C. M. Soukoulis Ames Lab. and Physics Dept. Iowa State University and IESL-FORTH & Materials Dept. - Heraklion, Crete Negative Index Materials: New Frontiers in Optics

Left-Handed Materials History: Permittivity , permeability  and index of refraction n negative Reversal of Snell’s Law, perfect focusing, flat lenses, etc. Impedance match z=√  and n =-1 >> a in LHM, while  a in PBG Both PBG and LHM exhibit properties not found in naturally materials Vision: Understanding the physics and the exotic properties of LHMs Perfect Lens. Near-field optical microscopy, nano-lithography Wireless and optical communications. RF sensing. Antenna and microwave device miniaturization ________________________________ Breakthroughs and new concepts in materials processing at nanoscale Search for new materials that exhibit  < 0 at THz or optical regime

Computational Methods Plane wave expansion method (PWE) R. Moussa, S. Foteinopoulou & M. Kafesaki Transfer matrix method (TMM) Th. Koschny & P. Markos Finite-difference-time-domain-method (FDTD) M. Agio, M. Kafesaki, R. Moussa, & S. Foteinopoulou Effective medium theories E. N. Economou, Th. Koschny, M. Kafesaki The LHM effort is in close collaboration with experiments (ISU, Crete, Bilkent, UCSD, Boeing) Karlsruhe

Collaborators Negative refraction in photonic crystals: E. N. Economou (Crete, Greece) S. Foteinopoulou and R. Moussa (Ames, USA) E.Ozbay (Bilkent, Turkey) Left-handed Materials P. Markos (Ames & Slovakia), T. Koschny (Crete & Ames) E. N. Economou, M. Kafesaki, & N. Katsarakis (Crete, Greece) G. Konstandinidis, R. Penciu and T. Gundogdu (Crete, Greece) E. Ozbay (Bilkent, Turkey) Lei Zhang J. Zhou & G. Tuttle (Ames, USA) D. R. Smith (UCSD, USA) M. Wegener (Karlsruhe) Boeing’s group (Seattle, USA)

Outline of Talk  Historical review of left-handed materials  Results of the transfer matrix method  Determination of the effective  eff and  eff Negative imaginary parts in  and   Periodic effective medium theory. Im  and Im  > 0  Electric and Magnetic Response of SRRs and LHMs Effective  p of the LHM is much lower than  p of the wires. There are “phony” LH peaks when  p <  m. It’s difficult to find a LH peak!  Negative n and FDTD results in PBGs (ENE & SF)  Experiments on negative refraction and superlenses (Ozbay)  Ongoing and future work  Concluding Remarks

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

“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

J. B. Pendry Evanescent wave refocusing: Perfect lensing

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 Resonances

Medium response Drude-Lorentz forms for  and 

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 d

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

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

Transfer matrix method to compute scattering amplitudes

continuum Homogeneous Effective Medium inversion

Generic LH related Metamaterials

Resonance and anti-resonance Typical LHM behavior PRL 93, (2004)

Closing the gaps of the SRRs,---> magnetic response disappears FORTH and Ames

PRL 93, (2004) Electric response of LHMs is the sum of wires and closed SRRs FORTH and Ames

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

Left-Handed Materials SRR Parameters: r 1 =2.5 mm, r 2 =3.6 mm, d=w=0.2 mm t=0.9 mm Parameters: a x =9.3 mm a y =9mm a z =6.5 mm N x =15 N y =15 w t d r1r1 r2r2

Transmission data for open and closed SRRs Bilkent, Crete & Ames Magnetic resonance disappears for closed SRRs

Bilkent, Crete & Ames Effective  p of closed SRRs & wires is much lower than  p of the wires.

