Gyrotron Inter nal Mode Converter ReflectorShapingfr om MeasuredField Intensity

Denison,D. R.,Chu,T. S.*, Shapiro,M. A., Temkin,R. J.

December1998

PlasmaScienceandFusionCenterMassachusettsInstituteof Technology

Cambridge,MA 02139

Abstract

Wepresenttheformulationandexperimentalresultsof a new approachto design-ing internalmodeconverterreflectorsfor high-power gyrotrons.Themethodemploysanumericalphaseretrieval algorithmthatreconstructsthefield in themodeconverterfrom intensitymeasurements,thusaccountingfor the true field structurein shapingthebeam-formingreflectors.An outlinefor designinga four-reflectormodeconverteris presentedandgeneralizedto thecaseof anoffset-fedshapedreflectorantenna.Therequisitephaseretrieval andreflectorshapingalgorithmsarealsodevelopedwithoutreferenceto specificmodeconverter geometry. The designapproachis appliedtoa 110GHz internalmodeconverter that transformstheTE22� 6 gyrotroncavity modeinto aGaussianbeamatthegyrotronwindow. Coldtestexperimentresultsof themodeconvertershow thata Gaussianbeamwith thedesiredamplitudeandphaseis formedat the window aperture. Subsequenthigh-power testsin a 1 MW gyrotronconfirmtheGaussianbeamobserved in cold tests.Thegeneraldevelopmentof theapproachand its validation in a quasi-opticalmodeconverter indicatethat it is also applica-ble to otherquasi-optical,microwaveapplicationssuchasradioastronomy, free-spacetransmissionlines,andmitre bendsfor overmodedwaveguides.

*Microwave Power ProductsDivision, CommunicationsandPower Industries,Palo Alto,CA 94303USA

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1 Intr oduction

Gyrotroninternalmodeconverterstransformthehigh-ordercircularwaveguidemodeof agyrotroncavity into a Gaussian-like modesuitablefor free-spacepropagation.A typicalmodeconverterconsistsof a waveguidesectionwith a radiatingaperturefollowed by aseriesof reflectors(or mirrors),asdepictedin Figure1. ThewaveguideconstituteseitheraVlasov [1] or rippled-wall [2] launcherthatdirectsthemicrowaveenergy radially througha wall aperture,separatingit from the spentelectronbeam. The radiatedwave is thenfocussedby a seriesof reflectorsthat alsoserve to guidethe microwave beamthroughalow-lossvacuumwindow andoutof thetube.

At thepower levelsof microwave radiationproducedby conventionalhigh-power ( �1 MW) gyrotrons,the electricfield intensityprofile over the window apertureprovestobe the limiting factor in the overall power handlingcapabilityof the gyrotron for long-pulseandCW operation. The microwave beamshapeon the window is constrainedbythethermalcharacteristicsof thewindow materialandcoolingconfiguration,aswell asbyedgediffractionlossesat thewindow aperture.Theinternalmodeconverterreflectorsmustbeshapedto provideafield profileatthewindow thataccommodatesthethermalpropertiesof thewindow materialandminimizesedgelosses.

Previousmodeconverterreflectordesignshave reliedon numericalsimulationsof thefields radiatedby the launcher. For instance,the rippled-wall launchercan be modeledusingcoupledmodetheory, which leadsto asetof simultaneousdifferentialequationsthataresolvednumericallyfor thefieldsinsidetheguide[3]. Theradiatedfield,computedfroma vectordiffraction integral, behavesasa quasi-Gaussianbeamthat canbe focussedintoa nearly-idealGaussianbeamat the window by usingdoubly-curved reflectorsdesignedfrom analyticexpressions[3, 4]. For morecomplicatedfinal beamshapes,suchasa flatpower profile over the window aperture,reflectorsynthesismethodsthat incorporatethesimulatedlauncherfields in the reflectorshapingareused[5]. Experimentalresultshavedemonstrated,however, thatthefield intensityonthegyrotronwindow oftendoesnotagreewith thedesignprofile [6]. This deviation from designresultsin a lower power handlingcapabilityof thewindow, limiting theoverall performanceof thegyrotron.

M2

M3

Launcher

M1

M4

Windowy

z

Figure1: ModeConverterSchematic.

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Cold test(low-power) measurementsof the field profile in the modeconverterrevealthat thefields radiatedby the launcherdiffer from thosepredictedby theory. This differ-encemayarisefrom machiningerrorsin the launcher, misalignmentof the launcherwithrespectto thereflectors,or anincompletetheoryfor thelauncher. Suchvariationsfrom theideal, theoreticalsituationsuggestthatany reflectordesignbasedon simulatedfieldswillnotproducethedesiredoutputfield in actualoperation.In orderto overcometheobserveddifficulties in forming thedesiredmicrowave beamshapein a gyrotron,we recentlypro-posedanew approachto reflectordesignthatincorporatesmeasuredfield intensitiesin thedesignprocess[7].

