Nuneral Network Laser Radar AO 1994

8/2/2019 Nuneral Network Laser Radar AO 1994

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Neural-network laser radarKeigo lizuka and Satoshi Fujii

A laser radar whose resolution is greater than 1 pums reported. We present the radar results when theyare used for such purposes as determining the size of a void inside a silicon wafer, profiling across-sectional pattern of an optical fiber, studying the birefringence of a lithium-niobate crystal, orfinding a fault in an optical guide in an optical integrated-circuit wafer. Neural-network theory was usedin processing the radar signal. Radar processing based on neural-network theory gave significantlysuperior resolution compared with Fourier-transform-based processing.

IntroductionThe resolution of the step-frequency radar increaseswith the range of frequency steps. Laser diodes canshift their oscillating frequency by tens of terahertz.The obtainable resolution of a radar with such a widerange of frequency steps is of the order of micrometers.This kind of high-resolution radar opens up newapplications in measurement in the field of optics.An immediate application of such an imaging radarwould be the prescreening of either silicon or GaAswafers to locate possible internal flaws. It could alsobe implemented in a device for profiling the crosssection of an optical fiber while the fiber is beingdrawn without interruption of production. It wouldbe possible to use the radar for locating faults even inoptical integrated circuits (IC's).One of the most popular optical-fiber fault locatorsto date is the optical time-domain reflectometer.1Extending the resolution of this method to an order ofmicrometers would be difficult. Obtaining a -pmresolution requires femtosecond pulses. Some ofthe delicate optical IC's might not withstand the peakintensity of such pulses.Two types of optical frequency-domain reflectome-ter2-5 have been proposed; one modulates the fre-quency of the amplitude modulation, and the other

K. lizuka is now with the Ontario Laser and Lightwave ResearchCenter, Department of Electrical Engineering, University ofToronto, Toronto, Canada M5S 1A4. S. Fujii now is with theCommunications Research Laboratory, Okinawa Radio Observa-tory, Nakagusuku-son, Nakagami-gun, Okinawa 901-24, Japan.Received 25 November 1992.0003-6935/94/132492-10$06.00/0.0 1994 Optical Society of America.

modulates the carrier frequency of the laser light.The former type does not necessitate coherent laserlight, but the obtainable resolution is limited to anorder of meters. While a resolution of a meter maybe sufficient for an optical-fiber fault locator, it is notgood enough for an optical IC fault locator.The latter type of optical frequency-domain reflec-tometer can achieve the micrometer resolution neededfor an IC fault locator; however, source coherencyrequirements are more demanding.Another interesting approach is the optical coher-ence-domain reflectometer,6' 7 which makes use of thefact that, if an incoherent light source is used for aMach-Zehnder interferometer, an interference pat-tern exists only at the location where the path fromthe source is exactly identical to that from thereference arm. The optical coherence-domain reflec-tometer, however, requires a moving stage and asource with a higher degree of incoherency. Thereported resolutions are of the order of 10 m.Principles of OperationThe neural-network laser radar uses a cw. Thefrequency of the transmitted wave is changed step-wise, and at each step of frequency the phase andamplitude of the wave scattered from the target aremeasured. One extracts the distance information byanalyzing the changes in phase and amplitude of thereceived wave as the frequency of the transmittedwave changes.Figure 1 shows the layout of the step-frequencyradar. The carrier frequency of the laser beam isstepped. The operating frequency f of the nthfrequency is

fn = o +nAf. (1)2492 APPLIED OPTICS / Vol. 33, No. 13 / 1 May 1994


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Xo Xkftime

Fig. 1. Block diagram showing the principle of a neural-network laser radar.where n = 0,1, 2, . . , N - 1, ois the frequency oinitial step, and Af is the frequency step width.amplitude of the transmitting light is E.The received signal is the sum of the sigscattered from all the scattering centers. Letscattering centers be located at

Xk = x 0 + kAxand their backscattering cross section be k.received signal Hn at frequency fn is

Hn= N- 1E fo+ nAfk=O k-s V kAx) .

