Absolute displacement measurements using modulation of thespectrum of white light in a Michelson interferometer

L. Montgomery Smith and Chris C. Dobson

A technique is presented for measuring small displacements by observing the frequency of spectral modula-tion of white light in a Michelson interferometer. An experiment is described in which the step size of a

stepper-motor-driven translation stage was measured by recording the spectrum of light output from aninterferometer and performing a cross-correlation calculation with theoretical spectra. Measurements made

using standard laboratory-quality optical equipment were accurate to within -10 nm for a range of over 100

/M.

1. Introduction

Since its invention in the 1880s, the Michelson inter-ferometer has been used to measure small spatial dis-placements. Standard laboratory experiments dem-onstrate methods for making such measurementseither by fringe counting or by zero path length deter-mination with white lightfringes.1 Laser illuminationof Michelson interferometers has been used to mea-sure small vibrational displacements2 3and other rapidmotions.4 More recently, white light Michelson inter-ferometry has been employed to study thicknesses andindices of refraction of thin films.5

These techniques have achieved accuracies of lessthan one wavelength of light over displacements rang-ing from one wavelength up to several hundred wave-lengths, however, measurement of displacements overa similar range with an accuracy of much less than onewavelength is usually not considered possible withsuch a simple apparatus. Fabry-Perot interferome-ters are capable of extremely accurate displacementmeasurements but are limited in their range and re-quire high-quality components and careful alignment.

However, a curious property of the spectrum of lightemerging from a Michelson interferometer can be ex-ploited to make accurate measurements of absolutespatial displacements over an extended range. Thepower spectrum of light resulting from the superposi-

The authors are with the University of Tennessee Space Institute,Center for Laser Applications, Tullahoma, Tennessee 37388.

Received 24 July 1988.0003-6935/89/163339-04$2.00/0.© 1989 Optical Society of America.

tion of the beams from the reference and variable pathlength arms is modulated in frequency space at a cycli-cal rate that is quite sensitive to the path length differ-ence. This spectral modulation is mentioned in Good-man6 in the discussion on cross-spectral purity instatistical optics, and the same principle has been ap-plied to electrical signals to measure the speed oflight.7 Vienot et al.

8 have applied this property usingan experimental setup similar to that described in thispaper to study surface roughness of reflectors, howev-er, no absolute displacement measurement was per-formed.

In this paper, an experiment is described that usedthe spectral modulation of white light in a Michelsoninterferometer to measure the small linear displace-ments of a stepper-motor-driven translation stage.For different displacements, spectra were recorded us-ing a spectrometer and optical multichannel analyzer(OMA). The actual displacement measurements weremade by performing a cross-correlation calculationbetween the experimental spectra and theoreticalspectra. The error analysis indicates that this tech-nique can be used to make absolute spatial measure-ments accurate to 10 nm over a range of -100 pim.

II. Theory of Operation

The displacement determination technique pre-sented here is based on measuring the spectrum oflight incident on the output detector of a Michelsoninterferometer. The dependence of the spectrum onthe displacement of one mirror can be calculated in thefollowing way. If the mirror in one arm is displaced asmall distance x, the time-varying electric or magneticfield component incident on the detector is given bythe superposition of the reference beam field u(t) andthe same field delayed by propagating the path lengthdifference 2x as

15 August 1989 / Vol. 28, No. 15 / APPLIED OPTICS 3339

UD(t) = au(t) - bu(t - 2x/c),

where a and b are constants inserted in the equation toaccount for possible differences in relative amplitudesbetween the reference and delayed fields, and c is thevelocity of light. The negative sign on the delayedfield component is due to the 1800 phase shift intro-duced by internal reflection within the beam splitter, aproperty that was present in the interferometer used toperform the experiment described in this paper, butwhich is not necessarily common to all Michelson in-terferometers.9

The energy spectrum of the field is found from theFourier transform of Eq. (1) as

UD(V) = U(v)[a - b exp(i4irxp/c)].

x

Spectrometer(2)

And the power spectrum of the light incident on thedetector is thus given by

GD(V) IUD(V)i2

= U(v)12[a2 + b2- 2ab cos(47rxP/c)]

= (a2 + b2 )G()[1 - v cos(47rxvlc)], (3)

where G(v) is the power spectrum of the input lightsource, and v is the visibility of the spectral modulationdefined by

2aba

2+ b2 (4)

Fig. 1. Schematic of configuration of experimental apparatus.

The multiplicative factor (a2 + b2) can be absorbedinto the power spectrum function and Eq. (3) can bewritten in terms of intensity as a function of wave-length , where I(X) = (a2 + b2)G(c/X), and ID(X) =GD(c/X) as

ID(X) = I(X)[1 - v cos(4irx/X)I. (5)

Equation (5) serves as the basis for measuring thedisplacement distance x. The spectrum of the lightsource I(X), and the relative amplitudes a and b can bemeasured with a spectrometer connected to the outputof the interferometer and blocking first one beam andthen the other. This leaves the displacement x as theonly unknown in the measured intensity spectrum."Knowledge of the form of the measured spectrum canthus be used to obtain an estimate of the displacementx.l

11. Description of the Experiment

This measurement technique has been successfullyapplied to the problem of determining the perfor-mance characteristics of a stepper-motor-driven opti-cal translation stage. This stage was specified by themanufacturer to have a 1 Am travel per 1/400th revolu-tion step and was electronically configured so that itwas capable of making ten microsteps of nominally 100nm each between each step. The major goal of theexperiment discussed in this section was to measurethese microstep displacements over a usable range ofthe translation stage.

