Wavelength-multiplexed phase-lockedlaser diode interferometer using a phase-shiftingtechnique

Takamasa Suzuki, Takaya Muto, Osami Sasaki, and Takeo Maruyama

We propose a new signal-processing method for eliminating measurement errors that occur in thewavelength-multiplexed phase-locked laser diode interferometer. The basic idea proposed here is a verysimple but effective way to improve measurement accuracy. With our scheme, the phase in the inter-ference signal is strictly shifted by 2p, which enables us to eliminate measurement errors. The equiv-alent wavelength L is 80 mm, and the measurement accuracy reaches ;Ly600. A step-heightmeasurement was also carried out in the experiment. © 1997 Optical Society of America

Key words: Interferometer, laser diode, phase lock, equivalent wavelength.

1. Introduction

Optical interferometry is useful in the precise mea-surement of microscopic vibration or surface rough-ness, because it is highly sensitivity and does notrequire physical contact. These days, many kinds ofoptical interferometers employing a laser diode ~LD!have been proposed and are in practical use, but themeasurable range of the conventional interferom-eters is limited by phase ambiguity caused by phasewrapping. To solve this problem, we need devicesthat perform well in optical systems or in signal pro-cessing. One such device is a wavelength-multiplexing method for an optical system. Thisidea was proposed in the early days of holographicinterferometry,1 but many kinds of two-wavelengthinterferometers ~TWI’s! have been proposed2–4 sincethen.

Recently, many kinds of LD’s have been developed,and they offer many different wavelengths. There-fore TWI’s that utilize two different LD’s are oftenproposed.5–7 The LD is small in size and has someattractive features, such as wavelength tunabilityand energy efficiency; thus it is suitable for this kindof interferometer. Usually, two different LD’s areused in the TWI to obtain two different wavelengths.

The authors are with Faculty of Engineering, Niigata Univer-sity, 8050 Ikarashi 2, Niigata 950-21, Japan.

Received 3 June 1996; revised manuscript received 17 October1996.

0003-6935y97y256196-06$10.00y0© 1997 Optical Society of America

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This type of TWI, however, is necessary to align theoptical axes of each LD and to stabilize the relativechange in wavelengths.8 To overcome these prob-lems, a TWI with a single LD has been proposed9 inwhich temporally multiplexed wavelengths wereused and the phase was detected by the heterodynemethod by using an acousto-optic Bragg cell.

On the other hand, we proposed a new type of TWIthat used a single LD, and we analyzed its errorsources.10 In that TWI a single LD is oscillated al-ternately with two wavelengths that are slightly dif-ferent. The initial phase in the interference signalcorresponding to each wavelength is locked at eachstable point by using a phase-locked technique.11–13

The error analysis clarified that a major error sourceother than external disturbance induced by unknownvibration was offset voltage caused by the total offsetvoltage of electrical devices in the feedback controlsystem.

In this paper, we introduce a new type of signalprocessing for feedback control. It is a simple buteffective way to eliminate influence from the offsetvoltage generated by electrical devices. This TWIenables us to measure absolute distance more accu-rately compared with the TWI that we previouslyproposed.10 The external disturbance is also elimi-nated by feedback control12,14 and a differential de-tection.15 Thus measurement accuracy reaches;Ly600. A step-height measurement was also car-ried out in the experiment.

In Section 2 we describe the construction of thisinterferometer and briefly the principle of wavelength-multiplexed interferometry. A detailed explanation

of eliminating errors introduced by the offset voltage isdescribed in Section 3. The experimental setup andresults and discussion are in Sections 4 and 5, respec-tively.

