Differential type of phase-locked laser diode interferometer free from external disturbance

Differential type of phase-locked laser diodeinterferometer free from external disturbance

Takamasa Suzuki, Osami Sasaki, Katsuhiro Higuchi, and Takeo Maruyama

A phase-locked laser diode interferometer with differential detection to eliminate external disturbance isproposed. In this interferometer, the measurements are implemented at two different points at the sametime. The surface profile that contains the disturbance is obtained at the scanned measuring point, andthe disturbance is obtained at the fixed measuring point. The exact profile is obtained by subtracting thelatter from the former. The limitations and characteristics are examined theoretically. The analyticalresults agree well with the experimental results. The repeated measurement accuracy is estimated to be

5 nm in this interferometer.

1. Introduction

Although various kinds of interferometer are used inthe manufacturing field for the inspection of opticalparts or surface profile measurement of the objectand so on, one of the disadvantages of a conventionalinterferometer is its weakness for external distur-bance. Recently, many kinds of interferometer thatuse a laser diode (LD) have been proposed'-3 that usethe tunability of the wavelength of a LD. In suchinterferometers it is easy to eliminate the externaldisturbances by means of feedback control4 5in whichthe tunability of the wavelength is also utilized. Butit is difficult for a phase-locked laser diode (PLLD)interferometer6 7 to introduce feedback control forthe elimination of disturbance because the tunabilityof the wavelength has already been used for theimplementation of phase locking.-10 Another wayto eliminate the disturbance is the incorporation ofdifferential detection for the phase term. In such asystem, the phase terms must be obtained at the sametime. One is obtained at a fixed point and gives theinformation of the disturbance. The other is ob-tained by scanning along the surface of the object, andit contains information about the surface profile ofthe object and the same disturbance such as thatdetected at the fixed point. Then, the surface profileis obtained by subtracting the former from the latter.

The authors are with the Faculty of Engineering, NiigataUniversity, 8050 Ikarashi 2, Niigata-shi 950-21, Japan.

Received 10 September 1991.0003-6935/92/347242-07$05.00/0.© 1992 Optical Society of America.

Since the PLLD interferometer incorporates a dis-crete-time-control system, phase locking is achievedsimultaneously at different measuring points in realtime with the time-sharing control. That is to say,simultaneous measurement at different points can beachieved.

In this paper we propose a differential type of PLLDinterferometer and examine the characteristics andthe measurement accuracy. It is shown that thesurface profile can be measured with high accuracyeven if external disturbance exists.

2. Principle

Phase detections at fixed measuring point p andscanned measuring point x are achieved by PLLDinterferometry. The setup of the PLLD interferom-eter is shown in Fig. 1.The Twyman-Green inter-ferometer is used as an optical system in which theoptical path difference between the two arms is 2Do.The LD is modulated sinusoidallyll-13 by the modula-tion current

Im(t) = a cos wet. (1)

The dc bias current Io, modulation current Im(t), andcontrol currents I(t, p) and I(t, x) are injected intothe LD by the LD modulator LM. I(t, p) and I(t, x)are obtained for fixed measuring point p and scannedmeasuring point x, respectively. Io determines thecentral wavelength X0 of the LD. The wavelength ofthe LD is changed by M3Im, where fB is the modulationefficiency of the LD and Iin is the injection current.The interference signals for pointsp and x are imaged

7242 APPLIED OPTICS / Vol. 31, No. 34 / 1 December 1992

EiT M

1 - A Z JUJ~I b

L2

t g /\ ~~Maskyi (tp) CCD image sensorYj (t )

Fig. 1. Experimental setup of a differential-type PLLD interfeieter: M, mirror; BS, beam splitter; L's, lenses; LD, laser diLM, laser diode modulator; FBC, feedback controller.

onto the charge-coupled device (CCD) image senso:follows 13 :

