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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 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 dm dd xx 0 0 0 0 yy 0 0 0 zz Pendry’s artificial plasma Motivation: metallic behavior at GHz frequencies Problem: the dielectric response is negatively (close to) infinite Solution: “dilute” the metal ne e 2 0m 2 p The electrons density is reduced nneff ne p2,eff r 2 a2 neff e 2 0 meff * The effective electron mass is increased due to self inductance Lowering the plasma frequency, Pendry, PRL,76, 4773 (1996) Simple analysis of 1D and 2D systems a Periodicity or inclusions much smaller than wavelength 2+1D or 1+2D (dimensions of variations) Effective dielectric response determined by filling fraction f 1D-periodic (stratified) 2D-periodic (nano-wire aray) 3D? a Averaging over the (fast) changing dielectric response 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 1 2 ll Note: parallel=ordinary For isotropic constituents effective fields Di i Ei Eeff Eave fE1 (1 f ) E2 a ll ll Deff Dave fD1 (1 f ) D2 Stratified metal-dielectric metamaterial: Parallel polarization ll E k a Boundary conditions E1 E2 E Eeff Eave fE (1 f ) E E Deff Dave fD1 (1 f ) D2 f 1 E (1 f ) 2 E eff E ll f1 (1 f ) 2 ll Stratified metal-dielectric metamaterial: Normal polarization ll E ll a D1 D2 D Eeff Eave fE1 (1 f ) E2 Deff Dave fD (1 f ) D D Eeff f D 1 (1 f ) D 2 D eff 1 f 1 (1 f ) 2 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 a ll 1 2 ll Note: parallel=extraordinary Nanowire metamaterial: Parallel polarization ll E E1 E2 E Eeff Eave fE (1 f ) E E Deff Dave fD1 (1 f ) D2 f 1 E (1 f ) 2 E eff E ll f1 (1 f ) 2 Nanowire metamaterial: Normal polarization polarization ll E • More complicated derivation • Homogenization (not simple averaging) • Assume small inclusions (<20%) • Maxwell-Garnett Theory (MGT) (metal nanowires in dielectric host) ( x y ) d (1 f ) m (1 f ) d (1 f ) m (1 f ) d Strongly anisotropic dielectric Metamaterial 0 0 0 0 0 0 0 ll ll ll f m (1 f ) d ( x y ) d ll 0 0 0 (1 f ) m (1 f ) d (1 f ) m (1 f ) d 0 ll 0 0 0 ll f m (1 f ) d 1 f 1 ll ll (1 f ) 2 For most visible and IR wavelengths m d ll 0, 0 Example: nanowire medium medium 60nm nanowire diameter Ag wires 110nm center-center wire distance 4 Al2O3 matrix 2 0 // -2 -4 -6 Effective permittivity from MG theory // ( z ) p m (1 p) d (1 p) m (1 p) d ( x y ) d (1 p) m (1 p) d -8 -10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 um 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 um 30 20 10 0 -10 -20 0.3 Broad band Wave propagation in anisotropic medium xx 0 0 0 0 yy 0 0 0 zz Uniaxial xx yy Maxwell equations for time-harmonic waves k H D k E 0 H D E 0 ( xx E x xˆ yy E y yˆ zz E x zˆ ) k k E 0 k H k02 ( xx Ex xˆ yy E y yˆ zz Ex zˆ) xx k02 (k y2 k z2 ) E x kxk y kxkz 2 2 2 kxk y yy k0 (k x k z ) k ykz E y 0 2 2 2 k k k k k ( k k ) E x z y z zz 0 x y z Det(M)=0, xx yy k x2 k y2 k z2 k k 0 x z 2 2 x 0 Wave propagation in anisotropic medium x 0 0 0 0 x 0 0 0 z 2 2 k k k z2 y 2 2 2 2 x 2 k x k y k z x k0 k0 0 x z Extraordinary waves (TM) Ordinary waves (TE) E H • 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 H E Extraordinary waves in anisotropic medium x 0 0 0 0 0 z 0 x 0 kz isotropic medium x z 1 kx 2 z kz kx 2 x k 2 0 k x k z k02 2 2 1.5 anisotropic medium For x<0 kx 2 z kz kz x z n n( ) ‘Hyperbolic’ medium kz 2 x k02 kx kx Energy flow in anisotropic