Wave Polarization

Abstract - In this section we will introduce the description of Transversal Electro-Magnetic waves (TEM waves or just EM waves), in terms of their field vectors. In the following we will introduce the polarization ellipse and discuss some special cases of polarization and the meaning of the polarization ratio.

keywords: TEM waves, EM waves, polarization ellipse, special cases of polarization, polarization ratio

For a plane wave propagating in an arbitrary direction $\vec{e}_{p}$, the electromagnetic wave system is made up of a set of coupled, time varying and mutually orthogonal electric and magnetic vector fields. The field vectors are perpendicular to the propagation direction (for a detailed derivation of this see Appendix A).

$$\vec{E}(r)= \hat{E}_0e^{ik\vec{e}_{p} \vec{r}}$$ (2.1)

$$\vec{H}(r)= \hat{H}_0e^{ik\vec{e}_{p} \vec{r}} \quad \quad with \hat{H}_0 = \frac{1}{\eta}\vec{e}_p \times \hat{E}_0$$ (2.2)

where $\vec{e}_{p}$ is a unit vector parallel to the propagation direction. As we can see from the equations above, the electric field vector is directly related to the magnetic field vector and the latter can always be calculated from the electric field vector. Therefore, we will only focus on the electric fields vector in the following. For a specific time t and point in space $\vec{r}$ the direction and the magnitude of the electric field vector $\vec{E}(r,t)$ is given by the real part of the complex harmonic field expression

$$\vec{E}{re}(r,t) = \Re(\hat{E}_{0}e^{i(\omega t + \vec{k}\vec{r})})$$ (2.3)

As shown above for a transverse electro-magnetic wave, the $\vec{E}_0$-vector oscillates in a plane perpendicular to the propagation direction. (Fig. 2.1)

Figure: $\vec{E}$-field of a plane EM wave oscillating in a plane perpendicular to the propagation direction $\vec{r}$

The vectorial nature of plane transverse electromagnetic (TEM) waves is called polarization and is independent of the chosen space coordinate system. As we shall see in the following, in the general case, the trace of the tip of the field vector $\vec{E}$ within a plane perpendicular to the propagation direction, is an ellipse, meaning the wave is elliptically polarized. Special cases of elliptically polarization are linear or circular polarization. Even though the polarization itself is independent of the coordinate system, for the description of polarization a coordinate system and a reference direction of propagation is needed. Due to the fact that the most fully polarimetric radar systems use two orthogonal linear polarized antennas, we introduce a Cartesian coordinate system, where z is the propagation direction. The electric field of a wave propagating in z-direction is located in the x-y plane and consists of a x-component $\vec{E}_x$ and a y-component $\vec{E}_y$ with the complex amplitudes $\hat{E}_{x/y}=E_{x0/y0}\cdot e^{i\delta_{x/y}}$

$$\vec{E}(z) = E_x(z)\vec{e}_x + E_y(z)\vec{e}_y$$ (2.4)

with

$$\vec{E}_x(z) = \vec{E}_x e^{ikz}=E_{x0} e^{ikz}e^{i\delta_x} \vec{e}_x$$ (2.5)

$$\vec{E}_y(z) = \vec{E}_y e^{ikz}=E_{y0} e^{ikz}e^{i\delta_y} \vec{e}_y$$ (2.6)

Therefore the instantaneous values of $\vec{E}_x(z,t)$ and $\vec{E}_y(z,t)$ are given by

$$\vec{E}_x(z,t) =\Re(\vec{E}_x(z)e^{i\omega t}) = E_{x0} \cos(\omega t - kz + \delta_x)$$ (2.7)

$$\vec{E}_y(z,t) =\Re(\vec{E}_y(z)e^{i\omega t}) = E_{y0} \cos(\omega t - kz + \delta_y)$$ (2.8)

In the general case where $E_{x0} \ne 0$, $E_{y0} \ne 0$ and $\delta_y - \delta_x = \delta_0 \ne 0$ we may rewrite (2.7) and (2.8) as

$$\vec{E}_x(z,t) =\Re(\vec{E}_x(z)e^{i\omega t}) = E_{x0} \cos(\omega t - kz)$$ (2.9)

$$\vec{E}_y(z,t) =\Re(\vec{E}_y(z)e^{i\omega t}) = E_{y0} \cos(\omega t - kz + \delta_0)$$ (2.10)

We now eliminate the term $(\omega t - kz)$ by expanding (2.7) and (2.8) into

$$\frac{\vec{E}_x(z,t)}{E_{x0}}\cos(\delta_0)=\cos(\omega t -kz )\cos(\delta_0)$$ (2.11)

$$\frac{\vec{E}_y(z,t)}{E_{y0}}=\cos(\omega t -kz)\cos(\delta_0)+\sin(\omega t -kz)\sin(\delta_0)$$ (2.12)

we find that

$$\frac{E_y(z,t)}{E_{y0}}-\frac{E_x(z,t)}{E_{x0}}\cos(\delta_0)=+\sin(\omega t -kz) \sin(\delta_0)$$ (2.13)

