Deformation-free form error measurement of thin, plane ...

Deformation-free form error measurement of thin,plane-parallel optics floated on a heavy liquid

Jiyoung Chu,1,2 Ulf Griesmann,1,* Quandou Wang,1 Johannes A. Soons,1

and Eric C. Benck3

1National Institute of Standards and Technology, Manufacturing Engineering Laboratory,Gaithersburg, Maryland 20899-8223, USA

2Samsung Electronics Co., 416 Maetan 3-dong, Yeongtong-gu, Suwon-si,Gyeonggi-do, South Korea

3National Institute of Standards and Technology, Physics Laboratory,Gaithersburg, Maryland 20899-8423, USA

*Corresponding author: [email protected]

Received 15 January 2010; revised 1 March 2010; accepted 4 March 2010;posted 4 March 2010 (Doc. ID 122798); published 29 March 2010

We describe a novel method for measuring the unconstrained flatness error of thin, plane-parallel pre-cision optics. Test parts are floated on high-density aqueous metatungstate solutions while measuringthe flatness error with an interferometer. The support of the flat optics by the uniform hydrostatic pres-sure at the submerged face of the flat optic eliminates flatness errors caused by mounting forces. A small,well characterized flatness error results from the bending of the floating flat by the hydrostatic pressuregradient at the edges. An equation describing the bending of thin, flat plates floating on a liquid is de-rived, which can be used to correct the flatness measurements of arbitrarily shaped plates. The methodcan be used to measure flatness errors of both nontransparent and transparent parts, and it is illustratedwith flatnessmeasurements of photomask blanks and substrates for extreme ultraviolet lithography. Therefractive index of a saturated aqueous lithium metatungstate solution was measured at 632:8nm andwas found to be close to the refractive indices of several low thermal expansion optical materials.

OCIS codes: 120.3180, 220.4840, 220.3740.

1. Introduction

A central problem in precision form metrology is theseparation of errors of the measurement instrument,for example, an interferometer, from the intrinsicform error of the surface under test. In addition, theform of optical precision surfaces is always affectedby the forces from mechanical fixtures holding them.Forces due to mechanical clamping, chucking invacuum or electrostatic chucks, and gravity causesurface deformations that must be either eliminatedor calibrated. This is particularly important for thin,flexible parts when form error measurements withlow uncertainty are desired.Measurement algorithms, sometimes called “abso-

lute tests,” for the separation of instrument errorsand mounting-induced errors from the form errors

of the part under test have been developed for manyyears. Well known examples are the three-flat testsused for the calibration of interferometer referenceflats (see, e.g., [1–3] and the references therein).However, error separation procedures cannot befound for all situations, and it is particularly difficultto characterize deformations of a surface due tomounting forces with an absolute test.

An example of thin flat optics that must be charac-terized with low uncertainty are photomask sub-strates and blanks for extreme ultraviolet (EUV)lithography (EUVL) at a wavelength of 13:5nm.Photomasks contain the layout of an integrated cir-cuit that is projected, usually demagnified 4∶1, ontothe wafer surface by an imaging system. EUVLphotomask substrates are square with a standardwidth of 152mm and a thickness of 6:35mm as

1 April 2010 / Vol. 49, No. 10 / APPLIED OPTICS 1849

shown in Fig. 1. Several low thermal expansionmaterials (LTEM) with coefficients of thermal expan-sion <0:03 × 10−6 K−1 are used to make EUVL photo-mask substrates. Within a quality area of 142mm×142mm the peak-to-valley flatness error of front andback sides of an unconstrained EUVL photomasksubstrate must be as low as 30nm for the highestquality substrates to achieve the very small patternplacement errors required in EUV lithography.These and other characteristics of EUVL mask sub-strates are described in a standard published by theSemiconductor Equipment and Materials Interna-tional (SEMI) organization [4].Coated but unpatterned photomask substrates are

termed “photomask blanks.” The flatness error ofphotomask blanks is strongly influenced by the coat-ings that are applied to the front and back surfacesbecause of coating stresses. The front surface ofEUVL photomask blanks receives a reflective multi-layer coating consisting of multiple Mo/Si layer pairs,a capping layer to reduce the diffusion of oxygen intothe reflective multilayer, and possibly additionallayers [4]. The back side coating facilitates the elec-trostatic chucking of the finished photomask.One way of determining the intrinsic flatness er-

rors of photomask blanks and substrates is to mountthe parts in a kinematic mount and calculate themounting-induced deformations using a numericalfinite-element analysis (FEA). The calculated defor-mation can then be subtracted from the measuredflatness error to yield the unconstrained flatnesserror. This approach was taken by Battula et al.[5], who have modeled the deformation of EUVL sub-strates with FEA and then sought to verify the mod-els using different substrate mounts. It appears thatthe chief problem of themodeling approach lies in thedifficulty of verifying the numerical model as long asthe unconstrained flatness error of the mask sub-strate cannot be measured.Evans et al. [6] have described a quasi-absolute op-

