effect of beat frequency on the measured phase of laser-diode heterodyne interferometry

Effect of beat frequency on the measuredphase of laser-diode heterodyne interferometry

Ribun Onodera and Yukihiro Ishii

An optical heterodyne interferometer with a frequency-ramped laser diode has been constructed. Theeffect of the beat frequency on the measured phase has been theoretically investigated in the frequencydomain and experimentally verified. Phase errors caused by the difference between the ramp frequencyand the beat frequency alter sinusoidally in accordance with the p periodicity of the interferogram. Theerror can be eliminated by the electronic calibration technique of the beat frequency. © 1996 OpticalSociety of America

Key words: Laser diode, optical heterodyne interferometer, phase-extraction analysis

1. Introduction

The capability of the direct frequency-modulationproperty of laser diodes ~LD’s! has been applied tovarious interferometric techniques.1 The opticalfrequencies of LD’s have been shifted by varying theinjection currents on an unbalanced interferometer tointroduce phase changes between two beams. Manyinterferometric methods based on the frequency-modulated continuous-wave ~FMCW! technique havebeen developed in which temporal carrier frequenciesare produced by ramping the LD currents. A het-erodyne signal-recovery scheme for interferometricsensors has been described.2 An optical-fiber sensorhas been constructed in which sin f- and cos f-inter-ferometer outputs are generated by LD frequencyswitching.3 An optical time-domain reflectometer inoptical-fiber systems has been developed.4 A varietyof limitations and noise sources in FMCW systemsusing frequency-ramped laser diodes have been dis-cussed.5 A fiber-optic interferometer has been pre-sented that uses dual time-multiplexed LD’s drivenby sample sinusoidal currents.6 A quasi-heterodynefiber sensor has been proposed in which a staircasephase modulation with a py2-step difference is auto-matically produced.7 A Doppler beat signal hasbeen separated from the drift component by the fre-

The authors are with the Department of Electronics, Universityof Industrial Technology, Sagamihara, 4-1-1 Hashimotodai,Sagamihara, Kanagawa 229, Japan.Received 3 November 1995; revised manuscript received 1 Feb-

ruary 1996.0003-6935y96y224355-06$10.00y0© 1996 Optical Society of America

quency shifting in laser Doppler velocimetry.8 AnLD-heterodyne interferometry for displacement mea-surement has been reported.9 A multiplexed fiber-optic sensor system has been presented that uses adual-slope ~triangular! frequency-modulated LD.10An optical heterodyne polarimeter has been describedfor dynamic studies of the space- and time-dependentpolarization state of light.11 A two-dimensional inter-ferogram with both spatial and temporal carriers hasbeen applied to the simultaneous recording of multi-ple-phase objects and to the extraction of them.12 Anew technique for selective imaging has been devel-oped that uses a serial-to-parallel conversion of inter-ference signals with an image dissector camera andelectronic tuning.13 A multiplexed fiber sensor hasbeen presented that permits simultaneous opticalsensing.14 A two-wavelength interferometerwith fre-quency-ramped LD’s that utilizes the fractional fringetechnique has been constructed.15 A stable two-fre-quency beam with orthogonal polarization has beenmade from a frequency-modulated LD.16 A two-wavelength LD interferometry has been proposed thatis based on the heterodyne detection with one phasemeter.17A general formulation of LD-heterodyne inter-

ferometry based on the FMCW technique has beenpresented.18 It has been experimentally shown thatthe phase error caused by the LD-power variation canbe eliminated with an amplitude-stabilized, frequen-cy-modulated LD source. In the LD-heterodyne in-terferometer, the amplitude of the ramp current isadjusted to equalize the beat frequency with themod-ulation frequency. Then the actual phase can bemeasured by the optical heterodyne method. In thispaper, we analyze the effect of the beat frequency on

1 August 1996 y Vol. 35, No. 22 y APPLIED OPTICS 4355

the measured phase of LD-heterodyne interferome-try. It has been investigated theoretically in the fre-quency domain and verified experimentally that asinusoidal phase error with p periodicity is caused bythe difference between a ramp frequency and a beatfrequency. A dynamic calibration technique hasbeen presented that can eliminate the phase error.