S(t, x) = S + S cos[z cos oct + CL(x) + (t)],

S(t, ) = S2 + Sp cos[z cos ot + (p) + (t)],

where z = (47r/O 2 )af3Do,

a(x) + 8(t) = 41Tr/[Xo + Xc(x)]1[Do + D(x) + d(t)],

ot(p) + 8(t) = 4Tr/[Xo + X,(p)]}[Do + D(p) + d(t)],

S1 and S2 are the dc components, and Sx and Spthe amplitudes of the ac components. The surfprofile of the object and the external disturbancerepresented by D(x) and d(t), respectively. Dtakes a constant value. (x) and a(p) are the phterm determined by D(x) and D(p). 8(t) is alsophase term determined by the disturbance d (t).the phases represented in Eqs. (4) and (5) are loc]to specified value a(x0) independently, the surfprofiles that contain external disturbance aretained as follows6'7:

D(x) + d(t) = (Do/Ao)II(t, x),

D(p) + d(t) = (DO/XO)I(t,p).

By subtracting Eq. (7) from Eq. (6), we can obtain,difference of the surface profile as

D(x) - D(p) = (Do/Xo)P[I(t, x) - I(t, p)]

= Kj[I(t, x) - I(t, p)],

where K = (DO/X0)P. Then the disturbance ismoved from the measurement results. Since D(pa constant, Eq. (8) represents the exact surface proof the object.

The PLLD interferometer is equipped with a feback controller (FBC). In Section 3 we explain Ioperation of the FBC in detail.

3. Operation of the Feedback Controller

A block diagram of the FBC is shown in Fig. 2.consists of a feedback signal generator and proportiial-integral (PI) controllers. The feedback signal gi

DCX) +o - -t x)

.,4[ Y _ (tx)

................................................................................................... ;...........................................

W f~~~~N

to LM ) . ~ P1

SW.....1. .........................

D (x)_ ~ r

Fig. 2. Block diagram of the FBC: SH's, sample-and-hold cir-cuits; PI's, proportional-integral controllers; SW, switch; AMP,amplifier; int, integrator.

erator uses four sample-and-hold (SH) circuits. Theoperation of these SH circuits is explained schemati-cally in Fig. 3. The modulation current I(t) isshown in Fig. 3(a). The interference signals S(t, p)and S(t, x), which are imaged onto the CCD imagesensor, are shown in Fig. 3(b). They are obtainedevery second period of Im(t) for the fixed measuringpointp and the scanned measuring point x, alternately.The signals yi show the outputs of the CCD imagesensor. Since the output of the CCD image sensor isproportional to the time integration of the incidentlight, the hatching at yi means that yi is an integralvalue. To simplify the explanation, we take note ofthe interference signal S(t, p) that was obtained forfixed point p. Since the CCD image sensor is readout synchronously with the modulation current I(t),the integral time of y1(t,p) lies between -T/4 andT/4. The SHI samples y1(t, p) at t = T/4 and holdsit during the period T as shown in Fig. 3(c), whereT = 2 /(oc. In the same way, the SH2 samples andholds 2(t, p) during the period T as shown in Fig.

(b)S(t)

(c)SHI._

[AT . '\ 3T 4 7/

S(t'p) . S( 'X)

t' l 2 t t ) \ \, ,

\7T 11T\137 1

i Y 1(t, p) Y 1 (t,x) !

(d)SH2__l 1 Y2 (t'p) 1Y2(ttx)

* ; I z Y~~~~Y(t, )i (e)SH3 j~=]H--,z>,,>,lD|4l~li]l'RE=

(g)CNT

Fig. 3. Generation of the feedback signals: (a) modulation cur-rent, (b) interference signals, time chart of the sample-and-hold(SH) circuits, (c) SH1, (d) SH2, (e) SH3, (f) SH4, (g) control (CNT)signal for switch SW.

1 December 1992 / Vol. 31, No. 34 / APPLIED OPTICS 7243

(7) (a), M MA A A A

- y

rnk S .rt

J,

3(d). The segments under yj's show the mainte-nance period of its signal. By subtracting y2(t, p)from y1(t, p) at t = 3T/4, the feedback signal for fixedpointp is obtained as follows7:

Y(t, p) = yl(t, p) - y2(t, P)

= Krj sin[cx(p) + 8(t)], (9)

where Krj is the amplitude of the feedback signal.The signal Y(t, p) is held between 3T/4 and 11T/4 bySH3 as shown in Fig. 3(e). Also the segment underY(t, p) shows the maintenance period of Y(t, p). Thegeneration process of the feedback signal for measur-ingpointx is the same as that of fixed pointp. Then,the feedback signal for measuring point x is obtainedas

