Download Electric and Magnetic Properties of Materials

Electric and Magnetic Properties of
Materials and Stealth Applications
(Chapter 7)
EC4630 Radar and Laser Cross Section
Fall 2010
Prof. D. Jenn
[email protected]
www.nps.navy.mil/jenn
November 2010
1
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Debye Model for Dielectrics (1)
The Debye model has been used to predict the interaction of EM waves with materials
since the early 1900s. Molecules are represented by the nucleus with a positive charge
center and the electron cloud, which has a negative charge center. In the absence of an
external field, the charge centers are coincident as shown below.
ELECTRON
CLOUD (-)
NUCLEUS (+)
THE CHARGE CENTERS ARE COINCIDENT
IN THE ABSENCE OF AN EXTERNAL FIELD
When an external field is applied, the charge centers separate.
The response of the

molecule is expressed in terms of a polarization
vector, P(t) .


P = ε o χ e Eext
ELECTRON
CLOUD
CHARGE
CENTER (-)
NUCLEUS (+)

Eext
Each molecule of material is essentially an oscillating dipole.
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
Debye Model for Dielectrics (2)
The separation is referred to as electronic polarization and χ e is the electric susceptibility.
It takes time for the molecules to respond to the impressed field. The time dependent form
of the polarization vector is
P (t ) =
Po e−t /τ

ε o χ e (0) Eext
where τ is the relaxation constant (about 10−15 second).
The Debye model is never seen in real materials, but it can be approached for single
particle non-interacting systems like gases. The assumptions are that all of the dipoles are
identical, independent, and relaxation times are the same. In fact, dipoles are spatially and
temporally coupled, relaxation times vary, and other types of polarization exist.
Other types of polarization:
Ionic: mutual displacement of the molecule charge centers (relaxation constant about
10−13 second)
Orientational: rotation of the molecule (relaxation constant about 10−11 second)
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
Debye Model for Dielectrics (3)
Typical behavior of the dielectric constant with frequency is shown below. The
relationship between the real and imaginary parts is not independent, but given by the
Kramers- Kronig formula. At characteristic (resonant) frequencies there is a rapid decrease
in ε r′ and sharp increase in ε r′′ .
ε′
UHF and
Microwaves
Dipolar
Infrared
Ultraviolet
Ionic
εo
Electronic
ω
ε ′′
ω
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
Nature of Magnetic Materials (1)
Accurate quantitative analysis requires quantum mechanics. A simple atomic model is the

Bohr model, where orbiting electrons are small current loops with magnetic moment m .

m
The magnetic moment is caused by:
1. electron orbit (electron orbiting the nucleus)
2. electron spin (electron spinning about its axis)
3. nuclear spin (nucleus spinning about its axis –
weak effect)
I

Bext
<

An external magnetic field Bext puts a torque on the atomic loops causing the dipoles to
align with or against the external field

The magnetization M is a measure of how theexternal magnetic field aligns the internal
dipole moments. In a linear medium M = χ m H where χ m is the magnetic susceptibility.
The permeability is defined in terms of the susceptibility




=
B µ=
H µo µr=
H µo (1 + χ m ) H
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
Nature of Magnetic Materials (2)
A broad classification of the magnetic properties of materials is as follows:
a. diamagnetic, small negative χ m
b. paramagnetic, small positive χ m
c. ferromagnetic, large positive χ m
Most materials have a very weak magnetization and can be considered non-magnetic
( µr = 1). Exceptions are materials such a iron, which have a very strong magnetization and
exhibit hysteresis.
a. Diamagnetic
 Materials
1. When Bext = 0 the net magnetic moment is zero (the spin and orbit components
cancel)
2. When Bext ≠ 0 there is a small net magnetic moment induced in a direction opposite

to Bext (negative χ m , µ r < 1 but close to 1)

3. When Bext is removed no magnetization remains
Diamagnetic materials have a negative χ m , the direction of the induced magnetic field is
opposite to the external field, but it is very small, and thus µr > 0 .
November 2010
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Department of Electrical & Computer Engineering
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Nature of Magnetic Materials (3)
b. Paramagnetic Materials

1. Spin and orbit components do not completely cancel, but the net m from atom to atom
is randomized
due to thermal agitation (thus paramagnetism
is temperature dependent)


