Wave propagation in structures with left-handed materials Ilya V. Shadrivov Nonlinear Physics Group, RSPhysSE Australian National University, Canberra, Australia In collaboration with: Yu. S. Kivshar, A. A. Sukhorukov, D. Neshev

Outline Introduction: Left-Handed Materials (LHM)Introduction: Left-Handed Materials (LHM) Interfaces and surface wavesInterfaces and surface waves Nonlinear properties of LHMNonlinear properties of LHM Nonlinear surface wavesNonlinear surface waves Giant Goos-Hänchen effectGiant Goos-Hänchen effect Guided waves in a slab waveguideGuided waves in a slab waveguide Photonic crystals based on LHMPhotonic crystals based on LHM Presentations and publicationsPresentations and publications

Left-handed materials Materials with negative permittivity ε and negative permeability μMaterials with negative permittivity ε and negative permeability μ Such materials support propagating wavesSuch materials support propagating waves Energy flow is backward with respect to the wave vectorEnergy flow is backward with respect to the wave vector

Frequency dispersion of LH medium Energy density in dispersive mediumEnergy density in dispersive medium Positivity of requiresPositivity of W requires LH medium is always dispersiveLH medium is always dispersive

D.R.Smith, W.J.Padilla, D.C.Vier, S.C.Nemat-Nasser and S.Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84, 4184 (2000) Metamaterial The first experiment on LH media Frequency range with ε<0, μ<0: 4 - 6Ghz Effective medium approximation

Negative refraction at LH/RH interface LH RH

Unusual lenses V. G. Veselago Soviet Physics Uspekhi 10 (4), (1968)V. G. Veselago Soviet Physics Uspekhi 10 (4), (1968) J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000) LH lens does not have a diffraction resolution limit Improved resolution is due to the surface waves

Surface waves of left-handed materials x z RHM LHM

Guided modes TE-polarization:TE-polarization: TM-polarization:TM-polarization: Solutions for guided modesSolutions for guided modes I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, submitted to Phys. Rev. E (2003)

Energy flux at the interface P=P 1 +P 2 Total energy flux z RHMLHM P1P1 P2P2 Forward waves: P > 0 Backward waves: P < 0

Existence regions of surface waves ParametersParameters No regions where TE and TM waves exist simultaneously Waves can be both forward traveling and backward traveling

Dispersion of TE guided modes Normalized frequency Normalized wave number Waves can posses normal or anomalous dispersion Dispersion depends on the dielectric permittivity of RH medium Dispersion control in a nonlinear medium?

Nonlinear properties of left-handed materials Metallic composite structure embedded into the nonlinear dielectricMetallic composite structure embedded into the nonlinear dielectric A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. Lett. 91, (2003)

Nonlinear dielectric properties of the composite Microscopic derivation in the effective medium approximationMicroscopic derivation in the effective medium approximation Contribution from nonlinear dielectric Contribution from metallic wires

Effective magnetic permeability q -q Effective medium approximation

Magnetic properties of composite material with self-focusing dielectric Frequency Magnetic permeability

Magnetic properties of composite material with self-defocusing dielectric Frequency Magnetic permeability

Effective magnetic permeability

Nonlinear properties management Composite completely filled by nonlinear dielectricComposite completely filled by nonlinear dielectric

Nonlinear properties management Only resonator gaps are filled by nonlinear dielectricOnly resonator gaps are filled by nonlinear dielectric

Nonlinear properties management Composite completely filled by nonlinear dielectric, but resonator gapsComposite completely filled by nonlinear dielectric, but resonator gaps

Nonlinear surface waves LHM: x nonlinear RHM nonlinear LHM z RHM: Self-focusing I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, submitted to Phys. Rev. E (2003)

Nonlinear dispersion of surface wave The dispersion is multi-valued Two different types of surface waves Normalized wave number Normalized energy flow

Localized polaritons V g - the group velocity δ - the group velocity dispersion  - the nonlinear coefficient LHMRHM P1P1 P2P2 Energy flow has a vortex structure

Energy flow in a pulse

Surface waves We have revealed the unusual properties of linear surface waves Both TE and TM modes exist at a LH/RH interface Modal dispersion can be either normal or anomalous Total energy flow can be either positive and negative Wave packets have a vortex structure of the energy flow

Goos-Hänchen effect A shift of the reflected beam from the position predicted by geometric optics Δ << Δ << width of the beam

