Lecture4.pdf

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

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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

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Lossy media We study lossy isotropic media, where D = d E,

J = σE,

B = µH

The conductivity is incorporated in the permittivity,   σ E J tot = J + jωD = (σ + jωd )E = jω d + jω which implies a complex permittivity c = d − j

σ ω

Often, the dielectric permittivity d is itself complex, d = 0d − j00d , 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!

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Characterization of lossy media

In a previous lecture, we have shown that a passive material is characterized by       ξ  ξ Re jω = −ω Im ≥0 ζ µ ζ µ For isotropic media with  = c I, ξ = ζ = 0 and µ = µc I, this boils down to c = 0c − j00c

µc = µ0c − jµ00c

⇒

00c ≥ 0

µ00c ≥ 0

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Maxwell’s equations in lossy media Assuming dependence only on z we obtain  ∂ (   ˆ × E = −jωµc H z ∇ × E = −jωµc H ∂z ⇒  ∇ × H = jωc E  ∂ z ˆ × H = jωc E ∂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 E ∂ 0 −jkc = −jkc 0 ˆ ˆ ∂z ηc H × z ηc H × z where the complex wave number kc and the complex wave impedance ηc are r µc √ kc = ω c µc , and ηc = c 7 / 46

The parameters in the complex plane √ For passive media, the parameters c , µc , and kc = ω c µc take p their values in the complex lower half plane, whereas ηc = µc /c is restricted to the right half plane. Im

11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111

Im

Re

11111111 00000000 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111 00000000 11111111

Re

Equivalently, all parameters (jωc , jωµc , jkc , ηc ) take their values in the right half plane. 8 / 46

Solutions

The solution to the system      ∂ E E 0 −jkc = −jkc 0 ˆ ˆ ∂z ηc H × z ηc H × z can be written (no z-components in the amplitudes E + and E − ) E(z) = E + e−jkc z + E − ejkc z   1 ˆ × E + e−jkc z − E − ejkc z H(z) = z ηc Thus, the solutions are the same as in the lossless case, as long as we “complexify” the coefficients.

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Exponential attenuation The dominating effect of wave propagation in lossy media is exponential decrease of the amplitude of the wave: kc = β − jα

⇒

e−jkc z = e−jβz e−αz

Thus, α = − Im(kc ) represents the attenuation of the wave, whereas β = Re(kc ) represents the oscillations. The exponential is sometimes written in terms of γ = jkc = α + jβ as e−γz = e−jβz e−αz where γ can be seen as a spatial Laplace transform variable, in the same way that the temporal Laplace variable is s = ν + jω.

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Power flow The power flow is given by the Poynting vector   ∗  1 1 −jβz−αz −jβz−αz ˆ × E0e P(z) = Re E 0 e × z 2 ηc   1 1 ˆ Re =z |E 0 |2 e−2αz = P(0)e−2αz 2 ηc∗

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Characterization of attenuation The power is damped by a factor e−2αz . The attenuation is often expressed in logarithmic scale, decibel (dB). A = e−2αz

⇒

AdB = −10 log10 (A) = 20 log10 (e)αz = 8.686αz

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) tan θ =

00d + σ/ω 00c = 0c 0d

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.

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Example of material properties From D. M. Pozar, Microwave Engineering: Material Beeswax Fused quartz Gallium arsenide Glass (pyrex) Plexiglass Silicon Styrofoam Water (distilled)

Frequency 10 GHz 10 GHz 10 GHz 3 GHz 3 GHz 10 GHz 3 GHz 3 GHz

0r 2.35 6.4 13. 4.82 2.60 11.9 1.03 76.7

tan θ 0.005 0.0003 0.006 0.0054 0.0057 0.004 0.0001 0.157

The imaginary part of the relative permittivity is given by 00r = 0r tan θ.

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Approximations for weak losses In weakly lossy dielectrics, the material parameters are (where 00c  0c ) c = 0c − j00c = 0c (1 − j tan θ)

µc = µ0

The wave parameters can then be approximated as   p 1 √ 0 kc = ω c µc ≈ ω c µ0 1 − j tan θ 2  r r  µc µ0 1 ≈ ηc = 1 + j tan θ c 0c 2 If the losses are caused mainly by a small conductivity, we have 00c = σ/ω, tan θ = σ/(ω0c ), and the attenuation constant r 1 p 0 σ σ µ0 α = − Im(kc ) = ω c µ0 0 = 2 ωc 2 0c 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 ω0 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 · 2π0 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  ω)  σ c =  − jσ/ω =  1 − j ω µc = µ The wave parameters can then be approximated as r r σ ωµσ √ (1 − j) kc = ω c µc ≈ ω −j µ = ω 2 r r r µc µ ωµ ηc = ≈ = (1 + j) c −jσ/ω 2σ This demonstrates that the wave number is proportional to rather than ω in a good conductor, and that the real and imaginary part have equal amplitude.