2. When Bext ≠ 0 the dipoles align themselves with Bext (positive χ m , µ r > 1 but close
to 1) 
3. When Bext is removed almost no magnetization remains
c. Ferromagnetic Materials
1. Large dipole moments are due to electron spin
2. Groups of adjacent atoms (domains) have dipole moments similarly aligned

3. The alignment of the domains can be random (therefore no magnetization) until Bext
is applied

4. When Bext is removed a net magnetization remains
Other categories:
Ferrimagnetism: Similar to ferromagnetism, except that the domains are anti-parallel and
do not quite cancel
Anti-ferrimagnetism: The domains are anti-parallel and completely cancel
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (1)
Materials can be classified in several ways:
1. Linear or nonlinear
2. Conducting or nonconducting
3. Dispersive or nondispersive
4. Homogeneous or inhomogeneous
5. Isotropic, anisotropic, or bianisotropic1
A generalized representation of matrix constitutive relations that covers all of these cases
(Tellegen representation of a bianisotropic medium) is
 D  ε
B = 
  ζ
ξ E
 
µ   H 
where the overbar denotes a 3-element column vector and the double overbar is a 3 by 3
ε xx ε xy ε xz 
 Dx 


 
matrix. For example, D =  D y  and ε = ε yx ε yy ε yz 


D 
ε
ε
ε
 z
 zx
zy
zz 
1
The prefex bi refers to the fact that D depends on the two fields E and H (and similarly B). Anisotropic signifies that D is not parallel to
E and B is not parallel to H.
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (2)
In a bianisotropic medium D depends on both E and B, and H to both E and B. For the
time-harmonic case (i.e., phasor representation) these quantities are complex and frequency
dependent.
The vast majority of natural materials are isotropic, or perhaps have limited anisotropy (for
example, some crystals in one or two dimensions). Almost any medium that is in motion is
bianisotropic, and therefore most of the past research has dealt with wave propagation in
uniformly moving media. (For example, plasmas created by hot jet exhaust or weapon
explosions.) However, in recent years artificial materials have been constructed with
complex behaviors, some even with negative permittivity and permeability.
E and B are the fundamental quantities (as illustrated by duality), so often the Boys-Post
representation is more useful
 D  ε p α p  E 


H  =
−1  B 
   β p µ p   
These new constitutive parameters (with subscript p) are related to the original ones as
follows:
ε =
ε p −α pµ pβ p , µ =
µ p, ξ =
α p µ p , and ζ =
−µ p β p
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (3)
Maxwell’s equations require that the following relationship be satisfied:
(
)
Trace ξ µ −1 + µ −1ζ =
0
Special cases:
a. In an isotropic medium the permittivity and permeability are scalars.
b. In a homogeneous medium the parameters are independent of position.
c. In an anisotropic medium, either or both the permittivity and permeability can be a 3
by 3 matrix (or tensor).
d. If all four matrices are diagonal (i.e., reduce to scalars), the medium is biisotropic.
1. A simple medium is linear (elements are independent of field strength), isotropic (no
directionality; diagonal matrix of scalars) and homogeneous (scalars are independent of
position in the medium).
 D  ε 0   E 
 B  =  0 µ  H 
 
  
ε r
ε =0

 0
November 2010
0
εr
0
0
 µr
0  ε o and µ =  0


ε r 
 0
0
µr
0
0
0  µo

µr 
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Monterey, California
General Constitutive Parameters (4)
2. A biaxial medium has scalars on the diagonal; for example:
ε x 0 0 
1 0 0 


ε =  0 ε y 0  ε o and µ = 0 1 0  µo


0 0 ε 
0 0 1 

z
A uniaxial medium has two of the three diagonal elements the same (e.g., ε x = ε y )
3. For a biisotropic or chiral1 medium, the 3 by 3 matrices are diagonal. Hence the
constitutive relations can be written with scalars





