Wave-Mechanics and Circuit Theory
for
Left Handed Maxwell Systems
Part-2

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Few Terms for Left Handed Maxwell Systems
NPV: Negative Phase Velocity
DNG: Double Negative Material
LHM: Left Handed Material
NGV: Negative Group Velocity
NRI: Negative Refractive Index
NGD: Negative Group Delay
PRI/NRI: Positive Refractive Index/Negative Refractive Index
METAMATERIALS: Composite structures made of naturally occurring materials,
specifically designed to obtain ‘anomalous’ Electromagnetic Properties, not found commonly
in nature.
LEFT HANDED MAXWELL SYSTEMS
is an Young Subject
recently born in XXI century-will make several new theories

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Right Handed Maxwell System
For
μ > 0,ε > 0
→
→
y
E
→
→
→ →
E(r, t) = E exp (− j k . r − jωt)
0
{
}
∇ × E = − jω μ H
→
→
→ →
∇ ×
H (r, t) = H exp (− j k . r − jωt)
0
{
}
H = jω ε E
→
→
→
Power flow
→
⎛
⎞
S = ⎜ E× H ⎟
→
→
x
⎝
⎠
→
→
→
→
S , k
→
k × E = + ω μ H
z
H
→
→
→
k × H = − ω ε E
k is propagation eigen mode (vector)
from which attenuation & phase
constants appear in TL theory.
→
→
k & S
are in same direction. Thus phase velocity & Poynting Vector are in same
direction. An effective medium explanation in large scales

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Left handed Maxwell System
→
For
μ < 0,ε < 0
E
→ →
E(r, t) = E exp (+ j k . r − jωt)
0
{
}
→
→
→
k × E = − ω μ H
→ →
→
→
→
H (r, t) = H exp (+ j k . r − jωt)
0
{
}
k × H
= + ω ε E
→
1
→
→
⎛
⎞
S =
⎜ E × H ⎟
→
2 ⎝
⎠
k
Power flow
→
∇ × E = + jω μ H
S
∇ × H = − jω ε E
→
Phase velocity
H
→
→
k
(hence the phase velocity)is opposite to the direction of power flow .
S
These system (LHM) support waves with phase velocity negative, or backward waves,
also are DNG Doubly Negative Material
…..AMBIDEXTEROUS……

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DNG Interpretation macroscopically
1.
We may admit that the properties of substance are actually not affected by
simultaneous change of sign of effective permeability & permittivity.
2.
It might be for permeability & permittivity to be simultaneously negative
contradicts some fundamental laws of nature. Therefore no substance of DNG
type exists.
3.
It could be admitted that substances with DNG type have some different
properties (different from substances with permeability & permittivity, positive)
Taking point 3 the result of backward wave where plane wave propagation
is, opposite to the direction of energy flow, does not follow from the wave
equation, which remains unchanged for DNG
⎛ E ⎞
⎛ E ⎞
2
2
∇
⎜
⎟ + k
⎜
⎟ = 0
⎝ H ⎠
⎝ H ⎠
Where
2
2
k
= ω μ ε
But comes individual Curl equations, can be formed by Left-Hand
→
→
→
∇ × E = + j ω μ H
k ×
E
=
− ω μ
H
∇ ×
→
→
→
H
= − j ω ε E
k ×
H
=
+ ω ε
E
Poynting vector tells direction of power flow (group-velocity) and wave vector
tells the direction of phase velocity. Opposite means “Backward-Wave”

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Is there fundamental contradiction with Maxwell,
For DNG substances?
The dot product of Poynting Vector and wave-vector is negative, for LHM
That is ;Poynting vector is
S
= E × H
S .k < 0
However, it is to be noted that there is no fundamental contradiction as
Maxwell’s equations do not indicate the dot product of the two must be
Positive, thus it is possible to have DNG and a Left Handed Maxwell System.
DNG system with and
μ < 0
give
ε < 0
s
n < 0 where
n = ±
ε μ
Negative refractive index gives interesting “ Canceling Scattering-Properties”
of other materials as gives reversal of path rays of EM signals.

