© 2005 by CRC Press 31 2 Plane Wave Propagation 2.1 PLANE WAVE DEFINITION In the previous chapter, we defined a plane wave as a wave whose characteristics depend on only one Cartesian coordinate. We also noted that the plane wave is excited 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 of large radius differs little from a plane and may be supposed to be a plane in the defined frames of space. Let us now refine the bounds of these frames. Ignoring unimportant details, the spherical wave may be described by the fol- lowing expression ( ε = 1): where the vector T does not depend on R and describes the wave polarization, amplitude, and distribution angle. Here, where x, y, and z are 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 by the condition r = . If we consider that, in some of the space, area r 2 << z 2 , then approximately: . (2.1) 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: (2.2) E = T e R ikR , R =++xyz 222 x+y 22 = 0 ET= + () exp ikz ikr 2 2z z Φ =+k kr z z 2 2 . TF1710_book.fm Page 31 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 32 Radio Propagation and Remote Sensing of the Environment By convention, the wave phase is considered to be constant in the stated plane if coordinates x and y change within such limits so that the value of kr 2 /2z < π , or: (2.3) So, the wave radiated by the real sources can be considered to be a plane within the Fresnel zone. Let us point out that the size of the fixed area increases according to the distance from the radiating source and can be rather large, which gives us the opportunity to analyze various wave phenomena within the frame of the plane wave approximation and to approach essential results with acceptable accuracy. Also, a plane wave is one of the simplest types of waves. 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 character- istics. The simplest is the case of homogeneous and isotropic media. In this case, as was shown in the first chapter, the electromagnetic field vectors satisfy the wave equation, Equation (1.13). It follows, then, that every component of electrical and magnetic 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 its parameters 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 u 1 and u 2 are defined from the exiting and boundary conditions. The first term in Equation (2.5) represents a wave propagating in the direction of positive 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 negative values of z. If all the sources are to the left along the z-axis and no obstacles are causing wave reflection, then the back wave has to be absent and we suppose that u 2 = 0. For simplification and convenience of further calculations, let us introduce the value: (2.6) r <=λρz F . du d ku 2 2 0 z 2 +=ε . uue ue ik ik =+ − 12 εεzz , nnin= ′ + ′′ = ε. TF1710_book.fm Page 32 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 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 plane wave can be represented as: (2.9) The value u 0 is called the initial wave amplitude , where (2.10) is the phase, and (2.11) is the coefficient of attenuation (absorption) of the wave. Because of the absorption dielectric, wave amplitude decreases with distance, according to the exponential law: (2.12) By convention, it is supposed that at depth z = d s , such that γ d s = 1, the field is practically faded. The value: (2.13) is called the depth of penetration or skin depth . It is a simple matter to determine the phase velocity from Equation (2.10): (2.14) ′ = ′ + ′′ + ′ () =+ ′ () n 1 2 1 2 22 εε ε εε ′′ = ′ + ′′ − ′ () = − ′ () n 1 2 1 2 22 εε ε εε uue i = − 0 Φγz . Φ = ′ −kn tz ω γ = ′′ kn uue= − 0 γz . d