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Devices and Materials for Thz Spectrosopy

Permanent Link: http://ufdc.ufl.edu/UFE0024735/00001

Material Information

Title: Devices and Materials for Thz Spectrosopy Ghz Cmos Circuits, Periodic Hole-Arrays and High-Frequency Dielectric Materials
Physical Description: 1 online resource (115 p.)
Language: english
Creator: Arenas, Daniel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: arrays, bismuth, cmos, detection, ftir, ghz, hole, infrared, periodic, pyrochlores, raman, sources, spectroscopy, terahertz
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is composed of three main projects, linked together by the THz region of the electromagnetic spectrum. In the first project, we detected the radiation from a silicon CMOS circuit, using a fourier transform interferometer. At the time of measurement, this 410 GHz circuit had the highest operating frequency for silicon integrated technology observed to date. The measured radiated power from the 410 GHz circuits was in the order of 0.01 microWatts. This circuit had radiated intensities comparable to those of commercially available black-body sources for the 200 - 400 GHz region. The high power and high emission per source area suggested possible spectroscopy applications. We also studied the optical properties of periodic hole-arrays with resonant frequencies in the THz region. Although the transmittance spectra of these structures have been extensively studied, here we present reflectance measurements that allow the analysis of the extinction/absorption spectra. The results were compared to predictions from the trapped-mode theory on the ohmic losses of these systems. Our results did not show the prediction of a suppression of the R + T spectra at the resonant frequency. Also, we studied the time-dependence of femtosecond pulses reflected from periodic hole arrays with resonant frequencies in the NIR region. Our results show that if the trapped modes theory is correct, then the lifetime of these modes are below 100 fs. Finally, in the third project, we studied the Raman active modes of various bismuth pyrochlores containing Zn, Mg, Ta and Nb, which have earned recent attention for high-frequency applications. The spectra of the four compositions are very similar, suggesting no major structural differences among these materials. The spectra were compared to those of other pyrochlores and specific discussions are offered for the assignment of each mode. Although there are clear differences between the spectra of these samples compared to other pyrochlores, these differences can be explained by the appearance of additional modes due to the relaxation of the selection rules (caused by the displacive disorder in the Bi pyrochlores). Some additional modes had frequencies close to modes in the IR data, and others had frequencies close to optically inactive modes calculated by computational work in the literature. The additional modes were tentatively assigned by this comparison. Finally, the existence of additional modes in the Raman spectra of all four compounds suggests no difference in the amount of disorder among these samples.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Daniel Arenas.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Tanner, David B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024735:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024735/00001

Material Information

Title: Devices and Materials for Thz Spectrosopy Ghz Cmos Circuits, Periodic Hole-Arrays and High-Frequency Dielectric Materials
Physical Description: 1 online resource (115 p.)
Language: english
Creator: Arenas, Daniel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: arrays, bismuth, cmos, detection, ftir, ghz, hole, infrared, periodic, pyrochlores, raman, sources, spectroscopy, terahertz
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is composed of three main projects, linked together by the THz region of the electromagnetic spectrum. In the first project, we detected the radiation from a silicon CMOS circuit, using a fourier transform interferometer. At the time of measurement, this 410 GHz circuit had the highest operating frequency for silicon integrated technology observed to date. The measured radiated power from the 410 GHz circuits was in the order of 0.01 microWatts. This circuit had radiated intensities comparable to those of commercially available black-body sources for the 200 - 400 GHz region. The high power and high emission per source area suggested possible spectroscopy applications. We also studied the optical properties of periodic hole-arrays with resonant frequencies in the THz region. Although the transmittance spectra of these structures have been extensively studied, here we present reflectance measurements that allow the analysis of the extinction/absorption spectra. The results were compared to predictions from the trapped-mode theory on the ohmic losses of these systems. Our results did not show the prediction of a suppression of the R + T spectra at the resonant frequency. Also, we studied the time-dependence of femtosecond pulses reflected from periodic hole arrays with resonant frequencies in the NIR region. Our results show that if the trapped modes theory is correct, then the lifetime of these modes are below 100 fs. Finally, in the third project, we studied the Raman active modes of various bismuth pyrochlores containing Zn, Mg, Ta and Nb, which have earned recent attention for high-frequency applications. The spectra of the four compositions are very similar, suggesting no major structural differences among these materials. The spectra were compared to those of other pyrochlores and specific discussions are offered for the assignment of each mode. Although there are clear differences between the spectra of these samples compared to other pyrochlores, these differences can be explained by the appearance of additional modes due to the relaxation of the selection rules (caused by the displacive disorder in the Bi pyrochlores). Some additional modes had frequencies close to modes in the IR data, and others had frequencies close to optically inactive modes calculated by computational work in the literature. The additional modes were tentatively assigned by this comparison. Finally, the existence of additional modes in the Raman spectra of all four compounds suggests no difference in the amount of disorder among these samples.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Daniel Arenas.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Tanner, David B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024735:00001


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First,Iwouldliketothankmyadvisor,ProfessorDavidB.Tannerforgivingmegreatopportunitiesinmygraduatecareer.Thankstohim,Ihavebeenfortunatetoworkinmanyprojectsandlearnagreatdeal.Hisadvice,teachingandcontagiousenthusiasmforphysicshavebeengreatlyvaluedandappreciated.AlsoIwouldliketothankmysupervisorycommittee:ProfessorArthurF.Hebard,ProfessorJuanNino,ProfessorDavidReitzeandProfessorPeterHirschfeld.Asagraduatestudent,Ihaveworkedinmanydierentprojectsandhavehadtheprivilegetomeet,collaborateandlearnfrommanywonderfulpeople.IwouldliketothankSinanSelcukandhisadvisorProfessorArtHebardforlettingmebepartoftheperiodicholearraysproject.ThankstoJinhoLee,TakahisaTokumotoandProfessorStephenMcGillinNHMFLforteachingmeandhelpingmeagreatdealinultrafastoptics.ProfessorKennethOforlettingmebepartofthe410GHzcircuitproject,andhisstudentsandmygoodfriendsEunyoungSeokandDonghaShim.ThankstoProfessorLevGasparovinUNFforhishelpintheRamanmeasurementandintroducingmetotheopticseldandteachingmewhenIwasanundergrad.ThankstoProfessorTomPekarekinUNFforhisinvaluableadviceinthelastsevenyears.OnceagainthankstoProfessorJuanNinoforhisteachingandpatienceinthebismuthpyrochloresproject;andhisstudentWeiQiu.AlsoimmensethankstoProfessorDavidSilvermanforlettingmebepartofhisCocrystalsprojectandhisstudentBaluAvaru.ThankstoJayHorton,MarcLink,EdStorch,RaymondFrommeyer,BillMalphurs,LarryPhelpsandRobHamersmaforallthehelpindesigning,repairingandbuildingequipment.Ilearnedagreatdealfromthem.ThankstoDr.RobertDeSerioandCharlesParksfortheirtutelagewhenIwasateachingassistant.Tomycolleaguesinthelab.NaveenMargankunte,mylabsenior,whomIlearnedsomuchinourlabandfromourtripstoNHMFL.DimitriosKoukis,whohasbeenthebestteammateanyonecouldwishtoworkwith.ThankstoNathanHeston,mygoodfriend 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 2DETECTIONOFRADIATIONFROMA410GHzCIRCUIT .......... 18 2.1Introduction ................................... 18 2.2ExperimentalProcedures ............................ 19 2.2.1CircuitDesign .............................. 19 2.2.2DetectionoftheRadiationUsinganInterferometer ......... 21 2.3ResultsandDiscussion ............................. 22 2.3.1DemonstrationoftheOperatingFrequency .............. 22 2.3.2EstimateoftheRadiatedPower .................... 22 2.4ConclusionsandFutureWork ......................... 24 3PERIODICHOLEARRAYS ............................ 29 3.1EnhancedTransmissionEect ......................... 29 3.2Theories ..................................... 30 3.2.1SurfacePlasmonsTheory ........................ 30 3.2.2DynamicalDiractionTheory ..................... 30 3.2.3TrappedModes ............................. 31 3.3ReectionandTransmissionStudiesofPeriodicHoleArrays ........ 32 3.3.1Motivation ................................ 32 3.3.2ExperimentalProcedures ........................ 32 3.3.3ResultsandDiscussion ......................... 33 3.4Ultra-FastOpticsStudyofPeriodicHoleArrays ............... 37 3.4.1Introduction ............................... 37 3.4.2ExperimentalProcedures ........................ 38 3.4.3ResultsandDiscussion ......................... 39 3.5ConsiderationsforNonlinearApplications .................. 39 3.6FutureWork ................................... 40 4RAMANSTUDYOFTHEPHONONMODESINBISMUTHPYROCHLORES 52 4.1Introduction ................................... 52 4.2ExperimentalProcedures ............................ 53 6

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............................. 54 4.3.1TentativeAssignmentofthe\Ideal"Modes .............. 54 4.3.2TentativeAssignmentofthe\Disorder"Modes ............ 56 4.4Conclusions ................................... 60 4.5FutureWork ................................... 61 APPENDIX AEXTRACTINGELECTROMAGNETICWAVESANDTHEOPTICALCONSTANTSFROMMAXWELL'SEQUATIONS ........................ 71 A.1Introduction ................................... 71 A.2ElectromagneticWavesandtheOpticalConstants .............. 72 BBOUNDARYCONDITIONS.TRANSMITTANCEANDREFLECTANCE ... 78 CRESPONSEFUNCTIONSANDKRAMERSKRONIGANALYSIS ....... 82 DMICROSCOPICMODELSFORTHEOPTICALCONSTANTS ......... 89 D.1TheDrudeModel ................................ 89 D.2LorentzianOscillators ............................. 92 EINTERFEROMETERS ............................... 95 E.1Derivation .................................... 95 E.2PropertiesoftheInterferogram ........................ 97 E.3Resolution .................................... 98 FSURFACEPLASMONS ............................... 101 REFERENCES ....................................... 108 BIOGRAPHICALSKETCH ................................ 115 7

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Table page 4-1RamanmodesandtentativeassignmentofBMN,BZN,BZTandBMT. ..... 62 4-2ComparisonbetweenIRmodesandRamanmodes. ................ 62 4-3ComparisonbetweentheRamanmodesofBMNandotherpyrochlores ..... 63 8