Y(t, x) = y1(t, x),- y 2 (t, x)

= Kr2 sin[a(x) + (t)],

a Laplace operation. Since D(p) is a constant, wetake note of only the disturbance d(t). The controlsystem for the scanned measuring point x is also thesame as that shown in Fig. 4. The transfer functionsGh(s), Ge(S), and G8(s) are SH circuit, PI controller,and the amplitude of the feedback signal Y(t, p),respectively. They are represented by

Gh(s) = [1 - exp(-Tcs)]Is,

Ge(s) = Kd[Kp + (Tls)],

G8(s) = K 1 ,

(11)

(12)

(13)

where T = 2T and K = Kri = Kr2. The transferfunctions Ga(s) and Gd(s) convert the control currentI,(t, p) and the disturbance d(t) to the phases au(t, p)and 5(t), respectively. They are represented by

Ga(s) = 4rrPDo/,o 2 ,

Gd(s) = 4rrlAo.(10)

where Kr2 is the amplitude f the feedback signal.Y(t, x) is held by SH4 between 7T/4 and 15T/4 asshown in Fig. (f). Thus the feedback signals areobtained by overlapping the signals that are then fedinto PI contTollers as shown in Fig. 2. The PIcontrollers are the same as those described in Ref. 7.The proportional gain Kp and the integral time T inPI controller 1 (PI1) are set to be the same as those ofPI contrqller 2 (I2). Thelr output is alternatelyintroduced to the LD through the LM each T periodby using the switch SW. The control sequence of SWis shown in Fig. 3(g). The output of PI1 and PI2 isfed to the LM when the control signal CNT is highand low, respectively. Since the feedback signals Ydo not change until' the next sampling point, the twofeedback loops act independently. Consequently, thesampling period TC of each feedback loop becomes 2T.The control currents I,(t, p) and I,(t, x) are introducedto the differential amplifier whose gain is Kc Thenthe exact surface profile can be obtained in real timeby Eq. (8).

4. Analysis of the Elimination of Disturbance

Now we discuss analytically the limitation of theelimination of disturbancein our system. Althoughthere are two feedback loops in the system, the blockdiagram of the control system for the fixed measuringpointp is shown in Fig. 4, where variable s represents

(14)

(15)

We suppose that the disturbance d(t) is sinusoidaland analyze the system by using the z transform.The z transform of the control current I,(t, p) is givenby 7

(16)Ip(z) = E(z)Y{Gh(s)Gc(s)1,

where

E(z) = -4Gd(s)Gs(s)D(s))/

[1 +Z[Gs(s)Gh(s)Gc(s)Ga(s)], (17)

and D(s) = jd(t)). SY f(t)) and Y{f(s)) denote theLaplace transform of f(t) and the z transform of f(s),respectively. If the disturbance d(t) is given by

d(t) = v sin od(t), (18)

the z transform of Eq. (18) is represented by

D(z) vz sin (WdT

Z) - 2z cos (odT, + 1(19)

Putting KdGa(s) = K2 and Gd(s) = K3, we can repre-sent Ip(z) by

I _( ) _ b1z + bo D(z) (20)a1z + aO

1 J-

Fig. 4. Block diagram of the discrete-time-control system for the scanned measuring point x.


A_ dX (TC)

l T !dp(kTc)

Fig. 5. Series of the measured data lined up along the evolution of the disturbance d (t).

where

a, = (1 + KlK2Kp)T,2,

ao = TI(TCKK 2 - TI(1 - KK2Kp))

b = KlK3TIKdKpTI, bo = KlK3 TIKd (Tc - KpT,).

By taking the inverse z transform of Ip(z), we canobtain the series of the control current Ip(kT,) for thefixed measuring points at each sampling point. Themeasured error dp(kT,) caused by the disturbance d(t)is calculated by

dp(kT) = (Do/Xo)PIp(kT).