kn s == ′′ 11 γ v c n Φ = ′ . TF1710_book.fm Page 33 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 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 case of low absorbed media. Than n ′ ≅ In this case, (2.16) In the opposite case of high absorbed media, ε′′ >> ε′ , and For this 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 and defines the direction of the wave propagation. Substitution in Maxwell’s equations gives: , (2.18) assuming current density j = 0 and density of charge ρ = 0. It follows from these equations that vectors E and H are orthogonal to each other and to the direction of wave propagation at real vector q . It is supposed in this case that ε = 0, which can occur in the case of plasma. Except, for instance, vector H from Equation (2.18), we may easily obtain the equation of dispersion: . (2.19) The fact that the modulus of vector q equals a complex number in the general case means that it is a complex vector itself; that is, q = q ′ + i q ′′ , where vectors q ′ and ′ = ′ = ′ λ πλ2 kn ′′′ ≅ ′′ ′ εεεand n /( ) . 2 v c c d c Φ = ′ ′ = ′ = ′′ ′ = ′ ′′ ε λ λ ε γ ωε ε ε ωε ,, , . 2 2 s ′ ≅ ′′ ≅ ′′ nn ε 2. vc c d c Φ = ′′ ′ ′′ = ′′ = ′′ 2 2 2 ε λ ε λγ ωε ωε ,= 2 s ,,. EH E H qr ,,= ⋅ 00 e i qE H qH qH E qE×     = ⋅ () =×     = −⋅ () =kk,, ,00ε q 2 = k 2 ε TF1710_book.fm Page 34 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Plane Wave Propagation 35 q ″ are real because their projections on the coordinates axes are real numbers. It follows from Equation (2.19) that: (2.20) Vectors q ′ and q ″ do not have to be parallel to each other, which means that for some plane waves (inhomogeneous plane waves) the planes of equal phase and equal amplitude do not coincide. Such waves do not correspond to the plane wave definition given at the beginning of this chapter because their different characteristics (the amplitude and the phase) depend on one coordinate. In homogeneous media, inho- mogeneous plane waves are not excited; however, they appear upon electromagnetic wave 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: . (2.21) 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 spatial than 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 aniso- tropic media is a tensor one. So, Equation (1.6) should be employed, and Equation (2.18) becomes: (2.23) Excluding vector H from these equations, we obtain: (2.24) ′ − ′′ = ′′ ⋅ ′′ () = ′′ qq 22 kk 22 2εε,.qq HE= ε S E q= ′ c k 2 8π , qE H qH qH D qD×     = ⋅ () =×     = −⋅ () =kk,, ,.00 qqE E D⋅ () − +=q 2 k 2 0. TF1710_book.fm Page 35 Thursday, September 30, 2004 1:43 PM Fourier integrals, which we have already used in Chapter 1, involves nothing more © 2005 by CRC Press 36 Radio Propagation and Remote Sensing of the Environment Let us suppose that the wave propagates along the z-axis, in which case q x = q y = 0, and we substitute q z = kn for the z-component of vector q . Then, the following system of equations can be obtained from Equation (2.24): (2.25) Furthermore, it is useful to exclude the z-component of electric field from this system so we have: (2.26) which, after substitution of this expression in the other two expressions of Equation (2.25), leads to a simpler system of equations: (2.27) Here, (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 require the determinant of the system to approach zero. The dispersion equation can be stated by the following expression, which is reduced to a biquadrate equation relative to refractive index n : (2.29) with the obvious solution: . (2.30) εε xx x xy y z EEE− () ++=n 2 0, εε ε yx x xx y yz z EEE+ − () +=n 2 0, εεε zx x zy y zz z E+ E+ E = 0 E EE z zx x zy y zz = − +εε ε , An C C Bn− () += +− () = 22 00EE E E xxyy yxx y ,. ABC= − = − = −ε εε ε ε εε ε ε ε xx xz zx zz yy yz zy zz xy xy xz ,, εε ε ε εε ε zy zz yx yx zx yz zz ,.C = − An Bn CC− () − () − = 22 0 xy yx nABABCC 12 2 2 1 2 4 , =+±− () +       xy yx TF1710_book.fm Page 36 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 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 will not develop the expression for tensor components of magnetic active plasma, as it may be found elsewhere (for example, in Ginsburg 12 from which we have taken some necessary expressions). Moreover, let us point out that waves are weakly absorbed 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 the electron concentration is N m 2 · 10 6 cm –3 , the maximal value of the plasma frequency is about 10 MHz. So, in microwaves the ratio: (2.32) always occurs. Now let us introduce the cyclotron frequency defined by the equality: (2.33) where H 0 is the strength of the magnetic field of Earth and has a value about 0.5 Oersted (Gauss). Therefore, the cyclotron frequency is equal to about 1.5 MHz, and 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 ω π ω π pp p ==≅⋅ 4 2 910 2 3 eN m fN, v =<< ω ω p 2 2 1 ω ω π HH H ==≅⋅ eH mc fH 0 6 0 2 28 10,., u =<< ω ω H 2 2 1 εεε β xx xy yx = − − = − = − 1 11 v u iv u u , cos , TF1710_book.fm Page 37 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 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 , C xy , and C yx 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 E z via the transversal components according to the equality: (2.39) εε β ε β xz zx yy = − = − − = − − () − iv u u vu u sin , sin , 1 1 1 1 2 εε ββ ε β yz zy zz == − = − − () − uv u vu u sin cos , cos 1 1 1 1 2 n vv vu u u v 12 2 224 1 21 21 4 1 , sin sin = − − () − () − ±+− ( ββ )) 2 2 cos . β nv u uu 12 22222 11 2 1 2 4 , sin sin cos .= − ++       βββ∓ nv= − = −≅−11 1 2 2 2 2 ω ω ω ω p 2 p . EEE zxy ≅−iv u usin [ cos ].ββ TF1710_book.fm Page 38 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 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 possibility of the existence 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 Ki uu 12 42 2 2 4 , cos sin cos sin .== +± E E x y β ββ β 2 2 cos sinββ>> u nn v unn v u oe12 1 2 11 2 1= ≅− − () = ≅− + () cos , cos .ββ KKi KK i 12 = ≅ = ≅− oe ,. TF1710_book.fm Page 39 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 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, Φ x and Φ y are initial phases of the x- and y-components of the field. So, amplitudes E x and E y 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 Ψ Fe = − () ≅ ωωβ 22c nnL vuL c o cos , Ψ F pH 2 == =⋅ ωω β ω β π 2 2 3 0 22 22 236 1 L c eNHL mc f cos cos . 00 4 0 2 NH L f cos . β u cos β Eqz- Eqz- xx x yy y =+ () =+ ˆ exp , ˆ expEEiiti iitiωωΦΦ (() . TF1710_book.fm Page 40 Thursday, September 30, 2004 1:43 PM [...]... by the angle ψ to the x-axis It is easy to define this angle with the equality: tg 2 = S2 , S1 (2. 47) where: ( ) ˆ ˆ S 2 = 2 E x E y cos Φ = 2 Re E x E ∗ , y 2 2 ˆ ˆ S1 = E 2 − E 2 = E x − E y x y (2. 48) b 2 = S 0 − S1 2 + S 2 2 (2. 49) The ellipse radii are defined as follows: a 2 = S 0 + S1 2 + S 2 2 , where 2 ˆ ˆ S0 = E2 + E2 = Ex + Ey x y 2 (2. 50) is the wave intensity The values S0, S1, S2, and. .. according to the formulas: © 20 05 by CRC Press TF1710_book.fm Page 42 Thursday, September 30, 20 04 1:43 PM 42 Radio Propagation and Remote Sensing of the Environment S0 = ( ) ( ) a 2 − b 2 cos 2 a 2 − b 2 sin 2 a2 + b2 , S1 = , S2 = , S 3 = ab , 2 2 2 (2. 53) and correspondingly the field components are calculated as: ˆ Ex = S 0 + S1 ˆ , Ey = 2 S 0 − S1 S , tan Φ = 3 2 S2 (2. 