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  1. Optical propertiesandInteraction of radiation with matter S.Nannarone TASC INFM-CNR & University of Modena

  2. Outline • Elements of Classical description of E.M. field propagation in absorbing/ polarizable media  Dielectric function • Quantum mechanics microscopic treatment of absorption and emission and connection with dielectric function Physics related to a wide class of Photon-in Photon-out experiments including Absorption, Reflectivity, Diffuse scattering, Luminescence and Fluorescence or radiation-matter interaction [some experimental arrangements and results, mainly in connection with the BEAR beamline at Elettra http://new.tasc.infm.it/research/bear/]

  3. Systems • bulk materials the whole space is occupied by matter • Surfaces  matter occupies a semi-space, properties of the vacuum matter interface on top of a semi-infinite bulk • Interfaces  transition region between two different semi-infinite materials Information [see mainly following lectures] • Electronic properties full and empty states, valence and core states, localized and delocalized states • Local atomic geometry /Morphology electronic states – atomic geometry different faces of the same coin Energy range Visible, Vacuum Ultraviolet, Soft X-rays)

  4. Synchrotron and laboratory sources/LAB Conceptually Shining light on a system, detecting the products and measuring effects of this interaction This can be done by Laboratory sources They cover in principle the whole energy range nowadays covered by synchrotrons (J.A.R.Samson Techniques of vacuum ultraviolet spectroscopy) • Incandescent sources • Gas discharge • X-ray e- bombardment line emission • Bremsstrahlung continuous emission sources • Higher harmonic source

  5. Synchrotron and laboratory sources / Synchrotron • Some well known features • Collimation • Intrinsic linear and circular polarization • Time structure (typically 01-1 ns length, 1 MHz-05GHz repetition rate) • Continuous spectrum, high energy  access to core levels • Reliable calculability of absolute intensity • Emission in clean vacuum, no gas or sputtered materials • High brilliance unprecedented energy resolution • High brilliance  small spot  Spectromicroscopy “The one important complication of synchrotron source is, however, that while laboratory sources are small appendices to the monochromators, in a synchrotron radiation set-up the measuring devices becomes a small appendices to the light source. It is therefore recommendable to make use of synchrotron radiation only when its advantages are really needed.” C.Kunz, In Optical properties of solids New developments, Ed.B.O.Seraphin, North Holland, 1976

  6. Radiation-Matter Interaction  Polarization and current induction in E.M. field Matter polarizes in presence of an electric field Result is the establishment in the medium of an electric field function of both external and polarization charges Matter polarizes in presence of a magnetic field Result is the establishment in the medium of a magnetic field function of both external and polarization currents The presence of fields induce currents • Mechanisms and peculiarities of polarization and currents induction in presence of an E.M. field • Scheme to calculate the E.M. field established and propagating in the material • Basis to understand how this knowledge can be exploited to get information on the microscopic properties of matter

  7. Basic expressions - Charge polarization and induced currents Polarization vectors

  8. -Ze- +Ze Charge and Magnetic/current polarization – closer look

  9. Induced currents Ze+ e- _ + Motion of charge under the effect of the electric field of the E.M. field but in an environment where it is present an E.M. field

  10. Expansion of polarisation Physical meaningElastic limit  the potential is not deformed by the field Linear and isotropic media Dielectric function Permeability function

  11. Linear versus non linear optics Formally linear optics implies neglecting terms corresponding to powers of the electric field Physicallyit meansE.M. forcesnegligible with respect to electron-nuclei coulomb attraction Nuclear atomic potential is deformed  not harmonic (out of the elastic limit) response  distortion  higher harmonic generation

  12. Dielectric function and response In very general way External stimulus Note is defined as a real quantity

  13. Summary material properties within linear approximation And Conduction under a scalar potential – Usual ohmic conduction Conduction in an e.m. field

  14. Maxwell equations in matter for the linear case Corresponding equations for vacuum case

  15. Wave equation - Vacuum Vacuum supports the propagation of plane E.M. waves with dispersion / wave vector energy dependence Wave equation - Matter Matter supports the propagation of E.M. waves with this dispersion Formally q is a complex wavevector Wave vector eigenvalue/dispersion depends on the properties of matter through    (all real quantities)

  16. Complex refraction index Absorption Phase velocity Real and imaginary parts not independent Absorption coefficient Lambert’s law

  17. Complex dielectric constant – Complex wave vector

  18. Supported/propagating E.M. modesdepend on the properties of matter through    The study of modes of the e.m. field supported/propagating in a medium and the related spectroscopical information is the essence of the optical properties of matter Relation between (r,t), (r,t) (r,t) or (q,) (q, ) (r ) and the properties of matter 1st part  Classical scheme / macroscopic picture 2nd part  Quantum mechanics / microscopic picture

  19. Spatial dispersion extension on which the average is made Note  0 wavevector does not mean lost of dependence on direction  anisotropic materials excited close to origin

