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The Strange Properties of Left-handed Materials C. M. Soukoulis Ames Lab. and Physics Dept. Iowa State University and Research Center of Crete, FORTH - Heraklion, Crete Outline of Talk • Historical review left-handed materials • Results of the transfer matrix method • Determination of the effective refractive index • Negative n and FDTD results in PBGs (ENE & SF) • New left-handed structures • Experiments on negative refraction and superlenses (Ekmel Ozbay, Bilkent) • Applications/Closing Remarks Peter Markos, E. N. Economou & S. Foteinopoulou Rabia Moussa, Lei Zhang & Gary Tuttle (ISU) M. Kafesaki & T. Koschny (Crete) What is an Electromagnetic Metamaterial? A composite or structured material that exhibits properties not found in naturally occurring materials or compounds. Left-handed materials have electromagnetic properties that are distinct from any known material, and hence are examples of metamaterials. Veselago We are interested in how waves propagate through various media, so we consider solutions to the wave equation. E 2 E em 2 t 2 (-,+) e,m space m n em (+,+) e k em (-,-) (+,-) Sov. Phys. Usp. 10, 509 (1968) Left-Handed Waves • If e 0, m 0 then vectors: • If e 0, m 0 then vectors: E, H , k is a right set of is a left set of E, H , k Energy flux in plane waves • Energy flux (Pointing vector): – Conventional (right-handed) medium – Left-handed medium Frequency dispersion of LH medium • Energy density in the dispersive medium e 2 m 2 W E H • Energy density W must be positive and this requires e 0; m 0 • LH medium is always dispersive • According to the Kramers-Kronig relations – it is always dissipative “Reversal” of Snell’s Law PIM RHM PIM RHM PIM RHM NIM LHM 2 1 (1) 2 1 (2) k S (1) (2) k S Focusing in a Left-Handed Medium RH RH RH RH LH RH n=1 n=1.3 n=1 n=1 n=-1 n=1 Left-handed Right-handed n=1 n=1 n=-1 n=1,52 n=1 n=1 Source Source M. Kafesaki Objections to the left-handed ideas Parallel momentum is not conserved S1 S2 A Causality is violated Fermat’s Principle Superlensing is not possible O΄ B Μ Ο ndl minimum (?) Reply to the objections • Photonic crystals have practically zero absorption • Momentum conservation is not violated • Fermat’s principle is OK ndl extremum • Causality is not violated • Superlensing possible but limited to a cutoff kc or 1/L Materials with e < 0 and m<0 g opposite to k g opposite to S u g S opposite to k 1 S S dn n d p c2 k S 8 k u g opposite to k 1 n 0 p c g , p k0 n em Photonic Crystals dn n , n ck d , 0 n 0 n n , , 0 p c g , p k0 n m e 2 2 E H u p k Super lenses 2 c 2 k||2 k 2 if k|| /c k is imaginary e ik r ~ e k r Wave components with decay, i.e. are lost , then Dmax l If n < 0, phase changes sign k k| | e ik r ~ e k r thus k if k imaginary k|| / c ARE NOT LOST !!! Resonant response 10 q E p 5 q p E 0 -5 0 0.5 1 1.5 / 2 0 2.5 3 E p q Where are material resonances? Most electric resonances are THz or higher. For many metals, p occurs in the UV Magnetic systems typically have resonances through the GHz (FMR, AFR; e.g., Fe, permalloy, YIG) Some magnetic systems have resonances up to THz frequencies (e.g., MnF2, FeF2) Metals such as Ag and Au have regions where e<0, relatively low loss Negative materials e<0 at optical wavelengths leads to important new optical phenomena. m<0 is possible in many resonant magnetic systems. What about e<0 and m<0? Unfortunately, electric and magnetic resonances do not overlap in existing materials. This restriction doesn’t exist for artificial materials! Obtaining electric response 2p e( ) 1 2 1 c e k 2 0 1.5 Drude Model /p e -1 -2 1 -3 0.5 E - - -4 -5 - - 0 1 /p 2 3 Gap 0 k 2c 2 1 2 2 d ln( d / r) d Le 0 2 p Obtaining electric response (Cut wires) 10 2 5 1.5 /p e 2p e( ) 1 2 20 0 Drude-Lorentz m 1 0.5 - - -10 - - k Gap -5 E c 0 1 /p 2 3 0 k Obtaining magnetic response To obtain a magnetic response from