Radio Propagation and Remote Sensing of the Environment - Chapter 2 pps

32 Radio Propagation and Remote Sensing of the EnvironmentBy convention, the wave phase is considered to be constant in the stated plane ifcoordinates x and y change within such limits s

2.1 PLANE WAVE DEFINITION

In the previous chapter, we defined a plane wave as a wave whose characteristicsdepend on only one Cartesian coordinate We also noted that the plane wave isexcited by a system of sources distributed uniformly on an infinite plane Because

a source of infinite size is an abstraction, the notion of a plane wave is also abstract

We also established in the previous chapter that, far from real sources (sources

of limited sizes), radiated waves can be considered to be spherical and that the phase

of radiated waves close to the pattern maximum is constant on a spherical surface

of radius R, where R is the distance from the source Locally, a spherical surface oflarge radius differs little from a plane and may be supposed to be a plane in thedefined frames of space Let us now refine the bounds of these frames

Ignoring unimportant details, the spherical wave may be described by the lowing expression (ε = 1):

fol-where the vector T does not depend on R and describes the wave polarization,

coordinates of the observation point Accordingly, the start of the coordinate system

is situated at the point where the radiator is located

Let us imagine that the pattern maximum is close to the direction defined bythe condition r = If we consider that, in some of the space, area r2

<< z2, then approximately:

It is now easy to set up the conditions such that the field depends on only coordinate

z In this case, the front of the spherical wave may be considered to be locally plane;thus, the wave is also thought to be locally plane The wave phase of Equation (2.1)changes in plane z = const due to the law:

32 Radio Propagation and Remote Sensing of the Environment

By convention, the wave phase is considered to be constant in the stated plane ifcoordinates x and y change within such limits so that the value of kr2/2z < π, or:

2.2 PLANE WAVES IN ISOTROPIC HOMOGENEOUS

MEDIA

The explanation of the plane wave concept provided above does not include all types

of waves, as the form of a plane wave depends on the propagation media istics The simplest is the case of homogeneous and isotropic media In this case,

character-as wcharacter-as shown in the first chapter, the electromagnetic field vectors satisfy the waveequation, Equation (1.13) It follows, then, that every component of electrical andmagnetic fields satisfies a scalar wave equation, thus we can examine the propagation

of any one and extend the results to others

Let us denote the chosen component field component as u Because all of itsparameters depend on one coordinate (for example, z), it must satisfy the equation:

(2.4)The common solution of this equation is:

(2.5)

where constants u1 and u2 are defined from the exiting and boundary conditions.The first term in Equation (2.5) represents a wave propagating in the direction ofpositive values of z These waves are usually referred to as direct The second term

in Equation (2.5) describes a back wave propagating in the direction of negativevalues of z If all the sources are to the left along the z-axis and no obstacles arecausing wave reflection, then the back wave has to be absent and we suppose that

Plane Wave Propagation 33

The value:

(2.7)

is called the refractive index (the term is borrowed from optics), and

(2.8)

is called the index of absorption

Taking into consideration the time dependence, the expression for the planewave can be represented as:

(2.9)The value u0 is called the initial wave amplitude, where

is called the depth of penetration or skin depth

It is a simple matter to determine the phase velocity from Equation (2.10):

u u e= i − 0

34 Radio Propagation and Remote Sensing of the Environment

At n′ > 1, which usually occurs, the phase velocity is less than the speed of light

In the case of plasma, when n′ < 1, the phase velocity is more the velocity of light.The wave number in dielectric k′ equals kn′, and the wavelength:

(2.15)

differs from the same in vacuum

Let us consider specific cases Very often ε′′ << ε′, which corresponds to the

(2.16)

In the opposite case of high absorbed media, ε′′ >> ε′, and Forthis case,

(2.17)

The plane wave may be introduced as follows:

in the case of an arbitrary direction of propagation Vector q is the wave vector anddefines the direction of the wave propagation Substitution in Maxwell’s equationsgives:

assuming current density j = 0 and density of charge ρ = 0 It follows from theseequations that vectors E and H are orthogonaltoeach other and to the direction ofwave propagation at real vector q It is supposed in this case that ε = 0, which canoccur in the case of plasma Except, for instance, vector H from Equation (2.18),

we may easily obtain the equation of dispersion:

Plane Wave Propagation 35

q″ are real because their projections on the coordinates axes are real numbers Itfollows from Equation (2.19) that:

(2.20)

