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2. Introduction to BRILLOUIN Spectroscopy SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume 2. Introduction to BRILLOUIN spectroscopy BRILLOUIN spectroscopy isis a well-established optical spectroscopic technique to investigate mechanical as well as magnetic properties of matter (e.g. SANDERCOCK 1982, KRÜGER 1989, HILLEBRANDS 2005). The technique bases on the phenomenon of inelastic laser light scattering, for whwhichich the incident photons are scattered by thermally driven elementary excitations in matter, like sound waves (acoustic phonons) or spin waves (magnons) (BRILLOUIN 1922, MANDELSTAM 1926, GROSS 1930, FABELINSKII 1968, CHU 1974, BERNE & PECORA 1976, DIL 19821982). Note that BRILLOUIN spectroscopy gives access to hypersonic waves and spin waves without exciting them explicitly by transducers as necessary e.g. for ultrasonic pulse-echo techniques. Since manifold introductory review articles and monographs on BRILLORILLOUINUIN scattering are available (FABELINSKII 1968, CHU 1974, BERNE & PECORA 1976, DIL 1982, SANDERCOCK 1982, KRÜGER 1989, HILLEBRANDS 2005) and since the current article focuses on scanning BRILLOUIN microscopy, we give only a short introduction to the theoretheoreticaltical background of BRILLOUIN scattering. In a first reading, the sections marked with an asterisk* 2.1 (except figure 2.1 and the related explications) and 2.4 can be skimmed over. 2.1. Scattering of laser light by thermally excited sound waves* The ththeoryeory of inelastic scattering of visible light by thermally excited hypersonic waves travelling through transparent matter will be recapitulated within a statistical framework (CHU 1974, BERNE & PECORA 1976). Remember that indeed the continuum of sound wavewavess within condensed matter can be correlated to spatio- temporal fluctuations of the mass density, or more generally, of the strain tensor components (AULD 1973). ConsiderConsider a monochromatic a monochromatic laser beam laser that beam can thatbe decomposedcan be decomposed into planar into waves,planar incidentwaves, incident on a di electricon a dielectric sample. sample. The electric The electric part of part each of incident each incident planar planar light wave light canwave be can described be described as: as: , (2.1) 11 98 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch.Arch. Sci.Sci. Nat.Nat. Phys.Phys. Math.Math. NSNS 46,46, SpecialSpecial volumevolume where denotes the wave vector in the medium, thethe positionposition vectorvector inin thethe sample’ssample’s coordinatecoordinate system,system, thethe wave’swave’s angularangular frequency,frequency, thethe electricelectric fieldfield amplitudeamplitude andand defines the direction of polarispolarisation.ation. For the sake of simplicity, we assumeassume thatthat within anisotropicanisotropic matter,matter, thethe incidentincident andand scatteredscattered electromagneticelectromagnetic waves represent eigenstates of the electromagnetic field. By this means we exclude effectseffects relatedrelated toto birefringence.birefringence. TheThe spatiallyspatially andand ttemporallyemporally fluctuatingfluctuating dielectricdielectric properties of the sample are described by the dielectric tensor .. IfIf thethe componentscomponents ofof thethe dielectricdielectric tensortensor atat opticaloptical frequenciesfrequencies fluctuatefluctuate onlyonly spatially,spatially, but not temporally, elastic light scatscatteringtering can occur. In that case, the scattered electromagneticelectromagnetic waveswaves generallygenerally propagatepropagate alongalong otherother directionsdirections thanthan thethe incidentincident one while the energy is conserved: andand ,, with denoting the scatteredscattered wavewave vector.vector. InIn thethe domaindomain of materials science, typical candidates for elasticelastic lightlight scatteringscattering processesprocesses areare polycrystallinepolycrystalline materialsmaterials andand nanocompositesnanocomposites possessing optical heterogeneities in the 100 nanometer to micrometer rangerange.. InelasticInelastic lightlight scatteringscattering occursoccurs ifif thethe componentscomponents of the dielectric tensor at optical frequenciesfrequencies do not only fluctuate spatially, but also temporally. In this context,context, twotwo aspectsaspects needneed consideration:consideration: thethe spatiallyspatially andand tempotemporallyrally fluctuatingfluctuating elementaryelementary excitationsexcitations ofof thethe samplesample thatthat couplecouple toto itsits opticaloptical propertiesproperties andand thethe