2. Theory

A. Measurement Principle of LD HeterodyneInterferometry

In the Michelson-type interferometer shown in Fig. 1the LD is modulated by a sawtooth injection currentwith period T 5 2pyvs and amplitude im. An opticalfrequency of the LD in the sawtooth waveformchanges in accordance with the current variation.The frequency tuning range is given by

Dv 5 2pbim, (1)

where b is the current-tuning ratio. The frequency-modulated laser beam enters the interferometer withan optical path length l. The lights returning fromobject mirror O and reference mirror MR that areshown by solid and dashed lines in Fig. 1, respec-tively, recombine at the beam splitter and interferewith each other. A beat frequency is generated bytime delay t between the object light and the refer-ence light, giving

vb 5 Dvt

T, (2)

where t 5 lyc and c is the speed of light. When Eq.~1! is used, Eq. ~2! becomes

vb 5 vs

blcim. (3)

Therefore we can set vb as an arbitrary frequency byadjusting the amplitude of themodulation current im.The interference signal over the modulation period T

Fig. 1. Experimental setup for a LD heterodyne interferometer.The tested phase is measured with the lock-in amplifier by theheterodyne method.

4356 APPLIED OPTICS y Vol. 35, No. 22 y 1 August 1996

is given by

s0~t! 5 IM@1 1 g cos~vbt 1 f!#, (4)

where IM is the bias intensity, g is the visibility, andf is the phase being tested. It is assumed that anintensity alteration of the interference signal withthe current variation has been removed with an elec-tric divider. The phase has been formulated as5

f 5 vct 1Dv

2Tt2, (5)

where vc is the optical frequency at the center of thesawtooth. By using the experimental conditions ofDvty~2p! . 1 and T .. t, we can express the mea-sured phase as

f 5 2pllc, (6)

where lc is the wavelength of the LD at the center ofthe sawtooth.The measured heterodyne signal can be considered

an infinite sequence of the sampled signal s0~t!, giv-ing

s~t! 5 FrectS tTD 3 s0~t!G p F1T 3 combS tTDG , (7)

where

rectS tTD 5 H10 for utu # Ty2,for otherwise,

the symbol p represents the convolution, and

1T

3 combS tTD 5 (n52`

`

d~t 2 nT!,

whose d~t! is a delta function. The signal s~t! is se-lectively amplified with a bandpass filter having acenter frequency vs and a bandwidth 2B. Then thetested phase is measured with a phase meter by het-erodyne detection. We will analyze further in thefrequency domain. The bandpass-filtered signalsf~t! is given by

sf~t! 512p *

2`

`

S~v!FrectSv 1 vs

2B D1 rectSv 2 vs

2B DGexp~ jvt!dv (8)

512p *

2vs2B

2vs1B

S~v!exp~ jvt!dv

112p *

vs2B

vs1B

S~v!exp~ jvt!dv.

S~v! is the Fourier transform of s~t! given by

S~v! 5 ^@s~t!#

5 F2pIM sincSv

vsD 1 pgIM exp~ jf!sincSv 2 vb

vsD

1 pgIM exp~2jf!sincSv 1 vb

vsDG 3 combSv

vsD ,(9)

where ^@a# denotes the Fourier transform of a andsinc~vyvs! 5 sin~pvyvs!y~pvyvs!. The second andthird terms in Eq. ~9! are dominant in the integral ofEq. ~8!. Then Eq. ~8! changes approximately withthese terms in Eq. ~9! to

sf~t! > DgIM cos~vst 1 f9!, (10)

where

D52

11 brUsin@p~12 br!#

p~12 br!U@~br cos f!2 1 sin2 f#,1y2 (11)

f9 5 tan21S1br tan fD , (12)

where br denotes a ratio of the beat frequency to themodulation frequency:

br 5vb

vs. (13)

If the amplitude of the modulation current is set as im5 cy~bl !, the beat frequency vb in Eq. ~3! is equal tothe modulation frequency vs, which results in br 5 1from Eq. ~13!. By substituting br 5 1 into Eqs. ~11!and ~12!, we obtain the filtered signal as

sf~t! 5 gIM cos~vst 1 f!. (14)

In this case we canmeasure the actual phase f by theheterodyne method.