tical test for separating the deformations of a trans-parent photomask substrate due to the mountingdevice, or chuck, from its intrinsic flatness errorusing a reversal method that can be implementedwith Fourier transform phase-shifting interferome-try. Using this method it was possible to separate

the mounting-induced deformations from the intrin-sic deformations of a mask substrate with the excep-tion of the component in the combined flatness errorwith fourfold symmetry that is odd under reflectionsat the y axis. For high-quality photomask substratesthis component is likely to be negligibly small. Themain limitation of this test is that it can only be ap-plied to transparent parts. The intrinsic flatness er-ror of a coated photomask blank cannot be separatedfrom mounting-induced deformations using the testby Evans et al. [6].

In this paper we describe a new method of separat-ing the intrinsic flatness errors of flat precision partsfrommounting-induced deformations that is suitablefor measuring the flatness of both photomask blanksand substrates. By floating the mask substrate orblank on a heavy, high density liquid, mounting-induced deformations are essentially eliminated,and the unconstrained flatness error of the part canbe measured directly. The unconstrained form errorcan then be used to determine the deformation ofthepart in amount or chuck.We illustrate themethodwith flatness measurements of EUVL photomaskblanks and substrates, but it is applicable to anyprecision flat for which a measurement of the uncon-strained form error is desired. In Section 2 the proper-ties of theheavy liquids thatwereused to float the testparts are summarized. In Section 3 themeasurementsetup is described, and several effects are analyzedthat can bias the measurement. Sections 4 and 5describe results of flatness error measurements ofEUVL mask blanks and substrates.

2. Properties of Metatungstate Solutions

Metatungstates are a group of tungstates (tungstencompounds containing oxoanions) that have polya-nions of the form ½H2W12O40�6− [7]. Lithium meta-tungstate (LMT, Li6½H2W12O40� · 3H2O) and sodiummetatungstate (SMT, Na6½H2W12O40� · 3H2O) arehighly soluble in water and other polar solvents.These metatungstate salts are only minimally toxicand can be used in a typical optical laboratory envi-ronmentwithout special equipment for handling toxicmaterials. The saturated aqueous solutions containabout 15% water and have densities close to 3 g=cm3

at room temperature. This density is substantiallyhigher than the density of many glasses and of lowthermal expansion materials (see Table 2) that areused to make photomask substrates. Aqueous solu-tions of LMT and SMT are available commerciallyfor applications requiring separation of materialswith different densities, e.g., for the separation of fos-sils from minerals or to separate valuable mineralsfrom sand and clay [8,9]. A similar heavy liquid, alithium polytungstate (LST), containing a proprie-tary mixture of polytungstates dissolved in water,is also available commercially. Data on physical prop-erties of the heavy liquids are scarce and had to becompiled from material safety data sheets and othertechnical information provided by the suppliers.Density and viscosity of the tungstate solutions at

Fig. 1. (Color online) Dimensions of an EUVL photomasksubstrate and coatings on an EUVL photomask blank.

1850 APPLIED OPTICS / Vol. 49, No. 10 / 1 April 2010

room temperature depend on the concentration. Wechose to use nearly saturated tungstate solutionswith densities of about 2:9 g=cm3. This density isabout 30% higher than the density of the photomaskmaterials and ensures sufficient buoyancy to preventthe top surface of a photomask from being submergedinadvertently in the fluid during handling. The prop-erties of the commercially available aqueous tung-state solutions that are most important for ourapplication are summarized in Table 1.The refractive index of a saturated aqueous

lithium metatungstate solution (LMT) at 2:95 g=cm3

was measured at 632:8nm, the wavelength of manyphase-shifting interferometers used for form mea-surements, using the minimum deviation methodas described by Burnett and Kaplan [10]. A hollowprism with plane parallel windows was filled withthe LMT solution, and the deviation angles of lightpassing through the prism at minimum deviationwere measured with a goniometer. Deviation angleswere measured for light from a helium–neon laser at632:834nm and for the red spectral line emitted by acadmium–argon low pressure discharge lamp at643:847nm. The measurements were made at the la-boratory room temperature of 16:6 °C. The refractiveindex of the saturated LMT solution at 632:8nmand its standard uncertainty, calculated from thestandard deviations of the angle measurements, is1:5888� 0:0002. Since the index n was measuredat two closely spaced wavelengths λ, an estimatefor the dispersion near 632:8nm could also be ob-tained: dn=dλ ¼ ð−7:5� 2:6Þ × 10−5 nm−1. The lineartemperature dependence of the refractive index,dn=dT, was not determined, because it was notpossible to control the laboratory room temperaturesufficiently well.The refractive index of the saturated LMT solution

at 632:8nm is much higher than the index of water(1.332, see [11]) and even slightly higher than therefractive indices of several low thermal expansionoptical materials. For comparison, Table 2 lists therefractive indices of several low thermal expansionoptical materials that were calculated from technicaldata provided by the glass manufacturers using atwo-term Cauchy formula interpolation. As long asthe density of the LMT liquid remains higher thanthat of a floating part, their refractive indices canbe matched by diluting the liquid with water.Stable metatungstate solutions are acidic [7] (see

