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Measurements with CapacitanceGage for Sub-Nanometer ThermalExpansion CharacterizationMTI Instruments contracted an independent Metrology expert to test and verify theperformance of a new high resolution capacitance gauge. The Accumeasure HD haspassed in house testing where its performance was tested against a commercial laserinterferometer (See Application Note). Optical Metrology Solutions was chosen basedon their extensive experience and proximity to MTI’s manufacturing plant.Optical Metrology Solutions(OMS) Task:OMS was tasked with designing a challengingexperiment to verify the accuracy and resolutionof the MTI Accumeasure HD capacitance system that could also be encountered in a typicalcustomer application. The experimental setupand results are as follows:Measurements with CapacitanceGage for Sub-Nanometer ThermalExpansion CharacterizationKevin HardingOptical Metrology SolutionsEXPERIMENTAL ABSTRACTThe ability to measure small thermal expansionsof precision components can be critical to theperformance of precision instruments such asoptical telescopes. This white paper exploresthe ability of MTI’s HD capacitance gage tomeasure expansion of optical elements in thesub-nanometer range.The image on the left demonstrates what can happen toastronomical images without temperature compensationDISCUSSIONSetupTo explore the ability to measure very smallthermal expansions of a ceramic material, weused a 6-millimeter thick, optical mirror with anunprotected gold coating. Many low expansionmaterials, such as Zerodur, are well characterizedmaterials used for critical optical mirrorapplications because of their low coefficientof thermal expansion of 0.1 x 10 -6. With a 6mmthick mirror, a change of 0.1 degrees C wouldcreate a change in mirror thickness of only 60pico-meters. For a glass mirror the expansion isonly about 6 to 10 nanometers for 0.1 degree Cchange. Stability at these levels is very difficultto achieve, but in some critical applications, suchchanges may cause an optical system to blur orMTI Instruments Inc. 1-800-342-2203 sales@mtiinstruments.com1

Measurements with Capacitance Gage for Sub-NanometerThermal Expansion Characterizationlose resolution. Capacitance probes have thepotential to measure such low changes over asmall measurement range. In order to explorethe ability to measure such a small change, wedeveloped a test fixture using a low expansionceramic material to hold the mirror and probewithin less than 50 microns of each other so asto be in range of the capacitance probe. The MTIcapacitance probe chosen was an ASP-50M-CTAwhich has a 50 um measuring range. The matingcapacitance amplifier is the MTI Accumeasure HDwhich is designed to have very high resolution.The probe and amplifier were linearized on an airbearing stage using a glass encoder scale with100nm accuracy.The setup is shown in FIGURE 1: The fixturestand consisted of a holder for the mirror whichsupported the mirror from the bottom leavingthe mirror free to expand upward. A similarholder supported the probe from the bottom,removing any expansion of the probe case fromthe measurement. The two support plates whereconnected with three rods made of the samevery low expansion ceramic material. The mirrorwas instrumented with three thermocouplesaround the exposed side of the mirror using heatconducting adhesive, as well as one thermistorused for feedback to a thermal electric deviceattached to the back of the mirror for heating.A ground wire needed for the operation ofthe capacitance probe was attached to thegold surface of the mirror using an electricalconducting adhesive.By providing local heating to the back of themirror and keeping the system isolated from room“breezes”, the mirror should slowly expand withlittle expansion of the ceramic support structure,permitting measurement of the mirror growthduring heating by the capacitance probe. Onlythe amount of the support structure betweenthe mirror back and the base of the probe wouldcontribute to any changes from the ceramicsupport. The thermal couple sensors monitorthe heat distribution in the mirror to verify thatuneven tilt does not occur, and also as an input tocalculating the theoretical expansion of the mirrorto compare against the reading from the probe.The system was checked in stages to verifyoperation of the various sensors. First just smallheating was applied to verify the mirror appearedto heat uniformly and that all thermocoupleswere operating consistently. The thermal coupleswere read using an Omega HH520, four channelhandheld data logger thermometer (the 4thchannel being used to monitor the air environmentsurrounding the structure). A box of 2-inch thickStyrofoam was used to isolate the whole test unitfrom room thermal variations. The heating wasdone using an Omega TE-8-0.45-1.3 thermalelectric module and TC-720 controller. An OmegaMP-3189 Thermistor was attached to the 4th or12 O’clock position (with the thermo couples at3, 6, and 9 o’clock positions) on the side of themirror for feedback to the thermo-electric controlunit. The whole system was placed on a vibrationisolated optics table using passively air isolation.ResultsAs a first simple test, the temperature of the mirrorwas raised about 1.5 degrees C then allowed to fallFIGURE 1.Mockup of the standto hold the probe andmirror (left and center)then instrumentedwith thermal couplesand thermal electricsystem (right).MTI Instruments Inc. 1-800-342-2203 sales@mtiinstruments.com2

