# Wave Polarimetry

Polarization refers to the trace of the extremity of an electric field vector as a function of time in a fixed space as seen back from the propagation direction as shown in Figure 2.1. If the extremity of the electric field vector traces linear, it is called a linearly polarized wave. If the trace is circular, the wave is circularly polarized. The shape of the trace can be linear, circular, elliptical, or even oriented elliptical. In addition, the clockwise or anticlockwise direction of the rotation enriches the directional polarization pattern. Each one of them represents the polarization state of the wave. It should be noted that polarization is not a field vector. There are several ways to represent the polarization state of the wave. In this chapter, we deal with wave polarization. Starting from Maxwell’s equation, we assess the fundamentals of the plane electromagnetic wave and how to represent polarization states using vector notation as well as geometrical parameters.

FIGURE 2.1 Trace of the extremity of time-varying electric field. (From Yamaguchi, Y., *Raclar Polarimetry from Basics to Applications: Radar Remote Sensing Using Polarimetric Information [in Japanese],* IEICE, 2007.)

## Plane Wave

The electromagnetic wave radiated from the source (antenna or scattering point) spreads spherically in a three-dimensional space as shown in Figure 2.2. The spherical equiphase surface expands according to range *r.* If the observation point *P* is located at r far from the source, the wave front surface can be approximated by a tangential plane. We assume r is much larger than the wavelength *Л.* The wave near *P* can be considered as a plane w'ave, indicating that the equiphase surface is plane. Therefore, wherever *r » A* is satisfied, we may regard the wave as a plane wave.

Since the plane wave plays the essential role for presentation of the polarization state of electromagnetic waves and can be derived from Maxwell’s equations, we start with Maxwell’s equation.

### Maxwell’s Equation and Wave Equation

Position vector can be written as r = ;r a _{v} + *у* a_{v} + *z* a., where a, ,a, ,a, are unit vectors in the *x, y, *and *z* directions, respectively. The electric field vector E(r, *t)* and the magnetic field vector H(r, *t),* as functions of position r and time *t,* satisfy the following Maxwell’s equation,

FIGURE 2.2 Plane wave approximation in the far field. (From Yamaguchi, Y., *Radar Polarimetry from Basics to Applications: Radar Remote Sensing Using Polarimetric Information [in Japanese],* IEICE, 2007.)

where, D is the electric flux density, В is the magnetic flux density, J is the electric current density, and *p* is the electric charge density. By taking the divergence of equation (2.1.2), the conservation law of charge can be obtained.

In homogeneous media, the macroscopic electrical property can be expressed by the dielectric permittivity *e,* the magnetic permeability *p,* and the electric conductivity *a.*

where the medium is also assumed to be isotropic. Equations (2.1.6) through (2.1.8) are called as constitutive relations. If the medium is anisotropic such as active ionosphere, *e* becomes the “tensor,” which induces the Faraday rotation.

The electric current density J is expressed by the sum of the conductive current J_{c} and the source current J_{s}.

Introducing the vector operator identity VxVxA = V(V-A)-V^{2}A, and operating the rotation operator on Equations (2.1.1) and (2.1.2), we obtain the vector wave equations for E and H.

Equations (2.1.10) and (2.1.11) are the general forms of wave equation. It would be desirable to determine a time-dependent polarization state presentation satisfying equation (2.1.10) and (2.1.11); however, no simple useful presentation was found to exist. Instead, we will treat plane wave propagation in a source-free medium with time-harmonic oscillation fields. This will allow a frequency domain treatment of the wave equation for which a unique monochromatic presentation of the polarization state exists.

### The Vector Wave Equation and Its Solution Using Phasor Representation

For a source-free medium (p = 0, J_{s} = 0), the right-hand side of (2.1.10) becomes zero.