With the constraint2AfAxN

V

f theThenalsthe

formN-i exp(j2nk N)-

(7)

En is the measured quantity, and Sk is the desired(2) information.Two methods of obtaining Sk were considered: (a)The fast-Fourier-transform (FFT) signal processing and(b) neural-network signal processing.Fast-Fourier-Transform Signal Processing(3) Since Eq. (7) is of the form of an inverse discreteFourier transform, one immediate method of obtain-ing Sk from En is the execution of the discrete Fouriertransform of En. The discrete Fourier transformprovides values of(4) SOWD 2) ..* Ski ..* SN-1-

N-1= NEi rE expk=Ox exp j2,r l

The left-hand side of Eq. (5) has onlyeter, whereas the sum of the firstright-hand side has only k as a pasecond factor in Eq. (5) has the form ofunction of the discrete Fourier transfLet us put

Nonzero values of Sk indicate the existence ofkAx scatterers at Xk = x0 + kAx. One of the attractions ofj4Tr -f FFT processing is that FFT algorithms are readilyJr -yv available, which simplifies the implementation.,k Another interesting feature of FFT processing is its(5) zooming capability, such as the zoom function of aV camera. The value of x0 in Eq. (2) can be arbitrarilyset to a desired value, and a higher resolution can be*n as a param- maintained in the x > x0 region. This zoom capabil-factor of the ity is desirable in applications such as the optical ICrameter. The chip fault locator, where the length of the input guidef the weighting of the IC chip is of the order of millimeters, but oneorm. desires the resolution inside the IC to be of the orderof micrometers.

En= Hnexp(-j4r x0),kAxSk = kEexp j4'Tr - fo . (6)

When Eqs. (6) are used, Eq. (3) assumes the simplified

Neural-Network Signal ProcessingThe neural network similar to the Hopfield type wasused to find Sk from E,,. Such a network is shown inFig. 2. It resembles an array of identical amplifiers.Two special features of the array are the nonlinearityof the input-output function of the amplifiers and theintensive network of feedback. The input-output

1 May 1994 / Vol. 33, No. 13 / APPLIED OPTICS 2493

Eq. (3) becomesHnexp( j4m vx0)

-- @

XN-1. . . .

:t=


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SN-Distance XN-1

Fig. 2. Neural network used for processing laser radar.

function of an amplifier is nonlinear, but the outputalways monotonically increases with the input signallevel. The output from each amplifier is fed back tothe inputs of all other amplifiers including to itselfthrough couplers with coupling constant TipThe output from the array is determined by thecoupling constants. The Ti. consists of N x Nconstants, which are in fact the same number as theweighting functions of the Fourier transform givenby Eq. (7). The weighting functions of the Fouriertransform are made so that twice the integral trans-form brings back the original function (except for thenegative sign in the variable). At issue here iswhether this special feature of the FFT is essentialwhen it is used for radar signal processing.On the other hand the values of Tij are calculatedfrom a given geometry of the radar so that thedifference between the measured and theoreticallycalculated values is minimized.Kirchhoff'scurrent law appliedat the input of theith amplifier in Fig. 2 gives

dU - U, N-1Ii - C dt - -TijVj=,dt R ~j=0 (8)

where U,and Viare the input and the output voltages,respectively, at the ith amplifier. Ii is the externalinput current from the ith terminal, and g(Uj) is theinput-output function of the amplifier and monotoni-cally increases with Uj. The negative sign was placedin front of the summation in Eq. (8) to indicate thenegative feedback explicitly.In the special case in which the input to only oneport is distinctively predominant and the couplingconstants are all positive real numbers, the status ofthe outputs from the array is clear; if the ith input islarge, the gain of the ith amplifier is shifted to a largervalue because of its nonlinearity, and the nonlinearityitself favorably raises the ith peak.Whereas a large output from the ith amplifier givesback a large negative feedback to all the ports toflatten their peaks, smaller outputs from other ampli-fiers cannot give back as large a negative feedback toflatten the ith peak, and the ith peak stands out.Thus a combination of the nonlinearity of theamplifier and the extensive negative feedback connec-tions produces an accentuated peak at the ith port.If, however, a multiple number of inputs are domi-nant, and Tij is arbitrary complex numbers, counter-acting actions exist among the ports and the mannerof convergence is not immediately obvious. The onlyknown fact8 is that the state of the array outputconverges so as to minimize the so-called Lyapnovenergy function L defined as

