electric and magnetic properties of materials and stealth...

November 2010 1

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

mailto:[email protected]

http://www.nps.navy.mil/jenn

November 2010 2

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).

ELECTRON CLOUD

CHARGE CENTER (-)

NUCLEUS (+)

E ext

exto eP Eε χ=

Each molecule of material is essentially an oscillating dipole.

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The separation is referred to as electronic polarization and eχ 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

ext(0)

/( )

o e E

tP t P eoε χ

τ−=

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)

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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 rε′ and sharp increase in rε′′ .

ε′

ε ′′

ω

ω

UHF andMicrowaves

Infrared UltravioletDipolar

Electronic

Ionic

oε

November 2010 5


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 .

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)

extBm

<I

An external magnetic field extB

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 mM Hχ=

where mχ is the magnetic susceptibility. The permeability is defined in terms of the susceptibility

(1 )o r o mB H H Hµ µ µ µ χ= = = +

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A broad classification of the magnetic properties of materials is as follows:

a. diamagnetic, small negative mχ b. paramagnetic, small positive mχ c. ferromagnetic, large positive mχ

Most materials have a very weak magnetization and can be considered non-magnetic ( 1rµ = ). Exceptions are materials such a iron, which have a very strong magnetization and exhibit hysteresis.

a. Diamagnetic Materials

1. When 0ext =B

the net magnetic moment is zero (the spin and orbit components cancel)

2. When 0ext ≠B

there is a small net magnetic moment induced in a direction opposite to extB

(negative mχ , 1<rµ but close to 1) 3. When extB

is removed no magnetization remains

Diamagnetic materials have a negative mχ , the direction of the induced magnetic field is opposite to the external field, but it is very small, and thus 0rµ > .

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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 0ext ≠B

the dipoles align themselves with extB

(positive mχ , 1>rµ but close to 1)

3. When extB

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 extB

is applied

4. When extB

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 8


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 EB H

ε ξ

ζ µ

=

where the overbar denotes a 3-element column vector and the double overbar is a 3 by 3

matrix. For example, x

y

z

DD D

D

=

and xx xy xz

yx yy yz

zx zy zz

ε ε ε

ε ε ε ε

ε ε ε

=

1The 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.

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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

1p p

p p

D EH B

ε α

β µ −

=

These new constitutive parameters (with subscript p) are related to the original ones as follows:

, , , and p p p p p p p p pε ε α µ β µ µ ξ α µ ζ µ β= − = = = −

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Maxwell’s equations require that the following relationship be satisfied:

( )1 1Trace 0ξ µ µ ζ− −+ = 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).

00

D EB H

εµ

=

0 00 00 0

r

r o

r

εε ε ε

ε

=

and 0 0

0 00 0

r

r o

r

µµ µ µ

µ

=

November 2010 11



2. A biaxial medium has scalars on the diagonal; for example: 0 0

0 0

0 0

x

y o

z

εε ε ε

ε

=

and 1 0 00 1 00 0 1

oµ µ =

A uniaxial medium has two of the three diagonal elements the same (e.g., x yε ε= )

3. For a biisotropic or chiral1 medium, the 3 by 3 matrices are diagonal. Hence the constitutive relations can be written with scalars

D E D E HB H B E H

ε ξ ε ξζ µζ µ

= + = → = +

Furthermore, the off-diagonal elements can be expressed as jξ χ κ= − and jζ χ κ= + . 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

0χ ≠ 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.

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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( ) zoE z xE e γ−=

where ( )( )o r o r o r r o r r r rj j jk jk j j jγ ω µε ω µ µ ε ε µ ε µ µ ε ε α β′ ′′ ′ ′′= = = = − − ≡ +

Version 4 (November 2009) 13



For simplicity, consider an isotropic medium ( ,r rε µ are scalars). Materials with 0rε > and

0rµ > are referred to as right-handed (RH) materials because the direction of power flow is according to the right-hand rule: W E H= ×

(W/m2). The propagation vector is also in the direction of W

ˆ

o r rk kk µ ε=

If it were possible to have both negative rµ and rε (double negative, DNG), then the direction of propagation would be given by

ˆ | || |o r rk kk µ ε= −

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

ˆ ˆ ˆ ˆ ˆˆ1 for RH materialsˆ ˆ ˆˆ ˆ ˆ1 for LH materials

ˆ ˆ ˆˆ ˆ ˆ

x e y e z e

p x h y h z h

x k y k z k

⋅ ⋅ ⋅+

= ⋅ ⋅ ⋅ = −⋅ ⋅ ⋅

where ˆ / | |e E E=

and ˆ / | |h H H=

are unit vectors in the directions of the fields.