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Figure page 2-1CircuitDiagram .................................... 26 2-2Emissionspectrumofthe410GHzcircuit. ..................... 26 2-3Observedemissionpeakforthe410GHzcircuit. .................. 27 2-4Emissioncomparisonbetweenthecircuitandblackbodysources. ......... 27 2-5Sourcecompartmentdiagram. ............................ 28 2-6Sourcecompartment.Collectingmirrorandaperture. ............... 28 3-1Aperiodicholearray. ................................. 41 3-2Transmittancespectraoftwoperiodicholearrays. ................. 41 3-3R+TspectraforZnSe. ............................... 42 3-4PredictedR+Tfromtrappedmodestheory. .................... 42 3-5RandTspectra.Dg=6m(ZnSesubstrate). .................. 43 3-6R+Tspectra.Dg=6m(ZnSesubstrate). .................... 43 3-7RandTspectra.Dg=8m(ZnSesubstrate). .................. 44 3-8R+Tspectra.Dg=8m(ZnSesubstrate). .................... 44 3-9Reectanceandtransmittancespectrafortwoperiodichole-arraysonaquartzsubstrate. ....................................... 45 3-10Sumofthereectanceandtransmittancespectraforthetwoperiodichole-arraysonaquartzsubstrate. ................................ 45 3-11Caricatureofasharppulsetransmittedthroughanarray. ............. 46 3-12Caricatureofabroadpulsetransmittedthroughanarray. ............ 46 3-13Caricatureofapulsewithtimewidthcomparabletothemodeslifetime ..... 47 3-14NHMFLSetup. .................................... 48 3-15Autocorrelationdataforthereectedpulsefromthesilverlm. ......... 49 3-16Autocorrelationdatafortheperiodicholearrays. ................. 50 3-17Comparisonoftheautocorrelateddataforthereectedpulsefromthesilverlmsandthevarioushole-arrays. .......................... 51 9

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................................. 64 4-2BMNRamanspectrum. ............................... 65 4-3BZNRamanspectrum. ................................ 66 4-4BMTRamanspectrum. ............................... 67 4-5BZTRamanspectrum. ................................ 68 4-6ComparisonofRamanspectrabetweenthefoursamples. ............. 69 4-7NormalmodesofalinearO-A-Omolecule. ..................... 70 A-1Boundaryconditionsataninterface. ........................ 77 C-1Complexplane .................................... 88 E-1Diagramofabasicinterferometer. .......................... 99 E-2Interferenceinaninterferometer. .......................... 99 E-3Caricatureofaninterferogram. ........................... 100 E-4Resolution. ...................................... 100 F-1Surfaceplasmonsattheinterface. .......................... 107 F-2TEandTMpolarizationforasurfacewave .................... 107 10

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1 2 ],multiferroics[ 3 ],manganites[ 4 ],nanostructures[ 5 ],heavyfermions[ 6 ],andothers.Theinfraredspectrumisseparatedintothreeregions(withvariousboundariesdependingonthedivisionscheme),thefar-infrared(10-700cm1;0.3-21THz;1000-15m),themid-infrared(700-4000cm1;15-2.5m),andthenear-infrared(4000-14000cm1;2.5-0.7m).The\THz"regionofthespectrumisnowusedtospecifythe0.3-3THzrange,althoughsomeauthorsmayextendthisdenitionto30THz[ 7 ].The0.3-3THzregionwasonceconsideredapoorlydevelopedregionofthespectrumduetothelackofintensesources[ 8 ].Blackbody(thermal)sources,themostcommoninspectroscopy,havelowintensityattheselowfrequencies.Foralongtime,thefrequenciesintheTHzregionwereconsideredtoofastforsolid-statecircuits,andtooslowforsolid-statelasers[ 9 ].OneoftherstalternativewaystogenerateTHzradiationbeganinthe60swiththeuseofnonlinearcrystalsfordierencefrequencygeneration[ 10 ]andparametricamplication[ 11 { 13 ].However,judgingfromtheliterature,theexplosioninTHzresearchandsourcesseemedtooccurinthemid80swiththeuseoffemtosecondlaserstoinduceTHzradiationfromvarioussystems;suchasphotoconductingstructures[ 14 { 20 ]andelectro-opticmaterials[ 21 22 ].ThegenerationofTHzradiationfromquantumcascadelaserswasalsoanimportanteldinthe90sandcontinuestobeso[ 23 24 ].Forthelastdecade,thenewexcitingandpowerfulsystemsforTHzradiationincludefreeelectronlasers(FEL)[ 9 ],synchrotronsources[ 25 26 ],andothersystemsthatalsouserelativisticelectronstogenerateTHzradiation[ 27 { 29 ]. 13

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30 ],andhencethedetectionofcertaincancercells[ 31 32 ].ThehightransmissionofTHzradiationthroughnon-metallicmediasuchasshoes,clothesandcardboard,hasalsomotivatedtheuseofTHzimagingfordetectionofconcealedweapons[ 33 34 ].Furthermore,thereiscontinuousresearchinthedetectionofdangerousmaterials,suchasexplosives[ 35 36 ],chemicalagents[ 37 38 ],andevenillicitdrugs[ 7 39 ].Areviewoftheseandotherapplications,aswellasotherTHzsourcesnotmentionedhere,isgivenbySiegelinreference[ 8 ].AlthoughthesenewTHzsourceshaveallowednewexcitingresearchinthisregion,theirsizesarebigandtheircostsarehigh.Forwidespreadapplications,itisdesirabletobuildmorecompact,andmoreimportantly,cost-ecientsourcesanddetectors.Onepossibilityistheuseofmainstreamsilicontechnology,suchasCMOS(complimentarymetal-oxidesemiconductors),tobuildfastcircuits.TherstprojectofmygraduatestudiesdealsinthisareaandispresentedinChapter2.ThischaptershowsthedetectionofTHzradiationfroma410GHzCMOScircuitequippedwithapatchantenna.ThecircuitwasdesignedbyDr.KennethO'sgroupintheSiliconMicrowaveIntegratedCircuitsandSystemsResearchGrouplocatedatUniversityofFloridaDepartmentofElectricalandComputerEngineering.And,itwasconstructedatTexasInstruments.Whenthecircuitwasconstructed,therewerenoavailablehigh-frequencyprobesabove325GHz[ 40 ]atthetime.Ourcontributiontothisprojectwastodemonstratetheoperating-frequencyofthecircuitbymeasuringitselectromagneticradiationusingaFourierTransformInterferometer.Atthetimeofmeasurement,thiscircuithadthehighestoperatingfrequencyofanycircuitfabricatedwithsiliconintegratedmainstreamtechnology;and,marksthersttimeforthirtyyearsthataCMOScircuitisfasterthan 14

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41 { 46 ].Periodicholearraysexhibitaneectwhereatresonantwavelengths,thetransmittancethroughthemetallmsexceedsthevaluepredictedbydiractionandgeometricoptics[ 47 ].Theexplanationofenhancedtransmissioninperiodichole-arraysremainscontroversialandseveraladditionaltheorieshavebeenproposed.AmongthesetheoriesistheTrappedModestheory,whichstatesthatataresonantfrequencynearthediractionthreshold,theelectromagneticeldsgettrappedinthesub-wavelengthsstructuresforacharacteristiclifetime[ 48 49 ].Thistheoryisfascinatingbecauseitsuggeststemporalmanipulationoflightandhasproposedexcitingapplicationsinnonlinearoptics[ 50 ].Thenonlinearopticspossibilitiesareimportantbecausetheycouldleadtoopticalsignalprocessing,wherethepurposeistocontrollightusinglight[ 44 ].Manyoftheapplicationsofthesestructures,includingnonlinearoptics,isrelevanttotheinfraredandTHzregionduetoourabilitytotunetheresonantfrequencyofthesestructuresbychangingtheperiodicityoftheholes.Inthiswork,wetrytotesttwopredictionsfromthetrappedmodestheory.Wemeasuredthetimecharacteristicsofa100fspulsereectedbyperiodichole-arrays,toseeifthelifetimeofthetrappedmodeswascomparabletothepulsetemporalwidth.TheotherpredictionstatesthatwhenEMmodesgettrappedattheresonantfrequency(redshiftedfromthediractionthreshold), 15

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49 ].Inthisworkwewereabletomeasurenotonlytransmittance,butalsoreectance,forperiodichole-arraysandtestedtheprediction.Chapter4isabouttheRamanstudyofphononmodesinbismuthpyrochlores.Bismuthpyrochloreshaveearnedrecentattentionforhigh-frequencyapplicationsthankstotheirlowloss,high-permittivityandgoodtemperaturestability[ 51 ].WepresenttheRamanspectraoffourdierentbismuthpyrochlores:Bi3=2ZnNb3=2O7(BZN),Bi3=2ZnTa3=2O7(BZT),Bi3=2MgNb3=2O7(BMN)andBi3=2MgTa3=2O7(BMT).ThepurposeofthisworkwastocomparehowtheRamanactivemodesbehavedacrossthefoursamplesandtocomparethemtootherpyrochlorestructures.Ourresultsshowthatspectraisoverallsimilarforthefourbismuthpyrochlores,butshowskeydierencestootherpyrochlorematerials.TheobservationofmorethanthesixRamanmodespredictedfromtheidealpyrochlorestructureconrmedthedisplacivedisorderinthebismuthpyrochlores.Comparisontotheinfrareddata[ 52 ]andcomputationalworkonotherpyrochlores[ 53 ]allowedidenticationoftheadditionalmodesduetotherelaxationoftheselectionrules.Besidestheprojectsmentionedinthisthesis,I'vehadtheopportunitytostudyothertypesofmaterialsusingIRspectroscopy[ 54 55 ].Inourlab,itiscommontocollaboratewithgroupsinterestedintheopticalpropertiesoftheirsamples.Inthisdissertation,theappendicesgiveabriefoverviewsoftheopticalconstants.Theseappendicesaremeanttoprovideapedagogicalintroductionforthebenetofyoungerstudentswhochoosetoreadthisthesis.AppendixArstshowshowtheopticalconstantsareobtainedfromMaxwell'sequationsandapropagatingwavesolution.Itshowswhytheycanbecomplexfunctions,andhowtheopticalconstantsareinter-related.Then,appendixBshowshowtheboundaryconditionsataninterfacebetweentwomediarelatetheopticalconstantstoreectanceandtransmittance.AppendixCshowsthederivationandtheuseofthe 16