Best LH peak in a left-handed material Losses: -0.3 dB/cm Bilkent, Crete & Ames Peak at f=4 GHz =75 mm much larger than size of SRR a=3.6 mm

Ames Lab. & Crete Electric coupling to the magnetic resonance APL 84, 2943 (2004)

Magnetic response at 100 THz, almost optical frequencies S. Linden & M. Wegener, Karlsruhe   10  

Magnetic response at 100 THz, almost optical frequencies S. Linden & M. Wegener, Karlsruhe

Magnetic response at 100 THz, almost optical frequencies S. Linden & M. Wegener, Karlsruhe

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)

Refractive index n Permittivity  Permeability  Im n > 0 Re n > 0 Im  < 0 ??? Re  > 0 Im  > 0 Re  < 0 Energy Losses Q in a passive medium are always positive in spite of the fact that Im  < 0 Q(  ) > 0, provided that Im n(  ) > 0 and Re z(  ) > 0

Band structure, negative refraction and experimental results Bilkent & Ames Negative refraction is achievable in this frequency range. f = 13.7 GHz  = 21.9 mm 17 layers in the x-direction and 21 layers in the y-direction PRL 91, (2003) Nature 423, 604 (2003)

2D field snapshot for two incoherent sources viewed from the image plane

Same as in previous slide but zoomed in

Coupling to surfaces waves can improve focusing

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

PRL 90, (2003) Negative refraction in photonic crystals

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)

Ongoing and future work Improvement of transfer matrix code. Off-normal incidence and non-uniform discetization. Improvement of the retrieval code and understanding of why Im  Im  < 0.  Effective medium theory. Effects of periodicity. Origin of losses.  Isotropic 2d and 3d designs of LHMs. Fabrication and testing.  Propose and fabricate LH structures at 94 GHz, THz and 10.6  m.  Superlattices of negative  and negative . Negative n? Objectives of the LHM effort  A better understanding of the physics of left-handed (LH) materials.  Improvement of the existing tools for modeling and simulating more complicated structures than can be done today.  Fabrication of LH-materials, using various approaches, materials and processes.  Testing the electromagnetic behavior of these materials.  Identifying several different applications where such materials can make a big contribution.

Conclusions Simulated various structures of SRRs & LHMs. Calculated transmission, reflection and absorption. Calculated  eff and  eff and refraction index. Im  Im  < 0. Periodic effective medium theory. Suggested new designs for left-handed materials and SRRs. A criterion was proposed for finding if a T peak is LH or RH. Magnetic response in 100 THz regime! (Experiment). Found negative refraction in photonic crystals. Low losses. Experimental demonstration of negative refraction and superlensing. Image of two points sources can be resolved by a distance of /3!!! Evanescent wave refocusing. Role of surface termination. Surface waves. $$$ 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)  N. Katsarakis, T. Koschny, M. Kafesaki, ENE and CMS, APL 84, 2943 (2004)  T. Koschny, M. Kafesaki, E. N. Economou and CMS, PRL 93, (2004)  Lei Zhang, G. Tuttle and CMS, Photonics and Nanostructures (accepted, 2004)

Material Response: Lorentz Oscillators Driven harmonic oscillator: Harmonic dependence: Ep q

Electric response of LHM Electric and magnetic response of SRR E and M response of LHM Electric response of wires Electric response of cut wires

1d single-ring SRR: retrieved Re n(  ) via cHEM inversion for different length of the unit cell: 6x10x9... 6x10x14

TMM simulated 1d single-ring SRR: retrieved Re n(  ) via cHEM inversion for different resonance frequencies Emulate small SRR gap: we fill the gap with dielectic, eg.eps=300 Vacuum case as before π/(N z  )

TMM simulated 1d single-ring SRR: retrieved eps(  ) and mu(  ) via cHEM inversion for different resonance frequencies Emulate small srr gap: we fill the gap with dielectic, eg. eps=300 Vacuum case as before No negative Im  and Im  are observed !