14



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:

2 2

1 1

2

1

sinsin

r ri

t r r

pp

µ εθθ µ ε

=

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 1 1, 0r rµ ε > ,

2 1r rε ε= − and 2 1r rµ µ= − (i.e., medium 2 is DNG). The reflection coefficient at

the boundary is 2 1

2 10η η

η η−

Γ = =+

and sin sint iθ θ= − , i.e., the boundary is transparent.

1 2

cos 0 sin0 1 0 1

sin 0 cos

i i

i i

p pθ θ

θ θ

−= − = =

−

e

h ik

1 1,µ ε

1 1,µ ε− −iθtθ

x

z

15


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

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

From “Recent Developments in Radar Absorbing Paints and the Zinc Oxide Tetrapod Whisker,” by Byron T. Caudle, et al.

16


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

/ /o oµ µ ε ε= 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 1sZ = .

Significance of / /o oµ µ ε ε= : / 1/ 0

/ / 1 r r

r ro o

o o r rµ ε

µ εη η µ ε ηη η µ ε η µ ε =

−− −Γ = = = →

+ + +

and 1sZ = :

11 01 s

sZs

ZZ =

−Γ = = →

+

17


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 eN electrons/m3 and the collision frequency ν /m3. For example, a standard fluorescent bulb has 11 310 /cmeN ≈ .

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.

18


Dielectric Constant of Plasma

The complex relative dielectric constant of the plasma is given by

( ) ( )

21 1

1p

r r rXj

jZ jω

ε ε εω ω ν

′ ′′= − = − = −− −

where 2

ep

o

N em

ωε

= is the plasma frequency, and 2

pXωω

=

, Z ν

ω= , 319.0 10m −= × kg

(electron mass), and 191.59 10e −= × C (electron charge).

The real and imaginary parts of the propagation constant are the attenuation and phase constants, respectively:

o r rj jkγ α β µ ε≡ + =

For a plasma 1rµ = . Separating into real and imaginary terms r r jε ε ε′ ′′= − gives

( )22

21

ωνεε

+−=′

m

eN

o

er and

( )2

2 2e

ro

N em

νεωε ν ω

′′ =+

19


Critical Frequency of Plasma

For the special case of negligible collisions, 0ν ≈ , the corresponding propagation constant

is 2

21 1po ojk jk X

ωγ

ω= − = −

There are three special cases of interest: 1. pω ω> : γ is imaginary and j ze β− is a propagating wave

2. pω ω< : γ is real and ze α− is an evanescent wave 3. pω ω= : 0γ = and this value of ω is called the critical frequency, cω which defines the boundary between propagation and attenuation of the EM wave. The intrinsic impedance of the plasma medium is

( )o

o jµη

ε ε ε=

′ ′′−

The magnitude the reflection coefficient at an infinite plane boundary between plasma and

free space, which is given by the formula o

o

η ηη η

−Γ =

+.

20


Reflection From Plasma

Reflection coefficient for a plane wave normally incident on a sharp plasma/air boundary ( 121 10eN = × /m3, 0ν = , dashed line in the plasma frequency, 8.9pf = 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).

100

101

102

103

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency, MHz

20*lo

g 10(|R

|)

21


Attenuation by Plasma

EM waves below the plasma frequency ( pω ω< ) are attenuated by the plasma at a rate

determined by the attenuation constant: ( )( ) ~ exp 1zoE z e zk Xα− = − −

The loss in decibels per meter (dB/m) is

( ){ }1020log exp 1ok X− − .

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.

10-1

100

101

102

-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency, MHz

Loss

in d

B/m

Ne=1012 /m3

Ne=1014 /m3

22


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 1510eN = /m3 and 1610 /m3 for collision frequencies of 0ν = /s,

710 /s, and 910 /s. A table of r rjε ε′ ′′− 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

23


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 ˆoB B z=

is

( )

2 2

2

2 2

2

2

1

/

1

pxx yy o

c

p c oxy yx

c

pzz o

j

ωε ε ε

ω ω

ω ω ω εε ε

ω ω

ωε ε

ω

∗

= = +

−

= =−

= −

where /c oeB mω = − is called the cyclotron frequency. A moving electron in a static magnetic field rotates with an angular velocity cω , even in the absence of an EM wave. If a wave at frequency cω 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.

