# Receivers Antennas and Signals Lecture Notes - lecture22 - Wave Propagation by YAdocs

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Wave Propagation
Molecular line absorption by gases:
• permanent electric dipole
C       O
(H2O, CO)
H
• permanent magnetic dipole
O     O               O        (O2)
H         • unpolarized (N2) (collision­
N      N
induced dipoles)
Quantized energy levels: Ei – Ej = hf
electronic states
vibrational states
rotational states
nuclear spin states

visible, UV, x-ray           IR - µW
r.f. → audio
Lec13a.3-1
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Molecular Lines in Gases
Quantized energy levels: Ei – Ej = hf

Probability of radiation = A + B ρf
“Einstein ‘A’ coeff.”               radiation intensity (energy density)
“Einstein ‘B’ coeff.”

Probability of absorption = Bρf
3       3                             Atmospheric absorption of
A B = 8πhf        c                  dB
radio waves at zenith (clear air)
100                                 O2     H2 O
O2
compete to control level     10
populations. In equilibrium,
1
temperatures are equal.

0                                           f (GHz)
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22.235      60        118      183           A2
Molecular Line Shape
Einstein “A” yields spontaneous emission, limiting state
lifetime T; intrinsic linewidth ≅ 1/T
Absorption             Electric                           random collisions
field                              change dipole
phase, orientation
t

f
∆ω = linewidth ∝ number of collisions/sec

standard temperature and pressure (STP) for O2, H2O < 1 THz

2    (
Doppler broadening has thermal 1 mv 2 ≅ 3 kT ,
2              )
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turbulent (random), and systematic components                        A3
Overlapping Spectral Lines
Superposition characterizes the cumulative absorption by independent
spectral lines, except for certain single molecules.

For a single molecule with collision-coupled states, total absorption is
generally greater than sum of line absorptions between coupled lines, and
less outside. Such coupled lines coherently “interfere” (e.g. 60-GHz oxygen
band).

interference-
α(f)       A+B
enhanced absorption          α(f)
A alone
B alone
0      ωo          f
interference-
reduced absorption

f
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Refractive Effects
ε(f)
α(f), resonance
absorption      The permittivity ε(f) of a medium is
ε εo                         related in part to the absorption
coefficient α(f) by the Hilbert transform;
α(f) is related to the imaginary part of ε(f).
1           fo               f

Atmospheric water vapor scale height = ~2 km
Atmospheric density scale height ≅ ~8 km
So humidity-based refractive effects are mostly a lower tropospheric
~
phenomenon (< 8 km).
Thermal inhomogeneities are turbulent in the boundary layer (first
few hundred meters or more) and near convective instabilities, and
are more layered at higher altitudes.
Humidity variations often dominate radio refraction, while density
variations dominate optical propagation.
Optical telescopes have ~1 arc-sec “seeing” on good nights (2” – 10”
in Boston is typical); the best mountain days may yield ~0.4 arc sec.,
Lec13a.3-5
1/12/01         where “seeing” is the blur spot size, not absolute refraction.           C1
Refractive Effects
The radio index of refraction n is given by: (n − 1) 10 = ( 79 T ) (p + 4800 e T )
6

where T is °K, and p and e are total and partial water vapor pressures (mb).
refraction           less dense
“duct” of humid air
ionosphere
dense                          dry
humid
over the horizon
Ducting can occur in cold or humid layers of air, or in under-ionized
ionospheric layers. Acoustic ducting can occur in cool or salty ocean layers.

Refractive seeing beyond the horizon can be ~ 30
>
arc minutes on RF, and less at optical frequencies.

Fading caused by interfering multipath: paths of different
length cause different frequencies to cancel out or “fade.”

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Ionospheric and Space Plasmas
Plasmas can have both neutral and ionized components.
(
The ionosphere has ne ≅ 107 − 1012 m−3 from ~50        )
to 5000 km altitude. Electron density ne (max) is ~100 - 400 km.
Plasma frequency:

ωp =
neq2 −1
m εo
rs  ( )
where m =
memi
me + mi
≅ me

ε = εo (1 − ωp ω2 )
2
q = electron charge

Evanescent waves only, if ω < ωp

Propagation delay:

1 − ( ωp ω) > c
2
phase velocity   vp = c                         for f > fp

Lec13a.3-7
group velocity   v g = c 1 − ( ωp ω) < c
2           (v g vp = c 2 here   )
1/12/01                                                                      C3
Refraction and Absorption by Plasmas
Refraction by plasmas:                                          y
vp
ω > ωp   refraction is governed by Snell’s Law                θt            t

sin θi sin θt = vpt vpi                                                z
ω < ωp   evanescent waves, total reflection           θi θi        vp
i

Absorption by plasmas:
transient electric dipole
emits and absorbs

# collisions ∝ n2 (weak in ionosphere)
e
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Magnetized Plasmas
x                x

⎡ K'     jK '' 0 ⎤     E          B
ε = εo ⎢− jK '' K '   0⎥⇒                                 -e
⎢                ⎥                                        z
⎢ 0
⎣         0 Ko ⎥ ⎦

y                   y

The EM interaction and Faraday rotation become strong near
the electron and ion cyclotron resonances, ωc = qBo m

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Scattering and Absorption by Dielectric Spheres
ε, sphere (e.g. raindrop)
EM
λo     εo
wave                                     D <<                    ⇒      "Rayleigh scattering"
D                     2π     ε
E                               λo
D>
                      ⇒      "Mie scattering"
2π
λo
D >>                    ⇒      Geometric scattering
2π
induced electric dipole                                    null
EM                                                                        pattern
Mie is multimode:
log (σs)                                              induced electric dipole

4πa2
πa2
slope = 4
0.25πa2           Geometric
Mie                   f
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Rayleigh                                                             C6
Scattering and Absorption by Dielectric Spheres
Rayleigh regime             ( λ >> 2πDε εo ) :             (constant ε)
σs ∝ ( a λ ) λ scattering cross-section, α scattering dBm
6   2
(    )
−1
∝f
4

σa ∝ ( a λ ) λ absorption cross-section, α absorption
3
(dBm ) ∝ f
−1        2


Cloud absorption < 85 GHz (Rayleigh region):
[0.0122( 291− T )− 6]
(
γ CLD nepers cm      ≅      )
m • 10
-1

λ
2

where m = g/m3 liquid water, λ = wavelength cm, T = K°

strong                         ice               
Albedo < 0.8, fmax scat. ≈ 100 − 150 GHz
scattering                                ∴ TB > 70K as seen from space

∆
updraft                              (Albedo = reflectivity, all angles)

cumulonimbus
cloud
rain
Rain attenuation > 30dB sometimes
Lec13a.3-11
1/12/01                                                                                    C7

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