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UNCERTAINTY IN MEASUREMENTS OF DIMENSIONAL CHANGESFROM AN INTERFEROMETRIC DILATOMETER

Dongmei Ren, Jimmie A. Miller, Kevin M. LawtonCenter for Precision Metrology

Department of Mechanical Engineering & Engineering ScienceUniversity of North Carolina at Charlotte

Charlotte, NC 28223, USA

INTRODUCTIONWith low expansion materials finding greatapplications in micro-lithographic systems andother systems requiring high dimensionalstability, accurate determination of thecoefficients of thermal expansion of suchmaterials becomes essential. In order to meetrequirements for high accuracy determinationsof the coefficients of thermal expansion (CTE) oflow expansion materials and measurements ofother dimensional instabilities, a nanoscaleabsolute interferometric dilatometer has beendeveloped at the University of North Carolina atCharlotte. This dilatometer is to measure avariety of samples of length up to 100 mm withan uncertainty of 2 nm for the measurement ofdimensional change and 1 ppb/K for CTEmeasurement in a temperature range of 293373 K. A novel double-pass heterodyneinterferometer has been designed for thisdilatometer. In this paper, the uncertainty of themeasurements of dimensional changes from the

interferometer is analyzed.

INTERFEROMETERThe working principle of the interferometer isshown in Figure 1 [1]. This interferometer has anearly balanced design, in which the nominaloptical lengths of the measurement path andreference path are equal except for the pathscorresponding to the sample length.

FIGURE 1. Interferometer

UNCERTAINTY ANALYSISThe main sources of uncertainty formeasurements of dimensional changes from thisinterferometer include the stability of the laserwavelength, nonlinearity of the system,uncertainty of the electronics, uncertainty of thesample length, and environmental changes if themeasurements are conducted in air.

Measurement EquationThe heterodyne interferometer determines thedimensional change of a sample by measuringthe relative phase change that occurs betweenthe optical path lengths of the interferometersmeasurement arm and reference arm. Thedisplacement information given by theinterferometer can be expressed as

nKkL

= (1)

where is the wavelength of laser in vacuum,

n is the refractive index of the measurementenvironment, K is the resolution of the

interferometer, and k is the electronic count.Considering the change in the unbalanced path,the dimensional change of the sample should be

dlnK

kl =

(2)

where dl is the change of the unbalanced

path. For the interferometer operated in air, the

major part ofdl results from environmental

changes. In this analysis, we use the modified

Edln equation [2] to correct the change ofrefractive index. The correction is given by

610)]10(05.0

)760(36.0)20(93.0[(

+

=

f

ptlln(3)

where l is the length under standard conditions(t = 20 C, p = 760 mmHg, f = 10 mmHg), t, p

and f are the air temperature (C), pressure

(mmHg) and humidity (water vapor pressure, in

M M

MSample QWPQWP

PBSPBS

HWPRAP

CC

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mmHg) of air. From equation (3), the dead-patherror caused by environmental changes can beexpressed as

6

0

00,

10)](05.0

)(36.0)(93.0[(

+

=

ff

ppttLl snd(4)

where 0t , 0p and 0f are the initialtemperature, pressure and humidity at the

beginning of the measurement, and sL is the

sample length. Another source of dl is the

change of laser wavelength during themeasurement, which is treated as uncertainty inthis analysis. So equation (2) becomes

6

0

00

6

0

10)](05.0

)(36.0)(93.0[(

}10)]10(05.0)760

(36.0)20(93.0[1{)(

+

+

+=

ff

ppttL

f

ptKnkl

s

(5)

where 0n is the refractive index of air at the

standard condition.