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40 ].The300GHz-3THzspectrum(dubbedtheTHzregion)hasbeenextensivelystudiedforuseinradars,remotesensing,advancedimagingandbio-agentandchemicaldetection[ 8 56 57 ].Tobringthepricedownoftheseapplications,itisdesirabletobuildtheTHZcircuitsusingthemain-streamsiliconintegratedtechnologyCMOS(Complimentarymetal-oxidesemiconductors)[ 40 58 ].ATHzcircuitcouldalsobeusefulinspectroscopyapplications,sinceblackbodysourceshavelowpowerintheTHzregionandtheirintensitydiesoatlongerwavelengths.Furthermore,thesecircuitscouldlaterbedesignedwithtunablefrequenciessothattheycanbeusedinspectroscopywithouttheuseofinterferometers.Inthiswork,wereportTHzradiationfroma410GHzCMOScircuit.Thiscircuithasthehighestoperatingfrequencyamongthosefabricatedusingcost-ecientsiliconintegratedtechnology.ItwasdesignedbyDr.KennethO'sgroupintheSiliconMicrowaveIntegratedCircuitsandSystemsResearchGrouplocatedattheUFElectricalandComputerengineeringdepartment(UF-EEL).Atthetimeofthecircuit'sdesignandconstruction(2007),highfrequencyprobescouldnotbeusedtomeasureitsoutputduetothelackofprobesavailableabove325GHz[ 40 ].Instead,theoperatingfrequencyoftheCMOScircuitwasdemonstratedbymeasuringitselectromagneticradiationfromanon-chippatchantennausinganinterferometer.AllopticalmeasurementswereperformedattheTannerlabinUF-Physics.Weestimatedthe410GHzradiationpowerofthecircuitataround0.001-0.01W.Atthesefrequency,thepowerfromthiscircuitiscomparabletocommerciallyavailableblackbodysourcesusedininterferometers.And,thesmaller 18

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2.2.1CircuitDesignAspreviouslystated,allthecreditincircuitdesigngoestoourcollaboratorsDr.EunyoungSeok,DonghaShimandP.I.Dr.KennethOfromUF-EEL.ThecircuitwasbuiltinNXPSemiconductorsandTexasInstruments.Figure 2.4 showsadiagramoftheircircuit.Thepush-pushoscillatorisequippedwithapatch-antennatoextractthesignal.TheresonantfrequencyofthecircuitisdeterminedbythecapacitancesofthetransistorsM1andM2andtheinductorsL1andL2.Asalloscillators,theamplier(i.e.transistor)limitsthemaximumfrequencyofthefundamental.Thisfrequencyisreferredtoasfmaxandcorrespondstothemaximumfrequencyatwhichthetransistorwillgiveanamplicationhigherthanunity.Forourcircuit,thisfrequencyisaround200GHz.Toincreasetheoperatingfrequency,thesecondharmonicofthefundamentalwasgeneratedbyusingapush-pushoscillatorarchitecture.Inthisdesign,thecross-coupleddesignofthetransistorsintheoscillatorcoreissuchthattheirsignalsare180degreesoutofphase.Unlikethefundamental,thesecondharmonicsignalgeneratedduetononlinearityofthetransistorsisinphase: 19

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TwoPMOStransistors(M3andM4)areusedtobiasthecore,andtwoquarterwavetransmissionlines(T.L1andT.L2)areusedtoisolatethesecomponentsfromthefundamentalandsecondharmonic.Detailsonthecircuit'sandpatchantennadesigncanbefoundinreferences[ 40 ]and[ 59 ]. 20

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2.3.1DemonstrationoftheOperatingFrequencyFigure 2-2 showstheresultsforthepowerspectrumofthecircuit,andthebackgroundspectrum(circuito).Thesignalat410GHzconrmstheoperatingfrequencyofthecircuit,althoughthefundamentalofthecircuitisalsoobservedat205GHz.Theremainingbaselineintensityabove10cm1inthebackgroundseemslikeblackbodyradiationfromthecircuitbeinghotterthanroomtemperature.Figure 2-3 showsthespectrumofthe410GHzcircuitneartheemissionpeak(thebackgroundhasbeensubtractedforthisgure).TheFWHM(FullWidthHalfMaximum)ofthepeakisaround0.1cm1whichcorrespondstotheresolutionofourmeasurements.Thissuggeststhatwecannotresolvethewidthoftheemissionpeakwithourinterferometrysetup. 2-4 showsacomparisonbetweenthepowerofthe410GHzradiation(substractedfromthebackground)tothepowerofthemercurylampatthesameconditions.Thespectrashowsthattheradiationpowerofthecircuitiscomparabletothatofacommerciallyavailablespectroscopysourceforthenarrow3GHzbandofemission.Toestimatethepowerfromamercurylampat410 22

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c2;(2{8)whereB(T)isthespectralradiancewithSIunitsofW/m2Hzsr(srreferstosteradians,theunitsofsolidangle),kBistheBoltzmannconstantequalto1.38x1023J/K,andisthefrequency.Ourmercuryarclamphasatemperaturecloseto5000K,whichyieldsaspectralradianceofaround2.6x1013W/m2Hzsrfor410GHz.Toestimatetheradiationareafromtheblackbodysourceandthecollectedsolidangle,werefertoFig. 2-5 and 2-6 .Thecollectingmirrorhasadiameterof70mmanditimagesthesourceintoanapertureofvaryingdiameter(Fig. 2-5 ).Sinceonlythelightthatpassesthisaperturegoesintotheinterferometer,theeectiveradiationareafromthesourceisapproximatelytheareaoftheaperture(whichhasa10mmmaximumdiameterfortheBruker113v): Area=(5mm)2=8105m2:(2{9)Tocalculatethecollectedsolidangle,weusethediameterofthecollectingmirroranditsdistancetothesource(Fig. 2-6 ): =2(1cos);(2{10)whereistheangleoftheradiationconeandisequalto: tan=35 240;(2{11)whichyieldsasolidangleof0.07sr.Thereforethepowerradiatedfromthemercurylampinthefrequencywidth(d)of3GHzisapproximately: 23

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2-4 suggeststhatthepowerofthecircuitishigherthan0.001W,butlowerthanourrstestimate(0.01W)usingthesensitivityofthedetectorreportedbythecompany.Theeciencyinemissionperunitareaisveryhighforthe410GHzsource.Themercuryarclamphasanareaofapproximately8x105m2whilethepatchantenna(200x200m2)forthe410Ghzcircuithasanareaintheorderof4x108m2.Thissuggeststhatthe410GHzsourceisintheorderof1000moreecientinemissionperunitarea.(Theequivalentofhavinga106Kblackbodysource). 24

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Circuitdiagramforthepush-pushoscillatorsystemwithanon-chippatchantenna. Emissionspectrumofthe410GHzcircuit. 26

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Observedemissionpeakforthe410GHzcircuit. Emissioncomparisonbetweenthe410GHzcircuitandthemercurylampandglobarlampnormallyusedinourBruker113vinterferometer. 27

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DiagramofthesourcecompartmentforthemercurylampusedinourBruker113vinterferometer.Notdrawntoscale! Dimensionsforthecollectingmirrorandapertureusedtocalculateradiationareafromthesourceandthecollectedsolidangle. 28

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60 ]showedthatforwavelengthslargerthantheholesizea,thetransmittancefallsoas: lima 1Ta !4:(3{1)However,Ebbesenetal.[ 47 ]discoveredthatforaperiodicarrayofsuchsub-wavelengthholes,thetransmittancespectracanexhibitlargetransmittancepeaks.Thiseectisevenmorepuzzlinginthateveninthegeometricopticslimit(a)thetransmittanceshouldnotexceedtheopenareafractionfoftheholes.Theopenareafractionofsquareholesonasquaregridisgivenby: Dg!2;(3{2)whereDgisthehole-separation(SeeFig. 3-1 ).Figure 3-2 showstwoexamplesfortheenhancedtransmittanceeectforperiodicholearrays.ThespectrashownareforpatternedsilverlmsonaZnSesubstratewithtwodierentperiodicitiesandopenareafractionof0.44(Fig. 3-3 showsthetransmittanceandreectanceofthesubstrate.)Thespectrashowthatthelocationoftheenhancedtransmittancepeakchangesfordierentperiodicitiesofthesample.Theenhancedtransmissioneecthasreceivedmuchattention[ 47 61 { 70 ]andhassuggestedmanyexcitingapplicationsinlightmanipulationandnonlinearoptics[ 44 50 71 72 ].However,theexplanationofthiseectremainscontroversial.And,althoughtheoriginal[ 47 ]andstillpopular[ 62 66 73 { 78 ]explanationbyEbbesenetal.attributedtheeecttosurfaceplasmons,otherexplanationshavebeenproposed.Inthenextsections,wewillmentiononlytwooftheothertheories:dynamicaldiractionandtrappedmodestheory(althoughthereaderissuggestedtosee 29

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79 { 83 ]foradditionaltheories).AlthoughIcannotoerinsightintothevalidityofthesetheoreticalorcomputationalarguments,thepurposeofthisworkwastoidentifypredictionsoeredbythesealternatetheoriesandtothenmeasurethem. 3.2.1SurfacePlasmonsTheoryTheenhancedtransmissioneectwasoriginallyattributedtotheinteractionoflightwithsurfaceplasmons(SPs)intheperforatedmetallm.Surfaceplasmonsareelectromagneticwavesconnedtotheinterfacebetweenapositivedielectricandanegativedielectric.Thewavepropagatesalongthesurfaceandcanonlycoupletolightwhenthesurfacehasaperiodicstructure.(SeeAppendixFforabriefdiscussiononsurfaceplasmons).ThemechanismproposedconsistsinexcitationofaSPinthetopofthelm,andareemissionoflightbyanSPatthebottomofthelm.Ebbesenetal.attributedthecausalroleofenhancedtransmissiontoSPsduetotwoimportantresults:One,usingangledependenttransmittancemeasurements,Ebbesenshowedthatthefrequencyofenhancedtransmissionversusthewavevectorkgivesadispersioncurvecharacteristicofsurfaceplasmons;and,theyshowedthatperiodicholearraysinGe(positivedielectric)didnotshowenhancedtransmission. 84 ]and[ 85 ]arguethatSPsdonotplayacausalroleinenhancedtransmission,andthattheeectislinkedtodiraction.Theinspirationforthistheoryisbasedondynamicaldiractiontheoryforx-rays.Perfectlyorientedcrystalsradiatex-raylightcoherentlyandcauseanomalouseectsatwavelengthsclosetothelatticeconstantofthecrystal(d).Ewald[ 86 ]createdthecoherentdynamicaldiractiontheorytoexplaintheseeectswhichwereunaccountedforbytraditionalkinematicdiractiontheory.ThetheoryconsistsinsolvingMaxwell'sequationsinaperiodicmedia,wheretheperiodicityofthedielectricfunction,(~r),iswrittenas: 30

