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Optical properties and Interaction of radiation with matter. S.Nannarone TASC INFM-CNR & University of Modena. Outline. Elements of Classical description of E.M. field propagation in absorbing/ polarizable media Dielectric function

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Optical propertiesandInteraction of radiation with matterS.NannaroneTASC INFM-CNR & University of ModenaOutlineElements 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 Elettrahttp://new.tasc.infm.it/research/bear/]Systemsbulk 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)Synchrotron and laboratory sources/LABConceptually Shining light on a system, detecting the products and measuring effects of this interactionThis 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 Synchrotron and laboratory sources / SynchrotronSome 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, 1976Radiation-Matter Interaction Polarization and current induction in E.M. fieldMatter polarizes in presence of an electric field Result is the establishment in the medium of an electric field function of both external and polarization chargesMatter polarizes in presence of a magnetic field Result is the establishment in the medium of a magnetic field function of both external and polarization currentsThe presence of fields induce currentsMechanisms 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 Basic expressions - Charge polarization and induced currentsPolarization vectors-Ze-+ZeCharge and Magnetic/current polarization – closer lookInduced currentsZe+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. fieldExpansion of polarisationPhysical meaningElastic limit the potential is not deformed by the fieldLinear and isotropic mediaDielectric functionPermeability functionLinear versus non linear opticsFormally linear optics implies neglecting terms corresponding to powers of the electric fieldPhysicallyit 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 generationDielectric function and responseIn very general wayExternal stimulusNote is defined as a real quantitySummary material properties within linear approximationAnd Conduction under a scalar potential – Usual ohmic conductionConduction in an e.m. fieldMaxwell equations in matter for the linear caseCorresponding equations for vacuum caseWave equation - VacuumVacuum supports the propagation of plane E.M. waves with dispersion / wave vector energy dependence Wave equation - MatterMatter supports the propagation of E.M. waves with this dispersion Formally q is a complex wavevectorWave vector eigenvalue/dispersion depends on the properties of matterthrough (all real quantities)Complex refraction indexAbsorptionPhase velocity Real and imaginary parts not independentAbsorption coefficientLambert’s lawComplex dielectric constant – Complex wave vectorSupported/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 matterRelation between (r,t), (r,t) (r,t) or (q,) (q, ) (r ) and the properties of matter 1st part Classical scheme / macroscopic picture2nd part Quantum mechanics / microscopic pictureSpatial dispersionextension on which the average is madeNote 0 wavevector does not mean lost of dependence on direction anisotropic materials excited close to originUnknowns 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 relationsSum rulesKramers – Kronig dispersion relationsUnder very general hypothesis including causality and linearityModels for the dielectric constant / Lorentz oscillatorMechanical dumped oscillator forced by a local e.m. field Neglecting the magnetic terme-Induced dipoleOut of phase – complex/dissipation – polarizability (Lorentzian line shape)Complex dielectric functionFrom Lorentz oscillator Dielectric functionLorentz oscillator Refraction indexLorentz oscillator Absorption Reflectivity Loss functionPhysics Difference between transverse and longitudinal excitationEEL spectroscopyOptical spectroscopyNon linear Lorentz oscillatorAnarmonic 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 Lorentz oscillator in a magnetic field 1/2x and y motions are coupledSolving for x and yLarmor frequencyLorentz oscillator in a magnetic field 2/2Lorentz oscillator in a magnetic field 1/3The dielectric function is a tensor[ Physically lost of symmetry for time reversal ]Wave equationEigenvalue equationPropagation in a magnetised medium 1/2Note≠ 0 in anisotropic mediawithPropagation in a magnetised medium 2/2Considering the medium with B||zElliptically polarized Rotation according to n+-n-Linear polarizedLongitudinal geometryN+ Right circular polarized wave N- Left circular polarized waveDichroism Magneto-optics effectsTwo waves propagating with two different velocities and different absorptionMagneto-optic effects e.g. Faraday and Kerr effects/geometriesDielectric tensorare in general tensorial quantitiesDielectric tensorScalar mediumMagnetized mediumLongitudinal and transverse dielectric constant 1/2Any vector field F can be decomposed into two vector fields one of which is irrotational and the other divergencelessIf a field is expanded in plane waves FT is perpendicular to the direction of propagation.Longitudinal and transverse dielectric