Metamaterials as Effective Medium

1
Metamaterials as Effective Medium

• Negative refraction and super-resolution

2
Previously seen in optical metamaterials

• Sub-wavelength dimensions with SPP
• Negative index
• Use of sub-wavelength components to create
  effective response
• Super-resolution imaging

3
Metamaterials as sub-wavelength mixture of
different elements
When two or more constituents are mixed at
sub-wavelength dimensions Effective properties
can be applied

• New type of artificial dielectrics
• Negative refraction in non-magnetic metamaterials
• Super-resolution imaging

4
Pendrys artificial plasma

• Motivation metallic behavior at GHz frequencies
• Problem the dielectric response is negatively
  (close to) infinite
• Solution dilute the metal

The electrons density is reduced
The effective electron mass is increased due to
self inductance
Lowering the plasma frequency, Pendry, PRL,76,
4773 (1996)
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Simple analysis of 1D and 2D systems

• Periodicity or inclusions much smaller than
  wavelength
• 21D or 12D (dimensions of variations)
• Effective dielectric response determined by
  filling fraction f

2D-periodic (nano-wire aray)
1D-periodic (stratified)
3D?
a

• Averaging over the (fast) changing dielectric
  response

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Stratified metal-dielectric metamaterial

• Two isotropic constituents with bulk
  permittivities
• Filling fractions f for e1,1-f for e2
• 2 ordinary and one extra-ordinary axes (uniaxial)
• 2 effective permittivities

Note parallelordinary

• For isotropic constituents
• effective fields

a
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Stratified metal-dielectric metamaterial
Parallel polarization
E
k
a
Boundary conditions
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Stratified metal-dielectric metamaterial Normal
polarization
E
a
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Nanowire metal-dielectric metamaterial

• Two isotropic constituents with bulk
  permittivities
• Filling fractions f for e1,1-f for e2
• 2 ordinary and one extra-ordinary axes
• 2 effective permittivities

Note parallelextraordinary
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Nanowire metamaterial Parallel polarization
E
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Nanowire metamaterial Normal polarization
polarization
E

• More complicated derivation
• Homogenization (not simple averaging)
• Assume small inclusions (lt20)
• Maxwell-Garnett Theory (MGT)

(metal nanowires in dielectric host)
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Strongly anisotropic dielectric Metamaterial
For most visible and IR wavelengths
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Example nanowire medium medium
60nm nanowire diameter
Ag wires
110nm center-center wire distance
Al2O3 matrix
Effective permittivity from MG theory
um
um
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Wave propagation in anisotropic medium
Uniaxial?
Maxwell equations for time-harmonic waves
Det(M)0,
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Wave propagation in anisotropic medium
Ordinary waves (TE)
Extraordinary waves (TM)
E

• Electric field along y-direction
• does not depend on angle
• constant response of ex

H
H
E

• Electric field in x-z(y-z) plan
• Depend on angle
• combined response of ex,ez

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Extraordinary waves in anisotropic medium
kz
isotropic medium
e1
kx
e1.5
anisotropic medium
Hyperbolic medium
kz
For exlt0
kz
kx
kx
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Energy flow in anisotropic medium
isotropic medium
kz
normal to the k-surface
e1
kx
e1.5
Indefinite medium
anisotropic medium
kz
kz
kx
and
are not parallel
and
are not parallel
Is normal to the curve!
kx
Complete proof in Waves and Fields in
Optoelectronics by Hermann Haus
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Refraction in anisotropic medium
kz
What is refraction?
e
e1
kx
e1.5
Conservation of tangential momentum
kz
Hyperbolicair
Negative refraction!
kx
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Refraction in nanowire medium medium
Ag wires
Al2O3 matrix
um
Effective permittivity from MG theory
um
Negative refraction for lgt630nm
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Refraction in layered semiconductor medium

• SiC
• Phonon-polariton resonance at IR

Negative refraction for 9gtlgt12mm
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Hyperbolic metamaterial phase diagram
Ag/TiO2 multilayer system
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Effective medium at different regimes
We choose propogation by
Xparallel Suitable for stratified medium
Xnormal (suitable for Nanowires)
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Conditions Normal-X direction (kxltltp/D)
Xnormal (suitable for Nanowires)
kz
kx

• Low loss
• moderate e values
• Limited by periodicity

• Low diffraction management
• diffraction management improves with em
• no near-0 e

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Conditions for Normal Z-direction
kr
kx

• Good diffraction management
• near-zero e
• Limited by ?

For large range of kx
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Effective medium with loss
(Long wavelengths)
Very low loss at low k Moderate loss at high k
High loss!
End of class
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Limits of indefinite medium for super-resolution

• Open curve vs. close curve
• No diffraction limit!
• No limit at all
• Is it physically valid?

kr
kx

• Reason approximation to homogeneous medium!
• What are the practical limitations?
• Can it be used for super-resolution?

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Exact solution transfer matrix
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Exact solution transfer matrix
(1) Maxwells equation
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Exact solution transfer matrix
(2) Boundary conditions

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Exact solution transfer matrix
(3) Combining with Bloch theorem

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Beyond effective medium SPP coupling in M-D-M

• gap plasmon mode
• deep sub-l waveguide
• symmetric and anti-symmetric modes

Metal
Metal
Symmetric kltksingle-wg
Antisymmetric kgtksingle-wg
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Beyond effective medium SPP coupling in M-D-M

• Abrupt change of the dielectric function
• variations much smaller than the wavelength
• Paraxial approximation not valid!
• Need to start from Maxwell Equations

• TM nature of SPPs
• Calculate 3 fields ?

Hamiltonian-like operator
Eigenmode problem

• Eigen vectors ? EM field
• Eigen values ? Propagation constants

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Plasmonic Bloch modes
Ag20nm Air30 nm l1.5mm
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Metamaterials at low spatial frequencies
The homogeneous medium perspective
Averaged dielectric response
Can be lt0
Hyperbolic dispersion!
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Metamaterials at low spatial frequencies
The homogeneous medium perspective
Averaged dielectric response
Can be lt0
Hyperbolic dispersion!
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Use of anisotropic medium for far-field super
resolution
Conventional lens

• Superlens can image near- to near-field
• Need conversion beyond diffraction limit
• Multilayers/effective medium?
• Can only replicate sub-diffraction image by
  diffraction suppression
• Solution curve the space

Superlens
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The Hyperlens

• Metal-dielectric sub-wavelength layers
• No diffraction in Cartesian space
• object dimension at input a
• Dq is constant
• Arc at output

Magnification ratio determines the resolution
limit.
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Optical hyperlens view by angular momentum

• Span plane waves in angular momentum base
  (Bessel func.)

• resolution detrrmined by mode order
• penetration of high-order modes to the center is
  diffraction limited

• hyperbolic dispersion lifts the diffraction
  limit
• Increased overlap with sub-wavelength object