Transcript EM theory and its application to microwave remote sensing

EM theory and its application to
microwave remote sensing
Chris Allen ([email protected])
Course website URL
people.eecs.ku.edu/~callen/823/EECS823.htm
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Outline
Plane wave propagation
Lossless media
Lossy media
Polarization and coherence
Fresnel reflection and transmission
Layered media
EM spectra, bands, and energy sources
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Plane wave propagation
Plane wave propagation through lossless and lossy media
is fundamental to microwave remote sensing.
Consider the wave equation and plane waves in
homogeneous unbounded, lossless media
Plane waves – constant phase and amplitude in the plane
Homogeneous – electrical and magnetic parameters do not vary with
throughout the medium
Beginning with Maxwell’s equations and assuming
a homogeneous, source-free medium leads to the
homogeneous wave equation
H
t
E
H  
t
E  
2 E
 E   2
t
2
where
E is the electric field vector (V/m) [note that bolded symbols denote vectors]
 is the medium’s magnetic permeability (H/m) [H: Henrys]
 is the medium’s permittivity (F/m) [F: Farads]
3
Plane wave propagation
Assuming sinusoidal time dependence

Er, t   Re Er e jt

where  is the radian frequency (rad/s)
r is the displacement vector
and Re{} is the real operator
E(r,t) satisfies the wave equation if
 2Er   2  Er   0
Using phasor representation (i.e., e.jt is understood) and
assuming a rectangular coordinate system, the solution has
the general form of
Er   E0 exp [ j (k x x  k y y  k z z)]
where E0 is a constant vector and
k 2  2    k 2x  k 2y  k 2z
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Plane wave propagation
A more compact form results from letting
k  xˆ k x  yˆ k y  zˆ k z
where k is the propagation vector, and
k = |k| is called the wave number (rad/m)
resulting in
Er   E0 exp [ j k  r]
Finally reintroducing the time dependence and expressing
only the real-time field component yields
Er, t   E0 cos  t  k  r 
This equation represents two waves propagating in
opposite directions defined by the propagation vector k
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Plane wave propagation
Rotating the Cartesian coordinate system such that the
z axis aligns with k yields
Er, t   E0 cos  t  k z 
representing two waves propagating in the + z and – z
directions.
The argument of the cosine function contains two phase
terms: the time phase, t, and the space phase, kz.
If we use the – part of the ± solution we have
Er, t   E0 cos  t  k z 
representing a wave whose constant phase moves in the
positive z direction. As time increases, z must increase to
maintain a constant phase argument.
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Plane wave propagation
The time phase component is characterized by  where
  2 f  2  T
f is the frequency (Hz) and T is the time period (s).
Similarly the space phase component depends on k where
k  2    
 is the space period (m), or wavelength, in the medium
which can also be expressed as

 1 f 

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Plane wave propagation
Consider now the electric field’s phase for a positivetraveling wave, i.e., t – kz.
A surface on which this phase is constant requires
 t  k z  constant
For any given time t, this surface represents a plane
defined by z = constant, on which both the phase and
amplitude are constant. As time progresses, this plane of
constant phase and amplitude advances along the z axis,
hence the name uniform plane wave.
The rate at which this plane advances along the z axis
is called the phase velocity, v (m/s)
dz 
1
v
 
dt k

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Plane wave propagation
Given an uniform plane E-field solution to the wave
equation, the H-field is found using Maxwell’s equations
H
E  
t
From the E-field component in the x-axis direction, Ex, is
found the H-field component in the y-axis direction, Hy, as
H y z, t  
E
k
E x 0 cos  t  k z   x 0 cos  t  k z 


where  is the intrinsic impedance () of the medium
   k   
Note that Ex and Hy are related through the intrinsic impedance similar
to how voltage and current in a circuit are related through Ohm’s law.
Note also the orthogonality of the E, H, and k vectors.
E  xˆ E x , H  yˆ H y ,
k  zˆ k
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Plane waves in a lossy medium
A lossy medium is characterized by its permeability, ,
permittivity, , and conductivity,  (S/m) [S: Siemens].
Maxwell’s equations for a source-free medium become
H
E
E  
,
H  E  
t
t
And the corresponding wave equation remains
 2Er   k 2 Er   0
where the wave number is
k   j     j  
Note that for a lossless medium, k is purely real when
 = 0 and both  and  are real
k
2  
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Plane waves in a lossy medium
For a lossy medium k is complex
k   j     j  
due to   0 or either  or  are complex
    j 
    j 
For lossless media the imaginary parts of the permeability
and permittivity are zero.
Non-zero imaginary terms ( > 0 and  > 0) represent
mechanisms for converting a portion of the electromagnetic
wave’s energy into heat, resulting in a loss of wave energy.
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Plane waves in a lossy medium
Consider the complex electric field plane wave propagation
along the positive z axis
Ez, t   E 0 e j t  k z 
whereas for the lossless case k was real, in a lossy
medium k is complex and is related to the propagation
factor or propagation constant,  (1/m), by
  j k  j  j    j  
    j
such that
Ez   E0 e  j k z  E0 e   z  E0 e    j  z
where  and  are real quantities and  is the attenuation
constant (Np/m) and  is the phase constant (rad/m)
[Np = Neper]
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Plane waves in a lossy medium
Clearly for a wave travelling along the +z axis
Ez   E 0 e
 z  j  z
e
as z increases, the magnitude of the electric field decreases.
The real time expression for the x-axis field component is
E x z, t   E x 0 e   z cos  t   z 
The attenuation constant is the real part of jk
  Re


