OPTIMIZED LASER TURRETS FOR MINIMUM PHASE
OPTIMIZED LASER TURRETSFOR MINIMUMPHASEDISTORTIONDr. G. N. VanderplaatsNASA Ames Research CenterMoffett Field, CA, 94035Dr. Allen E. FuhsNaval Postgraduate SchoolMonterey, CA, 93940andMr. Gregory A. BlaisdellStudent, Applied MathematicsCaliforniaInstituteof TechnologyPasadena, CA, 91125ABSTRACTPhase distortionlaserturretsdue to compressible,in subsonic or supersonicdeterminedby a two-dimensionalpropertiesare given by a FourierPhase distortionelevationsum became the objectiveshape of the turretseries;in a similarThe turretshape wasmanner, the flowseries.for propagationat severalcombinationsA sum was formed from the set of values,functionwas variedflow over small perturbationflow was calculated.Fourierwas calculatedand azimuth angles.inviscidfor an optimizationto providecomputer program.ofand thisTheminimum phase distortion.INTRODUCTIONFor many applicationsof a high energy lasermust be propagated with minimum phase distortion.on board an aircraft,the beamThe well known StrehleThe work reported in this paper was supported by the Air ForceACKNOWLEDGMENT.Weapons Laboratory, Kirtland Air Force Base, NM, and was monitored by LtCol KeithGilbert,Dr. Barry Hogge, and Captain Richard Cook.339
relation[l]gives the decrease in far fieldphase distortion.density,As a resultcompressibledensitybe the resultinviscidturretsindex of refractionof viscousas a consequence of the rmsof the dependence of index of refractionflow over laserand variableintensityon masscauses phase distortion.surroundingflow phenomena or inviscidan aircraftflow.The variablelaserturretmayThis paper focuses on theflow problem.In regard to the solutioncompressibleflow,is an importantthe turretseveralof the phase distortionoptionsconsideration.exist.Adaptivecan reduce significantlyshape.may be used.throughturretapproach would be to considerHigher order distortions,to remove by adaptivedesign,opticsthe turretopticsof the turretphase distortion.paper is to minimize phase distortionAn alternateLocationproblem due to inviscide.g.,shape ofThe approach taken in thisshape.opticsand turretand coma, are more difficultthan lower order distortions.shape should be modifiedCorrectcombined adaptiveastigmatismon the aircraftUsing thisso as to minimizehigherapproach toorderdistortions.a comment should be made about adaptiveIn passing,for compensation of atmosphericmirrordisplacementsIn contrast,requiresof a fractionthe adaptivemirroropticsdisplacementsAn analyticalturbulenceof a few wavelengthsflow fieldFuhs [3,4].A companion paper in thisanalyticalmodel.laserhas been describedThe turretat frequenciesfor compensation of lasercompressibleturretopticsof 25 kHz or so.turretphase distortionat frequenciesof a few Hertz.geometry and the associatedin the papers by Fuhs [2] and Fuhs andconferenceshape is described340Adaptiveand thermal blooming requiresof wavelengthmodel for describingoptics.proceedings[5] discussesby a two-dimensionalFouriertheseries.
