Plane Waves 1 Review dielectrics 2 Plane waves in the time domain 3 Plane waves in the frequency domain 4 Plane waves in lossy and dispersive media 5 Phase and group velocity 6 Wave polarization Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 1 / 25
I. Review dielectrics Simple medium: D = ɛe Dispersive medium: D = ɛ(ω)e Anisotropic medium: Permittivity as a tensor ɛ xx ɛ xy ɛ xz ɛ yx ɛ yy ɛ yz (1) ɛ zx ɛ zy ɛ zz Conducting medium: ɛ e = ɛ R j (ɛ I + σ/ω) (2) We will be assuming that B = µh throughout the course Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 2 / 25
II. Plane waves in the time domain Assuming that we are in a linear, isotropic, homogeneous, lossless, and time-invariant medium, Maxwell s source-free equations become H = ɛ E t (3) E = µ H t (4) E = 0 (5) H = 0 (6) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 3 / 25
Taking the curl of both sides of the first equation and using a vector identity results in We can also obtain 2 H = ɛµ 2 t 2 H (7) Both are vector wave equations 2 E = ɛµ 2 t 2 E (8) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 4 / 25
These involve a lot of terms, so let s simplify by assuming fields vary only along x 2 E x x 2 = µɛ 2 E x t 2 (9) 2 E y x 2 = µɛ 2 E y t 2 (10) 2 E z x 2 = µɛ 2 E z t 2 (11) Requiring E = 0 shows that E x could be at most a constant, set it to zero since it is not interesting. Remaining equations are of the form 2 f x 2 = 1 v 2 2 f t 2 (12) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 5 / 25
It is easy to show that these equations admit solutions of the form f = f 1 (x vt) + f 2 (x + vt) (13) where f 1 and f 2 are arbitrary twice-differentiable functions (D Alambert solutions). Accordingly, we find [ E = ŷ f 1 (x t ) + f 2 (x + t ] [ ) + ẑ f 3 (x t ) + f 4 (x + t ] ) µɛ µɛ µɛ µɛ and H = follows from Maxwell s equations ɛ µ [ ŷ (f 3 f 4 ) + ẑ (f 1 f 2 )] (14) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 6 / 25
These solutions represent traveling waves because a disturbance will move through space undistorted as time progresses. Functions of x vt propagate in the +x direction, while functions of x + vt propagate in the x direction. Furthermore, v = 1 µɛ (15) turns out to be the velocity of these waves. Given the direction of propagation of a pure traveling wave, ˆk, H = (ˆk E) /η (16) ) E = η (ˆk H (17) where the wave impedance η is given by η = ˆk E H = µ ɛ This is a TEM (transverse electromagnetic) field! (18) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 7 / 25
f 1 (x,t=0) f 1 (u) f 1 (x,t=2 (µ ε) 1/2 ) 1 0.5 (a) 0 0 1 2 3 4 x (b) 1 0.5 0 0 1 2 3 4 u (c) 1 0.5 0 0 1 2 3 4 x Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 8 / 25
H E k^ Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 9 / 25
Electromagnetic plane waves are ideal for information transmission in the simplest medium because time-domain signals propagate undistorted from one location to another D Alembert s solution, however, can be shown to be invalid if the medium is not simple - signals distort when propagated The bandwidth of a medium (i.e. the frequency spread of a signal which can be reliably transmitted) is thus related to the complexity of the propagation medium More simple media have larger bandwidths - free space has infinite bandwidth! Trying to remove medium effects (which are often unpredictable) usually is very difficult Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 10 / 25
III. Plane waves in the frequency domain For sinusoidal time-dependence, we need only specify the amplitude and phase of the fields as functions of space, this leads to phasor description E y (x, t) = Re [ E y (x)e jωt] (19) and we can work in terms of E y (x) (the phasor) only. A monochromatic plane wave traveling in the +x direction can be written as E = (ŷa + ẑb) e jkx (20) H = 1 η ( ŷb + ẑa) e jkx (21) with k = ω µɛ = ω v = 2π λ. Sinusoidal in time and space! Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 11 / 25
A plane wave traveling in an arbirtrary direction can be derived similarly as E = E 0 e j[kx x+ky y+kz z] = E 0 e jk r (22) H = 1 ωµ k E = 1 η ˆk E (23) where k = ˆxk x + ŷk y + ẑk z = ˆkk = ˆkω µɛ and E 0 ˆk = 0 (24) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 12 / 25
Plane wave examples Consider the following phasor plane waves in free space E = ˆxe j2πy H = ẑe jπz E = ˆxe j8π(y+z) E = (2ˆx + ŷ 3ẑ) e j2π(x+y+z)/ 3 Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 13 / 25
