A beautiful approach: “Transformational optics” [ several precursors, but generalized & popularized by Ward & Pendry (1996) ] warp a ray of light …by warping space(?) [ figure: J. Pendry ] Euclidean x coordinates transformed x'(x) coordinates amazing Solutions of ordinary Euclidean Maxwell equations in x' fact: = transformed solutions from x if x' uses transformed materials ε' and μ' Maxwell’s Equations constants: ε0, μ0 = vacuum permittivity/permeability = 1 –1/2 c = vacuum speed of light = (ε0 μ0 ) = 1 ! " B = 0 Gauss: constitutive ! " D = # relations: James Clerk Maxwell #D E = D – P 1864 Ampere: ! " H = + J #t H = B – M $B Faraday: ! " E = # $t electromagnetic fields: sources: J = current density E = electric field ρ = charge density D = displacement field H = magnetic field / induction material response to fields: B = magnetic field / flux density P = polarization density M = magnetization density Constitutive relations for macroscopic linear materials P = χe E ⇒ D = (1+χe) E = ε E M = χm H B = (1+χm) H = μ H where ε = 1+χe = electric permittivity or dielectric constant µ = 1+χm = magnetic permeability εµ = (refractive index)2 Transformation-mimicking materials [ Ward & Pendry (1996) ] E(x), H(x) J–TE(x(x')), J–TH(x(x')) [ figure: J. Pendry ] Euclidean x coordinates transformed x'(x) coordinates J"JT JµJT ε(x), μ(x) " ! = , µ ! = (linear materials) det J det J J = Jacobian (Jij = ∂xi’/∂xj) (isotropic, nonmagnetic [μ=1], homogeneous materials ⇒ anisotropic, magnetic, inhomogeneous materials) an elementary derivation [ Kottke (2008) ] consider× Ampére’s Law: (index notation) ε' Jacobian chain rule choice of fields E', H' in x' Cloaking transformations [ Pendry, Schurig, & Smith, Science 312, 1780 (2006) ] virtual space real space diameter D d ambient εa, μa x' ≠ x cloaking materials ε', µ' x' = x ambient εa, μa ambient εa, μa perfect cloaking: d → 0 (= singular transformation J) Example: linear, spherical transform R 1 r R1' linear radial scaling R2 cloak materials: = 0 at r=R1 [ note: no “negative index” ε, μ < 0 ] for R1'=0 Are these materials attainable? Highly anisotropic, even (effectively) magnetic materials can be fabricated by a “metamaterials” approach: [ Soukoulis & Wegener, Nature Photonics (2011) ] λ >> microstructure ⇒ “effective” homogeneous ε, µ = “metamaterial” Simplest Metamaterial: “Average” of two dielectrics effective dielectric is just some average, subject to Weiner bounds (Aspnes, 1982) in the large-λ limit: !1 !1 ! " !effective " ! a (isotropic for sufficient symmetry) λ >> a Simplest anisotropic metamaterial: multilayer film in large-λ limit eff Di !Ei Di ! ij = = = !1 E j E j ! Dj key to anisotropy is differing continuity conditions on E: E|| continuous ⇒ ε|| = <ε> D =εE continuous ⇒ ε = <ε–1>–1 a ⊥ ⊥ ⊥ λ >> a [ Not a metamaterial: multilayer film in λ ~ a regime ] e.g. Bragg reflection regime (photonic bandgaps) for λ ~ 2a is not completely reproduced by any effective ε, μ Metamaterials are a special case of periodic electromagnetic media (photonic crystals) a λ ~ a “Exotic” metamaterials [ = properties very different from constituents ] from sub-λ metallic resonances “split-ring” resonance magnetic resonator = pole in polarizability χ # ! ~ " ! (" 0 ! i"0 ) (Γ0 > 0 for causal, passive) Problem: more exotic often [ Smith et al, PRE (2005) ] = more absorption Problem: metals quite lossy @ optical & infrared .