© 2005 by CRC Press 85 4 Geometrical Optics Approximation 4.1 EQUATIONS OF GEOMETRICAL OPTICS APPROXIMATION In this chapter, we will discuss wave propagation problems in a medium with an arbitrary law of permittivity coordinate dependence; that is, we will assume that the permittivity has the form ε = ε ( r ) in the common case. The spatial variation slowness of the permittivity is assumed to be similar to that for the Wentzel–Kramers–Brillouin of permittivity at the wavelength scale. This property can be expressed as the inequality: (4.1) As in the WKB method, we will utilize a solution to Maxwell’s equations in the form of the asymptotic Debye series: (4.2) The value ψ is referred to as the eikonal value . We arrive at the system of connected equations: (4.3) after substitution of the Debye series in Maxwell’s equations; members of the same degree of k are equal to each other. Zero-order equations are a system of homogeneous linear algebraic equations. For the purpose of their nontrivial solution, it is necessary to reduce the determinant to zero. This requirement leads to the equation for ψ (the eikonal equation). We can obtain this equation fairly simply if H 0 is expressed through E 0 ; also, we must take into account the mutual orthogonality of vectors E 0 , H 0 , and ∇ψ that follows from ∇ <<ln .ε k E Er H Hr = () () = () () = ∞ = ∞ ∑ e ik e ik ik s s ik s s ss ψψ ,. 00 ∑∑ ∇ ×     += ∇ ×     − = ∇ ×     + ψε ψ ψε HE EH HE 00 00 1 00,, 110 110 = −∇× ∇ ×     − = −∇×[], [],HEHEψ TF1710_book.fm Page 85 Thursday, September 30, 2004 1:43 PM (WKB) approximation carried out in Chapter 3. We assume again a small change © 2005 by CRC Press 86 Radio Propagation and Remote Sensing of the Environment the equations of the zero-order approximation. Now we can easily show that the eikonal equation may be written down in the form: (4.4) The value: (4.5) represents the radiowave phase in the zero approximation, and eikonal ψ (in engi- neering terminology) represents the electrical length passed by the wave. We assume that the phases of components E 0 and H 0 do not depend on coordinates in the approximation. Furthermore, in this approximation, these vectors are believed to be real, including the initial wave phase at once in the wavelength. Certainly, small additions to this phase may be made by calculation of the following items of expansion. In the zero-order approximation, the power flow density: (4.6) is directed along lines of the eikonal gradient. This fact allows us to refer to the zero approximation as the geometrical optics approximation , which corresponds to the small wavelength conversion (hence the term optics ) and allows the wave propaga- tion laws to be formulated in the language of geometry. The validity of the geometrical optics approximation is defined by Equation (4.1). If, as before, the scale of permittivity change is designated Λ , then the Debye series is essentially expansion according to the inverse degree of large parameter k Λ . Other conditions will be formulated later. Let us assume, in the beginning, that the permittivity is a real value. We can define the vector of wave propagation by the formula: (4.7) where s is the unitary vector, which is orthogonal to the equiphase surfaces. The lines orthogonal to the surfaces of the eikonal constant value (to equiphase surfaces) are called rays . Vector s is tangential to the rays and describes the wave energy propagation direction. If τ is the length along the ray, then the ray equation has the form d r / d τ = s . Then, ∇ψ = s d / ψ / d τ , and the eikonal equation becomes the common differential equation: (4.8) ∇ () =ψε 2 . ϕψ= k SEH 000 =×     = ∇ cc 88 E 0 2 ππ ψ ∇ =ψεs, d d ψ τ ε= TF1710_book.fm Page 86 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Geometrical Optics Approximation 87 the solution of which is written in the form: (4.9) where ψ 0 is the initial value of the eikonal equation. Let us point out that use of the plus sign was determined by extraction of the square root in Equation (4.9). It is important to note that we are dealing with a direct wave. In the case of a backward wave, the minus sign should be used. It is clear that the eikonal form as a function of coordinates depends on the ray along which the integration is provided. We may use the orthogonal unitary vector system of the normal n and the binormal m (Figure 4.1). Their changes along the ray characterize its bending and torsion. The Frenet–Serre formulae: (4.10) are known from differential geometry. 