Chapter 4 Waves in Unbounded Medium 4. lectromagnetic Sources 4. Uniform plane waves in free space Mawell s equation in free space is given b: H rot = (4..) roth = (4..) div = (4..3) divh = (4..4) which is satisfied b electromagnetic filed in source free space. In this notation, the constitutive relations are used. D = (4..5) Chapter 4 Waves in Unbounded Medium
B = H (4..6) (4..)-(4..4) can be considered as simultaneous differential equations having two unknowns, i.e., and H. We will delete one of the unknowns from the four equations. Since and H have the same form, it can be done in the same manner. Here, we will deleted H first. H rot( rot) = rot( ) = (4..6) And using the vector identit, rot( rot) = grad( div) div = = (4..7) where indicates Laplacian to vector, and we have = (4..8) H H = (4..9) The time-harmonic representation of the wave equation (4..8) and (4..9) are give as: ω = (4..) H ω H = (4..) In the Cartesian coordinate, the Laplacian can be defined as: = i + i + i z (4..) and the vector notation in (4..8)(4..9) can be re-written as: + + + + + + z z z z = (4..3) = (4..4) = (4..5) It should be noticed that (4..3)-(4..5) are second-order differential equations of electrical field in time and space. This form of equation is generall refereed as wave equations. The time-varing electromagnetic field in free space have to satisf the wave equation of (4..3)-(4..5). 4.3 Plane-wave solution Assume the followings and solve (4..3)-(4..5). Assumption () : Wave propagates to z direction. Chapter 4 Waves in Unbounded Medium
() : The wave is uniform in the - plane. This is mathematicall identical to : = = B using the assumption (), (4..3)-(4..5) can be simplified as: = (4.3.) = (4.3.) z = (4.3.3) z Without loosing the generalit, we can rotate the coordinate and have: =, (4.3.4) Then we have non-zero solution of and z. Now we come back to the Mawell s equation. i i iz rot = i z = z H = = H i (4.3.5) t This equation indicates that H, H = H =, and z z roth = i i iz H i z = H = = i t t (4.3.6) shows that, =. Then we can find that onl z = (4.3.7) gives the non-zero plane wave solution. 4.4 General solution to the plane-wave equation The general solution to (4.3.7) is given b where = f( z ct) + f( z+ ct) (4.4.) 8 c = = 3 ( m/ s) (4.4.) Chapter 4 Waves in Unbounded Medium 3
is the velocit of light in free space, and f, fare arbitrar functions. f, findicates that the wave solution H propagate without deforming at the velocit of c. B substituting (4.4.) into rot = we obtain H = f ( z ct) f ( z+ ct) where η l q (4.4.3) η = 376 6 π is the intrinsic impedance of free space.. (4.4.4) The time-harmonic wave equation can be obtained from (4.3.7) as: + = ω + = k (4.4.5) ω k = ω = wave number (4.4.6) c and the general solution to (4.4.6) is given b jkz + jkz = Ae + Ae (4.4.7) (4.4.6) is called dispersion relation, which determines the wave number of the solution of the wave equation. B re-writing this time-harmonic notation in time-domain, we have jkz jkz j t (,) z t = Re Ae + A e + ω e = A cos( ωt kz) + A cos( ω t+ kz) (4.4.8) oc h t and this equation indicates a sinusoidal wave propagating to the z-direction at the velocit of c. The wavelength λ satisfies π λ = k (4.4.9) The figures show how the solution, which can now be recognized as a sinusoidal wave, propagates with time. Imagine we ride along with the wave. At what velocit shall we move in order to keep up with the wave? Mathematicall, the phase argument of the term cos( ωt kz) must be constant. That is, ωt kz = a constant (4.4.) Chapter 4 Waves in Unbounded Medium 4
So the velocit of propagation is given b dz dt From (4.4.6) we obtain the phase velocit = v = ω (4.4.) k v ω = = k (4.4.) The plane wave solution, which propagates to z-direction, then can be written as; = e jkz (4.4.3) jkz H = e (4.4.4) And the Pointing vector is S = z cos ( ωt kz) (4.4.5) η Notice that in all the above discussions, the solution in (4.4.7),(4.4.8) or equivalentl (4.4.3) are independent of a and coordinates. In other words for an observer anwhere in the - plane with the same value for z, the phenomena are the same. The constant phase front is defined b setting the phase in (4.4.7) equal to a constant. We have kz = C (4.4.6) where C is a constant defining a plane perpendicular to the z ais at z = C / k. We call waves whose phase fronts are planes plane waves. A plane wave with uniform amplitudes over its constant-phase planes is called a uniform plane wave. The power densit carried b a uniform wave is calculated b the time-average Ponting vector. S = Re{ H* } = z (4.4.7) Chapter 4 Waves in Unbounded Medium 5