medium isotropic medium kz k x k z k02 2 normal to the k-surface 2 1 kx 1.5 x z ‘Indefinite’ medium anisotropic medium kz x z kx 2 z kz kz 2 x k02 kx S and k are not parallel S Is normal to the curve! * Complete proof in “Waves and Fields in Optoelectronics” by Hermann Haus kx Refraction in anisotropic medium kz What is refraction? x 0 0 0 0 x 0 0 0 z kx 2 kz 2 2 c2 1 kx 1.5 Conservation of tangential momentum x 0, z 0 Sr , z kr , z H 02 0 x 2 0 Sr , x kr , x H 02 0 z 2 0 kz Hyperbolic air Negative refraction! kx Refraction in nanowire medium medium 4 Ag wires 2 0 Al2O3 matrix -2 -4 // -6 -8 -10 0.3 Effective permittivity from MG theory // ( z ) p m (1 p) d (1 p) m (1 p) d ( x y ) d (1 p) m (1 p) d 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 um 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 um 30 20 10 0 -10 -20 0.3 Broad band Negative refraction for >630nm Refraction in layered semiconductor medium •SiC •Phonon-polariton resonance at IR Negative refraction for 9>>12m // ( x y ) p m (1 p) d Hyperbolic metamaterial “phase diagram” kx 2 z kz x 0, z 0 2 x k02 ll f m (1 f ) d 1 f 1 x 0, z 0 (1 f ) x 0, z 0 2 x 0, z 0 dielectric Type I Type II Ag/TiO2 multilayer system Effective medium at different regimes We choose propogation by x 0, z 0 kz 2 2 kx2 0 x 2 z c X=normal (suitable for Nanowires) X=parallel Suitable for stratified medium propagation propagation m d // f m (1 f ) d (1 f ) m (1 f ) d d (1 f ) m (1 f ) d x m d • extreme material properties • Low-loss • epsilon near-zero • Broad-band • Diffraction management • resolution limited by periodicity • Resolution limited by loss x Conditions Normal-X direction (kx<</D) X=normal (suitable for Nanowires) propagation f x kz 2 2 kx2 0 2 ll c // m d 0 3 d m d 0 2 m 3 d 1 2 kz m 3 d 3 d 22 kz 2 2 kx 3 2 d 2 2 2 c c ll 3 d 2 c c • Low loss • Low diffraction management • moderate values • diffraction management improves with em • Limited by periodicity •no near-0 kx Conditions for Normal Z-direction propagation x kz 2 2 kx2 0 ll 2 c // m d 2 0 1 m d 0 2 m d k z 0 For large range of kx kr m d // 0 d kx • Good diffraction management • near-zero • Limited by ? Effective medium with loss… m m i m propagation x m d kz 2 2 kx2 0 ll 2 c // m d i m 2 3i m d d 2 2 d i m Im( kz ) Re( kz ) High loss! kz 2 2 kx2 0 2 c ll m d (Long wavelengths) // m d m 3 d m d 0 2 m 3 d 3 d 2 kz 2 2 kx2 2 ll c Very low loss at low k Moderate loss at high k Limits of indefinite medium for super-resolution Open curve vs. close curve No diffraction limit! No limit at all… kx 2 z kz 2 x x 0, z 0 k02 kr 2 kx2 k k z x k0 x z Is it physically valid? • Reason: approximation to homogeneous medium! • What are the practical limitations? • Can it be used for super-resolution? kx Exact solution – transfer matrix Z Unit Cell m ... d Am Cm Bm m A A M cell n 1 n Bn 1 Bn Am1 Bm1 Dm X X=nD X=nD+d U M (1,1) e ikm d m V M (1, 2) e W M (2,1) e 2 2 diel km2 i m2 kdiel cos k d sin k d diel d diel d 2 diel m kdiel km ikm d m ikm d m X M (2, 2) e X=(n+1)D 2 2 i m2 kdiel diel km2 sin kdiel d d 2 diel m kdiel km 2 2 i m2 kdiel diel km2 sin k d diel d 2 diel m kdiel km ikm d m km k02 m kz2 , kdiel k02 diel kz2 2 2 diel km2 i m2 kdiel cos k d sin k d diel d diel d 2 diel m kdiel km 2 2 diel km2 1 1 m2 kdiel K x arccos cos kdiel dd cos km d m sin kdiel d d sin km d m D 2 diel m kdiel km Exact solution – transfer matrix Z Unit Cell m ... d Am Cm Bm Dm m Am1 (1) Maxwell’s