Using the flow over a wavy walla Fourierdirectseriesnumericalexpliciton a circularcylindercan be found for the potentialintegrationanalyticalof the equationssolutionis obtained.as the basic solution,function.In contrastto aof motion of gas dynamics, anAs a result,computer time is significantlyless.At a plane normal to the beam sufficientlydistortionseriesis calculated.with ZemikethatvariousOne method to representpolynomialsthe coefficientsi.e.,A computer program [7,8]yieldingthe design variableswere treatedmirror.insidelaserturrettilt,focus,to a varietyfunctionas e values of designsubjectto constraints.For the case at hand,the turretof the laserConstraintsas maximum turretseriesto the magnitude of thegeometry.turretincludedAlsoprimary34 and EM in Figure 1 define locationwas the weighted sum of phase distortionand azimuth angles.is to use adirectlyof problems [9,10].and the locationas design variables;sets of elevationare relatedare the coefficientsThe objectivethe phase distortionhas been developed which can findthe turretthe phaseThe advantage of using a Zernikea minimum value of an objectiveThe program has been appliedgas density[6].in the seriestypes of distortion,variablesfar from the aircraft,mirrorof thefor severalthe maximum slope of theheight.PHASEDISTORTIONOpticalpath length,Li,is definedasbLwhere n is the index of refractiona and b are on the ray.i san(s)dsand s is distanceThe subscripti identifies341(1)along a particularthe ray.ray.The differencePointsin
opticalpath lengthfor two adjacentrays i and j can be calculated;Figure1.The phase distortion,the differencedividedseein opticalP, ispath lengthby the wavelength.Li - L.AP The index of refractionmass density,(2)is relatedtop, byp,n l K'--P(3)%L pOJ-TYPICAL sity,K’is a weak functionin the infrared,density.p, is the freestreamand psL is the sea levelThe form of equationused to highlightaltitude.Figure 1.Geometry for CalculationPhase Distortion.ofof flightis assumed to be isentropic;p, and density,introducedp, can be used.the pressurepm/p,,is a functionaltitude.thus the usual isentropicFurther,(3) wasthe dependence of n onThe ratioThe inviscidturretof wavelengthflow over the laserrelationcoefficient,between pressure,C , can bePwith the result(4)The pressurecoefficientLiepmann and Roshko [ll]for small perturbationas342axisymmetricflow is given by
C - 2u - v2Pwhere u is the perturbationradialvelocityvelocitysee equationin the freestreamwhich is normal to the fuselagecombined to give an integral(5)axis.for the phase distortion(7) of Reference[5].directionand v is theEquations(1) to (5) can bein terms of pressureWhen the potentialfunctioncoefficient;for the flow is known,both u and v can be calculated.METHODSTO COMPENSATEFOR PHASEDISTORTIONThe optionscompressibleopticsavailableflow over the laseris one technique.adaptivefor compensating for phase distortionturretWoltersdue to inviscidAdaptivewere mentioned in the Introduction.and Laffay[12] demonstratethe effectivenessofoptics.Another method to compensate for phase distortionof proper shape.a passiveThis is the approach of thistechnique.is discussedTurretThe polynomials,orthonormalTurretpaper.geometry constitutessections.phase distortionis to use Zernikepolynomialswhich are given in the paper by Hogge and Buttsset of functions.turretgeometry as a means to lessen phase distortionin the followingA method to representis to use a laserThe phase distortion[6],[1,61.are anisnP cj lAFj j(6)The summationand Fj is the jth Zernike polynomial.where A is a coefficientjTypicallyextends from 1 to n, where Fn is the highest order polynomial considered.n 10 is adequate.equationThe coefficient(6) by F and integrating4Aq is obtainedover the aperture343by multiplyingboth sides ofor beam cross section.All
terms in the summation vanish except for the term j q.Reference [5.] gives the resultsAn alternateopticsand turretand a formulais to combine adaptivegeometry.are more difficultHigher order phase distortionsoptics.Form an objectiveB where Wi is a weightingdistortion.The largerfactori,for the iththe value for B changes.the effectivenessdistortionsPhase is controlledcoefficientin ZernikeseriesAs the turretfor phaseshape isUsing COPES/CONMINcomputer program [7],through variationsof adaptiveof adaptiveof turretopticsopticsby mirrorthrough use of adaptivegeometry.theThe consequenceis enhanced since higheremploys segmented or deformabledisplacement.optics.Compensation for atmosphericof the adaptiveorderover the turretmap.locationstypes of adaptivein the mirrorof the mirroropticslow frequencyby turretregionplane.high frequencyCompensation for the adverse influenceresponse is dictatedcan beamplitude/frequencyoccurs in the low amplitude,occurs in the high amplitude,The frequencythe frequencymirrors.of the adverse phenomenon being overcometurbulenceopticsFurther,Hence, differentthought of as occupying differentmap.(7)is the value of Wi.motion is determined by the frequencyopticswhich isare minimized.The techniqueregionfunctionWAiicthe largervalue of B can be minimizedis that(19) offor A.Jmethod to compensate for phase distortionto compensate by adaptivevaried,Equationof flowof the adaptiveslew rates or aircraftmaneuver rates.ANALYTICALMODELThe linearizedpotentialequationfor axisymmetricflow is(8)344