2. Principle of Wavelength-Multiplexed Interferometry

The configuration of our TWI is shown in Fig. 1. ATwyman–Green interferometer consisting of a LD; alens, L; a beam splitter, BS; mirrors M1 and M2; anda CCD image sensor are used in the optical system.The optical path difference ~OPD! is 2D0~x! in thisinterferometer, where x denotes the coordinate alongthe one-dimensional CCD image sensor. A dc biascurrent I0, a sinusoidal modulating current,

Im~t! 5 a cos~vct 1 u!, (1)

and control current I1 or I2 is injected into the LDwith a laser diode modulator ~LM!. The control cur-rents, I1 and I2, are served alternately from the FBC.The FBC consists of a FBSG; a TRN; two proportional-integral controllers, PI1 and PI2; a SW; a TC; and aVI. The parameter r0 is the desired value of thecontrol system. The central wavelength l0 is deter-mined by I0. If the control current is zero, an inter-ference signal imaged onto the CCD image sensor isgiven by

S~t, x! 5 S1 1 S0 cos@z cos~vct 1 u! 1 a~x!#, (2)

where

z 5 4pabD0yl02 (3)

is the modulation depth and

a~x! 5 4pD0~x!yl0 (4)

Fig. 1. Experimental setup of the wavelength-multiplexed phase-locked LD interferometer by using a phase-shifting technique.Feedback current I1 or I2 generated by the feedback controller~FBC! is injected into the LD through the LD modulator ~LM!.The FBC consists of a feedback signal generator ~FBSG!, a feed-back signal transition controller ~TRN!, two PI controllers, a tim-ing controller ~TC!, a switch ~SW!, and a voltage-to-currentconverter ~VI!. OSC, oscillator.

is the initial phase of the interference signal. S1 andS0 are the dc component and the amplitude of the accomponent, respectively, and b is the modulating ef-ficiency of the LD. If the LD oscillates with differentwavelengths, l1 or l2, the phase given by Eq. ~4!changes slightly as follows:

a1~x! 5 4pD0~x!yl1, (5)

a2~x! 5 4pD0~x!yl2, (6)

where li 5 l0 1 bIi 5 l0 1 bKvVi ~i 5 1, 2!, Vi are theoutputs of PI1 and PI2, Kv is the gain of the VI. Thephase difference between a1~x! and a2~x! is given by

Da~x! 5 4pD0~x!yL, (7)

where

L 5 l1l2y~l1 2 l2! (8)

is the equivalent wavelength. Then absolute dis-tance D0~x! is given by

D0~x! 5L

4pDa~x!. (9)

By using control voltages, we rewrite Eq. ~9! as

D0~x! 5 CDa~x!

DV, (10)

where C ' l02y4pbKV, DV 5 V1 2 V2. Phases a1~x!

and a2~x! are locked at specified values by controllinginjection currents I1 and I2 by a phase-locked tech-nique. Differences Da and DV are then determinedaccurately, and absolute distance D0~x! can be ob-tained.

The CCD image sensor is read out every quarterperiod of the modulation current. Therefore we ob-tain four integral values:

pi~x! 5 *~Ty4!~i21!

~Ty4!i

S~t, x!dt ~i 5 1 , 4!, (11)

where T 5 2pyvc is a period of the modulation cur-rent. Signal pi~x! is sampled and additions and sub-tractions are executed for pi~x! in the FBSG. Thenwe can obtain feedback signals10

Fs@a~x!# 5 p1 1 p2 2 p3 2 p4 5 As sin a~x!, (12)

Fc@a~x!# 5 p1 2 p2 1 p3 2 p4 5 Ac sin a~x!, (13)

where As and Ac are functions of both modulationdepth z and initial phase u of the modulation current.

