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
Lec13a.3-1
1/12/01
IR - µW
visible I.R.
r.f. → audio
A1
Molecular Lines in Gases
Quantized energy levels: Ei – Ej = hf
Probability of radiation = A + B ρf
radiation intensity (energy density)
“Einstein ‘A’ coeff.”
“Einstein ‘B’ coeff.”
Probability of absorption = Bρf
A B = 8πhf
3
c
3
Atmospheric absorption of
radio waves at zenith (clear air)
dB
100
O2
Collisions and radiation
compete to control level
10
populations. In equilibrium,
kinetic and radiation
1
temperatures are equal.
Lec13a.3-2
1/12/01
O2
H2 O
H2 O
0
f (GHz)
22.235
60
118
183
A2
Molecular Line Shape
Broadening: intrinsic, collisional, Doppler
Einstein “A” yields spontaneous emission, limiting state
lifetime T; intrinsic linewidth ≅ 1/T
Absorption
Electric
field
random collisions
change dipole
phase, orientation
t
f
∆ω = linewidth ∝ number of collisions/sec
“Pressure broadening” or “collision broadening” ≅ 2 GHz at
standard temperature and pressure (STP) for O2, H2O < 1 THz
(
Lec13a.3-3
1/12/01
)
Doppler broadening has thermal 1 mv 2 ≅ 3 kT ,
2
2
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).
α(f)
A+B
interferenceenhanced absorption
A alone
B alone
interferencereduced absorption
α(f)
0
ωo
f
f
Lec13a.3-4
1/12/01
A4
Refractive Effects
ε(f)
ε εo
1
fo
α(f), resonance
absorption
The permittivity ε(f) of a medium is
related in part to the absorption
coefficient α(f) by the Hilbert transform;
α(f) is related to the imaginary part of ε(f).
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.
Lec13a.3-5
1/12/01
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.,
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 )
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
6
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.”
Lec13a.3-6
1/12/01
C2
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:
( )
ε = εo (1 − ωp2 ω2 )
neq2 −1
memi
where m =
rs
ωp =
≅ me
m εo
me + mi
q = electron charge
Evanescent waves only, if ω < ωp
Propagation delay:
Lec13a.3-7
1/12/01
1 − ( ωp ω) > c
2
phase velocity
vp = c
group velocity
v g = c 1 − ( ωp ω) < c
2
for f > fp
v g vp = c 2 here
(
)
C3
Refraction and Absorption by Plasmas
Refraction by plasmas:
ω > ωp
ω < ωp
θt
refraction is governed by Snell’s Law
sin θi sin θt = vpt vpi
evanescent waves, total reflection
Absorption by plasmas:
Lec13a.3-8
1/12/01
y
vp
t
z
θi θi
vp
i
transient electric dipole
emits and absorbs
# collisions ∝ n2e (weak in ionosphere)
C4
Magnetized Plasmas
Faraday rotation, bi-refringence
x
jK '' 0 ⎤
⎡ K'
ε = εo ⎢− jK '' K '
0⎥⇒
⎢
⎥
0 K o ⎥⎦
⎢⎣ 0
y
E
x
B
-e
z
y
The EM interaction and Faraday rotation become strong near
the electron and ion cyclotron resonances, ωc = qBo m
Lec13a.3-9
1/12/01
C5
Scattering and Absorption by Dielectric Spheres
ε, sphere (e.g. raindrop)
EM
wave
E
D <<
D
D >
λo
εo
2π
ε
λo
2π
λo
D >>
2π
induced electric dipole
⇒
"Rayleigh scattering"
⇒
"Mie scattering"
⇒
Geometric scattering
null
dipole re-radiation
pattern
EM
Mie is multimode:
log (σs)
induced electric dipole
4πa2
πa2
slope = 4
0.25πa2
Mie
Lec13a.3-10
1/12/01
Rayleigh
Geometric
f
C6
Scattering and Absorption by Dielectric Spheres
Rayleigh regime
( λ >> 2πDε εo ) :
(constant ε)
(
)
(dBm ) ∝ f
σs ∝ ( a λ ) λ scattering cross-section, α scattering dBm
−1
σa ∝ ( a λ ) λ absorption cross-section, α absorption
−1
6
2
3
∝f
4
2
Cloud absorption < 85 GHz (Rayleigh region):
[0.0122( 291− T )− 6]
m
•
10
γ CLD nepers cm
≅
2
λ
where m = g/m3 liquid water, λ = wavelength cm, T = K°
(
strong
scattering
-1
ice
)
Albedo < 0.8, fmax scat. ≈ 100 − 150 GHz
∴ 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