Left-handed materials:
Transfer matrix method studies
Peter Markos and C. M. Soukoulis
Outline of Talk
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What are Metamaterials?
An Example: Left-handed Materials
Results of the transfer matrix method
Negative n and FDTD results
New left-handed structures
Applications/Closing Remarks
What is an Electromagnetic Metamaterial?
¾A composite or structured
material that exhibits
properties not found in
naturally occurring materials
or compounds.
¾Left-handed materials have
electromagnetic properties that
are distinct from any known
material, and hence are
examples of metamaterials.
Electromagnetic Metamaterials
Example: Metamaterials based on repeated
cells…
Material Properties
¾ All EM phenomena can be explained by Maxwell’s
four Equations.
¾ When materials are present, we average over the
detailed properties to arrive at two parameters: m and e.
¾ Quantities e and m are typically positive.
¾ When e and m are both negative, there are dramatically
new properties.
Why create metamaterials?
• The electromagnetic response of naturally
occurring materials is limited. Metamaterials
can extend the material properties, while
simultaneously providing other advantageous
properties (e.g., strength, thermal, conformal)
• Possibility of materials with “ideal”
electromagnetic response over broad frequency
ranges, that can be customized, made active, or
made modulable or tunable.
Electromagnetic waves interact
with materials…
•
•
•
•
•
•
•
Lenses
Gratings
Modulators
Switches
Antennas
Multiplexers
Sources
Related Terms
• Artificial Dielectrics
• Effective Medium Theory: Connects the
macroscopic electromagnetic properties of a
composite to its constituent components, whether
atoms, molecules, or larger scale sub-composites.
Results in e and m.
• Photonic Band Gap Periodic structures,
periodicity is on the order of the wavelength.
• Metamaterials
What’s New?
• Advances in Computational Ability
• Advances in Fabrication Ability
• Market Needs
An Example: Left-handed Materials
• Predict structure and properties
• Simulate and optimize structures
• Fabricate structures
• Measure and confirm properties
• New material properties!
Veselago
We are interested in how waves propagate through various
media, so we consider solutions to the wave equation.
e,m space
(-,+)
k = w em
(-,-)
2
∂
E
2
— E = em 2
∂t
n = em
(+,+)
(+,-)
Sov. Phys. Usp. 10, 509 (1968)
Left-Handed Media: Introduction
In 1964, V. G. Veselago contemplated the consequences
of a medium having simultaneously negative permeability
and permittivity. Out of the three possibilities:
1. Such a condition is precluded by other physical laws.
2. Such a condition is possible, but has no effect on wave
propagation, since em is positive.
3. If such a condition occurs, it makes a considerable
difference in wave propagation!
Out of the three possibilities, it is the third that is
predicted.
Fundamental constraints on e and m
While we may observe a certain range of values for
the material parameters, the only fundamental
limit seems to be set by causality, which imposes
the∂(conditions:
we r )
e r (w Æ • ) = e / e 0 Æ +1
>1
∂w
∂(wmr )
>1
∂w
mr (w Æ • ) = m / m0 Æ +1
There appears to be no restriction on negative material
constants, other than that frequency dispersion is required.
Left-Handed Media: Introduction
In the propagation of electromagnetic waves, the
direction of energy flow is given by a right-hand rule,
involving E, H, and S:
S= E¥H
The propagation, or “phase” velocity, is usually also
determined by a right hand rule:
∂B
—¥E= ∂t
k ~ E ¥ B = mE ¥ H
Thus, when e<0 and m<0, the medium is Left-Handed!
Left-Handed Media: Consequences
Group and phase velocities reversed.
A Left-Handed medium can be interpreted as having a
negative refractive index, n.
sin q I = ± n sin qT
n= e◊ m
Snell’s law still holds, but the n must be interpreted
accordingly:
Unusual and non-intuitive geometrical optics result!
Negative Refractive Index
Fw 0 2
meff (w ) = 1 - 2
w - w 02 - iwG
w 2p
e eff (w ) = 1 - 2
w
1
n(w ) =
w
(w
2
(
- w b2 ) w 2 - w 2p
(w
2
- w 02 )
)
“Reversal” of Snell’s Law
Right Handed Media
kT=nkI
Left Handed Media
Focusing in a Left-Handed Medium
RH
RH
RH
RH
LH RH
n=1
n=1.3
n=1
n=1
n=-1 n=1
Metamaterials Extend Properties
w 2p
e (w ) = 1 - 2
w
J. B. Pendry
w 2p
m(w ) = 1 - 2
w - w 02
First Left-Handed Test Structure
UCSD, PRL 84, 4184 (2000)
Transmitted Power (dBm)
Transmission Measurements
Wires alone
Split rings alone
m>0
e<0
m<0
e<0
m>0
e<0
e<0
Wires alone
4.5
5.0
5.5
6.0
Frequency (GHz)
6.5
7.0
UCSD, PRL 84, 4184 (2000)
A 2-D Isotropic Structure
UCSD, APL 78, 489 (2001)
Example of Utility of Metamaterial
The transmission coefficient
is an example of a quantity
that can be determined
simply and analytically, if the
bulk material parameters are
known.