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84 ].Treacyetal.[ 85 ]arguedthatholearraysarebasicallythesamesystemexceptwithnegativedielectricconstantsasopposedtotheclosetounitydielectricinx-raydiraction.Theirtheoryandcalculationsalsosuccesfullyexplaintheenhancedtransmittanceeect.Theelegantpartofthistheoryisthatitarguesthatenhancedtransmission,anomalousx-raydiractionincrystals,andevengapsinphotoniccrystalsareallbasicallythesameeect.Fordiscussionsonphotoniccrystals,pleasereference[ 41 87 88 ]. 48 { 50 89 { 92 ]statesthatataresonantfrequency,wheretheenhancedtransmittanceeectoccur,theelectromagneticeldsgettrappedinthevicinityoftheholesanddecaybyemittingnearlymonochromaticlight.Thecharacteristicdecaytimeisrelatedtotheinverseofthewidthofthetransmittancepeak.Similartothedynamicaldiractiontheory,thetrappedmodestheoryarguesthatETeectispurelygeometricandnotduetosurfaceplasmons.Theircalculationsconsistonfullysolvingthetime-dependentMaxwell'sequationsforradiatingboundaryconditions.Theirresultsshowaresonantfrequencynearthediractionthresholdswheretheelectromagneticmodesgettrappedandenhancedtransmittanceoccurs.Thereareseveralpredictionsoeredbythetrappedmodestheory[ 49 50 ].Wewillstateheretheonesrelevanttothisdissertation: 1. Ohmiclossesandthereforeabsorptionshouldincreaseattheresonantfrequencyduetothe\longer"exposureofthemodestodissipativeprocesses.Thisohmiclosses(orabsorptive)peakshouldalsodependontheopen-areafractionofthearray. 2. Trapping(ordelay)oflightinsideperiodicholearraysoccursattheresonantfrequency. 3. Thetrappingoflightin2Dstructurescanleadtousefulapplicationsinnonlinearoptics. 31

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3.3.1MotivationAsstatedintheprevioussection,thetrappedmodestheorystatesthatlightgetstrappedinthevicinityoftheholesattheresonantfrequency,andthendecaysbyemission.Theypredictthatatthisfrequency,electromagneticeldsareexposedtoenergy-lossmechanismsforalongertime.Theircomputations[ 49 ]showadipinthespectraofthesumofthetransmittanceandreectanceneartheresonantwavelength(SeeFig. 3-4 ).Thisdipisattributedtoabsorptionduetoohmiclosses,andisthereforeredshiftedfromthediractionthreshold(redshiftedsimilarlytotheenhancedtransmittancepeak).Asof2006,whenthisprojectgotstarted,therewasnotmuchworkdoneonthereectanceofperiodicholearrays,andthereforedataontheextinctionorabsorptionofthesesampleswaslimited.Thelackofreectancedatawasperhapsduetothehigherdicultyofmeasuringreectanceinsteadoftransmission.Inthiswork,wepresentreectanceandtransmittancedatafortwosetsofsamplesfabricatedbydierentmethodsandmeasuredwithdierentequipmentatdierentspectra.FortheNIR-VISregion,weshowdatafromsilverlmsgrownonquartz,andwealsoshowdatafortheFIRregionforsilverlmsgrownonZnSe. 93 ].Thereectanceandtransmittancedataof 32

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FIR:AglmsonZnSe.Figure 3-3 showsthereectanceandtransmittanceoftheZnSesubstrateandtheirsumR+Tspectra.Asshownbythisgure,theabsorptionintheZnSesubstrateisverysmallforfrequenciesbetween650-4000cm1butbecomesverystrongforfrequenciesbelow650cm1.Thereisastrongphononmodearound200cm1andthethicknessofthematerial(1cm)makesthetransmittanceverysensitivetosmallabsorptioncoecients.TheindexofrefractionoftheZnSesubstratenswas2.4andthisvaluewasconrmedbyboththereectanceandtransmittancemeasurement.Fornormalincidence,theenhancedtransmittancepeakoccursnearthediractionthreshholdassociatedwiththeindexofrefractionofthesubstrate: 33

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3-5 showourresultsforthereectanceandtransmittanceofDg=6mholearrays.Thediractionthresholdsandthestrongesttransmittancepeakareobservednear700cm1,and,thenextthresholdislocatedaround1000cm1.Thereectancedatashowedthedipassociatedwiththetransmittancepeakandthediractionthresholds.Ithasbeenpredictedandobservedthatarrayswithlargeropenareafractionsfhavelargerwidthstransmittancepeaks[ 49 94 ].Theopenareafractionofthesesamplesisgivenby: 49 ].Ourdatasupportspreviousresultsfortherelationbetweenwidthandopenareafraction.Forthef=0.25sample(leftofFig. 3-5 )thefullwidthhalfmaximum(FWHM)is40cm1and70cm1forourf=0.44sample(right).ThemainpurposeofthereectancemeasurementswastotestthepredictionofthetrappedmodestheorythatadipoccursintheR+Tspectra(correspondingtoapeakintheabsorption)duetoohmiclosses.Thetheoryalsostatesthatsmalleropenareafractionsamplesshouldhavemorepronounceddipssincethemodesarelongerlived.Figure 3-6 showsourresultsfortheR+TspectraoftheDg=6msamples.Forsimplicity,weplotR+Tinsteadofabsorptionorextinction,becauseatwavelengthssmallerthanthediractionthresholdnsDg,bothabsorptionanddiractionlossesarepossible: 34

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49 ]computationsshowsgoodagreementintheoverallshapeoftheR+Tspectra.However,althoughourresultsshowadipintheR+Tspectra,thisdipislocatednearthediractionthresholdandnotredshiftedattheresonantfrequency.Therefore,thisdipintheR+Tspectracouldbeattributedtodiractionlosses,andnottoohmiclosses.AnothersetofsampleswithperiodicityDgof8mandopenareafractionsof0.25and0.44werealsostudied.Thetransmittancepeakofthesesamplesarelocatednear540cm1(Figure 3-7 )wheretheabsorptionoftheZnSebecomesnoticeable.Thereforetheanalysisofthesesampleshastobemorecareful.Figure 3-8 showstheR+Tspectraforthesetwosamples.Theresultsforthesearraysisthesameasfortheprevioussamples.TheoverallshapeoftheR+Tspectraagreeswithcomputation,butthedipisseennearthediractionthresholdandnotredshiftedneartheresonantfrequency. 3-9 (right)showsthereectanceandtransmittancespectraforaperiodicholearrayonquartzwithperiodicityof0.8mandopenareafractionof0.25.Theindexofrefractionofquartzis1.4andtheabsorptioninthisregionisnegligible.Thediractionthresholdforthisperiodicityisobservedandexpectedtobearound9000cm1.Thereectancedataisalsoconsistentwiththetransmittancedata.TheR+TspectraisshowninFigure 3-10 (right).ThereisasmalldipinR+Tattheresonantfrequency,butsimilarlytotheZnSesamples,thereisamuchstrongerdipnearandblueshiftedfromthediractionthreshold.Similarresultswereobtainedforlmswithperiodicityof1m.Inconclusion,forvarioussamplesfortwodierentfabricationandopticalmeasurement,theoverallshapeoftheR+Tspectraagreeswithcomputations.However,wedonothaveadirectobservationofthetrappedmodestheorypredictionduetothefactthat 35

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36

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3.4.1IntroductionThemainpredictionfromthetrappedmodestheoryisthatlightgetstrappedinsidetheperiodicholearraysandthenisremittedwithacharacteristicdecaytime2/.Totestthisprediction,westudiedthetimedependenceofatransmittedorreectedpulsefromaperiodichole-array.Figure 3-11 showsacaricatureofthereasoning.ThepulseisGaussianintime,withadurationsetbythepropertiesofthelasersource.Ifthelifetimeofthemodesislargerthanthepulsewidth,thenthetransmittedorreectedpulsewillbebroadenedwithanexponentiallydecayingtail,characteristicofthemodelifetime.However,ifweuseapulsethatistoolong,thenweexpectthepulsetoremainthesame(Fig. 3-12 ).Ifthelifetimeofthemodesiscomparabletothedurationofthepulse(Fig. 3-13 ),thenthedetectionofthiseectwouldrelyonthefasterdecayoftheGaussianpulseinsteadofanexponentialdecayprocess.Tostudypulsesthisshort(100fs),wehadtouseautocorrelationbecausewecannotmeasuretheirtemporalprolewithequipmentsuchasstreakcameras.Inautocorrelation,abeamsplitterisusedtosplitthebeam,whicharethenmixedinanonlinearcrystaltoobtainthesum-frequency.Thisnonlinearprocessmakessignaltonoise(S/N)averyimportantissue.Also,theautocorrelationofthepulsemustbesymmetricandtherefore,weloseinformationaboutthepulse(forexample,wewouldloseinformationonthesharpleftsideofthepulseinFig. 3-11 );however,theautocorrelatedpulseshouldstilllookbroader.TheDg=6msamples(Fig. 3-5 )ontheZnSesubstrateshadwidths50cm1(1.5THz).TheQofthesesamples10andbasedonthewidthweexpectedthelifetimestobearound100fs.However,femtosecondpulsesintheFIRregionarelimitedandareonlyavailableforspecializedfreeelectronlasersinnationallabssuchasJeersonLab(Virginia,USA).IntheNIRregion,however,chirpedpulseampliers(CPA)andopticalparametricamplifers(OPA)canachieve100fspulses.ACPA-OPAsetupisavailableintheUltrafast-OpticsCellintheNationalHighMagneticFieldLab(NHMFL).Forthe 37