√

ω

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Skin depth The skin depth of a good conductor is 1 δ= = α

r

2 1 =√ ωµσ πf µσ

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 ∞ σ J (z) dz = σE 0 e−γz dz = E 0 Js = γ 0 0 Thus, the surface current can be expressed as 1 Js = E0 Zs

air

E0

metal

J(z) = σE0 e−γz z

where the surface impedance is γ α + jβ α 1 Zs = = = (1 + j) = (1 + j) = σ σ σ σδ

r

ωµ (1 + j) = ηc 2σ 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

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Generalized propagation factor For a wave propagating in an arbitrary direction, the propagation factor is generalized as e−jkz → e−jk·r Assuming this as the only spatial dependence, the nabla operator can be replaced by −jk since ∇(e−jk·r ) = −jk(e−jk·r ) Writing the fields as E(r) = E 0 e−jk·r , Maxwell’s equations for isotropic media can then be written ( ( −jk × E 0 = −jωµH 0 k × E 0 = ωµH 0 ⇒ −jk × H 0 = jωE 0 k × H 0 = −ωE 0 21 / 46

Properties of the solutions Eliminating the magnetic field, we find k × (k × E 0 ) = −ω 2 µE 0 This shows that E 0 does not have any components parallel to k, and the BAC-CAB rule implies k × (k × E 0 ) = −E 0 (k · k). Thus, the total wave number is given by k 2 = k · k = ω 2 µ It is further clear that E 0 , H 0 and k constitute a right-handed triple since k × E 0 = ωµH 0 , or H0 =

k k 1ˆ × E0 × E0 = k ωµ k η

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Preferred direction

ˆ is not along the z-direction (which could be What happens when k the normal to a plane surface)? ˆ and z I There are then two preferred directions, k ˆ. 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).

I

When the E-vector is orthogonal to the plane of incidence, we have transverse electric polarization (TE).

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TM and TE polarization

From these figures it is clear that the transverse impedance is Ex A cos θ ηTM = = 1 = η cos θ Hy ηA ηTE =

Ey = −Hx

B 1 η B cos θ

=

η cos θ 24 / 46

Transverse wave impedance ˆ corresponds to the angle of The transverse wave vector kt = kx x incidence θ as kx = k sin θ The transverse impedance is ˆ ), E t = Zt · (H t × z

η ˆx ˆ+ ˆy ˆ Zt = η cos θx y cos θ } {z | isotropic case

The transverse wave impedance can be generalized to bianisotropic materials by solving the eigenvalue problem from last lecture kz ω



Et ˆ Ht × z

 =

         0 −ˆ z×I tt ξ tt I 0 Et · − A(kt ) · · ˆ×I ˆ I 0 ζ tt µtt 0 z Ht × z

ˆ ]. The eigenvalue and studying the eigenvectors [E t , H t × z kz /ω = n/c0 corresponds to the refractive index. 25 / 46

Complex waves When the material parameters are complexified, we still have kc2 = k · k = ω 2 c µc with a complex wave vector k = β − jα

⇒

e−jk·r = e−jβ·r e−α·r

The real vectors α and β do not need to be parallel.

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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

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The plane wave monster

So far we have treated plane waves, which have a serious drawback: I

Due to the infinite extent of e−jkz z in the xy-plane, the plane wave has infinite energy.

However, the plane wave is a useful object with which we can build other, more physically reasonable, solutions.

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Finite extent in the xy-plane We can represent a field distribution with finite extent in the ˆ + ky y ˆ ): xy-plane using a Fourier transform (where kt = kx x  ∞ ZZ  1   E t (x, y; z) = E t (kx , ky ; z)e−jkt ·r dkx dky    (2π)2  −∞

∞ ZZ      E (k , k ; z) = E t (x, y; z)ejkt ·r dx dy   t x y −∞

The z dependence in E t (kx , ky ; z) corresponds to a plane wave E t (kx , ky ; z)e−jkt ·r = E t (kx , ky ; 0)e−jkt ·r e−jkz z The total wavenumber for each kt is given by k 2 = ω 2 µ and k 2 = |kt |2 + kz2 = kx2 + ky2 + kz2

⇒

kz (kt ) = (k 2 − |kt |2 )1/2 29 / 46

Initial distribution Assume a Gaussian distribution in the plane z = 0 E t (x, y; 0) = Ae−(x

2 +y 2 )/(2b2 )

The transform is itself a Gaussian 2

2

E t (kx , ky ; 0) = A2πb2 e−(kx +ky )b

2 /2

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Paraxial approximation The field in z ≥ 0 is then 1 E t (x, y; z) = Ab2 2π

∞ ZZ

2

2

2 /2−j(k x+k y)−jk (k )z x y z t

e−(kx +ky )b

dkx dky

−∞

The exponential makes the main contribution to come from a region close to kt ≈ 0. This justifies the paraxial approximation kz (kt ) = (k 2 − |kt |2 )1/2 = k(1 − |kt |2 /k 2 )1/2   1 |kt |2 |kt |2 4 4 =k 1− + O(|k | /k ) = k − + ··· t 2 k2 2k

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Computing the field Inserting the paraxial approximation in the Fourier integral implies 1 E t (x, y; z) ≈ Ab2 2π

∞ ZZ

2

2

b2

−∞

= where F 2 = b2 − jz/k =

z

e−(kx +ky )( 2 −j 2k )−j(kx x+ky y)−jkz dkx dky

1 jk (z

Ab2 −(x2 +y2 )/(2F 2 ) −jkz e e F2

+ jkb2 ) = q(z)/(jk).