B. Effect of Beat Frequency on LD HeterodyneInterferometry

In Subsection 2.A the phase f can bemeasured by LDheterodyne interferometry in the condition of vb 5 vs.We analyze the effect of the beat frequency on theheterodyne interferometry.In Fig. 2 we show the numerical calculation of D as

a function of br by using Eq. ~11!. It is clear from Eq.~11! that the function D has the periodicity of p.Dave in the figure is taken as the average of the max-imum value and the minimum value of D with thevariation of phase f. In Fig. 2, Dave is 0 dB for br 51~vb 5 vs!, which means that the filtered signal pre-serves its amplitude gIM. For br 5 2, 3, and 4, thebandpass-filtered signals are completely annihilatedbecause the beat frequencies become an integral mul-tiple of the modulation frequency vs. We see fromthe figure thatDave for the noninteger br has a certain

value. Therefore the interference beat signal hasthe same spectrum component as the modulation fre-quency and can be applicable to the heterodyne de-modulatedmethodwhereas the beat frequency differsfrom the modulation frequency.Next we discuss the phase error caused by the

difference between the beat frequency vb and themodulation frequency vs. From Eq. ~12! one obtainstan f9 2 tan f 5 ~1ybr 2 1! tan f, and the equationsin~f9 2 f! 5 ~1ybr 2 1!sin f cos f9 can be derived.If f9 . f, the phase error is given to a first approxi-mation

Df ; f9 2 f . 21 2 br2br

sin 2f. (15)

It is shown from Eq. ~15! that the phase error variessinusoidally with p periodicity and vanishes in theideal case of br 5 1.

3. Experiment

The experimental setup for the LD-heterodyne inter-ferometer is shown in Fig. 1. The light source is aGaAlAs LD ~Hitachi HL7801E! with an operatingwavelength of 780 nm at a 51-mA current. An au-tomatic temperature-controlled ~ATC! circuit regu-lates the LD temperature within 20 6 0.03 °C tominimize the effect of the temperature on mode fluc-tuation. The injection current of the LD is modu-lated by the sawtooth waveform with the frequencyvsy2p 5 49.6 Hz. The amplitude of the modulationcurrent im is set to 2.5 mA tomake the beat frequencyequal to the modulation frequency vb 5 vs, and it isvaried during the experiment to change vb, i.e., br.The laser light is collimated by a 43 objective with anumerical aperture of 0.1. Then the frequency-mod-ulated laser light is coupled into a Twyman–Greeninterferometer with an optical path length l 5 60mm. The beat signal is generated from the interfer-ence between the reference and the object waves.The intensity of the interference signal is detected bya photodiode ~PD! and is fed to a lock-in amplifier

Fig. 2. Numerical calculation of the average amplitude Dave of abandpass-filtered signal as a function of br. The beat signal of theLD-heterodyne interferometer has the same spectrum componentas the modulation frequency.


through an electric divider. The divider eliminatesan intensity alteration of the signal with the currentchange by using the output of a monitor photodiode~M-PD! assembled with the LD. This procedure re-moves the phase error caused by the LD-power vari-ation.18 The known displacement of the optical pathlength l is given by moving the object mirror at aconstant speed with a voltage-controllable piezoelec-tric transducer ~PZT!. The tested phase is detectedby the lock-in amplifier having a time constant of 100ms. The measured phase f given by Eq. ~6! is dis-played on an x–t recorder, which illustrates displace-ment of the optical path length l.Figure 3 shows the experimental results of the

measured phases for br 5 0.7 ~top!, br 5 1.0 ~middle!,and br 5 1.3 ~bottom!. The vertical axis is the opti-cal path length l that is introduced by the displace-ment of the object mirror attached to the PZT. Thephase in the middle figure linearly changes from 0 to2p in which the actual phase can be measured. Thephase errors with p periodicity are illustrated in thetop and bottom figures. The amplitude of the phaseerror for 2f 5 3py2 is measured as 0.28 rad ~top! and20.16 rad ~bottom!. We can calculate the amplitude

Fig. 3. Tested phases measured for ~top! br 5 0.7, ~middle! br 51.0, and ~bottom! br 5 1.3. The phase errors with p periodicitycan be seen in the top and bottom figures. The errors are causedby the difference between the beat frequency vb and the modula-tion frequency vs.


of the phase error from Eq. ~15! for 2f 5 3py2 as 0.21rad for br 5 0.7 and 20.12 rad for br 5 1.3, whichshows good agreement with the experimental results.