exp(- ikd )
ts =
1ˆ
1Ê
z + sin( nkd )
cos( nkd ) z¯
2Ë
z=
m
e
n = me
2
w ep
e (w ) = 1 - 2
w - w e20 + iGe0
2
w mp
m(w ) = 1 - 2
w - w m2 0 + iGm 0
UCSD and ISU, PRB, 65, 195103 (2002)
Measurement of Refractive Index
UCSD, Science 292, 77 2001
Measurement of Refractive Index
UCSD, Science 292, 77 2001
Measurement of Refractive Index
UCSD, Science 292, 77 2001
Transfer matrix is able to find:
• Transmission (p--->p, p--->s,…) p polarization
• Reflection (p--->p, p--->s,…)
s polarization
• Both amplitude and phase
• Absorption
Some technical details:
• Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24
• Length of the sample: up to 300 unit cells
• Periodic boundaries in the transverse direction
• Can treat 2d and 3d systems
• Can treat oblique angles
• Weak point: Technique requires uniform discretization
Structure of the unit cell
EM wave propagates in the z -direction
Periodic boundary conditions
are used in transverse directions
Polarization: p wave: E parallel to y
s wave: E parallel to x
For the p wave, the resonance frequency
interval exists, where with Re meff <0, Re eeff<0
and Re np <0.
For the s wave, the refraction index ns = 1.
Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm
Typical permittivity of the metallic components: emetal = (-3+5.88 i) x 105
Structure of the unit cell:
r
w
EM waves propagate
in the z-direction.
Periodic boundary
conditions are used
in the xy-plane
dc
SRR
g
LHM
Left-handed material: array of SRRs and wires
10
[GHz]
9
8
0
Resonance frequency
as a function of
metallic permittivity
7
6
V complex em
5
1000
10000
|
m|
{ Real em
Dependence of LHM peak on metallic permittivity
Transmission
10
0
10
−2
10
−4
10
−6
10
−8
10
−10
10
−12
−588000
−300000+588000 i
1+588000 i
1 +168000 i
1 + 68000 i
−38000 + 38000 i
1+38000 i
1+18000 i
6
7
8
9
Frequency [GHz]
The length of the system is 10 unit cells
10
Dependence of LHM peak on metallic permittivity
Transmission peak
Dependence of LHM peak on L and Im em
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
−7
−588000
−300000+588000 i
1+588000 i
1+168000 i
1+68000 i
1+38000 i
1+18000 i
1+8000 i
−8
10
0
1
2
3
4
5
6
7
8
Length of the system
9
10 11
Transmission depends on the orientation of SRR
10
0
10
0
# of unit cells
# of unit cells:
−2
10
−4
10
−6
10
−8
10
−10
10
−12
N=5
N=10
N=15
N=20
10
Transmission
Transmission
10
Re(n)<0
N=5
N=10
N=15
N=20
−2
10
−4
10
−6
10
−8
SRR:
3x3x3 mm
Re(n)<0
Df=0.8GHz
7.5
8
8.5
9
9.5
Frequency [GHz]
Df=1.2GHz
10
10.5
8
9
10
11
Frequency [GHz]
12
13
Transmission properties depend on the orientation of the SRR:
¾Lower transmission
¾Narrower resonance interval
¾Lower resonance frequency
¾Higher transmission
¾Broader resonance interval
¾Higher resonance frequency
Determination of effective refraction index from
transmission studies
3
emetal=(−3+5.88 i)x10
5
4
emetal=(−3+5.88 i)x10
3
1
Im (n) = 10
0
−1
−2
−3
Re n
Im n
−4
−5
Re(n) < 0
8
8.5
9
Frequency [GHz]
9.5
10
5
Size of the unit cell:
3.3 x 3.67 x 3.67 mm
−2
Refraction index
Refraction index
2
2
1
−2
Im (n) = 10
0
−1
−2
−3
−4
Re n
Im n
Re(n) <0
9
9.5
10
10.5
11
11.5
12
Frequency [GHz]
From transmission and reflection data, the index of refraction n was calculated.
Frequency interval with Re n<0 and very small Im n was found.
How do the Left-handed EM waves look like?
0.2
1.0
f=11 GHz n=−0.378 + 0.008 i
0.0
−0.1
−0.2
200
225
250
275
300
System length [#unit cells]
Real part of the transmission
Real part of the transmission
f=9.8 GHz n=−3.26 + 0.0155 i
0.1
0.5
0.0
−0.5
−1.0
0
100
200
300
System length [#unit cells]
1.0
Real part of the transmission
f=10.5 GHz n=−1.31 + 0.005 i
0.5
0.0
−0.5
−1.0
200
225
250
275
System length [#unit cells]
300
Propagation through 300
unit cells was simulated.
The system length is 1.1 m.
Transmission in the LH
material is very good.