Table 1), which can be a problem if the surface of apart in contact with the solution is susceptible to cor-rosion. EUVL photomask blanks are unlikely to be

affected by the metatungstate solutions, becausethe rear surface of a mask blank is coated with siliconnitride, a material known to be highly corrosionresistant [12,13].

While liquids with lower viscosity are more desir-able for the primary application of heavy liquids,mineral separation, a higher viscosity proved to bean advantage for the flatness measurements of maskblanks and substrates because a fluid with higherviscosity dampens residual motions of the photo-mask blanks more effectively, as is described in moredetail in the following section. For the measurementsdescribed in this paper, the blanks and substrateswere floated on LMT solution at a density of2:95 g=cm3.

3. Measurement Procedures

The flatness measurements of EUVL photomaskblanks and substrates were made with the“eXtremely accurate CALibration InterferometeR”(XCALIBIR) at the National Institute of Standardsand Technology (NIST). XCALIBIR is a versatilephase-shifting interferometer operating at a wave-length of 632:8nm. The interferometer has a horizon-tal optical axis, and test parts are usually mountedvertically (on their edge). For the flatness measure-ments of floating photomask blanks and substrates,the interferometer was reconfigured so that the re-ference flat of the interferometer could be mountedabove the photomask in horizontal orientation. Aschematic of the XCALIBIR setup for the photomaskblank and substrate measurements is shown inFig. 2. The photograph inset in Fig. 2 shows the sup-port structure for the fold mirror and the referenceflat. On the left side of the photograph the collimatorlens assembly is visible, which creates a collimated

Table 1. Properties of Aqueous Tungstate Solutions at Room Temperature

Name Abbreviation Sodium Metatungstate (SMT) Lithium Metatungstate (LMT) Lithium Polytungstate (LST)

Formula Na6½H2W12O40� · 3H2O Li6½H2W12O40� · 3H2O proprietaryDensity ρ ½g=cm3� 2.95 2.85pH ≈3 ≈4 ≈4Viscosity μ ½mPa · s� 20� 1 36� 1 12� 1n at 632:8nm and 16:6 °C 1:5888� 0:0002

Table 2. Densities and Refractive Indices of Several Low ThermalExpansion Materials at 632.8 nm

Name Density ½g=cm3� Refractive Index

Fused silica (generic) 2.201 1.45701 (see [21])Corning ULE 7972a 2.21 1.4840Schott ZERODURa 2.55 1.5404Ohara CLEARCERAM-Za 2.53 1.5477aDisclaimer: The full description of the procedures used in this

paper requires the identification of certain commercial productsand their suppliers. The inclusion of such information should inno way be construed as indicating that such products or suppliersare endorsed by NIST or are recommended by NIST or that theyare necessarily the best materials or suppliers for the purposesdescribed.


test beam of just over 300mm diameter. The alumi-nized fold mirror has a diameter of 350mm andsends the test beam downward to the reference flat,which is visible below the fold mirror in Fig. 2. Thedownward beam is vignetted by the fold mirror, re-sulting in the noncircular field of view shown inFig. 3. The coating of the photomask blanks is highlyreflective in the visible, and poor fringe contrastwould result if the blank was measured against anuncoated reference flat. A reference flat coatedwith absorptive antireflection coating similar to aClapham–Dew coating [14] was used for the mea-surements of both coated photomask blanks and un-coated substrates. This resulted in excellent fringecontrast for the highly reflective photomask blanksand acceptable fringe contrast for the uncoated sub-strates. XCALIBIR is equipped with a tunable singlefrequency diode laser, and phase measurements

were made by shifting the wavelength of the diodelaser. For an interferometer resonator of length tthe free spectral range Δλ is

Δλ ¼ λ22nt

; ð1Þ

where n is the refractive index of the medium fillingthe resonator and λ is the interferometer wavelength.A change in wavelength corresponding to the freespectral range is equivalent to a phase shift of 2π.During wavelength shifting the laser power is mon-itored and held constant to prevent measurementerrors due to intensity variation.