IV. Plane waves in lossy and dispersive media Since lossy media are dispersive in general, signals clearly cannot propagate undistorted in a lossy medium since different frequency components propagate at different velocities! It is easiest to work in the frequency domain since things are simple for individual frequencies. 2 H = ω 2 µɛh (25) is now the vector wave equation in for phasors, and assuming variations only along x for simplicity gives 2 H y x 2 = ω 2 µɛh y (26) 2 H z x 2 = ω 2 µɛh z (27) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 14 / 25
These linear differential equations have exponentials as their solution H = (ŷm + ẑn) e jkx + (ŷp + ẑr) e jkx (28) The corresponding electric field follows where E = k ωɛ [ ( ŷn + ẑm) e jkx + (ŷr ẑp) e jkx] (29) k = ω µɛ (30) is now complex! Careful in choice of sign for square root! Complex k indicates we have both phase variation (real part of k) and amplitude decay (imaginary part of k) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 15 / 25
A plane wave propagating in any direction can be written as E = E 0 e jk(x cos α+y cos ζ+z cos δ) = E 0 e jk r (31) H = k E/ωµ = ˆk E/η (32) where k = ˆkk = ˆkω µɛ if all components of k are in phase with one another. Actually all that is required to solve the differential equations is k k = ω 2 µɛ (33) which allows for more general behaviors, as we will see in Chapter 6. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 16 / 25
V. Phase and group velocity Although plane waves in lossy media attenuate in space, we can still identify a phase associated with each point and thus a wavelength λ = 2π/k R (34) The phase velocity, v p of a wave determines how rapidly a point of constant phase moves, and is given by v p = dx dt ωt k R x=const = ω k R (35) However, in a dispersive medium, different signal frequencies will travel with different velocities, so how can we define an effective signal velocity? Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 17 / 25
(a) E y (x,t=0) E y (x,t=t 1 ) 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 x (b) 0 1 2 3 4 x Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 18 / 25
A convenient quantity is the group velocity, given by v g = dx dt t dk R dω (ωc)x=const = 1 dk R dω (ω c) (36) which is an effective signal velocity derived through a linear assumption for k versus ω, as can be shown through the modulated sine wave example of the notes. However, when signal distortion is large, the concept of group velocity becomes unclear - from which points of the signal should velocity be measured? - so caution should be used in describing signal velocities in dispersive media in a non-dispersive medium! v g = v p (37) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 19 / 25
VI. Wave Polarization Our example phasor field was E(x) = (ŷa + ẑb) e jkx (38) What does this look like in the time domain? E(x, t) = Re [ E(x)e jωt] { = Re [ŷa + ẑb]e jωt jkx} (39) Letting we get A = Ae ja B = Be jb (40) E(x, t) = ŷae k I x cos(ωt k R x + a) + ẑbe k I x cos(ωt k R x + b) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 20 / 25
Setting r = Ae k I x s = Be k I x (41) (42) and u = ωt k R x (43) this simplifies to E y (x, t) = r cos(u + a) (44) E z (x, t) = s cos(u + b) (45) which are the equations of an ellipse! Linear and circular polarizations are special cases. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 21 / 25
z z z x x t=0 t= /2 t=3 /2 y y x (a) Elliptical polarization Polarization Linear polarization (b) Linear Polarization Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 22 / 25
Polarization examples Consider the following phasor plane waves in free space E = ˆxe j2πy (linear) H = (ˆx + jŷ)e jπz (circular) E = (ˆx + 2(ŷ ẑ)/ 2)e j8π(y+z) (linear) E = (2ˆx + jŷ (2 + j)ẑ) e j2π(x+y+z)/ 3 (eliptical) Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 23 / 25
Poynting Vector We can also derive relations from Maxwell s equations that decribe the transfer of energy due to electromagnetic fields. The relevant quantity is the Poynting Vector which for phasor fields is computed as S = 1 [ 2 Re E H ] The amplitude of the Poynting vector has units of Watts per square meter, and is the power per unit area carried by the electromagetic field. The direction of the Poynting vector indicates the direction of power flow. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 24 / 25
Poynting Vector for a plane wave Consider a phasor plane wave with k = k R jk I = ˆkk: E = E 0 e jk r The Poynting vector then has the form { } S = ˆkRe 1 E 2η 0 2 e 2k I r if all components of k have the same phase. In a lossless medium this reduces to S = ˆk 1 E 2η 0 2 In both cases the direction of power flow is the direction of propagation of the plane wave. Levis, Johnson, Teixeira (ESL/OSU) Radiowave Propagation August 31, 2018 25 / 25