29 The value ρ is the ray curvature radius, and χ is its torsion. The vectors s , n , and m are the basis of the curved-line coordinate system formed by the ray ensemble and equiphase surfaces. This system is often referred to as the ray coordinates . Equation (4.9) is the eikonal equation solution in the ray coordinates system. Let us use the eikonal equation in Equation (4.8) to calculate ρ and χ . We must take the gradient of both parts to obtain: . Thus, it follows that: (4.11) Equation (4.11), together with the first Frenet–Serre equation, allows us to determine the radius of ray curvature: (4.12) ψτ ψ ετ τ τ () =+ ′ () ′ ∫ 0 0 d , FIGURE 4.1 The orthogonal uni- tary vector system. n m s d d d d d d sn n s m m n τρ τ ρ χ τ χ==−− =,, d d s ε τ ε () = ∇ d d s ss τ εε= ∇−⋅∇ () ln ln . 1 2 22 ρ εε= ∇ () −⋅∇ () ln ln .s TF1710_book.fm Page 87 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 88 Radio Propagation and Remote Sensing of the Environment If angle α between the direction of the ray and the direction of the permittivity is introduced, then: (4.13) Further, it is simple to establish that: (4.14) We must now derive equations for fields E 0 and H 0 . First of all, let us point out that the wave is transversal in zero-order approximation, so its components may be represented in the form: (4.15) On the basis of Equation (4.15) and after some not very complicated calculations, 25 we can define the conditions that connect the electrical field components directed along the normal and along the binormal: (4.16) Returning to the local cylindrical coordinate system: , (4.17) we can easily obtain from Equation (4.16) the transfer equation: (4.18) 1 ρ εα= ∇ln sin . nss ss m = ∇−⋅∇ ()       = ⋅∇ ()       = ρε ερ εln ln ln , [[] ln , . sn s n m ns m ⋅ = ⋅∇     = ⋅ = ⋅⋅∇ ()     ρε χ τ d d EnmH sE nm 000 =+ = ×     = − () EE EE nm nm ,.εε 220 2 εεεχε ε ∇ + ∇     ⋅ () + ∇⋅ += ∇ + EE E E EE nn n m m () ,ss mmmn ∇     ⋅ () + ∇⋅ − =() .εε χεssEE20 EE n ==EE 0m0 cos , sinϑϑ ∇⋅ () = ∇⋅ =εE 0 2 sS0 TF1710_book.fm Page 88 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press Geometrical Optics Approximation 89 and the equation of torsion: (4.19) Equation (4.18) conveys the energy conservation law. The solution to Equation (4.19) may be written as: (4.20) which describes the law of wave polarization elliptical rotation without changing its form (Rytov’s law). We may rewrite Equation (4.18) in another form by using Equation (4.7). Then, in the ray coordinates, , and the solution is obvious: (4.21) This expression can be written down in another form by using the ray divergence. Let us insert the ray coordinates ξ , η , and τ . Coordinate τ is directed along the rays, while the other two coordinates are orthogonal to it and are directed, for example, along the vectors of normal and binormal. The Jacobian of the transition from Cartesian coordinates (x,y,z) to ray coordinates is given by the formula: (4.22) The ray tube is defined as a ray family passing through the area d ξ d η near a point with coordinates ( ξ , η , τ ). The square of the surface element perpendicular to the s ⋅∇ () − =ϑχ0. ϑϑ χτ τ τ =+ ′ () ′ ∫ 0 0 d , d d E E 0 0 τ ψ ε + ∇ = 2 2 0 EE 00 τ ψτ ετ τ τ () = () − ∇ ′ () ′ () ′         ∫ 0 1 2 2 0 exp d D ξητ ∂ ∂ξ ∂ ∂η ∂ ∂τ ∂ ∂ξ ∂ ∂η ∂ ∂τ ∂ ∂ξ ∂ ∂η ∂ ,, () = xxx yyy zzz ∂∂τ . TF1710_book.fm Page 89 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 90 Radio Propagation and Remote Sensing of the Environment direction of the rays (equiphase surface) equals Dd ξ d η . The volume of the ray tube element equals Dd ξ d η d τ . Applying vector analysis to Equation (4.8), it is a simple matter to obtain: (4.23) We will use this formula for the volume bounded by the ray coordinates and will reduce the volume to zero; moreover, we take into account that ( n · s ) = 0 at the sides of the tube. We then have: or (4.24) Instead of Equation (4.21), we obtain: . (4.25) This result is transparent from the physical point of view: Due to the law of energy conservation, the field amplitude changes together with changes in the cross section of the ray tube. The second condition of the geometrical optics approximation validity follows from Equation (4.25). It is not valid where D ( τ ) = 0. Areas where the Jacobian is reduced to zero are called caustics and round out the ray family. In WKB approximation, corresponds to the caustics plane. For more details about caustics, refer to Kravtsow and Orlov. 