5. Experiments

A. Experimental Setup and Control ParametersThe experimental setup is shown in Fig. 1. A 5-mWGaAlAs LD is used as the light source. The centraloperating wavelength Xo is 790 nm, and the modula-tion efficiency p is 6 x 10-3 nm/mA. The opticalpath difference 2Do was set to 100 mm. The fre-quency of phase modulation lIT was 7 kHz, which islimited mainly by the acquisition time of SHI andSH2. Their acquisition times were 0.5 jis. Themeasured settling time of the system was 2 ms.The size of the photodetector of the CCD image

\ ..(21)

Since the sampling in the feedback loop for point x isdelayed just T/2 compared with that in the feedbackloop for point p as shown in Fig. 3, for the analysis ofpoint x it is necessary to use the modified z transformas follows:

D(z, 1/2) =v(z + l)sin[Wd(T,/2)]Z2- 2z cos owdT, + 1

5

-,

SJ

(22)

The z transform of the control current and themeasured error for the scanned measuring point x aregiven by

IX(z) = - + °D(z, 1/2),a1z + ao

dx(kT,) = (D0/Xo)F3Ix(kT,),

(23)

(24)

respectively. Since the series of obtained data islined up along the evolution of the disturbance d(t) asshown in Fig. 5, by using Eqs. (21) and (24) wecalculated the differential error ed(kT) by

ed[(2 n - 1)T] = (dp(nTc) - fdx[(n - 1)TJ]+ d(nTc)1/2),

ed(2nT) = (dp(nTc) + dp[(n + )Tc]1/2

- d.(nTc)),

0 20 40 60 80t (msec)('a)

60

40°

20

- -2 0

-40

-60

E

-,

S

(25)

where n = 1, 2, 3, .... The characteristic of theelimination of disturbance can be examined by usingEq. (25). If the disturbance can be eliminated com-pletely, ed(kT) becomes- all zero.

-1 OL0

0 20 40 60 80t (msec)

(b)

20 40 60 80t (msec)(c)

100

100

100

Fig. 6. Disturbances measured at (a) point p, (b) point x, (c) theoutput of the differential amplifier.


dp(TC)

Idp[(k+l)TC] 'II

sensor is 9 x 14 pim, and the photodetectors arearranged at intervals of 14 [um. The 50 elements ofthe photodetectors in the CCD image sensor wereused to detect the interference signal. The remain-ing elements were covered with black paper to elimi-nate the useless light. The image of the surface ofthe object was formed onto the CCD image sensorwith a magnification of 3.0, so the spatial interval ofthe measuring points was 4.7 ,m.

The differential gain Kd, the proportional gain Kp,and the integral time T. of the PI controller were3.3 x 10-5, 2.0, and 0.1 ms, respectively. The sam-pling time T, = 2T was 0.29 ms.

B. Characteristics of the Elimination of Disturbance

It is considered that the limitation of the eliminationof disturbance depends on the amplitude and thefrequency of the disturbance. Since the amplitudeK1 of the feedback signal depends on the reflectivityon the surface of the object, it is also expected that theratio of the reflection at the measuring points affectsthe limitation of the elimination of disturbance. Toclarify these points, we replaced the object by a mirrormounted on the piezoelectric transducer (PZT) in the

experimental setup shown in Fig. 1. The limitationof the elimination of disturbance was examined byvibrating the mirror on the PZT with the sinusoidalsignal given in Eq. (18). Both measuring pointswere fixed. The phase change introduced by thePZT is regarded as the disturbance. The remainingerror that could not be eliminated was estimated bycalculating its rms value E.

Figures 6(a) and 6(b) show the disturbances mea-sured at points x and p, respectively. Amplitude vand the frequency fd = (Jd/ 2 Tr of the disturbance were50 nm and 100 Hz, respectively. The remainingerror or the difference between these disturbances isshown in Fig. 6(c). The sinusoidal vibration was stillobserved after differential detection. But its ampli-tude became - 1/10. The rms values E of the re-maining error were measured for various conditionsas shown in Fig. 7. The plotted data were obtainedexperimentally. The solid and dashed curves werecalculated theoretically for various modulation fre-quencies by using Eq. (25). Figure 7(a) shows E forvarious amplitudes of the disturbance. Frequencyfdwas 100 Hz. Error E is proportional to amplitude vof the disturbance. Although the error could not be

20

215

W

10

5

0

10

0 50 100V

(a)

_8 -EC I

6 _

4 .

