Fringe-counting techniqueused to lock a suspended interferometer

Fabrizio Barone, Enrico Calloni, Rosario De Rosa, Luciano Di Fiore, Francesco Fusco,Leopoldo Milano, and Guido Russo

We implement a digital fringe-counting technique to measure in real time the relative mirrordisplacement of a suspended Michelson interferometer with modulated optical path length for oscillationsmuch larger than the laser wavelength (). This provides the proper error signal for a servo mechanismthat reduces the relative displacement within X/2. The implemented technique does not require extraoptics or polarizers and thus can be used for interferometric gravitational wave detectors as a startingprocedure to get the system locked.

Key words: Digital fringe counting, suspended interferometer, gravitational wave detectors.

Introduction

The mirrors and main optical components of aninterferometric gravitational wave (GW) detector' aregenerally suspended to a seismic attenuator that is asingle or multiple pendulum. In this way the mir-rors are well isolated from the seismic vibrations inthe measurement band of the antenna (that rangesfrom 10 Hz to a few kilohertz). Such a suspendedmirror undergoes large displacements at the suspen-sion normal mode frequencies, making it very diffi-cult, if not impossible, to lock the interferometer onthe working point (generally a dark fringe) by usingthe usual coherent-detection technique for error sig-nal extraction. Generally the problem is solved bydamping to ground the suspended mirror." 2

However, this can reintroduce some seismic noisebecause the detector used for measuring the mirroroscillations is also driven by seismic displacement,and this is transferred to the mirrors by the feedbackitself. For example, for the sevenfold seismic suspen-sion prototype developed in Pisa, with a dampingsystem acting on six degrees of freedom, a residualoscillation amplitude of the test mass of 3.4 jim isreported.2

The authors are with the Istituto Nazionale di Fisica Nucleare,Sezione di Napoli and the Dipartimento di Scienze Fisiche dell'Uni-versita degli studi di Napoli "Federico II," Pad. 20 Mostra d'Ol-tremare, 80125 Napoli, Italy.

Received 3 November 1992; revised manuscript received 25 May1993.

0003-6935/94/07119404$06.00/0.3 1994 Optical Society of America.

For the present prototype detectors the residualrelative mirror displacement is small, because thetypical wavelengths of seismic vibrations are, at thesuspension normal mode frequencies, larger than theinterferometer arm length (10-40 m), and the mir-rors undergo almost coherent motions. The situa-tion will be more severe for a long-baseline interferom-eter with mirrors that are 3 km apart as a result ofuncorrelated seismic noise. In this case the seismicvibrations must be added incoherently and the re-sidual relative displacement can be of the order ofseveral laser wavelengths (). This can be too muchfor locking the interferometer if the feedback (FB)bandwidth is not much larger than the oscillationfrequency, as in the case of detectors designed forlow-frequency detection.3 4 This is because whenthe oscillation amplitude is much larger than , themirror spends, close to the right position in which aproper error signal is provided by the demodulator, atime much shorter than the oscillation period, andthe FB may not be fast enough to lock the interferom-eter during this time. This is difficult if we aretrying to lock a Fabry-Perot interferometer, owing tothe finesse of the cavity. In the following sections wedescribe a fringe-counting technique that providesthe right relative mirror displacement x(t) in the casein which it is much larger than X, with a resolution ofX/4, directly from the interferometer output detectorfor either a Michelson or a Fabry-Perot cavity; x(t)can be used as the error signal for a servo mechanismthat reduces the relative mirror displacement withinone wavelength, which is a good starting point for theusual locking technique.

1194 APPLIED OPTICS / Vol. 33, No. 7 / 1 March 1994

Fringe-Counting System

Fringe-counting techniques are widely used for met-rological applications5 6 and permit the measurementof macroscopic lengths with interferometric resolu-tion. The output power of a Michelson interferom-eter is

PoP= 2 (1+Ccos), (1)

where P0 is the input power, + is the relative phase,and C is the fringe contrast. Here is generallytime dependent and is a function of the relativemirror displacement x(t):

4rr= + x(t).

(Here we neglect laser frequency fluctuations andrefractive-index variations.) When one mirror isdisplaced for a length L larger than , the outputdetector is crossed by a number N of dark and brightfringes that is four times the number of laser wave-lengths included in L. The output signal, properlyamplified, can be converted into a transistor-transis-tor logic signal or converted by an analog-to-digitalconverter (ADC) and processed by a digital computerto give N and then a measure of the length L.

Because the cosine is an even function it is notpossible to recover the direction of the motion, whichshould be known a priori or measured separately.When more than one fringe is present in the field ofview, it is possible to use two photodetectors slightlydisplaced; in this way not only the fringe intensity butalso the displacement of the fringe pattern can bemeasured and an arbitrary mirror displacement canbe recovered. Other metrological techniques re-quire the use of extra polarizing optics along theinterferometer arms.