j    j   ,
Np / m
The phase constant is the imaginary part of of jk
  Im


j    j   ,
rad / m
Note: (Neper/m)  8.686 (dB/Neper) = (dB/m)
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Plane waves in a lossy medium
In a lossy medium, the intrinsic impedance is also complex
  j  

giving rise to a non-zero phase relationship between the E
and H field components.
While a medium’s loss may be due to its conductivity, or
the imaginary components of permittivity or permeability, in
most microwave remote sensing applications the
magnetization loss () is negligible.
Exceptions include the ferrous-rich sands found in Hawaii and soils on
the Martian surface.
Therefore the  term will be neglected from now on.
A material’s permeability is usually specified relative to that
of free space, o, (o = 4  10-7 H/m), as  = r  o and
typically r = 1
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Plane waves in a lossy medium
A medium’s loss may be due to its conduction loss
( > 0 but finite) or its polarization loss ( > 0).
Conduction loss is gives rise to a conduction current
JC   E
A / m 
2
where electrical energy is
converted to heat energy
due to ohmic losses.
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Plane waves in a lossy medium
Polarization loss is due to a displacement current, similar to
current through a capacitor. For an ideal dielectric, equal
amounts of energy are stored and released during each
cycle. For lossy dielectrics, some of the stored energy is
converted into heat.
E
JD  
A / m2
t


The imaginary part of
the displacement
current is in-phase
from the E field, and
hence contributes
to real energy loss.
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Plane waves in a lossy medium
For a sinusoidal time variation,
there is a transition frequency,
t = 2  ft, where these two
current components are equal
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Plane waves in a lossy medium
For dielectric materials, these two loss mechanisms are
often combined into a single imaginary component as


    j    


The permittivity of dielectric materials is usually specified
relative to the permittivity of free space, o, where
(o = 8.854  10-12 F/m) as
  o  r  o r  jr 
where
r    o
   
r 
o
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Plane waves in a lossy medium
Often instead of specifying the r, a material’s loss tangent
tan  where
tan   r r
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Plane waves in a lossy medium
Clearly since the loss mechanism due to conductivity is
frequency dependent, a medium’s loss characteristics may
also be frequency dependent.
At low frequencies where the conductivity introduces signficant loss, the
intrinsic impedance and phase velocity will be frequency dependent.
At frequencies where the conductive loss is dimished, the intrinsic impedance
and phase velocity will become frequency independent.
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Plane waves in a lossy medium
Low-loss media (i.e., tan   1)
For the case of low-loss media, the expressions for v, , ,
and  can be simplified to be

v 

1  1   
 
1  
   8    
1

 
1
o o
r

c
r
where c = 3  108 m/s

 


1

j




 
2   

o
o

1
 o
r
r
where o = 120    377 
 

  2
2 
 1   2 
     
     1  
 8     
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Plane waves in a lossy medium
High-loss media (i.e., tan  >> 1)
For the case of high-loss media, the expressions for , 
and  can be simplified to

1 j
 
    f 
2
Also, when an electromagnetic wave impinges on a
conducting medium, the field amplitude decreases
exponentially with depth.
At a depth termed the skin depth,  (m), the field amplitude
is e-1 of its value at the surface, where
1
 

