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Metamaterials as Effective Medium

Metamaterials as Effective Medium. Negative refraction and super-resolution. Previously seen in “optical metamaterials”. Sub-wavelength dimensions with SPP Negative index Use of sub-wavelength components to create effective response Super-resolution imaging. d d. d m.

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Metamaterials as Effective Medium

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  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. dd dm 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. 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 polarization E k a Boundary conditions

  8. Stratified metal-dielectric metamaterial: Normal polarization E a

  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 E

  11. 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)

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

  13. Broad band Example: nanowire medium medium 60nm nanowire diameter Ag wires 110nm center-center wire distance Al2O3 matrix Effective permittivity from MG theory um um

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

  15. Wave propagation in anisotropic medium Extraordinary waves (TM) Ordinary waves (TE) 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 medium kz isotropic medium e=1 kx e=1.5 anisotropic medium ‘Hyperbolic’ medium kz For ex<0 kz kx kx

  17. Energy flow in anisotropic medium isotropic 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

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

  19. Broad band Refraction in nanowire medium medium Ag wires Al2O3 matrix um Effective permittivity from MG theory um Negative refraction for l>630nm

  20. Refraction in layered semiconductor medium • SiC • Phonon-polariton resonance at IR Negative refraction for 9>l>12mm

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

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

  23. propagation x Conditions Normal-X direction (kx<<p/D) X=normal (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

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

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

  32. metal dielectric z x 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

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

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

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

  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. dd dm 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.

  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

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