24


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 410 . The effective dielectric constant is

Wires

3-D grid of thin wires approximates a plasma

eff

2

2 2

2

1 pr

o pj a

r

ωε

ε ωω ω

πσ

= − +

where 2

22ln( / )p

ca a r

πω = , r is wire radius, a is

the grid and σ is the wire conductivity.

25



0.5 1 1.5 2-80

-60

-40

-20

0

20

40

60

80

100

120

rε′

rε′′

Frequency, GHz

Example: effective dielectric constant of a 3-dimensional grid of wires1: Wire radius: 610r −= m Grid wire spacing: 5a = mm Conductivity: 73.65 10σ = × S/m

1 Pendry, Holden and Stewart, “Extremely low frequency plasmons in metallic mesostructures,” Physical Review Letters, Vol. 76, No 25, June 1996, p. 4773.

26



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,

d

s

r

Metal ringswith gaps

+ + + + +

+ + + + +

- - - - -

- - - - -

.outI

inI

ik

iE

iH

d

s

r

Metal ringswith gaps

+ + + + ++ + + + +

+ + + + ++ + + + +

- - - - -- - - - -

- - - - -

..outI

inI

ik

iE

iH

1Composites are a mix of natural materials that are combined or processed to obtain specific properties or characteristics.

which is diamagnetic behavior. The electric field causes charge separation, which results in a polarization vector, thus changing the permittivity. 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.

27



Effective permeability of the CPR1

2 2

eff

2 2 3

/1 2 31o o

r a

jr Cr

πµ ρω µ π ω µ

= −+ −

ρ = resistivity of the metal (ohms/m), 2lno sCd

επ

= =

capacitance/m of two

parallel strips, a = lattice spacing in plane of rings, = spacing between sheets of rings

aa

2 3

4 3 3

10 m, 2 10 m,

10 m, 10 m, 2 10 m

a

d s r

− −

− − −

= = ×

= = = ×

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15-20

-10

0

10

20

30

40

rµ′

rµ′′

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.

28


Examples of Metamaterials

Left: from Markos and Soukoulis, Transmission Studies of LH Materials, Physical Review Center and right: Physics Today

29



Reflection from a DNG slab, showing the phase front reversals:

Early time Late time

(from www.fdtd.com (XFDTD software result)

30


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 pathsRay 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)

31



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 0χ = (reciprocal) and 0.44κ = . Often χ is expressed in terms of a new parameter ϑ such that sinχ ϑ= .

From Lindell, Sihvola, Tretyakov, and Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media, Artech House, 1994.

32



Example: For plane boundary between two bi-isotropic materials with complex intrinsic impedances 1η and 2η , the reflection coefficients for the co-and cross-polarized cases are (See reference on the previous page for details.):

2 22 1 1 1 2 1 2

2 21 2 1 2 1 2

cos(2 ) 2 sin sin2 cos( )c

η ϑ η η η ϑ ϑη η η η ϑ ϑ

− −Γ =

+ + +

and 2 1 2 1 1 22 21 2 1 2 1 2

2 cos ( sin sin )2 cos( )x

η ϑ η ϑ η ϑη η η η ϑ ϑ

−Γ =

+ + +

Example: Medium 1 is free space 1 0 1 1( , sin 0)η η χ ϑ= = = and medium 2 is a reciprocal chiral medium 2( 0κ ≠ and 2 2sin 0)χ ϑ= = , the equations reduce to the Fresnel formulas for isotropic media

( )( )( )

22 222 2

2 22 2 22 22

oo oc

o o oo o

η ηη η η ηη η η η η ηη η η η

−− −Γ = = =

− + ++ +

and 0xΓ = .

