Sinusoidal phase modulating interferometry for surface

profile measurement

Osami Sasaki and Hirokazu Okazaki

We propose a sinusoidal phase modulating (SPM) interferometer in which a reference wave is easily phase

modulated with a sinusoidally vibrating mirror. The surface profile is derived from the frequency compo-

nents of the detected signal. In experiments, CCD image sensors are used to detect the time-varying intensity

distribution produced by interference. The surface profiles of diamond-turned aluminum disks and gauge

blocks were measured. A measurement accuracy of 0.5-1.0 nm can be obtained with SPM interferometry.

I. Introduction

Improvement of the measurement accuracy in opti-cal interferometry is essential if it is to be used in themeasurement of optical surface profiles. Among sev-eral new methods developed, heterodyne interferome-tryl and fringe scanning interferometry used in a digi-tal wave front measuring interferometer2 are usefultechniques to achieve a measurement accuracy of theorder of 1 nm. In heterodyne interferometry, howev-er, a device which shifts the optical frequency is re-quired. As a frequency shifter, rotating quarterwaveplates, 1 rotating gratings, 3 Bragg cells,4 and a beam-splitter plate moving at a constant speed5 were used.A Zeeman split He-Ne laser was utilized in anotherheterodyne interferometer. 6 In fringe scanning inter-ferometry, the reference mirror must be displaced withan accuracy of about one-tenth of the wavelength. Incontrast to the phase modulations required in hetero-dyne interferometry and fringe scanning interferome-try, sinusoidal phase modulation using a sinusoidallyvibrating mirror is very simple.

In this paper we propose an interferometry using asinusoidal phase-modulated reference wave called si-nusoidal phase modulating (SPM) interferometry.The principle of SPM interferometry is described, andthen the most suitable frequency, amplitude, andphase of the sinusoidal phase modulation are given.In experiments, a CCD image sensor is used to detect atime-varying intensity distribution produced by inter-ference. The measured results for the surface profiles

The authors are with Niigata University, Faculty of Engineering,8050 Ikarashi 2, Niigata-shi, Japan.

Received 5 December 1985.0003-6935/86/183137-04$02.00/0.© 1986 Optical Society of America.

of diamond-turned aluminum disks and gauge blocksshow the usefulness of the SPM interferometer.

11. Principle

A. Signal Detected in a SPM Interferometer

Figure 1 shows a SPM interferometer. The refer-ence wave is phase modulated with a sinusoidally vi-brating mirror attached to a piezoelectric transducer(PZT). The vibration of the mirror is represented by

A(t) = a cos(wct + 6). (1)

The surface of an object is imaged onto a photodetec-tor. The distribution of the phase difference betweenthe object and reference waves on the photodetector isdenoted by a. The roughness of the object's surface isgiven by (X/4-r)a, where X is the wavelength of light.We consider the multiplicative noise nM(t) which rep-resents the fluctuation of phase a in time caused by thevibration of the optical devices; it has an effect on thedetermination of the frequency of sinusoidal phasemodulation. Then the ac component of the signaldetected with a photodetector is written as

s(t) = so cos[z cos(wct + 0) + a + nM(t)I, (2)

where z = (4-r/X)a. The Fourier transform of Eq. (2) isgiven by

F(w) = 5fs(t)}

= (1/2) (- 1)m K 2m[N(w - 2mo,) exp(ja)m=-+

+ N* (-co + 2mxc) exp(-ja)]

+ (1/2j) Z (-1) m K 2mi1JN[ - (2m - 1)wcI exp(ja)m=-

- N*[-w + (2m - 1)w,] exp(-ja)},

where

(3)

15 September 1986 / Vol. 25, No. 18 / APPLIED OPTICS 3137

a cos (ct+ 6) 1

LaserLens

Mask

CCD image sensor

10

0.1

0.01

Fig. 1. Sinusoidal phase modulating interferometer. Signal s(t) isdetected with a CCD image sensor.

Km = J,,,f(z) exp(jm0),

N(co) = 5rlexpUnM(t)]J,

Jlto is the I ml th order Bessel function, and * indicates acomplex conjugate. Assuming that the following con-dition is satisfied:

N(w) = 0 H c, (4)

we obtain

F(2mc) = (-lWlINsO cos(a + 3)J27,,4 (z) exp(j2m0),

F[(2m - 1)wj = (-l)'|Nlso sin(a + 3)Ji2 m.-.(z)

X expU(2m - 1)0] m = 0,+1,+2,..., (5)

where N(O) = IN exp(j3).