Substituting the modified (2.12)

$$\sin(\omega t -kz)=\sqrt{1-\left(\frac{E_x(z,t)}{E_{x0}}\right)^2}$$ (2.14)

into (2.13) and rearrange the terms we get

$$\left(\frac{E_y(\vec{r},t)}{E_{y0}}\right)^2 + \left(\frac{E_x(\v... ...c{r},t) E_x(\vec{r},t)}{E_{y0} E_{x0}} \right)\cos(\delta_0) = \sin^2(\delta_0)$$ (2.15)

which is the equation of an ellipse with an inclination angle $\psi$ (see Fig. 2.2) such that

$$\tan(2 \psi) = \frac{2 E_{x0} E_{y0}}{E_{x0}^2 - E_{y0}^2} \cos(\delta_0)$$ (2.16)

Figure 2.2: Polarization ellipse in x-y-plane, rotation angle $\psi$, ellipticity angle $\eta$, and auxiliary angle $\alpha$, for a wave traveling in $\vec{e}_z$-direction (out of page)

The polarization may also be described by the geometrical properties of the ellipse, which are the inclination angle $\psi$ and the ellipticity angle $\chi$

$$\tan(\chi) = \pm \frac{a_{\xi}}{a_{\eta}}$$ (2.17)

where $2 a_\eta$ and $2a_\xi$ are the major and minor axes of the ellipse. $\chi$ specifies the shape of the ellipse as well as the sense of rotation of the E-vector. The polarization is left handed for $\chi > 0$ and right handed, for $\chi < 0$ for an observer looking in the direction of propagation.

The polarization angles $\psi$ and $\chi$ are related to the wave parameters $E_{x0}, E_{y0}$ and $\delta$ by [Born70]

$$\sin(2\chi) = \sin(2\alpha)\sin(\delta)$$ (2.18)

$$\tan(2\psi) = \tan(2\alpha)\cos(\delta)$$ (2.19)

The auxiliary angle $\alpha$ as shown in Fig. 2.2 is defined as

$$\tan(\alpha) = \frac{E_{y0}}{E_{x0}}$$ (2.20)

Special Cases of Polarizations

• Circular Polarization
  If both amplitudes are equal $E_{x0} = E_{y0}$ and the phase difference $\delta_0 = \pm \pi/2 + 2m\pi$, where $m = 0, \pm 1, \pm 2, \pm 3, ...$ we have circular polarization. When $\delta_0 = - \pi/2 + 2m\pi$ the polarization states are defined as being right hand circularly (clockwise looking in the direction of propagation [IEEE79]), and likewise when $\delta_0 = + \pi/2 + 2m\pi$ it leads to left hand circularly polarized states (counterclockwise rotation).

• Linear Polarization
  In this case the phase angles are equal ( $\delta_x = \delta_y \rightarrow \delta_0 = 0$) and the trace of the tip of the field vector $\vec{E}_x(z,t)$ is a straight line. The modulus of $\vec{E}_x(z,t)$ is then

  $$\vert\vec{E}_x(z,t)\vert = \sqrt{E_x^2(z,t) + E_y^2(z,t)} = \sqrt{E_{x0}^2+E_{y0}^2}\cdot cos(kz +\omega t - \delta_0)$$ (2.21)

  
  
  In this special case the tangent of the inclination angle $\psi$ simplifies to the ratio of the two components $\vec{E}_y$ and $\vec{E}_x$

  $$\tan(\psi) =\frac{E_{y0}}{E_{x0}}$$ (2.22)

  
  

Polarization Ratio

In the previous section the polarization state was described by two parameters, the inclination angle $\psi$ and the ellipticity angle $\chi$. The same information can be expressed by a complex number, the complex polarization ratio $\rho$ [Agrawal89], which is the complex ratio between orthogonal components for a given basis, e.g. { $\vec{e}_x , \vec{e}_y$}

$$\rho = \frac{E_{y0}}{E_{x0}} \cdot e^{i(\delta_y - \delta_x)} = \frac{\cos(2 \chi) \sin(2 \psi) + i \sin(2 \chi)}{1 - \cos(2 \psi)\cos(2 \chi)}$$ (2.23)

For a given complex polarization state, we can find an orthogonal polarization state $\rho_\perp$ given by the orthogonality condition

$$\rho \rho_\perp^* = -1$$ (2.24)

where $\rho_\perp^*$ is the complex conjugate of the orthogonal polarization state of $\rho$ [Krogager93]. These polarization states play an important role in the description of polarization as we shall see later in this treatment.