DesignoftheLARES-2Arrayby David Arnold

Smithsonian Astrophysical Observatory, Cambridge, MA (retired) Contents:1.Abstract2.Designgoal3.Polarizationasymmetry4.Centroidvsvelocityaberration5.Thermalsimulations6.Summary1.Abstract:The design proposed for LARES-2 uses 1.0 inch uncoated COTS (Commercial Off-The-Shelf) cube corners with no intentional dihedral angle offset. The cubes are held in a floating mount that virtually eliminates conductive heat transfer to the cubes. The design minimizes the effect of thermal gradients and manufacturing errors in the dihedral angle offsets. The isothermal transfer function should be very close to the actual performance in orbit. Testing of a set of 10 COTS cubes shows good optical performance. The cubes are inexpensive and are manufactured in bulk. Simulations show that the systematic range errors should be on the order of a half millimeter. 2. Design goal: The design goal for the GGOS program is one millimeter accuracy. The present geodetic satellites using 1.5 inch cube corners cannot provide this level of accuracy. Thermal effects are minimized by the use of uncoated cubes and a floating mount to prevent conductive heat transfer to the cube corners. However, thermal testing of the 1.5 inch LARES cubes shows significant residual thermal distortion of the diffraction pattern. The cubes provide sufficient cross section. But the thermal distortion causes unmodeled changes in the range correction.

Test of LARES- Prism No. 51 , (in cavity)

20 deg C 150 deg C 150 deg C

deep space sun simulator D.Spano, PHD , La Sapienza Univ. di Roma, 2012

3.Polarizationasymmetry.Inanuncoatedcubethereisaninteractionbetweendihedralangleoffsetsandphasechangesduetototalinternalreflection.Thisresultsinanasymmetricaldiffractionpatterniflinearpolarizationisused.Thepatternhascircularsymmetryforcircularpolarization.Theasymmetrycanbevirtuallyeliminatedifnodihedralangleoffsetisused.Theoffsetof1.25arcsecinthe1.5inchcubesisnecessarytoaccountforvelocityaberration.If1.0inchuncoatedcubesareusedthediffractionpatterniswideenoughtoaccountforvelocityaberrationwithouttheneedforadihedralangleoffset.Simulationshavebeendonecomparing1.5and1.0inchuncoatedcubesforLARES-2.Thefirstdesignuses2041.5inchcubesona200mmradiussatellite.Theseconddesignuses3031.0inchcubesona202mmradiussatellite.Therangecorrectionandcrosssectionmatricesareirregularatasingleincidenceangleonthesatelliteforbothcircularandlinearpolarization.Thematriceshavebeenaveragedover2520orientationsofthesatellite.WhenthisisdoneacircularpatternshowsupforcircularpolarizationandanasymmetricpatternforlinearpolarizationThemagnitudeoftherangecorrectionisdifferentforthetwodesignsbecausetheopticalpathlengthisdifferentinthedifferentsizecubesandtheradiusofthesatelliteisslightlydifferent.

Centroidmatrices1.5inchcubes

Circularpolarization Linearpolarization

1.0inchcubes

Circularpolarization Linearpolarization

Circularpolarizationproducesacircularpatternforbothsizecubes.Thepatternisnotcompletelycircularforthe1.5inchcubesevenwithaveragingovermanyincidenceangles.Thepatternfor1.0inchcubeshasgoodcircularsymmetry.Linearpolarizationproducesanasymmetricpatternforthe1.5inchcubeswithadihedralangleoffsetof1.25arcsec.Forthe1.0inchcubeswithnodihedralangleoffsetthepatternisnearlycircularwithlinearpolarization.Thereisaverysmallremainingasymmetrythatislessthantheaccuracygoalforthesatellite.

Crosssectionmatrices1.5inchcube

Circularpolarization Linearpolarization

1.0 inchcube

Circularpolarization LinearpolarizationThepatternwithlinearpolarizationisasymmetricforthe1.5inchcubesbutsymmetricforthe1.0inchcubes.Themaximumandminimumvaluesofthecentroidhavebeencomputedaroundcirclesofincreasingradiusinthefarfield.Theasymmetryhasbeencomputedasthemaximumminustheminimum.Thisdifferencehasbeenplottedvsthemagnitudeofthevelocityaberration.

ComparisonoftheasymmetryinthecentroidRed=1.5inch,Green=1.0inch

Withthe1.0inchcubestheasymmetryislessthan.5mm.4.Centroidvsvelocityaberration.Theproposeddesignminimizesthevariationofthecentroidwithvelocityaberrationbychoosingthesizeofthecubetoplacethevelocityaberrationonapeakinthepattern.Theresultsareplottedinthefiguresbelow.Thevelocityaberrationvariesbetweenabout32and40microradians.Linearverticalpolarizationisused.