constant 2/2Optics EELS/e- scatteringThe 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.Transverse and longitudinal modes 1/3Propagating waves and excitation modes of matter are two different manifestation of the same physical situationPlasmon 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 modesTransverse and longitudinal modes 2/3Searching for modes eigenvectors of Transverse modes PolaritonsThe quantum particles are coupled modes of radiation field and of the elementary excitations of the system, called Polaritons including transverse (opical) phonons, excitons,….Longitudinal modesthe quantum particles are coupled modes of radiation field and of the elementary excitations of the system: Plasmons, longitudinal opical phonons, longitudinal excitons,….Transverse and longitudinal modes 3/3Polarization wavesSum rules for the dielectric constantExamples of sum rulesOf use in experimental spectra interpretationQuantum theory of the optical constantsMacroscopic optical response Microscopic structureTransition probabilityGround state HRADIATION + HMATTER perturbed by radiation-matter interactionTwo approaches fully quantum mechanics semi classical Three processes Absorption Stimulated emission Spontaneous emission O ° OO° O °°OTerm neglected for non relativistic particlesMicroscopic description of the absorption and emission processSystemRadiationmi,eimj,ejInteraction Hamiltonian HI Effect of the interaction on the states of the unperturbed HR + HI MatterHamiltonian of a charged particle in E.M. fieldParticle radiation interactionMatter Hamiltonian + perturbation HamiltonianProblem to be solvedEigenstate and eigenvector of the matter radiation system in interactionImportant notesThe 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 methodsTransition between states of ground state due to the perturbation termThe effect of perturbation HI on the eigenstates of H0Obtained by time dependent perturbation theoryMatrix elements 1/3The evolution of the state m is obtained calculating the matrix elementSystem states under perturbation due to Changes of photon occupation and matter (f.i. electronic) stateMatrix elements 2/3It is found that for photon mode k, only contribute linear terms to matrix elements+1 photon emission -1 photon absorptionProbability of transition of the system from state Matrix element 3/3Spontaneous emissionStimulated emissionSpontaneous emission present only in quantum mechanics treatmentTransition probabilitiesIntegrating in time from 0 to infinity for the transition probabilitiesper unit time for probability of finding the state in a state n, at thermodynamic equilibrium Dielectric function and microscopic propertiesDissipatedpowerMicroscopic expression of the dielectric functionPhysical meaning Sum of all the absorbing channels at that photon energyNote dissipation originates from non radiative de-excitation channelsIntuitive meaning of the expression for absorption coefficientEn’N(E) density of states(Number of states/eV)Joint Density Of States N(E) N(E’) EnDipole approximationMatrix elements of position operatorSemi classical approachNote that the same result can be obtained by considering the transition probability between quantized states of the matter system under the effect of classical external perturbation of the E.M. field with given by the same expression of This semi classical approach gives identical results for absorption and stimulated emission probabilities, but does not account for spontaneous emissionSelection rules for Hydrogen atomGeneric light polarizationSelection rules/2For radiation polarized along z[linear polarized light ]Expressions valid in any central fieldHydrogen - Selection rules Circular polarizationCalculation of matrix elements - Optical properties of matterThe basic step in calculation involves many particles wavefunctionsBorn - Oppenheimer approximationNuclear motions separated from electronic motionsOne electron descriptionOne electron WF Solution of motion in an average potential generated by all other electronsDielectric function in one electron approximationCase of crystalsK reduced vector within the Brillouin zoneCrystal states E(k) Joint Density of States - JDOSPhenomenology of absorptionInterband transitions direct/indirect Intraband absorption Phonon contribution Core/localized (e.g. molecular) level absorption Local field effects - Local (Lorentz) field correctionsDecay and relaxation of excited statesProbability of relaxation/decay of excited state as integral on all the spontaneous emission channels of field and matter statesAs a consequence the dependence of Im () has to be modifiedLorentzian broadening function substituted for by Lorentzian curve (e.g. see Lorentz oscillator)Lorentzian broadeningExploitation of emission / radiative decayTotal/Partial yield measurement of absorption through electron (Secondary, Auger,..) and photon (fluorescence, luminescence,…) yieldsDe-excitation spectroscopies Fluorescence Luminescence XEOL Auger electron and photon induced – Selection rules and surface sensitivity matrix element ~ constant I Density of states and 3Boundaries reflectivityFrom material filling the whole space to material with boundaries and