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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 New type of artificial dielectrics Negative refraction in non-magnetic metamaterials Super-resolution imaging dmdm d When two or more constituents are mixed at sub-wavelength dimensions Effective properties can be applied

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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 Lowering the plasma frequency, Pendry, PRL,76, 4773 (1996) The electrons density is reduced * The effective electron mass is increased due to self inductance

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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 a 1D-periodic (stratified) 2D-periodic (nano-wire aray) Averaging over the (fast) changing dielectric response 3D?

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Stratified metal-dielectric metamaterial Two isotropic constituents with bulk permittivities Filling fractions f for 1,1-f for 2 2 ordinary and one extra-ordinary axes (uniaxial) 2 effective permittivities a For isotropic constituents effective fields Note: parallel=ordinary

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

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Stratified metal-dielectric metamaterial: Normal polarization a E

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Nanowire metal-dielectric metamaterial Two isotropic constituents with bulk permittivities Filling fractions f for 1,1-f for 2 2 ordinary and one extra-ordinary axes 2 effective permittivities Note: parallel=extraordinary

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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 (<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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Effective permittivity from MG theory Al 2 O 3 matrix Ag wires Broad band um Example: nanowire medium medium 60nm nanowire diameter 110nm center-center wire distance

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

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Wave propagation in anisotropic medium Ordinary waves (TE) Extraordinary waves (TM) E H H E Electric field along y-direction does not depend on angle constant response of x Electric field in x-z(y-z) plan Depend on angle combined response of x, z

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Extraordinary waves in anisotropic medium kxkx kzkz kxkx kzkz isotropic medium anisotropic medium kxkx kzkz ‘Hyperbolic’ medium For x <0

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Energy flow in anisotropic medium kxkx kzkz kxkx kzkz isotropic medium anisotropic medium kxkx kzkz normal to the k-surface andare not parallel ‘Indefinite’ medium * Complete proof in “Waves and Fields in Optoelectronics” by Hermann Haus andare not parallel Is normal to the curve!

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Refraction in anisotropic medium What is refraction? kxkx kzkz kxkx kzkz Hyperbolic air Conservation of tangential momentum Negative refraction!

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Effective permittivity from MG theory Al 2 O 3 matrix Ag wires Broad band um Refraction in nanowire medium medium Negative refraction for >630nm

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Refraction in layered semiconductor medium SiC Phonon-polariton resonance at IR Negative refraction for 9>>12m

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

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

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Conditions Normal-X direction (k x << /D) x propagation X=normal (suitable for Nanowires) Low loss moderate values Limited by periodicity kxkx kzkz Low diffraction management diffraction management improves with em no near-0

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Conditions for Normal Z-direction x propagation kxkx krkr For large range of k x Good diffraction management near-zero Limited by ?

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Effective medium with loss… x propagation (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? kxkx krkr 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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(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 Metal Symmetric: k

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Beyond effective medium: SPP coupling in M-D-M TM nature of SPPs Calculate 3 fields Eigenmode problem: Hamiltonian-like operator: Eigen vectors EM field Eigen values Propagation constants z x metaldielectric Abrupt change of the dielectric function variations much smaller than the wavelength Paraxial approximation not valid! Need to start from Maxwell Equations

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Plasmonic Bloch modes K x = /D Magnetic Tangential Electric 1 Kx=Kx= Magnetic Tangential Electric 0.97 1 1 Ag=20nm Air=30 nm =1.5 m

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Metamaterials at low spatial frequencies The homogeneous medium perspective Averaged dielectric response Hyperbolic dispersion! Can be <0

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Metamaterials at low spatial frequencies The homogeneous medium perspective Averaged dielectric response Hyperbolic dispersion! Can be <0

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Use of anisotropic medium for far-field super resolution 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 Conventional lens Superlens

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Metal-dielectric sub-wavelength layers No diffraction in Cartesian space object dimension at input a is constant Arc at output dmdm d The Hyperlens 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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