A tray machined from a black acetal plastic filledwith LMT was used to float the EUVL photomaskblanks and substrates. The bottom of the tray wassandblasted to prevent specular reflections fromthe tray bottom in the measurements of transparentphotomask substrates. Figure 4 shows an EUVLphotomask blank floating on a tungstate solution.The internal width of the tray is 175mm, and the cor-ners are rounded with a radius of 10mm. The stan-dard width of photomasks is 152mm. The gap of11:5mm on all sides of the photomask was consid-ered sufficiently large to minimize effects due tothe surface tension of the liquids on the flatness errorof the parts under test. In preparation for a flatnessmeasurement, between 90mL and 100mL of freshtungstate fluid was slowly poured into the tray toavoid formation of air bubbles and the trapping ofbubbles at the bottom of the tray. Bubbles that occa-sionally were found on the liquid surface after thetray was filled were suctioned off with a syringe.The photomask blank or substrate was then loweredonto the liquid by hand. Initially, two plastic pins ateach corner of the photomask blank as shown inFig. 4 were used to prevent lateral movement ofthe floating parts. Two pairs of pins were laterremoved to investigate effects of the pins on the flat-ness error of the blanks. For the measurements de-scribed in this paper two pairs of restraining pinsat diagonally opposing corners were used. The space

Fig. 2. (Color online) Schematic of the XCALIBIR setup forphotomask blank and substrate measurements. The inset photo-graph shows the actual setup of fold mirror and reference flat.

Fig. 3. (Color online) Flatness error of the reference flat shown inFig. 2. The black square indicates the approximate size and loca-tion of the photomask blanks and substrates in the field of view ofthe interferometer.

Fig. 4. (Color online) EUVL photomask blank floating on alithium metatungstate (LMT) solution.


between the pins and the photomask blank is a fewtenths of a millimeter, and no force is exerted by thepins on the photomask blanks or substrates. Effectsdue to the surface tension of the heavy liquid werenot observed. For example, flatness measurement re-sults were identical when the tops of the restrainingpins were above or below the fluid level. For a volumeof 90mL of LMT liquid in the tray, the distance be-tween the photomask and the bottom of the tray is1:8mm and the photomask is immersed in the liquidto a height of 4:8mm. Measurements were typicallymade shortly after lowering a part onto fresh LMTliquid to avoid problems resulting from water eva-poration and crystal formation between the edge ofa part and the restraining pins. Crystal growthwas observed when a part was left in the liquid foran extended period of time, for example, overnight.In the following subsections the residual deforma-

tion of a floating mask blank or substrate due to thehydrostatic pressure gradient in the fluid and themagnitude of thermal gradient effects is estimated.A calibration procedure for the horizontally mountedinterferometer reference flat is also discussed, be-cause it is an important contributor to the measure-ment uncertainty. Fluid dynamic effects in the heavyliquid were considered negligible. Test measure-ments with pure water in the tray, which is much lessviscous than the heavy liquids, indicated no surfacestructures or disturbances that would suggest acous-tic excitation of waves or currents in the liquid.

A. Hydrostatic Pressure Effects

While the hydrostatic pressure at the bottom of afloating photomask blank or substrate is nearlyconstant, a pressure gradient exists at the edges asshown in Fig. 5. The pressure gradient results in auniformly distributed edge torque M that bendsthe flat. The deformation resulting from the pressuregradient at the plate edge can be calculated analyti-cally for thin plates. For a flat plate of thickness twith density ρg floating on a liquid with density ρl,the liquid reaches a height zl relative to the neutralplane of the flat:

zl ¼ t

�ρgρl

−12

�: ð2Þ

The pressure distribution pðzÞ exerted by the liquidon the edge of the flat is

pðzÞ ¼�

0 if z ≥ zlρlgðzl − zÞ if z < zl

; ð3Þ

where g is the gravitational acceleration. The edgetorqueM per unit of edge length at the neutral plane(z ¼ 0) of the plate equals

M ¼ −

Zt=2

−t=2pðzÞzdz ¼ −

Zzl

−t=2ρlgðzl − zÞzdz

¼ −gt3ρg� ρ2g6ρ2l

−ρg4ρl

�: ð4Þ

The transversal deformation wðx; yÞ of a thin, rectan-gular plate (t ≪ L), subjected to a uniform edgetorque M, and where the maximum deformationwm is small compared to the thickness of the plate(wm ≪ t), is described by the equation [15]

wðx; yÞ ¼ −M

2Dð1þ νÞ ðx2 þ y2Þ; ð5Þ

where D is the flexural rigidity,

D ¼ Et3

12ð1 − ν2Þ ; ð6Þ

E the elastic modulus of the plate material, and ν itsPoisson ratio. Combining Eqs. (4)–(6) leads to anequation describing the deformation of a thin platefloating on a liquid by the hydrostatic pressuregradient at the edge:

wðx; yÞ ¼ 6ð1 − νÞgρgE

� ρ2g6ρ2l

−ρg4ρl

�ðx2 þ y2Þ: ð7Þ

It is worth noting that the deformation in Eq. (7) isindependent of the plate thickness. The maximum(peak-to-valley) deformation wm of a square flat withwidth L is given by

wm ¼ 3ð1 − νÞgρgE

� ρg4ρl

−ρ2g6ρ2l

�L2: ð8Þ

For a square plate of width L ¼ 152mmmade from alow thermal expansion material with elastic modu-lus E ¼ 67:6GPa, density ρg ¼ 2:21 g=cm3, andPoisson’s ratio ν ¼ 0:17, floating on a liquid with adensity of ρl ¼ 2:95 g=cm3, the maximum deforma-tion due to the hydrostatic pressure gradient atthe edges is 1:7nm. This result was corroboratedby a finite-element model of the floating photomaskblank. Calculating the uncertainty of the maximumdeformation is complicated by the lack of uncertaintydata for the material parameters, but an estimate ofthe typical uncertainty can be obtained based onplausible assumptions for the uncertainties of thevariables in Eq. (8). When it is assumed that the den-sity of the plate ρg, the elastic modulus E, and thewidth L of the plate are known with relative uncer-

Fig. 5. (Color online) Mechanical parameters of a thin plate float-ing on a liquid. The pressure gradient due to hydrostatic pressureat the edge of the plate is indicated in red.


tainties of 1%, and Poisson’s ratio ν of the plate ma-terial and the density of the liquid ρl are known withrelative uncertainties of 5%, and assuming Gaussianuncertainty propagation, the uncertainty for themaximum deformation is 0:05nm. For many applica-tions a small deformation wm ¼ ð1:7� 0:05Þnm cansimply be neglected or it can be considered a contri-butor to the measurement uncertainty. If necessary,the deformation due to the hydrostatic pressure gra-dient wðx; yÞ can be subtracted from the measuredflatness error.The distribution of the torque around the edge of a

floating plate caused by the hydrostatic pressure gra-dient is independent of the shape of the plate, and itcan be shown that Eq. (5) is valid for any shape [15].Many optical components have a circular shape, andthe maximum (peak-to-valley) deformation of thin,circular plates with radius R by hydrostatic pressuregradients at the edge is described by an equationsimilar to Eq. (7):

wm ¼ 6ð1 − νÞgρgE

� ρg4ρl

−ρ2g6ρ2l

�R2: ð9Þ

In the derivation of Eq. (8) it was assumed that thesubmerged face of the floating plate is perfectly flat.This assumption is only approximately valid, be-cause the flat has a flatness error and the hydrostaticpressure will vary across the bottom of the plate. Apeak-to-valley flatness error δz results in a pressuredifferential δpz ¼ ρlgδz. The ratio δpz=pð−t=2Þ of thepressure differential to the pressure at the bottomof the plate is about 2 × 10−5 for a peak-to-valleyflatness error of 100nm. The deflection of the platecaused by the pressure differential due to the flat-ness error was considered negligible.A further nonuniformity of the hydrostatic pres-

sure is caused by the spherical nature of the gravita-tional force. Gravity is directed toward the center ofthe earth, which results in a slightly nonuniformdistribution of the gravitational force component nor-mal to the flat surface. When it is assumed that theplate is rigid, which yields the worst-case pressuredifferential, the pressure differential δpg ¼ ρlδgðzl þt=2Þ resulting from the variation in the gravity com-ponent normal to the plate surface is small. For a152mm wide photomask blank floating on LMT,the ratio δpg=pð−t=2Þ of the pressure differential tothe pressure at the bottom of the plate is about1:7 × 10−8. This pressure differential is much smallerthan the pressure differential due to the flatnesserror, and the effect is negligible.

B. Effects of Temperature Gradients

Next to the hydrostatic pressure a temperature gra-dient can result in a deformation of a floating photo-mask blank or substrate. A temperature differencebetween the liquid and the surrounding air will re-sult in a thermal gradient dT=dz in vertical directionin the glass, which causes a deformation with con-stant curvature. The deformation is approximatelystress free, and a simple one-dimensional model

can be used to estimate the radius of curvature R.When one considers the difference Δl in the thermalexpansion of two parallel fibers of the material par-allel to the faces with length l, coefficient of thermalexpansion α, and separated by a distance w, thedifference Δl of the fibers due to the temperaturegradient is

Δl ¼ lαwdTdz

: ð10ÞThe length difference Δl of the fibers due to aconstant radius of curvature R is also described by