I

q(z) = z + jz0 is known as the q-parameter of the beam.

I

z0 = kb2 is known as the Rayleigh range.

The final expression for the beam distribution is then 2

E t (x, y; z) ≈

2

x +y A − e 2b2 (1−jz/z0 ) e−jkz 1 − jz/z0 32 / 46

Beam width The power density of the beam is proportional to −x

e

2 +y 2 2b2

Re

1 1−jz/z0

and the beam width is then p b b B(z) = q = b 1 + (z/z0 )2 =q 1+jz/z0 1 Re 1−jz/z Re 1+(z/z 2 0 0) where z0 = kb2 . For large z, the beam width is B(z) → bz/z0 =

z , kb

z→∞

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Beam width

The beam angle θb is characterized by B(z) 1 tan θb = = z kb Small initial width compared to wavelength implies large beam angle. 34 / 46

How can beams be used? Beams can be an efficient representation of fields, determined by three parameters: I Propagation direction z ˆ I Polarization A(ω) I Initial beam width b(ω)

High frequency propagation in office spaces (Timchenko, Heyman, Boag, EMTS Berlin 2010).

Raytracing in optics, FRED

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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

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The Doppler effect Classical formulas: fb =

fb =

The relativistically correct formula is r   v c−v fb = fa ≈ fa 1 − c+v c0

where

c c = fa λb c − va

c − vb c − vb = fa λa c

v=

vb − va 1 − va vb /c20

Lots more on relativistic Doppler effect in Orfanidis. Do not dive too deep into this, it is not central material in the course. 37 / 46

Negative material parameters Passivity requires the material parameters c and µc to be in the lower half plane, Im

11111111111111 00000000000000 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111 00000000000000 11111111111111

Re

This means we could very well have c /0 ≈ −1 and µc /µ0 ≈ −1 for some frequency. value for p What is then the appropriate p √ k = ω c µc = k0 (c /0 )(µc /µ0 ) = k0 (−1)(−1), k = +k0

or

k = −k0 ? 38 / 46

Negative refractive index Simple solution: consider all parameters in the right half plane and approach the negative axis from inside the half plane, using the standard square root (with branch cut along negative real axis): p jkc = (jωc )(jωµc ) Im

Im

Im

ǫ µ

jωǫ jωµ Re

Re

Im

jk =

jωǫ · jωµ Re

√

Im

n=

jωǫ · jωµ Re

jk jk0 Re

The refractive index is then n=

p jkc = −j (jc /0 )(jµc /µ0 ) = −1 jk0

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Consequences of negative refractive index With a negative refractive index, the exponential factor ej(ωt−kz) = ej(ωt+|k|z) represents a phase traveling in the negative z-direction, even ˆ 12 Re( η1∗ )|E 0 |2 is though the Poynting vector 12 Re{E × H ∗ } = z c still pointing in the positive z-direction. I

The power flow is in the opposite direction of the phase velocity!

I

Snel’s law has to be “inverted”, the rays are refracted in the wrong direction.

First investigated by Veselago in 1967. Enormous scientific interest since about a decade, since the materials can now (to some extent) be fabricated. 40 / 46

Negative refraction

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Realization of negative refractive index

I

Artificial materials, “metamaterials”

I

Periodic structures

I

Resonant inclusions

I

Small losses required

I

Theoretical and practical challenges

I

Very hot topic since about 10 years

To describe the structure as a material usually requires microstructure a  λ, which is not easily achieved.

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Band limitations

If the negative properties are realized with passive, causal materials, they must satisfy Kramers-Kronig’s relations (∞ = lim (ω)) ω→∞

Z ∞ 00 0  (ω ) 1 p.v. dω 0 0−ω π ω −∞ Z ∞ 0 0 1  (ω ) − ∞ 00  (ω) = − p.v. dω 0 π ω0 − ω −∞

0 (ω) − ∞ =

These relations represent restriction on the possible frequency behavior, and can be used to derive bounds on the bandwidth where the material parameters can be negative.

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Example: two Lorentz models ²(!) 4 2 0

²s Im

²m =1.5 ²1 !

Re 0.1

1

10 ²m =-1

-2 1

j²(!)-²m j

0.8 0.6 0.4 0.2 0

j²(!)+1j

j²(!)+1j

j²(!)-1.5j 0.1

1

! 10

An m between s and ∞ is easily realized for a large bandwidth, whereas an m < ∞ is not. With fractional bandwidth B: ( 1/2 lossy case B max |(ω) − m | ≥ (∞ − m ) ω∈B 1 + B/2 1 lossless case 44 / 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

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Conclusions

I

Lossy media leads to complex material parameters, but plane wave formalism is the same as in lossless media.

I

At oblique propagation, the transverse fields are most important.

I

The paraxial approximation can be used to describe beams. The beam angle depends on the original beam width in terms of wavelengths.

I

The Doppler effect can be used to detect motion.

I

Negative refractive index is possible, but only for very narrow frequency band.

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