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 dm dd

4. Pendry’s 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)

5. Simple analysis of 1D and 2D systems
Periodicity or inclusions much smaller than wavelength
2+1D or 1+2D (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

6. 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: parallel=ordinary
For isotropic constituents
effective fields a

7. Stratified metal-dielectric metamaterial: Parallel polarizationk a
Boundary conditions

8. Stratified metal-dielectric metamaterial: Normal polarization

9. 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: parallel=extraordinary

10. Nanowire metamaterial: Parallel polarization

11. Nanowire metamaterial: Normal polarization polarization
More complicated derivation
Homogenization (not simple averaging)
Assume small inclusions (<20%)
Maxwell-Garnett Theory (MGT)
(metal nanowires in dielectric host)

12. Strongly anisotropic dielectric Metamaterial
For most visible and IR wavelengths

13. Example: nanowire medium medium
60nm nanowire diameter
Ag wires
110nm center-center wire distance
Al2O3 matrix
Broad band
Effective permittivity from MG theory um
60nm nanowire diameter, 110nm center-center wire distance, push the broad band to shorter WL um 13

14. Wave propagation in anisotropic medium
Uniaxial
Maxwell equations for time-harmonic waves
Det(M)=0,

15. 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

16. Extraordinary waves in anisotropic mediumkz
isotropic medium
e=1 kx
e=1.5
anisotropic medium
‘Hyperbolic’ medium kz
For ex<0 kz kx kx

17. Energy flow in anisotropic mediumkz
normal to the k-surface
e=1 kx
e=1.5
‘Indefinite’ medium
anisotropic medium kz kz kx
and
and
are not parallel
are not parallel
Is normal to the curve! kx
* Complete proof in “Waves and Fields in Optoelectronics” by Hermann Haus

18. Refraction in anisotropic medium
What is refraction? kz e
e=1 kx
e=1.5
Conservation of tangential momentum kz
Hyperbolic air
Negative refraction! kx

19. Refraction in nanowire medium medium
Ag wires
Broad band
Al2O3 matrix um
Effective permittivity from MG theory
60nm nanowire diameter, 110nm center-center wire distance, push the broad band to shorter WL um
Negative refraction for l>630nm 19

20. Refraction in layered semiconductor medium
SiC
Phonon-polariton resonance at IR
60nm nanowire diameter, 110nm center-center wire distance, push the broad band to shorter WL
Negative refraction for 9>l>12mm 20

21. Hyperbolic metamaterial “phase diagram”
dielectric
Type I
Type II
Ag/TiO2 multilayer system

22. Effective medium at different regimes
We choose propogation by
X=normal
(suitable for Nanowires)
X=parallel
Suitable for stratified medium x
propagation
extreme material properties
epsilon near-zero
Diffraction management
Resolution limited by loss
Low-loss
Broad-band
resolution limited by periodicity x
propagation

23. Conditions Normal-X direction (kx<<p/D)
X=normal
(suitable for Nanowires) x
propagation kz kx
Low loss
moderate e values
Limited by periodicity
Low diffraction management
diffraction management improves with em
no near-0 e

24. Conditions for Normal Z-directionx
propagation kr kx
Good diffraction management
near-zero e
Limited by ?
For large range of kx

25. Effective medium with loss…x
propagation
(Long wavelengths)
High loss!
Very low loss at low k
Moderate loss at high k
End of class

26. 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?

27. Exact solution – transfer matrix

28. Exact solution – transfer matrix
(1) Maxwell’s equation

29. Exact solution – transfer matrix
(2) Boundary conditions

30. Exact solution – transfer matrix
(3) Combining with Bloch theorem

31. Beyond effective medium: SPP coupling in M-D-M
“gap plasmon” mode
deep sub-l “waveguide”
symmetric and anti-symmetric modes
Metal
Metal
Symmetric: k<ksingle-wg
Antisymmetric: k>ksingle-wg
Now you can bring the waveguide very close as the interaction is screened through the metal

32. Beyond effective medium: SPP coupling in M-D-Mz x
metal
dielectric
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

33. Plasmonic Bloch modes Kx=p/D Kx=0 Ag=20nm Air=30 nm l=1.5mm 1 1 0.97
Magnetic
Tangential
Electric
0.97 1 -1
Kx=p/D
Magnetic
Tangential
Electric -1 1
Ag=20nm Air=30 nm l=1.5mm

34. Metamaterials at low spatial frequencies
The homogeneous medium perspective
Averaged dielectric response
Can be <0
To make it more quantitative, we can use a different perspective for metamaterials, of homogeneous medium that applies for spatial frequencies that are much smaller than the Bloch(?) wave-vector pi/D.
For exaqmple, coupling light from air is well within this approximation and results in effective medium with new material properties. Also referred to as “indefinite medium”
Hyperbolic dispersion! 34

35. Metamaterials at low spatial frequencies
The homogeneous medium perspective
Averaged dielectric response
Can be <0
To make it more quantitative, we can use a different perspective for metamaterials, of homogeneous medium that applies for spatial frequencies that are much smaller than the Bloch(?) wave-vector pi/D.
For exaqmple, coupling light from air is well within this approximation and results in effective medium with new material properties. Also referred to as “indefinite medium”
Hyperbolic dispersion! 35

36. 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

37. The Hyperlens dm dd 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.

38. 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