B. Signal Processing to Obtain Phase aIf we know the values of z and 0, we can obtain a + 3

from F(w,) and F(2wC) using the relationf-1 - l [JF(,o,)1JJl(z)J] sgn-Re[F(.,)J(z) cosol

[IF(2w)I/iJ2(z)JI sgnJ-Re[F(2co,)JJ 2 (z) cos20 )

where

sgnJXJ = I X>0f= 1 X < 0,

and Re y] is the real part of y. Phase A is the measure-ment error in phase a, and its statistical characteristicscan be made clear by the theoretical analysis which willbe reported in another paper. 7

The value of z can be determined from the relationR = IF(3Coc)I/IF(.,)J = 1J3(z)/4Ji(z)J, (7)

if the value of z exists in a limited region, for example, z< 3.5, as shown in Fig. 2.

The value of 0 is the argument of F(cv,), so thatargJ-F()} = arglJl(z) exp(j0) sinaj

to a>0, (8)t + r a < (8

where J(z) is assumed to be positive. Although thevalue of 0 is uncertain depending on the sign of a, thisuncertainty can be eliminated by measuring the knownsurface profile of the object. If the measured value of

0 2 4 6

Fig. 2. Relationship between amplitude z and the ratio R = IJ3(z)/J(Z)

is different from the real value by r, the measuredsurface profile becomes concave for the convex surfaceof the object.

The values of z and 0 can be measured independentof the multiplicative noise in the condition of Eq. (4).

C. Sinusoidal Phase Modulation

We discuss the most suitable frequency, amplitude,and phase of the sinusoidal phase modulation. Thelower limit of the angular frequency c is given by Eq.(4). Phase 0 should be zero or r to determine accurate-ly the signs of sina and cosa in Eq. (6) even if there is anerror in the measured value of 0. It is expected thatphase a can be most accurately measured at the mini-mum value of z which satisfies the relation of J(z) =J 2 (z). This relation leads to the most suitable value z= 2.63. By theoretically analyzing the effect of theadditive noise contained in signal s(t), it can be provedthat z = 2.63 is the most suitable amplitude.7

Ill. Experiments

A. Signal Detection with CCD Image SensorsSignal s(t) is detected with a CCD image sensor.

Detection of a time-varying intensity distribution withCCD image sensors was reported in Ref. 8. The sam-pled data of the output of a CCD image sensor are givenby

g(mAt) = I s(t + mAt)dt m = 0,1. M- 1, (9)J-TF/2

where T is the period of charge storage, At is thesampling interval, and M is the number of samplepoints. Representing the discrete Fourier transformof s(mAt) and g(mAt) with F(lAf) and G(lAf), respec-tively we have the relation

F(lAf) = G(lAf)[7rlTa/MAt)/sin(-7rlTa/MAt)], (10)

where Af = 1/MAt. The vibration frequency of thereference mirror or the frequency of the sinusoidalphase modulation wc/27r is 220 Hz. The sampling in-terval At, which is equal to the period of the chargestorage T is (2i7ric)/16 = 0.248 ms, and the number ofsample points M is 128.

B. Measurements of 1-D Surface ProfilesThe object is a diamond-turned aluminum surface.

A 1-D CCD image sensor operated with a clock pulse of-290 kHz is employed. The forty elements of the

3138 APPLIED OPTICS / Vol. 25, No. 18 / 15 September 1986

lPZT 100

.: 1

-. s\ I

2

n,0

.1440

20

CE 10

-1 0

-in

880 1 320

1Af (Hz)

Fig. 3. Amplitude of discrete Fourier transform of signal s(mAt)

obtained from the output g(mAt) of the CCD image sensor.

- .v Ds 1t00 200Distance on Surface (pLm)

Fig. 6. Surface profile of a diamond-turned aluminum disk mea-sured with a Talystep instrument.

c100

-10

0 100 200Distance on Surface (am)

300

Fig. 4. Tilt and aberration of the reference wave front measured by

selecting the light near the optical axis at the focal plane of the lens. Fig. 7. Surface profile of a gauge block measured using a 2-D CCD

20

E

10

-10

E 4

0

-4

0 100 200 300

Distance on Surface (pm)

Fig. 5. Surface profile of a diamond-turned aluminum disk ob-

tained by subtracting the profile of Fig. 4 from the raw measuredprofile.