conductors, we need to induce solenoidal currents with a time-varying magnetic field Introducing A A metal metal ring diska gap into the is is also weakly ring creates a weakly diamagnetic resonance to diamagnetic enhance the response + - - + H Obtaining magnetic response m( ) 1 F 2p 2 2 0 3 c m k 2 2 1.5 /mp m 1 0 1 Gap -1 0.5 -2 -3 0 1 /mp 2 0 k Metamaterials Resonance Properties 2p e 1 2 J. B. Pendry 2p m 1 2 02 First Left-Handed Test Structure UCSD, PRL 84, 4184 (2000) Transmitted Power (dBm) Transmission Measurements Wires alone Split rings alone m>0 e<0 m<0 e<0 m>0 e<0 e<0 Wires alone 4.5 5.0 5.5 6.0 Frequency (GHz) 6.5 7.0 UCSD, PRL 84, 4184 (2000) Best LH peak observed in left-handed materials Transmission (dB) 0 SRR Wire CMM -10 -20 -30 -40 -50 3 4 5 6 7 Frequency (GHz) t r1 d r2 Bilkent, ISU & FORTH w Single SRR Parameters: r1 = 2.5 mm r2 = 3.6 mm d = w = 0.2 mm t = 0.9 mm A 2-D Isotropic Structure UCSD, APL 78, 489 (2001) Measurement of Refractive Index UCSD, Science 292, 77 2001 Measurement of Refractive Index UCSD, Science 292, 77 2001 Measurement of Refractive Index UCSD, Science 292, 77 2001 Boeing free space measurements for negative refraction n PRL 90, 107401 (2003) & APL 82, 2535 (2003) Transfer matrix is able to find: • Transmission (p--->p, p--->s,…) p polarization • Reflection (p--->p, p--->s,…) s polarization • Both amplitude and phase • Absorption Some technical details: • Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24 • Length of the sample: up to 300 unit cells • Periodic boundaries in the transverse direction • Can treat 2d and 3d systems • Can treat oblique angles • Weak point: Technique requires uniform discretization Structure of the unit cell EM wave propagates in the z -direction Periodic boundary conditions are used in transverse directions Polarization: p wave: E parallel to y s wave: E parallel to x For the p wave, the resonance frequency interval exists, where with Re meff <0, Re eeff<0 and Re np <0. For the s wave, the refraction index ns = 1. Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm Typical permittivity of the metallic components: emetal = (-3+5.88 i) x 105 Generic LH related Metamaterials Typical LHM behavior m e p p m a/c a/c m m a/c Resonance and anti-resonance m LHM Design used by UCSD, Bilkent and ISU LHM SRR Closed LHM T Substrate GaAs eb=12.3 f (GHz) 30 GHz FORTH structure with 600 x 500 x 500 mm3 T and R of a Metamaterial exp(ikd) ts 1 1 cosnkd z sin nkd 2 z z d rs ts exp(ikd)i(z 1 / z)sin( nkd) / 2 UCSD and ISU, PRB, 65, 195103 (2002) m e n me 2ep e 1 2 2e0 ie 0 2mp m 1 2 2m0 im0 Inversion of S-parameters 1 2m 1 1 2 2 n cos 1 r t kd 2t kd e ik z, n ik te re ik d UCSD and ISU, PRB, 65, 195103 (2002) 1 r t 2 z 2 1 r t 2 2 n e z m nz Effective permittivity e and permeability m of wires and SRRs e 1 2 p 2 UCSD and ISU, PRB, 65, 195103 (2002) m2 m 1 2 20 i Effective permittivity e and permeability m of LHM UCSD and ISU, PRB, 65, 195103 (2002) Effective refractive index n of LHM UCSD and ISU, PRB, 65, 195103 (2002) New designs for left-handed materials eb=4.4 Bilkent and ISU, APL 81, 120 (2002) Bilkent & FORTH Photonic Crystals with negative refraction. Triangular lattice of rods with e=12.96 and radius r, r/a=0.35 in air. H (TE) polarization. Same structure as in Notomi, PRB 62,10696 (2000) PRL 90, 107402 (2003) CASE 1 CASE 2 Photonic Crystals with negative refraction. g g Equal Frequency Surfaces (EFS) Schematics for Refraction at the PC interface EFS plot of frequency a/l = 0.58 Experimental verification of negative refraction a Lattice constant a=4.794 mm Dielectric constant=9.61 R/a=0.329 Frequency=13.698 GHz square lattice E(TM) polarization Bilkent & ISU Band structure, negative refraction and experimental set up Negative refraction is achievable in this frequency range for certain angles of incidence. Bilkent & ISU Frequency = 13.7 GHz l= 21.9 mm 17 layers