Vectors q′ and q″ do not have to be parallel to each other, which means that forsome plane waves (inhomogeneous plane waves) the planes of equal phase and equalamplitude do not coincide Such waves do not correspond to the plane wave definitiongiven at the beginning of this chapter because their different characteristics (theamplitude and the phase) depend on one coordinate In homogeneous media, inho-mogeneous plane waves are not excited; however, they appear upon electromagneticwave propagation in inhomogeneous media

Equations (2.18−19) permit us to state that the complex amplitudes of vectors

E and H are connected with the equality:

Poynting’s vector of a plane wave in the general case is defined by the formula:

(2.22)

from which we may conclude that the plane wave is directed along a ray orthogonal

to the plane of uniform phase

In conclusion, we have shown that investigating fields with respect to spatialthan expansion of the fields with respect to plane waves

2.3 PLANE WAVES IN ANISOTROPIC MEDIA

As we have already pointed out, the connection between vectors D and E in tropic media is a tensor one So, Equation (1.6) should be employed, and Equation(2.18) becomes:

aniso-(2.23)Excluding vector H from these equations, we obtain:

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Fourier integrals, which we have already used in Chapter 1, involves nothing more

36 Radio Propagation and Remote Sensing of the Environment

Let us suppose that the wave propagates along the z-axis, in which case qx = qy =

0, and we substitute qz = kn for the z-component of vector q Then, the followingsystem of equations can be obtained from Equation (2.24):

(2.28)

Because Equation (2.27) is a system of linear uniform algebraic equations relative

to the components of an electric field, the conditions of the nontrivial solution requirethe determinant of the system to approach zero The dispersion equation can bestated by the following expression, which is reduced to a biquadrate equation relative

to refractive index n:

(2.29)with the obvious solution:

xz zx zz

yy

yz zy zz

zz

yx yx

zx yz zz, C = −

Plane Wave Propagation 37

Let us now apply the common expressions obtained for the case of plasma, which

is of special interest because of radiowave propagation in the ionosphere We willnot develop the expression for tensor components of magnetic active plasma, as itmay be found elsewhere (for example, in Ginsburg12 from which we have takensome necessary expressions) Moreover, let us point out that waves are weaklyabsorbed in the ionosphere because microwaves are discussed throughout this book;therefore, we neglect the absorption to avoid complicating the problem

Let us introduce some definitions The value:

(2.31)

is called the plasma frequency Here, N is the concentration of electrons in plasma,

e = 4.8 · 10–10 CGS electrostatic system (CGSE) is the electron charge, and m = 9.1

· 10–28 g is its mass For the ionosphere of Earth, where the maximal value of theelectron concentration is Nm 2 · 106 cm–3, the maximal value of the plasma frequency

is about 10 MHz So, in microwaves the ratio:

(2.34)

exists in the microwave region

We shall suppose that the magnetic field of Earth lies in the z0y plane at angle

β to the z-axis, which, we recall, coincides with the direction of wave propagation.The components of the permittivity tensor are described by:12

iv u u

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38 Radio Propagation and Remote Sensing of the Environment

(2.35)

in the chosen coordinate system The substitution of these expressions in Equation

(2.28) permits us to calculate the values A, B, Cxy, and Cyx and then to obtain an

expression for the refractive index:

(2.36)

Having two solutions for the refraction index means that two types of waves occur

in magnetic active plasma: the ordinary one, to which the (+) sign corresponds in

Equation (2.36), and the extraordinary one, for which the (–) sign would be chosen

Equations (2.32) and (2.34) can be used to represent Equation (2.36) more simply as:

(2.37)

It is often supposed that u = 0 for ultra-high-frequency (UHF) and microwave

regions, in which case the ordinary and extraordinary waves do not differ, and only

one wave exists in the plasma and has the index of refraction:

(2.38)

Later, we will define more precisely when it is sufficient to use the approximation

for wave propagation in the ionosphere, but for now we will say only that refraction

index n < 1 in this approximation, which means that the phase velocity of waves in

plasma is greater than the velocity of light

Equation (2.26) allows us to express the longitudinal component of field Ez via

the transversal components according to the equality:

2

1 2 2

ωω

ωωp

Plane Wave Propagation 39

The expression presented here shows that the longitudinal components of waves are

smaller than the transversal ones; therefore, waves in the ionosphere at high enough

frequencies can be considered transversal in all events

Finally, the polarization coefficient is an important characteristic which is

described by the relation:

