Electromagnetic Wave Propagation Lecture 4: Propagation in lossy media, complex waves Daniel Sj¨oberg Department of Electrical and Information Technology September 13, 2012 Outline 1 Propagation in lossy media 2 Oblique propagation and complex waves 3 Paraxial approximation: beams (not in Orfanidis) 4 Doppler effect and negative index media 5 Conclusions 2 / 46 Outline 1 Propagation in lossy media 2 Oblique propagation and complex waves 3 Paraxial approximation: beams (not in Orfanidis) 4 Doppler effect and negative index media 5 Conclusions 3 / 46 Lossy media We study lossy isotropic media, where D = dE; J = σE; B = µH The conductivity is incorporated in the permittivity, σ J = J + j!D = (σ + j! )E = j! + E tot d d j! which implies a complex permittivity σ = j c d − ! Often, the dielectric permittivity d is itself complex, d = d0 jd00, due to molecular interactions. − 4 / 46 Examples of lossy media I Metals (high conductivity) I Liquid solutions (ionic conductivity) I Resonant media I Just about anything! 5 / 46 Characterization of lossy media In a previous lecture, we have shown that a passive material is characterized by ξ ξ Re j! = ! Im 0 ζ µ − ζ µ ≥ For isotropic media with = cI, ξ = ζ = 0 and µ = µcI, this boils down to = 0 j00 00 0 c c − c c ≥ µ = µ0 jµ00 ) µ00 0 c c − c c ≥ 6 / 46 Maxwell's equations in lossy media Assuming dependence only on z we obtain 8 ( @ E = j!µ H <> z^ E = j!µcH r × − c @z × − H = j!cE ) > @ r × : z^ H = j!cE @z × Nothing really changes compared to the lossless case, for instance it is seen that the fields do not have a z-component. This can be written as a system @ E 0 jk E = − c @z η H z^ jkc 0 η H z^ c × − c × where the complex wave number kc and the complex wave impedance ηc are r µc kc = !pcµc; and ηc = c 7 / 46 The parameters in the complex plane For passive media, the parameters c, µc, and kc = !pcµc take p their values in the complex lower half plane, whereas ηc = µc/c is restricted to the right half plane. ÁÑ ÁÑ Êe Êe Equivalently, all parameters (j!c; j!µc; jkc; ηc) take their values in the right half plane. 8 / 46 Solutions The solution to the system @ E 0 jk E = − c @z η H z^ jkc 0 η H z^ c × − c × can be written (no z-components in the amplitudes E+ and E ) − jkcz jkcz E(z) = E+e− + E e − 1 jkcz jkcz H(z) = z^ E+e− E e ηc × − − Thus, the solutions are the same as in the lossless case, as long as we \complexify" the coefficients. 9 / 46 Exponential attenuation The dominating effect of wave propagation in lossy media is exponential decrease of the amplitude of the wave: jkcz jβz αz k = β jα e− = e− e− c − ) Thus, α = Im(k ) represents the attenuation of the wave, − c whereas β = Re(kc) represents the oscillations. The exponential is sometimes written in terms of γ = jkc = α + jβ as γz jβz αz e− = e− e− where γ can be seen as a spatial Laplace transform variable, in the same way that the temporal Laplace variable is s = ν + j!. 10 / 46 Power flow The power flow is given by the Poynting vector 1 jβz αz 1 jβz αz ∗ P(z) = Re E0e− − z^ E0e− − 2 × ηc × 1 1 2 2αz 2αz = z^ Re E0 e− = P(0)e− 2 ηc∗ j j 11 / 46 Characterization of attenuation 2αz The power is damped by a factor e− . The attenuation is often expressed in logarithmic scale, decibel (dB). 2αz A = e− A = 10 log (A) = 20 log (e)αz = 8:686αz ) dB − 10 10 Thus, the attenuation coefficient α can be expressed in dB per meter as αdB = 8:686α Instead of the attenuation coefficient, often the skin depth (also called penetration depth) δ = 1/α is used. When the wave propagates the distance δ, its power is attenuated a factor e2 7:4, or 8:686 dB 9 dB. ≈ ≈ 12 / 46 Characterization of losses A common way to characterize losses is by the loss tangent (sometimes denoted tan δ) + σ=! tan θ = c00 = d00 c0 d0 which usually depends on frequency. In spite of this, it is often seen that the loss tangent is given for only one frequency. This is acceptable if the material properties vary only little with frequency. 