Uncertainty budgetThe uncertainty of the measurement ofdimensional change can be estimated byconsidering the uncertainty sources individuallyand then combining them. According to the lawof propagation of uncertainty [3], the combined

standard uncertainty of the estimate of l inequation (5) can be determined by the positive

square root of the combined variance )(2

luc ,which is given by

=

=

+

+

+

+

+

+

+

+

+

=

10

1

2

22

0

22

0

0

22

0

0

22

0

2222

2222

0

22

0

222

)(

)()()()(

)()()()(

)()()()(

)()()()(

)()()()()(

i

i

s

s

c

lu

LuL

lfuf

l

pup

ltu

t

l

fuf

lpu

p

l

tut

lku

k

l

nun

lu

llu

(6)

where )(u , )( 0nu , )(tu , )(pu , )( fu , )( 0tu ,

)( 0pu , )( 0fu , )( sLu and )(ku are standard

uncertainties of , 0n , t, p , f, 0t , 0p , 0f ,

sL and k, respectively. )(2 lui represents the

i th term in the equations. In equation (6), alluncertainty components are assumed to beindependent.

Uncertainty of wavelengthThe variance in equation (6) associated withwavelength uncertainty is

)()()( 2221 uLlu =

The accuracy of vacuum wavelength of the laserhead, Zygo ZMI 7722, is 0.16 ppm. Thestability of vacuum wavelength of the laser is10 ppb over the lifetime of the laser head and0.5 ppb over one hour. Assuming that theseuncertainties have normal distributions, thestandard uncertainty caused by the accuracy is

Lu

L

lu aa8

00

,1 103.5)()(

==

and the standard uncertainty caused by thelong-term stability of wavelength, in which theinfluence of wavelength stability to theunbalanced path is included, is

sss

s LuLL

lu 900

,1 103.3)()(

=

Since the dimensional changes to be measuredare typically smaller than 0.1 mm, theuncertainty due to wavelength is mainly in theform of the influence of wavelength instability tothe dead-path length.

Uncertainty of refractive index of air understandard conditionsThe refractive index of air under the standardcondition can be calculated using the modifiedEdln equation with an uncertainty of

8103 (2). So the standard uncertainty

associated with 0n in equation (6) is

LnunLlu8

002 105.1)()(

==

This uncertainty term is negligibly small formeasurements of small dimensional changes.

Uncertainty of the detection electronicsThe variance associated with the uncertainty ofelectronics in equation (6) is

)()()( 2223 kunK

lu

=

The standard uncertainty caused by electronicsof the ZMI 4004 measurement system isassumed to equal the least resolution of the

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interferometer, which is 4096 for a double-

pass interferometer. So the resulting standarduncertainty is

15.0)(3 =lu nm.

Uncertainties of environmental parametersWhen using the modified Edln equation tocorrect for environmental changes, theuncertainties in determining the air temperature,pressure and humidity cause measurementuncertainties. The variances in equation (6)

associated with t, p and f are given by

)(1093.0)()(212222

4 tuLLlu s

=

)(1036.0)()( 2122225

puLLlus

=

)(1005.0)()( 2122226 fuLLlu s

=

Assuming that the uncertainties, which includeboth measurement uncertainties and non-

uniformities, of the temperature, pressure andhumidity are 0.05 C, 0.1 mmHg and 0.5 mmHg,respectively, and they have normal distributions,

the resulting standard uncertainties, )(4 lu ,

)(5 lu and )(6 lu will be dL8106.1 ,

dL8102.1 and dL

9105.8 , respectively.

Similarly, the variances associated with 0t , 0p

and 0f are given by

)(1093.0)( 0212222

7 tuLlu s =

)(1036.0)( 0212222

8 puLlu s

= )(1005.0)( 0

212222

9 fuLlu s

=

Since the dimensional change is negligibly smallcompared to the sample length, theenvironmental-induced uncertainties mainlyarise from the correction of the dead-path error.For measurements in vacuum, theseuncertainties can be neglected.

Uncertainty of sample lengthThe variance associated with the sample lengthin equation (6) is

)(10)](05.0)(36.0)(93.0[)(

2122

0

00

2

10

sLuffppttlu

+

=

If the maximum changes in air temperature,pressure and humidity during the measurement

are 1 C, 10 mmHg and 10 mmHg, respectively,the standard uncertainty estimated by the root-sum-square of these items is

)(108.3)(6

10 sLulu

=

Assuming that the uncertainty of the samplelength is 0.1 mm and it has a normaldistribution, this term of standard uncertainty isapproximately 0.13 nm.