3. Error-Reduction Method

Figure 2 shows the feedback signals used in our TWI.In the previous method10 we used two feedback sig-nals, F1~a! and F2~a!, whose phase difference was p, asshown in Fig. 2~a!, where coordinate x is omitted tosimplify the following explanations. In this case,feedback signal F1~a! is primitive, because the secondfeedback signal F2~a! is generated by inverting only

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F1~a!. Because the phase was controlled at each sta-ble point, P1 and P3, which are intersections betweenthe ground level and the region of positive inclinationon the feedback signal, phase difference Da was p.This previous method, however, contained one prob-lem or the generation of the measurement error causedby the offset voltage. From a theoretical analysis,10

we clarified that the offset voltage appears only in thefeedback signal sin a by the intensity modulation inthe LD. But some offset voltage was observed exper-imentally also in the feedback signal cos a. We con-cluded that the offset voltage in the feedback signal cosa came from the electrical circuit. If primitive feed-back signal F1~a! contains positive offset voltage VO,the negative offset voltage is contained in the secondfeedback signal F2~a! as shown in Fig. 2~b!. Thenphase difference Da becomes larger than p by 2da,which induces a measurement error.

If the phase of the feedback signal changes period-ically, we can distinguish the offset voltage in theelectric circuit from the required feedback signal, andwe can remove it electrically. But the phase of thefeedback signal is a function of the coordinate on theCCD image sensor, and the feedback signal does notchange periodically. So electrical cancellation of theoffset voltage is impossible.

We proposed here a way to reduce measurementerrors caused by offset voltage. Our idea for errorreduction is very simple: Phase locks are achievedon the same feedback signal F1~a! as shown in Fig.

Fig. 2. Feedback signals in ~a! the ideal case and ~b! the actualcase that contains offset voltage VO. Although the phase differ-ence between P1 and P3 in ~b! becomes larger than that in ~a!, thephase difference between P1 and P2 is the same in both cases.

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2~a!. Then the phase difference Da between the sta-ble points, P1 and P2, becomes 2p. In this case, thephase difference Da is accurately 2p even if the offsetvoltage remains on the feedback signal as shown inFig. 2~b!. Moreover the equivalent wavelength L be-comes half of that in the previous paper, because Dadoubles in Eq. ~9!. When the same feedback signalsare used as F1~a! and F2~a!, however, two phases arelocked at the same stable point, for example, P1.Therefore we must shift one phase to another stablepoint, P2, leaving the other at P1. It is difficult tomove the phase by 2p at one time, without the sta-bility of the control system deteriorating. Thereforethe phase should be moved gradually.

To achieve this process, we use cos a and sin a asprimitive feedback signals. Since there is a phasedifference of py2 between them, we can shift thephase by py2. Such feedback signals are obtainedeasily from the output signal of the CCD image sen-sor10 as shown in Eqs. ~12! and ~13!. The proposedprocedure is shown in Fig. 3. Only changes in feed-back signal F2~a!, while another feedback signalF1~a! is always cos a, are shown.

In the first stage shown in Fig. 3~a!, cos a is com-monly used as feedback signals F1~a! and F2~a!.Then two phases are locked a1 at the same stablepoint, P1, on the feedback signals shown in Fig. 2. Inthe second stage, F2~a! changes from cos a to sin a asshown in Fig. 3~b! and the phase moves to a2. In thethird stage, we use 2cos a as feedback signal F2~a! asshown in Fig. 3~c!. Then the phase is moved grad-ually from a2 to a3. In this way, the phase ischanged by py2 in phase for each step. Finally, inthe fifth stage, F2~a! returns to the initial feedbacksignal cos a as shown in Fig. 3~e!, and the phasesettles at the desired stable point, P2. Therefore thephase difference becomes 2p at the final stage.

4. Experimental Setup

The experimental setup is shown in Fig. 1. The op-tical path difference 2D0 of the Twyman–Green in-

Fig. 3. Phase-shifting process according to the transition of feed-back signal F2~a!. Only the transition of feedback signal F2~a! isshown.