ε
ξ
D
E
=
D εE +ξH
 
 
=
→





B
H 
=
+
ζ
µ
B
E
H
  ζ µ   
Furthermore, the off-diagonal elements can be expressed as ξ= χ − jκ and ζ= χ + jκ .
The quantity κ is the chirality parameter, and it measures the degree of “handedness” of
the medium (κ is real for a lossless medium). χ is the magneto-dielectric parameter, and if
χ ≠ 0 the medium is nonreciprocal. In a nonreciprocal medium, a permanent electric
dipole is tied to a permanent magnetic dipole by a non-electromagnetic force.
1
In the Russian literature the term gyrotropy, referring to the gyromagnetic characteristics of the medium, is often used instead of chirality.
November 2010
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Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (5)
The important issues dealing with any material are:
1. The behavior of propagating waves in the medium
2. Symmetry conditions that must be satisfied by the constitutive relations
3. Time reversal and spatial inversion
4. Applicability of reciprocity, image theory, and duality
What does it mean to have negative permeability or permittivity?
• The induced polarization and magnetization vectors must be anti-parallel (opposite) to
the original definitions.
• The susceptibilities must be sufficiently large to drive the permittivity and permeability
negative.
Examine plane wave propagation in an isotropic material

ˆ oe −γ z
E ( z ) = xE
where
=
γ jω =
µε jω µo µ r ε=
jko µ=
jko ( µ r′ − j µ r′′ )(ε r′ − jε r′′ ) ≡ α + j β
oε r
rε r
November 2010
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Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (6)
For simplicity, consider an isotropic medium ( ε r , µ r are scalars). Materials with ε r > 0 and
µ r > 0 are referred to as right-handed (RH) materials because the direction of power flow
  
2
is according to the
right-hand
rule:
(W/m
). The propagation vector is also in
W
=
E
×
H

the direction of W

ˆ
k = kk
o µrε r
If it were possible to have both negative µr and ε r (double negative, DNG), then the
direction of propagation would be given by

ˆ
k = − kk
o | µ r|| ε r |
which is given by the left-hand rule. This is called a left-handed (LH) material. The
“handedness” parameter p of a medium is given by the determinant
xˆ ⋅ eˆ
p = xˆ ⋅ hˆ
xˆ ⋅ kˆ
yˆ ⋅ eˆ
yˆ ⋅ hˆ
zˆ ⋅ eˆ
+1 for RH materials
zˆ ⋅ hˆ = 
 −1 for LH materials
ˆ
zˆ ⋅ k
yˆ ⋅ kˆ
 
 
where eˆ = E / | E | and hˆ = H / | H | are unit vectors in the directions of the fields.
Version 4 (November 2009)
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Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (7)
Implications
of
 a left-handed material:

1. W and k are in opposite directions
2. The group velocity is negative
3. The Doppler shift is reversed (an approaching source has a negative Doppler shift)
4. The index of refraction is negative relative to that of a vacuum. Snell’s law must be
amended:
sin θi p2
=
sin θt p1
µr2 ε r2
µr1ε r1
5. Convex and concave lenses change roles when rays impinge from infinity.
Example: Consider a plane wave incident on a plane interface between two media where
µ r1 , ε r1 > 0 , ε r2 = −ε r1 and µ r2 = − µ r1 (i.e., medium 2 is DNG). The reflection coefficient at
η2 − η1
the boundary=
is Γ
= 0 and sin θ t = − sin θ i , i.e., the boundary is transparent.
η2 + η1
cosθi
p1 =
− p2 = 0
− sin θi
0 − sin θi
1
0 =
1
0 cosθi
µ1, ε1
θi
ê
ĥ
kˆi
x
θt
z
− µ1, −ε1
14
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Department of Electrical & Computer Engineering
Monterey, California
Radar Absorbing Material (RAM)
RAM is generally considered to be a coating applied to a target surface to reduce its RCS.
It can also be applied to reduce
electromagnetic interference (EMI).
Desirable mechanical and electrical
properties include:
• Thin
• Lightweight
• Durable, low maintenance
• High attenuation for large RCS
From “ Recent Developments in Radar Absorbing Paints and the Zinc Oxide
Tetrapod Whisker,” by Byron T. Caudle, et al.
reduction
• Broadband
• Angle independent
Most RAM is a mixture of materials such
as polymers, carbon and other
nanoparticles or “whiskers.”
RAM applied to a ship
for EMI reduction
15
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Monterey, California
Theorems on Absorbers
Basic theorems on absorbers due to Weston, “Theory of Absorbers in Scattering,” IEEE
Trans. on Antennas & Prop., Vol. 11, No. 5, Sept. 1963:
1. If a plane electromagnetics wave is incident on a body composed of material such that
µ / µo = ε / ε o at each point, then the backscattered field is zero provided that the incidence
direction is parallel to an axis of he body about which a rotation of 90 leaves the shape of
the body, together with its material medium invariant.
2. If a plane wave is incident on a body composed of material such that the total field
components satisfy the impedance boundary condition, and if the surface is invariant under
a 90 rotation, the backscattered field is zero if the direction of incidence is along the axis of
symmetry and Z s = 1.
Significance of µ / µo = ε / ε o :
=
Γ
and Z s = 1 :
η − ηo
=
η + ηo
µ / ε − ηo
=
µ / ε + ηo
=
Γ
Zs −1
=
Zs + 1
µr / ε r − 1
→ 0
µr =ε r
µr / ε r + 1

