Plane Wave Representation

Abstract - In this part we discuss how to describe a plane wave with vectors. The two common approaches are the Stokes vector representation and the Jones vector representation. For the Stokes vector representation we will show, how the representation can be visualized by means of the Poincaré sphere.

keywords: Stokes vector, Jones vector, Poincaré sphere,

Stokes Vector Representation

In 1852 Stokes introduced a set of parameters in order to characterize the polarization state. The representation consists of four Stokes parameters $g_0,g_1,g_2$ and $g_3$ which are defined as

$$\vec{g}(\vec{E}) = \left[\begin{array}{c} g_0 g_1 g_2 g_3 \end{array}\right]= ... ...\\ 2 E_{v0}E_{h0}\cos(\delta) 2 E_{v0}E_{h0}\sin(\delta) \end{array}\right]$$ (2.33)

where $E_v$ and $E_h$ denotes the vertical and horizontal component (see section 4.5) of the electric field vector $\vec{E}$. $g_0$ is proportional to the total intensity of the wave, $g_1$ represents the difference between the vertically and horizontally polarized intensity and expresses the amount of linear horizontal or linear vertical polarization of the wave. $g_2$ and $g_3$ jointly represent the phase difference between the h and v component of the electric field and can be regarded as the amount of right or left circular polarization of the wave. [Collett93].

This representation is commonly referred as a Stokes vector and is valid for completely and partially polarized waves. Even though the Stokes vector consists of 4 parameters, in the case of completely polarized waves, only three of them are independent because the sum of the polarized components $g_1$,$g_2$,$g_3$ equals the total intensity of the wave.

$$g_0^2 = g_1^2 + g_2^2 + g_3^2$$ (2.34)

For partially polarized waves not all the intensity is contained in the polarized components and therefore, the total intensity is bigger than the sum of the polarized components.

$$g_0^2 > g_1^2 + g_2^2 + g_3^2$$ (2.35)

For completely polarized waves the stokes parameters can be expressed as

$$\vec{g} = \left[\begin{array}{c}g_0 g_0 \cos(2\psi)\cos(2\chi) \\ g_0 \sin(2\psi)\cos(2\chi) g_0 \sin(2\chi) \end{array}\right]$$ (2.36)

where $\psi$ is the inclination angle from (2.18) and $\chi$ is the ellipticity angle from (2.19).

The state of polarization of a completely polarized plane wave can be mapped uniquely to a point P on the surface of a sphere with radius $g_0$, called the Poincaré sphere (Fig. 2.4) by regarding $g_1,g_2$ and $g_3$ as the Cartesian coordinates of P [Descamps51], [Descamps73], [Boerner91]. $2\psi$ defines the latitude and $2\chi$ the longitude of P. As discussed above the sign of $\chi$ determines the handedness or orientation of the polarization state. Therefore the upper hemisphere ($\chi > 0$) displays left handed polarizations, while the lower hemisphere ($\chi < 0$) displays right handed polarizations. The poles represent circular polarizations and in the equatorial plane we find linear polarization.

Figure 2.4: Poincaré sphere visualization of polarization states. linear polarization along the equator, circular polarization at the poles, elliptical polarization: the upper and lower hemisphere

As stated above (2.34) is only valid for completely polarized waves. For the general case we have to use [Huynen87]

$$g_0^2 \ge g_1^2 + g_2^2 + g_3^2$$ (2.37)

Introducing the degree of polarization p

$$p=\frac{\sqrt{\sum_{i=1}^3 {g_i}}}{g_0}$$ (2.38)

the Stokes vector can be decomposed into a completely polarized and an unpolarized component. If we define a we can express the Stokes vector as a sum of a completely polarized and a completely unpolarized component

$$\vec{g} = \left[\begin{array}{c} g_0 g_1 g_2 g_3 \end{array}\right]= ... ...2\psi)\cos(2\chi) p \sin(2\psi)\cos(2\chi) p \sin(2\chi)\end{array}\right]$$ (2.39)

This representation yields two particularly convenient features.

a)

the orthogonal polarization states are located on diametrically opposite points of the Poincaré sphere.

b)

all four parameters can be derived from intensity measurements, which, especially for optical applications, is a great advantage.

Jones Vector Representation

While the Stokes vector is still widely used for optical applications, another representation exists and is used in this treatment. Instead of a representation in a three-dimensional real space, a representation in a two-dimensional complex space is preferred for radar remote sensing applications as mentioned in sections 1.3 and 1.6. In this representation of a monochromatic plane wave, the E field is written, as a linear combination of two arbitrary orthonormal polarization states $\vec{e}_a$ and $\vec{e}_b$ weighted by their corresponding complex amplitudes $E_a$ and $E_b$ as already stated in (2.7) for the case of { $\vec{e}_x , \vec{e}_y$}-polarisation states.

$$\vec{E} = E_a \vec{e}_a + E_b \vec{e}_b$$ (2.40)

or as 2-dimensional complex vector $\vec{E}_{ab}$ as introduced by Jones [Jones41]

$$\vec{E}_{ab} = \left[\begin{array}{c} E_a E_b \end{array}\right]= \left[\begin{array}{c} E_{a0} e^{i\delta_a} E_{b0} e^{i\delta_b} \end{array}\right]$$ (2.41)