equation Bm1 X X=nD X=nD+d X=(n+1)D Am eikm ( x mD ) Bm eikm ( x mD ) H ( x) Cm eikd ( x dmetal mD ) Dm eikd ( x dmetal mD ) ikm ( x m 1 D ) Bm1 e km ( x m1 D ) Am1 e mD x mD d metal mD d metal x m 1 D m 1 D x m 1 D d metal km k02 m kz2 , kdiel k02 diel kz2 i A0ikm eikm x B0ikm e ikm x metal i E ( x) C0ikd eikd ( x dmetal ) D0ikd e ikd ( x dmetal ) diel i A1ikm eikm ( x D ) B1ikm e km ( x D ) metal 0 x d metal d metal x D D x D d metal Exact solution – transfer matrix Z Unit Cell m ... m d Am Am1 Cm Bm (2) Boundary conditions Bm1 Dm X X=nD H (x d X=nD+d metal metal ) H (x d E( x d eikm dm ik d ikm e m m metal 1 ikd diel ) E(x d X=(n+1)D metal metal ) e ikm dm ikm e ikm dm metal ) 1 metal 1 A0 B ikd 0 diel 1 1 eikm dm ikd ikm eikmdm diel metal eikm dm ikm eikmdm metal A0 eikm dm B0 e ikm dm C0 D0 1 A0ikm eikm dm B0ikm e ikm dm C0ikd D0ikd C0 ikd D diel 0 1 A0 C0 B D 0 0 diel Exact solution – transfer matrix Z Unit Cell m ... d Am Cm Bm m Am1 (3) Combining with Bloch theorem Bm1 Dm X X=nD X=nD+d X=(n+1)D Am Am1 M cell Bm Bm1 eiK x D Am Am1 B B m m 1 U eiK x D det W M cell e 0 X eiK x D V iK x D e U eiK x D Am 0 Bm W iK x D Am 0 X eiK x D Bm V UX U X i 1 2 2 2 2 2 diel km2 1 1 m2 kdiel K x arccos cos kdiel dd cos km d m sin kdiel d d sin km d m D 2 diel m kdiel km Beyond effective medium: SPP coupling in M-D-M • “gap plasmon” mode • deep sub- “waveguide” • symmetric and anti-symmetric modes Metal Symmetric: k<ksingle-wg Metal Antisymmetric: k>ksingle-wg Beyond effective medium: SPP coupling in M-D-M metal dielectric • Abrupt change of the dielectric function • variations much smaller than the wavelength • Paraxial approximation not valid! •Need to start from Maxwell Equations z x • TM nature of SPPs • Calculate 3 fields 1 E 1 H H , E c t c t Eigenmode problem: • Eigen vectors EM field • Eigen values Propagation constants ~ Ex ( x, z) Ex ( x)eiz ~ E z ( x , z ) E z ( x ) e i z ~ H y ( x, z ) H y ( x)eiz Hamiltonian-like operator: ˆ M ( x) ( x) ( x) ~ ~ ( H y , E x )T 2 0 k 1 0 ˆ M ( x) ˆ k 0 H ( x) 0 1 Hˆ ( ) k02 x x Plasmonic Bloch modes Kx=/D Kx=0 1 1 1 Magnetic Tangential Electric Magnetic Tangential Electric -1 0.97 -1 Ag=20nm Air=30 nm =1.5m kz k0 k x / k0 Metamaterials at low spatial frequencies The homogeneous medium perspective D k D Averaged dielectric response Can be <0 // ( z ) p m (1 p) d (1 p) m (1 p) d ( x y ) d (1 p) m (1 p) d kx kz 2 2 z x c 2 2 Hyperbolic dispersion! Metamaterials at low spatial frequencies The homogeneous medium perspective D k D Averaged dielectric response Can be <0 // ( z ) p m (1 p) d (1 p) m (1 p) d ( x y ) d (1 p) m (1 p) d kx kz 2 z x c 2 2 3 2.5 2 1.5 1 2 Hyperbolic dispersion! 0.5 0 0.5 1 1.5 2 2.5 3 3.5 Use of anisotropic medium for far-field super resolution Conventional lens Superlens can image near- to near-field Superlens Need conversion beyond diffraction limit Multilayers/effective medium? Can only replicate sub-diffraction image by diffraction suppression Solution: curve the space The Hyperlens X Z r dm dd • Metal-dielectric sub-wavelength layers • No diffraction in Cartesian space • object dimension at input a • D is constant D a r •Arc at output kr 2 k 2 r ll 0 R A RD a r Magnification ratio determines the resolution limit. k0 2 Optical hyperlens view by angular momentum • Span plane waves in angular momentum base (Bessel func.) e ikx m im i J ( kr ) e m m • 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