Acousto Optic

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acousto optic

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DiractionatUltrasonicWavesFebruary7,2005DiractionatUltrasonicWavesContents1 Introduction 32 ProductionofUltrasound 33 PropagationofSoundWaves 83.1 Solution of the Wave Equation for Periodical Excitation . . . 114 DiractionofLightatUltrasoundWaves 125 Intensitydistributionoftheinterferencepattern 166 Apparatus 226.1 Diraction from Sonic-Waves . . . . . . . . . . . . . . . . . . 226.2 The Striae method . . . . . . . . . . . . . . . . . . . . . . . . 227 Problems 237.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2 General Problems . . . . . . . . . . . . . . . . . . . . . . . . . 232DiractionatUltrasonicWaves1 IntroductionThetermultrasound refers tomechanical oscillations, whosefrequenciesvary between 16 kHz, the upper audio limit of the human ear, and 1010Hz.A human ear can only hear sound in the range of approx. 16 to 16,000 Hz.Oscillationsintheregionbelow16Hzarereferredtoasinfrasound(e.g.earthquakes)andoscillationswithfrequenciesabove1010Hzarereferredtoashyper-sound. Thisupperfrequency-limitationfortheultrasoundisdeterminedbytheatomiccongurationofmatter. Thewavelengthoftheultrasoundinthisregionisintheorderofasmall multipleofthelatticeconstant a ( 103a); inthenext regionof hyper-sound, inwhichallthethermal movements of atoms andmolecules occur, strongquantum-mechanical eects have to be taken into account (phonon1theory). Belowtheseveryhighfrequencies of 1010Hzthematerial canbetreatedas acontinuumi.e. thelawsofclassical acousticsapplytoit, whichoriginallyonly dealt with the problem of the sound within the hearing range.Therearefewphysical applicationsforultrasound, themostimportantlythedeterminationofelasticityconstantsfrommeasurementsofthespeedof sound. Frommeasurements of thespeedof soundandabsorbability,thestructural natureofthematerial canbedeterminedwithinthescopeofmicroscopictheories. Todaytheapplicationsofultrasoundhavegrownoutofthenarrowareaofphysics; tobementionedaboveall isthesonar(found mainly on ships), the non destructive testing of material and medicaldiagnosticsinthehumanbody. Anultrasoundmicroscopeisalsounderdevelopment nowadays. Apart from these passive applications there is alsoanumber of activeapplications wheretheoscillationenergyis usedforperforming work processese.g. cleaning (ultrasonic baths), welding plasticsand treatment of ceramic materials.2 ProductionofUltrasoundThe oscillating systems used for the production of ultrasound waves shouldbe suitable for working with high frequencies. This means that all oscillatingsystems with spring and mass separated, which are used for the productionof sound in the hearing range, can not be used to produce ultrasound sincewecannotincreasetheireigenfrequenciesaboveacertainvalue. Instead,1soundparticles3DiractionatUltrasonicWavesFrequencyWave lengthA few inter-atomicintervals10-110111010109108107106105104103102101100InfrasoundAcousticrangeUltrasoundHypersoundIn solid s m c / 4000 m cm 1 . 0 25 In air s m c / 330 =mm m 20 20 Earthquake wavesm 100 For example: the wave lengthof green light is m 5 . 0 Figure 1: Sonic regions and typical wave lengthintheultrasoundrange, continuawhichareabletooscillateareusede.g.cavitieslledwithgasorliquidandsolidbodiesintheformof platesorbars. Inthesesystemstheelasticityofthematerial playstheroleofthespring, and the density together with the geometric properties play the roleofthemassi.e.thespringandmassaredistributedcontinuouslyovertheoscillator.Onecansimulateultrasonicoscillationwithfrequenciesuptoabout100kHzwithpurelymechanicoscillators. Examplesofmechanical oscillatorsare gas and liquid lled whistles, which work on the same principle as a setofwindinstruments. Fortheproductionofnon-sinusoidaloscillationsthehole-siren can be used.4DiractionatUltrasonicWavesOf far grater importance than mechanical oscillators are the electromechan-ical oscillators. Asthenameimplieselectricenergyisconvertedintome-chanical oscillation energy. In this group we can ndPiezoelectric converters.Magnetostrictive converters.Electrodynamic converters.Electrostatic converters.For the production of higher frequencies magnetostrictive and piezoelectricconverters are the most important.Magnetostrictiveconverters operate according to the Magnetostrictioneect. IfabarfromaFerromagneticmaterial,mostlyfromNi,ismagne-tized, itslengthlwillvaryslightlybyl, becausethemagneticmomentisalignedinthedirectionoftheeldandthusaectsthedeformationofthe lattice in the crystal. For aN-bar the relatively big changellamountsto 2.5 105inamagneticeldof 1Tesla(=14 105m/A). ApplyingoutsideforcestoanalreadymagnetizedN-barwillchangethemagnetiza-tionofthebar, whichinturnwillcauseavoltagesurgeinthebar. Thisvoltagesurgecanbemeasuredwhenaninductioncoiliswrappedaroundthe bar. Ultrasound can thus beproduced and measured using thesecon-vertersandgenerators. Magnetostrictiveoscillatorsaremostlysuitabletothe production of intense sound levels, up to 200 kHz.The piezoelectric converter is todays