1N-1N-1 - 1 i fN-1iL=-2 a X umi - IX i + R I 9 (V)d(9)where g-i(V) is the value of U when expressed interms of V. Note here that energy is a hypotheticalquantity and we are not referring to energy measuredin joules.If an expression that represents the error betweenthe experimentally measured values and the theoreti-cally calculated values is reformulated to be of a formthat is similar to the Lyapnov energy function, thestatus of the outputs converges to the values of theminimum error. This characteristic of the Hopfieldneural network is utilized to find the outputs thatoptimumly fit with the theoretical results.Thus below we find first the error function andthen reformulate it in the form of the Lyapnov energyfunction to find the values of Tij for our present case.Once the values of Tij are found, the numericalsolution of the differential equation, Eq. (8), whichrepresents the function of the circuit in Fig. 2, givesthe converging values of Vi hat minimize the error.Short-hand notation is used, namely,

n = exp j2,r N} (10)2494 APPLIED OPTICS / Vol. 33, No. 13 / 1 May 1994

with

*= V0 V, V2C

$9 SI SC XOt Xl x2


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where the summation sign has been removed withthe understanding that the appearance of the samesubscripts twice means the summation with respectto that subscript from 0 to N - 1. Equation (7),which is the measured quantities at frequencies fo,fi,f2,... ) fn*... * fN-l are now rewritten by use of Eq.(10) as

Eo= eOkSk,E= elkSk,E2= e2kSk,

E= enkSk,

eOkelke2k

ek =enk

e(N- l)k

(15)

Equation (14) is recast into the form of Eq. (9).The Euclidean norm is the sum of the squares of allmatrix elements:

EN-, = e(N-l)kSk. (11)Equations (11) can be represented in vector notationas

1 N-1 N-1Q- E I PnmI2n=O m=O1 N-1 N-1= - IX I IS, 121ISjIIei t ej 2=O j=O

N-1- I S,12ei'ei.i=O (16)

The covariance matrix P of Eq. (12) iseOkSkelkSk

P=EEt= enkSke(N- l)kSk

X (eOk*Sk*, elk*Sk*, ... enk*Sk*, e(N-l)k*Sk*)-

The derivation of Eq. (16) from Eq. (14) is shown inAppendix A by an example with N = 2. Before wecompare Eqs. (9) and (16), the third term of Eq. (9)needs special attention.A simpler input-output function g(U) of a neuron(12) is desirable from the viewpoint of computation. Wefound that the function of such a simple shape asshown in Fig. 3 serves our purpose. It is expressedasAU U> 0g(U)= 0 U< 0' (17)

where A is the gain factor of the amplifiers. WithEq. (17) the third term of Eq. (9) becomesN1 Vi 1 1 N-1RizoJogV)dV IX V 2 . (18)R 0 j 0 2 RA 0

Equation (9) can now be rewritten as(13) 1N-1 N-1L=2 Ii=O j=OWe now show that the Euclidean norm Q of thedifference between the covariance matrix obtainedexperimentally and theoretically can be reformulatedto the Lyapnov energy function so that the Hopfield-type neural network can be used for optimization.9Q is expressed as

1 N-i 2QIXF I7 lekek=O (14)where ek is a vector defined by the kth target, 11-11denotes the Euclidean norm, and the bar over Pindicates the measured values. The elements of thevector are phases at different frequencies for the kth

0

Fig. 3. Input-output function of a neuron.

