# Radio Wave Polarization

Here is a simplified explanation of polarization and Stokes parameters.

A single electromagnetic wave can be thought of as an oscillating electric and magnetic field. The direction in which the electric field oscillates is the polarization direction. The polarization angle can have values between 0 and 180 degrees.

### Monochromatic Waves

In the following text, $\delta$ refers to the phases of electric field vectors and $\phi$ to their polarization angle. $E$ denotes an electric field and $V$ the voltage of this field.

The electric field of a harmonic, monochromatic plane wave at a fixed location in space can be described by electric field components in the x- and y-directions:

$E_x(t) = A_1\,e^{i\omega t}$ and

$E_y(t) = A_2\,e^{i\omega t},$

which are two linearly polarized waves with orthogonal polarization directions and the circular frequency $\omega$. The complex amplitudes $A_1$ and $A_2$ are $A_1 = a_1 e^{i\delta_1}$ and $A_2 = a_2 e^{i\delta_2}$, with the real amplitudes $a_1$ and $a_2$ and phases $\delta_1$ and $\delta_2$. The absolute phases of $A_1$ and $A_2$ are not important; only the relative phase $\delta = \delta_2 − \delta_1$ matters so that:

$A_1 = a_1$ and

$A_2 = a_2\,e^{i\delta}$.

The electric field vector of arbitrary polarization then reads:

$\mathbf{E}(t) = \binom{A_1}{A_2}\,e^{i\omega t} = \left(a_1 \mathbf{e}_x + a_2 \mathbf{e}_y e^{i\delta}\right) e^{i\omega t}$

in which $\mathbf{e}_x$ and $\mathbf{e}_y$ are unit vectors of a Cartesian coordinate system.

The electric field vector $\mathbf{E}(t)$ is a complex number. It is understood that this is the analytic representation of a plane harmonic wave. To obtain a physically meaningful quantity, for instance the voltage V(t) within the radio receiver, one has to take the real part of the above equation:

$V(t) = \mathfrak{Re}\left[\left(A_x \mathbf{e}_x + A_y \mathbf{e}_y \right) e^{i\omega t} \right]$.

In general, the imaginary part of the analytic signal does not physically exist. Also, whenever nonlinear operations are applied to the electric field vector such as squaring, etc., the real parts must be taken first and the operation is applied to these alone. This, however, is not necessary if the time average of a quadratic expression is required.

The polarization or position angle $\phi$ of the electric field vector is defined by:

$\tan\phi = \frac{E_x}{E_y} = \frac{a_2}{a_1}$.

With the amplitudes $a_1$ and $a_2$ expressed by the amplitude $a_0$ of the initial wave and the polarization angle $\phi$:

$a_1 = a_0\cos\phi$ and

$a_2 = a_0\sin\phi$,

the polarization state of a linearly polarized wave is completely described by $a_0$ and $\phi$ in terms of two linear polarization components.

### Stokes Parameters

Three independent parameters are needed to describe the polarization state of the initial vector wave. In the case of linear polarization components, these are the amplitudes $a_x$ , $a_y$ and the relative phase $\phi$. A practical way of expressing these parameters is by the use of the so-called Stokes parameters. The following relation exists-by definition-between Stokes parameters and the amplitude and phase of the polarization components:

$I = a_x^2 + a_y^2$

$Q = a_x^2 - a_y^2$

$U = 2 a_x a_y \cos \delta$

$V = 2 a_x a_y \sin \delta$.

Stokes I is the total power signal received through both hands of polarization. Stokes Q and U are the two (x and y) components of the linearly polarized fraction of the signal, and Stokes V represents the circularly polarized component.

The amplitude (polarized intensity PI) of the polarized component can be calculated as follows:

$\mathrm{PI} = \sqrt{U^2 + Q^2}$

and the polarization angle PA of the linearly polarized component:

$\mathrm{PA} = \arctan\left(\dfrac{Q}{U}\right)$

The fractional (percentage) polarization PP is given by:

$\mathrm{PP} = \mathrm{PI} / I$

### Partial Polarization

A single electromagnetic wave is fully polarized. In nature, however, electromagnetic radiation is produced by a large ensemble of radiators, producing incoherent waves. Incoherent radiation may still show a statistical correlation between the polarization components. This can be interpreted as partial polarization. Stokes parameters are then given by time averages:

$I = \left<a_x^2\right> + \left<a_y^2\right>$

$Q = \left<a_x^2\right> - \left<a_y^2\right>$

$U = 2 \left<a_x a_y \cos \delta\right>$

$V = 2 \left<a_x a_y \sin \delta\right>$.

### Polarization Measurements

A polarimeter does not measure the amplitudes and phase differences of the polarization components directly; it rather detects time-averaged products of the two components, such as, for example, the following product of the x- and y-components:

$\left<V_x V_y\right> = \lim\limits_{T'\to\infty} \frac{1}{4T'} \int_{-T'}^{T'} \left(E_x + E_x^*\right) \left(E_y + E_y^*\right) dt$.

With

$\frac{1}{2T'} \int_{-T'}^{T'} e^{2i\omega t} dt = \frac{T}{4\pi T'}\sin 2\omega T' \approx 0$

for $T' \gg T$ the time-averaged product becomes:

$\left< V_x V_y \right> \propto A_x A_y^* + A_x^* A_y = a_x a_y \cos \delta$

and hence a measure for Stokes U. Using the analytic representation, this equation can be written as:

$\left< V_x V_y \right> = \mathfrak{Re} \left(E_x E_y^*\right) = a_x a_y \cos\delta$.

Time averaging the other possible products gives other Stokes parameters, which leads to the following variant of the definition of Stokes parameters:

$I = \left< E_x E_x^* \right> + \left< E_y E_y^* \right>$

$Q = \left< E_x E_x^* \right> - \left< E_y E_y^* \right>$

$U = 2 \, \mathfrak{Re} \left< E_x E_y^* \right>$

$V = 2 \, \mathfrak{Im} \left< E_x E_y^* \right>$.

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