  20. Unknowns and equations    (real quantities) are the unknowns related with the material properties (r,t) is close to unity at optical frequencies  magnetic effects are small (not to be confused with magneto-optic effects: i.e. optics in presence of an external magnetic field) Generally a single spectrum – f.i. absorption – is available from experiment (An ellipsometric measurement provides real and imaginary parts at the same time. It is based on the use of polarizers not easily available in an extended energy range) Real and imaginary parts are related through Kramers – Kronig relations Sum rules

  21. Kramers – Kronig dispersion relations Under very general hypothesis including causality and linearity

  22. Models for the dielectric constant / Lorentz oscillator Mechanical dumped oscillator forced by a local e.m. field Neglecting the magnetic term e- Induced dipole Out of phase – complex/dissipation – polarizability (Lorentzian line shape)

  23. Complex dielectric function From

  24. Lorentz oscillator Dielectric function

  25. Lorentz oscillator Refraction index

  26. Lorentz oscillator Absorption Reflectivity Loss function Physics Difference between transverse and longitudinal excitation EEL spectroscopy Optical spectroscopy

  27. Non linear Lorentz oscillator Anarmonic potential • induced dipole at frequency  and 2  • the system is excited by a frequency  but oscillates also at frequency 2  • re-emitting both  and 2 

  28. Lorentz oscillator in a magnetic field 1/2 x and y motions are coupled Solving for x and y Larmor frequency

  29. Lorentz oscillator in a magnetic field 2/2

  30. Lorentz oscillator in a magnetic field 1/3 The dielectric function is a tensor [ Physically lost of symmetry for time reversal ]

  31. Wave equation Eigenvalue equation Propagation in a magnetised medium 1/2 Note≠ 0 in anisotropic media with

  32. Propagation in a magnetised medium 2/2 Considering the medium with B||z

  33. Elliptically polarized  Rotation according to n+-n- Linear polarized Longitudinal geometry N+ Right circular polarized wave N-  Left circular polarized wave Dichroism Magneto-optics effects Two waves propagating with two different velocities and different absorption Magneto-optic effects e.g. Faraday and Kerr effects/geometries

  34. Dielectric tensor are in general tensorial quantities

  35. Dielectric tensor Scalar medium Magnetized medium

  36. Longitudinal and transverse dielectric constant 1/2 Any vector field F can be decomposed into two vector fields one of which is irrotational and the other divergenceless If a field is expanded in plane waves FT is perpendicular to the direction of propagation.

  37. Longitudinal and transverse dielectric constant 2/2 Optics  EELS/e- scattering The description in terms of longitudinal and transverse dielectric function is equivalent to the description in terms of the usual (longitudinal) dielectric function and magnetic permeability. They are both/all real quantities together with conductivity. They combine together to forming the complex dielectric constant defined here.

  38. Transverse and longitudinal modes 1/3 Propagating waves and excitation modes of matter are two different manifestation of the same physical situation Plasmon is a charge oscillation at a frequency defined by the normal modes oscillation produces a field  only a field of this kind is able to excite this mode _ + • Modes can be transverse or longitudinal in the same meaning of transverse and longitudinal E.M. field • searching for transverse waves is equivalent to searching for transverse modes

  39. Transverse and longitudinal modes 2/3 Searching for modes  eigenvectors of Transverse modes  Polaritons The quantum particles are coupled modes of radiation field and of the elementary excitations of the system, called Polaritons including transverse (opical) phonons, excitons,…. Longitudinal modes the quantum particles are coupled modes of radiation field and of the elementary excitations of the system: Plasmons, longitudinal opical phonons, longitudinal excitons,….

  40. Transverse and longitudinal modes 3/3 Polarization waves

  41. Sum rules for the dielectric constant Examples of sum rules Of use in experimental spectra interpretation

  42. Quantum theory of the optical constants Macroscopic optical response Microscopic structure Transition probability Ground state HRADIATION + HMATTER perturbed by radiation-matter interaction • Two approaches • fully quantum mechanics • semi classical • Three processes • Absorption • Stimulated emission • Spontaneous emission

  43. O ° O O° O °°O Term neglected for non relativistic particles Microscopic description of the absorption and emission process System Radiation mi,ei mj,ej • Interaction Hamiltonian HI • Effect of the interaction on the states of the unperturbed HR + HI Matter

  44. Hamiltonian of a charged particle in E.M. field

  45. Particle radiation interaction Matter Hamiltonian + perturbation Hamiltonian Problem to be solved Eigenstate and eigenvector of the matter radiation system in interaction

  46. Important notes The solution is found by a perturbative method • it is assumed here – formally - that the problem in absence of interactions has been solved. • In practice this can be done with more or less severe approximations. • The calculation of the electronic properties of the ground state is a special and important topic of the physics of matter Many particles state Generally obtained by approximate methods

  47. Transition between states of ground state due to the perturbation term The effect of perturbation HI on the eigenstates of H0 Obtained by time dependent perturbation theory

  48. Matrix elements 1/3 The evolution of the state m is obtained calculating the matrix element System states under perturbation due to Changes of photon occupation and matter (f.i. electronic) state

  49. Matrix elements 2/3 It is found that for photon mode k, only contribute linear terms to matrix elements +1  photon emission -1 photon absorption Probability of transition of the system from state

  50. Matrix element 3/3