The following paperreportsthe developmentandexperimentalvalidationof this ap-proachto internal modeconverter reflectordesign. Section2 outlinesthe basicdesignprocedurefor a four-reflectorinternalmodeconverter. Thereflectorsaretreatedasphase-correctingsurfaces,which permitsa generalizationof the designapproachto arbitraryquasi-opticalsystems.An algorithmfor retrieving phasefrom intensitymeasurementsisdiscussedin Section3, while Section4 extendsthephaseretrieval algorithmto theprob-lem of shapingphase-correctingsurfaces.Section5 appliesthe proposeddesignmethodto a 110GHz internalgyrotronmodeconverterwith a diamondwindow. We presentex-perimentalresultsfor bothcold testsandhot teststhatverify theefficacy of theproposedmethod.

2 Outline of ReflectorDesignApproach

As discussedin Section1, the measuredoutputbeamof gyrotronmodeconvertersoftendiffers from the theoretically-predictedfields. This discrepancy betweentheoryandex-perimentappearsto originatewith the launcherfields. For example,considera 110GHzgyrotroninternalmodeconverterthatusesarippled-wall launcherandfour reflectors(Fig-ure1) to transformtheTE22� 6 cylindrical modeinto a flat-topbeamat theoutputwindow.Figure2 shows thetheoretically-computedfield intensityat theplaneof thethird reflectorsurface(M3 in Figure1), while Figure3 givesthemeasuredfield intensityover thesameplane.In eachfigure,thez-axisorigin is atthebeginningof thelauncher, asdepictedin Fig-ure1. Althoughthetwo patternsaresimilar, therearesignificantdifferencesin thesizeandshapeof theindividual intensitycontoursthatultimatelyresultin theobserveddifferencesbetweentheoreticalandexperimentalwindow electricfield intensityprofiles[6].

To accountfor theactualfieldsin themodeconverter, weproposethefollowing proce-durefor designingthereflectorsbasedon intensitymeasurements[7, 8]:

1. Designandbuild thelauncherto produceaGaussian-likebeam.

2. Measurethefield radiatedby thelauncheranddesignthefirst doubly-curvedreflectorby fitting anelliptical Gaussianbeamto themeasuredpattern.

3. Measurethefield intensityfollowing thefirst reflectoranddesignthesecondreflectorbasedonabest-fitGaussian.

4. Measurethe field intensityfollowing the secondreflectorandretrieve the phaseofthebeamto reconstructthefull field structureof thewave.

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Figure2: Theoreticalfield intensityover a planeat theReflector3 position. Contoursofconstant

�Ex� 2 areat3 dB intervalsfrom peak;the � 3 dB and � 24 dB curvesarelabeled.

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Figure3: Measuredfield intensityover a planeat the Reflector3 position. Contoursofconstant�Ex � 2 areat3 dB intervalsfrom peak;the � 3 dB and � 24 dB curvesarelabeled.

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5. Usethereconstructedfield afterthesecondreflectorandthedesiredfield structureonthewindow asinput to areflectorshapingprocedureto determinethesurfaceprofilesfor reflectorsthreeandfour.

6. Simulatereflectorsthreeandfour in anumericalelectromagneticscodeto verify thedesign.

Thesestepscanbegeneralizedto thecaseof shapingreflectorsin anoffset-fed,dual-reflectorantenna.Steps(1) – (3) definethefeed,andtheremainingstepsinvolve shapingthe reflectorantennasfrom measurementsof the feed. Theabove outline thenreducestotheproblemsof retrieving phasefrom intensitymeasurementsandreflectorshaping.

3 PhaseRetrieval fr om Intensity Measurements

At high frequencies( � 100GHz), thedifficultiesof directly measuringphaseleadto thestipulationthat we recover the phasefrom field intensitymeasurements.This phasere-trieval problemwasconsideredby Katsenelenbaum[9] in termsof aniterativedetermina-tion of phasecorrectors;Gerchberg andSaxton[10] (evidently independently)introducedasimilar approachfor themorerestrictivecaseof retrieving phasefrom anobjectfunctionandits Fourier transform.Thegeneralcaseof two or morearbitrarymeasurementplanesis presentedasamodifiedGerchberg-Saxtonalgorithmby AndersonandSali [11], andweoutlinea similar methodbelow. Althoughtheformulationis in termsof a scalarfield, wecanalwaysdecomposethebeamin freespaceasa sumof threelinearfield vectors.Theadditionalissuesof attainability, uniqueness,convergenceof thesolution,andexperimentalvalidationof this inverseproblemhavebeenthesubjectof muchstudy;for asurvey see[9]– [16].