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

Electric and Magnetic Response of SRRs and LHMs Electric and Magnetic Response are independent. One can change the magnetic response without changing the electric response. GHz and THz magnetic response in artificial structures! The SRR has strong electric response. It’s cut-wire like. Effective electric response of LHM is the sum of wire and SRR. Effective  p of the LHM is much lower than  p of the wires. There are “phony” LH peaks when  p <  m PRL 93, (2004)

LH materials will exhibit a negative index of refraction in 1D, 2D and 3D The change in phase of propagating waves in the LH frequency band will evolve in time with opposite sign of the change in phase for waves in a RH band. By constructing LHMs, the magnitudes of the effective index n and surface impedance Z at a chosen frequency can be separated designed over an appreciable continuous range, with the algebraic sign of n going from positive to negative values. Match both Z and n of free space with LHMs, i.e. Z=+1, while n=-1! LH materials can reconstitute evanescent wave components in space (passively, without active components). Thus, the superposition of reconstituted evanescent components can result in refocusing of “point sources” below the traditional far-field diffraction limit! Some Significance, and Unique Properties of LHMs

Left-handed materials (LHM), as well as photonic crystals (PC) are composite metamaterials whose properties are not determined by the fundamental physical properties of their constituents but by the shape and distribution of specific patterns included in them. LHM have the unique property of having both the effective permittivity and the effective permeability negative. The aim of the research is the theoretical understanding, analysis, development, fabrication and testing of LHM, and also the investigation of their feasibility for applications. Background and goals The LHM effort is in close collaboration with experiments (ISU, Crete, Bilkent, UCSD, Boeing) Karlsruhe

More generally… The general response of a material is a sum over oscillators This implies a low frequency electrical permittivity: Insulator Conductor

More generally The permittivity and permeability must be causal analytic functions, implying Kramers-Kronig relations hold:

Electric coupling to the magnetic resonance The second electric coupling hinders the appearance of LH behavior. Problem in higher dimensions For higher dimensions  More symmetric structures are required H perp at the magnetic and the electric resonance, for normal incidence E kH

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

C. M. Soukoulis Ames Lab. and Physics Dept. Iowa State University Negative Index Materials: New Frontiers in Optics

Intermediate summary: continuum homogeneous effective material (cHEM) ● cHEM inversion basically works, we find length-independent(!) effective material behavior  Re n(  ) seems to be cut-off at Brillouing zone.  Discrepancy between n(  ) and z(  ): where is the resonance?  Resonance/anti-resonance coupling.  Negative imaginary parts in  or   Deformed resonances, i.e. unexpected shallow negative   What is all this structure at higher frequencies? but problems:

Going to multi-gap structures (1) (a)better than (b) (wider SRR dip); (c) better than (d) (stronger dip); (e) like the conventional SRR but weaker dip (for large separation) Problem: Increase of  m (  m close to  0 ) Gaps act like capacitors in series:  m 2 (n gaps) ~ n  m 2 (1 gap) Reason: requirement for higher symmetry, for use in 3D LH structures a) b) c) d)e)

Going to multi-gap structures (2) Solution: Make the gaps smaller or change the design Improvements?  Up to a point Only the left one

Promising multi-gap structures from 1D study a) b)b) (a): Detailed study on progress (in 1D) (b): Not studied in detail yet (c): Good LH T 3D structures a) b)b) c) Best combination: (b)+(c) c)

Two-sided SRR Structures: No coupling to Electric Field

Two-sided SRRs do not have coupling to electric field

Electric coupling to the magnetic resonance The external electric field excites the resonant circular currents, i.e. the SRR resonance Reason: system asymmetry Result: Electric resonance close to SRR magnetic resonance It happens also for the in-plane propagation E H perp at the magnetic resonance, for normal incidence E APL 84, 2943 (2004)

Analytic model for the electric and magnetic response of SRRs

Analytic model of the electric and magnetic response of LHMs PRL 93, (2004)

Photonics and Nanostructures (accepted, 2004)

TMM simulated 1d single-ring off-plane LHM: retrieved Re n(  ) and Im n(  ) via cHEM inversion Re n(  ) Im n(  )

TMM simulated 1d single-ring off-plane LHM: retrieved  (  ) and  (  ) via cHEM

Model: Effective periodic material (PEM)

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)