1
f 
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Plane waves in a lossy medium
General lossy media
For media that is neither high loss nor low loss, the
simplifying approximations  and  do not apply.
For these cases we have
  k o Im
  k o Re

r
 
 k  1  tan 2   1  2

 
 r  k  1  tan 2   1  2

 
12
12
where ko is the free-space wave number
and o is the free-space wavelength
2
ko 
where  o  c / f
o
It is sometimes useful to refer to a medium’s refractive
index, n, where
n 2   r  r  j r or n   r
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Plane waves in a lossy medium
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Plane waves in a lossy medium
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Polarization of plane waves and coherence
For a +z-axis propagating uniform plane wave, the E-field
components must lie in the xy plane
Ez, t   E x 0 xˆ cos  t   z   E y 0 yˆ cos  t   z   y 
For any point on the xy-plane, the E-field varies with time.
The wave’s polarization is associated with
the curve the tip of the E-field vector traces out.
A straight line indicates a linear polarization
(i.e., y = 0 or 180) with tilt angle 
Ez, t   E x 0 xˆ  E y 0 yˆ cos  t   z  
  tan 1 E yo E xo 
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Polarization of plane waves and coherence
A circle indicates circular polarization
Ez, t   E x 0 xˆ cos  t   z   E y 0 yˆ cos  t   z   y


where Exo = Eyo and y =  90
Left-hand circular polarization (LCP), which results
when  y = +90, has the E-field rotating once
each cycle in the direction the fingers of the
left hand point when gripping the z-axis
with the thumb in the +z direction.
Right-hand circular polarization (RCP), which
results when  y = -90, has the E-field rotating
in the same direction as the fingers when
gripping the z-axis with the right hand
so that the thumb is in the +z direction.
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Polarization of plane waves and coherence
An elliptical pattern indicates elliptical polarization
Ez, t   E x 0 xˆ cos  t   z 
 E y 0 yˆ cos  t   z   y 
where Exo and Eyo > 0 and y  0, 90, or 180
yet these variables remain constant over time.
A wave is unpolarized when the amplitude
and phase relationships of the orthogonal
components are time varying.
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Polarization of plane waves and coherence
Signals produced by single-frequency or multifrequency
transmitters are typically completely polarized
Signals emitted by physical objects, irregular terrain, or
inhomogeneous media are usually broadband and are
composed of many statistically independent waves with
different polarizations
If there is no correlation between the component waves of
such a signal it is incoherent or unpolarized
Between these two extremes (completely polarized and unpolarized)
are the partially-polarized signals that result when
polarized signals are scattered by random targets
While characterization of completely polarized plane waves
is fairly straightforward, the characterization of these
partially-polarized signals is more challenging
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Polarization of plane waves and coherence
An analysis technique to evaluate the state of polarization
or degree of coherence of a plane wave involves the
magnitude of the normalized cross-correlation of the x and
y phasor components as
 xy  E x E
*
y
Ex
2
Ey
2
where * denotes the complex conjugate and  denotes the
average operator over some finite time interval T
1
  lim
T  T

T
0
 dt
For completely polarized signals xy = 1 while for incoherent
or unpolarized signals xy = 0.
For partially-polarized or partially-coherent signals
0 < xy < 1
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Polarization of plane waves and coherence
To evaluate the degree of polarization (DOP), another
measure of a signal’s polarization, involves the Stokes
parameters
S0  E x  E y
2
2
S2  2 Re E x E *y
S1 
Ex  Ey
2
2
S3  2 Im E x E *y
The DOP is found by
DOP 
S12  S22  S32
S0
where DOP = 1 for a completely polarized signal, a DOP = 0
for an unpolarized signal, and for partially-polarized signals
0 < DOP < 1
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Polarization of plane waves and coherence
The Poincaré sphere is a tool for visualizing the continuum
of polarization states.
Derived from the Stokes parameters, the sphere maps
linear polarizations on the equator (LVP: linear vertical pol.; LHP:
linear horizontal pol.) and the right (RCP) and left (LCP) circular
polarizations at the north and south poles.
Points opposite one another on the sphere
represent orthogonal polarizations.
Points representing completelypolarized signals lie on the sphere’s
surface while points representing
partially-polarized waves appear
within the sphere.
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Electromagnetic phenomena
Speed of light
c: speed of light in vacuum (2.99792458  108 m/s  3  108 m/s)
n: refractive index of material (n  1)
v: speed of light in material (m/s), v = c/n
Wavelength and frequency
f: frequency (Hz)
: wavelength (m)
o: wavelength in free space (m)
f: bandwidth in frequency (Hz)
: bandwidth in wavelength (m)
In vacuum, (n = 1, v = c)
c
c
c  f  o ,
o  ,
f 
,
f
o
In a medium, (n  1, v  c)
v
c
 