Iterative phaseretrieval is accomplishedwith the following algorithm. Supposewehave two measurementplanes,perpendicularto the z-axis in a Cartesiancoordinatesys-tem,with a wave propagatingroughlyparaxially, or quasi-optically, alongthez-axis. Themeasurementplanesare locatedat z1 andz2 with z1 � z2, andthe (for now continuous)measuredamplitudeovereachplaneis denotedA1 x y� andA2 x y� , respectively.

Thetotal scalarfieldson thetwo measurementplanesarethenwrittenas

u1 x y z1 � � A1 x y� eiφ1 � x � y� u2 x y z2 � � A2 x y� eiφ2 � x � y� (1)

whereφ1 x y�� φ2 x y� aretheexactphasesontheirrespectiveplanes.Thefieldsarerelatedby theplanewaveexpansionas

u2 x y z2 ������� 1 � ei � z2 � z1 ��� k2 � kx2 � ky

2 � ��� u1 x y z1 ����� (2)

where � is the Fourier transformand � � 1 its inverse. Our earlier requirementthat thebeampropagateroughlyparaxiallystemsfromthefactthatin thecaseof discretelysampleddataunderpracticalmeasuringconditions,the rangein k-spacefor the Fourier kernel islimited by the samplingrate, and we cannothopeto recover modal componentsof thewave that have large valuesof kx andky. Furthermore,we plan to treat the reflectorsas

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phase-correctingsurfacesin Section4, andthatdecisionnecessarilylimits ourapproachtoquasi-opticalbeams.

Thephaseis retrievedby iteratively solving(2) for thephasefunctionsφ1 ! φ2 with themeasuredamplitudesasweights.For compactness,we have droppedtheexplicit variabledependencein the field functionsu ! A ! andφ. Beginning with an initial phaseguessonthefirst plane,φ1 " 0# , wecomputeu2 " 0# with u1 " 0#�$ A1eiφ1 % 0& ! wherethesuperscriptdenotesthefield afterthenth iteration.Clearly, thecalculatedamplitudeonplane2, A2 " 0# , will notequalthemeasured(correct)amplitudeover thatplane(unlessthe initial phaseguesswastheactualphaseon plane1). Wecanrepresenttheerroraftereachiterationas

E " n#1 ' 2 $)(S1 * 2 +A1 ' 2 , A " n#1 ' 2 + 2dS1 ' 2 ! (3)

wherethesubscripts1, 2 referto thecorrespondingplane.For thenext stageof theiteration,we replacethecomputedamplitudeon plane2 with

the measuredamplitudeA2 to producea modifiedfield on plane2: u2 " 0# $ A2eiφ2 % 0&.- Wethenpropagatethis field backto plane1 via (2), with indexes1 and2 interchanged.A newfield is constructedon plane1 usingthecomputedphaseandmeasuredamplitudeon thatplane.Theprocessis repeatedN timesuntil theerrorE " N # (3) is lessthansomeprescribedvalue.

Theadvantageof theabovephaseretrieval algorithmlies in evaluating(2) for thecaseof discretely-sampledamplitudes(or field intensity I , whereA $0/ I ). Eachiterationin-volvestwodiscreteFouriertransformsandonecomplex multiply. Usingatwo-dimensionalFastFourier transform[17] we canrapidly compute(2). We have implementedthis ap-proachin an original FORTRAN codethat hasbeenbenchmarked againstsimulatedandmeasureddata. In practice,the error (3) tendsto decreasein stageswith a numberofplateausbeforethe error reachessomeabsoluteminimum (seee.g.[13]). Our studiesofsimulateddataindicatethat overcomingthe local minima correspondsto improving theresolutionof fine detailsof the field structure,particularlyat low intensity. Additionally,theconvergenceof thesolutiondependsonthedistancebetweenmeasurementplanes,withlargerplaneseparationsimproving therateof convergence.However, if thebeamchangessignificantlyover a relatively shortdistance,thenthemeasurementdatastill containsuffi-cient informationfor thephaseretrieval, at theexpenseof needingmoreiterations.Giventhesefactors,the ability to performmany (100 – 1000) iterations,madefeasibleby ourimplementationof the phaseretrieval algorithm, is crucial to obtainingan accuratefieldreconstruction.

4 Reflector Shaping

In Section2 weintroducedtheconceptthatthelauncherandfirst two reflectorsin themodeconverterconstitutea feedin a conventionaloffset-fed,dual-reflectorantenna.Extendingthisanalogy, wecanview theshapingof reflectorsthreeandfour asthegeneralproblemofbeamsynthesisin antennadesign— givenaspecifiedfeed,shapethereflectorsto producethedesiredoutputbeam.

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∆1

Aei ϕ1 Ae Ae Ae

∆2

i ϕ1sc i ϕ2

sc i ϕ2

Figure4: Reflectorshapinggeometry.