Centroidvsvelocityaberration

1.5inchcubeRed=average,Green=minimum,Blue=maximum

1.0inch

Theaverage(redcurve)forthe1.5inchcubechangesby.74mmfrom32to40microradians.However,theasymmetryofthepatterncausessignificantrangeerrors.Thechangeoftheredcurveis.47mmforthe1.0inchcubewithverylittleasymmetry.Thisiswithintheaccuracygoalofonemillimeter.Acorrectioncouldbeappliedasafunctionofvelocityaberration.5.ThermalsimulationsTwothermalsimulationshavebeendoneinItalytocompareafloatingmount(Case11)withonewherethereisstrongconductivecontact(Case12)betweenthemountandthecubecorner.Theoutputofthethermalsimulationsisathree-dimensionalmatrixgivingthetemperaturedistributioninthecube.Ihaveusedthetemperaturematrixtocomputethediffractionpatternundervariousconditionsofpolarizationanddihedralangles.Thefirststepistodoaraytracingtogetthephasefrontduetothethermalgradients.Thenextstepistoaddphaseduetodihedralangleoffsets,totalinternalreflection,andpolarizationoftheincidentwave.Threesimulationshavebeenrunforeachcase.Thereferencecaseiswithadihedralangleoffsetof1.25arcsecwithnothermalgradient.Withnothermalgradientthecrosssectionvsvelocityaberrationisthesameforeither+1.25or-1.25arcsec.Thephasefrontduetoathermalgradientmaybeeitherprimarilyconcaveorprimarilyconvex.Thephasefrontforadihedralangleoffsetiseitherconcaveorconvexdependingonthesign.Adihedralangleoffsetcaneitherpartiallycanceltheeffectofathermalgradientoraddtoit.InthefigurebelowforCase11thecrosssectionisaveragedaroundcirclesofincreasingradiusinthefarfieldandplottedvsthemagnitudeofthevelocityaberration.

Case11Floatingmount

Red=isothermal+or-1.25dihedralGreen=thermalgradient,+1.25dihedralBlue=thermalgradient,-1.25dihedral

CurvemicroradcrosssectionBlue32.0000000.707174Red32.0000000.687217Green32.0000000.660673Theratioofthemaximumandminimumcrosssectionat32microadiansisBlue/Green=1.07.Thisisaverysmallchange.

Diffractionpatternsforthethreecurvesinthefigureabove.Thermalgradient,Dihedral-1.25Isothermal,Dihedral1.25

Thermalgradient,Dihedral+1.25

Thediffractionpatternsdonotshowlargedifferences.Thecentralintensityislargerfor-1.25arcsecandsmallerfor+1.25arcsec.

Case12Pressureonthecube

Red=isothermal+or-1.25dihedralGreen=thermalgradient,+1.25dihedralBlue=thermalgradient,-1.25dihedral

CurvemicroradcrosssectionBlue32.0000000.768003Red32.0000000.687217Green32.0000000.505958 Theratiobetweenthemaximumandminimumcrosssectionat32microradiansisBlue/Green=1.52.Thefractionalchangeis7.5timesgreaterthanwiththefloatingmount(Case11).

DiffractionpatternsforthethreecasesThermalgradient,dihedral-1.25Isothermal,dihedral+or-1.25

Thermalgradient,dihedral+1.25

Thepatternsshowsignificanteffectsduetothethermalgradientparticularlyinthecentralintensity.Thecentralpeakisneverobserved.However,thechangesinthevelocityaberrationrangefrom32to40microradiansaresignificant.6.Summary.Theuseofsmallcubeseliminatestheneedfordihedralangleoffsets.ThisallowstheuseofinexpensiveCOTScubes.Thesmallcubesproduceamuchmoreaccurateisothermalrangecorrection.Theisothermalrangecorrectionwillbeveryclosetotheactualrangecorrectioninorbitsincethesmallcubessignificantlyreducetheeffectofthermalgradients.Specifically,thebenefitsare:1.Thereismoreuniformcoverageofthesurfacewithsmallervariationswithincidenceangle.2.The1.5inchcubesaretoolargeforthevelocityaberrationandrequireddihedralangleoffsets.Thisproducesa"lumpy"diffractionpatternthatcausesvariationsinrangewithinthefarfielddiffractionpattern.

3.Thereisaninteractionbetweendihedralangleoffsetsandthephasechangesduetototalinternalreflectionthatproducesanasymmetricaldiffractionpatternwhenlinearpolarizationisused.4.The1.0inchcubesprovidethenecessarybeamspreadtoaccountforvelocityaberrationwithouttheneedfordihedralangleoffsets.Thisalsoremovestheasymmetryinthediffractionpatternwithlinearpolarization.5.Thediffractionpatternwithoutdihedralangleoffsetsissmootherthanthepatternswithoffsets.6.Thediffractionpatternofanuncoatedcubehasaringofspotsaroundthecentralpeak.Thesizeofthecubecanbechosentoputthevelocityaberrationonthisringofspotsratherthanonaslopeinthediffractionpattern.Thisreducesthevariationoftherangecorrectionwithvelocityaberration.Thisringofspotsisaverystablepartofthediffractionpatternthatdoesnotchangemuchduetovariousperturbations.7.Thermaleffectsincreaseassomepower(approximatelythe4thpower)ofthesize.Thereductioninsizefrom1.5to1.0inchesappearstoreducevariationsinthecrosssectionbyaboutafactorof5or6.8.EliminatingthedihedralangleoffsetmakesitpossibletouseCOTS(CommercialOff-The-Shelf)cubesthatareinexpensiveandavailablequickly.TestingbyLudwigGrunwaldtshowsthattheopticalqualityofthesecubesisasgoodascustommadecubeswithdihedralangleoffsetsthatareexpensiveandtimeconsumingtomanufacture.9.TherearesmallunintentionaldihedralangleoffsetsinCOTScubesthataregenerallylessthanonearsecbutcanbeuptotwoarcsec.Theeffectofapositive(>90deg)offsetisintheoppositedirectionfromtheeffectofanegative(<90deg)offset.Sincethemeanoffsetiszerothepositiveoffsetstendtopartiallycanceltheeffectofthenegativeoffsets.10.Thermalsimulationsshowthattheeffectofthermalgradientsina1.0inchcubeisverysmallwithafloatingmount.11.Afloatingmountrequiresleavingasmallgapbetweentheringandthecube.Thiscouldpotentiallyresultindamagetothecubeduetovibrationsduringlaunch.Vibrationtestingwithaverylargegapshowednodamagetothecube.