 o
f
nf
f 
c 
2o
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Fresnel reflection and transmission
Now consider the electromagnetic interactions as a plane
wave impinges on a planar boundary between two different
homogeneous media with semi-infinite extent
Properties of interest include reflection, refraction,
transmission
Solutions are found by satisfying the
requirement for the continuity of
tangential E and H fields across the
boundary
Simplifying assumptions:
•
•
•
•
Plane wave propagation
Smooth, planar interface infinite in extent
Linear, isotropic refractive indices
Semi-infinite media
34
Fresnel reflection and transmission
Snell’s law
n1:
n2:
i:
r:
t:
refractive index of medium 1 (incidence side)
refractive index of medium 2
incidence angle (measured from surface normal)
reflected angle (measured from surface normal)
transmitted angle (measured from surface normal)
Requirement for continuity of tangential
E and H fields across the boundary yields
Reflected
r  i
Transmitted n 1 sin  i  n 2 sin  t
sin  t
n1

n2
sin  i
35
Fresnel reflection and transmission
Fresnel equations
• Predicts reflected and transmitted vector field quantities at a plane
•
•
•
•
•
interface
Satisfies requirement for continuity of tangential E and H fields
across the boundary
Consequently the interaction is polarization dependent
Arbitrarily polarized incident plane wave
can be decomposed into two orthogonal
linear polarizations: perpendicular () and
parallel (//)
Separate solutions for perpendicular and
parallel cases
Perpendicular and parallel refer to E-field
orientation with respect to the plane of incidence
(plane containing incident, reflected, and transmitted rays)
• These same polarizations have other names as well
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Fresnel reflection and transmission
Perpendicular (horizontal) case
Reflection coefficient (relates to field strength)
(sometimes represented by , , or r).
n cos i  n 2 cos  t
Er
R 
 1
Ei
n1 cos i  n 2 cos  t
sin i   t 
 

sin i   t    0
i
 2 cos i   1 cos  t
 2 cos i   1 cos  t
Transmission coefficient (relates to field strength)
2 n 1 cos  i
Et
T 

Ei
n 1 cos  i  n 2 cos  t

2 sin  t cos  i
sin ( i   t )

i  0
2  2 cos  i
 2 cos  i   1 cos  t
Note that 1 + R  = T
37
Fresnel reflection and transmission
Perpendicular (horizontal) case
Reflectivity (relates to power or intensity)
  R 
2
Transmissivity (relates to power or intensity)
(sometimes represented by T)
Recos  t  /  2 
 
T
Recos  i  / 1 
2
or
   1  
Note that += 1 which satisfies the conservation of energy
38
Fresnel reflection and transmission
Parallel (vertical) case
Reflection coefficient (relates to field strength)
n cos i  n1 cos  t
Er
R // 
 2
Ei
n 2 cos i  n1 cos  t
 1 cos i   2 cos  t
tan i   t 


tan i   t    0  1 cos i   2 cos  t
i
Transmission coefficient (relates to field strength)
E
2 n 2 cos i
T//  t 
E i n 2 cos i  n1 cos  t
2  1 cos i
2 sin  t cos i


sin i   t  cosi   t    0  1 cos i   2 cos  t
i
Note that 1 + R // = T//
39
Fresnel reflection and transmission
Parallel (vertical) case
Reflectivity (relates to power or intensity)
//  R //
2
Transmissivity (relates to power or intensity)
 // 
or
Re 2 cos  t 
Re 1 cos i 
T//
2
 //  1  //
Note that //+//= 1 which satisfies the conservation of energy
40
Fresnel reflection and transmission
Special cases
Normal incidence (i = 0)
i = r = t = 0, cos  = 1
n1  n 2
R //  R  
n1  n 2
2n 1
T//  T 
n1  n 2
Brewster angle
B: angle where reflection coefficient for parallel (vertical) polarized field
goes to zero
i.e., at  = B, // = 0, // = 1 (note polarization dependence)
n2
tan  B 
n1
41
Fresnel reflection and transmission
Special cases
Critical angle
C: incidence angle at which total internal reflection occurs (for n1 > n2)
i.e., at C, // =  = 1, // =  = 0 (note polarization independence)
sin C  n 2 n1
Evanescent waves exist in medium 2, with imaginary propagation
coefficients meaning they decay rapidly with distance z from the
boundary.
E(z)  Ei e z
2


