Fiber-optic acoustic Fourier transducer for audiosound processing

Gan Zhou, Louis Bintz, and Dana Z. Anderson

We demonstrate a fiber-optic acoustic transducer operating in the audio-frequency regime. The device ismade of an array of 120 multimode optical fibers and a photorefractive novelty filter. Each fiber in thearray acts as a cantilevered mechanical resonator. The resonant frequencies of the fibers logarithmicallysample the acoustic spectrum from approximately 100 Hz to 5 kHz. Laser light is injected into all fiberssimultaneously and is reflected from the end of each fiber. An optical novelty filter extracts the acousticinformation from the reflected light. The output of the novelty filter is essentially a Fourier transform ofthe acoustic signal. The background intensity in the transducer output corresponds to a driving amplitudeof approximately 50 A. We describe holographic storage of complex sound patterns that use a LiNbO3crystal and an acoustic transducer.

IntroductionIn the earliest theory of hearing that was proposed byHelmholtz,' the human acoustic transducer, namely,the cochlea, is assumed to be a collection of mechani-cal resonators. These resonators perform a frequencyanalysis and transmit the intensities of each fre-quency channel to the brain as the strength of thecorresponding nerve discharges. Bekesy's investiga-tion of the cochlea has shown that the cochlea doesperform a frequency analysis more or less. Differentacoustic frequencies excite different parts of thebasilar membrane.2 5 Such a Fourier representation isone approach toward capturing the phonetic featuresof speech, and it provides valuable information aboutspeech articulation. Thus the first step in almost anyspeech processing task is to produce the Fouriertransform of an acoustic signal. In this paper wereport on a fiber-optic acoustic Fourier transduceroperating in the audio-frequency regime that pro-duces the short-time transform directly.

A cantilevered optical fiber is a mechanical resona-tor. Its resonant frequency w', depends on its elasticconstants and its physical size. In particular w isinversely proportional to the square of the length ofthe fiber. Thus an array of cantilevered optical fiberswith different lengths can serve as a spectrum ana-lyzer. Figure 1 illustrates such an array that uses aphotorefractive novelty filter for optical readout. 9

The authors are with the Department of Physics and JointInstitute for Laboratory Astrophysics, University of Colorado andNational Institute for Standards and Technology, Boulder, Colo-rado 80309-0440.

Received 24 June 1991.0003-6935/92/111740-05$05.00/0.© 1992 Optical Society of America.

The resonant frequencies of the fibers logarithmicallysample the acoustic spectrum from approximately100 Hz to 5 kHz. Sound signals are received by eachfiber in the array. Laser light is injected into all fiberssimultaneously and is reflected from the end of eachfiber. The reflected light passes through the photore-fractive novelty filter. The novelty filter suppressesthe optical carrier and leaves only the acousticallyimposed sidebands. The output image is essentially ascrambled Fourier transform of the driving signal(scrambled because we do not retain the spatialordering of the fibers). An acoustic signal processingdevice named SCEPTRON'0 also takes advantage ofmechanical resonances of optical fibers, but the opti-cal processing is done with white light and photo-graphic masks. The phase information is lost, and thedevice is not adaptive.

In our transducer the sensing optical fibers work inthe reflection mode. The free ends of the fibers aresilver coated. This permits us to image the fixed endsand obtain a well-defined output plane. A position onthe output plane represents an acoustic frequency.Furthermore this relationship between spatial loca-tion and temporal frequency is independent of theamplitude of the driving signal.

The phase-sensitive detection of the novelty filter isnecessary because bending an optical fiber causesmostly phase modulation on its reflected light." Inaddition the novelty filter is adaptable to any slow(depending on the time constant of the photorefrac-tive crystal) environmental perturbations such asthose caused by temperature and pressure changes.This ensures that undesired variations in the fibermodes are not detected at the output; therefore thetransducer has good long-term stability.