2

101

25

20

15

10

5

0

102

(b)

100SE

50

0

+

-;x -50

-100150Onm)

100

C 50

0

xcn -50

103(Hz)

0 0.5 1.0 1.5 2.0R

(c)

Fig. 7. Dependence of the rms value E of the remaining error on(a) the amplitude of the disturbance, (b) the frequency of thedisturbance, (c) the reflective ratio R. 0, Experimental plots;solid and dashed curves, theoretical calculations.

-1 nn0

100

C 50

0

n -50

-100

0 50 100 150 200x (urm)(a)

C

250

50 100 150 200 250x(b)

(lpm)

0 50 100 1 50 200x (pm)(c)

250

Fig. 8. Surface profiles measured with (a) a Talystep instrument,(b) a conventional PLLD interferometer, (c) differential-type PLLDinterferometer.


1/T=7 kHz00 \10 kHz _

~~~ \ ~~~.

.- � I . . . I - . . . -1/T=7 kHz

0

__ I- 10 kHz

'_ 0... 20 kHz :

. . . . . . . . I . . .

- . . . . . . . . . . . . . . ... . . . . . .

I . I . I , I I . I . . . . . . . . . I - - - - .

200

eliminated completely, the reduced rate for error Egets large according to the increase of modulationfrequency. Figure 7(b) shows error E for variousfd's. Amplitude v of the disturbance was 50 nm.The values calculated from Eq. (25) agree well withthe measured results. Error E gets large accordingto the increase of the frequency until fd 400 Hz.But it decreases at the region of fd > 400 Hz. It isthought that the control system cannot follow thedisturbance for fd > 400 Hz because the settling timeis 2 ms. Error E was measured by changing thegain Kr1. The reflective ratio R is defined as

R = KrI/Kr 2, (26)

where Kr2 is a constant. Amplitude v and the fre-quencyfd were 50 nm and 100 Hz, respectively. Themeasured results are shown in Fig. 7(c), where errorE does not have a minimum value at R = 1.0 becauseof the discrepancy in sampling time and the detectedamplitude of the disturbance. But error E is below- 5 nm near R = 1.0.

Since the timing of the sampling at the fixed

E

+

x0;

40

20

0

-20

-40

40

2

rn

xE

20

0

-20

-40

40

20

0

-20

-40

0 50 100 150 200x (lpm)(a)

0 50 100 150 200x (pm)(b)

0 50 100 150 200x (urm)(c)

measuring point p is different from that at thescanned measuring point x, it is difficult to eliminatethe disturbance completely. But if the interval timebetween the sampling at point x and that at point pwould be reduced by heightening the modulationfrequency, the disturbance could be eliminated muchmore.

C. Measurements of the Surface ProfileWe show examples of the measurement of the surfaceprofile. The objects are diamond-turned aluminumdisks.

First, the disk whose roughness and cutting pitchwere 100 nm and 50 pm, respectively, was mea-sured. Figure 8(a) shows the surface profile mea-sured with a Talystep instrument. It shows a peri-odic structure determined by the cutting conditions.The surface profile measured at the scanned measur-ing point x is shown in Fig. 8(b). It is equivalent tothe surface profile measured with a conventionalPLLD interferometer. Since it contains the externaldisturbance, the shape is different from that shown inFig. 8(a). Figure 8(c) shows the surface profile that

40

E

20

+- -20

-40250

40

a 20

0

'5-20

-40250

40

20

0

x0

250

Fig. 9. Experimental results: (a) surface profile that containsthe external disturbance measured at the scanned measuring pointx, (b) external disturbance measured at the fixed measuring pointp, (c) exact surface profile obtained by differential detection.

-20

-40

0 50 100 150 200x (p1m)

(a)

0 50 100 150x (Pm)

(b)

200

250

250

0 50 100 150 200 250x (urm)(c)

Fig. 10. Experimental results obtained after a few minutes byusing the same conditions as in Fig. 9; (a), (b), and (c) correspond toFigs. 9(a), 9(b), and 9(c), respectively.


. . . ' ' . . . 1 1 . . . . I . . . .

I . . . . . . . . . . .

I ' . . . . . ' ' . .

.... I................. _I

-