The ( ) sign is for subsonic flow,The quantityand yieldsMach number.the perturbationLPgThe velocitiesappearingThe boundary conditionsare discussedsign is for supersonicflow.6 iswhere M, is the freestreampotentialand the (-)r is the perturbationvelocitiesv gla!w yae;(10)in equation(5) can be obtainedfrom equation(10).for equation(8) and a wavy wallon a circularcylinderin Referencefor the wavy wall.;The potential[4].A solutionThe solutionfor the flow is constructed.is obtainedis the basic functionThe turretis representedfor one spatialfrequencyfrom which a Fourierseriesby two polynomialsKf(x) 1 c ZkXkk land(12)To obtainequationa symmetric turret(12).only even values of j are used inin the e-direction,In terms of f(x)and f(O),the turretgeometry isNx, 0) R. Ef(x)f(e)where R is the radialEquationdistance(13) is representedin the Fourierseriesto the surfaceby a Fourierfor the potentialFigure 2 shows the geometry.of the turretis 2R.Figure 3 is an artist's(13)of the turretseriesor fuselage.which leads to the coefficientsflow.The maximum turretThe meaning of emax is thatconcept of the laser345turret.f(e)heightis E, and the lengthis zero for(81 emax.Two comments are applicableto
MirrorFigure 2.Figure 3.Geometry of a Small PerturbationFirst,Figure 3.Artist'sthe turretwillcanopies can be manufactured!of wings, blade antennas,Laser Turreton a CircularConcept of Laser Turret.become operationalat some futureSecond, the model does not includeand similarFuselage.items.346date when laserperturbationeffects
Figure 4 illustratesthe beam relativethe coordinateto the aircraft.system used to describeA Cartesiancoordinateorientedas shown.system.The beam is at azimuth angle 4 and elevationThe Z-axisforms the polarthe directionofsystem k, Y, Z isaxis for a sphericalcoordinatey.LASER BEAM\AXIS OF 1BEAM I\-&,/-\‘\,’\LASER TURRET.//’\-(/-1)-‘.--c-- Iq-AXESFigure 4.CoordinateThe design variablesSystem for Directionbecome G and %j'reduce the number of independentof Laser Beam Propagation.Conditionsdesign variables.R( emax,x) R(e,J?) R.at [8[ Omax and 1x1 RTypicalconditionsare(14)and- aR 0ax x !2-;a.Rae 9 emax347 O(15)
The interceptbetween a ray and the turretproblem in analyticalAn iterationgeometry.surfaceis a particularlyscheme was used to finddifficultthe interceptas follows:.a.At a given pointb.The iterationIn equation(l),startsat a and ends at b.mirrorsurface.turret[13].The startThe end of integration,so thatadditionala portionHence the densitythe airthe turretis a factorto the turretcould be generated by the air withinb, is at athe turret.window is assumed to be of zero thicknessthein the phasewere uniform,In thisby anSince integrationof the ray between a and b is withinwithinexternalofintegrationchange in the value of the integral.surface,Even ifR RT.by Vanderplaatsthe integrationa negligibleat the mirrorturret.component of thein detailfar from the laseramount 6s yieldsdistortion.is describeda, is at the primarysufficientlylaserthe radial%*to find a value of s' such rtscalculateFor these values of x and 8, calculateturretpoints s' on the ray,a phase distortionstudy,the turretso as to be distortionless.OPTIMIZATION OF LASER TURRETSHAPEFor any given azimuth, , and elevationray in the beam can be calculatedAt a specifiedbeam orientation,using equation(7) of Referenceangularlocations,optimum overallsquaringi.e.,using the center[5] at two radiala measure of totaltypicallylocationsQ of Figure 1 occurs every 45'.severalof phase distortionperformance,348Furthermore,and summing over allaswillbe calculatedfor each of eightbeam orientationsS, is obtainedof anyray of the beam as a reference.the phase distortionsystem performance,the valuesy, the phase distortionangle,to provideare considered.Byrays and orientations,
(16)@The variablez is definedzY11'1; z givesthe radialin Figurelocationwithinthe beam.is a weighting function.Values of W are determined from mission studies.wOYcpYFor a particularmission, the laser beam may be pointed most of the time at aparticularis larger.obtaineddirection,particularFor more extensiveThe objectiveproper combinationmentioned entcan be definedfunctiongivingThe design lesfromlocationwithinangles.Figureshave beenare as follows:cM%fturretpresentedin this%paper and Reference [5] has beenfor given azimuth and5 to 7 are phase distortionmaps.the windward side of the beam is at the top of the map.from the equationthe%Tijcoded in FORTRANto produce maps of phase distortionelevationS by determiningTo summarize, the design variablescapabilitycan befor Wthe function is Y;the probabilitythe beamwas to minimizeof design variables.densityW@Yby @ and y.of the optimizationmirrorThe analysisFor that direction,functionprobabilityin the directionvalues of 9 and y.missionso that a meaningfula two-dimensionalpointsi.e.,For allThis factthree mapscan be determinedfor phase distortionL.La,co(Sj -si Pt P,(17)fsSi‘j349
Figure5.Phase DistortionFigureMap.6.Azimuth, (I 0'; Elevation,M, 0.5.Y 45'.@cm 1.0. Cosine Turret.Phase DistortionAzimuth, @ 0';Map.Elevation,Y 45O. M, 2.0.Pt'P,, 1.0. Cosine Turret.'0.60Figure7.Phase DistortionMC0 2.0.Map.Azimuth,pt/pw 0.309.350 I O"; Eleva tion,OptimizedTurret.y 90'.