The experimental apparatus used in this applicationis shown schematically in Fig. 1. White light from anincandescent bulb was collimated and passed through

a small commercially manufactured Michelson inter-ferometer. The slit of a 0.25-m monochromator wasplaced at the output of the interferometer so that thesignal observed originated from a narrow slice of therecombined beams. The intensity of the dispersedlight was measured with a 1000 pixel detector arrayaffixed to the monochromator and interfaced via theOMA to a microcomputer that was used for data acqui-sition and storage. The system was calibrated forwavelength using a mercury vapor lamp, giving anaccuracy of -1 nm. White light fringes were observedat the monochromator slit and the live spectrum dis-play showed clearly the modulation sequence as themovable mirror of the interferometer was passedthrough zero path difference.

The fixed mirror of the interferometer was removedand mounted on the translation stage and both thestage and the interferometer were bolted to the table,which was air floated. In this way, translation of thestage resulted in a change in path difference and thusspectral modulation, and the movable mirror was ". . .used to provide zero path length difference for anychosen initial stage position."

The spectrum window and the wavelength resolu-tion were functions of the spectrometer dispersion andthe OMA detector array. For the measurements, thegrating was chosen to give a range of from 530 to 830nm, corresponding to a dispersion of 12 nm/mm, as thispermitted the observation of several modulation extre-ma for even near-zero positions and still provided suf-ficient resolution for the large number of extrema asso-ciated with the larger displacements.

3340 APPLIED OPTICS / Vol. 28, No. 15 / 15 August 1989

(1)

_,

40000

INTENSITy

20000

500 600 700 800 9WAVELENGTH (NM)

Fig. 2. Example of unmodulated white light spectrum.

0SS

N 0.4-

1000 1500 2000 2500 3000DI SPLACEMENT (NM)

Fig. 4. Normalized cross-correlation function between theoretical

spectrum and experimental spectrum shown in Fig. 3. Peak value is0.9966 at a displacement of 1977 am.

N0

X10 5

1.

INTENSITY

0.5

500 60 7 0 . . 8

5'00 600 700 800 900WAVELENGTH (NM)

Fig. 3. Example of modulated white light spectrum.

Long exposure times (10 to 30 s) were used to aver-age over temporal variations in the spectrum caused bymechanical vibrations and air turbulence. The mono-chromator slit width was -10 Am and the slit wascovered for a dark count at the start of each set ofmeasurements. Since the zero-path (unmodulated)spectrum was available at any path length by blockingone arm of the interferometer, the acutal interferencespectra were begun at small nonzero arm differences.The stage was driven in a single direction, usually inincrements of one microstep between observations andspectral data were acquired in sets of ten over a 1-Mmrange. The displacement generated by one such mi-crostep was known from the manufacturer to be sub-micron, but the pole configuration of the stepper mo-tor was designed for 1-Arm step sizes, and the individualmicrosteps had not been measured previously.

IV. Experimental Results

A. Data Reduction

The estimate for the displacement was determinedfrom the measured spectrum using the following datareduction method. The spectra ID(X) and I(X) weremeasured at discrete wavelengths Xn over a finite spec-tral interval, n = 1,2,. .. ,N. By computing a sampledtheoretical spectrum as a function of estimated dis-placement x, where

ID(Xn) = (a2 + b2 )I(Xn)[1 - v cos(47r/Xn)*; n = 1,2,.. .,N, (6)

and comparing it with the measured spectrum bymeans of a normalized cross-correlation function givenby

N

EDn) dn)

N N

\/n1l n1l

(7)

an estimate for the displacement was determined asbeing that value of for which r(x) was maximized. Inthis way, all the information within the measured spec-trum was utilized, and the effects of measurementuncertainty were reduced.

This procedure is outlined in Figs. 2-4. Figure 2shows the white light spectrum obtained by blockingone of the arms of the interferometer. (The smallamount of modulation present in this spectrum couldhave arisen from interference in the achromatic colli-mating lens, although this is not certain.) In Fig. 3, anexample of a modulated spectrum resulting from thesuperposition of the reference and delayed fields isshown. The normalized cross-correlation function forthis particular case is shown in Fig. 4, having a peakvalue of r(i) = 0.9966 at x = 1977 nm.