4. Calibration Method of the Beat Frequency

In this section we describe an electronic calibrationmethod that equalizes the beat frequency with themodulation frequency that can eliminate the phaseerror. Figures 4 and 5 show a schematic and a tim-ing chart, respectively, of the electronic calibrationcircuit. A ramp generator in Fig. 4 produces thesawtooth waveform for LDmodulation by the use of astandard clock ~CLK!. The modulation period isfixed to be 256 times as large as the CLK period thatis illustrated in Fig. 5. An interference beat signal ismeasured by a photodiode ~PD! and is normalized byan electric divider. The interference signal biased tozero is used to detect zero-crossing points by the com-parator shown in Figs. 4 and 5. A counter gating inFig. 4 produces a half-period of the interference sig-

Fig. 4. Block diagram of the electronic calibration method forequalizing the beat frequency with the modulation frequency.

Fig. 5. Timing chart for the calibration circuit in Fig. 4 of the beatfrequency.

nal by taking first and second zero-crossing points.If the half-period is set to the CLK period as a factorof 128, which corresponds to the half-modulation-pe-riod, the beat frequency vb becomes equal to the mod-ulation frequency vs. According to Eq. ~3!, we canchange the half-period of the interference signal pyvbby adjusting the amplitude of the modulation currentim. Let us show how to make an ideal amplitude ofthemodulation current im to setvb 5 vs. The logicalAND of the CLK and the output of the counter gatingare taken as shown at the bottom of Fig. 5, whichproduces the CLK over the half-period of the inter-ference signal. The number of the CLK is countedby a binary counter and is converted to a feedbacksignal with a digital-to-analog converter ~DyA!. Thefeedback signal is fed to a multiplier in the LD driverfor every modulation interval, T 5 20 ms. If thenumber of the CLK over the half-period is less than128 ~the left-hand side of Fig. 5!, a small feedbacksignal is fed to the multiplier, which causes decre-ments in the amplitude of the modulation current im.In the case of a CLK number of more than 128, theamplitude im increases. Finally we can automati-cally obtain the ideal amplitude im to equalize thebeat frequency vb with the modulation frequency vs.Figure 6 shows the variation of the injection cur-

rent and the corresponding interference beat signalswith the feedback off ~top! and on ~bottom! in whichan initial amplitude of the modulation current im isset as br 5 1.3. It is clear from the figure that thebeat frequency becomes equal to the modulation fre-quency by the feedback operation.Figure 7 shows the experimental result of the mea-

sured phase for br 5 1.3. The optical path length l inEq. ~6! is given by the movement of the object mirrorby the PZT as the same in Fig. 3. The electroniccalibration circuit was operated over the period fromFeedback On to Feedback Off in Fig. 7. The mea-

Fig. 6. ~Upper traces! sawtooth modulation currents and ~lowertraces! beat signals when the feedback loop is off and on.

sured phase error with p periodicity can be removedwhen the feedback is on, which demonstrates an im-provement in measurement accuracy.

5. Conclusion

We have presented an analysis of the LD-heterodyneinterferometry by considering the effect of beat fre-quency on the measured phase. The phase errorcaused by the difference between the modulation fre-quency and the beat frequency, which varies sinusoi-dally with p periodicity, has been theoretically andexperimentally investigated. The electronic calibra-tion method has been applied to the LD-heterodyneinterferometer in which the amplitude of the modu-lation current is adjusted to make the beat frequencyequal to the modulation frequency. The experimen-tal results show the usefulness of the method in LDinterferometry.

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effect of beat frequency on the measured phase of laser-diode heterodyne interferometry

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