most frequently used sonic gener-atoranddetector. Comparedtothepreviouslydiscussedtechnologies,farhigher frequencies can be achieved (in the MHz range).The piezoelectric (or pressure-electric) eect was discovered in 1880 by theCurie brothers. With some crystals,when subjected to pressure or tensilestress in special crystallographic directions, electrical charges are realest oncertain crystal surfaces. The produced charges are proportional to the pres-sure or the stretch applied. The sign of the charges changes if for examplea compression alters into a dilation.The reversed piezoelectric eect was detected soon after in 1881. The samegroup of crystals , when put between two electrodes with a potential dier-ence, reacts by deformation. The direct piezoelectric eect is used for detec-5DiractionatUltrasonicWavestion of ultrasound waves and the reversed piezoelectric eect (Electrostric-tion) is used for their production. When placing an alternating voltage ontwo condenser plates, between which the crystal is located, the crystal willoscillate according to the frequency of the alternating voltage. The lengthvariationof thecrystal isproportional tothepiezoelectricmoduledandtheelectrical tensionputonit. Becausepiezoelectriccrystalsarealwaysanisotropic, d is a tensor and l also depends on the direction of the appliedelectric eld relative to the crystal axes. Putting a eld in parallel to a mainaxis we obtainl = djjU1(for quartz at low frequenciesd11 = 2.3 1012m/V ).For the receptor, the situation is analogous. The amplitude of the producedchange of pressure (by the sonic wave) is proportional to the tension on thecondenserplates. However, theproportionalityconstantsaredierentinthis case. The received voltageU2 isU2 = hii lwhereh is the deformation constant (for quartz with low frequenciesh11 =4.9 109V/m). For the same length variation, the voltagesU1andU2thusdier. The proportionU2U1for identical l is described by the square of thecoupling coecientkkii = _dii hiiIn general the coupling coecient at low frequencies is smaller than 1. How-ever, with resonance support and low damping the coupling coecient canbecome nearly 1.All crystals that show the piezoelectric eect have several similar qualities,theyisolatewellandhaveoneormorepolaraxes. A180turnofapolaraxis does not result in the same state. The piezoelectric eect now appearsin the directions of the polar axes. Crystals on which the piezoelectric eectcanbeobservedareforexample: lithiumsulfate,tourmaline,zincblende,Seignette-salt and tartaric acid.In a quartz crystal, which we want to study as an example, we have threepolar axes. Quartz has the chemical formulaSiO2and it forms hexagonalcrystals. EverySi-atom has four positive elementary charges and everyO-atom two negative elementary charges. Figure 2(a) shows a structural cellof quartz.6DiractionatUltrasonicWaves(a)(b) Si +O -O -O -Si +Si +X 1 X 3 X 2 ----+ -- +++++ Si +O -O - O -Si + Si +Staggerd point ofthe + and -charges Dipol momentFigure 2: A structural cell of quartzX1, X2 and X3 are the polar axes. If we now apply, for instance, pressure indirection of the X1 axis , we reach the situation depicted in gure 2(b) wherewe see the resulting shift of the atoms and charges on the surface. Qualita-tively the same shift can be achieved, if a thrust is applied perpendicularlyto theX1 axis (transverse piezoelectric eect).Apartfromthisgroupofpiezoelectricsinglecrystalsthereisalsoaseriesof ferroelectricsubstances. Withthesessubstancesthedipolemomentisnot only produced by applying pressure or tension, but the electric dipolesalready exist within the crystal unit cell, similar to the magnetic momentsexistine.g. Fe. Forthisreasonferroelectricmaterialshaveaveryhighdielectric constant. Applying an electric eld at a high temperature alignsthese electrical dipole moments in the same way that a magnetic eld alignsthemagneticdipolemomentsinFe. Whenaferromagneticmaterial inan electric eld is cooled down below a certain temperature, known as theCurrie-temperature, the aligned electric dipole moments freeze. The out-come is a permanent macroscopic electric dipole. This modication remainstoalargeextent, aslongasthetemperatureofthesampledoesnotriseabove the Curie-temperature. Above the curie temperature the polarizationdisappearsirreversibly, i.e. thesamplemustbepolarizedagainusinganelectric eld.In contrast totheexample of aquartz piezo-crystal,ferromagnetic sound-converters are not necessarily single crystals. Poly-crystaline materials aresucient, which can be produced in an inexpensive way and in various forms7DiractionatUltrasonicWaves(plates, tubes, hollowspheres)bymeansofsintering. Examplesforthesematerialsare: leadzirconatetitanate(PZT), bariumtitanate, leadmetaniobium and lithium niobium.In this experiment a PZT oscillator is used. PZT is a mixture of PbZrO3 andPbTiO3. Thepiezoelectricconstantdependsonthemixingproportionofthe two materials and can vary within certain limits. The Curie-temperaturefor the PZT is about 250C.3 PropagationofSoundWavesIfpressureortension2isappliedtoamec