Onetechniquefor shapingthe reflectorsis to treatthemasphase-correctingsurfaces;thatis, thepolarizationandamplitudeof theincidentwave remainunchangeduponreflec-tion. This approximationholds for a wide classof quasi-opticalmicrowave engineeringproblems;the designof mirrors, lenses,andresonatorsin Gaussianopticsreliesheavilyon modelingelementsasphasecorrectors(e.g.[4]). Anotheradvantageof this treatmentis thatwe canusethephaseretrieval algorithmdescribedin Section3 to definethephasecorrectingsurfaces[18, 19, 20], unifying our designapproachunderasingleconstruct.

To seetherelationshipbetweenphaseretrieval andreflectorshaping,considerthege-ometryof Figure4. The two phase-correctingsurfacesarecastasequivalentthin lenseswith phase-correctingfunctions∆1 1 x 2 y3 and∆2 1 x 2 y3 . Thebeamis propagatingalongthez-axis, perpendicularlyto eachlenssurface. The incidentbeamis denotedasA1eiφ1 andtheoutputbeamis A2eiφ2. WeseethattheamplitudesA1 andA2 maybeassociatedthroughthepair of self-consistentphasesφsc

1 andφsc2 , which aredeterminedby applyingthephase

retrieval algorithmto A1 andA2. Thenthephasecorrectingfunctionrequiredto transformthe incidentphase,φ1 2 into the first self-consistentphase,φsc

1 2 is givenby ∆1 4 φsc1 5 φ1.

Similarly for thesecondphase-correctingfunctionwehave∆2 4 φ2 5 φsc2 .

In the above procedurewe seethat the first phasecorrectortransformsthe incidentphasesuchthatthebeamamplitudeat thesecondphasecorrectorwill bethedesiredoutputamplitude,A2, andthesecondcorrectorsynthesizesthedesiredoutputphase.An interest-ing consequenceof this observationis thatwe canproducea specifiedbeamintensity, butnotphase,usingonly onereflector. Thissituationis relatedto basicGaussianopticswherewe canusea singlesphericalmirror or lens to transforma Gaussianbeaminto eitherabeamwith aspecifiedwaistsizeor with aspecifiedminimumwaistposition(focal length),but notboth.

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5 Experimental Resultsfor a 110GHz Gyrotron

5.1 Overview

Our applicationinvolves retrofitting the last two reflectors(M3 andM4 in Figure 1) oftheexisting modeconverterin a 1 MW, 110GHz gyrotron. Theexisting modeconvertertransformsthe TE226 6 cavity modeinto a beamwith a flat power profile acrossa 10-cm-diameterdouble-disksapphirewindow. Thenew modeconverterusesa 5-cm-diameterdi-amondwindow to improvethepower-handlingcapabilitiesof thegyrotron,andnew modeconverterreflectorsarerequiredto producea Gaussianbeamon the window. The speci-fied beamminimumwaist is 1.52cm andshouldoccurat thewindow aperture,providinga flat phaseover thewindow. This waistsizecorrespondsto a theoreticaltransmissionof99.6%of thepower in themicrowave beamthroughthewindow aperture.Theserequire-mentsof circular beamshape,flat phase,andhigh transmissionefficiency at thewindowarestringent,andthey providea rigoroustestof theproposedreflectordesign.

5.2 FeedField Measurements

Thefirst two reflectorsin themodeconverterwerepreviouslydesignedastoroidal(doubly-curved) surfacesbasedon simulatedfields from the launcher, obviating the needin thisapplicationfor Steps(1) – (3) in Section2. Thepre-existinglauncherandtoroidalreflectorsthusform a feedwhoseradiatedfield wemeasureaccordingto Step(4).

Thefeedfield intensitymeasurementswereperformedat CommunicationsandPowerIndustries(CPI) usinga three-axismotorizedscannerbuilt by the University of Wiscon-sin [5]. Thereceiving hornwasanopen-endedwaveguidemountedon thescannerandfedto a heterodynereceiver. The feedstructurewasplacedexternalto thegyrotrontubeandthe scannerwasorientedto make planarscansasshown in Figure5. The launcherwasexcitedby a Gunndiodesourcewhoseoutputwastransformedinto theTE226 6 modeby acoaxialmodetransducer. The modepurity of this transduceris estimatedto be approxi-mately99 %; a detaileddiscussionof modepurity in a similar transducerdesignedfor theTE156 2 modeis givenin [21].