Giant Goos-Hänchen effect Δ ~ Δ ~ width of the beam

Giant Goos-Hänchen effect LHM Excitation of forward surface wave results in a positive shift Excitation of backward surface wave results in a negative shift

Resonant beam shift

Energy flow at a negative beam shift Vortex surface wave excitation RHM LHM Air

Possible application Measure the angle of resonant shiftMeasure the angle of resonant shift Determine surface mode eigen wave numberDetermine surface mode eigen wave number Calculate LH material parametersCalculate LH material parameters

Negative refractive index waveguide x z L-L Slab thickness 2L RHM LHM

Guided modes in Negative Refractive Index Waveguides TE-polarization:TE-polarization: TM-polarization:TM-polarization: Solutions for guided modesSolutions for guided modes I. V. Shadrivov, A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. I. V. Shadrivov, A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. E. 67, (2003)

TE Modes Dispersion of guided modes Fastslow Fast and slow modes Fundamental mode may be absent Normal and anomalous dispersion

TM Modes Dispersion of guided modes Fastslow Fast and slow modes Fundamental mode may be absent Normal and anomalous dispersion

Sign-varying energy flux LHM RHM P1P1 P2P2 P2P2 P=P 1 +P 2 Total energy flux

Energy flow in a pulse LHMRHM P1P1 P2P2 P2P2

Negative refractive index waveguide We have revealed the unusual properties of guided modes in negative refractive index waveguides such as –Both fast and slow modes exist in –Both fast and slow modes exist in LH slab waveguide – –Modal dispersion can be either normal or anomalous – –Total energy flow can be either positive or negative – –Wave packets have a double vortex structure of the energy flow

Unusual lenses V. G. Veselago Soviet Physics Uspekhi 10 (4), (1968)V. G. Veselago Soviet Physics Uspekhi 10 (4), (1968) J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)

Periodic structure with a left-handed material Photonic band gap appears in periodic structures with zero averaged refractive index. J. Li, L. Zhou, C. T. Chan, and P. Sheng, Phys. Rev. Lett. 90, (2003)

Reflection from periodic structures Bragg resonant reflection φ 2 - φ 1 = 2πm, m=1,2,3… φ2φ2 φ1φ1 RHM 1 RHM 2 φ2φ2 φ1φ1 LHM RHM Nonresonant reflection φ 2 - φ 1 = 0

Band gap in 1D photonic crystal Transfer Matrix Trace Z-components of wavevectors in right- and left-handed media Wave impedances of right- and left-handed media KxKx K KzKz RHM LHM a b

Band gap structure Gap Wave propagation Frequency Transverse wavenumber I.V. Shadrivov, A.A. Sukhorukov and Yu.S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003)

Beam shaping Spatial filter Incident Reflected Transmitted Gaussian beam oblique incidence Vortex beam normal incidence Vortex beam oblique incidence

Transmission properties of the layered structure. We have analyzed transmission properties of a layered periodic structure with left-handed materialsWe have analyzed transmission properties of a layered periodic structure with left-handed materials We have shown the existence of narrow angular transmission resonances embedded into a wide band gapWe have shown the existence of narrow angular transmission resonances embedded into a wide band gap We have suggested applications of the transmission resonances for the beam shapingWe have suggested applications of the transmission resonances for the beam shaping

Oral presentations Ilya Shadrivov, Andrey Sukhorukov, and Yuri Kivshar, Guided waves and beam transmission in layered structures with left-handed materials, K22.010, APS March Meeting, March 3-7, 2003 Austin, Texas, USA Ilya Shadrivov, Andrey Sukhorukov, and Yuri Kivshar, Guided modes in negative refractive index waveguides, CMM5, CLEO/QELS, June 1-6, 2003, Baltimore Maryland, USA Ilya Shadrivov, Andrey Sukhorukov, and Yuri Kivshar, Beam shaping by a periodic structure of left-handed slabs, QThN3, CLEO/QELS, June 1-6, 2003, Baltimore Maryland, USA I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, Surface Polaritons of Nonlinear Left-Handed Materials, EB2-6-MON, CLEO/Europe – EQEC 2003, June, 2003, Munich, Germany

Related publications I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. E 67, (2003) A. A. Zharov, I. V. Shadrivov and Yu. S. Kivshar, Phys. Rev. Lett. 91, (2003) I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, Appl. Phys. Lett. 82, 3820 (2003) I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, submitted to Phys. Rev. E I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, Submitted to Appl. Phys. Lett.