Fig. 4. Block diagram of thefeedback signal transition con-troller TRN. It consists of twoanalog switches, SW1 and SW2;a sign converter; transition-timing control logic; and twosample-hold circuits, SyH1 andSyH2.

terferometer is ;80 mm. A GaAlAs laser diode isused as a light source. Its maximum output powerand central wavelength l0 are 5 mW and 780 nm,respectively. The modulation efficiency b of the LDis 6.8 3 1023 nmymA. The frequency of the phasemodulation vCy2p was set to 2 kHz. The interval ofthe photodetector and the readout frequency of theCCD image sensor are 14 mm and 2 MHz, respec-tively. The TC determines the switchover period ofthe feedback signals, F1~a! and F2~a!. It alternatelychanges the two feedback signals every 20 periods ofthe modulation current, i.e., the feedback signal ischanged every 10 ms. Because the integral time ofthe integrator in the PI controller is 3.9 ms, 10 ms issufficient time for the phase lock to settle. The co-efficient KV is 0.17 mAyV. The desired value r0 wasset to zero to control the phase at the most appropri-ate position at which the inclination or the gain of thefeedback signal is maximum. The control voltagesare sampled by the analog-to-digital converter ~AD!,and D0~x! is calculated from Eq. ~10!. Because thephase difference Da is set to 2p in this experiment,the equivalent wavelength L was calculated as 80mm from Eq. ~9!.

One of the most significant parts of Fig. 1 is a TRNthat controls the transition of the feedback signals.A block diagram of the TRN is shown in Fig. 4. Itconsists of two analog switches, SW1 and SW2; a signconverter; two sample-hold circuits, SyH1 and SyH2;and a transition-timing control logic. The feedbacksignal, F1~a! 5 cos a, is always fed into PI1, whereasthe changing feedback signal, F2~a!, is fed into PI2.One of two switches in SW1 and one of two in SW2are closed to fulfill the sequence shown in Fig. 3.SW1 and SW2 select one feedback signal and the signof the feedback signal, respectively. To obtain feed-back signal sin a in Fig. 3~b!, for example, thetransition-timing control logic switches on SW1b andSW2a. SyH1 holds F1~a! until feedback signal F2~a!is obtained in the FBSG.

5. Experimental Results

First, the control voltages corresponding to Fig. 3 areobserved when small external vibrations are added.They are shown in Fig. 5. The lower traces are ob-tained with F1~a!. In the first stage, the two feed-back signals are the same; they are the cos a’s shownin Fig. 3~a!. The evolutions of two control voltagescompletely overlap each other, as shown in Fig. 5~a!.

When one feedback signal F2~a! is changed from cosa to sin a in the second stage, the difference voltageDV appears between the control voltages as shown inFig. 5~b!.

As one feedback signal is being changed in order,such as from sin a to 2cos a and from 2cos a to 2sina, the difference DV between two control voltagesbecomes large. The increased voltage should be thesame theoretically at each stage, but it is not thesame. This is mainly caused by the offset voltage insin a discussed in Section 3. The feedback signal sin

Fig. 5. Observed control voltages corresponding to the feedbacksignal shown in Fig. 3. The difference DV becomes large as thefeedback signal is changed.

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a theoretically contains an offset component inducedby the intensity modulation10 in the LD, whereas thefeedback signal cos a does not contain such an offsetcomponent.

The fluctuation was observed in DV only in Fig.5~b!. We consider that the offset voltage or the re-maining electrical resistance after the switch isclosed in the FBC in Fig. 1 generated the error volt-age, and it contributed to the fluctuation of DV, be-cause the voltage level of the upper trace was almostzero and the signal-to-noise ratio was not as good.But this fluctuation is not as large, and the followingcontrols in Fig. 5 do not suffer. Besides the errorsfrom this source are eliminated with our method,because the phase-lock is achieved by the same feed-back signal at the final stage. This is not a fatalerror in our system.