First we describe the theory of the operation of the

1740 APPLIED OPTICS / Vol. 31, No. 11 / 10 April 1992

Outu

~~~~> No~~~~~~~~~~~~velty Filter

Fiber Sensor

Fig. 1. Schematic drawing of the setup of the acoustic transducer.The fixed ends of the fibers are imaged after the novelty filter. Theobtained output image is a scrambled Fourier transform of theacoustic signal that is driving the fiber array.

transducer, the design and fabrication of the fiberarray, and the method to impedance match whendriving the array. Then we present experimentalresults that characterize the transducer. We alsodescribe holographic storage and the recall of complexsounds by using an acoustic transducer.

Theory of OperationAn optical fiber cantilever can be modeled as aEuler-Bernoulli beam. The resonant frequency of thefundamental mode is given by'2

3.52 IEI\1"2(Oyij-|) (1)

where L is the length of the fiber beam, p is thevolume density of the beam, A is the cross-sectionalarea, E is Young's modulus, and I is the moment ofinertia. In the case of a beam with a circular-shapedcross section with radius R, I = rrR4/4. The higher-order mode frequencies are not multiples of thefundamental frequency. For example, the frequencyof the second-order mode is 6.26 times that of thefundamental. This is basically because the equationof motion contains a fourth-order derivative in space;thus it is not the usual Helmholtz wave equation. Inthe design we regard the fundamental vibration modeas the sensing mode. Higher-order modes cause somedegree of cross talk in the output of the device.

A single vibrating optical fiber as shown in Fig. 2undergoes periodic deformations and changes in itsrefractive index, which result in periodic phase modu-lation of the reflected laser light. The amount ofphase modulation ld4 of a given fiber is a function ofthe driving frequency w and the driving amplitude A,i.e.,

= x(w, A)sin(wt). (2)

This periodic phase modulation is detected by anadaptive interferometer called the novelty filter. Thenovelty filter utilizes the two-beam coupling processin a photorefractive BaTiO, crystal. 9 In small-signalconditions the transfer function of such an opticalnovelty filter is7

H(s) = exp. 1 r12

Piezoelectric

x-axis

Fig. 2. Fiber array driven by a piezoelectric transducer. Thebending of a fiber causes its reflected light to be amplitude andphase modulated.

where s is a complex frequency in the Laplace trans-form of the input field, F is a gain factor that can be aslarge as approximately 65/cm, I is the interactionlength of the BaTiO, crystal, and T is the timeconstant of the crystal. When a fiber is not vibrating,the reflected light from it writes an index grating withthe pump beam in the BaTiO, crystal. The gratingcouples the energy of the fiber output light into thepump beam. Thus the output image is dark. However,when the optical fiber is vibrating, the reflected lightfrom it is phase modulated, as described by Eq. (2),and the energy coupling in the BaTiO, crystal isreduced. The reflected light from the vibrating fiber isthen transmitted through the crystal. Thus we ob-serve a bright spot at the output plane. Using thetransfer function [Eq. (3)] and assuming thatX(A, ) << ar, we obtain the steady-state outputintensity:

I..t(w, A) = Iin[exp-R2) + 4X2(w, A)sin'( t +-R/2 (4)

where Iin is the input light intensity and o is again thedriving frequency. The output intensity depends onthe driving frequency through X(A, w) and has thesame characteristic resonance.

To drive a fiber cantilever in an efficient manner,the driver (which in our case is a piezoelectric trans-ducer) must be mechanically impedance matched tothe fiber. The fiber cantilever has maximum imped-ance near the clamped end (x = 0) and minimumimpedance at the free end (x = L). Let KF(x) denotethe effective spring constant at position x of a cantile-vered fiber beam, let KD denote the spring constant ofthe driver, and let Q be the fiber beam's mechanical Qfactor. By assuming that the resonance of the driveris far above the resonance of the fiber cantilever, wecan derive the condition of impedance match betweenthe driver and a fiber beam,

KF(X) = QKD, (5)

where KF(x) is infinity at x = 0 and is minimum at x =L. It can be approximated as 2

(3) KF(x) = 3EIlx', (6)

10 April 1992 / Vol. 31, No. 11 / APPLIED OPTICS 1741