Nonlinearity of the interferometerAn uncertainty source that is not included inequation (6) is the nonlinearity of theinterferometer, which is caused by opticalmixing. The nonlinearity of the interferometerwas measured using the thermal translationmethod [4], as shown in Figure 2. If theuncertainty is treated conservatively as having arectangular distribution, the standard uncertaintyis

43.0375.0)(11 =lu (nm)

-1.0

-0.5

0.0

0.5

1.0

1 2 3 4 5

Displacement (wavelength)

Nonlinearity(nm)

FIGURE 2. Nonlinearity of the interferometer

Other uncertaintiesThe correction of refractive index of air using themodified Edln equation has a relative

uncertainty of8

103

(2). The resultingstandard uncertainty is approximately

ssLLlu 88

12101.22/1032)( ==

Combination of the uncertaintiesThe sources of uncertainty and theircontributions to the measurement uncertainty ofthe interferometer are summarized in Table 1.The combined standard uncertainty of theinterferometer can be obtained by calculatingthe root-sum-square of the individual standarduncertainties. The combined standarduncertainty in the laboratory environment, wherethe temperature, pressure and humidity are 22.9 0.5 C, 747 5 mmHg and 40 20 %, iscalculated and listed in the column of in air inthe table. This interferometer is designed beoperated in a vacuum chamber with thetemperature control of 20 0.1 C. Thecombined standard uncertainty under suchcondition is also calculated and is listed in Table1 in the column of in vacuum.

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TABLE 1. Summary of standard uncertainties of the interferometer

for measurements of dimensional changes

CONCLUSIONSThe analysis of uncertainty has shown that,although the dead-path lengths have beenreduced to the sample length by the balancedinterferometer design, the uncertaintyassociated with the correction for dead-patherror is the most significant source of uncertaintyfor measurements in air. It is necessary tooperate the interferometer in a vacuum chamberin order to achieve the proposed accuracy. Twokey components for further improving themeasurement accuracy of the interferometer invacuum are to reduce the nonlinearity resulting

from the optical mixing and to improve thestability of laser wavelength. It can be concludedthat this double-pass interferometer should beable to measure dimensional changes with anuncertainty of 2 nm (3), which meets theaccuracy requirement of the interferometricdilatometer.

ACKNOWLEDGEMENTSThis work was supported by the NationalScience Foundation under Grant No. 0216420.The Center for Precision Metrology is supportedas an NSF Industry/University CooperativeCenter under Grant No. 0331950.

REFERENCES[1] Ren D. Optical Measurements of

Dimensional Instability. Ph.D dissertation.University of North Carolina at Charlotte.2006.

[2] Birch K, Downs M. An Updated Edln

Equation for the Refractive Index of Air.Metrologia. 1993; 30: 155-162.

[3] Guide to the Expression of Uncertainty inMeasurement. ISO. 1995.

[4] Zhang L. Fiber-coupled HeterodyneInterferometer. Ph.D dissertation. Universityof North Carolina at Charlotte. 2003.

Source of uncertainty

Standard uncertainty

)( ixu ix

f

)( lui (nm)

In air In vacuum

Wavelength stability )(u 3.3 ppb )( LLs 0.33 0.33

Electronics )(ku 1 K0 0.15 0.15

Environment

Temperature

Pressure

Humidity

Initial temperature

Initial pressure

Initial humidity

Edln equation

Total

)(tu

)(pu

)( fu

)( 0tu

)( 0pu

)( 0fu

0.017 C

0.03 mmHg

0.17 mmHg

0.017 C

0.03 mmHg

0.17 mmHg

sL8105.1

610)(93.0 LLs610)(36.0 LL

s

610)(05.0 LLs61093.0

sL

61036.0 sL61005.0 sL

2

1.6

1.2

0.85

1.6

1.2

0.85

2.1

3.72 0

Sample length )( sLu 0.033 mm6108.3 0.13 0

Nonlinearity 0.43 nm 1 0.43 0.43

Dilatation measurement

=

=12

1

2 )()(i

ic lulu 3.8 0.6