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**Metamaterials as Effective Medium**

Negative refraction and super-resolution

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**Previously seen in “optical metamaterials”**

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

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**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

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**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)

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**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

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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: parallel=ordinary For isotropic constituents effective fields a

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**Stratified metal-dielectric metamaterial: Parallel polarization**

k a Boundary conditions

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**Stratified metal-dielectric metamaterial: Normal polarization**

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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: parallel=extraordinary

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**Nanowire metamaterial: Parallel polarization**

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**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)

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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 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

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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 e=1 kx e=1.5 anisotropic medium ‘Hyperbolic’ medium kz For ex<0 kz kx kx

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**Energy flow in anisotropic medium**

kz 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

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**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

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**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

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**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

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**Hyperbolic metamaterial “phase diagram”**

dielectric Type I Type II Ag/TiO2 multilayer system

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**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

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**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

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**Conditions for Normal Z-direction**

x propagation kr kx Good diffraction management near-zero e Limited by ? For large range of kx

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**Effective medium with loss…**

x propagation (Long wavelengths) High loss! Very low loss at low k Moderate loss at high k 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) Maxwell’s 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: k<ksingle-wg Antisymmetric: k>ksingle-wg Now you can bring the waveguide very close as the interaction is screened through the metal

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**Beyond effective medium: SPP coupling in M-D-M**

z 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

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**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

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**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

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**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

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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 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.

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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

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