where 0 < x < L. From Eqs. (5) and (6) we obtain theexpression for the impedance-matched driving posi-tion on the cantilevered fiber beam:

(3E1DxO= \-K< 7

KD can be estimated from the properties of thepiezoceramic driver. The Q factors of our fiber beamsare typically from 20 to 30. An estimate gives us x0 =200 pm.

Fiber ArrayThe fiber array is fabricated with multimode opticalfibers of 100-110-pm core-cladding size. The Young'smodulus E of quartz glass is 5.4 x 1010 N/m2, itsdensity p = 2.4 g/cm3. Thus, according to Eq. (1), thefrequency range from fmin = 100 Hz to fma = 5000 Hzrequires fiber lengths from Lm. = 2.7 cm to L =0.38 cm. The human ear is known to have higherresolution at the lower acoustic frequencies,2 and wearrange our fiber array so that it samples the spec-trum in a similar way. In fact we choose the samplingdensity function D ( f ) to be

D( f )- 8[n(f)] 1 (fmi. f fma)

TIME (0.5 ms/div)Fig. 4. Lower curve: the driving signal. Upper curve: the outputintensity after the novelty filter. The driving amplitude A = 0.02pum. Clearly the output intensity contains the second harmonic ofthe driving signal.

where N is the total number of fibers. Using Eqs. (2)and (4) we obtain the distribution of fiber length interms of their indices n;

L, = Lm_ exp[-b(n - 1)/2] 1 < n < N. (10)

(8)

where n( f ) is the number of fibers whose resonancesare below or equal to the frequency f; b is a parameterto be determined. It follows that

( =) 1 + (N- 1) ln(f) - ln(fm m)n~f) 1 + N-i)ln(fm_,) - ln(fmin)

100%

(9)

80%~

0

3IoP..

60%

40%

20% .

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8A

Driving Amplitude (m)

Fig. 3. Response of the transducer with respect to the drivingamplitude. I is the time-averaged optical intensity from theresonating fiber after the novelty filter; I, is the reflected intensityfrom the fiber when the loss pump of the novelty filter is blocked; Ais the driving amplitude.

(c) _

Fig. 5. Frequency response of the acoustic transducer. Thephotographs are taken after the novelty filter. The acoustic imagescorrespond to (a) no signal applied at the driver, (b) a pure toneapplied, and (c) a complex sound with five frequencies applied.

1742 APPLIED OPTICS / Vol. 31, No. 11 / 10 April 1992

(7)

Fig. 6. Experimental setup for storing sounds in LiNbO3 crystal. An electro-optic modulator provides the necessary acoustic sidebands onthe reference beam to write the hologram. Several holograms are recorded by angle encoding the reference wave: M, mirrors; L, lenses; BSand PBS, beam splitters and polarizing beam splitters; X/2, half-wave plate; EOM, electro-optic modulator.

with b = ln( fmax/ fmin)/ (N - 1). Here L0 is the lengthof the nth fiber.

The response time of the transducer is determinedby the Q factors of the optical fibers. As statedpreviously Q is typically from 20 to 30. Thus theresponse time of the transducer is from 250 ms at100 Hz to 5 ms at 5 kHz.

ExperimentsWe have implemented an array with 120 opticalfibers. As shown in Fig. 1 the ends of the fiber beamsconstitute the negative exponential curve given byEq. (10). The free ends of the fibers are coated withsilver by chemical deposition. The fiber array isdriven by a piezoelectric monomorph driver, as shownin Fig. 2.

The first experimental setup is as in Fig. 1. TheAr-ion laser operates at 514.5 nm. A laser beam fromit is injected into the array of 120 optical fibers. Theinjected light is reflected from the free end of thefibers and passes through the photorefractive noveltyfilter. We first fix the frequency of the driving signaland look at the output response of the resonatingfiber as the driving amplitude is changed. The mea-sured curve is as shown in Fig. 3, where A is thedisplacement of the piezoelectric transducer and Iout /Iin is the time-averaged percentage output intensityfrom the resonating fiber. Here in is the reflectedintensity from the fiber. The background intensity in

out corresponds to a displacement of 50 A at thepiezoelectric transducer.

At the part of the curve where A << 0.1 pum, I isquadratically dependent on A (the optical field islinearly dependent). The quadratic dependence canbe seen from Eq. (4). Assume that the phase-modulation amplitude X(w, A) = aA when A is small;then from Eq. (4)

Io= exp(-rl) + 4t 2A2 sin2t +-).in O

(11)