→ 0
Z s =1
16
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Department of Electrical & Computer Engineering
Monterey, California
Plasma Absorbers and FSS
A plasma can be generated from neutral molecules that are separated into negative
electrons and positive ions by an ionization process (e.g., laser heating or spark discharge).
A Lorentz plasma is a simple model in which the electrons interact with each other only
through collective space-charge forces. The positive ions and neutral particles are much
heavier than the electrons, and therefore the electrons can be considered as moving through
a continuous stationary fluid of ions and neutrals with some viscous friction.
The propagation characteristics of electromagnetic waves in a uniform ionized
medium can be inferred from the equation of motion of a single “typical” electron. This
model would be rigorous if the ionized medium was comprised entirely of electrons that do
not interact with the background particles (neutrals and ions) and posses thermal speeds
that are negligible with respect to the phase velocity of the EM wave. Such a medium is
called a cold plasma.
In the absence of a magnetic field, the important parameters for a cold plasma are the
electron density N e electrons/m3 and the collision frequency ν /m3. For example, a
standard fluorescent bulb has N e ≈ 1011 /cm3 .
Plasma exhibits behavior of a frequency selective surface (FSS). Waves below the
critical frequency are reflected; those above the critical frequency pass. The attenuation of
the plasma can be controlled by the collision frequency.
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Dielectric Constant of Plasma
The complex relative dielectric constant of the plasma is given by
ωp
X
=
εr =
ε r′ − jε r′′ =
1−
1−
ω (ω − jν )
(1 − jZ )
2
ωp 
N ee2
ν
is the plasma frequency, and X = 
,
,=
where ω p =
m 9.0 × 10−31 kg
Z
=

mε o
ω
 ω 
(electron mass), and=
e 1.59 × 10−19 C (electron charge).
2
The real and imaginary parts of the propagation constant are the attenuation and phase
constants, respectively:
γ ≡ α + jβ =
jko µr ε r
For a plasma µr = 1. Separating into real and imaginary terms ε=
r ε r′ − jε ′′ gives
ε r′ = 1 −
Nee2
(
εomν +ω
2
2
)
and ε r′′ =
N ee2ν
(
ωε o m ν 2 + ω 2
)
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Department of Electrical & Computer Engineering
Monterey, California
Critical Frequency of Plasma
For the special case of negligible collisions, ν ≈ 0 , the corresponding propagation constant
is
γ= jko 1 −
ω 2p
ω
= jko 1 − X
2
There are three special cases of interest:
1. ω > ω p : γ is imaginary and e− j β z is a propagating wave
2. ω < ω p : γ is real and e−α z is an evanescent wave
3. ω = ω p : γ = 0 and this value of ω is called the critical frequency, ωc which defines the
boundary between propagation and attenuation of the EM wave.
The intrinsic impedance of the plasma medium is
η=
µo
ε o (ε ′ − jε ′′)
The magnitude the reflection coefficient at an infinite plane boundary between plasma and
η − ηo
.
free space, which is given by the formula Γ =
η + ηo
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Reflection From Plasma
Reflection coefficient for a plane wave normally incident on a sharp plasma/air boundary
( N e = 1× 1012 /m3, ν = 0 , dashed line in the plasma frequency, f p = 8.9 MHz). From the
figure it is evident that at frequencies below the plasma frequency, the plasma is a good
reflector. A plasma has the characteristics of a frequency selective surface (FSS).
0
-10
-20
10
20*log (|R|)
-30
-40
-50
-60
-70
-80
-90
-100
0
10
1
2
10
10
3
10
Frequency, MHz
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Attenuation by Plasma
EM waves below the plasma frequency ( ω < ω p ) are attenuated by the plasma at a rate
(
determined by the attenuation constant: E ( z ) ~ e−α z =
exp − zko X − 1
The loss in decibels per meter
(dB/m) is
{ (
0
-5
)}
20log10 exp − ko X − 1 .
Ne=1012 /m3
-10
Loss in dB/m
Loss is plotted for several
electron densities. This shows
that plasma can be a good
absorber once the EM wave
enters the plasma medium, a
characteristic that has been
exploited in the design of plasma
radar absorbing material (RAM)
for stealth applications.
)
-15
-20
Ne=1014 /m3
-25
-30
-35
-40
-1
10
0
1
10
10
2
10
Frequency, MHz
21
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Plasma RAM
Sample calculation:
For an EM wave frequency of f = 1 GHz, compute the dielectric constant and attenuation in
dB/m of a plasma with N e = 1015 /m3 and 1016 /m3 for collision frequencies of ν = 0 /s,
107 /s, and 109 /s.
A table of ε r′ − jε r′′ is given below. Also shown is the loss in dB/m for a wave propagating
through the plasma.
/s
/s
/s
(
/m3
MHz)
0 dB/m
0.012 dB/m
1.183 dB/m
(
/m3
MHz)
0 dB/m
0.263 dB/m
23.5 dB/m
22
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Plasma in a Magnetic Field
If there is a static magnetic field present, the plasma medium becomes
 anisotropic. The
permittivity matrix of a plasma in the presence of a magnetic field B = Bo zˆ is