C0


(.~~ + -Nij + 8zJ j E-Ovi. (19)Input U

I


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Now we compare Eq. (15) with Eqs. (19):Tij= Ieitej1-8__ARIi = eitPei,vi= S,12,Vj = ISi2 . (20)

If the values of the parameters given in Eqs. (20) areused in the neural network shown in Fig. 2, theoutput values are such that the Euclidean norm ofthe error function is minimized.In conclusion the purpose of the neural-networkprocessing procedure isEqs. (10) and (20) andmeasured P, (2) eithershown in Fig. 2 with thecalculated above or to relydifferential equation, Eq.obtained above. In this

(1) to calculate Tij fromto calculate Ii from theto implement the circuitvalues of the componentson a computer to solve the(8), with the parameterspaper we took the latterapproach, and we solved the differential equationnumerically by the Runge-Kutta method. 10We should emphasize that once the coupling con-stants Ti3 are calculated, they do not have to bechanged according to the type of target; only thevalues of Ii have to be calculated from the covariancematrix whose elements are obtained from the mea-sured Envalues.Layout of the Experimental ApparatusDetails of the imaging-radar layout are shown in Fig.4. An external-cavity-type cw laser diode was used

Wave meterAOA

Laser 7BS -

fn

0Step freq.generator

as a source. The wavelength could be varied from1.5 to 1.6 pm, which corresponds to 12.5 THz offrequency shifting. By means of an acousto-opticmodulator (AOM) he laser beam is split into a probe(object) beam, shown as a solid line, and a referencebeam, shown as a dashed line. The AOM not onlysplits the beam into two but also shifts the carrierfrequency of the reference beam by the driver fre-quency of 40 MHz from that of the probe beam. Theprobe beam is focused into the target by way of anonpolarizing beam splitter. The return beam scat-tered from the scatterers in the target takes the sameroute and is deflected by the nonpolarizing beamsplitter into the photodiode mixer.The reference beam whose carrier frequency isshifted from that of the probe beam by 40 MHz isguided by two mirrors into the photodiode mixermentioned above. The mixer diode puts out a 40-MHz cw signal whose amplitude is proportional to thereflection from the target and whose phase is thesame as that of the probe beam. The advantage ofsuch a detection scheme is that no matter how muchfrequency of the light source is shifted, all one needsis a 40-MHz amplifier. The amplitude and phaseinformation at each step of frequency are fed into theprocessor.A He-Ne visible-light laser beam is also put into thesystem for easier alignment of the object beam withthe target. The target is mounted on an electronicprecision translator stage for easy alignment. Twovisible-range stereoscopic microscopes and an IRcamera were found useful for coupling the objectbeam into the optical guide on the optical IC.A microscope objective focused the probe beam to a

Xk-A l

Fig. 4. Block diagram of neural-network laser radar: BS, beam splitter; AOM, acousto-optic modulator.

2496 APPLIED OPTICS / Vol. 33, No. 13 / 1 May 1994


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- DiscreteFourier Transtorm. Neural network processing

0.8 2

05 4

0.0(a)

Probe beam

(b)

7

H- 321 AFig. 5.target. Radar display when a glass plate with a void was used as a

specific region of interest. A large numerical-aperture microscope lens, however, distorts the rela-tive intensities of the scattering along the distance.The intensity of the scattered wave at the distancethat the probe beam is focused is highlighted.At each step of the frequency the wavelength of theprobe beam was measured by a seven-digit waveme-ter.(N-1) Af = 12.5THz

The first target was a quartz glass plate with a27-pm-thick air void. We made the void by inserting27-p1m-thickspacers between two glass plates.Figure 5 shows the display of the radar output.The vertical axis is the scattering intensity and thehorizontal axis, the distance. The range of fre-quency (N - 1)Af was 12.5 THz, and the number ofsteps was N = 512. The distance that appears in thedisplay depends on the wavelength in the medium.The index of refraction of each layer is necessary tofind the exact thickness of the layers. The locationsof the four peaks clearly correspond to the interfacesbetween air and glass. We show the display obtainedby FFT processing with a Blackman Harris windowby the solid curve and that obtained by the neuralnetwork by the dotted curve. A significant reductionof the sidelobes is observed in the result processed bythe neural network.Figure 6 demonstrates the power of neural-net-work processing over the FFT processing. Using thesame quartz glass target, we repeated the measure-ments with a reduced frequency range (N - )Af =4.23 THz and compared the measured result with(N - 1)Af= 12.5 THz.The top two graphs in Fig. 6 are the resultsprocessed by the FFT with these two frequencyranges. The resolution of the discrete-Fourier-transform processing is given from Eq. (4) by