in the x-direction and 21 layers in the y-direction Superlensing in photonic crystals Image Plane FWHM = 0.21 l Distance of the source from the PC interface is 0.7 mm (l/30) Subwavelength Resolution in PC based Superlens The separation between the two point sources is l/3 Subwavelength Resolution in PC based Superlens Power distribution along the image plane The separation between the two point sources is l/3 ! Controversial issues raised for negative refraction Among others 1) What are the allowed signs for the phase index np and group index ng ? PIM NIM 2) Signal front should move causally from AB to AO to AB’; i.e. point B reaches B’ in infinite speed. Does negative refraction violate causality and the speed of light limit ? Valanju et. al., PRL 88, 187401 (2002) Photonic Crystals with negative refraction. Photonic Crystal vacuum FDTD simulations were used to study the time evolution of an EM wave as it hits the interface vacuum/photonic crystal. Photonic crystal consists of an hexagonal lattice of dielectric rods with e=12.96. The radius of rods is r=0.35a. a is the lattice constant. QuickTime™ and a BMP decompressor are needed to see this picture. We use the PC system of case1 to address the controversial issue raised Time evolution of negative refraction shows: The wave is trapped initially at the interface. Gradually reorganizes itself. Eventually propagates in negative direction Causality and speed of light limit not violated S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, 107402 (2003) Photonic Crystals: negative refraction The EM wave is trapped temporarily at the interface and after a long time, the wave front moves eventually in the negative direction. Negative refraction was observed for wavelength of the EM wave l= 1.64 – 1.75 a (a is the lattice constant of PC) Conclusions • Simulated various structures of SRRs & LHMs • Calculated transmission, reflection and absorption • Calculated meff and eeff and refraction index (with UCSD) • Suggested new designs for left-handed materials • Found negative refraction in photonic crystals • A transient time is needed for the wave to move along the - direction • Causality and speed of light is not violated. • Existence of negative refraction does not guarantee the existence of negative n and so LH behavior • Experimental demonstration of negative refraction and superlensing • Image of two points sources can be resolved by a distance of l/3!!! $$$ DOE, DARPA, NSF, NATO, EU Publications: P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002) P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002) D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002) M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, APL 81, 120 (2002) P. Markos, I. Rousochatzakis and C. M. Soukoulis, Phys. Rev. E 66, 045601 (R) (2002) S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, 107402 (2003) S. Foteinopoulou and C. M. Soukoulis, Phys. Rev. B 67, 235107 (2003) P. Markos and C. M. Soukoulis, Opt. Lett. 28, 846 (2003) E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and CMS, Nature 423, 604 (2003) P. Markos and C. M. Soukoulis, Optics Express 11, 649 (2003) E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and CMS, PRL 91, 207401 (2003) T. Koshny, P. Markos, D. R. Smith and C. M. Soukoulis, PR E 68, 065602(R) (2003) PBGs as Negative Index Materials (NIM) Veselago : Materials (if any) with e < 0 and m< 0 em>0 Propagation k, E, H Left Handed (LHM) S=c(E x H)/4 opposite to k Snell’s law with g opposite to k n em Flat lenses Super lenses < 0 (NIM) 0.33 mm w t»w t t=0.5 or 1 mm w=0.01 mm 0.33 mm l=9 cm 3 mm 0.33 mm 3 mm ax Periodicity: ax=5 or 6.5 mm ay=3.63 mm az=5 mm Polarization: TM y E x B y z x Number of SRR Nx=20 Ny=25 Nz=25 ax=6.5 mm t= 0.5 mm Transmission (dB) 0 -10 -20 -30 SRR Wire LHM -40 -50 -60 7 8 9 10 11 12 13 14 Frequency (GHz) Bilkent & ISU APL 81, 120 (2002)