ε=
xx ε=
yy ε o 1 +

ε=
xy
∗
ε=
yx
ωp


ωc2 − ω 2 
jω 2p (ωc / ω ) ε o
ωc2 − ω 2
 ω 2p 
=
ε zz ε o 1 − 2 
 ω 
where ωc = −eBo / m is called the cyclotron frequency. A moving electron in a static
magnetic field rotates with an angular velocity ωc , even in the absence of an EM wave. If
a wave at frequency ωc enters the medium, it is synchronized with the electron motion,
and will continue to push the electrons to higher velocities. All energy is extracted from
the wave and no propagation occurs.
Electric and magnetic fields can be used to confine the plasma.
23
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Artificial Materials (1)
In general, artificial materials refer to materials that do not occur naturally. Depending on
the breadth of the definition, composites may or may not be considered as artificial
materials. Artificial materials are built around inclusions (added structures or elements)
that are small in scale compared to the wavelengths at which the material is designed to
operate. Inclusions are generally man-made structures like rings, helices, wires, spheres,
discs, etc. They may be distributed periodically or randomly, depending upon the desired
electromagnetic properties.
Collective oscillations of electrons (plasmons) occur in conductors as well as plasmas. A
plasma can be simulated by a three dimensional array of wires. Confining electrons to thin
wires effectively enhances their mass by a factor of 104 . The effective dielectric constant
is
ε reff = 1 −
Wires
3-D grid of thin wires approximates a plasma
ω 2p

jε o a 2ω 2p 

ω ω +
2 

πσ r 

2π c 2
where ω p = 2
, r is wire radius, a is
a ln(a / r )
the grid and σ is the wire conductivity.
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Artificial Materials (2)
120
Example: effective dielectric
constant of a 3-dimensional grid
of wires1:
100
80
60
40
Wire radius: r = 10−6 m
Grid wire spacing: a = 5 mm
Conductivity:=
σ 3.65 × 107 S/m
ε r′′
20
0
ε r′
-20
-40
-60
-80
0.5
1
1.5
Frequency, GHz
2
1
Pendry, Holden and Stewart, “Extremely low
frequency plasmons in metallic mesostructures,”
Physical Review Letters, Vol. 76, No 25, June 1996,
p. 4773.
25
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Artificial Materials (3)
A coplanar ring (CPR) is an example of an inclusion that influences both the effective
(macroscopic) permittivity and permeability of a material. The magnetic field induces
currents on the rings. According to Lentz’s law, the induced currents oppose the external
field,
which is diamagnetic behavior. The
s
electric field causes charge separation,
which results in a polarization vector,
+++++
Metal rings
thus changing the permittivity.
with gaps
-----

Ei
d
r
.
kˆi
+++++

Hi
I in
I out
If the rings are arrayed in one dimension
(laid out on a plane), the material will
have an anisotropic behavior. Isotropic
properties are achieved by having 3-d
inclusions, as shown below.