In equationinterceptA.s. is the distance from the surface of the primary mirror to theJThe laser radiation has wavelength,of ray j with the laser turret surface.(17),C along ray j exceeds that along ray i, a positivePRefer to Figure 2 or Figure 4.to the value of phase distortionoccurs.When the pressurecontributionFor an elevationfuselage,spositive3coefficientangle of y 45' and the laser si.Consequently,on the windward side.wavefrontthe term (s. - si)pt in equation (17) is alsoJA positive phase distortion,P, means a lag offor ray j compared to wavefrontThe lasercompatibleturretwithbeam in the plane of symmetry of theof ray i.geometry and the associatedthe generalpurpose optimizationflow were coded in subroutineformprogram.COPES/CONMIN[7]The COPES/CONMINprogram solves the design problem of the followingMinimizesubjectform:& (18)to the constraintsGj(z)LO,j l.(19)and is definedThe vectorthe design variablessummarized above.that were consideredat one time oranotherduringdirection,equation(15).objectivethe study were the maximum slope of the turretno discontinuitythe linearizedcontinuousflow equationsand constraintfirstfunctionsat turretfuselageThe slope was restricted,that both functionslinearThe constraintsare the constraints.by equation-where F(z) is calledGj(z)the objective function of design variables,x, contains.mderivatives.for largeris somewhat arbitrary;must be continuousin the streamwiseand the conditionsofat the most, to a value of 0.3 sincebecome inaccuratefunctionsboundary,(16).functionsIn general,of G.351values.The choice ofthe only restrictionof the design variables,F(z) and Gj(z)may be any linear;,iswithor non-
TWOEXAMPLESOF LASER TURRETDESIGNTwo design examples are presentedand the second being for supersonicTable I.withinCalculationshere,flow.the firstbeing for subsonic flowThe design conditionswere conducted for six beam orientationsare listedand sixteenthe beam.Table I.Design ConditionsAERO-OPTICSM, M03 y h Case 1Case 2Mach numberRatio of heat capacitiesWavelength of laser radiationDensity ratioConstant for index of refraction0.52.01.43.8 micronsPJPSL 0.3K' 0.00023GEOMETRY ROL &M S OR Fuselage radiusSpacing of turretsMirror locationTurret lengthTurret heightMirror radiusMaximum angle extent1.05.01.1252.0E 0.2Rm 0.050 60 maxof turretGEOMETRICBOUNDARYCONDITIONSf(x)- 1.001.001.00df(x)/dx"'max 1.000varies0352f,(e)01.0df(S)/devaries0inrays
-.--e---.-. .- .-. . 7.------. --p-----.Table I Continued.-Design Conditions--- ---BEAMORIENTATIONS.- - Beam NumberAzimuth,9, degreesElevation,000459090---y, degrees4590120453060PHASEDISTORTION CALCULATIONPOINTS.- -L.- 3 - ---Rays defined by all combinations of:radius within beamz/R 0.025, 0.0500angle within beamTl 0, 45, 90, . . . 315Note: All rays are shown in Figure 1. F-.-.-F-T jY.--e--. - - -Ye-- -CONSTRAINTIN SLOPE., .-- .--.- . r-3-- - .-. --- - .for0.3- q (0.30e o- .- .- . .As a reference,for turretflow over a cosine-shapedturretwas calculated.The equationsgeometry weref(x) 1.0 - 0.50(f)2 0.0625(fj4f(e) 1.0 - 1.824(;)2 0.832( max(20)and353)4ILBX(21)
lllllllllllIllIlll II II I I I IllIPhase distortionequationsmaps are shown in Figures(20) and (21).The perturbationin the plane of symmetry of the fuselage5 and 6 for the turretvelocities,The radialperturbationat the turretperturbationvelocity,surface.Hence v is identicalvelocity,u, is different8 and 9.9 is for the supersonicv, is dictatedbyu and v, were calculatedand are shown in Figuresis for the subsonic flow example, and FigurespecifiedFigure 8flow example.by the boundary conditionin both Figures8 and 9.The axialfor subsonic flow as compared to supersonicu.viFigure 8.flow.PerturbationVelocitiesfora Cosine Turret in SubsonicFlow.In subsonic flow,supersonicflow,FigurePerturbationVelocity fora Cosine Turret in Supersonic Flow.the maximum value of u occurs at x 0, whileat x 0, the value of u is zero.is compressed (u 0) on the forwardleeward side of the turret,9.In supersonicor windward side of the turret;the flow is expanded (u 0).354flow,inthe flowon the