Weperformedmeasurementsovernineplanesnearthepositionof thethird reflector, asshown schematicallyin Figure5. Theplanesare14cm 7 14cmwith a64point 7 64pointsamplinggrid thatcorrespondsto asamplingdensityof 0 8 8λ. Figure3 showsthemeasuredfield intensityon the planeof the third reflector, andFigure6 shows the measuredfieldintensity11.9cm from thethird reflectorplane.Thebeamis propagatingparaxiallyalonga line 25o off they-axis,out of thepage,andhas 9�: 25 dB cross-polarization,indicatingthe scalarphaseretrieval is suitablefor this case.Furthermore,the beamshapechangessignificantlyover themeasurementrange— from Figure3 to Figure6 —, sodespiteourrelatively closeplanespacing(on theorderof tensof wavelengths)we canstill expectanaccuratefield reconstructionaccordingto thediscussionin Section3.

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z

yLauncher

M1

M2

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Beam

Figure5: Schematicof scangeometryfor the rigid launcher-toroidal reflectorfeedstruc-ture.

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Figure6: Measuredfield intensityon a planelocated11.9 cm in y from the Reflector3position.Contoursof constant;Ex ; 2 areat3 dB intervalsfrom peak;the < 3 dB and < 21dBcurvesarelabeled.

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5.3 PhaseRetrieval

The measurementsdetailedin Section5.2 wereinterpolatedonto 128 point = 128 pointgrids to provide a 0 > 4λ grid spacingin orderto avoid half-wavelengthundersamplingnu-mericalartifactsin thephaseretrieval. In addition,we modifiedthebeamshapeandposi-tion to temporarilycreateabeampropagatingnormalto themeasurement(or observation)planes.Specifically, theprofileof thebeamwascontractedby afactorcosθ in z (θ is thein-cidenceanglewith respectto theplanenormal)to accountfor thebeamdistensionoverthemeasurementplanes,andthegeometriccentersof thebeamswerealignedalongacommonpropagationaxis.Sucha transformationrelievesthephaseretrieval algorithmof retrievingboththenominalphasestructureandphasetilt of thebeam.Obliquebeampropagationisaccomplishedby addinga phasetilt aposteriorito thenominalphasestructuredeterminedby thephaseretrieval.

The phasewasretrieved in 500 iterationswith threemeasurementplaneslocatedap-proximately25λ apart.Thereconstructedfield waspropagatedto anindependent(i.e. notusedin the reconstruction)measurementplaneto verify the phaseretrieval, andwe ob-servedgoodagreementbetweenthemeasuredandsimulatedamplitudeson thatplane.

5.4 Reflector Shapingand Simulation

We shapeda pair of reflectorsusingtheproceduredescribedin Section4 with the recon-structedfield from Section5.3 asthe incidentwave, A1eiφ1, andthe desiredGaussianasoutput,A2eiφ2.

Oneimportantaspectof thereflectorshapingprocedurenot mentionedin Section4 isthefactthatthephysicalsizesof theactualreflectorsurfacesmustbetakeninto accountintheshaping.For instance,the third reflector(M3 in Figure1) is limited in z — alongthetubeaxis— to approximately8.5cm; this dimensionis largeenoughto interceptboththemain beamandthe ? 24 dB sidelobein Figure3, but it is significantlymorenarrow thantheobservationplane.To accountfor thesmallersizeof thephysicalsurface,wesimplysetall field amplitudesoutsideof thesurfaceperimeterto zero. Thebenefitof incorporatingthe final reflectordimensionsin the shapingprocedurebecomesapparentif we considerinsteadthe caseof usingthe wholeobservation planein the designandthenforming thesurfaceboundaryafterwards.Theresultis thatourassumedfield overtheentireobservationplanewill be truncatedby a rectangularwindow whosepulsewidth is thephysicalwidthof thereflector. Theradiatedfield revealsthiswindowing asthefamiliarconvolutionof thedesiredfield patternwith asincfunction,which leadsto unwantedsidelobes.

The final reflectorsurfaceshapesareshown in Figure7. For the third reflector, thebeamis incidentfrom below. Physically, weseethattheelongationin thex-directionof thepatternin Figure3 will befocussedinto themainportionof thebeamby thesharpcurvatureevidentfor x @ 0 in thereflector’sprofile. Therandomsurfacefluctuationsaroundtheedgeof thereflectorarisebecausethephasein thatnegligible-amplituderegionis indeterminate.The beamradiatedfrom the third reflectorpropagatesto the fourth reflector, alsoshownin Figure7. The surfaceis stronglysphericalto focusthe incidentwave into the desiredGaussianbeamat thewindow aperture.

Thereflectorsweresimulatedin aphysicalopticscode[3] usingthereconstructedfeed

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Figure7: Top: Third reflectorprofile. Bottom:Fourthreflectorprofile.