Δl ¼ wlR: ð11Þ

Combining Eqs. (10) and (11) yields an expression forthe curvature 1=R of the plate:

1R

¼ αdTdz

: ð12Þ

The resulting flatness error (sag) for a square plate ofwidth L is

s≃L2

212R

¼ 14L2αdT

dz: ð13Þ

Eq. (13) can be used to estimate the worst-caseflatness error of the floating photomask blank orsubstrate due to temperature gradients. The roomtemperature in the XCALIBIR laboratory is con-trolled at 20 °C. The temperature fluctuates by�0:05 °C around the set temperature. The coefficientof thermal expansion for the photomask material isapproximately 0:03 × 10−6 K−1, and the width of thephotomask is 152mm. The photomask thickness is6:35mm, and a worst-case thermal gradient of0:1K=6:35mm can be assumed. Using Eq. (13), thisthermal gradient leads to a flatness error of 2:8nm.In practice the thermal gradient is likely to be muchsmaller because the poor heat transfer from air to thelow thermal expansion substrate material will delayand reduce the creation of a thermal gradient in thematerial when the air temperature changes.

Another potential cause of a temperature gradientin the photomask is evaporative cooling of the heavyliquid during a measurement. A simple experimentwas conducted to estimate the magnitude of theevaporative cooling. A small plastic beaker with50mm diameter was filled with 20mL of saturatedLMT solution at 2:95 g=cm3 density. A thermistortemperature sensor was used to measure the tem-perature of the liquid. It was found that the tempera-ture of the liquid dropped at a rate of about 0:006 °C=min. Any effect of the cooling would first be observedat the edges of a photomask blank or substrate float-ing in the tray shown in Fig. 4, because the liquidunder the part is not exposed to air. No indicationof significant changes in the form error near the edgeof a floating blank or substrate was found in repeatmeasurements. Figure. 6 shows the repeatability,pixel by pixel, of ten measurements that were made


over a period of about 30 min. It does not show a uni-form worsening of the repeatability near the edges ofthe photomask blank.

C. Interferometer Calibration

Acritical part of themeasurement is the calibration ofthe horizontal reference surface. The reference flathad previously only been calibrated in vertical orien-tation, and it cannot be assumed that the referencesurface remains unchanged after remounting the flathorizontally. An additional complication is that wellknown three-flat calibration procedures (see, e.g.,[3]) could not be applied in this case because mechan-ical fixtures for rotating the flats in horizontal orien-tation were not available, and a reliable method forcalibrating the sag of a horizontally mounted flat isnot known. The reference flat was calibrated usingoverlapping measurements of a small flat with verylow flatness error. The small flat is visible in Fig. 2to the right of the support structure for reference flatand fold mirror. This flat has a diameter of 150mmand a known peak-to-valley flatness error of 3:3nm.The parabolic component (power and astigmatism)in the flatness error has a magnitude of only about1nm peak-to-valley. This flat was calibrated withthe normal vector of the reference surface in a hori-zontal plane. It was then assumed that changingthe orientation of the reference flat such that the nor-mal vector is vertical results in a negligible change inthe flatness error, because the flat is very thick (ap-proximately 50mm). The small flat was measured atseven overlapping locations that cover the wholeaperture of the interferometer. At each location theflat wasmeasured five times and after eachmeasure-ment the flat was rotated by about 72° to approxi-mately average out the flatness error withoutrotation symmetry in the small flat. All measure-

ments were then stitched together, and the flatnesserror map of the reference flat shown in Fig. 3 wasobtained. The flatness error of the reference flat issubtracted from the flatness error maps shown inSections 4 and 5.

4. Photomask Blanks

For the photomask blank flatness measurements,the tray holding the heavy liquid and the blank(shown in Fig. 4) was placed on the optical tablebelow the reference flat. The distance between theblank and the reference flat was about 10 cm. Fora typical EUVL photomask blank, fringes like thoseshown in Fig. 7 were then observed with the interfe-rometer. The measurements are complicated by theresidual motion of the floating blank, which in thecase of our experiment was caused by slow move-ments of the optical table of XCALIBIR. While theblanks and substrates are prevented from movingsideways, the movement of the optical table intro-duces a gradually changing tilt of the floating blankrelative to the reference flat. The fringes, like thoseshown in Fig. 7, will slowly drift and lead to samplingerrors in the phase measuring which, in turn, resultsin characteristic errors in the flatness measurementsat twice the spatial frequency of the fringes. Anotherpotential error source is the high reflectivity of thephotomask blank coating that can result in multiplereflections between test surface and reference flat,resulting in higher harmonics in the fringe signal.We found that, despite the special coating on the re-ference flat, the observed fringes were not perfectlysinusoidal.