This is not the case for the suspended mirrorinterferometers used as GW antennas. In this casethe mirror displacement is periodic and the interfer-ometer is highly symmetric and carefully aligned, sothe intensity of the interference pattern is almostuniform along the beam section. In the followingsection we propose a digital fringe counter that, whenjoined to a demodulator, permits real-time measure-ment of the relative mirror displacement for a sus-pended interferometer.

Determination of the Direction of Motion

In the case of a Michelson interferometer the outputpower P is given by Eq. (1), where +0 = 4rALO/X is theaverage phase and ALo is the average armlengthdifference.

Generally one obtains the error signal to lock theinterferometer on a dark fringe ( = r, mod 2Tr) byapplying phase modulation and a coherent-detectiontechnique. When the pulsation is flm and the modu-

(2)

lation index is m, the modulated phase is

m = )o + A x(t) + m sin(flmt). (3)

Equation (1) can be expanded in terms of a Besselfunction as -

P (l-1 + C~cos[ko + A x(t)Jo(m)

+ 2 E J 2 n(m)cos(2nfmt)- 2 sin Ihon=1

X z J 2 n+1(m)sin[(2n + 1)fQmt] .n=O

+ X (t)]

(4)

The signal at the output photodetector is propor-tional to P; the low-frequency component is

.. Vd = A1 O + CJ0(m)cos[+0 ± x(t)JJ, (5)

and the signal demodulated at frequency fm is

Vm = -A2 PoCJi(m)sin[ o + A x(t)1, (6)

where Al and A2 are two positive constants.The time derivative of Vd is

VdC =-Al - CJO(m)sinL o + -x(t) A(t). (7)

One can obtain the relative mirror velocity directly bycalculating the ratio

VDC= 2AJm) (t) K-(t)Vm -4 2 J 1 *m

(8)

with K as a positive constant. It should be pointedout that the ratio of Eq. (8) cannot be used directly asan error signal for a feedback controlling the mirrorposition because it is divergent when sin + = 0, andhence Vm = 0; furthermore, the division can increasethe detection noise. Nevertheless Eq. (8) providesthe sign of (t), and it can be used to obtain thedirection of motion when a fringe counter is used.It then allows one to extract the actual mirrordisplacement.

Implemented Algorithm

We have implemented an algorithm that performsthe technique described above with the use of a digitalsystem based on a VME bus. Two 12-bit ADC's areused to acquire the signals Vd, and Yim. In thestarting phase we measure the maximum and mini-mum amplitude Vow: and Vmin, corresponding to thebright and dark fringes, respectively. The range ofthe signal is then divided into three parts; and twothresholds, V, and V,,, are defined (as for transistor-

1 March 1994 / Vol. 33, No. 7 / APPLIED OPTICS 1195

8.0 A- A_ r_ A N

-8. 0 I Sac 8.0

I=A I

(a)V

(b)

0.0 Sac 8.0

Fig. 1. (a) Input signal simulating a mirror oscillation. (b)Corresponding output of the fringe counter.

600 .

-60 . I V

(a)

Sec 4.0

H A_ -) .. -/ ~ (b)

0.0 Sac 4.0

Fig. 3. Measured interferometer (a) output and (b) recovereddisplacement in number of fringes, with the mirror free to oscillatealong the optical axis.

transistor logic signals) that allow us to know if theinterferometer goes from a bright fringe to a dark one(or vice versa). Associated with the signal, a statusvariable S (spin) is assigned. It is S = 1 if the signalis above the threshold Vu, S = -1 if it is below VI, andremains unchanged in the intermediate case.

At this point the algorithm begins the fringe-counting and signal-extraction process; at each sam-pling step VdC and V/m are acquired, and Vd, is calcu-lated and checked if (a) the signal is crossing theupper threshold V, coming from a dark fringe(S = -1) or (b) the signal is crossing the lowerthreshold VI coming from the bright fringe (S = 1).

In both cases the spin is changed (S - -S) and afringe is counted, i.e., a variable N, which is propor-tional to the number of fringes and then to the mirrordisplacement, is changed by ± 1 according to

N = N + (Vdc/Vm)/ Vdc/Vm . (9)

If neither of the two conditions is fulfilled, both Sand N remain unchanged. The numeric variable Nis amplified and converted by means of a digital-to-analog converter (DAC), thereby providing an ana-

logic signal proportional to the mirror displacementx(t).

Because the ratio Vdc/Vm is calculated when thesignal Vdc is crossing one of the two thresholds, theymust be chosen in such a way that the two signals arelarge compared with the noise; the solution that wechoose is

V1 = Vmin + 025(Vm - min)

which corresponds to + = +2 rr/3 (mod 2Tr), and

Vu Vmin + 0.75(Vm. - Vmin),

which corresponds to 4 = ± Tr 3 (mod 2rr).We performed a preliminary test of the digital

fringe counter by simulating the output of a Michel-son interferometer with phase modulation by using asignal generated by a digital computer and sent to aDAC. This signal is demodulated by a lock-in ampli-fier and then processed with the system describedabove. The displacement used for the simulation isthe sum of two sinusoidal motions at frequencies 1and 2.5 Hz and a maximum amplitude of 10 fringespeak to peak; Fig. 1 shows the input signal and theone recovered by the fringe counter.