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Figure 8: Simulatedfield intensity on the window plane. The window centeris at z A37B 4 cm,x A 0 cm. Contoursof constantCEx C 2 areat 3 dB intervalsfrom peak;the D 3 dBand D 21 dB curvesarelabeled.

fieldsasthe initial field distribution. Thephysicalopticscodeis independentof thecodeusedin the phaseretrieval/reflectorshaping,andthuswill reveal any systemicerrors(ifthey exist) in thephaseretrieval code.Thesimulatedfield intensitypatternon thewindow,shown in Figure8, is in fact the desiredGaussianbeam. Integratingthe power over theobservationplanerevealsthat99.5%of thebeampower will passthroughtheapertureascomparedto 99.6%for anidealGaussianwith awaistsizeof 1.52cmin the5 cmaperture.Figure9 comparestheintensityprofileof thedesiredGaussianbeamto thesimulatedbeam.Thesimulatedbeamexhibits a nearly-idealGaussianprofile with a waist sizeof 1.55cm— 0.03cm larger thanthe1.52cm design.Thesidelobeto the right of beam,on the E zside,correspondsto unrecoveredsidelobepower originally generatedby thefeed,asseenin Figure3.

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Figure9: DesiredGaussian(solid) andSimulated(dashed)beamintensitiesalongthe z-axisat thewindow plane.Thewindow centeris at z F 37G 4 cm.

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5.5 Experimental Results— Cold Tests

Theshapedreflectorswerefabricatedfrom solidcopperandmountedon thefeedstructurefor cold testmeasurementsof theoutputfield. Usingthesamesource/receiverarrangementfrom the feedfield intensityscans(Section5.2), we measuredthe output field intensitybefore,on, andafter thewindow plane. Figure10 shows thefield patternon thewindowplane.Themeasuredbeamis a well-formedGaussianwith a size,shape,andpositionthatagreewell with thesimulatedbeamin Figure8. Thecross-polarizedcomponentis approx-imately 30 dB below the peakof the main polarization,confirmingour assumptionof ascalarfield. Themeasuredbeamis slightly elliptical with a waistsizein z of 1.6cm andawaistsizein x of 1.7cm. Thelargerbeamwaistof 1.7cm amountsto only a 0 H 66λ devia-tion in beamradiusfrom design.An idealGaussianbeamwith thesewaistparameterswilltransmit99%of thebeampower throughthe5 cm window aperture;dueto thepresenceof somelow sidelobepower, theintegratedvaluefor themeasureddatais 98%.Thisvalueis acceptablefor high power gyrotronoperation,andrepresentsa 1% error in thedesign.This closeagreementof measuredbeamwaistsizeandtransmittedpower to thespecifieddesignparametersindicatethatour reflectorshapingapproachworksverywell.

To further examinethe Gaussianquality of the beam,we comparean ideal Gaussianintensity profile to that of the measuredbeam. The measuredand theoreticalintensityprofiles along z are given in Figure 11. We note the measuredbeamhasan excellentGaussianprofile thatmatchesthe idealbeamover therangeof appreciableintensity. TheI 21dB sidelobeto theright of themainbeamappearsbecausethesidelobeincidentonthethird reflector(seeFigure3) is not fully reflectedinto themainbeam.This sidelobemaybe unrecoverablebecauseit is propagatingat a differentanglethanthemain beam,mostlikely theresultof spuriousmoderadiationfrom thelauncher.

The Gaussiannatureof the beamcanalsobeverifiedby consideringthe evolution ofthe wave profile with distance.Figure12 shows the beamintensitycontourson a planelocated60 cmfrom thewindow aperture.Thefield is Gaussianwith awaistsizeof 3.9cmin both x andz; the theoreticalwaist size(assuminga minimum waist at the window of1.6cm) is 3.6cm. This0.3cmdivergencein beamsizeover the60cm(220λ J propagationdistanceis practicallynegligible, andwe seethat themeasuredbeambehavesasa nearly-idealGaussianbeam.

We have shown explicitly, usingthemeasuredfield intensities,that theoutputbeamisa well-formedGaussianwith parameterscloseto thoseof thedesign.We canextendthedataanalysisby employing thephaseretrieval algorithmto roundout thestudy. With inputfield intensitiesonplaneslocated10cmbefore,at,and40cmbeyondthewindow positionin y, weretrievedthephaseoverthewindow aperture,andthisphaseis shown in Figure13alongthez-axis.SincethedesignGaussianbeamhasits minimumwaistat thewindow, weexpectthephasethereto beflat. This is indeedthecasefor thereconstructedoutputfieldphaseof Figure13. The mild slopein the phasearisesbecausethe beamis propagatingat anangleof 0 H 2o in the y I z plane,which leadsto a small beam-centeroffset in z (seeFigure12).

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Figure10: Measuredfield intensityon the window plane. The window centeris at z K37L 4 cm,x K 0 cm. Contoursof constantMEx M 2 areat 3 dB intervalsfrom peak;the N 3 dBand N 21 dB curvesarelabeled.