The best fit to experimental data gives a = 7.4rad/ Am.

Figure 4 shows the time dependence of the outputintensity compared with the driving signal. We clearlysee that when A is small Iout contains the secondharmonic of the driving frequency, as predicted byEq. (11). We also observed 10% intensity modula-tion on the light that is reflected from a vibratingfiber when the driving amplitude A approaches 1 p1m.

On the output image plane we monitor the re-sponse of the transducer as different frequencies areapplied at the driver; this is illustrated in Fig. 5. Theacoustic images correspond to (a) no signal applied,(b) a pure tone applied, and (c) a complex sound withfive frequencies applied. The mapping between thespot location and the acoustic frequency is random-ized because the bunched ends of the fibers in the

10 April 1992 / Vol. 31, No. 11 / APPLIED OPTICS 1743

WzW

0uj

I-I0L-C]

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

_ _ _ '#~L H-.. ._

. .. =. TI (0 . . . . .

TIME (0.5 s/div)Fig. 7. Recognition of two complex sounds. Each channel on theoscilloscope monitors the diffracted light intensity from one of thetwo holograms. The two sounds, each lasting 0.3 s, are appliedconsecutively to the driver with a silence of 1.9 s inserted betweenthem. The peaks are 20 dB above the backgrounds. The rise timeof the diffracted intensity is due to a low-pass filter in the detectionsystem.

transducer are not ordered, although the free endsare.

As an example of the usage of the acoustic trans-ducer, we implemented a scheme to store sounds in asingle LiNbO, crystal. The experimental setup is as inFig. 6. When a sound is received by the transducer itgenerates a characteristic vibrational pattern on thefiber array and a corresponding modulation on thefiber modes. After the novelty filter the optical carrieris suppressed, and only the acoustic sidebands areleft. The acoustic image thus obtained is used to writea hologram with a reference beam of light. Thehologram then acts as a matched filter for the soundand allows us to recognize a previously stored sound.

The reference wave must also contain the sameacoustic sidebands to write the hologram with thetransducer output. (One possible way to record astationary hologram by different frequency beams isto write the hologram in an ac field.'3) In our demon-stration an electro-optic modulator is used to providethe sidebands on the reference wave. The electro-optic modulator is used as an amplitude modulator.The direction of the output polarizer is perpendicularto the input polarization, so that the optical carrier issuppressed at the output. The electro-optic modula-tor and the acoustic transducer are simultaneouslydriven by the same sound; thus a grating is formed onthe holographic crystal. By angle encoding the refer-ence wave, one can store several holograms in a singleLiNbO3 crystal.

We have written two angle-encoded hologramswith two sounds. Each sound has a stationary spec-trum and is synthesized with five different acousticfrequencies. Figure 7 is the experimental result of therecognition of the two sounds. The two oscilloscopetraces indicate the diffracted light intensities from

the two holograms. We see that each channel iscorrelated with one of the stored sounds; the presenceof a sound at the input is indicated by a peak (20 dBabove the background) in diffracted light intensity atthe corresponding channel. Notice that on the oscillo-scope trace the rise time (- 0.07 s) in diffracted lightintensity is from a low-pass filter in the detectionsystem. It is not due to the response time of theacoustic transducer.

ConclusionsWe have demonstrated a fiber-optic acoustic trans-ducer. The device can be used to interface soundsignals with a holographic optical processing system.We have described and implemented a holographicscheme of storage and recall of sounds with station-ary spectra. The same type of acoustic transducer canbe implemented in a few other ways. For example, onecan replace the fiber array with reflection-coatedmetal fingers having the right resonant frequencyrange. In this case the displacement sensitivity of thetransducer is improved.

We gratefully acknowledge the support of U.S.Army Research Office grants DAA L03-87-K-0140and DAA L03-91-G-0312.

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