Length dependence of the transmission
Transmission
10
0
10
−2
10
−4
10
−6
10
−8
10
−10
10
−12
8.00
8.50
8.70
9.00
9.30
9.50
10.00
0
5
10
15
System length [# of unit cells]
20
Outside the resonance interval,
transmission tpp decreases
exponentially as the length
of the system increases.
However, it never decreases
below a certain limit. This is
due to the non-zero transmission
of the p wave into the s wave:
tpp=tpp(0)+tps tss tsp + …
The first term, tpp(0) decreases exponentially, the second term does
not depend on the system length.
Anisotropy is important outside the LH interval
1
f=9.7 GHz
0.5
p wave
0
−0.5
Real (t)
−1
1
p wave x 100
0.5
0
−0.5
p wave has the frequency of s wave!
−1
1
0.5
s wave
0
−0.5
−1
0
10
20
30
40
50
System length [# unit cells]
Transmission tpp decreases
exponentially only for short
system length. For longer
systems, tpp=const.
There is nonzero transmission
tps of the p wave into the s
wave and back.
The s wave propagates through
the structure without decay.
Re ns =1, Im ns=0
tpp=tpp(0)+tps tss tsp
Therefore, only data for short
systems are relevant for the
estimation of refraction
index of the p wave.
Dependence on the incident angle
10
0
10
0
−2
10
−4
10
−6
10
−8
10
−10
10
−12
Incident angle [p]
LHM, 0
LHM, 0.05
LHM, 0.10
SRR, 0
SRR, 0.05
8
8.5
9
9.5
Frequency [GHz]
10
Transmission
Transmission
Incident angle [p]
10
10.5
Transmission peak does not depend
on the angle of incidence !
This structure has an additional
xz - plane of symmetry
10
−2
10
−4
10
−6
10
−8
10
LHM, 0
LHM, 0.05
LHM, 0.15
−10
9
10
11
12
Frequency [GHz]
13
Transition peak strongly depends
on the angle of incidence.
Dependence of the LHM T peak on the Im eBoard
0
10
−2
10
−4
10
−6
10
−4
10
−8
10
10
In our simulations, we have:
0
10
Transmission
Transmission peak
10
Re eBoard=3.4
−12
6
7
8
Frequency [GHz]
9
−8
10
−3
10
−2
10
Im eBoard
−1
10
0
Periodic boundary condition,
therefore no losses due to
scattering into another direction.
Very high Im emetal therefore
very small losses in the metallic
components.
Losses in the dielectric board are crucial for the transmission
properties of the LH structures.
Another 1D left-handed structure:
0
−2
10
−4
10
−6
10
−8
10
0
10
−2
10
−4
10
−6
10
−8
Unit cell:
10
−10
10
−12
10
−14
Transmission
Transmission
10
10
5x3.3x5 mm
SRR
LHM
6
7
8
Frequency [GHz]
9
emetal=(−3+5.88)x10
5
SRR
LHM
10
10
−10
8
8.5
9
9.5
Frequency [GHz]
10
Both SRR and wires are located on the same side of the dielectric board.
Transmission depends on the orientation of SRR.
Photonic Crystals with negative refraction.
Photonic Crystals with negative refraction.
Photonic
Crystal
vacuum
FDTD simulations were used to study the time evolution of an EM
wave as it hits the interface vacuum/photonic crystal.
Photonic crystal consists of an hexagonal lattice of dielectric rods
with e=12.96. The radius of rods is r=0.35a. a is the lattice constant.
Photonic Crystals with negative refraction.
t0=1.5T
T=l/c
Photonic Crystals with negative refraction.
Photonic Crystals with negative refraction.
Photonic Crystals with negative refraction.
Photonic Crystals: negative refraction
The EM wave is trapped temporarily at the interface and after a long time,
the wave front moves eventually in the negative direction.
Negative refraction was observed for wavelength of the EM wave
l= 1.64 – 1.75 a (a is the lattice constant of PC)
Conclusions
• Simulated various structures of SRRs & LHMs
• Predicted how resonance transmission frequency depends on
the parameters of the system
• Calculated transmission, reflection and absorption
• Calculated meff and eeff and refraction index (with UCSD)
• Analyzed transmission properties of 2d and 3d structures
• Found negative refraction in photonic crystals
DOE, DARPA, NSF
Publications:
ƒ P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002)
ƒ P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002)
ƒ D. R. Smith, S. Schultz, P. Markos and C.M.Soukoulis, Phys. Rev. B 65, 195104 (2002)
ƒ M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, Appl. Phys. Lett. (2002)
ƒ P. Markos, I. Rousochatzakis and C. M. Soukoulis, submitted (2002)
ƒ S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, submitted (2002)
Metamaterials: New Material
Responses for Applications
¾Compact cell phone antennas
¾Improvements to base station antenna
¾Multiple degree-of-freedom antennas for MIMO
¾Active materials for beam steering/smart antennas (SDMA)
¾Phased array antennas
¾Military applications