Measurement errors resulting from the residualmotion of the photomask blank and higher harmonicsin the fringeswere reduced by over two orders ofmag-

Fig. 6. (Color online) Repeatability map (standard deviation) ofthe blank flatness error calculated from 10 repeated measure-ments of the photomask blank flatness.

Fig. 7. Interferometer fringes of a photomask blank floating on aheavy liquid.


nitude by using a phase shifting algorithm that is in-sensitive to sampling errors and to higher harmonicsin the fringe signal. The phase and amplitude re-sponse of the13-samplephase shiftingalgorithmwith60° sampling increment that was designed for thismeasurement using the characteristic polynomialtheory by Surrel [16] is shown in Fig. 8. This figureshows the first and second alias of the phase and am-plitude response of the phase shifting algorithm. Likeall algorithms with a sampling increment of 60°, thisalgorithm has a normalized Nyquist frequency of 3.Thealgorithm isnot sensitive to theharmonic contentof the fringe signal up to the 4thharmonic (blue line inFig. 8). The phase response (red line in Fig. 8) is flat atthe fundamental harmonic, which makes the algo-rithm insensitive to sampling errors due to thetime-varying tilt of the floating mask blank. Table 3lists the sampling coefficients for the 13-sample algo-rithm in the format introduced by Surrel [16]. Alsolisted is the loss factor η (see [17]) for the phase shift-ing algorithms used in the photomask blank andsubstrate flatness measurements.Figure 9 is a false color map of the flatness error of

the same EUVL photomask blank shown in Fig. 7.The form error of the reference flat is subtracted.The peak-to-valley flatness error within the qualityarea is approximately 100nm. Ten measurementswere made to evaluate the short-term repeatabilityof the measurements. Figure 6 shows a map of thepixel-by-pixel repeatability (one standard deviation)of the mask blank measurement in Fig. 9. This imagereveals that the effects of the sampling errors due tothe motion of the mask blank could not be completelyeliminated, but the resulting spurious midfrequencyripple leads to an uncertainty of the flatness errormeasurement well below 1nm for most of the photo-mask blank area. It must be emphasized that theripple is not a feature of the measurement methodbut of the implementation.

The long-term repeatability was assessed by com-paring two flatness error measurements of the sameEUVL photomask blank made three weeks apart.The difference of the two measurements of the blankwith the flatness error in Fig. 9 is shown in Fig. 10.The peak-to-valley of the difference is about 2:5nm,and the root-mean-square (rms) of the differencemap in Fig. 10 is 0:6nm. The largest differences inrepeat measurements were consistently found at theedges of the photomask blanks. This may be causedby the heat transfer into the mask blanks when theywere loaded into the tray by hand. The circular fea-tures in the difference map Fig. 10 are residuals ofthe averaging procedure that was used for the cali-

Fig. 8. (Color online) Phase response (red) and normalized sen-sitivity (blue) of the 13-sample phase shifting algorithm with60° sampling steps used for the photomask blank measurements.FaðνÞ and FbðνÞ are the real and imaginary spectral transfer func-tions of the phase shifting algorithm.

Fig. 9. (Color online) Flatness error of the photomask blank(reference error removed).

Fig. 10. (Color online) Difference of two measurements of thephotomask blank of Fig. 7 made three weeks apart.


bration of the reference flat (see Subsection 3.C);they will not be present when an appropriate full-aperture calibration method is used.

5. Photomask Substrates

The method described in Section 4 for the flatnessmeasurement of mask blanks can also be used tomeasure the flatness error of uncoated substrates.Substrate flatness measurements are a challenge be-cause they are transparent and, as indicated inFig. 11, several fringe systems are seen in the inter-ferometer. In addition to the interference of light re-flected by the reference surface and the front surfaceof the mask substrate, additional fringes are ob-served, which result from interference of reflectionsfrom the substrate’s front and back surfaces and frominterference of reflections from reference flat andsubstrate back surface. When the back surface ofthe photomask substrate is in contact with the heavyliquid, the fringe contrast of the unwanted fringes ismuch reduced, because the refractive index of theheavy liquid is close to the refractive index of the sub-strate material. If the refractive index of the photo-mask substrate is known, the heavy liquid can bediluted with water until its index matches the indexof the substrate material, which eliminates theunwanted reflections. Since the material of thephotomask substrate sample was unknown, the sub-strate was floated on the saturated LMT solution andwavelength shifting interferometry with a phaseshifting algorithm that is insensitive to the fringesinvolving a reflection from the back surface of thesubstrate was used to filter out the unwanted fringes.