strengthstrength ofof thisthis coupling.coupling. TheThe spatiospatio--temporaltemporal elementaryelementary excitationsexcitations ofof thethe samplesample cancan bebe divideddivided intointo relaxationalrelaxational fluctuations,fluctuations, likelike fluctuationsfluctuations ofof thethe entropy,entropy, andand propagating fluctuations, like fluctuationsfluctuations of the mass density (BERNE & PECORA 1976). The corresponding modes are either diffusive or propagating ones. Sound waves (also denoted as acoustic phonons) are generally induced by thermally excitedexcited fluctuationsfluctuations ofof thethe strainstrain tensortensor components,components, whichwhich degeneratedegenerate inin idealideal liquidsliquids toto massmass densitydensity fluctuations.fluctuations. TheThe elastoelasto--optical coupling between the strain fluctuationsfluctuations andand thethe opticaloptical fluctuationsfluctuations isis describeddescribed byby thethe PockelsPockels tensortensor ((NYE 1972, VACHER & BOYER 1972).1972). ForFor mostmost materials,materials, thethe elastoelasto--opticaloptical coupling for shearshear strainstrain isis muchmuch weakerweaker thanthan thatthat forfor longitudinallongitudinal strainstrain ((NYE 1972, VACHER & BOYER 1972). The spatiallyThe andspatially temporally and fluctuatingtemporally dielectricfluctuating tensor dielectric at optical tensor frequencies at optical can befrequencies described can as: be described as: 12 109 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume , (2.2) with denoting the spatially and temporally averaged dielectric tensor and the fluctuating part of the dielectric tensor. ThThee transversely polarized phonons are responsible for the off-diagonal entries of . The directions of polarization of the incident planar light wave and of the chosen inelastically scattered light wave yield the components according to: . (2.3) These dielectric fluctuations can be described by the spatio-temporal autocorrelation function of , with V being the scattering volume and the considered time interval: (2.4) The dynamic structure factor of the scattered light component travelling in the -direction results from the spatio-temporaltemporal FOURIER transformation of the autocorrelation function of : , (2.5) with designating the acoustic pphonon’shonon’s wave vector and (with being the angular frequencies of the involved scattered electric fields; see section 2.2) its angular frequency. The spectral power density , wwhichhich is closely related to the dynamic structure factor, is experimentallyexperimentally accessible by BRILLOUIN spectroscopy. The factor depends on the intensity of the incident light. As will be delineated below in a rough overview, the BRIRILLOUINLLOUIN lines of the spectral power density are intimately related to the hypersonic properties of the sample, since these lines are due to dielectric fluctuations induced by thermal fluctuations of the strain tensor components (SOMMERFELD 1945, GRIMVALL 191986,86, LANDAU & LIFSHITZ 1991). As describedAs described in KONDEPUDI in KONDEPUDI & PRIGOGINE & P (1998),RIGOGIN theE (1998),thermally the excited thermally fluctuations excited fluctuationsof the strain oftensor the straincomponents tensor developcomponents according develop to accordingthe same lawsto the as same macroscopic laws as macroscopicdeformations deformationsof small amplitude. of small Within amplitude. the frameWithin of the linear frame res ofponse linear theory, response the 13 1110 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial ScanningScanning Brillouin Brillouin microscopy microscopy Arch. Arch. Sci. Sci. Nat. Nat. Phys. Phys. Math. Math. NS NS 46, 46, Special Special volume volume straintheory, the strain can be related can beto relatethe thermallyd to the thermally excited excitedfluctuating fluctuating elastic elasticforce densityforce density via the via elastic the elasticsusceptibility susceptibility tensor tensor , according, according to: to: . (2.6)(2.6) ConsideringConsidering thethe storingstoring andand dissipativedissipative partsparts ofof thethe elasticelastic interactioninteraction byby thethe symmetricsymmetric forthforth rankrank tensorstensors ofof thethe elasticelastic modulimoduli andand viscositiesviscosities (with(with k,k, l, l, m m,, n n = = 1, 1, 2, 2, 3), 3), thethe strainsstrains evolveevolve withwith timetime likelike dampeddamped oscillators.oscillators. TheThe componentscomponents ofof thethe inverseinverse elasticelastic susceptibilitiessusceptibilities correspondcorrespond to to ( (LLANDAUANDAU & & L LIFSHITZIFSHITZ
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