----1
Composites are a mix of natural materials that are combined or
processed to obtain specific properties or characteristics.


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Artificial Materials (4)
a= 10−2 m, = 2 × 10−3 m,
Effective permeability of the CPR1
π r 2 / a2
µeff = 1 −
2 ρ
3
− 2 2
1+ j
ω r µo π ω µoCr 3
ρ = resistivity of the metal (ohms/m),
ε o  2s 
=
C =
ln   capacitance/m of two
π d
parallel strips, a = lattice spacing in
plane of rings,  = spacing between
sheets of rings
d = 10−4 m, s= 10−3 m, r= 2 × 10−3 m
40
30
20
µ r′
10
0
µ r′′

-10
a
a
-20
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
Frequency, GHz
1
Pendry, Holden, Robbins and Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. on Microwave
Theory and Techniques., Vol 47, No. 11, Nov. 1999, p.2075.
27
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Examples of Metamaterials
Left: from Markos and Soukoulis,
Transmission Studies of LH
Materials, Physical Review
Center and right: Physics Today
28
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
General Constitutive Parameters (8)
Reflection from a DNG slab, showing the phase front reversals:
Early time
Late time
(from www.fdtd.com (XFDTD software result)
29
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Cloaking Using Metamaterials
•
•
•
•
EM cloaks are used to hide objects from sensors
The field at the input plane is transferred to the output plane
To mimic free space the time delay and phase shift for all paths must be equal
DNG materials can be used to increase the phase velocity for the longer paths
Ray paths
Above: From Pendry, Schurig and Smith, Science
312, 1780, 2006
Right: Liang, et al, “The physical picture and the
essential elements of the dynamical process
for dispersive cloaking structures,” Applied Physics
Letters, 92, 131118 (2008)
30
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Artificial Materials (5)
The helix is another element used in artificial materials. The magnetic response is
maximum when the magnetic field is parallel to the helix axis. If large number of
randomly oriented small helices are added to a material, its macroscopic properties will be
isotropic (i.e., no preferred direction). For this sample χ = 0 (reciprocal) and κ = 0.44 .
Often χ is expressed in terms of a new parameter ϑ such that χ = sin ϑ .
From Lindell, Sihvola, Tretyakov, and Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media, Artech
House, 1994.
31
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Artificial Materials (6)
Example: For plane boundary between two bi-isotropic materials with complex intrinsic
impedances η1 and η2 , the reflection coefficients for the co-and cross-polarized cases are
(See reference on the previous page for details.):
η22 cos(2ϑ1 ) − η12 − 2η1η2 sin ϑ1 sin ϑ2
Γc =
η12 + η22 + 2η1η2 cos(ϑ1 + ϑ2 )
and
2η cosϑ1 (η2 sin ϑ1 − η1 sin ϑ2 )
Γ x = 22
η1 + η22 + 2η1η2 cos(ϑ1 + ϑ2 )
Example: Medium 1 is free space=
=
ϑ1 0) and medium 2 is a reciprocal
(η1 η0=
, χ1 sin
chiral medium (κ 2 ≠ 0 and=
χ 2 sin
=
ϑ2 0) , the equations reduce to the Fresnel formulas
for isotropic media
η22 − ηo2
=
Γc
=
2
ηo + η22 + 2ηoη2
and Γ x =
0.
2
−
η
η 2 − ηo
(η=
)
2
o
(η2 − ηo )(η2 + ηo ) η2 + ηo
32
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Artificial Materials (7)
Example: Dallenbach layer using bi-isotropic material the reflection coefficients for the coand cross-polarized cases are
Γc = 2
(ηo2 − η22 )sin 2 (k2t cosϑ2 ) − ηo2 cos 2 ϑ2
ηo cos 2 ϑ2 − (ηo2 + η22 )sin 2 (k2t cosϑ2 ) + jηoη2 cosϑ2 sin(2k2t cosϑ2 )
Γx = 2
2ηoη2 sin ϑ2 sin 2 (k2t cosϑ2 )
ηo cos 2 ϑ2 − (ηo2 + η22 )sin 2 (k2t cosϑ2 ) + jηoη2 cosϑ2 sin(2k2t cosϑ2 )
where the intrinsic impedance of the bi-isotropic medium is η2 , and k2 is the propagation
constant. It can be shown that when
sin(k2=
t cosϑ2 ) cosϑ2 / 1 − (η2 / η1 ) 2
Γ c =0 and Γ x =
1 (See reference on the previous page for details.) The layer acts as a
“twist polarizer.” The boundary completely reflects the incident wave in the crosspolarized component. This effectively reduces the RCS of the target when the threat radar
