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Metamaterials as Effective Medium Negative refraction and super-resolution.

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Presentation on theme: "Metamaterials as Effective Medium Negative refraction and super-resolution."— Presentation transcript:

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  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

4 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

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

6 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

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

8 Stratified metal-dielectric metamaterial: Normal polarization a E

9 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

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 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

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

15 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

16 Extraordinary waves in anisotropic medium kxkx kzkz   kxkx kzkz isotropic medium anisotropic medium kxkx kzkz ‘Hyperbolic’ medium For  x <0

17 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!

18 Refraction in anisotropic medium What is refraction? kxkx kzkz    kxkx kzkz Hyperbolic air Conservation of tangential momentum Negative refraction!

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

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

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

22 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)

23 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 

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

25 Effective medium with loss… x propagation (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? kxkx krkr Reason: approximation to homogeneous medium! What are the practical limitations? Can it be used for super-resolution?

27 Exact solution – transfer matrix

28 (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 Metal Symmetric: kk single-wg “gap plasmon” mode deep sub-  “waveguide” symmetric and anti-symmetric modes

32 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

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

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

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

36 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

37 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.

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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