4.5 Polarization The uniform plane wave discussed in the previous section is (,) zt = cos( t kz) (4.5.) ω Tracing the tip of the vector at an point z will show that the tip alwas stas on the ais with maimum displacement. We thus conclude that the uniform plane wave is linearl polarized. Now we consider a plane wave with the following electric-filed vector. j kz φ a = ae + be ( ) j( kz φb) (4.5.) The real time-space vector in (4.5.) has and components: = acos( ωt kz+ φ a) (4.5.3) = bcos( ωt kz+ φ ) (4.5.4) b where a and b are real constants. To determine the locus of the tip of vector in the - plane as a function of time at an z, we can eliminate the variable ( ωt kz) from (4.5.3)(4.5.4) to obtain an equation for and. ()Liner Polarization φ = φ a φ b =, π (4.5.5) b a =±( ) (4.5.6) ()Circular Polarization π φ = φa φb = ± A = b = (4.5.8) a (4.5.7) and = acos( ωt kz+ φ a) (4.5.9) = bsin( ωt kz+ φ ) (4.5.) + = a (4.5.) a Chapter 4 Waves in Unbounded Medium 6
(3)lliptical Polarization The wave (4.5.) is ellipticall polarized when it is neither linearl nor circularl polarized. Consider the case φ = π / and A= b/ a =. = acos( ωt kz+ φ a) (4.5.) = asin( ωt kz+ φ ) (4.5.3) B eliminating t ields a + = a a (4.5.4) Let the comple electric filed be jkz = ( + ) e (4.5.5) jφ = Ae (4.5.6) then, all the polarization status can be epressed b using ( A, φ ). The figure shows the polarization chart in - plane or comple = Ae jφ plane. Chapter 4 Waves in Unbounded Medium 7
4.6 Plane wave in dissipative media The most material has electrical conductivit, and its effect is characterized b the conductivit σ. Ohm s law states that J = σ c (4.6.) where J c denotes the conducting current. From Ampere s law, roth = J + jω D (4.6.) we can see that current densit J can embod two kinds of currents- namel, the source current J and the conducting current J c. Then, in the conducting medium, the Ampere s law becomes Thus we define σ roth = jω ( j ) + J ω (4.6.3) = σ j (4.6.4) ω the conductivit becomes the imaginar part of the comple permittivit. Then the Mawell s equation in a conducting medium devoid of an source can be written as rot = jω H (4.6.5) roth = jω (4.6.6) divh = (4.6.7) div = (4.6.8) where is the comple permittivit. The wave equation derived from (4.6.5)-(4.6.8) can be written as ω = (4.6.9) The set of plane-wave fields is still a solution of Mawell s equations. Namel, where k = e j z (4.6.) jkz e (4.6.) H = k = ω (4.6.) is the dispersion relation derived from the wave equation and η = (4.6.3) is the intrinsic impedance of the isotropic media. Note that k and η are now comple numbers. Chapter 4 Waves in Unbounded Medium 8
Because k and η are now comple numbers, we can rewrite these parameters as: k = kr jki (4.6.4) η = ηe (4.6.5) Substitution of (4.6.4) in (4.6.) and (4.6.) ields = = kz I jkrz e e kz I jkrz j H = e e e φ (4.6.7) (4.6.6) jωt The instantaneous value is (,) z t = Re( e ) and from (4.6.6) we find I = e kz cos( ωt k z) (4.6.8) R The above equation represents a wave traveling in the z direction with a velocit equal to v, where v = ω (4.6.9) k R As the wave travels, the amplitude is attenuated eponentiall at the rate k I nepers per meter. We define a penetration depth d p such that, when kz I = kd I p =, the amplitude of the electric field shown in (4.6.8) will deca to /e of its value at z=. d p = (4.6.) k I For conducting media, we have σ R I k = ω j = k jk ω kr ω σ = + ± ki ω (4.6.) (4.6.) where σ / ω is called the loss tangent of the conducting media. () Waves in Non-dissipative Media In material, where the electrical conductivit is zero, the wave behaves similar to that in free space. However, due to and, the speed of the wave is given b v = / (4.6.3) () Waves in Slightl Conducting Media A slightl conducting medium is one for which σ / ω <<. The value of k in (4.6.) can be approimated b Chapter 4 Waves in Unbounded Medium 9
k j σ σ = ω ω ( j ) ω = ω (4.6.4) and we find k R = ω (4.6.5) k I = σ (4.6.6) In this condition, the penetration depth is given b d p = σ (4.6.7) (3) Waves in Highl Conducting Media A highl conducting medium is also called a good conductor, for which σ / ω >>. In this case, the wave number k can be approimated b k j σ σ = ω = ω ( j) ω (4.6.8) Therefore, the penetration depth becomes d p = δ (4.6.9) ωσ The smbol δ signifies that d p is so small it is better called the skin depth δ. For a good conductor, the conducting current concentrates on the surface and ver little flows inside the conductor. This phenomena is called the skin effect. In the above discussion, we used the comple permittivbit defined b (4.6.4). However, if we substitute (4.6.4) into (4.6.9), we have ω j ωσ = (4.6.3 If we neglect the third term in (4.6.3) his is a wave equation given in (..) and if we neglect the second term, this is a diffusion equation given in (.3.8). The loss tangent σ / ω defines the contribution of the second and the third terms:. It means that, the in Nondispersive media, the electromagnetic wave is governed b the wave equation, and in highl conducting media, it is governed b a diffusion equation. The time harmonic solution of wave equation gives the unified solution to the wave equation, when in highl conducting media, but the phsical properties of electromagnetic wave is quite different. Chapter 4 Waves in Unbounded Medium