VAx= 2fN' (21)and the resolution is expected to be reduced to one

(N-1) A f = 4.23THz

FFT a 0.6a)0 7U) 0.2)-

ono1.0 _0.8 i

.L ICNeural ) 0.6Network O 0.C= 0.4 0Z -cn 0.2k

0.00

-20 I i I

1~~ I,200 0.0 200

Fig. 6. Comparison of the radar displays with FFT processing and with neural-network processing.

1 May 1994 / Vol. 33, No. 13 / APPLIED OPTICS 2497-

Distance arbitrary-

uLr


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third when the frequency range is reduced by onethird. The case with (N - 1)Af = 4.23 THz can nolonger resolve the center peaks, and the total numberof peaks is reduced from four to three.The two lower curves show the results when neural-network processing was applied to these two fre-quency ranges. The lower right curve was madewith (N - 1)Af = 4.23 THz. The center peak, whichcould not be resolved as two peaks with FFT process-ing, can now be resolved as two peaks with neural-network processing. The widths of the peaks of thelower right curves are sharpened further when thefrequency range is raised to 12.5 THz, as shown in thefigure at the bottom left. To quantify the improve-ment in the resolution, we compare the widths of thepeaks at the distance of -200 units shown on thegraph compared for FFT processing and neural-network processing. The ratio of the widths is 6,which means that neural-network processing yields afactor-of-6 improvement over FFT processing, andthe resolution of the radar with neural-networkprocessing in this medium is 1 jim.Next Fig. 7 shows the result when an optical fiber isused as a target. The fiber is an oversized 250-jim-diameter, step-index, multimode fiber. The indicesof refraction of the core and cladding layers are 1.59and 1.52, respectively. The probe beam was incidentnormal to the wall of the cladding layer. A possibleapplication of such a radar is for profiling the crosssection of an optical fiber while the optical fiber isbeing drawn from the nozzle. The production is notinterrupted for the sake of measurement.Figure 8 shows the results when the radar was used

1.0 | | l- DiscreteFourier Transform

. * Neural networkprocessing

*S 0.6-

0020 0

0.0 .. ...........-200 0 200(a) Distance (arbitrary)

CoreProbe beamCladding

(bI20A(b)~ ~ 250mFig. 7. Radar display when an oversized optical fiber was used as atarget.

0.5

I.- 0.5C0

,. 0.00.5

0.00.0 200 400 Distance (arbitrary)

Fig. 8. Demonstration of the change in the apparent thickness ofan anisotropic crystal as the crystal is rotated: x, direction ofpropagation of the probe beam; E, electric field of light; C, opticalaxis of the crystal.

to measure the birefringence of a crystal. A 500-,jm-thick lithium-niobate (LiNbO3) crystal was used as atarget. The probe beam was incident perpendicularto the surface that contained the crystal axis. Theeffective thicknesses were measured as the crystalwas rotated around the incident probe beam. Figure8(a) shows the case in which the direction of thepolarization of the probe beam is parallel to thecrystal axis (the e wave is excited). Figure 8(b) showsthe case in which the crystal is rotated so that thecrystal axis is at 45 deg from the direction of thepolarization of the probe beam (both the e and owaves are excited). Figure 8(c) shows the case inwhich the crystal is rotated so that the crystal axisbecomes perpendicular to the direction of the polariza-tion of the probe beam (the o wave is excited). Theordinary index of refraction n0 of the LiNbO3 crystalis no = 2.2113, whereas the extraordinary index ofrefraction ne is ne = 2.1361 at = 1.6 jm.li Thethickness of the same crystal measured by the ordi-nary wave appears longer (in terms of wavelength inthe medium) than that measured by the extraordi-

Fig. 9. Indicatrix of a uniaxial crystal.