n 1 sin  i 2  n 22
42
Fresnel reflection and transmission
Normal incidence reflection
coefficient for some typical
geological contacts
43
Fresnel reflection and transmission
Example #1
Consider the case where a plane wave impinges on a
planar boundary between homogenous
ice ( = 3.14, n = 1.78) and air ( = 1, n = 1).
From the formulas presented earlier
Normal reflectivity = -11.0 dB
Normal transmissivity = -0.4 dB
Critical angle, C = 34.4°
Brewster angle, B = 29.4°
44
Fresnel reflection and transmission
Fresnel reflection and transmission coefficients vs.
incidence angle at an ice-air boundary (ice = 3.14, air = 1)
45
Fresnel reflection and transmission
Reflectivity and transmissivity expressed in linear units vs.
incidence angle at an ice-air boundary (ice = 3.14, air = 1)
46
Fresnel reflection and transmission
Reflectivity and transmissivity expressed in decibels vs.
incidence angle at an ice-air boundary (ice = 3.14, air = 1)
47
Fresnel reflection and transmission
Example #2
Consider the case where a plane wave impinges on a
planar boundary between homogenous
ice ( = 3.14, n = 1.78) and rock ( = 5, n = 2.24).
From the formulas presented earlier
Normal reflectivity = -19.1 dB
Normal transmissivity = -0.1 dB
Critical angle, C = NA
Brewster angle, B = 51.6°
48
Fresnel reflection and transmission
Fresnel reflection and transmission coefficients vs.
incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)
49
Fresnel reflection and transmission
Reflectivity and transmissivity expressed in linear units vs.
incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)
50
Fresnel reflection and transmission
Reflectivity and transmissivity expressed in decibels vs.
incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)
51
Layered media
Now consider the case of multiple, planar layers and the
associated composite reflection and transmission
characteristics.
Consider first the simplest case, a single layer of thickness
d1 sandwiched between two semi-infinite layers.
z
y
x
R
Layer 0, 0  1, 0
z=0
Layer 1, 1  1, 1
z = -d1
Layer 2, 2  1, 2
T
52
Layered media (1 of 5)
53
Layered media (2 of 5)
54
Layered media (3 of 5)
55
Layered media (4 of 5)
56
Layered media (5 of 5)
A similar approach can be developed for the V-polarized
case.
57
Layered media – example (1 of 4)
Multilayer example: 1-layer case
58
Layered media – example (2 of 4)
Wavenumbers in media 1 and 2 are
k z1  2 1  k 02 sin 2 0
k z 2  2  2  k 02 sin 2 0
Matching tangential E-field components at each interface
requires
A1 e j k z1 d1  C1 e j k z1d1  A 2 e j k z 2 d1  C2 e j k z 2 d1
Matching tangential H-field components at each interface
requires
k z 0 A0  C0   k z1 A1  C1 



k z1 A1 e j k z1 d1  C1 e j k z1 d1  k z 2 A 2 e j k z 2 d1  C2 e j k z 2 d1

59
Layered media – example (3 of 4)
These boundary matching conditions can be written in
matix form as
 j k z1 d1
A


 0
A1 e
 C   B01 
j k z1 d1 
 0
 C1 e

A1 e  jk z1 d1 
A 2 e  j k z 2 d 2 
 B12 

j k z1 d1 
jk z 2 d2 
 C1 e

 C2 e

which leads to
T e  j k z 2 d 2 
1

R   B01B12 
 
 0 
j k z1d1



e
1
k z1

 
 1 
2  k z 0  R 01 e j k z1 d1
R 01 e  j k z1 d1  1  k z 2   T e  j k z 2d1 
 
  1 
 j k z1d1
 j k z 2 d1 
k z1  R 12Te
e

 2
60
Layered media – example (4 of 4)
And the R variables in the matrices are defined as
k z 0  k z1
R 01 
k z 0  k z1
k z1  k z 2
R12 
k z1  k z 2
To find the reflection and transmission coefficient for this
1-layer structure, we solve for R and T
61
Electromagnetic spectrum
62
Radio spectrum
63
Atmospheric transmission at radio frequencies
64
Reserved frequencies and band designations
65
Generic energy band diagram
66
Quantum energy levels
Photon contains energy related to the electromagnetic frequency,  (Hz)
E = h
where h is Planck’s constant
h = 6.625 × 10-34 J s
h = 4.136 × 10-15 eV s
[Note: 1 eV = 1.6 × 10-19 J ]
At 1 GHz ( = 109 Hz),
E = 4.136 × 10-6 eV or 6.625 × 10-25 J
At 10 GHz ( = 1010 Hz),
E = 4.136 × 10-5 eV or 6.625 × 10-24 J
Therefore at 1 GHz, a 10-mW signal contains more than
1.5 × 1022 photons
For comparison, one photon of visible light (  500 nm,  = 600 THz)
E = 2.5 eV or 4 × 10-19 J
Therefore, a 10-mW signal would contain more than 2.5 × 1016 photons
67
Relating EM band frequencies and wavelengths
to mechanisms
68
EM source mechanisms
69