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Figure11: Gaussianbeamwith a z-waist of 1.6 cm (solid) andMeasuredbeam(dashed)intensityalongthez-axisof thewindow plane.Thewindow centeris at z O 37P 4 cm.

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Figure12: Measuredfield intensity60 cm from thewindow plane. Contoursof constantQExQ 2 areat3 dB intervalsfrom peak;the R 3 dB and R 21 dB curvesarelabeled.

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Figure13: Reconstructedphaseof themeasuredbeamover thewindow aperturealongthez-axis.

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5.6 Experimental Results— Hot Tests

The completemodeconverterwasassembledandplacedinside the gyrotron for hot, orhigh-power, testingwith thediamondwindow atCPI.Thetubeoperatedat650kW for 1.6spulselengthsandat 940 kW for 0.001s pulselengthswith an averagepower of 50 kW.Thediamondwindow performedwell throughoutthetesting,indicatingthatthemicrowavebeamis well-matchedto thewindow aperture.Preliminaryinfraredcamerameasurementswere also madethat confirm the Gaussianshapeof the output beam. Figure 14 showsnormalizedintensitycontoursof an IR camerameasurement12.7 cm from the window.Thedimensionsin Figure14 arebasedon estimatedbeamsizeat thatplaneposition.Theprecisesizeandpositionof thebeamareunknown becauseof difficultiesassociatedwithdetermininga referenceframein theclosedloadsystem.However, we caninfer from theGaussianshapeof the intensity contoursand the fact that the beampassesthroughthewindow with very low lossthat thehigh-power microwave beamhascharacteristicscloseto thoseof thecold testresultsdiscussedin Section5.5.

6 Conclusions

Wehavepresentedageneralmethodfor designingreflectorsin quasi-opticalsystemsbasedonphaseretrieval from intensity-onlymeasurements.Themethodreliesona fastphasere-trieval algorithmthat is alsousedto shapephase-correctingsurfaces.Following thebasicdevelopmentof necessarycomponents,we appliedthemethodto the specificcaseof de-signinggyrotroninternalmodeconverterreflectorsto producea specifiedGaussianbeamonthegyrotronvacuumwindow. Thefield intensityradiatedby thefeed(launcherandtwotoroidal reflectors)wasmeasuredandthe phasereconstructedfrom thosemeasurements.The reconstructedwave thenservedasinput to the reflectorshapingroutine,from whichwe derived the reflectorsurfaceshapes.The reflectorswere thenmountedin the modeconverterandwe measuredthe final outputbeam. Examinationof the outputbeampa-rametersshowed that the designmethodworksextremelywell, producinga beamwith awaist sizethat differedfrom designby only 0 S 66λ. We showed that approximately98%of the power in the beamwill exit the window apertureandthat beyond the aperturethebeamevolvesasa nearly-idealGaussianwith theprescribedflat phaseprofile at thewin-dow plane. High-power testingof the modeconverter indicatesthat the outputbeamiswell-matchedto thediamondwindow, whichshouldenablelong-pulseoperationof thegy-rotron.Theexperimentalvalidationof theproposeddesignapproachalongwith its generalformulationin termsof quasi-opticalphase-correctionprovide incentive for applyingthemethodto otherareasof microwave engineeringsuchasradioastronomyandhigh-powermicrowavestructuredesign.

7 Acknowledgements

TheauthorsgratefullyacknowledgeDr. Kevin FelchandDr. Steve Cauffmanat Commu-nicationsand Power Industriesfor their help in conductingthe experimentsreportedin

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-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6

x (c

m)

z (cm)

-3

-24

Figure14: Infraredcamerameasurementof the gyrotronmicrowave beam12.7cm fromthe diamondwindow. Normalizedintensity contoursare shown at 0.1 incrementsfrompeak.

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this paper, andProf. RonaldVernonandhis researchgroupat the University of Wiscon-sin for helpful discussionsand providing the RF receiver components.This work wassupportedby the Departmentof Energy, Office of FusionEnergy Sciences,contractDE-FC02-93ER54186.

References

[1] S.N. Vlasov, L. I. Zagryadskaya,andM. I. Petelin,“Transformationof a whisperinggallery mode,propagatingin a circular waveguide, into a beamof waves,” RadioEngineeringandElectronicPhysics, vol. 12,no.10,pp.14 – 17,1975.

[2] G. G. Denisov, A. N. Kuftin, V. I. Malygin, N. P. Venediktov, D. V. Vinogradov, andV. E. Zapevelov, “110 GHzgyrotronwith built-in high-efficiency converter,” Interna-tional Journal of Electronics, vol. 72,no.5 and6, pp.1079–1091,1992.

[3] M. Blank, K. Kreischer, andR. J. Temkin, “Theoreticalandexperimentalinvestiga-tion of a quasi-opticalmodeconverterfor a 110-GHzgyrotron,” IEEE TransactionsonPlasmaScience, vol. 24,no.3, pp.1058–1066,1996.

