Electromagnetics P16-1 Lesson 16 Plane Waves in Homogeneous Media Introduction v v v By eq’s (15.5-7), E and H in a charge-free ( ρ = 0 ), nonconducting ( J = 0 ), simple (linear, homogeneous, isotropic) medium are governed by the vector wave equations: 2 v 2 v 2 v 1 ∂ E 2 v 1 ∂ H 1 ∇ E − 2 2 = 0 , ∇ H − 2 2 = 0 , where u p = . u p ∂t u p ∂t µε The general solutions are waves propagating with phase velocity u p . A plane wave is a particular solution to the vector wave equation, where every point on an v infinite plane perpendicular to the direction of propagation has the same electric field E and v magnetic field H (in terms of magnitude and direction, see Fig. 16-1). Fig. 16-1. A general plane wave propagates in +z-direction illustrated at some time instant. The v v fields E and H are constant throughout the plane z = z1 , but may differ from those on another plane z = z2 . Time-harmonic (sinusoidal) plane waves are of particular importance because: 1) The mathematical treatments are greatly simplified. 2) Superposition of time-harmonic plane waves remains a solution to the (linear) vector wave equation. They can be used as a basis to describe general electromagnetic waves. Edited by: Shang-Da Yang Electromagnetics P16-2 E.g. Superposition of time-harmonic plane waves of different frequencies and common direction of propagation can describe plane wave packets. Superposition of time- harmonic plane waves of the same frequency and different directions of propagation can describe Gaussian beam. 3) It is a good approximation for far-field EM waves (e.g. sunlight received on the earth), and waves whose cross-sections have a linear dimension much larger than the wavelength (e.g. typical laser beam). 16.1 Plane Waves in Vacuum Most simplified time-harmonic (sinusoidal) plane waves v The vector phasor E of a time-harmonic E-field is satisfied with: 2 v 2 v ∇ E + k E = 0 (15.13) where k = ω u p [eq. (15.15)] denotes the wavenumber. For simplicity, consider a time-harmonic plane wave propagating in the z-direction and the electric field is polarized in the x-direction, ⇒ v v E = a x E x (z) . (16.1) Note that eq. (16.1) does represent a plane wave, for any point on an infinite plane z = z0 v (perpendicular to the direction of propagation az ) must have the same electric field v a x E x (z0 ) . In this particular case, eq. (15.13) reduces to a scalar ordinary differential equation: d 2 E x + k 2 E = 0 (16.2) dz 2 x In the absence of physical boundary, the general solution to eq. (16.2) consists of two counter- propagating waves: + − jkz − + jkz Ex (z) = E0 e + E0 e , (16.3) Edited by: Shang-Da Yang Electromagnetics P16-3 + + jφ + − − jφ − where E0 = E0 e and E0 = E0 e are complex amplitudes corresponding to time-harmonic waves propagating in the +z and −z directions, respectively. <Comments> + + − jkz 1) As in transmission lines [eq’s (4.7), (4.8)], phasor Ex (z) = E0 e describes a time-harmonic wave with velocity u p = ω k because: + + jωt + + Ex (z,t) = Re{}Ex (z)e = E0 cos(ωt − kz + φ ) z ω 1 is a function of variable τ = t − . By eq. (15.15), u = = , which is the ()ω k p ω µε µε same as eq. (15.7). − − + jkz 2) As in transmission lines, reflected wave Ex (z) = E0 e can emerge if the medium is nonhomogeneous along the z-direction (i.e. ε or µ changes with z ). Characteristics of plane waves v The electrical field of a plane wave propagating in an arbitrary direction ak is of the form (Fig. 16-2): v v v v − jk⋅rv v v v − jk⋅rv E = eE0e = ()ax cosα + ay cos β + az cosγ E0e (16.4) v (1) The phase evolution term kz in eq. (16.3) is generalized by k ⋅rv , where v v v v v k = axkx + ay k y + az kz = ak k v v v v is the wavevector, and r = ax x + ay y + az z is the position vector of the observation point. (2) v The direction of electric field (polarization direction) ax in eq. (16.1) is generalized by a unit vector: Edited