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3-9 )resultsinlifetimesaround10-100fs.Wecarriedouttheultra-fastopticsmeasurementsforfoursampleswithresonantwavelengthsnear1600nm.Ourresultsshownobroadeningofthepulseandsuggestthatthelifetimeofthetrappedmodesinthesesamplesisbelow100fs. 3-14 showsadiagramofthesetup.Femtosecondpulseswithfrequency12500cm1(375THz;800nm)atarepetitionrateof1KHzwereobtainedfromaClark-MXRCPA-2001chirpedpulseamplier.Tobringthisfrequencyneartheresonantfrequencyofoursamples,weusedaTOPAS4/800OPAandhalvedthefrequencyto6250cm1(188THz;1600nm).TheautocorrelationofpulsesfromtheOPAshowedthatthewidthofthesepulseswerearound140fs.Thenormal-incidentreectedbeamwasmeasuredbyusingabeamsplitterinthegeometryshowninFig. 3-14 .Allfourarraysmeasuredwereinthesamesilverlm;andthelmwastransversallydisplacedfromtheincidentbeamtochangebetweendierentarraysandtomeasurethesilverlmasreference.TheautocorrelatorsetupusedtomeasurethetimeproleofthereectedpulseisshowninFig. 3-14 .Abeamsplitterwasusedtosplitthebeamintotwopaths,andonepathcontainedamovableretro-reectorusedtochangethepathlength.Then,usingalens,thetwobeamswerefocusedintoaBBO(BaB2O4)nonlinearcrystal.Thesum-frequency(SF)generatedbeamwasthenfocusedintoaphotodiodedetectorandthissignalwasusedastheautocorrelateddata.Thelaserpowerincidentintothesamplewaskeptbelow1mW.The1mJ/cm2highlaseruencewasnecessarytoobtaingoodS/Nfromtheautocorrelationsetup.Afterthelaser 38

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3-15 showstheautocorrelationdataforthebeamreectedfromthesilverlm.TheFWHMoftheautocorrelationpulseisaround200fs,whichgivesapulsewidthof140fs.Figure 3-17 showstheresultsfortheperiodicholearraysofvaryinggeometrycomparedtothesilverpulse.Theresultsshownobroadeningofthepulseoranyappreciablehighersignalforthelongertimedelays.ThereforeourresultssuggestthatiftheEMmodesdogettrappedinoursamples,theyhavealifetimelessthan100fs. 50 ].Instandardnonlinearmaterials,theinteractionlengthoftheEMeldshastobemanyordersofmagnitudehigherthanthewavelength,andthereforethematerialshavetobethick.Thetrappedmodestheoryenthusiastsarguethatforholearrays,thereisnorequirementforalonginteractionlengthbecausetheeldsinteractforalongtimebeforeradiating.Theircomputations[ 50 ]report105enhancementofnonlineareectsonperiodicholearrayslledwithnonlinearmaterials,andtheenhancementisattributedtothelongerinteractiontimeoftrappedmodes[ 50 ].Unfortunately,themanuscriptdoesnotreportthelifetimeofthemodesfortheirsimulation.Here,wepresentasimplistic(andperhapsnaive)insightintohowlongthelifetimeofthesemodesshouldbeforpossibleapplications.AtypicalBBOcrystal(liketheoneusedinthisworktondthesum-frequencygeneration)isabout5mmthick,andcanhaveecienciesintheorderof10%forsecondharmonicgenerationinthevisible.Thisinteractionlengthof5mmisintheorderof1000timesthewavelengthandtranslatestoaninteractiontimeofaround16ps.Basedonthiscalculation,wewouldconcludethattheperiodichole-arrayswecanbuildwithresonantfrequenciesaroundthevisibleregionwithQfactorof10andlifetimeslessthana100fswouldbeterriblefornonlineareects 39

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40

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Diagramofapatternedholearray.Thewhitespacesdenoteemptyholes.Theholesizeisdenotedasaandthehole-spacingasDg. TransmittancespectrafortwosilverlmsonaZnSesubstrate.Thehorizontallinerepresents0.44,theopenareafraction. 41

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ReectanceandtransmittancespectrafortheZnSesubstrate(Left).Thesum(R+T)isshownintheright. Sumofthereectanceandtransmittancecalculatedbytrappedmodestheory.Thisgureisborrowedfromreference[ 49 ]. 42

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Reectanceandtransmittancespectrafortwoperiodichole-arrayswithperiodicityDg=6monaZnSesubstrate.adenotesholesizeandDgperiodicity. Sumofthereectanceandtransmittancespectraforthetwoperiodichole-arraysonaZnSesubstratewithperiodicityDg=6m. 43

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Reectanceandtransmittancespectrafortwoperiodichole-arrayswithperiodicityDg=8monaZnSesubstrate.adenotesholesizeandDgperiodicity. Sumofthereectanceandtransmittancespectraforthetwoperiodichole-arraysonaZnSesubstratewithperiodicityDg=8m. 44

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Reectanceandtransmittancespectrafortwoperiodichole-arraysonaquartzsubstrate. Sumofthereectanceandtransmittancespectraforthetwoperiodichole-arraysonaquartzsubstrate. 45

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Thoughtexperimentforthetimedependenceofapulsetransmittedorreectedfromaperiodichole-array.Thisisthecasewherethelifetimeofthemodesislargerthanthepulsewidth. Thoughtexperimentforthetimedependenceofapulsetransmittedorreectedfromaperiodichole-array.Thisisthecasewherethelifetimeofthemodesismuchsmallerthanthepulsewidth. 46

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Thoughtexperimentforthetimedependenceofapulsetransmittedorreectedfromaperiodichole-array.Thelifetimeofthemodesiscomparabletothepulsewidth. 47

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ExperimentalsetupatNHMFLfortheautocorrelationofreectedpulsesfromperiodichole-arrays.SFreferstothesumfrequencypulse,andBBOreferstotheBaB2O4nonlinearcrystalthatgeneratestheSF.BSreferstobeamsplitters. 48

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Autocorrelationdataforthereectedpulsefromthesilverlm. 49

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Autocorrelationdataforthereectedpulsefromfourperiodichole-arrays.TheholesizeisdenotedbyaandtheperiodicitybyDg. 50

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Comparisonoftheautocorrelateddataforthereectedpulsefromthesilverlmsandthevarioushole-arrays. 51

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95 96 ],buthaveearnedrecentattentionforhigh-frequencylterapplicationsthankstotheirlowloss,highpermittivity,andgoodtemperaturestability[ 51 ].Thepyrochlorestructure(Fig. 4-1 )isdescribedasconsistingofinterpenetratingnetworksofBO6octahedraandA2O0chains[ 97 ]anditisassignedtothespacegroupFd3m.ThenominalcompositioncanbewrittenasA2B2O7orasA2B2O6O0,withthelatterformuladierentiatingtheoxygenintheA-O20chains.ThepyrochlorefamilyisfascinatingbecausetheAandBsitescanbeoccupiedbyabroadrangeofelementsthatcangiverisetoagreatvarietyofphysicalproperties.Inthebismuthpyrochlore,Bi1:5Zn0:92Nb1:5O6:92(BZN),theAsiteismostlyoccupiedbyBiandtheBsitebyNb;whileZnpartiallyoccupiesbothsites.Itisimportanttonotethatintheliterature,cubicBZNistypicallydescribedashavingtheexpectednominalcompositionofBi1:5Zn1:0Nb1:5O7.However,phaserenementstudies[ 98 ],havedemonstratedpartialsubstitutionofZnintheA2O0network(witharesultingoxygendeciencyaspresentedabove)tosatisfythecrystallochemicalbalancebetweenionicbonding,latticestrainandchargebalance.Inaddition,theBZNstructurehasbeenshowntodierfromanidealpyrochlorestructurethroughrandomdisplacementsoftheAandO0ions[ 98 ].Manybismuth-basedmaterialshavebeeninvestigated,butonlythespectrumofBZN[ 99 ]wasknownuntilrecently,whenChenetal.investigatedtheinfraredmodesofBZNalongwiththreeadditionalsystems:Bi3=2ZnTa3=2O7(BZT),Bi3=2MgNb3=2O7(BMN)andBi3=2MgTa3=2O7(BMT)[ 52 ].Itisofgreatinteresttostudythevibrationalspectraofthesematerials,becausetheyprovideuniquematerial-dependentinformationaboutdefectsorimpurities,thecrystallographicordering,andtheorderingandorientationofdipoles.However,infraredspectroscopycanonlydetectthosevibrationalmodeswhichhaveanetdipolemomentchange:in 52

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52 100 ].ThemainpurposeofthisworkwastocomparehowtheRamanactivemodesbehavedacrossfourbismuthpyrochloreshavingdierentconstituents,andtocomparethemwithotherpyrochlore-structuredmaterials.TheresultsshowthattheRamanspectraareonbalancequitesimilarforthebismuthsamples.Eachsampleshowsmorethanthesixmodespredictedfortheidealpyrochlorestructure,conrmingthedisplacivedisorderinthebismuthpyrochlores.TheRamanmodesweobservedareassignedtospecicnormalmodesbyreferencetotheliterature[ 101 { 107 ].Furthermore,theresultswerecomparedtotheworkbyFischeretal.wherethefrequencyofRaman,infrared,andopticallyinactivemodesofCd2Nb2O7werecalculatedbyabinitiocalculations[ 53 ].Thiscomparisonoersinsightintotheoriginoftheadditionalmodesduetodisorder.OurRamanspectrawerealsocomparedtotheinfrareddatabyChenetal.[ 52 ]andthecomparisonalsosuggeststhatsomeoftheextramodesareduetodisorder. 108 109 ]wasusedforsampleprocessing.RoomtemperatureRamanspectraweremeasuredwithaT64000JobinYvontripleRamanspectrometerequippedwithaliquid-nitrogen-cooledback-illuminatedCCDdetector.Weusedthe488nmand501nmlinesoftheAr+ionlasertoexciteRamanscattering.Themeasurementsweredonewiththelaserpoweronthesamplenotexceeding6kW/cm2andwithanaccumulationtimeof20seconds.Thespectraweretakeninthebackscatteringgeometry;thescatteredlightwasnotpolarized 53

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110 ]yieldssixRamanactivemodes(R)andseveninfraredactive(IR)modes: =A1g(R)+Eg(R)+4F2g(R)+7F1u(IR)+F1u+4F2u+2F1g+3A2u+3Eu; wheretheAandBcationsareplacedonaninversioncenter,andallofthesixRamanmodesinvolvemotionofoxygensatomsonly.Figures( 4-2 4-3 4-4 4-5 )showtheRamanspectraforeachsamplealongwithindividuallorentzianoscillatorsusedforeacht.Figure( 4-6 )showsthefourspectrawithscaledandshiftedintensitiesforeaseofcomparison.TheappearanceofmorethansixRamanmodesinallfoursamplesconrmedtheadditionaldisorderorionicdisplacementsfromtheidealatomicpositionsinthepyrochlorestructureintheinvestigatedbismuthbasedsamples.However,itwasreasonabletoexpectthatmanyofthemodesfromtheidealpyrochlorestructurewouldstillbepresent,andtheassignmentofthese\ideal"modeswasdonebyreferencingpreviousliteratureondiversepyrochlores[ 101 { 104 106 107 111 112 ].TableIshowsthefrequenciesofthevariousobservedbandsforthefoursamplesalongwiththeassignmentofmodes.Itisnottrivialtoassigneachbandtoaspecicstretchingorbendingvibration,sincebothVanderborreetal.[ 101 ]andBrownetal.[ 102 ]showthatthereismixingofdierentvibrationsforaparticularband.Theirworkestimatesthecontributionofeachvibrationalmodetoaobservedmodebycalculatingthepotentialenergydistribution.TableIlistsonlythemostsignicantlycontributingvibrationalmode. 101 ]. 54