[4] H. A. Haus,Wavesand Fields in Optoelectronics. EnglewoodClif fs, NJ: Prentice-Hall, 1984.

[5] J. A. Lorbeck and R. J. Vernon, “A shaped-reflectorhigh-power converter for awhispering-gallerymodegyrotronoutput,” IEEETransactionsonAntennasandProp-agation, vol. 43,pp.1383– 1388,Dec.1995.

[6] K. Felch,M. Blank, P. Borchard,T. S. Chu,J. Feinstein,H. R. Jory, J. A. Lorbeck,C. M. Loring, Y. M. Mizuhara,J. M. Neilson, R. Schumacher, and R. J. Temkin,“Long-pulseand CW testsof a 110-GHzgyrotron with an internal, quasi-opticalmodeconverter,” IEEE Transactionson PlasmaScience, vol. 24,no.3, pp.558–569,1996.

[7] D. R. Denison,T. Kimura, M. A. Shapiro,andR. J. Temkin, “Phaseretrieval fromgyrotronnear-field intensitymeasurements,” in ConferenceDigest– TwentySecondInternationalConferenceon InfraredandMillimeter Waves, pp.81 – 82,1997.

[8] D. R.Denison,M. A. Shapiro,andR.J.Temkin,“Reflectorantennashapingfromfeedfield intensitymeasurements,” in ConferenceDigest– USNC/URSINational RadioScienceMeeting, p. 258,1998.

[9] B. Z. KatsenelenbaumandV. V. Semenov, “Synthesisof phasecorrectorsshapingaspecifiedfield,” RadioEngineeringand Electronic Physics, vol. 12, pp. 223 – 230,1967.

[10] R. W. Gerchberg andW. O. Saxton,“A practicalalgorithmfor the determinationofphasefrom imageanddiffractionplanepictures,” Optik, vol. 35,no.2, pp.237– 246,1972.

23

[11] A. P. AndersonandS. Sali, “New possibilitiesfor phaselessmicrowave diagnostics.Part1: Error reductiontechniques,” IEE Proceedings, vol. 132,Pt.H, pp.291– 298,Aug. 1985.

[12] L. S. Taylor, “The phaseretrieval problem,” IEEE Transactionson AntennasandPropagation, vol. AP-29,pp.386– 391,Mar. 1981.

[13] J. R. Fienup,“Phaseretrieval algorithms: A comparison,” AppliedOptics, vol. 21,no.15,pp.2758– 2769,1982.

[14] R. H. T. Bates,“Fourierphaseproblemsareuniquelysolvablein morethanonedi-mension.I: Underlyingtheory,” Optik, vol. 61,no.3, pp.247– 262,1982.

[15] A. V. Chirkov, G.G.Denisov, andN. L. Aleksandrov, “3D wavebeamfield reconstruc-tion from intensitymeasurementsin a few crosssections,” OpticsCommunications,vol. 115,pp.449– 452,1995.

[16] T. Isernia,G.Leone,andR.Pierri,“Radiationpatternevaluationfromnear-field inten-sitieson planes,” IEEE Transactionson AntennasandPropagation, vol. 44, pp. 701– 710,May 1996.

[17] W. H. Press,S.A. Teukolsky, W. T. Vetterling,andB. P. Flannery, NumericalRecipesin Fortran: TheArt of ScientificComputing. New York: CambridgeUniversityPress,seconded.,1992.

[18] A. A. Bogdashov, A. V. Chirkov, G. G. Denisov, D. V. Vinogradov, A. N. Kuftin,V. I. Malygin, andV. E. Zapevalov, “Mirror synthesisfor gyrotronquasi-opticalmodeconverters,” InternationalJournal of Infrared and Millimeter Waves, vol. 16, no. 4,pp.735– 744,1995.

[19] G. G. Denisov, A. V. Chirkov, D. V. Vinogradov, V. I. Malugin, A. A. Bogdashov,V. I. Belousov, N. L. Alexandrov, andV. E. Zapelov, “Phasecorrectorsynthesisandfield measurementsfor gyrotronquasi-opticalwave beams,” in ConferenceDigest–TwentiethInternationalConferenceonInfraredandMillimeter Waves, pp.483– 484,1995.

[20] Y. Hirata,Y. Mitsunaka,K. Hayashi,andY. Itoh, “Wave-beamshapingusingmultiplephase-correctionmirrors,” IEEETransactionsonMicrowaveTheoryandTechniques,vol. 45,pp.72– 77,Jan.1997.

[21] C. P. Moeller, “A coupledcavity whisperinggallerymodetransducer,” in ConferenceDigest – SeventeenthInternational Conferenceon Infrared and Millimeter Waves,pp.42 – 43,1992.

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