Measurements with Capacitance Gage for Sub-NanometerThermal Expansion CharacterizationFIGURE 2. Mirror Expansion vs TimeFIGURE 3. The change of the mirror thickness closelyfollows the temperature change.naturally. The heating controller uses pulse widthmodulation for regulating the heating which wasfound to introduce some noise into the stability ofthe temperature ramp. By turning off the heating,the thermal noise was minimized. A graph of themirror expansion versus time is shown in FIGURE2 and expansion versus temperature in FIGURE 3.Note: The capacitance sensor is measuring thegap from the probe to the mirror so as the mirrorcontracts with the temperature drop, the gapincreases proportionally and the gap is what isplotted here. The measured change of the mirrorsurface contains whatever the expansion factoris from the mounting structure, as well as anyexpansion of the probe tip which is very close tothe mirror surface.closely followed the temperature change in anearly linear fashion. The total displacement wasabout 90 nanometers for 1 degree C change.The effective change was about 8.9 nanometersper 0.1 degree C change. The total time of thechange was about 3 minutes. For the 6-millimeterthick mirror, this represented an effective thermalcoefficient of about 15 x 10 -6 /C, which is closeto the typical number for glass.As expected, the measured effective mirrorexpansion (contraction for temperature decrease)We attempted to make small temperature stepchanges using a dwell at each level of about15 minutes. Even at this level, the temperaturecontroller tended to oscillate (PWM modulation)in order to produce the desired temperatureintroducing noise. FIGURE 4 shows the typicaltemperature fluctuation for steps of 0.1 C.FIGURE 4.The thermal noiseis pronouncedeven with 0.1C stepchanges.MTI Instruments Inc. 1-800-342-2203 sales@mtiinstruments.com3

Measurements with Capacitance Gage for Sub-NanometerThermal Expansion CharacterizationFIGURE 5.Analysis of rangechanges fortemperature goingdown at 0.05 degreeC step changes.FIGURE 6. Temperaturefluctuations for controllerin oscillation.The graph and analysis of each level for a 0.05C change is shown in FIGURE 5. The standarddeviation average was around 0.16 nanometerswith a variance in the step size of 1.36 nanometers.It proved difficult to obtain steady temperaturechanges over an extended length of time. To tryto get around this limitation, the controller wasset to make a small oscillation in temperaturewhich resulted in the least thermal variationnoise. The temperature profile is shown in FIGURE6 and the resulting displacement and analysis inFIGURE 7. A least noisedata set from setting thetemperature controllerinto oscillation showconsistent steps andnoise levels on averagearound 60 picometers,representing the noiselevel of the sensor.MTI Instruments Inc. 1-800-342-2203 sales@mtiinstruments.com4

Measurements with Capacitance Gage for Sub-NanometerThermal Expansion CharacterizationFIGURE 8.Temperature Oscillations 0ver0.04 degree C simulatinginstrument stability test.Step Oscillations over 0.04CFIGURE 9.Effective Optical elementthickness variations due totemperature changes with anaverage 1 sigma value of 66picometers. The thicknesschange is clearly measurable.FIGURE 7. This minimum temperature controllernoise situation suggests a noise floor for the sensorof around 60 picometers with step sizes under ananometer clearly measurable with good signal tonoise ratio.A second set of oscillations over 0.04 degreesare shown in FIGURE 8 AND 9. This level ofsensitivity would be typical of what might beneeded to characterize optical elements for aprecision instrument such as might be used inaerospace applications.The average standard deviation of about 69picometers is consistent with previous runs. Thisnumber represents the lower limit on what rangechanges could be detected by the sensor, withclearly resolvable changes available in the 300picometer range.ConclusionsThese results conclusively demonstrate thecapability of the MTI capacitance HD amplifier, anda 50um range probe to measure the very smallMTI Instruments Inc. 1-800-342-2203 sales@mtiinstruments.com5