Maxwell’s microscopic equations (gaussian units): Classical theory: Quantum theory:

Maxwell’s macroscopic equations Macroscopic charge density and current averaged over a volume ΔV, where a 0 3 << ΔV << (2πc/ω) 3 Gauss: Ampère: Faraday: Gauss' law for magnetism:

Purely transversal Currents and charge densities: External sources + internal sources We can distinguish three types of macroscopic internal sources: Conduction by free charges, polarization (‘bound charge) and magnetization

Gauss: Ampère: Magnetic field strength Gauss: Ampère: External field

Properties of the Medium, Linear Response to an externally applied electric field in homogeneous matter: Plane waves External currents are zero inside sample, Homogeneous sample: Ampère’s law:

Induced current: free charges+polarization+magnetization

Current response to an externally applied electric field in homogeneous matter:

Kramers Kronig Relations

Transverse EM+matter waves: Polaritons Polaritons: Transverse polarized waves of Matter & EM field Wave equation Substituton of this solution in the wave equation provides the dispersion relation:

It is often convenient to use the optical constant in this expression, which has a real and imaginary part: Note, that n>0 and k>0. Also Im(ε)>0, but it is possible to have Re(ε)<0. If Im(ε)=0 and Re(ε) 0, but there is no dissipation! The polariton solitions in the solid have the form

Case study: The Drude model

Optical techniques Polarizer Sample Au evaporator Polarizer Sample Analyzer ellipsometry reflection Optical conductivity  1 (  i    sample transmission

1) In most cases only information can be obtained for q << 1/a 0 Experimental ways to measure 2) can be found by means of optical refraction, reflection, absorption, and polarization analysis.

Transverse EM+matter waves: Unless specified otherwise, we will from now on assume that

Reflection and transmission at a vacuum-sample interface EiEi ErEr EtEt

Often the experiment provides the reflected intensity instead of the amplitude, and the phase of the reflected signal is in general difficult to measure. The reflection coefficient is: Kramers Kronig Relations are often used to get the phase of the reflectivity

Example I : pure Bi

E (meV) 0

Reflectivity at an oblique angle P-polarization: E p is Parallel to the plane of reflection b c a EpEp HsHs HsHs EpEp

Reflectivity at an oblique angle S-polarization: E s is Senkrecht to the plane of reflection b c a HpHp EsEs EsEs HpHp Senkrecht (german) = Perpendicular

NbN Optically isotropic Normal incidence grazing incidence. Angle = 80 0 p-polarized light

Grazing incidence. Angle = 80 0 p-polarized light

Josephson Coupled Planes d d C C L L Josephson Plasma Resonance at

 Re  0  1 Reflection normal to ac-plane  0 1 Grazing incidence reflection of ab-plane

La 2-x Sr x CuO 4 Tl 2 Ba 2 CuO 6

Spectroscopic ellipsometry: Measurement of |r p /r s | and  p -  s   (  )  i   (  )  - self normalizing technique (no reference is required) - measures directly both real and imaginary parts of the dielectric function

Spectroscopic ellipsometry b c a P A0A0

I) II) polariseranalyser Ellipsometrie A0A0 2γ Ellipsometry technique

polariseranalyser Ellipsometrie A0A0 2γ Ellipsometry technique I) II)

Spectroscopic ellipsometry Aspnes theorem b c a P A0A0 Aspnes theorem:

Bi2212

Pseudo ab-plane dielectric function ab-plane dielectric function corrected for c-axis admixture

Experiment and ab-initio calculations

Thick wedged films:

M.U. Gruninger, 1999 PhD Thesis YBa 2 Cu 3 O 6 Weakly absorbing excitations in insulating YBa 2 Cu 3 O 6 No absorbtive features in R(  ) Absorbtive features in T(  )

Optical Transmission

Thin films:

NbN d=400 nm 9 K 13 K 9 K 13 K 18 K

Fused quartz KRS5 NdGaO 3

SrTiO 3 Sr Transmission

THz time domain measurements Fabry-Perot etalon source detector

THz time domain measurements Fabry-Perot etalon source detector

THz transmission of SrTiO delay line (mm) intensity (a.u.) Time domain

THz transmission of SrTiO wavenumber (cm -1 ) delay line (mm) intensity (a.u.) transmission Time domain Frequency domain Fourier transformation

Drude-Lorentz fit with RefFIT

Transmission 50 Direct measurement of the polariton  (q) relation