is linearly polarized.
33
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Self Induced Transparency (1)
The modern view is that media have a far more complex EM relaxation behavior than
previously realized. Much of this has arisen from research involved with ultra-short pulse
lasers interacting with materials. New theories have been devised. The most promising in
the Dissado-Hill model that takes all of the spatial and temporal factors into account:
• Individual polarized molecules (dipoles) have a homogeneous lifetime, To .
• In the coupled environment, the dipoles have an inhomogeneous lifetime, Tc , that can be
greater than or less than To . The inhomogeneous lifetime depends on the number of
other dipoles and their distances, as well as their relaxation times.
• Absorption of a wave passing through a material takes time. If To > Tc then energy
extracted from the wave as it passes through the material can be returned back to the
wave.
This condition is called self-induced transparency. The wave can penetrate the medium
without loss and therefore any radar absorbing material would be useless. This effect may
have been observed at optical frequencies (interpretation of the data is in question).
34
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Self Induced Transparency (2)
I
Self-induced transparency occurs when the waveform duration T satisfies To > T > Tc .
Then the wave penetrates the medium without loss. Coherency of the wave is maintained.
Energy is extracted by the dipoles from the first half of the wave. The extracted energy is
returned during the wave during the second half if the homogeneous lifetime is not
exceeded.
The area theorem is a statement of this condition: Efficient penetration of an absorbing
material occurs when the area under the energy vs. time curve of the wave in the material
satisfies ∫ Edt = 0 and ∫ | E |2 dt ≠ 0 . An example is shown below:
ENERGY EXTRACTED
DURING FIRST HALF
T
t
ENERGY RETURNED
COHERENTLY THE
SECOND HALF
New insight into the behavior of materials has given rise to the concept of “crafting”
waveforms for specific materials. That is, waveforms are designed to efficiently penetrate
a specific material.
35
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Precursors (1)
Examine the transmitted wave that has a very narrow pulse:
ENVELOPE
CARRIER
TIME
When a conventional waveform passes through a material, the waveform out of the
material is a time delayed replica of the waveform at the input. (We assume that the
waveform has a long pulse width compared to the relaxation time of the material.) The
group velocity vg is usually taken as the velocity of energy propagation in the material.
(Neglecting any distortions due to dispersion.)
CARRIER
ENVELOPE
TIME
PROPAGATION DELAY
THOUGH M ATERIAL
TIM E
REFERENCE
TIME
td
36
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Precursors (2)
The group velocity is less than the phase velocity, u p =
2π f
β
, which in turn is less than the
velocity of light in a vacuum (except for anomalous cases). Below is shown the dielectric
constant vs. frequency for a
typical material ( ε = ε ′ − jε ′′ ).
UHF and
Note that high frequencies
Microwaves
ε′
travel faster than low
Dipolar
Infrared
Ultraviolet
frequencies because
Ionic
up =
1
µε ′
.
Precursors are features in
waves transmitted through
media due the ultra-fast rise
and fall times of the pulse
envelope. They occur because
the transferal of energy is not
instantaneous.
εo
Electronic
ω
ε ′′
ω
37
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Specular RAM Design
Radar absorbing material (RAM) application takes on many different configurations. A
conventional Dallenbach layer has a thickness of λ/4. Assume that the plane wave is
normally incident. The design parameters are the thickness, t, and constitutive parameters
of the medium µr =
µr′ − j µr′′ , and ε r =
ε r′ − jε r′′
t
GROUND PLANE
µ, ε
DIELECTRIC/M AGNETIC
M ATERIAL
µ, ε
Z
t
Note that we are assuming
an isotropic material. More
specialized and advanced
designs may be comprised
of anisotropic materials.
in