The so called Jones vector contains the complete information about the polarization ellipse, except the handedness, i.e. two plane waves propagating in opposite directions have the same Jones vector representation. In order to compensate this lack of consistency the Jones vector can be completed by using the subscripts "$_+$" and "$_-$" , where subscript "$_+$" denotes a wave propagating in $\vec{k}$ direction and subscript "$_-$" a wave propagating in $-\vec{k}$ direction [Lüneburg95].

$$\vec{E}_+ (\vec{r},t)= \Re\{\vec{E}_+ e^{i(\omega t - \vec{k} \cd... ...vec{E}_- (\vec{r},t)= \Re\{\vec{E}_- e^{i(\vec{k} \cdot \vec{r} + \omega t)} \}$$ (2.42)

The vectors $\vec{E}_\pm$ are denoted as directional Jones vectors [Graves56] where $\vec{E}_+$ is the complex conjugate of $\vec{E}_-$ and vice versa. The conjugation of the Jones vectors is equivalent to the change of the sign of the difference angle $\delta= \delta_b - \delta_a$ and thus to a change of the handedness of the polarization state. Using the polarization ratio from (2.23) the Jones vector can also be expressed as

$$\vec{E}_{ab} = \left[\begin{array}{c} E_a E_b \end{array}\right]= E_a \left[\begin{array}{c} 1 \rho \end{array}\right]$$ (2.43)

Change of Polarization Basis

The polarization ratio $\rho$ in (2.47) depends on the actual basis in which the quantities are given. In general, any orthonormal set of elliptically polarized states can form a polarization basis. Therefore, as we change from one basis to another, also the representation of the wave changes. If { $\vec{e}_{x1},\vec{e}_{x2}$} and { $\vec{e}_{y1},\vec{e}_{y2}$} are two orthonormal polarisation bases, the E-field $\vec{E}$ is represented as

$$\vec{E} = E_{x1}\vec{e}_{x1}+E_{x2}\vec{e}_{x2} = E_{y1}\vec{e}_{y1}+E_{y2}\vec{e}_{y2}$$ (2.44)

with the corresponding Jones vectors

$$\vec{E}_{x1,x2} = \left[\begin{array}{c} E_{x1} E_{x2} \end{ar... ...ad \vec{E}_{y1,y2} = \left[\begin{array}{c} E_{y1} E_{y2} \end{array}\right]$$ (2.45)

The transformation of a 2-element complex vector into another is given via a $2 \times 2$ complex matrix $[U]_{2 \times 2}$, so that

$$\vec{E}_{y1,y2} = [U]_{2 \times 2} \vec{E}_{x1,x2}$$ (2.46)

As a consequence of the preservation of the energy of the wave, the matrix $[U]_{2 \times 2}$ is unitary. With the polarization ratio $\rho_{x}$ of $\vec{E}_{x1,x2}$ the basis vectors of the new basis { $\vec{e}_{y1},\vec{e}_{y2}$} are given as [Krogager95a], [Krogager95b], [Auermann98]

$$\vec{e}_{y1} = e^{i\delta_{y1}} \frac{1}{\sqrt{1+\rho_{x}\rho_{x}^*}}\left[\begin{array}{c} 1 \rho_{x} \end{array}\right]$$ (2.47)

$$\vec{e}_{y2} = e^{i\delta_{y2}} \frac{1}{\sqrt{1+\rho_{x}\rho_{x}^*}}\left[\begin{array}{c} - \rho_{x}^* 1 \end{array}\right]$$ (2.48)

$\delta_{y1}$ and $\delta_{y2}$ represent the phase reference for the new basis states. It can be proved, that $\delta_{y1} = - \delta_{y2}$ [Papathanassiou99]. The transformation matrix $[U]_{2 \times 2}$ is the projection of the new basis { $\vec{e}_{y1},\vec{e}_{y2}$} onto the original basis { $\vec{e}_{x1},\vec{e}_{x2}$}. This means that the columns of the transformation matrix $[U]_{2 \times 2}$ are the vectors of the new basis [Strang88] $\vec{e}_{y1}$ and $\vec{e}_{y2}$ expressed in the original basis { $\vec{e}_{x1},\vec{e}_{x2}$}, with $det([U]_{2 \times 2}) = 1$.

$$[U]_{2 \times 2} = \frac{1}{\sqrt{1+\rho \rho^*}} \cdot \left[\be... ...^{-i\delta_{y1}} \rho e^{i\delta_{y1}} & e^{-i\delta_{y1}} \end{array}\right]$$ (2.49)

Where $\rho$ is the polarization ratio of E in the { $\vec{e}_{x1},\vec{e}_{x2}$} basis. The phase reference angle $\delta_{y1}$ is not important for the determination of the polarimetric parametres of the new basis states and is usually set as $\delta_{y1} = 0$ in radar polarimetry [Kong86], [Kennaugh52].