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