27 Up to this point, it was assumed that permittivity is the real value. Let us now turn to the more realistic case of weak absorption in the media and, related to this, complex permittivity. We can still use Equation (4.4), but the eikonal itself must now be complex (i.e., ψ = ψ′ + i ψ′′ ). It is apparent that the eikonal imaginary part describes wave attenuation due to absorption and, being multiplied by the wave number, is equal to the coefficient of extinction. The separation of real and imaginary parts in Equation (4.4) leads to a pair of equations concerning ∇ψ′ and ∇ψ′′ . It is difficult to find the solution of these equations, particularly because it is necessary to know the angle between ∇ψ′ and ∇ψ′′ . ∇ = ∇⋅ ∇ = ⋅∇ () = ⋅ () ∫∫ 23 3 2 2 ψψ ψεddd d SS rrnrnsr() .  VVV ∫∫ ∇ () =+ () + − () 2 ψτξητ τ τεττ τετξηDddd D d d D dd[()()] ∇ = () ()       2 ψ ε τ ετ τ d d Dln . EE 00 τ ε ετ τ () = () () () () () 00 0 D D TF1710_book.fm Page 90 Thursday, September 30, 2004 1:43 PM particular, the turning point, which we mentioned in Chapter 3 with regard to the © 2005 by CRC Press Geometrical Optics Approximation 91 We will now consider a simple but common case of small absorption in the sense that ε′′ << ε′ and ( ∇ψ′′ ) << ( ∇ψ′ ) 2 . The pair of equations then acquires the form: (4.26) The first equation is solved as before and the second one is transformed to the form: . Hence, it follows that: (4.27) We will now briefly address the case of anisotropic media, including the iono- sphere. In this case, the eikonal equation is broken down into two equations — one for ordinary and one for extraordinary waves: 28 (4.28) These waves, generally speaking, can be considered to be independent if the length of the beating between them is much smaller than the scale of the medium inhomo- geneities. The beating length is estimated by the value: (4.29) The substitution of specific values ( f = 10 8 Hz, N = 2 · 10 6 cm 3 , H 0 = 0.5 Oersted) gives the estimation l ≈ 10 km. It would seem that the independence of ordinary and nonordinary waves can be broken with increasing frequency and, correspond- ingly, with increasing beating length. This is not so, however, because in this case the wave relation coefficient decreases with increases in frequency; 28 therefore, ordinary and extraordinary waves are practically always independent for the ultra- high-frequency and microwave bands for the ionosphere of Earth. The ray trajectories of ordinary and extraordinary waves practically coincide, because their refractive indexes differ little in the range of waves being considered here, which allows us to develop a formula to calculate the polarization angle rotation value due to the Faraday effect: ∇ ′ () = ′ ∇ ′ ∇ ′′ () = ′′ ψε ψψε 2 2,. d d ′′ = ′′ ′ ψ τ ε ε2 ′′ () = ′′ ′ () ′′ () ′ ∫ ψτ ετ ετ τ τ 1 2 0 d . ∇ () = ∇ () =ψψ oo 2 2 2 2 nn ee ,. l c nn cf ff f NH e = − () ≅ = ⋅ − 21310 2 2 42 0 π ωβ opH cos . . TF1710_book.fm Page 91 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 92 Radio Propagation and Remote Sensing of the Environment (4.30) Finally, we will calculate the Doppler frequency shift for wave propagation in an inhomogeneous medium. For this purpose, let us refer back to Equation (2.97) and rewrite it as follows: (4.31) It is easy to see that the Doppler shift value is proportional to the velocity component directed along the ray (ray velocity). 4.2 RADIOWAVE PROPAGATION IN THE ATMOSPHERE OF EARTH The atmosphere of Earth can be considered, in the first approximation, as a spher- ically layered medium where the permittivity is a function of the radius beginning at the center of the Earth. We do not include in our consideration here the changes in atmospheric parameters along the surface of the Earth that take place at the transition from day to night (light to shadow), along frontal zones with significant changes of air temperature, and so on. It should be supposed that ε = ε ( R ), ∇ε = R / R ( d ε / dR ), etc. It is believed, that, on average, 1/m in the troposphere near the surface of Earth; therefore, the geometrical optics approxima- tion is highly accurate for the given wave range. The vertical gradient in the iono- sphere is even smaller, so applying the geometrical optics approximation is still appropriate. Although the permittivity of air does differ from unity, the difference is insignificant and we can assume it to be equal to unity without causing problems. We can prove rather easily the permanency of vector along the ray trajectory; hence, we can make the statement that in the case of a spherically layered medium the ray trajectories are plane curves. The product: (4.32) is invariant along the ray, where the constant η is determined from the initial conditions. If, for example, a ray left Earth at angle α 0 , then , where the radius of the Earth a ≅ 3 established on the ray prolongation until the surface of the Earth. Equation (4.32) is often referred to as Snell’s law for spherically layered media. We will now consider the situation when a ray passes by the surface of Earth ε → 1, and R sin α → p , where p is the aimed distance (a term borrowed from the theory of particle scattering). In this case, a ray turning point occurs at R = R m , where α(R m ) = π/2. Ψ Fo 2 = − () ′ ≅ ′ () ′ () ′ ω τ π ττβ 2 2 3 22 0 c nnd e mc f NH e cos τττ ττ () ′ ∫∫ d . 00 ωψε d kk= −⋅∇ () = −⋅ () vsv. gddR ε ε= ≅− ⋅ − 810 8 ε[]Rs× εαηRR R () () =sin , ηε α= 00 a sin TF1710_book.fm Page 92 Thursday, September 30, 2004 1:43 PM 6.4 · 10 km (see Figure 4.2). The starting point may be (Figure 4.3). Far from the atmosphere of Earth, (R) © 2005 by CRC Press Geometrical Optics Approximation 93 It is convenient, in our case, to write the eikonal equation for the spherical coordinate system with the center coinciding with the center of the Earth. This system can be chosen in such a way as to take into account the plane character of the ray trajectories so that the eikonal will not depend on the azimuthal angle ϕ. The eikonal equation can be written as: (4.33) in our chosen system of coordinates. Proceeding according to the variable separation method, we will seek a solution as the sum: (4.34) Combining Equations (4.33) and (4.34) and taking into account the initial conditions, we obtain: (4.35) where R 0 and θ 0 are the coordinates of the initial ray point. The sign before the integral is chosen depending on the type of ray branch: the plus sign for the ascendant branch and the minus sign for the descendent branch. For geometrical reasons, (4.36) FIGURE 4.2 Radio propagation from the surface of the Earth. FIGURE 4.3 Propagation radio wave along the surface of the Earth. a α ξ R P R m ξ ∂ψ ∂ ∂ψ ∂θ ε R R R       +       = () 2 2 2 1 ψθψ ψθ θ RR R ,. () = () + () ψθηθθ ε η RR R dR R R ,, () = − () ± ′ () − ′ ′ ∫ 0 2 2 0 Rd dR R R θ α η ε η == ± − tan 2 2 TF1710_book.fm Page 93 Thursday, September 30, 2004 1:43 PM © 2005 by CRC Press 94 Radio Propagation and Remote Sensing of the Environment and (4.37) The rule for choosing the appropriate sign is the same as for the previous case. As a result, we now have: (4.38) It follows from Equation (4.38) that: Taking into account Equation (4.21), the equality can be derived as: (4.39) As before, the result obtained corresponds to the WKB approximation. m with the caustic surface and additional caustic phase shift occurs. 12,19 Without going into detail regarding the calculation, we should point out that, in this case, (4.40) and cos α approaches zero close to the turning point; the amplitude, formally calculated in the geometrical optics approximation, tends to infinity, which empha- sizes once again that the geometrical optics approximation is inapplicable in areas close to caustics. θθ η ε η − =± ′ ′ − ′ ∫ 0 2 2 2 0 dR R R R R . ψθ ε εη R RRdR RR R R ,. () =± ′′ () ′ ′ () ′ − ∫ 22 0 ∇ = −       − 2 2 2 2 2 1 2 ψ ε η ε η d dR R R . EE 00 R RR RR () = () − () −         () = εη εη ε 0 2 0 2 22 1 4 0 000 0 cos cos . α εαR () () E 0 ψθ ε ε ε ε R RRdR RR p RRdR R , () = ′′ () ′ ′ () ′ − + ′′ () ′ ′ ( 22 )) ′ − − ∫∫ Rp R R R R mm 22 2 0 π , TF1710_book.fm Page 94 Thursday, September 30, 2004 1:43 PM When the ray passes through point R = R (Figure 4.3) it comes into contact [...]... September 30, 20 04 1 :43 PM 98 Radio Propagation and Remote Sensing of the Environment where ( ) ∞ Z s 0 = 2s 0 e 2 s0 ∫e −s2 s0 = ds, s0 ae cos α 0 , 2H T (4. 53) and ae 8500 km is the so-called equivalent radius of the Earth At s0 >> 1, Z(s0) ≅ 1, and ( ) ξ α0 ≅ ε0 −1 tan α 0 , 2 (4. 54) which corresponds to the plane-layered troposphere approximation At s0 . September 30, 20 04 1 :43 PM © 2005 by CRC Press 88 Radio Propagation and Remote Sensing of the Environment If angle α between the direction of the ray and the direction of the permittivity. September 30, 20 04 1 :43 PM © 2005 by CRC Press 94 Radio Propagation and Remote Sensing of the Environment and (4. 37) The rule for choosing the appropriate sign is the same as for the previous case 30, 20 04 1 :43 PM for the rays shown in Figure 4. 3. In particular, if R and R are sufficiently large, the © 2005 by CRC Press 96 Radio Propagation and Remote Sensing of the Environment (4. 44) Angle