was measured with a differential-type PLLD inter-ferometer. The measuring points on the surface ofthe object are different from those in Fig. 8(a). Theshapes shown in Figs. 8(a) and 8(c) agree well witheach other.

Next, we measured another disk. Its roughnesswas 40 nm and the cutting pitch was 35 pum.The surface profile measured at the scanned measur-ing point x and the external disturbance measured atthe fixed measuring point p are shown in Figs. 9(a)and 9(b), respectively. The surface profile measuredwith differential detection is shown in Fig. 9(c). Itwas obtained by subtracting Fig. 9(b) from Fig. 9(a)with a differential amplifier. The periodic structurewhose period is 35 pum is observed in Fig. 9(c).The same disk was measured after a few minutes; theresults are shown in Fig. 10. The surface profilethat contains the external disturbance is shown inFig. 10(a). Its shape is different from that in Fig.9(a) for the sake of the disturbance shown in Fig.10(b). The rms value of the difference between thetwo measured surface profiles shown in Figs. 9(a) and10(a) is 20 nm. The surface profile measured withthe subtraction of the disturbance is shown in Fig.10(c). The shapes shown in Figs. 9(c) and 10(c)agree well, and the repeated rms measurement accu-racy is - 5 nm.

6. Conclusions

A modified-type PLLD interferometer or differential-type PLLD interferometer has been described. Ithas been shown that the external disturbance can beeliminated with differential detection in this inter-ferometer. The limitation for the elimination ofdisturbance is examined theoretically by using the ztransform. The analytical results agree well withthe experimental results. The limitation depends onthe amplitude and the frequency of the disturbanceand the reflective ratio of the surface of the object.The rms repeated measurement accuracy could bereduced from -20 to -5 nm in this interferometer.

Moreover, the rms value E of the remaining error canbe reduced much more by heightening the modula-tion frequency.

References1. K. Tatsuno and Y. Tsunoda, "Diode laser direct modulation

heterodyne interferometer," Appl. Opt. 26, 37-40 (1987).2. Y. Ishii, J. Chen, and K. Murata, "Digital phase-measuring

interferometry with a tunable laser diode," Opt. Lett. 12,233-235 (1987).

3. J. Chen, Y. Ishii, and K. Murata, "Heterodyne interferometrywith a frequency-modulated laser diode," Appl. Opt. 27,124-128 (1988).

4. T. Yoshino, M. Nara, S. Mnatzakanian, B. S. Lee, and T. C.

Strand, "Laser diode feedback interferometer for stabilizationand displacement measurements," Appl. Opt. 26, 892-897(1987).

5. 0. Sasaki, K. Takahashi, and T. Suzuki, "Sinusoidal phasemodulating laser diode interferometer with a feedback controlsystem to eliminate external disturbance," Opt. Eng. 29,1511-1515 (1990).

6. T. Suzuki, 0. Sasaki, and T. Maruyama, "Phase locked laserdiode interferometry for surface profile measurement," Appl.Opt. 28, 4407-4410 (1989).

7. T. Suzuki, 0. Sasaki, K. Higuchi, and T. Maruyama, "Phase-locked laser diode interferometer: high-speed feedback con-

trol system," Appl. Opt. 30, 3622-3626 (1991).8. D. T. Moore, R. P. Murray, and F. B. Neves, "Large aperture ac

interferometer for optical testing," Appl. Opt. 17, 3959-3963(1978).

9. G. W. Johnson, D. C. Leiner, and D. T. Moore, "Phase-locked

interferometry," Opt. Eng. 18,46-52 (1979).10. H. J. Matthews, D. K. Hamilton, and C. J. R. Sheppard,

"Surface profiling by phase-locked interferometry," Appl. Opt.25, 2372-2374 (1986).

11. 0. Sasaki and H. Okazaki, "Sinusoidal phase modulatinginterferometry for surface profile measurement," Appl. Opt.25,3137-3140 (1986).

12. 0. Sasaki and H. Okazaki, "Analysis of measurement accuracyin sinusoidal phase modulating interferometry," Appl. Opt.25, 3152-3158 (1986).

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Differential type of phase-locked laser diode interferometer free from external disturbance