Experimental Test

The fringe-counting technique has been tested on asemisuspended Michelson interferometer; the experi-

0.0

60. 0

Fig. 2. Experimental setup: BS, beam splitter; PD, photodiode;PS, position-sensing device; PZT, piezoelectric transducer; A, loopamplifier and current driver; M, mirror.

N

-60 .0

00 1 1 C 1 I 1 I0

) __ ~~~~_ I =_

(a)

(b)

Y 0.0 Sac 200m

Fig. 4. Same as Fig. 3 but with a shorter record length (200 ms).

1196 APPLIED OPTICS / Vol. 33, No. 7 / 1 March 1994

m0.0 see 200.

0

0.0 see

is reduced to plus or minus one fringe ( X/2 250nm peak to peak), as is shown in Fig. 5.

(a)

(b)

200m

Fig. 5. Measured interferometer (a) output and (b) recovereddisplacement in number of fringes, with the control loop closed.The oscillation is reduced to plus or minus one fringe.

mental setup is sketched in Fig. 2. The source is anAr+ laser (514 nm). One of the two mirrors ismounted on an aluminum mass, suspended by foursteel wires; the other mirror is glued onto a piezoelec-tric transducer that is used to modulate the opticalpath length and then the relative phase. On thebottom of the suspended mass four permanent mag-nets are glued, and the mass position and orientationare controlled by four coils fixed to ground. Thesuspended mirror is not damped to ground, so it isfree to oscillate at the pendulum resonance frequency.The alignment of the interferometer is provided by anoptical lever that measures, by means of a position-sensing device, the position of a He-Ne laser beamreflected by the mirror surface: the signal is fedback to two of the coils. The optical path length ismodulated at 5 kHz, and the output signal is demodu-lated by a lock-in amplifier. The low-frequency sig-nal and the demodulated one are processed with thefringe-counting algorithm.

The system successfully recovered the mirror dis-placements, with oscillations as large as more than100 fringes peak to peak. Figure 3 shows the photo-diode output and the recovered displacement (innumber of fringes) when the mirror is free to movealong the optical axis. The large signal at 1.3 Hz isthe pendulum resonance. Figure 4 plots a shorterrecord length (200 ms) in which the fringe variation isbetter resolved.

The signal has been used for a derivative FB on themirror position; when the loop is closed the oscillation

Conclusion

We developed and experimentally tested a (digital)fringe-counting technique that is able to recover therelative displacement of two suspended mirrors in thepresence of oscillations much larger than X. Thepeculiarity with respect to other counting techniqueswidely used in metrology is that this technique doesnot require any extra optics or detectors. It onlyuses the output signal of the photodiode used tomeasure the interference and a phase modulation.For this reason this method can be used for reducingthe relative displacements of the mirrors of an inter-ferometric GW detector, in a preoperation phasebefore the locking is acquired.

A further application can be implementation oflocal controls to damp to ground the residual mirroroscillation, in a configuration similar to the setupwith one mirror fixed to ground such as the one wehave tested. A possible improvement that we plan toimplement in the immediate future is the possibilityof a resolution smaller than one fringe, found byadding the inversion of cost + to the algorithm.

References and Notes

1. For a review of GW interferometric detectors see, for example,The Detection of Gravitational Waves, D. G. Blair, ed. (Cam-bridge U. Press, Cambridge, 1991).

2. C. Bradaschia, R. Del Fabbro, L. Di Fiore, A. Di Virgilio, A.Giazotto, H. Kautzky, V. Montelatici, and D. Passuello, "Firstresults on the electronic cooling of the Pisa seismic noisesuper-attenuator for gravitational wave detection," Phys. Lett.A 137, 329-333 (1989).

3. The Virgo Project, a proposal by a collaboration of 38 physicistsof the Italian-French large base interferometric antenna Virgofor GW detection, June 1989, of which F. Barone, L. Di Fiore, L.Milano, and G. Russo are members.

4. A. Augurio, F. Barone, E. Calloni, L. Di Fiore, L. Milano, G.Russo, and S. Solimeno, "Automatic control system for mirrorsalignment of interferometric antenna Virgo," presented at theSixth Marcel Grossman Meeting on General Relativity, Kyoto,Japan, 23-29 June 1991.

5. A. Sona, "Laser in metrology: distance measurements," inLaser Handbook, F. T. Arecchi and E. 0. Shulz-Dubois, eds.(North-Holland, Amsterdam, 1972), Vol. 2, pp. 1457-1486.

6. H. J. Tiziani, "Optical methods for precision measurements,"Opt. Quantum Electron. 21, 253-282 (1989).

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2 .

-2.

Conclusion