Table I summarizes one set of measurements of dis-placements made with this technique to determine theaccuracy of the nominal 100-nm microstep size of the

Table I. Example Set of Displacement Measurements

Nominal Measured Measureddisplacement displacement microstep

(nm) (nm) (nm)

1000 1049 -1100 1170 1211200 1411 2411300 1613 2021400 1757 1441500 1977 2201600 2114 1371700 2237 1231800 2339 1021900 2444 1052000 2523 79

15 August 1989 / Vol. 28, No. 15 / APPLIED OPTICS 3341

1), i l l , .! l l

n.

r

.u

rX =

n.

translation stage. As can be seen from the data, themeasurements made in this manner indicate that themicrostep size varies widely even over a limited 1 rmregion. Many displacement measurements weremade in this way for displacements ranging from 500nm to over 100 gim. In all cases, a distinct and uniquepeak in the cross-correlation function was found tooccur, indicating that no problem with ambiguity ex-isted.

B. Error Analysis

While a rigorous error determination of the expectedaccuracy of this measurement technique and data re-duction method is quite difficult, the following analy-sis can be used to place approximate error bounds onthe displacement estimates. The detected intensityspectrum ID(X,) can be written

ID(Xn) = Irue)(XA) + en, (8)

where Irue)(Xn) is the true value of the spectral inten-sity incident on the detector, and where en is the mea-surement uncertainty, or noise, corrupting the data ateach discrete wavelength. Each en is assumed to be azero-mean random variable. Similarly, the theoreti-cal spectrum can be written

ID(Xn) = Irue)(Xn) + An, (9)

where 6,, is the error in the calculated spectrum due toerror in the estimated displacement. If the estimateddisplacement is written = x + Ax, where x is the truedisplacement and Ax is the error in the estimate, forsmall Ax, Eq. (5) can be used to approximate thetheoretical spectrum by

ID(X ) Irue)(X ) + 4rvAx I(X) sin(47r!/. (10)

Subtracting Eq. (10) from Eq. (8), squaring bothsides, summing over n, and making the approxima-tions,

N5E2 _ (N )aY2, (11)n=

n= 1

N 47rvAxE X I(Ad sin(47ix/Xn)fn _ ° (12)

result in the following relation:

(N -1)af + (AX)2 X A I2 (Xn) sin2 (47rxlAn)

N

-E [ID(Xn) D(Xn)]2, (13)n=l

from whence it follows thatN

n=1[ID(n) - Dn)2

(Ax)2 < N

Y. (4rv/XAn)2I2(X,,) sin (4,rt/X,,)

Thus, the error in the displacement estimate is approx-imately bounded by a term that depends on the squareerror between the theoretical and experimental spec-tra, the visibility of the modulation, the intensity of thewhite light spectrum, and the sinusoidal modulation ofthe spectrum. Note that this bound is the value for Axthat would result if no noise were present on the dataand the error in displacement were the only source ofdiscrepancy between the theoretical and observedspectra.

Application of Eq. (14) to several sets of data result-ed in Ax being bounded by values ranging from 10 to 15nm. In light of the loose upper bound approximationof Eq. (14), an experimental accuracy of 10 nm appearsto be a justifiable claim for this experiment.

V. Conclusions

In this paper, a simple method for making absolutedisplacement measurements has been presented thatuses the frequency space modulation of the spectrumof white light in a Michelson interferometer. Theexperiment that was performed to test the applicabil-ity of this technique measured the microstep displace-ments of a stepper-motor-driven optical translationstage and was conducted with standard laboratoryquality optical components. Data reduction via crosscorrelation between theoretical and experimentalspectra proved to be a very usable and accurate meth-od for obtaining an estimate of the displacement. Anaccuracy of 10 nm over a range of 100 Aim or moreappears to be achievable in most laboratories andcould be improved with the use of higher quality opti-cal equipment and a more controlled environment.

References

1. C. H. Palmer, Optics Experiments and Demonstrations (JohnsHopkins, Baltimore, 1962), pp. 147-157.

2. H. A. Deferrari and F. A. Andrews, "Laser Interferometric Tech-nique for Measuring Small-Order Vibration Displacements," J.Acoust. Soc. Am. 39, 979-980 (1966).

3. H. A. Deferrari, R-A. Darby, and F. A. Andrews, "VibrationalDisplacement and Mode-Shape Measurement by a Laser Inter-ferometer," J. Acoust. Soc. Am. 42, 982-990 (1967).

4. R. Keller, R. Salath6, T. Tschudi, and C. Vourmard, "MichelsonInterferometer for Detection of Fast Displacements of Less thana Quarter-Wave over Small Areas," Appl. Opt. 14, 1616-1620(1975).

5. P. A. Flourney, R. W. McClure and G. Wyntjes, "White-LightInterferometric Thickness Gauge," Appl. Opt. 11, 1907-1915(1972).

6. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp.191-192.

7. W. P. Alford and A. Gold, "Laboratory Measurements of theVelocity of Light," Am. J. Phys. 56, 481-484 (1958).

8. J-C. Vienot, J-P. Goedgebuer, and A. Lacourt, "Space and TimeVariables in Optics and Holography: Recent ExperimentalAspects," Appl. Opt. 16, 454-461 (1977).

9. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1980), pp. 300-301.

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