TRANSM ISSION LINE
EQUIVALENT CIRCUIT
The propagation constant in this material is
=
γ jω =
µε jω µo µr ε=
j βo µ=
j βo ( µr′ − j µr′′ )(ε r′ − jε r′′ ) ≡ α + j β
oε r
rε r
where=
βo 2=
π / λ ω µoε o and λ is the free space wavelength.
38
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Specular RAM Design
There are two approaches to selecting the layer parameters, and they are discussed in
Section 7.5.4:
1. Matched characteristic impedance method:
Make µr = ε r everywhere in the material and µr′′ and ε r′′ large enough so that the
attenuation constant α provides sufficient round trip attenuation.
Dallenbach layer with equal electric and magnetic loss tangents.
39
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Specular RAM Design
2. Matched wave impedance method:
Make the wave impedance at the input of the equivalent transmission line circuit
equal to that of free space. Equivalently, make the net reflection coefficient at the
front face zero
Zin − Z o
=
Γ
= 0
Zin + Z o
where
Z + Z d tanh(γ t )
.
Zin = Z d L
Z d + Z L tanh(γ t )
and Z d = µ / ε is the impedance of the coating layer. If the backing material is a
PEC then Z L = 0 and Zin = Z d tanh(γ t ) so that
Zd
| Γ=| 0 → Z d tanh(γ t ) − Z o= 0 →
tanh(γ t=
) 1
Zo
40
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Universal Curves
There are six degrees of freedom in designing the Dallenbach layer: t , λ , µr′ , µr′′ , ε r′ and ε r′′ .
This can be reduced to four by normalizing the permittivity and permeability by t / λ
a
=
t
t
t
t
=
ε r′ b =
ε r′′ x =
µr′ y
µr′′
λ
λ
λ
λ
In terms of the new variables, the transcendental equation becomes
j
x − jy
tan 2π (a − jb)( x − jy ) =
1
a − jb
(
)
Now a set of curves can be drawn up where the abscissa is x and the ordinate is y, for
constant values of a and b. The figure on the next page (Figure 7.25 in the book) shows
such a set of curves. The loss tangents are defined in Section 7.3.1 (also see the Appendix
A.9).
41
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Universal Curves
42
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Bandwidth of RAM Treatments
We want to find the conditions on the
dielectric constant and permeability in
order to maximize the bandwidth ∆f ,
that is, the range of frequencies over
which the reflection coefficient is no
greater than a specified value, Γo .
|Γ|
1
∆f
Γo
f
fc
Consider a center frequency fc we can write and a second frequency f ′ > fc . The formula
for the reflection coefficient is (see book for details,
=
A β=
t , B β ′t )
=
Γo
B− A
2
=
1 + tan
( B)
2 tan( B )
j ( B − A) 1 − (ε r / µr ) 2π ( f ′ − fc )t | ε r − µr |
=
2λ ′
2 ε r / µr
Solving this for the bandwidth:
=
BW
2( f ′ − fc ) ∆f λc
2 | Γo |
= = =
fc
fc ∆λ π | ε r − µr | t / λ ′
43
Naval Postgraduate School
Department of Electrical & Computer Engineering
Monterey, California
Bandwidth of RAM Treatments
Now assume: 1. the layer is thin t << λc ,
2. the bandwidth small compared to the center frequency ∆λ << λc ,
3. | ε r |>>| µr | , which is true for most natural materials, and
4. 2π t ε r µr / λc ≈ π / 2 .
These assumptions give the approximation
The bandwidth of a single layer Dallenbach
layer can be increased by increasing the
32 Re( µr )t
| Γo |
∆λ ≈
permeability of the layer. Generally,
π
wideband absorbers require a high
permeability.
Typical MAGRAM specs (ARC Technologies, http://www.arc-tech.com/pdf/datasheets/MAGRAM/UD-12300-1.pdf):
Description: Flexible, broadband, carbonyl iron loaded urethane rubber based microwave absorber tuned for mode
suppression in an enclosed cavity, signal isolation and surface current attenuation. This product is electrically non-conductive.
It is provided with pressure sensitive transfer adhesive (PSA) on the back side.
Sheet Size: 24" x 24" (60.96 x 60.96cm)
Part Size: Can be die cut or waterjet cut to many Configurations.
Thickness: 0.094" +/- 0.005 (2.38mm +/- 0.127)
Color: Gray
Temperature Range: -60F to +275 (-51C to 135C)
Flammability Rating: UL94-HB (file number E204422)
Specific Gravity: 3.5 to 4.2
Far Field Reflection Loss Performance (NRL Arch): >17 dB @ 4 GHz
44