Left Handed Maxwell SystemsSAMEER Shantanu DasRR&PSReactor Control Division, B.A.R.C. Mumbai-400085shantanu@barc.gov.in
The LHM TeamSAMEER Calcutta: 1.
Dr A. L. Das (Director SAMEER)
2.
Arijit Majumder (Scientist SAMEER Principal-Investigator)
3.
Paulami Sarkar (Scientist SAMEER)
4.
Amitesh Kumar (Research Scientist SAMEER)
5.
Saugata Chaterjee (Reseach Scientist SAMEER)
University of Calcutta:Prof. Subal Kar (IRPE Calcutta Univ. CU; Consultant & Guide)
BARC:Shantanu Das (Scientist RRPS/RCND-BARC; Principal-Collaborator)
Funded by
BRNS Department of Atomic Energy under MoU between
SAMEER- CU-BARC for
three years “Setting up Programme on Left Handed Maxwell Systems” Rs. 110,00,000. Project MoU
signed on January, 2010.
General exposition & Utility ofThe Left Handed Maxwell SystemsPart-1
Wave propagation is reversedn = 1
n = −1
n = 1
No natural material (so-far) known to behave as Left Handed material.
The concept will fabricate and demonstrate this phenomena by artificially structured &
fabricated using natural materials FR4, RT-DUROID, copper wires-rings to have artificial
material “Meta-Materials”, where the effective permittivity & permeability will behave as
‘negatives'; and the refractive index as ‘negative value’.
The idea of FLAT surface focusing.
Bending of electromagnetic energy in wrong way through LHM
Poynting Vector opposite to Wave-Propagation Vector with negative phase velocity in LHM
→
Eθ = θ
−
ri→
n = 1
−
kiθ = θ
n = 1
→
→
→
⎛
⎞
S = ⎜
E ×
H ⎟
rii⎝
⎠
→
LHM-Backward waveHRegion –II (negative phase velocity)ε < 0, μ < 0
SPP-regionRegion-I (positive phase velocity)RHM-Forward waven = 1
ε > μ
iθ
0, > 0
i→
Eω
d ω
→
v=
;
v=
pgkkd k→
→
→
⎛
⎞
S = ⎜
E ×
H ⎟
→
⎝
⎠
HWith negative unity relative permittivity & permeability the interface
impedance is just of free-space, so at the interface no-reflection wave.
μ
rη = η
= η = 120π = 377 Ω
0
0
ε
rBut the refractive index is negative unity. An Ideal Case
Negative Refractive Index NRIA very preliminary explanation the detailed will be carried out later in other section
n ( ω ) =
ε ( ω ) μ ( ω )
rrIn LHM these parameters are negative for certain frequency ε
< 0 , μ < 0
rrε (ω ) < 0
( ) =
( )
jπ
ε ω
ε ω
errrμ (ω ) < 0
( ) =
( )
jπ
μ ω
μ ω
errrπ
n ( ω ) =
ε ( ω ) μ ( ω ) .
je= − ε ( ω ) μ
( ω )
rrrrLimiting to EM up to 100GHz, presently. Optical magnetism for IR and beyond not considered. Negative
n is illusive in
optical regime. The imaginary part of permittivity and permeability in those regions will be rather large, will manifest
majorly as ohmic, and radiation loss. Also at HF the sub nano structures becomes less than skin depth; whereas in uW
ranges metals can be regarded as PEC and skin depth is smaller than characteristic size of EM lattice.
μ 1 − 2
η
= η
∠ η
=
1 − 2
1 − 2
1 − 2
ε
though seems same across, perhaps needs to be re looked at?
1 − 2
There is definitely a different explanation with respect to impedance, perhaps wave impedance may also have sign change?
Materialsμ
rENGDPSε
rDNGMNG
Wave-Mechanics RevisitedThe Wave Equation obtained from Maxwell’s equation is
2
∇
2
E +
k E and
= 0
is same for
positive or negative wave-vector. Where
2
2
k= ω μ ε The solution is
.
e
jk rEE−
=
0
for quasi-static case.
EIs constant vector perpendicular to
k0
2
2
2
2
k =
k +
k +
kxyzThis is plane wave as none of its variable change in plane perpendicular to wave-vector
k = β −
jα
is complex wave-number
. β is propagation coefficient
, α is attenuation
.Wave in
z-direction is: − β −α
E=
E jzzee i.e. wave declines exponentially in the
z.
z0
This exponential decay is expected and true in all cases of forward wave. If losses are nil
then α = 0
The wave number is
k, then equal to propagation coefficient β
Wave impedance
μ
Eμ Free space wave
0
η =
=
π
rimpedance
η =
= η
1 2 0
0
0
ε
Hε
0
rPhase velocity is velocity with which single frequency wave travels is
v= ω /
k =
c /
μ ε
p hrrBut single frequency wave does not carry information, can be carrier for the information
.
Velocity of a group of frequencies that do carry information is group
velocity and is
d ω
v=
gd kFor free isotropic space
v=
cω =
p hdispersion relation is
k No
cchange in velocity
or the wave-number with the frequency. Refractive index also remains same with frequency
.n =
μ ε = 1
0
rr
Interpretation macroscopically Left Hand cross ProductFor the Left Handed Maxwell (LHM) Media 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
⎛
⎞
⎛
⎞
2
E2
E∇ ⎜
⎟ +
2
2
k⎜
⎟ = 0
k= ω μ ε
where
⎝
H ⎠
⎝
H ⎠
But comes individual Curl equations, can be formed by Left-Hand
for the LHM left handed curl equations are the following:
∇ ×
→
→
→
E= +
j ω μ
Hk ×
E= − ω μ
H∇ ×
H= −
j ω ε
E→
→
→
k ×
H= + ω ε
EPoynting vector tells direction of power flow (group-velocity) and wave vector
tells the direction of phase velocity. Opposite means “Backward-Wave”
For the RHM, wave propagation phase velocity is in direction of energy flow:
→
→
→
→
→
∇ ×
E = −
j ω μ
Hk ×
E= + ω μ
H→
→
∇ ×
→
→
→
H=
j ω ε
Ek ×
H= − ω ε
E
Traveling Waves in LHM and perfect imaging!!1 Propagating-Traveling Wave (Bulk wave)
±
jk zFor Large wave length
z2
2
For
k =
k−
kk <
k wave is:
A ez0
xx0
z = 0
z/
d= 2
z = 3
d / 2
z = 2
dn = 1
−
n = 1
n = 1
Image-PlaneObject-Planeε = 1
− ,μ = 1
−
rrxzyMedium-1 Medium-2 Medium-31.
The amplitude of propagating waves are always same as losses are neglected
2.
The phase goes forward by
k d / 2 in medium-1
z3.
The phase goes backward by in medium-2
k dz4.
The phase goes forward by
k d / 2 in medium-3
z5.
The total phase change is zero, phase is restored. Transfer function is flat for all
propagating wave-numbers, or spatial spectral frequencies.
6.
The perfect EM image is thus possible (ideally).
7.
Zero EM (optical) path is only possible with negative refractive index (NRI)
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