(2.40)

at high enough frequencies Because u is small, the condition

is true over a wide range of angles and leads to quasi-longitudinal propagation at

UHF and microwave ranges Thus, the approximations:

(2.41)

are correct for both wave types The polarization coefficients are defined as follows:

(2.42)

Hence, the ordinary and extraordinary waves are circularly polarized with directions

of rotation opposite those of the polarization planes

2.4 ROTATION OF POLARIZATION PLANE (FARADAY

EFFECT)

The possibilityof theexistence of two types of waves in magnetized plasma results

in some specific effects, one of them being rotation of the plane of polarization,

known as the Faraday effect Let us imagine that a linearly polarized plane wave is

incident on a layer of magnetized plasma A plane with invariable linear polarization

is not able to propagate in the plasma considered here, and, as we have just

estab-lished, only the existence of circular polarized waves is possible, both ordinary and

special waves They are excited at the plasma input, adding in such a way that their

sum is equal to the linearly polarized incident wave (taking into account, of course,

the processes of reflection and penetration at the plasma boundary) If the phase

velocities of ordinary and extraordinary waves are the same, then a wave with

invariable linear polarization would propagate; however, in this case, the velocities

are different, which means, for instance, that the electrical vectors of ordinary and

extraordinary waves turn in opposite directions at different angles This difference

in angle rotation leads to rotation of the summary polarization vector, the electrical

one, at an angle, and is known as the Faraday effect The described rotation differs

in essence from the rotation of electrical (and, of course, magnetic) vectors of circular

polarized waves in that it rotates with the field frequency at each point of space In

x y

40 Radio Propagation and Remote Sensing of the Environment

this case, the polarization direction is left unchangeable at each point of the space

and changes only during transition from point to point in the wave propagation

direction

Elementary calculations show that the value of the summary wave rotation angle

is:

(2.43)

where L is the distance passed by the wave in plasma On the basis of this formula,

it is easy to establish that the Faraday angle of rotation is proportional to half of the

phase difference of ordinary and extraordinary waves when they pass distance L

By using our expressions for u and v in Equations (2.32) and (2.34), we obtain:

(2.44)

The estimations carried out for the ionosphere of the Earth show that the angle of

Faraday rotation is sizeable even at frequencies of hundreds of megahertz, and it

should be taken into consideration when designing radio systems of this range

Measurement of the plane polarization angle rotation can be used for estimating the

electron content, as the magnetic field strength of the Earth is known

The reduced formulas help to answer the question of when we should take into

consideration the terms with in Equation (2.41) If the Faraday angle is

small, then the difference between ordinary and extraordinary waves is insignificant;

otherwise, it is necessary to take this difference into account, at least, while analyzing

polarization phenomenon

2.5 GENERAL CHARACTERISTICS OF POLARIZATION

AND STOKES PARAMETERS

Linear and circular polarization, as discussed previously, are particular cases In this

section, we will consider general characteristics of polarization and interpolate

parameters that describe these characteristics with sufficient complexity Let us

choose the z-axis as the wave propagation direction We shall assume that the waves

are completely transversal with the components of the electrical field:

(2.45)

Here, Φx and Φy are initial phases of the x- and y-components of the field So,

amplitudes Ex and Ey can be considered as real values If we write Equation (2.45)

in the view of real expressions and take the real part of the right part and exclude

ΨF= ω( − e) ≅ω β

v uL c

o

cos,

2 2

3 0

2 2

L c

cos

Plane Wave Propagation 41

the phase qz – ωt, then it is easy to establish that the summary field vector E = Exex +

Eyey is elliptically polarized The ellipse of polarization is described by the equation:

(2.46)

where Φ = Φx – Φy is the phase difference of the vector E components The large

axis of ellipse is inclined by the angle ψ to the x-axis It is easy to define this angle

with the equality:

(2.47)

where:

(2.48)The ellipse radii are defined as follows:

(2.49)where

(2.50)

is the wave intensity

The values S0, S1, S2, and

(2.51)

are called Stokes parameters and characterize the polarization property of transversal

plane waves It is easy to be convinced of the truth of the relation:

(2.52)which takes place for coherent waves The problem of combined coherent wavesand noise radiation will be examined later

The ellipse radii a and b and the angle of inclination, ψ, are defined at polarization

measurements Stokes parameters are calculated according to the formulas:

x

y y

S3=2E Eˆ ˆ sinx y Φ=2Im( )E Ex *y

S02=S12+S22+ ,S32