H i .I

n LI--

-I I i


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0.2

02

o.1

U

0.0~

Distance (arbitrary)

End face Score Optical guideFig. 10. Radar display when an optical guide on an optical IC isused as a target.

nary wave. A comparison between the curves inFigs. 8(a) and 8(c) verifies this fact. The ratio of theindices of refraction is nol/ne= 1.035, and the ratioobtained from the experimental data is 1.033 0.004.Good agreement was obtained even with such a thinsample.Referring to the optical indicatrix shown in Fig. 9,we review the law of propagation in a birefringentcrystal. When the light is incident on an anisotropiccrystal along any direction ON of propagation, onlytwo directions of polarization are permitted; they arethe directions of the minor and major axes of ellipsesD1 and D2. No other direction of polarization ispermitted. 12

Figure 10 shows the result when the radar wasused for locating a score made on an optical IC wafer.Several straight single-mode optical guides were madeon a LiNbO3 wafer. The guides were scored at thesame location a few times perpendicularly to theoptical guides. The cross section of the score mea-sured by a profiler is shown at the top left of Fig. 10.The width of the score is 100 jim, and the depth is20 im and located 300 jim from the endface ofthe wafer. The middle graph shows the radar display.A corresponding microscope photograph is shown atthe bottom of Fig. 10 to verify the locations of thepeaks in the display. The peak midway between thefront and back edges of the score may be due to theirregularity of the score caused by repeated scoring.ConclusionsA laser radar that employs neural-network signalprocessing was constructed, and its performance wasevaluated. The resolution of the radar by the neural-network processing was demonstrated to be at least 6times better than that processed by FFT. The reso-lution of the radar with neural-network processing isless than 1 im in the lithium-niobate crystal. A fewpossible applications such as a flaw locator for asilicon wafer, cross-sectional profiling of an opticalfiber, determination of the birefringence of a crystal,and a fault locator for optical IC's have been explored.Appendix AThe derivation of Eq. (16) from Eq. (14) for the case oftwo frequency steps with two scatterers is as follows:

P = EEt= ekSk )(eok*Sk*,lk*Sk*),

witheOkSk= eSo + eolSl,elkSk = e1oSo+ e11S1,

L (eooSo+ eoiSi)(eoo*So* + eoi*Si*) (eooSo+ eoiSi)(eio*So* + ell*Sl*)1(eloSo + elSi)(eoo*So* + eoi*Si*) (eloSo + e1iSi)(eio*So* + eln*Si*)] '- I SoI 2 + IS, 2 +SOS1*eooeo1*+SSo*eoeoo*I 2eOe0O*+ IS1I2 enjeol*+SoS*eloeol*+SoS,*elleoo*

Let us focus on the twin peaks in Fig. 8(b) at thelocation of the back wall of the crystal. One of thetwin peaks in Fig. 8(b) lines up with the e-wave peakin Fig. 8(a), and the other twin peak in Fig. 8(b) linesup with the o wave in Fig. 8(c). The appearance ofthe twin peaks, rather than a single broader peak,confirms the above-mentioned law of propagation in abirefringent crystal.

ISOi2eooelo*+ IS i2eolell* +SoS*eooell* +SSo*eolelo4IS012+IS1J2+SoS1*eioe11*+SSo*e1elo* 1

Note that aside from the e valuesISo - S 12 = ISo12 ISl 2 - (SOS,* + S 1SO*) > 0,

ISo 2 + IS1 12> SoS,* + SlSO*.The difference between the left- and right-hand sidesof the equation above becomes even larger if either S0or S1 has a value of zero. In the case of quantized



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points with a relatively small number of scatterers,the left-hand side becomes much larger than theright-hand side:P=ISOI2(1 eooelo*) 1(eP= IS1 eoeo* I ) + IS 12ellel*

= IS12eoeOl + IS12elell1,/

1= X ISk 12 ekekt,k=O

where e, e, and ek are phasor vectors associatedwith the zeroth and first target, respectively, and aredefined aseooe = elo ' (el eok