DV that appeared at the second stage or in Fig. 5~b!is smaller than expected, indicating that the phasedifference between F1~a! and F2~a! is smaller thanpy2. In this case, the locked phase a2 in Fig. 3~b! isshifted left. Therefore it is obvious that the feed-back signal sin a had a positive offset voltage. Onthe contrary, DV that appeared at the fourth stage orin Fig. 5~d! is larger than expected, indicating thatthe phase difference is larger than py2, and thelocked phase a4 in Fig. 3~d! was shifted right by thenegative offset voltage. Therefore the differencevoltage DV in Fig. 5~d! is not exactly double that inFig. 5~b!.

On the other hand, the difference in voltage DVthat appeared in the final stage or in Fig. 5~e! almostdoubles compared with that in Fig. 5~c!, because thephase difference in the feedback signals in Fig. 5~e! istwice as large as that in Fig. 5~c!. It shows that thefeedback signal cos a does not contain much offsetvoltage. The time required from the initial stage inFig. 5~a! to the final stage Fig. 5~e! is 40 ms.

Next we investigated measurement accuracy bymeasuring the displacement of mirror M2. The mir-ror was mounted on the x-axis stage and moved alongthe optical axis from D0 ; 42 mm to ;47.5 mm indiscrete 0.5-mm increments with a micrometer.Measurements were implemented by keeping thephase difference at 2p. Difference voltage DV wasdetected by subtracting one feedback signal from theother. Then the external disturbance was removedsufficiently. The measured result is shown in Fig. 6.The experimental values agree quite well with thetheoretical line. The standard deviation from thetheoretical line is 0.12 mm rms. The accuracy of theabsolute distance measurement is therefore con-cluded to be ;Ly600 experimentally. This high ac-curacy was achieved with double error-reducingtechniques, that is, feedback control12,14 and differ-ential detecting techniques.15

Finally, we measured the one-dimensional stepheight. The object was a block gauge whose thick-ness was ;2 mm. The measured result is shown inFig. 7. Circles show the measured values, and theyare interpolated with a solid line. The distance wasmeasured along the x axis on the surface of the object

6200 APPLIED OPTICS y Vol. 36, No. 25 y 1 September 1997

in discrete 14-mm increments. The step height wasaccurately measured in this interferometer. Themeasurement error along the x axis was 0.16 mmrms.

With this interferometer, we used the wavelengthtunability of the LD. But the current range in whichwe can obtain a linear wavelength change is re-stricted to ;10 mA in this type of LD in our experi-ment. Therefore the minimum equivalentwavelength L becomes ;8.95 mm theoretically. Inthe direct phase modulation of the LD, however, alarge OPD is necessary to satisfy the appropriatemodulation depth, z 5 2.45, with a small modulationcurrent that does not induce the intensity modulationin the interference signal. In our experiment theamplitude of the modulation current and the OPDwere 0.44 mA and ;80 mm, respectively. On theother hand, the maximum L is unlimited because thewavelength difference Dl can be set as small as wewant. But usually the LD has a finite coherentlength, so we cannot lengthen the OPD without limit.

Consequently, the minimum L and maximum Lare restricted by the initial OPD, which is decided bythe amplitude of the modulation current, and thecoherent length of the LD, respectively, but L is very

Fig. 6. Measured absolute distance. The object mirror M2 wasmoved in discrete 0.5-mm increments.

Fig. 7. Measured step height of the block gauge whose thicknesswas ;2 mm. The measured values are represented by circles.

large compared with that in the other interferometer.The feasible measuring range in distance is then Ly2,which is similar to that in the conventional inter-ferometer.

6. Conclusions

A new signal-processing method has been introducedinto the wavelength-multiplexed phase-locked LD in-terferometer. The fundamental error caused by off-set voltage can be eliminated in a simple way. Theprinciple and experimental results have been de-scribed. The advantage of this interferometer isthat there is no need to consider errors caused byoffset voltage in the feedback signals. Because onlyone LD is required in our interferometer, the opticalsystem is very simple despite the use of multiplewavelengths. The measurement error is estimatedto be ;Ly600 rms with the experimental measure-ment.

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