54) Equation (2. 52) allows... Equation (2. 52) should be replaced by: { ˆ S 0 2 − S1 2 − S 2 2 − S 3 2 = 4 E 2 N y x 2 ˆ + E2 Nx y 2 + Nx Ny 2 − 2 2  ˆ ˆ −  N x N y cos γ  −  N x N y sin γ  − 2E x E y N x N y cos Φ − γ       ( ) (2. 61) It is not difficult to establish that, in the general case, S 0 2 ≥ S1 2 + S 2 2 + S 3 2 (2. 62) Therefore, the coefficient m= S1 2 + S 2 2 + S 3 2 S0 2 (2. 63) defines the degree of wave coherence,... p ) 3 (2. 86) 2 By assuming that ω >> ωp for the ionosphere, then: τ= ωp ω 3 2 2z 1 = 3 c f 2 e 2 Nz 2. 92 ⋅ 10 2 Nz = 3 π 2mc f 2 (2. 87) And the pass band is: ∆F = © 20 05 by CRC Press 13.6 Nz 3 f 2 (2. 88) TF1710_book.fm Page 50 Thursday, September 30, 20 04 1:43 PM 50 Radio Propagation and Remote Sensing of the Environment 2. 7 DOPPLER EFFECT In all the cases considered above, the sources and receivers... field, and it is supposed that 〈Nx〉 = 〈Nx〉 = 0 Because of the statistical character of the problem, the Stokes parameters should be considered as statistically averaged values Then, S0 = Ex 2 + Ey 2 ˆ ˆ = E2 + E2 + Nx x y 2 + Ny 2 S1 = E x 2 − Ey 2 ˆ ˆ = E2 − E2 + Nx x y 2 − Ny 2 , (2. 57) , (2. 58) ∗ ˆ ˆ S 2 = 2 Re E x E ∗ = 2 E x E y cos Φ + 2 Re N x N y , y ∗ ˆ ˆ S 3 = 2 Im E x E ∗ = 2 E x E y sin Φ + 2. .. consider the case when ϑ = 0, which occurs when the source and the receiver of radiation are moving toward each other Then, © 20 05 by CRC Press TF1710_book.fm Page 52 Thursday, September 30, 20 04 1:43 PM 52 Radio Propagation and Remote Sensing of the Environment ω = ω′ 1+ β , 1− β (2. 93) which occurs when the frequency of the received wave is more than the frequency of the radiation In the opposite case of. .. E 0, t 2 ∞ dt = 2 ∫ −∞ ( ) g Ω 2 dΩ (2. 79) TF1710_book.fm Page 48 Thursday, September 30, 20 04 1:43 PM 48 Radio Propagation and Remote Sensing of the Environment is proportional to the total signal energy The corresponding calculation gives the result: ∞ Kd = ∫ g (Ω) −∞ 2 ( 2 2 cos Ω τ ∞ ∫ g (Ω) 2 4 ) dΩ (2. 80) dΩ −∞ A comparison of the shape of the signal passing at a definite distance in the medium... 3 = 2 E x E y sin Φ = 2 Im E x E * y ) (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 2 2 S 0 = S 12 + S 2 + S 3 , (2. 52) which takes place for coherent waves The problem of combined coherent waves and noise radiation will be examined later The ellipse radii a and b and the angle of. .. + v  2  v  )   t′ − 1 +  1− β 2  1− β 2   t= 1 t′ +  ,   (2. 89) ( r′ ⋅ v ) c2 1–β 2 (2. 90) Here, r and t are the radius vector of the point and the time in the non-dotted coordinate system, respectively, and r′ and t′ are the same values in the dotted system; β = v/c To be more certain, we will assume that source is at the beginning of the dotted coordinate system and moves together... within the interval ±1 τ Therefore, ∞ ∫ ( ) g Ω ∞ 2 ( 2 2 cos Ω τ 4 ) () dΩ ≅ g 0 2 ∞ ∫ ( 2 2 cos Ω τ −∞ 4 ) dΩ = 2 g 0 τ () 2 The estimation: ∞ ∫ g(Ω) 2 2 d Ω ≅ g(0) ∆Ω ∞ is applicable for the lower integral As a result: Kd ≅ 2 . = 2 2 2 Φ iin , 2 Φ tg S S 2 2 1 ψ = , SS 21 2 22= = () = − = − ∗ ˆˆ cos Re , ˆˆ EE E E xy xy x 2 y 2 x EE EΦ EE y 2 . aS S S bS S S 2 01 2 2 22 01 2 2 2 =+ + =− +,. S 0 2 2 =+= + ˆˆ EE x 2 y 2 xy EE S 3 22 == () ˆˆ sin. of the statistical character of the problem, the Stokes parameters should be considered as statistically averaged values. Then, (2. 57) (2. 58) (2. 59) (2. 60) S ab S ab S ab 0 22 1 22 2 22 2 2 2 2 2 = + = − () = − () , cos , sinψψ ,,. = − 1 1 2 22 2 2 2 c vc c p ,,  22 3 2 () .  τ ω ω π == = ⋅ − p c f eN mc N f 3 2 3 2 22 3 2 129 2102z z z 2 . . ∆F N f= 13 6 3 2 . . z TF1710_book.fm Page 49 Thursday, September 30, 20 04 1:43