Resultsfor the two examples are summarized in Table II.of the coefficientsturrets.2 and 8 are listedfor both the initialUsing these values of 2 and %, the laserappear in Figures10 to 12.Table II.turretsFigure 10 is the cosine-shapedSummary of Laser TurretIn Table IIand optimizedthe valueslaserhave been drawn andturretused as reference.Design timized TurretsCase 1 (Subsonic)Case 2 0282**Design Variable-A---VALUE OF OBJECTIVE FUNCTION, S, AND DENSITY RATIOQuantityInitialSubsonic SupersonicSp,k36.020.7Optimized TurretsTurret2.69Case 1 (Subsonic)Case 2 (Supersonic)31.220.71.550.3094355
Figure12.Optimized Laser TurretFigure 11 is the optimizedlaserturretlaserfor supersonicflow.turretfor Supersonicfor subsonic flow.Comparing FiguresM, 2.0.Figure 12 is the optimized10 and 12, very littlebetween the cosine-shapedand the optimizedseen.to Table II shows the optimizedHowever, referenceturretFlow.for supersonicturretdifferenceflow can behas odd powersfor f(x);note that a1 0.2651, a3 - 0.1326, and a5 0.0166.For the calculationssummarized in Table II, all the weightingwere unity.function,laserUsing the COPES/CONMINoptimizationS, definedturretfunction,(16),was reduced from 36.02 to 31.22 for thefor subsonic flow.S, for the laser2.69 to 1.55.turret,optimizedby equationThe reductionturretvalues WQvcomputer program, the objectiveThe reductiondesigned for supersonicis 42 per cent.is 13 per cent.flow was reduced fromAlso note that a densityp,, less than ambient helps to reduce the phase distortion;357The objectivewithinthesee Table II.
EXAMPLEOF LASER TURRETGIVING LEAST AND WORSTPHASEDISTORTIONTo illustratethe range of values of the objectiveobtainedby varyingdesigned.Detailsthe best turret;the initial'the turretyieldedof 890 per cent.The worst turretThe best turretchange of 690 per cent.The cross sectionsare shown in FigureS 0.0115.S, thatboth the best and worst laserare given in Table III.and the worst turretturretTable IIIgeometry,functions,13.can beturretwereof the initialturret,For the case at hand,gave S 0.0918 which is ahas S 0.0012 which is an improvement in SThe range from the worst to the best is 0.0918/0.0012 78.5.A. for the Zernike polynomials.The phaseJdistortioncan be represented by equation (6) using A. from Table III.TheJreader should compare Aj for the initialturret with the other two turrets.Thebest turrethas the coefficientshas a slightlyFor the laserturretlargervalue for A4.givingis slightlyCompared to the two examples of the previouswere optimizedazimuth,allthe worst distortion,except for A . The focus coefficient4has littlesignificance.sectionThe value of A9 is reduced greatly.coefficientssmaller.section,The average value Althe turretsThe beam directionfor only one beam direction.are increasedin thiswas at an0 I 45' and an elevation,y 45 .COMPUTERCODEFOR LASER TURRETOPTIMIZATIONAn extensiveFuhs [X4].calculatesfieldcomputer code has been writtenThe computer code is based on Referencesthe opticalsurroundingthe optimum turretand controland versatilepath lengtha laserturretshape yieldingcodes are thoroughly[2,3,5,7,8,13].and phase distortionin compressibleflow.arisingFurther,358in ReferenceandThe programfrom the densitythe program findsThe optimizationminimum phase distortion.discussedby Vanderplaats[14].Sample data input and
Table III.Summary of rage value52InitialTurretfor Phase DistortionLaser Turret a-5.9513-034.2853-03-0.02840Mach number .500Flightaltitude sea levelTurretheight/fuselageBeam radius/fuselageElevationradiusradius .200 .05angle 45OAzimuth angle 45Osample outputReference[14]are given.constitutesThe materiala user'sis presentedin sufficientmanual for LASTOP.359detailso that
lIIlIlllllllllllllllll I(a)INominal yieldingleastFigure13.has valueObjectivedistortion.distortion.Cross SectionfunctionObjectiveShape of Turrets360of 0.0115.functionfunctionhas valuehas valueof 0.0918.of 0.0012.in the Plane of Symmetry.