The basic concepts of measuring the flatness errorof parallel plates with wavelength shifting interfero-metry are well known (see, e.g., de Groot [18] orBurke et al. [19]). For the flatness measurementthe mask substrate was positioned so that the airgap between reference flat and front surface of thesubstrate was approximately twice the optical thick-ness nt of the mask substrate as shown in Fig. 11.Since the photomask substrate material was notknown, a refractive index of 1.5 was assumed. Whenthe wavelength is changed by one free spectral rangefor the cavity between the front and back surfaces ofthe substrate, it will be changed by exactly two orthree free spectral ranges for the other two cavities.At the detector three frequencies are observed duringwavelength shifting with exact frequency ratios:ν1∶ν2∶ν3 ¼ 1∶2∶3. The signal containing the desiredinformation about the substrate flatness is the 2ndharmonic ν2. It can be measured with a phase shift-ing algorithm that is sensitive at the 2nd harmonic ν2and insensitive to the fundamental frequency ν1 andits 3rd harmonic. The algorithm must also have astationary phase response at the frequency ν2. Thephase and amplitude response of the seven-samplephase shifting algorithm with 60° sampling steps,which was designed for the measurement, is shownin Fig. 12. This algorithm is closely related to theLarkin–Oreb algorithm [20], and its samplingcoefficients are listed in Table 3. The reason a muchless robust phase shifting algorithm with only seven

Fig. 11. (Color online) Frequencies observed in wavelengthshifting during photomask substrate measurements.

Fig. 12. (Color online) Phase response (red) and normalizedsensitivity (blue) of the seven-sample phase shifting algorithmwith 60° sampling steps used for the photomask substratemeasurements.

Table 3. Coefficients of Phase Shifting Algorithms for Blank andSubstrate Flatness Measurements

Samples Sampling coefficients η (see Ref. [17])

13 ½−1;−8;−11;10;43;46;0;−46;−43;−10;11;8;1�ffiffi3

p ½1;0;−9;−18;−9;18;34;18;−9;−18;−9;0;1�108ffiffiffiffiffiffiffiffiffiffi18239

p ≃ 0:8

7 ½−1;−3;3;0;−3;3;1�ffiffi3

p ½1;−1;−1;2;−1;−1;1�6

ffiffi3

pffiffiffiffiffiffi119

p ≃ 0:95

Fig. 13. (Color online) Flatness error of the top surface of a photo-masksubstrate floatingona lithiummetatungstatesolution(LMT).


samples was used for the mask substrate measure-ment is the limited tuning range of the laser. The freespectral range of the shortest interferometer cavitybetween the front and back surfaces of the masksubstrate was almost as large as the maximum tun-ing range of the laser, which meant that phaseshifts >2π could not be realized and phase shiftingalgorithms with better phase response and bettersuppression of undesired frequencies, which requirelarger phase shifts, could not be used. The result of aflatness measurement of a floating EUVL masksubstrate is shown in Fig. 13.

6. Conclusion

We have developed and demonstrated a new methodfor measuring the flatness error of thin, plane-parallel flat plates free from deformation by amount-ing fixture that is applicable to both nontransparentand transparent parts. Instead of mounting the flatsusing a mechanical fixture, they are floated on a sa-turated aqueous lithiummetatungstate solution dur-ing the measurement. This sidesteps the problem ofhaving to separate the intrinsic flatness error of theplate from the flatness error introduced inadvertentlyby forces from themounting fixture. Inmost cases thebending of the floating plate by the edge moment dueto the hydrostatic pressure gradient at the edges willbe negligible. A simple analytical model was derivedto calculate the deformation due to the edge moment,which allows the measurements to be corrected forthe hydrostatic pressure deformation. The newmeth-od is particularly valuable for flatnessmeasurementsof coated photomask blanks for EUV lithography be-cause no other measurement procedures are knownfor separating the intrinsic flatness error of theblanks from mounting induced deformations. Thehigh refractive index of the lithiummetatungstate so-lution also simplifies the flatness measurement oftransparent substrates. The liquid can be diluted tomatch the refractive index of the photomask sub-strate material to eliminate unwanted reflectionsfrom the back surface of the substrates as long asthe density of the liquid remains higher than thatof the test part. This obviates the need for wavelengthshifting interferometry, and the flatness error of thefront surface can be measured with conventionalphase shifting interferometers.

We gratefully acknowledge the support and assis-tance by Jaewoong Sohn (SEMATECH) who encour-aged us to develop a method for deformation-freemeasurements of EUVL photomask blank flatnesserrors. We thank John P. Lehan (NASA GoddardSpace Flight Center and University of MarylandBaltimore County) for relating to us his experiencewith heavy liquids for measuring the form errorsof optics under conditions of simulated weightless-ness. We are grateful to John H. Burnett, NIST,for advice regarding the refractive index measure-ments, and to Jan Burke (CSIRO Materials Scienceand Engineering, Australia) for introducing us to the

characteristic polynomial theory for designingphase-shifting algorithms.

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