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103 ]andBiYTi2O7(520cm1)[ 104 ].TheA1gmodefortheNbbasedpyrochloreCd2Nb2O7isat509cm1andpredictedat482cm1[ 53 ].Thismodedoesnotseemtovarygreatlybetweenpyrochlores:523cm1(Y2Ti2O7),489cm1(Ti2Mn2O7),511cm1(In2Mn2O7),512cm1(Tb2Ti2O7),489cm1(Ti2Mn2O7),498cm1(La2Zr2O7),sincethevarianceintheA1gfrequencyforallthesamplesmentioned,includingthebismuthpyrochlores,is3%.Therewas,however,asystematicincreaseinfrequencyof3percentfortheA1gmodeforBMToverBMNandBZToverBZN.IntheA1gmode,theBatomdoesnotmoveandthusthefrequencyofthemodeshouldonlydependonthesquarerootoftheforceconstant.The3percentincreaseinTasamplessuggestsaforce-constantratioof1.06forTaoverNb.ThisresultcorroboratesworkbyWangetal.whereithasbeenreportedthatintheoctahedron,oxygenbindstightertoTathantoNbasmuchasa1.10force-constantratio[ 113 ].Furthermore,thewidthsofthemodeswerealsolowerforBZTandBMT(70cm1)thanforBZNandBMN(100cm1).TheresultsforA1garecorroboratedbythemodelocatedaround428cm1whichhadthesametrend,withfrequencies3percenthigherfortheTasamplesthantheNbsamples,andwidthssmallerforBZTandBMT(45cm1)thanforBZNandBMN(88cm1).ThismodewastentativelyassignedbycomparingtheF2gmodeatCd2Nb2O7calculatedat441cm1[ 53 ]andobservedat422cm1,andotherpyrochloresGd2Ti2O7(455cm1),Tb2Ti207(452cm1),In2Mn2O7[ 102 ](442cm1)andthebismuthbasedYBiTi2O7(451cm1).TheEgmodehadanoppositetrend;frequenciesweresignicantlyhigher(>10percent)forBZNandBMN(310cm1)thanforBZTandBMT(345cm1).Intheliterature,thebandassignedtoEgcanhaveasignicantlyvaryingfrequency:250cm1(Cd2Re2O7)[ 106 ],297cm1(BiYTi2O7),312cm1(Y2Ti2O7),327cm1(Ti2Mn2O7), 55

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101 ];forthemanganitesitisat292cm1[ 102 ];and,thelowestreportedmodesareabove210cm1forY2Ti2O7,BYTi2O7[ 104 ]andCd2Re2O7[ 106 ].AnexceptionisTb2Ti2O7,wherethereisabandat173cm1assignedtoanF2gmode.IntheBZNliterature,thismodeseemstobediculttoassign:InRef.[ 113 ]itisassignedasthesamenormalvibrationF2gof255cm1(wherethe255cm1bandbelongstoaZn-Ostretch,andthe180cm1belongstoBi-Ostretch) 56

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113 ],whileotherworkhasassignedittoaF2gbandseparatethanthe255cm1F2gmode[ 114 ].Weproposeanalternativetentativeassignment.WeproposethisbandisanormallyRaman-inactive/IRactivemodethatappearsintheRamanduetothedisplacivedisorderoftheAsiteintheBipyrochlores.Wecanrecallthatbasedonsymmetry,theselectionrulesresultinsomevibrationalmodesbeingopticallyinactive.Foraninversion-symmetriccenter,themutualexclusionrulestatesamodecanbeIRactiveorRamanactive,butneverboth.Fromrandomdisplacementdisorder,wecanexpectnewpreviouslyinactivemodestoappearinbothIRandRamanspectra,butalsothatnormallyIR-onlymodesappearintheRamanspectraandviceversa.Foroursamples,theinfrareddataofthesesamples[ 52 ]showbandswithveryclosefrequenciestothosereportedherefortheRaman(compareTableIandII).Furthermore,thewidthoftheRamanbandsincm1are(71and75)forBMNandBZNand58forBZT,andfortheIRmodes,thewidthsare84,84forBMNandBZNand68forBZT.ThissuggestionisalsocorroboratedbythecalculationsonCd2Nb2O7byFischeretal.,whichgivesaF1umodeat190cm1,andnoF2gmodesbelow265cm1.Giventhesomewhatunusualassignmentproposedhere,anextendeddiscussionispresented.ItisimportanttorecallthatsincetheAandBsitesareplacedatinversioncentersintheidealpyrochlorestructure,themutualexclusionrulestatestheactiveRamanmodesareinactiveintheIRandviceversa[ 115 { 117 ].Themutualexclusionruleisageneralresultfromsymmetryandgrouptheory,butfordescriptivepurposes,considersmallvibrationsinthelinearO-A-OmoleculeshowninFigure( 4-7 ):allmodesareeitherRamanactiveorIRactive[ 118 119 ].Thesymmetricstretchingmodeisnotinfraredactivebecausethenetchangeindipolemomentiszero.ThismodehoweverisRamanactivebecausethestretchingofeachbondyieldsapositivechangeinpolarizability(apositivechangeinpolarizabilityresultsfromanincreaseinbondlength).AsimilaranalysiswouldshowthatQ2,theantisymmetricstretch,wouldbeIRactivebutnot 57

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113 114 ].ForTl2Mn2O7,In2Mn2O7andLa2Zr2O7thereisanassignedF2gmodeat512,548and590cm1respectively[ 101 102 ].InCd2Re2O7thereisnobandinthisregion.InY2Ti2O7andGd2Ti2O7abandnear570cm1hasbeenattributedtotheF2gmode,butthebandisnotpresentforallpreparationmethods[ 104 ].BiYTi2O7showstwobandsat588and612cm1,wheretherstisassignedasF2gandthesecondtoleftoverTiO2rutile.ForTb2Ti2O7thereisabandnear582cm1,butitisnotknownifthisbandortheir452cm-1bandistheF2gmode.ForCd2Nb2O7,Ref[ 53 ]calculatesnoF2gmodesinthisregion,twoopticallyinactivemodesat579cm1(F2u)and617cm1(F1g).Foroursamples,thesuggestionthatthisbandisnotanormalF2gmodeissupportedbythesimilarbandfoundintheIR(seeTableIandII).Ourresults, 58

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114 ]andalsotoastretchingmodeoftheNb-Obond[ 113 ].Theovertoneassignmentisthemostcommonforotherpyrochlores:La2Zr2O7(743cm1)[ 101 ];In2Mn2O7(700cm1)andTl2Mn2O7(750cm1);Cd2Re2O7(700cm1).Basedonlatticedynamiccalculations,Maczkaetal.[ 107 ]suggestthatnofundamentalF2gstretchingmodeshouldexceed600cm1forT2Ti2O7andthereforeassignhighermodestoovertones.However,Fischer'scalculationspredicta883cm1F2gmodeforCd2Nb2O7.InBi2Hf2O7andBi2Ti2O7theauthorsassociatemodesinthe650-800cm1regionwithoctahedralB-Ostretchingmodes.And,inRef[ 104 ]themodesaround700cm1areleftunassignedforGd2Ti2O7,Y2Ti2O7andBiYTi2O7.Therefore,assigningthismodebasedontheliteratureisnottrivial.ForBMN,BMT,BZNandBZT,thehighintensityofthismodesuggestsitismorelikelyafundamentalmoderatherthanatwo-phononscatteringprocess(overtone).ItisalsopossiblethatthismodeisanormallyRamansilentmodeaswell.AsfortheNb-Ostretchsuggestion,thesystematicincreaseinfrequencyforBMNoverBMTandsimilarlyforBZNoverBZTsuggeststhatitwouldbeanassymetricstretchingmode,wheretheBcationmovesaswell(basedontheassumptionfromtheA1gmodeandRef.[[ 113 ]]thatTabondsstrongerthanNb).Then,the3%increasefromNbsamplesoverTasamplesisexpectedfromthereportedforceconstantratio(kTa/kNb=1.10)forTaoverNbandthereducedmassratio()oftheBO6octahedron(1.15). (!Ta)2 59

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53 ]andcorroboratestheircalculation.However,basedonthelowamplitudeandhigh-frequencyofthemodeitcouldbearguedthatthemodeisanovertoneaswell.Itisalsointerestingthattheinfraredspectraofthesesamplesalsoshowan850cm1modeforNbsamplesbutnotforTasamples[ 52 ].Thereisyetanotherinterpretationforthismodethen.Chenetal.arguedthatthismodeappearsintheIRduetothevibrationoftheunequalbondlengthO-A-Obond(wheretheunequalityinbondsisduetothedisplacedAcation).TheyarguethatthemodeappearsinBZNandBMNandnotinBZTandBMT,becausetheTasampleshavelessdisplacivedisorder,duetodecreaseinthelonepaircharacterofBi3+bylone-pairhybridizationwithTa.TheargumentisbasedinthelargerelectronegativityofTaoverNb.However,theRamanspectraofthesefoursamplessuggestthereisnodierenceindisorderbetweentheNbsamplesandtheTasamples.BothNbandTasampleshadmodesthatappearedduetotherelaxationoftheselectionrules. 60

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53 ]calculations.Finally,theexistenceofadditionalmodesintheRamanspectraofallfourcompoundssuggestsnodierenceintheamountofdisorderbetweenNbandTasamples. 61