In the following section and throughout the book, we shall adopt the harmonic time phasor definition. All the field quantity is assumed to have time-harmonic oscillation *e ^{jtt}"* and expressed as,

where A(r) is the phasor representation of A(r,r). A(r) is a complex-valued vector as a function of position r, and independent of time *t.*

Now, we clarify here the relation of A(r,/) and A(r). If we let instantaneous field vector as A(r,/), it can be decomposed into

On the other hand, the measurable quantity should be real-valued, and it can be written as

*0mx,0my,Qmz* are phases of the *x-, y-, z-* component, respectively. The subscript *m* refers to “measured.” The x-component leads to the following relation:

Re{»j- implies to take the real part of • Therefore, the measured value of the x-component can be expressed by the product of * A_{x}* and

*Similar expressions can be applied to the*

**e**^{Ja}”.*y-*and z-compo- nents. Therefore, the vector A can be related to the following equation:

where

If we use phasor notation **A(r) **(2.1.13) in Maxwell’s equation, the equation becomes simple, that is, independent of time. The field vector becomes a function of position r only. By assuming time harmonics *e' ^{al},* the time derivative becomes

*jco.*After calculating simple algebra and finding solution

**A(r),**the actual field quantity can be obtained from

**A(r,r)**= Re

**jA(r)e-'**These are the advantages of using the phasor notation.

^{<0}'|.Now, the vector wave Equations (2.1.10) and (2.1.11) in the source-free space becomes in the phasor notation as

where *к* is called the “wave number” and is defined as

The above equation is called the wave equation or Helmholtz equation.

#### Separation of Variables

Each scalar component of Equations (2.1.19) and (2.1.20) should satisfy the wave equation. For example, the equation for *E _{x}* becomes

Now we try to solve this second-order partial differential equation. It is anticipated that the solution of this equation is a function of a;, *y,* and z. Assuming the solution is in the form of *E _{x} = X* (

*x)Y(y)Z(z*), substitute it into (2.1.22) and divide by

*X(x)Y(y)Z(z)■*Then the next equation comes out.

From this equation, we notice that each term should be independent of each other and must be a constant. Therefore, we put each term as

The constants *k _{x},k_{y},k,* must satisfy,

For each variable, we have the second-order differential equation and its solution.

Therefore, *E _{x} = X (x)Y (y)Z(z)* can be multiplied as
a>

*C*amplitude coefficients

Since similar solutions can be obtained for *E _{y}* and £,, the vector form E(r) can be written as
where

Finally, the electric field vector E(r,r) as a function of space position and time can be obtained by the real part of the phasor multiplied by the time factor *e ^{Ja}',*

#### Physical Interpretation of the Solution

Now, let’s check for a moment how the mathematical solution is related to physical phenomena. We know that the term *at* - к • r in exp[y'(<»r - к • r)] is a phase. For simplicity, we assume |E_{0}| = 1 and the wave is propagating toward the *r* direction so that к • r = *kr,* and then (2.1,30b) becomes just a cosine function, cos *(at* - *kr).*

Figure 2.3 shows how cos *(at* - *kr)* changes with time *t.* When *t =* 0, it starts from 1. As time goes, the shape moves toward the right direction. We focus on a black dot •, which has a phase *0* = *at - kr.* If black dot • does not change its position with respect to time just like wind surfing, the next equation holds,

Since *%* represents velocity, the phase • moves toward the positive *r* direction with the speed of v = *f.* Therefore, we can consider:

exp[ *j(at* — к • r)] represents a wave propagating toward + *r* direction, ехр[ /(<у/ + к • r)] represents a wave propagating toward *-r* direction.

It is very important to pay attention to the sign in front of к ■ r because it plays the key role for polarization transformation or polarimetric analysis. Historically, in the field of optics, the expression

exp (/k r) has been used from the outset. So, the wave expression exp [/(kr-&)/)] represents wave propagating in the *+r* direction [1]. On the other hand, exp (*jcot*) has been used in engineering field, and exp [ *j(cot* - к • **r)] **represents wave propagating in *+r* direction. Therefore, the sign in front of к • **r **is opposite between optics and engineering. In order to avoid confusion, it is convenient to assume that *i = -j,* or *j =* -j for both academic communities.

The phase of the electric field vector (2.1.27) is constant if k - **r **= *k _{x}x+k_{y}y+k,z =* const. Since the equation к

**r**= const, represents a two-dimensional plane, the phase is constant on the plane as shown in Figure 2.4. If the equiphase is a plane, it is called a plane wave. Therefore (2.1.30) represents a plane wave.

In Figure 2.4, к is taken in the propagation direction. The position vectors r_{0}, r_{b} r_{2}, which satisfies к г, = к r_{2} = к r_{0}, spans a plane orthogonal to к. This plane is called a transverse plane. Since r_{0} is taken in the same direction as k, the phase change is the largest in this direction.