---

For experimentation and studies on negative refraction-
salient points
1 . Dispersion is the basic concept. In any Wave phenomena one must
know about plot of wave number VS frequency, diagram.
ω − k
The
‘wave number’, (k) is called wave number, complex propagation-
constant, wave vector.
2. Talking about dispersion necessitates the introduction of forward
and “backward” waves. In classical electrodynamics the properties
of “backward-waves” rarely were emphasized or studied.
3.
In LHM the “back-ward” wave has risen to fame!!
4. Very often we shall have to look at the LHM phenomena both from
point of view of “field-theory” and “circuit-theory”. SRR-WA from
field theory and PLTL from circuit theory, in realizing hardware of
LHM.
5.
The building blocks in most of the cases are “resonant-elements”
much smaller in wavelength of EM Wave shining it.

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Anomalous Dispersion
This is what is seen in our negative refraction systems with LHM. The anomalous
dispersion is old phenomena, from 19th century it is observed, when absorption
spectra of various materials were studied in optical region, where variation of
refractive index close to absorption peaks show slope as negative.
n
ω
To have NGV can this negative slope be sufficient ? Not necessarily.
c
c
c k
n =
=
=
v
ω
ω
p
k
d n
c k
c
d k
c ⎛
k
d k ⎞
= −
+
=
−
+
⎜
⎟
2
d ω
ω
ω d ω
ω ⎝
ω
d ω ⎠
d ω
c
v
=
=
g
d k
d n
n + d ω
Not only slope dn/dw should be negative
but should be negatively large.
Possible to have NGV/NGD in PLTL

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Newton’s expression for electrical conductivity
The most basic is the equation of motion of Newton, for an electron of mass m and charge e,
with a damping term in electric
τ
field E is:
⎛
e
E
d v
v ⎞
v =
m
+
= e E
⎜
⎟
⎝
m
1
d t
τ ⎠
j ω + τ
With temporal variation re-written as , i.e. temporal
d / d t →
j ω
form varying as exp( jω t )
and the current density as which may be written as
J = σ E
2
N e τ
E
J = N e v =
m
1 + ω τ
Where
2
σ
N e τ
0
σ =
; σ
=
0
1 +
j ω τ
m
Maxwell’s equations are
∇ × H = J + jω D
∇ × H = J + jω ε E = jω ε
E
e f f
∇ × E = − jω B
∇ . D = ρ ; ∇ . B = 0
D = ε E , ; B = μ H
− 1 2
− 7
ε
= 8 . 8 5 × 1 0
A s / V m ; μ
= 4 π × 1 0
V s / A m
0
0

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The effective epsilon in presence of current and effective epsilon for wire medium
Write the first Maxwell equation RHS as
J +
j ω
ε E = jω ε
E
e f f
σ
with which effective is defined.
ε
But
J
=

0
E ; which when substituted above
e f f
+ ω τ
we find that
1
σ
1
0
ε
= ε +
e f f
j ω 1 +
j ω τ
a
In low frequency limit we get:
ε
= ε − jσ
/ ω
e f f
0
In high frequency limit we get :
2
2
2
2
ε
= ε (1 − ω
/ ω
) ; ω
= N e / ε m
e f f
0
p
p
0
(by taking )
ε = ε 0
An incident electric field, E parallel to the wires, separated by gap a will give
E
E a
Z
= R
+ jω L
I =
w
w
w
Z w
Average current density A/cm*cm, in this unit cell that has an area is:
Using above method of finding the effective epsilon
1
E
ε
= 1 +
ε ( R
+ jω L ) a
J
=
r
0
w
w
a v
j ω
+
ω
Defining now
( R
j
L
) a
w
w
2
ω
= 1 / ε a L
2
ω
p
0
w
, we may write
p
, where losses
ε
= 1 +
r
are characterized as time constant
2
ω
− j ( ω / τ )
w
τ
= L
/ R
w
w
w
Expression for
2
R
= a / π r σ ; L
= ( μ
/ 2 π ) l n ( 2 a / r ) − 0 . 7 5
w
w
0
w
0
[
w
]
7
a = 6 m m ; r
= 0 . 0 3 m m ; σ
= 5 . 8 × 1 0 S / m
w
0
f
= 8 . 7 3 G H z
p
− 8
τ
= 2 . 2 4 × 1 0
s
w

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