COMMENTSAND CONCLUSIONSA versatilelaserturreton a fuselageanalysisand computer program has been developed which optimizesgeometry to obtainof circularexceed the perturbationminimum phase distortion.cross section.Turretallowed by the linearizedcomputer code is describedin ReferenceThe turretsslope is limitedequations(16),thatso as not tofor the flow.The[14].Examples have been given which show the decrease in objectivein equationare locatedfunction,Scan be achieved.REFERENCES1.M. Born and E. Wolf,2.A. E. Fuhs, "Distortionof Laser Turret Optics Due to AircraftMainstreamFlow," Journal -of Optical Society -of America, 66, p. 1137, 1976.3.A. E. Fuhs and S. E. Fuhs, "Phase DistortionDue to Airflow over aHemispherical Laser Turret,"Naval Postgraduate School Report NPS-69FU76101,September, 1976.4.A. E. Fuhs and S. E. Fuhs, "Phase Distortionat High Subsonic Mach Numbersfor a Small PerturbationLaser Turret,"Proceedings of Electra-OpticalSystemsDesign Conference--1976,New York, pp. 9-19, publishzby IndustrialandScientificConference Management, Inc., 222 W. Adams St., Chicago, IL, 60606.5.AllenglowPrinciplesof Optics,Pergamon Press, New York, 1964.E. Fuhs and Susan E. Fuhs, "Optical Phase Distortionover Laser Turrets,"Paper This Conference.Due to Compressible6.C. B. Hogge and R. R. Butts, "Frequency Spectra for the Geometric Representationof Wavefront DistortionsDue to Atmospheric Turbulence,"IEEE Transactions -onAntennas and Propagation,Vol. AP-24, pp. 144-154, 1976.7.Garret N. Vanderplaats,"CONMIN--A FORTRANProgram for ConstrainedMinimization,"NASA TM X-62282, August, 1973.8.Garret N. Vanderplaats,"The Computer for Design and Optimization,"-in Applied Mechanics, AMD Vol. 18, ASME, Dec., 1976.9.Garret Vanderplaats and Allen E. Fuhs, "Aerodynamic Design of a ConventionalWindmill Using Numerical Optimization,"Journal of Energy, 1, pp. 132-134, 1977.10.FunctionComputingS. E. Fuhs, G. N. Vanderplaats,and A. E. Fuhs, "Land Contouring to OptimizeWind Power," AIAA 16th Aerospace Sciences Meeting, 1978, Paper 78-279.361
IIIIIllllllllllIllI11.H. W. Leipmann and A. E. Puckett, Introductionto Aerodynamics --of aCompressible Fluid, John Wiley and Sons, New Yoz, 1947. See Chapter 10.12.D. J. Wolters and P. J. Laffay, "Mainstream Flow Effects on F-15 TurretMcDonnell AircraftCompany Report MDCA3179, January 3, 1975.13.G. N. Vanderplaats,"InviscidFlow over Turrets; Optimum Turret Shape,"Lecture 5A, Laser Aerodynamics, a Short Course at the Air Force WeaponsLaboratory,Kirtland Air Force Base, NM, April 11-22, 1977.14.G. N. Vanderplaats and A. E. Fuhs, "LASTOP - A Computer Code for LaserTurret Optimizationof Small PerturbationTurrets in Subsonic or SupersonicFlow," Naval Postgraduate School, Technical Report NPS 69-77-004, December, 1977.362Optics,"
Figure 2 shows the geometry. The maximum turret height is E, and the length of the turret is 2R. The meaning of emax is that f(e) is zero for (81 emax. Figure 3 is an artist's concept of the laser turret. Two comments are applicable to 345Cited by: 1Publish Year: 1980Author: G. N. Vanderplaats, A. E. Fuhs, G. A. Blaisdell
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