2Q = 2{IQoo12 + IQ0112 + IQ1oI2 + IQ1112}1 1 1= E |Prn | - Sk I2{FooeOk*eOkm=0 n=0 k=0+ PO*eOkeOk* POleOk*elk+ P*eOkelk+ PlOelkOk+ lo*elkeok*+ Pllelk elk+ Pll*elkelk }

1 2 1 2+ ISk |2 eOkeOk* + I I Sk 2eOkelk*k=O k=O

1+ I Sk 2 elkeOk*k=O

= F+ G + H.Next the Euclidean norm is calculated:

1 1 2Q = PT_ E Sk 2ekektk=O

1 221Qoo 12= Po - I I h I2 eokeOk*k=O

1= IPool2I Po I ISk I2eok *eokk=O

1 1- oo* |Sk 2 eOkeOk* + Ik=O k=O

1 221 Qol 2 = o I |Sk eOkelkk=O

1= IPO- 2 Po I Sk 12eOk*elkk=O

1 1- poy* ISk |2 eOkelk* + I

k=O k=O1 22 1Q1oI = P10- I Sk 2elkeokk=O

P 1 2 - p10 ISkI2elk*eOkk=O1 1

- Po* IX I=Sk elkeok* + I1 2

1 2 = Pll - 12 lkl21~ ~ -1 _l p11 Sk e lk lk= k

1 1- Pl-l,* 1SkIekelk + Xk=0 k=0

2ISk 12eokeok* ,

2ISk 1eOkelk* ,

2ISk 12elkeOk* ,

2I k 2elkelk* ,

We assume the following to be Hermitian:0* P 00 , P01 * = P01 , P 10* P 10 , P11 * =

1G = -2 ISkI 2{Pooe0k*eok le0k*elkk=O+ PlOelk*eOk llelk elk}

= -2 I |SkI2{e0k*(P00e0k+ P(lelk)k=O+ elk*(PloeOk+ Pllelk)}

1 1( Po~e~k+ POlelk'\=-2 '~ (~k ek*k=O Pl0e0k+ Pllelk/= -2 I |Sk 2 (eok* elk*)(P 1 0 P1 1 )k=00 i, l

G = -2 IX Skl 2 ektPek.k=ONext H is obtained:

1 1H = IX IX ISE 2eoieoi* S 1 eoj*eoji=0 j=o

1 1+ eX S o2eoieli* Sj 2eoj*elji=o j=01 1

+ eXSi I2 eoieoi*Sj 1eij*eoji=0 j=01 1+ I I ISI 2elie1i* Sj 2elj*eji=o j=0

1 1= z IX| jI2 Ilj 2[eoieoj*(eoi*eojei*elj)o=0j=o

+ elieV*(eoi*eoj + eli*eV)]


2 1 2+ I ISk12elkelk*k=0


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1 1=E Ei=0 j=01 1

i=o =o

ISiI2 ISj 2(eoi*eoj + eli*elj)(eoieoj* + elielj*)

XSiI 2 1Sj121eoi*eoj + eli*eljI 21 1

= IX ESiS 1210Sj1 (eoi*i=0 j=o1 1

i=o =o

eli*) (eoi)2Si 121Sj2 eitej 12.

The authors are grateful to I. Kitano of NipponSheet Glass, T. Sueta, M. Izutu, and H. Nishihara ofOsaka University, Y. Sakauchi of Sanyo ElectricCompany, and J. Minowa of Sumitomo Cement Com-pany for providing us with various kinds of target totry with our fault locator. They are also grateful toY. Imai for suggesting that we use the fault locator asa device to determine the anisotropy of a crystal.K. Kawashima of Optical and Radio CommunicationResearch Laboratories was helpful in preparing thesamples. H. Murata of Furukawa Electric Companysuggested the possible use of a monitor during thefabrication of the fiber. They acknowledge MaryJean Giliberto for assistance in preparing the manu-script.The authors are grateful to T. Manabe and H.Shimodahira of ATR for technical discussions onneural-network theory and Y. Furuhama of ATR forenthusiastically supporting this project.

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