matter-vacuum interfacesReflectivity - Measure of the reflected intensity as a function of incident intensity Fresnels relations based on boundary conditions of fields linkreflected intensity with dielectric function Reflectivity from a semi-infinite homogeneous materialSurface planeNormal to surfaceModellisation of surfaces and interfacesMultiple boundariesS and p reflectivityDiffuse scattering 1/2Small and/or rough objectsScattered wavescattering mediumIncident fieldTerm neglected if dimensionsInhomogeneous filling of spaceScalar theory of scattering(single Cartesian component)Defining:Scattering potentialDiffuse scattering 2/2Born approximationThe scattering amplitude is the Fourier transform of the scattering potentialInverting F(r) n(r) ConclusionsClassical scheme Introduction of the dielectric functionMicroscopic (quantum mechanics) treatment of emission and absorptionRelation between macroscopic dielectric function (measured quantity) and microscopic propertieshttp://www.gfms.unimore.it/Calculation of ij elementsSource35x 1012T320eV r = 1 m 0.1% BW4.5243.5//13Eoy N/C2.502x 10121.5-11Eoz N/C0.5-20-2-1.5-1-0.500.511.52y (mrad)-3-2-1.5-1-0.500.511.52y (mrad)Source: 3.3 m of arc, 3.1 mx 3.3 m vertical x horizontaltwo fields – vertical and horizontal – out of phase of ±/2according to the sign of take off angle (J.Schwinger PR 75(1949)1912)Electric fields≈ 103 photons/bunch - bunch duration ≈ 20 ps320eV r = 1 m 0.1% BWPolarimetry 100 eV ellipseSelector fully open: Zc= 45 mm, Zg = 1 mmS1=-0.9 S2=0.011 S3=-0.068Ey=0.95 Ez=0.04 =-1.4 Ellipticity, =0.04 Polarization selector position: Zc = 34 mm, Zg = 41 mm (aperture 4 mm) S1=-0.97 S2=0.011 S3=0.082 Ey=0.98 Ez=0.04 =-1.44 Ellipticity, =0.33 Polarization selector position: Zc = 31 mm, Zg = 31 mm (aperture 14 mm) S1=-0.77 S2=0.08 S3=-0.57 Ey=0.93 Ez=0.31 =1.43 P2Helicity selectorBPMGAS CELLP1EXIT SLITSMONOTransport and conditioning opticsSource 4 m HxV Mirrors in sagittal focusing reduction of slope errors effects in the dispersion planeIntensity monitorLight spotEnergy range 3- 1600 eVEnergy resolution E/E ≈ 3000 (peak 5000) at vertical slit (typically 30 μm) x 400 μm (variable)Variable divergence (maximum, variable) 20 m vert x horellipticity variable horizontal/vertical (typically in the range 1.5 – 3.5, Stokes parameters (normalized to the beam intensity) S1 0.5 - 0.6, S2 0 - 0.1, S3 0.75 -0.85 )helicity variable (typical value for rate of circular polarization P or S3 0.75 – 0.95)plane-grating-plane mirror monochromator based on the Naletto-Tondello configurationExamples and experimental arrangements at BEAR (Bending magnet for Absorption Emission and Reflectivity)Bulk materials SurfacesInterfacesAbsorptionReflectivityFluorescenceLuminescence – XEOLDiffuse scatteringExperimental arrangementsBEAR (Bending magnet for Emission Absorption Reflectivity) beamline at ElettraExperimental/scattering chamberDetectione- analyser /photodiodes(2 solid angle)VIS LuminescencemonochromatorGoniometers M,A 0.001°A 0.01°S 0.05°C 0.1°(Positive -Differentially pumped joints)Sample manipulator6 degree of freedomRotation around beam axis any position of E in the sample frameOptical constants of rare hearths See f.i. Mónica Fernández-Perea, Juan I. Larruquert, José A. Aznárez1, José A. Méndez Luca Poletto, Denis Garoli, A. Marco Malvezzi, Angelo Giglia, Stefano Nannarone, JOSA to be publishedultra-thin deposited filmsburied interface spectroscopy Devices of use in spectroscopy ZInterfaces & surface physics in periodic structures(multilayer optics)See also posterP III 26ML : Artificial periodic stack of materials(Optical technology Band pass mirrors)BRAGGAt Bragg Z dependent standing e.m. field establishes both inside the structure and at the vacuum-surface interface modulated in amplitude and positionScanning throughBragg peakIn energy or angleSpectroscopy of interfacesPhysics of mirror/ReflectionStanding waves & excitationSiMoSiLocal modulation of excitationPhotoemission, Auger, fluorescence, luminescence etc..Cr2O3(6 Å)Cr(15 Å)Sc(25 Å)Cr/Sc Cr-Oxide interface (As received )573 eVX 60Qualitative analysis-Opposite behavior of Cr and ScDifferent chemical states of the buried Sc Two signalsfrom oxygen: one bound to Cr at the surface, the second coming from the interface - Carbon segregation at the interfaceMo(39.6 Å)Ru(15 Å)Si(41.2 Å)Ru (Clean) -Si buried interfaceAngular scan through the Bragg peakat 838 eVX 40silicideModel – Ru-Si interface Interface morphology Calculation of e.m. field inside Ml Photoemission was calculated, (Ek= h - EB) Minimum position and lineshape depend critically on the morphology profile RuRu-SiMo/Si61.2 %58.4 %Wavelength [nm]Mo-Si ML & i.f. roughnessMotivation role of ion kinetic energy and flux during ML growthML (P 8 nm, 0.44) Performance- R (10°)Ion assistanceIons EK: 5 eV (1st nm), 74 eV > 1nm Controlled activation of surface mobilityPerformance & diffuse scatteringPerformance differences are to be related to interface quality Diffuse scattering around the specular beam was measuredKS=Ki + qZ + q// f.i. Stearns jAP 84,1003diffuse scattering - MLsI.f.roughnes produces diffuse scattering around the specular beamI.f.roug

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