Wave propagation in structures with left-handed materials

1
Wave propagation in structures with left-handed
materials

• Ilya V. Shadrivov
• Nonlinear Physics Group, RSPhysSE
• Australian National University, Canberra,
  Australia
• http//rsphysse.anu.edu.au/nonlinear/
• In collaboration with
• Yu. S. Kivshar, A. A. Sukhorukov, D. Neshev

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Outline

• Introduction Left-Handed Materials (LHM)
• Interfaces and surface waves
• Nonlinear properties of LHM
• Nonlinear surface waves
• Giant Goos-Hänchen effect
• Guided waves in a slab waveguide
• Photonic crystals based on LHM
• Presentations and publications

3
Left-handed materials

• Materials with negative permittivity e and
  negative permeability µ
• Such materials support propagating waves
• Energy flow is backward with respect to the wave
  vector

4
Frequency dispersion of LH medium

• Energy density in dispersive medium
• Positivity of W requires
• LH medium is always dispersive

5
The first experiment on LH media

• 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

Effective medium approximation
Frequency range with elt0, µlt0 4 - 6Ghz
6
Negative refraction at LH/RH interface
RH
LH
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Unusual lenses
LH lens does not have a diffraction resolution
limit Improved resolution is due to the surface
waves

• V. G. Veselago Soviet Physics Uspekhi 10 (4),
  509-514 (1968)
• J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)

8
Surface waves of left-handed materials
z
RHM
LHM
x
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Guided modes

• TE-polarization
• TM-polarization
• Solutions 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)
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Energy flux at the interface
z
Total energy flux
RHM
LHM
P1
P2
PP1P2
Forward waves P gt 0 Backward waves P lt 0
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Existence regions of surface waves

• Parameters

• No regions where TE and TM waves exist
  simultaneously
• Waves can be both forward traveling and backward
  traveling

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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?

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Nonlinear properties of left-handed materials

• Metallic composite structure embedded into the
  nonlinear dielectric

A. A. Zharov, I. V. Shadrivov, and Yu. S.
Kivshar, Phys. Rev. Lett. 91, 037401 (2003)
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Nonlinear dielectric properties of the composite

• Microscopic derivation in the effective medium
  approximation

Contribution from nonlinear dielectric
Contribution from metallic wires
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Effective magnetic permeability
Effective medium approximation
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Magnetic properties of composite materialwith
self-focusing dielectric
Frequency
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Magnetic properties of composite materialwith
self-defocusing dielectric
Frequency
18
Effective magnetic permeability
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Nonlinear properties management

• Composite completely filled by nonlinear
  dielectric

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Nonlinear properties management

• Only resonator gaps are filled by nonlinear
  dielectric

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Nonlinear properties management

• Composite completely filled by nonlinear
  dielectric, but resonator gaps

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Nonlinear surface waves
RHM
LHM
Self-focusing
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)
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Nonlinear dispersion of surface wave

• The dispersion is multi-valued
• Two different types of surface waves

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Localized polaritons
Vg - the group velocity d - the group velocity
dispersion ? - the nonlinear coefficient
Energy flow has a vortex structure
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Energy flow in a pulse
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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

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Goos-Hänchen effect
A shift of the reflected beam from the position
predicted by geometric optics ? ltlt width of the
beam
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Giant Goos-Hänchen effect

• ? width of the beam

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Giant Goos-Hänchen effect
LHM
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Resonant beam shift
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Energy flow at a negative beam shift
RHM

• Vortex
• surface wave excitation

Air
LHM
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Possible application

• Measure the angle of resonant shift
• Determine surface mode eigen wave number
• Calculate LH material parameters

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Negative refractive index waveguide
z
RHM
RHM
LHM
x
L
-L
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Guided modes in Negative Refractive Index
Waveguides

• TE-polarization
• TM-polarization
• Solutions for guided modes

I. V. Shadrivov, A. A. Sukhorukov and Yu. S.
Kivshar, Phys. Rev. E. 67, 057602-4 (2003)
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Dispersion of guided modes

• Fast and slow modes
• Fundamental mode may be absent
• Normal and anomalous dispersion

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Dispersion of guided modes

• Fast and slow modes
• Fundamental mode may be absent
• Normal and anomalous dispersion

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Sign-varying energy flux
PP1P2
Total energy flux
38
Energy flow in a pulse
39
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 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

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Unusual lenses

• V. G. Veselago Soviet Physics Uspekhi 10 (4),
  509-514 (1968)
• J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000)

41
Periodic structure witha 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, 083901 (2003)
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Reflection from periodic structures
Bragg resonant reflection f2 - f1 2pm,
m1,2,3
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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
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Band gap structure
I.V. Shadrivov, A.A. Sukhorukov and Yu.S.
Kivshar, Appl. Phys. Lett. 82, 3820 (2003)
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Beam shaping
Gaussian beam oblique incidence
Vortex beam normal incidence
Vortex beam oblique incidence
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Transmission properties of the layered structure.

• We 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
  gap
• We have suggested applications of the
  transmission resonances for the beam shaping

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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,
  22-27 June, 2003, Munich, Germany

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Related publications

• I. V. Shadrivov, A. A. Sukhorukov, and Yu. S.
  Kivshar, Phys. Rev. E 67, 057602-4 (2003)
• A. A. Zharov, I. V. Shadrivov and Yu. S. Kivshar,
  Phys. Rev. Lett. 91, 037401 (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.