13 / 46 Example of material properties From D. M. Pozar, Microwave Engineering: Material Frequency r0 tan θ Beeswax 10 GHz 2.35 0.005 Fused quartz 10 GHz 6.4 0.0003 Gallium arsenide 10 GHz 13. 0.006 Glass (pyrex) 3 GHz 4.82 0.0054 Plexiglass 3 GHz 2.60 0.0057 Silicon 10 GHz 11.9 0.004 Styrofoam 3 GHz 1.03 0.0001 Water (distilled) 3 GHz 76.7 0.157 The imaginary part of the relative permittivity is given by r00 = r0 tan θ. 14 / 46 Approximations for weak losses In weakly lossy dielectrics, the material parameters are (where ) c00 c0 = 0 j00 = 0 (1 j tan θ) c c − c c − µc = µ0 The wave parameters can then be approximated as p 1 k = !p µ ! µ 1 j tan θ c c c ≈ c0 0 − 2 r r µc µ0 1 ηc = 1 + j tan θ c ≈ c0 2 If the losses are caused mainly by a small conductivity, we have c00 = σ=!, tan θ = σ=(!c0 ), and the attenuation constant r 1 p σ σ µ0 α = Im(kc) = ! c0 µ0 = − 2 !c0 2 c0 is proportional to conductivity and independent of frequency. 15 / 46 Example: propagation in sea water A simple model of the dielectric properties of sea water is 4 S=m c = 0 81 j − !0 that is, it has a relative permittivity of 81 and a conductivity of σ = 4 S=m. The imaginary part is much smaller than the real part for frequencies 4 S=m f = 888 MHz 81 2π · 0 for which we have α = 728 dB=m. For lower frequencies, the exact calculations give f = 50 Hz α = 0:028 dB=m δ = 35:6 m f = 1 kHz α = 1:09 dB=m δ = 7:96 m f = 1 MHz α = 34:49 dB=m δ = 25:18 cm f = 1 GHz α = 672:69 dB=m δ = 1:29 cm 16 / 46 Approximations for good conductors In good conductors, the material parameters are (where σ !) σ = jσ=! = 1 j c − − ! µc = µ The wave parameters can then be approximated as r σ r!µσ k = !p µ ! j µ = (1 j) c c c ≈ − ! 2 − rµ r µ r!µ η = c = (1 + j) c ≈ jσ=! 2σ c − This demonstrates that the wave number is proportional to p! rather than ! in a good conductor, and that the real and imaginary part have equal amplitude. 17 / 46 Skin depth The skin depth of a good conductor is 1 r 2 1 δ = = = α !µσ pπfµσ For copper, we have σ = 5:8 107 S=m. This implies · f = 50 Hz δ = 9:35 mm f = 1 kHz δ = 2:09 mm f = 1 MHz δ = 0:07 mm f = 1 GHz δ = 2:09 µm This effectively confines all fields in a metal to a thin region near the surface. 18 / 46 Surface impedance Integrating the currents near the surface z = 0 implies (with γ = α + jβ) Z Z 1 1 γz σ J s = J(z) dz = σE0e− dz = E0 0 0 γ Thus, the surface current can be expressed as 1 aiÖ E0 J = E s 0 ÑeØaÐ Z γz s J(z)= σE0e− z where the surface impedance is γ α + jβ α 1 r!µ Z = = = (1 + j) = (1 + j) = (1 + j) = η s σ σ σ σδ 2σ c 19 / 46 Outline 1 Propagation in lossy media 2 Oblique propagation and complex waves 3 Paraxial approximation: beams (not in Orfanidis) 4 Doppler effect and negative index media 5 Conclusions 20 / 46 Generalized propagation factor For a wave propagating in an arbitrary direction, the propagation factor is generalized as jkz jk r e− e− · ! Assuming this as the only spatial dependence, the nabla operator can be replaced by jk since − jk r jk r (e− · ) = jk(e− · ) r − jk r Writing the fields as E(r) = E0e− · , Maxwell's equations for isotropic media can then be written ( ( jk E = j!µH k E = !µH − × 0 − 0 × 0 0 jk H = j!E ) k H = !E − × 0 0 × 0 − 0 21 / 46 Properties of the solutions Eliminating the magnetic field, we find k (k E ) = !2µE × × 0 − 0 This shows that E0 does not have any components parallel to k, and the BAC-CAB rule implies k (k E0) = E0(k k). Thus, the total wave number is given by× × − · k2 = k k = !2µ · It is further clear that E0, H0 and k constitute a right-handed triple since k E = !µH , or × 0 0 k k 1 H = E = k^ E 0 !µ k × 0 η × 0 22 / 46 Preferred direction What happens when k^ is not along the z-direction (which could be the normal to a plane surface)? I There are then two preferred directions, k^ and z^. I These span a plane, the plane of incidence. I It is natural to specify the polarizations with respect to that plane. I When the H-vector is orthogonal to the plane of incidence, we have transverse magnetic polarization (TM).