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TheRamanmodesofBMN,BZN,BMTandBZTareshownalongwiththeirtentativeassignment.Theclassassignments(i.e.F2g)aremeantforcomparisontotheidealpyrochlorestructure. 78727669EuandorF1u:O'-A-O'Bend 150148150151Eu:O-A-OBend 186185184F1u:A-BO6stretch 236223255243F2g:A-O'Stretch 350310343309Egand/orF2g:O-B-OBend 414428418433Eg:O-B-OBend 513530528541A1g:SymmetricBO6elongation 603619612622F2gand/orF1g:B-OStretch 781759762742B-OStretch 819805804809Overtone 862860F2gorOvertone Table4-2. TheinfraredmodeswithclosefrequenciestothoseoftheRamanareshown(WorkfromChenetal.)ChenassignsthismodetoO-A-Obending. 868381O'-A-O'Bend 149142145O-A-OBend 178*178178192A-BO6stretch 367336340303A-Ostretch 599642624639B-Ostretch 850850ShortA-Obondstretch 62

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ComparisonbetweentheRamanmodesofBMNandotherpyrochlores BMNCd2Nb2O7(calc.)[ 53 ]YBiTi2O7[ 104 ]Bi2Ti2O7[ 103 ]Gd2Ti2O7[ 104 ]Tb2Ti2O7[ 107 ]La2Zr2O7[ 101 ]In2Mn2O7[ 102 ]Tl2Mn2O7[ 102 ]Cd2Re2O7[ 106 ] 7869(Eu)and71(F1u)76(LongA-Ostretch)n.o.n.o.n.o.n.o.n.o.n.o.100 150133(Eu)n.o.n.o.n.o.n.o.n.o.n.on.o.150 186190(F1u)n.o.n.o.n.o.173(F2g)n.o.n.o.n.o.180 236265(F2g)220(F2g)230211(F2g)210(F2g)238(F2g)292(F2g289(F2g)240(F2g+Eg) 350300(Eg),332(F2g),360(F1u)297(F2g)360312(F2g)310(F2g),330(Eg)307(F2g)346(Eg)327Eg320Eu 513482(A1g)523(A1g)550519(A1g)512(A1g)490(A1g)510(A1g)510(A1g)510(A1g) 603617(F1g)588(F2g),600(TiO2rutile)600560(F2g)582(F2g?)591(F2g)548(F2g)510(F2g)n.o. 781n.o.725780(B-Ostretch)708672(over-tone)743(over-tone)700(over-tone)700(over-tone)700(over-tone) 810,850883(F2g)n.o.n.o.n.o.n.o.n.o.n.o.n.o.n.o.

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Pyrochlorestructure.RedatomssignifyoxygenintheBO6octahedraandblueatomssignifyoxygenintheO'-A-O'chain.TheAandBcationsaredepictedbyyellowandgreenrespectively,whilethewhiteatomsshowvacancies. 64

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RamanspectrumofBMN(black)withthetting(red)tothedata.Theindividuallorentzianoscillators(blue)usedinthetarealsoshown. 65

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RamanspectrumofBZN(black)withthetting(red)tothedata.Theindividuallorentzianoscillators(blue)usedinthetarealsoshown. 66

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RamanspectrumofBMT(black)withthetting(red)tothedata.Theindividuallorentzianoscillators(blue)usedinthetarealsoshown. 67

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RamanspectrumofBZT(black)withthetting(red)tothedata.Theindividuallorentzianoscillators(blue)usedinthetarealsoshown. 68

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RamanspectraofBMN,BZN,BMTandBZT.Theirintensitieshavebeenscaled(by1,1.14,1.64,and2.46,respectively)andshiftedforeaseofcomparison 69

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ThenormalmodesofvibrationofalinearO-A-Omoleculeareshown(top),alongwiththemodesofanonlinearmolecule(bottom).TheIRandRamanmodesarelabeled. 70

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120 121 ]:(CGSunits) @t=4 c~J(A{1) @t=0(A{2) 71

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A{1 and A{2 )thatinterrelatetheelectricandmagneticeld.Firstwetakethecurlofequation A{1 : A{2 : c@ @t(1 @t+4 c~J):(A{9)Forneutrallychargedmaterials,iszeroandequation A{9 simpliesto 72

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c2@ @t(@~D @t+4 c~J) (A{10) c2@2~E @t2+4 c@~E @t: Forvacuum,thisequationbecomesthesimpleandfamiliarwaveequation @t2=0;(A{12)anditssolutionisanunattenuatedpropagatingwaveintheform A{12 ,weobtainthedispersionrelationinvacuum A{11 willattenuatethepropagatingwave.Thesolutionwillbeacomplexpropagationconstant: c2!:(A{16)Fromnowon,wejustdeneacomplexdielectricfunction: 73

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^=1+i4^ !;(A{18)wheretheirrealandimaginarypartsarereferredtobysubscripts1and2: ^=1+i2 ^=1+i2 Itisusefultodenecomplexmaterialparameters.Nowwecanuseeitherthedielectricfunctionorthecomplexconductivitytodescribeasystem,becauseonepropertycanbeobtainedfromtheother.Moreimportantly,responses(i.e.currentsandpolarization)arenotalwayslocalintime,meaningtheyaredependentonthedrivingeldatprevioustimesandcanthereforehaveadierentphasetothedrivingeld.Thedenitionofcomplexconductivitysumsupthisstatement.ForamaterialinuencedbyanelectriceldE0ei!t: ^Jei!t=1E0ei!t+i2E0ei!t=1E0ei!t+2E0ei!t=2;(A{21)therealpartoftheconductivityisassociatedwiththecurrentgeneratedinphasewiththeelectriceld,andtheimaginarypartdescribesa90degree-out-of-phasecurrent.Therefractiveindex^Nisalsoacomplexfunction,anditisdenedusing: ^k=! c^N(!);(A{22)andwiththisdenition: 74

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A{23 intothesolutionoftheattenuatedpropagatingwaveinsidethematerial A{14 : A{3 and A.1 tellusthatforahomogeneous,neutrallychargedmedium,thepropagatingwavesolutionsyields: 75

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A{2 andobtain: I(z)=Re[EH]E02e2(!)! cz;(A{34)theexponentialdecayoftheintensityischaracterizedbytheabsorptioncocient: c:(A{35) 76

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Boundaryconditionsataninterface.Electriceldsfortheincident,reectedandtransmittedpropagatingwavesolutionsattheinterfacebetweentwomedia. 77

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A{30 and A{32 ,whichrelatethemagneticeldtotheelectriceld,andequations B{1 and B{2 ,weobtain A-1 ),andthattherealandimaginarypartsofNaregivenbynandasinequation A{23 .ThenegativeterminN11Ercomesfromequation A{30 and B{2 andthefactthatthereectedwavehasanegativekvalue.Nowwecansolveforthereectionamplitudecoecient: 78

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^t12=Et B{4 .Nonmagneticmaterials: 79

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B{6 forrealN1 ^r=Er 80

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122 ] ^t=^t12^t23ei[1+(^r12^r23e2i)+(^r12^r23e2i)2+:::]:(B{19)Thisinniteseriesreducesto ^t=^t12^t23ei c^Nd=n! cd+i 122 123 ]and[ 124 ]forthematrixmethod.Noticethattherearetwounknownsinthedeterminationoftherefractiveindex,therealandimaginarypart.Therefore,twoseparatemeasurementswouldberequired(i.e.TandR)toobtaintheopticalconstants.However,certainsamplescanbealmostcompletelyabsorbingorcompletelyreective(i.e.metallic)sothatonlythereectancefromtheair-sampleinterfacecanbemeasured(Wecallthissingle-bouncereectance).Inthiscase,theequationforsinglebouncereectanceisvalidbecausetheretherearenointernalreections.But,itleavesuswithtwounknownsandonlyonemeasurablequantity.OnetechniquetoovercomethisproblemisKramersKronigAnalysis,whichconsistsinawidefrequencymeasurementofthereectancetoobtainthephasechangeinreectancefromequation B{5 .ThefollowingsectionexplainsKKanalysis. 81

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^X(t)=Z1^G(tt0)^f(t0)dt0;(C{1)^X(t)isaresponseofthesystem(i.e.currentorpolarization),^f(tt0)istheexternalstimulus(i.e.electriceld),and^G(tt0)istheresponsefunction.Examplesofresponsefunctionsaretheconductivityandthesusceptibility ^J(t)=Z1^(tt0)^E(t0)dt0;(C{2) ^P(t)=Z1^(tt0)^E(t0)dt0:(C{3)Theintegralsinequations C{2 and C{3 statethatresponsesaregenerallynonlocalintime.Responsesdependonstimuliappliedatprevioustimes,thesamewaythevelocityofaparticledependsonpreviousforces(ie.afallingobject'sdependenceonhowlongithasbeenfalling).Moreimportantly,theseequationsshowthattheresponseofasystemwon'talwaysbeinphasewiththestimulus.Thisoutofphasepossibilityisanotherjusticationfortheuseofcomplexresponsefunctions.Equation C{1 canberewritteninasimplerformbywritingthefouriertransformsofX,Gandf: ^X(!)=^G(!)^f(!); where 82

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ThisFouriertransformequationimpliesthat ^G(!)=^G(!); whichisanusefulrelationfortheoddnessandevennessoftherealandimaginarypartsofcomplexfunctions.Wewillusethisrelationlater.Causalityputsanimportantrestrictiononresponsefunctions.Itstatesthataresponse^Gattcannotdependonfuturetimes: ^G=0;fortt0<0: WewillquicklyderivethatthisrestrictioncausestheKramersKronigrelations: ^G1(!)=1 ^G2(!)=1 Toderivetheserelationsbetweentherealandimaginarypartsofaresponsefunction,werstmap^Gonthecomplexplanebyrewritingequation C{5 usingbothrealandimaginaryfrequencies: ^G(!)=Z1^G(tt0)ei!1(tt0)e!2(tt0)d(tt0):(C{11) 83

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C{11 imposesthatforpositive!2,theterm(tt0)mustbepositiveaswell.Similarly,fornegative!2,(tt0)mustbenegative.But,causality, C{7 ,statesthat^G=0fornegative(tt0);thereforearesponsefunctionthatobeyscausalityandequation C{11 isrestrictedtotheupperhalfofthecomplexplane(seeFig. C-1 ).Thepurposeofthislongrantistomap^GanduseCauchy'stheoremtorelatethevalueof^Gatacertainfrequency!0totheotherfrequenciesaroundit.FirstweuseCauchy'stheoremforananalyticfunction^G(!)[ 120 ]: C-1 showstheclosedloopwewilluseintheintegral.Wedrawasemicirclearoundthefrequencyofinterest,!0,withaninnitesimalradius",andanoutersemicirclewithradiusR.Wecandrawtheboundaryoftheintegralthiswaybecauseweknowthat^G(!)iszerofornegativeimaginaryfrequencies(goodoldcausality).Weevaluatethewholeintegral,whichshouldbezeroaccordingtoCauchy'stheorem: ^G(!0)=P1 84