33



Example: Dallenbach layer using bi-isotropic material the reflection coefficients for the co-and cross-polarized cases are

2 2 2 2 22 2 2 2

2 2 2 2 22 2 2 2 2 2 2 2

( )sin ( cos ) coscos ( )sin ( cos ) cos sin(2 cos )

o oc

o o o

k tk t j k t

η η ϑ η ϑη ϑ η η ϑ η η ϑ ϑ

− −Γ =

− + +

22 2 2 2

2 2 2 2 22 2 2 2 2 2 2 2

2 sin sin ( cos )cos ( )sin ( cos ) cos sin(2 cos )

ox

o o o

k tk t j k t

η η ϑ ϑη ϑ η η ϑ η η ϑ ϑ

Γ =− + +

where the intrinsic impedance of the bi-isotropic medium is 2η , and 2k is the propagation constant. It can be shown that when

22 2 2 2 1sin( cos ) cos / 1 ( / )k t ϑ ϑ η η= −

0cΓ = and 1xΓ = (See reference on the previous page for details.) The layer acts as a

“twist polarizer.” The boundary completely reflects the incident wave in the cross-polarized component. This effectively reduces the RCS of the target when the threat radar is linearly polarized.

34


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).

35


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

ENERGY RETURNED COHERENTLY THE

SECOND HALF

T t

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.

36


Precursors (1)

Examine the transmitted wave that has a very narrow pulse:

CARRIERENVELOPETIME

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

ENVELOPETIME

TIME

TIME REFERENCE

PROPAGATION DELAY THOUGH MATERIAL

td

37


Precursors (2)

The group velocity is less than the phase velocity, 2p

fu πβ

= , 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 ( εεε ′′−′= j ). Note that high frequencies travel faster than low frequencies because

up =1µ ′ ε

.

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.

ε′

ε ′′

ω

ω

UHF andMicrowaves

Infrared UltravioletDipolar

Electronic

Ionic

oε

38


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 , and r r r r r rj jµ µ µ ε ε ε′ ′′ ′ ′′= − = −

t

GROUND PLANE

µ, ε

DIELECTRIC/MAGNETIC MATERIAL

t

µ, ε

Zin

TRANSMISSION LINE EQUIVALENT CIRCUIT

Note that we are assuming an isotropic material. More specialized and advanced designs may be comprised of anisotropic materials.

The propagation constant in this material is

( )( )o r o r o r r o r r r rj j j j j j jγ ω µε ω µ µ ε ε β µ ε β µ µ ε ε α β′ ′′ ′ ′′= = = = − − ≡ +

where 2 /o o oβ π λ ω µ ε= = and λ is the free space wavelength.

39


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 r rµ ε= everywhere in the material and rµ′′ and rε ′′ large enough so that the attenuation constant α provides sufficient round trip attenuation.

Dallenbach layer with equal electric and magnetic loss tangents.

40


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

inin

0oo

Z ZZ Z

−Γ = =

+

where

intanh( )tanh( )

L dd

d L

Z Z tZ ZZ Z t

γγ

+=

+.

and /dZ µ ε= is the impedance of the coating layer. If the backing material is a PEC then 0LZ = and in tanh( )dZ Z tγ= so that

| | 0 tanh( ) 0 tanh( ) 1dd o

o

ZZ t Z tZ

γ γΓ = → − = → =

41


Universal Curves

There are six degrees of freedom in designing the Dallenbach layer: , , , , and r r r rt λ µ µ ε ε′ ′′ ′ ′′ . This can be reduced to four by normalizing the permittivity and permeability by /t λ

r r r rt t t ta b x yε ε µ µλ λ λ λ

′ ′′ ′ ′′= = = =

In terms of the new variables, the transcendental equation becomes

( )tan 2 ( )( ) 1x jyj a jb x jya jb

π−− − =

−

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).

42


Universal Curves

43


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, oΓ .

| |Γ

oΓ

1

f

f∆

cfConsider a center frequency cf we can write and a second frequency cf f′ > . The formula for the reflection coefficient is (see book for details, ,A t B tβ β′= = )

2 ( ) 1 ( / ) 2 ( ) | |1 tan ( )2 tan( ) 22 /

r r c r ro

r r

j B A f f tB A BB

ε µ π ε µλε µ

− − ′ − −−Γ = + = =

′

Solving this for the bandwidth:

2( ) 2 | || | /

c c oc c r r

f f fBWf f t

λλ π ε µ λ

′ − Γ∆= = = =

′∆ −

44


Bandwidth of RAM Treatments

Now assume: 1. the layer is thin ct λ<< ,

2. the bandwidth small compared to the center frequency cλ λ∆ << , 3. | | | |r rε µ>> , which is true for most natural materials, and 4. 2 / / 2r r ctπ ε µ λ π≈ .

These assumptions give the approximation

32Re( ) | |r

otµλ

π∆ ≈ Γ

The bandwidth of a single layer Dallenbach layer can be increased by increasing the permeability of the layer. Generally, 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

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