by: Shang-Da Yang Electromagnetics P16-4 v v v v e = ax cosα + ay cos β + az cosγ v v v v where {}α, β,γ are the included angles between e and {ax ,ay , az } , respectively. For any v v point r on an infinite plane perpendicular to the direction of propagation ak (intensity & v v v v v − jk⋅rv phase front, Fig. 16-2), k ⋅r = kr cosθ = constant , ⇒ E = eE0e is a constant vector. Therefore, eq. (16.4) does represent a plane wave. Fig. 16-2. A plane wave propagates in an arbitrary direction on the xz-plane. v v v The three vectors E , H , k of a time-harmonic plane wave are mutually orthogonal in a charge-free simple medium. Proof: (1) In a charge-free (ρ = 0) simple medium, eq. (7.8) reduces to: v ∇ ⋅ E = 0 . v v v v (2) By the vector identity ∇ ⋅ (ψA) =ψ (∇ ⋅ A)+ A⋅ (∇ψ ) (where ψ , A are arbitrary scalar and vector fields, respectively), eq. (16.4) reduces to: v v v v v v − jk⋅rv − jk⋅rv v v − jk⋅rv v − jk⋅rv ∇ ⋅ E = ∇ ⋅ (eE0e )= E0e ()∇ ⋅ e + e ⋅∇(E0e )()= e ⋅[E0∇ e ] . (3) By the formula of gradient in Cartesian coordinate system, v v v v v v − jk⋅r v ∂ v ∂ v ∂ − j(kx x+k y y+kz z) v v v − jk⋅r v − jk⋅r ∇()e = ax + ay + az e = − j()axkx + ay k y + az kz e = − jke . ∂x ∂y ∂z Edited by: Shang-Da Yang Electromagnetics P16-5 v v v v v − jk⋅rv − jk⋅rv v v v v (4) ∇ ⋅ E = e ⋅ []E0 ()− jke = − jkE0e ()e ⋅ ak = 0 , ⇒ e ⊥ ak , i.e., v v E ⊥ k (16.5) v v v v If E has been known, one can derive H by ∇ × E = − jωµH [eq. (15.9)]: (5) By the formula of curl in Cartesian coordinate system, v v v ax ay az v −1 v −1 H = ()∇ × E = ∂ ∂x ∂ ∂y ∂ ∂z v v v jωµ jωµ − jk⋅rv − jk⋅rv − jk⋅rv E0e cosα E0e cos β E0e cosγ v v v ax ay az −1 = − jkx − jk y − jkz v v v jωµ − jk⋅rv − jk⋅rv − jk⋅rv E0e cosα E0e cos β E0e cosγ v v v −1 v v k × E av × E = ()− jk × E = = k , ⇒ jωµ ωµ ()ωµ k v v av × E H = k , (16.6) η ωµ ωµ µ where η = = = is the intrinsic impedance of the medium: k ω µε ε µ η = (Ω) (16.7) ε v v v v For vacuum, η0 = µ0 ε 0 = 377 ()Ω . Eq. (16.6) implies that H ⊥ k and H ⊥ E . v v v Combined with eq. (16.5), ⇒ E , H , k of a time-harmonic plane wave are mutually v v orthogonal in a charge-free simple medium. The fact that both E and H are perpendicular v to the direction of propagation ( ak ) makes the plane wave a transverse electromagnetic (TEM) wave. v v + v + − jkz v v Example 16-1: (1) If E = ax Ex (z) = ax E0 e , ⇒ k = az k . By eq. (16.6), ⇒ v v + + + v a × a E (z) E (z) + E (z) v − − + jkz H = z x x = av x , where H (z) = x . (2) If E = av E (z) = av E e , ⇒ η y η y η x x x 0 Edited by: Shang-Da Yang Electromagnetics P16-6 v v − − − v v − a × a E (z) E (z) − E (z) k = −av k , H = z x x = av x , where H (z) = x . (3) If z η y −η y −η v v + + v + + − jkz v v az × ay Ey (z) Ey (z) E = av E (z) = av E e , ⇒ k = av k , H = = av , where y y y 0 z η x −η E + (z) H + (z) = y . x −η <Comments> 1) Intrinsic impedance η [eq. (16.7)] of plane wave is analogous to the characteristic impedance Z 0 [eq. (2.8)] in transmission lines: v v + v a × E + V v v H = k ↔ I = implies { E ↔ v , H ↔ i }. η Z0 µ L η = ↔ Z = implies { µ ↔ L , ε ↔ C }. ε 0 C 2) Not all TEM waves are plane waves. E.g. Coaxial transmission lines support TEM waves v v v v v v ( E //ar , H // aφ , k //az ) with r − dependent (non-uniform) fields. What is the state of polarization of time-harmonic plane waves? By eq. (16.4), the electric field phasor of a time-harmonic plane wave propagating in the +z-direction is: v v v − jk⋅rv v v − jkz E = eE0e = (ax cosα + ay cos β )E0e . At z = z0 , the time-dependent electric field vector: v v jωt v E(z0 ,t) = Re{E(z0 )e }= eE0 cos(ωt − kz0 ) v will traverse a line of length 2E0 along e in one period T = 2π ω (Fig. 16-3a). In general, the electric field of a time-harmonic plane wave propagating in the +z-direction can v v have x- and y-components ( E ⊥ k is still satisfied), and their superposition will traverse a tilted ellipse (Fig.