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C{8 and C{9 .Noticethatthisresultisduetocausality;iftheintegralinthelowerhalfofthecomplexplanewerenotzero,thenwewouldhavenointerestingresult.ToeliminatethenegativefrequenciesintheKKequations,weusethepropertiesofthecomplexconjugateof^Ginequation C{6 andshowthat ^G1(!)=^G1(!) (C{15) ^G2(!)=^G2(!): UsingtheevenpropertyofG1andtheoddnessofG2,weobtainthemoreusefulKKrelations: ^G1(!)=2 ^G2(!)=2! Z10^G1(!0) Therefore,fortheconductivityresponsefunctionweget: ^1(!)=2 ^2(!)=2! Z10^1(!0) Asawordofcaution,wedonotassumethattheKKrelationsworkforanycomplexquantityandassumethatwecanequatetherealandtheimaginarypartsbyequations C{8 and C{9 foranyarbitrarycomplexfunction.Forexample,togettheKKrelations 85

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^1(!)1=2 ^2(!)=2! Z10^1(!0) ThesewordsofcautionaremostrelevantforthederivationoftheKKrelationsbetweenReectanceRandthephasechange.Thestartingresponsefunctionisthereectionamplitude^r,wheretheresponseisErandthestimulusisEi ^r=jrjei;(C{24)wherejrj2isthemeasurablesingle-bouncereectance.Wetakethelogarithmofequation C{24 toseparate`nrastherealpartandastheimaginarypart.ThederivationofthisKramersKronigrelationforthelogarithmicfunctionisdicultbecausergoestozeroatinnitefrequenciesandlnnrwouldthen'blowup'.ThereaderisreferencedtoWooten'sOpticalPropertiesofSolids[ 124 ]forthederivation.Theresultis: Z10lnkr(!0)klnkr(!)k andthesearetheimportantrelationsweusethemostinthiswork.ThemainpointisthatbymeasuringonequantityRoverabroadspectra,wecanobtainasecondvariable 86

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^r=j^rjei=(1n)i C{25 and C{26 spanallfrequencies.Experimentally,thisisimpossibleandrequiresthatweestimate(guess)thebehavioratthelowerorhigherendsofthespectrum.Fortunately,thedenominatorintheKKintegral, C{8 ,suggeststhatG1atafrequency!dependsmostlyonG2atfrequenciescloseto!.Furthermore,weestimatethelowerandhigherendsofthespectrumbyusingphysicalmodelsthatexplainthebehavioroftheopticalconstants.TheestimationforfreeelectronsandboundchargesfromtheDrudeandLorentzmodelwillbediscussedinthenextsection. 87

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Closedloopintegrationaroundafrequencypoint!0inthecomplexplane.CausalityrequiresGtobezerointhebottomhalf. 88

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120 124 ]. dt;(D{1)wherethersttermistheEMeld,andthesecondtermrepresentsthedampingtermwithascatteringrateequaltotheinverseoftherelaxationtime.IntheDrudemodel,theforcearisingfromthemagneticeldisignoredduetoitssmallervaluecomparedtotheelectricforce.Toincorporate\notsofree"electrons,thedrudemodelreplacesthemassmbyaneectivemassm,toestimatesmallelectron-latticeandelectron-electroninteractions.Thesolutiontoequation D{1 usingmis: 89

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^(!)=Nee2 m ^(!)=0 125 126 ]: m(D{6) D{7 canberewrittenas: 90

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D{7 and D{9 ,wecanseethatgoodconductors(metals)havenegative1atfrequenciesmuchlowerthantheplasmafrequency.Fortherefractiveindex,weuseequation D{8 andignorethefastdying2.Weobtain: lim!!1RDrude(!)!p2 1+(!)20(!):(D{13)Asforthedielectricfunction,therealpartapproachesaconstantnegativevaluewhiletheimaginarypartdiverges: 91

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lim!!0RDrude(!)1[2! 0]1=2:(D{17) D{1 ,arestoringforce(k~r)thatkeepstheelectronbounded: dtm!02~r=md2~r dt2;(D{18)andthesolutionis: ^(!)=1+!p2 ^(!)=! 92

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^1(!)=1+!pj2(!0j2!2) (!0j2!2)2+!22j1+!pj2 j=!pj2 ^(!)=1+Xj!pj2 lim!!0Lorentz=1+Xjj=Constant:(D{25)ormoreimportantly,thatthedielectricfunctionwillberealandconstantinthislimit.Thereforethereectanceduetoboundchargesshouldstayaconstanttowardslowerfrequencies.Forhighfrequencies,wecanseefromtheconductivityterm: ^(!)=! 93

PAGE 94

94

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E-1 ).Theintensityatthedetectorismeasuredasafunctionofpathlengthtoobtaintheinterferogram.TheFouriertransformoftheinterferogramthengivesyouthespectrum(intensityvsfrequency).ByreferringtoFigure E-2 ,wecanseethatiftheelectriceldintensityatthesourceisE0withpropagationvectork,thentheelectriceldafterrecombinationatthebeamsplitteris: 95

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E{5 ,thatthecoherence(interference)termcomesfromthecos(!x)term.And,foraperfectbeamsplitter: 2(E{6)thewholebeam(intensityI0)wouldbemeasuredatthedetectoratazeropathdierence.Intheincoherentlimit(xapproachesinnity),thetotalintensityisjustthesumofallintensities: limx!1Itotal=Z102I0(!)Rb(!)Tb(!)d!:(E{7)Thistermrepresentsthebaselineoftheinterferogram(seeFigure E-3 ).Inpractice,wemeasuretheinterferogramandsubstractthebaselineintensity: 96

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2+(!2+!) 2)d!;(E{12)therstdeltafunctiondropsthedummy!2andtheseconddeltafunctionyieldszero(fortherearenonegativefrequenciesontheintegral).Ournalresultrelatestheintensityspectratothemeasuredinterferogram: 2I0(!)Rb(!)Tb(!)=Zx=1x=Is(x)ei!xdx:(E{13) Symmetry.IftheinterferogramIs(x)isperfectlysymmetric,thenequation E{13 becomesacosinetransformandI(!)isreal.However,iftheinterferogramisnotperfectlysymmetric(likethenoisyhanddrawinginFigure E-2 ),thenthespectrumwilllooklike: 2I0(!)Rb(!)Tb(!)=Zx=1x=Ieven(x)cos(!x)dxiZx=1x=Iodd(x)sin(!x)dx;(E{14)whichgivesthespectrumaphase 97

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2jImirror(!)jeiRb(!)Tb(!):(E{16) E-4 ,andthewaves!2,!3and!4whichareexactreplicasofpoorlydrawnwave1,butshrunk.Forthetwowaveswithdistinctfrequencies(!1and!2),thesignalscouldberesolvedeasilybecausetheirsumwilldisplayinterference.However,forthewaveswithclosefrequencies(!3and!4),wewouldhavetoscanfarthertoseeinterference.Ifwescantooshortly,thenwejustseeasumofthesamewaveandwecan'tresolvethem.Thisisanalogoustoabeatwave.Forasourcewithtwofrequencies!1and!2,theinterferogramwillgiveyou cos(!1x)+cos(!2x)(E{17)Thebeatwaveis: cos((!2!1) 2x);(E{18)toseeaminimum,youhavetoscantill: (!2!1) 2xmax= 98

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Diagramofabasicinterferometer. Additionoftheelectriceldfromthetwodierentpaths.Theamplitudeoftheelectriceldofthetopmirrorisshownasitprogressesthroughthepath. 99

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Caricatureofaninterferogram(intensityvs.pathdierence).Thelimitasxgoestoinnitygivesthebaseline. Caricatureoftheresolutionbetweentwowaves.Alongerscanisneededtoobservedestructiveinterferencebetweentwowavesofcloserfrequency. 100

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F-1 ): @t=@ @x(Ex;Ey;Ez)e(ikxx!t)e1z=i!~E(F{5) @x=@ @x(Hx;Hy;Hz)e(ikxx!t)e1z=ikx~H(F{6) @y=0(F{7) 101

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F-2 ).We'dliketoexplorehowthesesurfacewavesbehaveforthesetwocases.ForTM,wheretheelectriceldisintheplaneofincidence, (F{13) 102

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(F{15) sincetheboundaryconditionsstatethat: Now,letsrelatetheelectriceldtothemagneticeldbyusingMaxwell's A{1 withnosurfacecurrent: c@~E @t:(F{19)Tosolvethisequationwecanuse: c@~E @t;(F{20) (F{24) 103

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@t=i1!Ex~xi1!Etopz~z;(F{27)then,bottom: @t=i2!Ex~xi2!Ebotz~z;(F{29)andequatexcomponentsforboth: F{32 showsthatthepropagatingwaveforthesurfaceisonlypossiblefortheinterfacebetweenanegativedielectric(metal)andapositivedielectric(i.e.airorquartz): A{11 weobtainedbackinAppendixA,andusethepropertiesforthecurlofETMlistedabove: c2@2~E @t2;(F{34) 104

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(k2x+22)~Ebot=2!2 F{35 from F{36 andalotofalgebragives: cr A{2 : @t(F{40)Referencingequations F{23 through F{25 ,andrememberingthatEyandHxarecontinuous: 1 cHx~x+i! cHtopz~z(F{43) 105

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cHx~x+i! cHbotz~z;(F{44)weobtain: Sinceispositive,HxonlymakessenseifEyiszero.Thiscombinedwithequations F{46 through F{48 resultineverycomponentbeingzero: 106

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Electriceldsforaconnedsurfacewavepropagatinginthexdirection.Topandbottomsolutions. DiagramsfortheelectricandmagneticeldcomponentsofTMandTEpolarizationsofasurfacewave. 107

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DanielArenaswasborninColombiain1983.HemovedtotheU.S.Aaftergraduatingfromhighschool.HeattendedEdisonCommunityCollegeinNaples,FLandthentransferredtotheUniversityofNorthFloridainJacksonville.Bychance,hetooksomeupper-levelphysicsclasses,lovedthem,andendedupswitchinghismajortophysics.AftergettinghisB.S.,hewasacceptedintoUniversityofFloridain2004andjoinedtheTannerLabinhissecondyear.Inthelastveyears,hehasmetwonderfulpeopleandmadegreatfriends. 115