In lossless and isotropic media, the wave number becomes,

where v is the velocity of electromagnetic wave in the medium, and A is the wavelength. The naming of wave number comes from that is, how many A exists in the range *In.* If the medium has relative dielectric constant *s _{n}* the wave number becomes

The subscript 0 is added for the values in the free-space,

In the dielectric medium, the wavelength becomes shorter, and the velocity becomes slow because of *e _{r} >* 1.

**FIGURE 2.4 **Constant phase plane. (From Yamaguchi, Y., *Radar Polarimetry from Basics to Applications: Radar Remote Sensing Using Polarimetric Information [in Japanese],* IEICE, 2007.)

### Transverse Electromagnetic Wave

Here, we examine the relation between the electric field E(r) and the magnetic field H(r) in the free- space. The constitutive parameters become e = £_{0}, *ц = /j*_{0}, and *a =* 0. Maxwell’s equation (2.1.1) becomes in the phasor notation

Substituting the electric field E_{0}(r) (2.1.27) propagating to the +r direction into (2.1.35), we can obtain the following:

Therefore, equation (2.1.1) becomes
^{л} R R

If we normalize к as к = *т*-г = —_{;} : unit vector, equation (2.1.36) can be written as

|k| *G>y]c _{0}fj_{0}*

where г?о is the intrinsic impedance in the free-space,

Similarly from equation (2.1.2), we obtain,

In addition, V • D = 0 and V • В = 0 derive the following relations:

The vector relations among Equations (2.1.37) through (2.1.42) can be visualized as shown in Figure 2.5. E_{0} and ;/_{0}Hn have the same magnitude and are orthogonal to each other. They are also orthogonal to the propagation direction k. Since E_{0} and H_{0} reside in the same transverse plane perpendicular to the propagation direction, the wave is called the transverse electromagnetic (ТЕМ) wave.

FIGURE 2.5 Vector relations of ТЕМ wave propagation. (From Yamaguchi, Y., *Radar Polarimetry from Basics to Applications: Radar Remote Sensing Using Polarimetric Information [in Japanese],* IEICE, 2007.)

For the electric field E| (2.1.27) propagating in the -r(-k) direction, we have similar vector relations,

In this case, the vector relations of equation (2.1.43) become the right one shown in Figure 2.5. E, and t)_{0}H| are in the same magnitude and are orthogonal to each other and perpendicular to the -k direction. It is easy to understand these relations in a graphical way rather than in mathematical formulations.

To sum up the properties of ТЕМ waves in both propagation directions, E and H are orthogonal to each other. The propagation direction is E x H.

The propagation direction к can be taken arbitrarily. If we choose the propagation direction as the z-axis in the rectangular coordinate, к becomes к = *ka..* In this case, E and H are laid in the *x-y *plane as shown in Figure 2.6, and the *x-y* plane becomes transverse plane for the wave.

Note that the polarization of electromagnetic waves is defined for the electric field vector E only. Since the magnetic field vector H is orthogonal to E, no similar definition is required in order to avoid confusion.

### TEM Wave Power

Next, we consider the power of a plane wave. According to Poynting theorem, the power is represented by the product of the instantaneous electric field E and the instantaneous magnetic field H, which is denoted as the Poynting vector S as

S is a function of space r and time *t,* and therefore is not a phasor representation. Assuming the wave is propagating in the z-direction, and using Equations (2.1.30) and (2.1.45), we obtain the following form:

The time averaging of the instantaneous power through a constant z-plane yields the net flow of power,

IF|-

From this equation, we can notice that the power is represented by and is carried out by (E_{0},H_{0}) and (E_{b}H|) independently.

If we use phasor representation for power expression, it is convenient to employ a complex Poynting vector P defined as

where the superscript * denotes complex conjugation. Since E(r) and H(r) are complex valued, we can write them as

P can be expressed as

Since the instantaneous field vectors are given by the instantaneous Poynting vector S can be expressed as

Although the expression of P and S is different, the averaging over time yields the following relation:

This equation indicates that the time averaging of S is equal to half of the real part of P. Therefore, without integral calculation, we can obtain the electromagnetic power (net flow) by the complex Poynting vector P using phasor notation.

For a plane wave propagating in the z-direction, the time average power flow is given by

and is proportional to IE I^{2}. If E has an *x-* and у-component, the power can be decomposed into the sum.

It is a simple vector relation; however, it is a very important from the radar point of view. Power is the most fundamental radar parameter for which we will see in the following chapter.