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)

 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. What is an Electromagnetic Metamaterial?

Veselago We are interested in how waves propagate through various media, so we consider solutions to the wave equation. (+,+)(-,+) (-,-)(+,-)  space Sov. Phys. Usp. 10, 509 (1968)  

Left-Handed Waves Ifthenis a right set of vectors: Ifthenis a left set of vectors:

Energy flux (Pointing vector): –Conventional (right-handed) medium –Left-handed medium Energy flux in plane waves

Frequency dispersion of LH medium Energy density in the dispersive medium Energy density W must be positive and this requires LH medium is always dispersive According to the Kramers-Kronig relations – it is always dissipative

“Reversal” of Snell’s Law 11 22 11 22 PIM RHM PIM RHM PIM RHM NIM LHM (1)(2)(1)(2)

n=-1n=1 RH LH RH n=1.3n=1 Focusing in a Left-Handed Medium

Left-handed Right-handed Left-handed Right-handed SourceSource n=-1 n=1,52 n=1 M. Kafesaki

 Objections to the left-handed ideas S1S1 S2S2 A B O΄ Μ Ο Causality is violated Parallel momentum is not conserved Fermat’s Principle  ndl minimum (?) Superlensing is not possible

Reply to the objections Photonic crystals have practically zero absorption Momentum conservation is not violated Fermat’s principle is OK Causality is not violated Superlensing possible but limited to a cutoff k c or 1/L

Materials with  < 0 and  <0 Photonic Crystals opposite to,

Super lenses is imaginary  Wave components with decay, i.e. are lost, then  max  If n < 0, phase changes sign ifimaginary ARE NOT LOST !!! thus

Resonant response q E p q E p q E p

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., MnF 2, FeF 2 ) Metals such as Ag and Au have regions where  <0, relatively low loss

Negative materials  <0 at optical wavelengths leads to important new optical phenomena.  <0 is possible in many resonant magnetic systems. What about  <0 and  <0? Unfortunately, electric and magnetic resonances do not overlap in existing materials. This restriction doesn’t exist for artificial materials!

Obtaining electric response Drude Model - E Gap

Obtaining electric response (Cut wires) Drude-Lorentz E Gap

Obtaining magnetic response To obtain a magnetic response from conductors, we need to induce solenoidal currents with a time-varying magnetic field A metal disk is weakly diamagnetic A metal ring is also weakly diamagnetic Introducing a gap into the ring creates a resonance to enhance the response H

Obtaining magnetic response Gap

Metamaterials Resonance Properties J. B. Pendry

First Left-Handed Test Structure UCSD, PRL 84, 4184 (2000)

Wires alone  <0 Wires alone Split rings alone Transmission Measurements Frequency (GHz) Transmitted Power (dBm)  >0  <0  >0  <0  >0  <0 UCSD, PRL 84, 4184 (2000)

Best LH peak observed in left-handed materials Bilkent, ISU & FORTH w t d r1r1 r2r2 Single SRR Parameters: r 1 = 2.5 mm r 2 = 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,

Measurement of Refractive Index UCSD, Science 292,

Measurement of Refractive Index UCSD, Science 292,

Boeing free space measurements for negative refraction PRL 90, (2003) & APL 82, 2535 (2003) n

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 N x x N y x N z : 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 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  eff <0, Re  eff <0 and Re n p <0. For the s wave, the refraction index n s = 1. Typical permittivity of the metallic components:  metal = ( i) x 10 5 EM wave propagates in the z -direction Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm

Generic LH related Metamaterials

Resonance and anti-resonance Typical LHM behavior

f (GHz) T 30 GHz FORTH structure with 600 x 500 x 500  m 3 Substrate GaAs  b =12.3 LHM Design used by UCSD, Bilkent and ISU LHM SRR Closed LHM

T and R of a Metamaterial UCSD and ISU, PRB, 65, (2002) d

z, n Inversion of S-parameters d UCSD and ISU, PRB, 65, (2002)

Effective permittivity  and permeability  of wires and SRRs UCSD and ISU, PRB, 65, (2002)

Effective permittivity  and permeability  of LHM

UCSD and ISU, PRB, 65, (2002) Effective refractive index n  of LHM

 b =4.4 New designs for left-handed materials Bilkent and ISU, APL 81, 120 (2002)

Bilkent & FORTH

Photonic Crystals with negative refraction. Triangular lattice of rods with  =12.96 and radius r, r/a=0.35 in air. H (TE) polarization. Same structure as in Notomi, PRB 62,10696 (2000) CASE 1 CASE 2 PRL 90, (2003)

Photonic Crystals with negative refraction. gg gg Equal Frequency Surfaces (EFS)

Schematics for Refraction at the PC interface EFS plot of frequency a/ = 0.58

Experimental verification of negative refraction a  Lattice constant a=4.794 mm Dielectric constant=9.61 R/a=0.329 Frequency= GHz square lattice E(TM) polarization Bilkent & ISU

Band structure, negative refraction and experimental set up Bilkent & ISU Negative refraction is achievable in this frequency range for certain angles of incidence. Frequency = 13.7 GHz  = 21.9 mm 17 layers in the x-direction and 21 layers in the y-direction

Superlensing in photonic crystals FWHM = 0.21 Image Plane Distance of the source from the PC interface is 0.7 mm ( /30)

Subwavelength Resolution in PC based Superlens The separation between the two point sources is /3

Subwavelength Resolution in PC based Superlens The separation between the two point sources is /3 ! Power distribution along the image plane

Controversial issues raised for negative refraction PIMNIM Among others 1) What are the allowed signs for the phase index n p and group index n g ? 2) Signal front should move causally from AB to AO to AB’; i.e. point B reaches B’ in infinite speed. violate causality speed of light limit Does negative refraction violate causality and the speed of light limit ? Valanju et. al., PRL 88, (2002)

Photonic Crystals with negative refraction. 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  = The radius of rods is r=0.35a. a is the lattice constant. Photonic Crystal vacuum

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, (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 = 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  eff and  eff 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 /3!!! $$$ DOE, DARPA, NSF, NATO, EU

Publications:  P. Markos and C. M. Soukoulis, Phys. Rev. B 65, (2002)  P. Markos and C. M. Soukoulis, Phys. Rev. E 65, (2002)  D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Phys. Rev. B 65, (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, (R) (2002)  S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, (2003)  S. Foteinopoulou and C. M. Soukoulis, Phys. Rev. B 67, (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, (2003)  T. Koshny, P. Markos, D. R. Smith and C. M. Soukoulis, PR E 68, (R) (2003)

PBGs as Negative Index Materials (NIM) Veselago : Materials (if any) with  < 0and  < 0   > 0  Propagation  k, E, H Left Handed (LHM)  S=c(E x H)/4  opposite to k  Snell’s law with < 0 (NIM)   g opposite to k  Flat lenses  Super lenses

x y z t w t»w t=0.5 or 1 mm w=0.01 mm l=9 cm 3 mm 0.33 mm Periodicity: a x =5 or 6.5 mm a y =3.63 mm a z =5 mm Number of SRR N x =20 N y =25 N z =25 axax 0.33 mm Polarization: TM E B y x

a x =6.5 mm t= 0.5 mm Bilkent & ISU APL 81, 120 (2002)