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Metamaterials as Effective Medium

Metamaterials as Effective Medium. Negative refraction and super-resolution. Strongly anisotropic dielectric Metamaterial. For most visible and IR wavelengths. Limits of hyperbolic medium for super-resolution. Open curve vs. close curve No diffraction limit! No limit at all…

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Metamaterials as Effective Medium

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  1. Metamaterials as Effective Medium Negative refraction and super-resolution

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

  3. Limits of hyperbolic 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?

  4. Exact solution for stratified medium – transfer matrix

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

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

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

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

  9. Effective medium vs. periodic multilayer

  10. Effective medium vs. periodic multilayer

  11. Metal Metal Metal Metal Surface Plasmons coupling in M-D-M • symmetric and anti-symmetric modes • anti-symmetric mode cutoff • single mode “waveguide” • deep sub-l Metal Metal H-field

  12. Surface Plasmons coupling in M-D-M • “gap plasmon” mode • deep sub-l “waveguide” • symmetric and anti-symmetric modes • No cut-off through the metal Metal Metal Symmetric: k<ksingle-wg Antisymmetric: k>ksingle-wg

  13. Plasmonic waveguide coupling • No counterpart in dielectrics! Metal Metal High contrastDielectric WG – E-field confinement nlow=1 nhigh=3.5 Diffraction limit Michal Lipson, OL (2004)

  14. High spatial frequencies with low loss? • No limitations on the proximity • deep sub-l “waveguide” • symmetric and anti-symmetric modes Metal Metal Multi-layer plamonic metamaterial

  15. Modes in M-D multilayer – beyond EMA Maxwell Equations Maxwell Equations Time-harmonic solution • Sub-WL scale layers • Strong variation in the dielectric function (sign and magnitude) • Paraxial approximation Is not valid!

  16. Linear modes in M-D multilayer TM mode

  17. Modes in M-D multilayer Looking for SPATIAL eigenmodes (not varying with propagation) An “eigenvalue” problem First-order equation for the vector

  18. Kx=p/D Kx=0 1 1 1 Magnetic Tangential Electric Magnetic Tangential Electric -1 -1 0.97 Plasmonic Bloch modes Spatial frequency limited by periodicity large K available even far from the resonance

  19. Kx=p/D Kx=0 1 1 1 Magnetic Tangential Electric Magnetic Tangential Electric -1 -1 0.97 Plasmonic Bloch modes Symmetric in dielectric Symmetric in dielectric Symmetric in metal Antisymmetric in metal Same symmetry as H • Symmetry opposite to H

  20. ↔ y d 2 2 d x d 2 k º z D d 2 k x Anomalous Diffraction and Refraction Normal diffraction Anomalous diffraction Negative refraction without actual negative index Diffraction Direction of energy

  21. d,r<<l K<<p/d 2D analog: Metal Nanowires array Podolskiy, APL 89, 261102 2006 • Show anomalous properties in all directions • Broad-band response • Large-scale manufacturing Averaged dielectric response Hyperbolic dispersion!

  22. Single band Metal-dielectric multilayers – dispersion curve

  23. Periodic metal-dielectric composites Dispersion relation • At longer wavelengths metal permittivity grows (negatively) • Less E-field in the metal • Less loss • Less coupling (tunneling) • Less diffraction • Dkz decreases • Resolution limited by the period • kx/k0 increases • Short l • strong coupling • Large wavenumber • Broad range • Longer l • weak coupling • moderate wavenumber • Large Bloch k-vectors • lower loss

  24. 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 near-field to near-field • Solution: curve the space Superlens

  25. 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.

  26. Maxwell’s equations in cylindrical coordinates TM solution Isotropic case .

  27. Maxwell’s equations in cylindrical coordinates Separation of variables: Solution given by Bessel functions • penetration of high-order modes to the center is diffraction limited .

  28. Maxwell’s equations in cylindrical coordinates .

  29. 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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