Metamaterials. Peter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China

Transcription

1 University of Osnabrück, Germany Lecture presented at APS, Nankai University, China Spring 2012

2 are produced artificially with strange optical properties for instance negative dielectric permittivity *and* negative magnetic permeability split ring resonator SRR Snellius law and more Same for a material Applications

3 ISBN Graduate Texts in Physics Continuum Physics This small book on the continuously distributed matter covers a huge field. It sets out the governing principles of continuum physics and illustrates them by carefully chosen examples. These examples comprise structural mechanics and elasticity, fluid media, electricity and optics, thermoelectricity, fluctuation phenomena and more, from Archimedes principle via Brownian motion to white dwarfs., pattern formation by reaction-diffusion and surface plasmon polaritons are dealt with as well as classical topics such as Stokes formula, beam bending and buckling, crystal optics and electro- and magnetooptic effects, dielectric waveguides, Ohm s law, surface acoustic waves, to mention just some. The set of balance equations for content, flow and production of particles, mass, charge, momentum, energy and entropy is augmented by material, or constitutive equations. They describe entire classes of materials, such as viscid fluids and gases, elastic media, dielectrics or electrical conductors. We discuss the response of matter to rapidly oscillating external parameters, in particular the electric field strength of light, in the framework of statistical thermodynamics. An appendix on fields and a glossary round off this bird s-eye view on continuum physics. Students of physics, engineering and related fields will benefit from the clear presentation of worked examples and the variety of solution methods, including numerical techniques. Lecturers or advanced students may profit from the unified view on a substantial part of physics. It may help them to embed their research field conceptually within a wider context. Hertel 1 Continuum Physics Graduate Texts in Physics Continuum Physics Physics On sale

4 Drude model harmonically oscillating field E(t) = Ẽ e iωt elastically bound damped electron m{ẍ + Γ ẋ + Ω 2 x} = qe Fourier transform it x = q Ẽ m ω 2 iγ ω + Ω 2 polarization for N electrons per unit volume P = Nq2 Ẽ m ω 2 iγ ω + Ω 2 magnetization M negligibly small

5 Dielectric permittivity susceptibility χ defined by P = χ(ω)ɛ 0 Ẽ generalized Drude model χ(ω) = f a Ωa 2 ω 2 iγ a a ω + Ωa 2 sum over all resonance frequencies Ω a with weights f a (oscillator strength) a variant of this is also known as Sellmeier s formula permittivity function ɛ(ω) defined by D = ɛ(ω)ɛ 0 Ẽ ɛ(ω) = 1 + χ(ω) permittivity of natural materials, like solids or liquids

6 If you want strange optical effects... you must provide for strange, not just elastically bound damped electrons. are arrays of resonating circuits the dimension of which is small if compared with the wavelength can be realized easily for microwaves metamaterials require advanced nano-technology such as self-assembling as of today: science fiction

7 The split ring resonator. Inductance L and capacitance C. Resonance frequency is Ω = 1/ LC.

8 Maxwell s equations Maxwell s equations for plain waves f(t, x) = f e iωt e ik x curl of the magnetic field k H = ωɛ(ω)ɛ 0 Ẽ curl of the electric field k Ẽ = ωµ(ω)µ 0 H k = kˆk and Ẽ = Ẽê wave number, propagation direction, amplitude, polarization vector solution H = kẽ ˆk ê ωµµ 0

9 Maxwell s equations ctd. dispersion relation k 2 c 2 = ω 2 ɛ(ω)µ(ω) or, with k 0 = ω/c k = n(ω) k 0 where n(ω) = + ɛ(ω)µ(ω) Poynting vector S = 2 Re E H S = 2n Ẽ 2ˆk cµµ 0 n real if ɛ > 0 and µ > 0 this is a forward n also real if ɛ < 0 and µ < 0 meta instead of natural material

10 y ˆk i ˆk r n I = ǫ I µ I n II = ǫ II µ II α I α I α II x ˆk t Incident, transmitted and reflected beam at interface of normal materials

11 Continuity requirements let us study perpendicularly polarized light only E = E z components do not vanish y > 0: incident+reflected; y < 0: transmitted E i = E i e ik 0n I (sin α i x cos α i y) E r = E r e ik 0n I (sin α r x + cos α r y) E t = E t e ik 0n II (sin α t x cos α t y) tangential components must be continuous E i + E r = E t incident and reflected beam angles are equal, Snell s law n I sin α i = n I sin α r = n II sin α t H x, H z and µh y must be continuous satisfied if n I cos α I µ I (E i E r ) = nii cos α II µ II E t

12 we discuss an ideal ɛ = 1 and µ = 1 and its plane interface with vacuum refractive indexes of both media are n I = n II = 1 all angles are equal as before: E r + E i = E t different: E r E i = E t therefore E i = 0 and E r = E t the formerly reflected beam is now incident the formerly incident beam vanishes

13 y ideal S i ǫ = µ = 1 ǫ = µ = +1 α α x vacuum S t Power flow of a a beam which passes from an ideal to vacuum.

14 2w w w 2w A parallel slab of ideal material. It maps the upward object at the left into an equally large upward image at the right. No diffraction limitation!

15 consist of regularly arranged LC Realized for microwaves, still science fiction for optics progress in nano technology expected in particular, self-assembling metamaterials show positive and negative permittivities and permeabilities if both are negative - in a certain frequency range - the energy flows counter to wave propagation metamaterials may be strange behavior at interfaces between normal and metamaterial example: a new microscope not restricted by the diffraction limit