offset angle is large the normal vector to the polarization plane can no longer be taken as the echo arrival angle without a correction.
The polarization ellipse of a monochromatic composition of the two characteristic waves propagating in the same direction is spanned by two frame vectors. These two frame vectors determine a polarization plane that wobbles as the wave propagates. On the wobbling polarization plane, the tip of the electric field traces out an ellipse as a function of time. This makes the normal vector of the polarization plane either nutating (sense of rotation unchanged) or rolling (sense of rotation alternating). Thus, the use of normal vector as an indicator of wave propagation direction must be done with care.
In the cases X ≪ 1 and Y ≪ 1, polarization plane rolling effect is not easily perceivable since the angle between the two planes on which the two frame vectors reside respectively is very small. Even though, awareness of its existence may improve our understanding of the abruptly changed sense of polarization rotation along the propagation path as being caused by the projection of the electric field on the wave front of a continuous rolling process.
One of the two frame vectors shrinks into the origin when the propagation direction tends to the steady magnetic field direction and when the two modes have the equally amplitude (i.e., ∣w∣ = 1). Then only a single frame vector exists and polarization becomes linear with its electric field direction rotating along the propagation path. This is then reduced to the classical Faraday rotation case.
 This work was partially supported by the National Science Foundation under grants ATM-97-13435 and ATM-00-03418.