Distance on Surface x (pm)

(a)

4.5

photodetectors in the CCD image sensor are used todetect the time-varying intensity distribution of thelight. The remaining elements are covered with blackpaper to receive no light. The size of the photodetec-tor of the CCD image sensor is 9 X 14 gim, and thephotodetectors are arranged at 14-,gm intervals. Theimage of the surface of an object is formed on the CCDimage sensor with a magnification of 1.79, so that thespatial interval of the measured points is 7.8 gum andthe lateral resolution is -5 ,gm. The values of z and 0

are 2.63 and or, respectively. The sampled data of M =128 for one measured point are processed in a micro-computer to obtain phase a or roughness r = (X/4ir)a.

The amplitude of F(lAf) obtained from g(mAt) isshown in Fig. 3. This figure indicates that the condi-tion about the relationship between the frequency ofthe sinusoidal phase modulation and the multiplica-tive noise, given by Eq. (4), is satisfied. The tilt andaberration of the reference wave were measured byselecting the light near the optical axis at the focalplane of the lens. The result is shown in Fig. 4. Figure5 shows the surface profile of the object which is ob-

100 200

Distance on Surface y (pm)

(b)

Fig. 8. Surface profiles along the lines directed toward (a) the x axis

and (b) the y axis, taken from Fig. 7.

tained by subtracting the profile shown in Fig. 4 fromthe raw measured profile. Figure 6 shows the surfaceprofile measured with a Talystep instrument whichhas a lateral resolution of 0.2 gim. The two measuredsurface profiles shown in Figs. 5 and 6 indicate thesame properties about the roughness of the diamond-turned surface. The measurements of the same objectwere repeated at intervals of a few minutes. The rmsof the difference between the two measured results was-0.5 nm.

15 September 1986 / Vol. 25, No. 18 / APPLIED OPTICS 3139

,. ..... ... ...... .... .... . ... . . .T--.-- ,

-

..

To

1300

Fig. 9. Measured surface profile of another gauge block whoseroughness is smaller than that of Fig. 7.

C. Measurements of 2-D Surface ProfilesSurface profiles of gauge blocks are measured using

a 2-D CCD image sensor. The 30 X 30 elements of thephotodetectors in the CCD image sensor are used todetect the intensity distribution of the light. Thefrequency of the clock pulse is -8 MHz. The photode-tectors are arranged at intervals of 23.0 and 26.8 gm ina plane. The magnification of the image is 2.875, sothe intervals of the measured points are 8.0 gim in thedirection of the x axis and 9.3gm in the direction of they axis. Figure 7 shows the measured profile of a gaugeblock in which the tilt and the aberration of the surfacecontained in the raw data are eliminated. The surfaceprofiles along some lines directed toward the x axis andthey axis, taken from Fig. 7, are shown in Figs. 8(a) and(b), respectively. The measured surface profile of an-other gauge block having a smaller roughness is shownin Fig. 9. The repeatability in these measurementswas -0.7 nm.

IV. Conclusion

We have proposed a SPM interferometer in whichthe reference wave is easily phase modulated with asinusoidally vibrating mirror. The most suitable fre-quency, amplitude, and phase of the sinusoidal phase

modulation are given. In the experiments, CCD imagesensors were used to detect the time-varying intensitydistribution. The repeatability in the measurementsof the surface profiles of diamond-turned aluminumdisks and gauge blocks was -0.5-1.0 nm.References

1. R. Crane, "Interference Phase Measurement," Appl. Opt. 8, 538(1969).

2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A.D. White, and D. J. Brangaccio, "Digital Wavefront MeasuringInterferometer for Testing Optical Surfaces and Lenses," Appl.Opt. 13, 2693 (1974).

3. K. D. Stumpf, "Real-Time Interferometer," Opt. Eng. 18, 648(1979).

4. N. A. Massie, "Real-Time Digital Heterodyne Interferometry: aSystem," Appl. Opt. 19, 154 (1980).

5. B. Bhushan, J. C. Wyant, and C. L. Koliopoulos, "Measurementof Surface Topography of Magnetic Tapes by Mirau Interferome-try," Appl. Opt. 24, 1489 (1985).

6. G. E. Sommargren, "Optical Heterodyne Profilometry," Appl.Opt. 20, 610 (1981).

7. 0. Sasaki and H. Okazaki, "Analysis of Measurement Accuracy inSinusoidal Phase Modulating Interferometry," Appl. Opt. 25,000(1986), same issue.

8. 0. Sasaki and H. Okazaki, "Detection of Time-Varying IntensityDistribution with CCD Image Sensors," Appl. Opt. 24, 2124(1985).

0

3140 APPLIED OPTICS / Vol. 25, No. 18 / 15 September 1986