AD-A105 929

OPTICAL PROCESSING OF ULTRASONIC WAVES

Department Of Mechanical EngineeringHouston, TX

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U.&, 0"~u d WNWNational Technical Infrnutice Service

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Final Report

to the

Office of Naval ResearciContract: N00014-80-C-059I

OPTICAL PROCESSING OF ULTRASONIC WAVES

submitted, by

Copy available to DTIC does notpeimit fully legible zeproduction

Bill D. CookDepartment of Mechanical Engineering

Cullen College of EngineeringUniversitý uf Houston Central Campus

Houston, Texas 77004

This document has been appr,-, 9dfor public release and sale; its

REPRODUCED By di-tbibution is unlimited.NATIONAL TECHNICAL

INFORMATION SERVICEUS DEPARYMENI OF COMMERCE

SPRINGFIELD, VA. 22161

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THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DTIC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOTREPRODUCE LEGIBLY.

During the year prior to the contract with ONR, the

principal investigator had a similar contract. During

that year the principal investigator had developed the

theory and algorithms of the prediction of the scattering

of light by sound and the inversion procedure of

investigating the sound fields from the scattered optical

data. For the purpose of this report, these theories

will be known as the direct theory and the inversion

theory.

During the Spring of 1980, the principal investigator

trained two students, Mr. Charles Fray and Mr. John R. Laflin

in the aspects of acousto-optic interaction.

The objective of this contract was to allow the

principal investigator and his graduate students to work

with colleagues in the Physical Acoustics Branch of the

Naval Research Laboratory, Washington, DC to implement

algorithms on their computer data acquisition system.

During the stay at NRL, Mr.. Fray, in conjuction

with others, developed a modelling system which would

precidt the colored schlieren patterns of ultrasonic

fields. The output of this model was by colored

television display of computed values. The work of

Mr. Fray will be continuted by those at NRL.

O110eOpy

INSPEcTrco

-2--

Mr. Laflin pursued the inversion problem. That is,

he developed a computer based experimental system to

acquire acous'o-optic data and process it to reveal the

complicated near field of an ultrasonic transducer.

The principal investigator directed the students,

collaborated on a new theory for tomographic processing

acousto-optic data, and generally supported the Physical

Acoustic Branch with theory and concepts in acousto-optics

and scattering of sound.

During the extension period of the contract from

September 1980 to May 1981, substantial progress has

been made in the graphic routines associated with Doth

the direct theory and inversion theory.

Attachment A is a preprint of a manuscript resulting

from the tomographic work which was presented I:o the

Acoustic Imaging conferences, Monterey, CA, Spring 1981.

Attachemnt B and C are abstracts of papers presented at

the Fall 1980 meeting of the Acoustical Society of

America, Los Angeles, CA.

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`_3 ABSTRACT ,

3_•The principles of computeriied transverse tomography can be .I_applied to the acousto-optic reconstruction of the local sound 33pressure of an ultrasonic field. 1 For sufficiently narrow beantsof ultrasound in the low megahertz region, the total opticallphase retardation of an interrogdting light beam crn be consi-

aedered as a projection of thce sound field pressure. (Fouriet be3i !techniques for the numerical recdnstruction of the pressure field

-yield as intermediate steps a Fourier domain associated with theangular spectrum of plane waves comprising the sound field.) Con-- 3

- equently the sound field can ýe reconstructed in other regionsthan the plane of interrogation. In this work we discuss two al . .

-ternative methods for acquiring data. One method builds the'VI. IFourier domain along radial spokes which is Inconvenient for

3 ;numerical processing by DFFT algorithms. The other procedure bu-ilds the Fourier domain in a nearly rectangular format compatibleWith two-dimensional DFFT algorithms. With this latter method,"t is possible to evaluate the pressure along a line with linited'data and a one-dimensional DFFT.

• Work supported in part by Physical Acoustics Center Program atthe Naval Research Laboratory, Code 5130, Washington, D.C. .

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2 2

__ ~INTRODUCT.ION ____

A sound field of low ultras"nic power, low ultr!sonic, fre _-jquency, and narrow beam width behaves as a4 optical phase grat-

l ing. Collimated light passing through such a sound field ezperi-

8ences an optical phase retardation proportional to the -i ocalsound pressure, integrated over ihe light path.. This constit4Ite3 'I -e_~

Lo j 7projection'! of the sound field and nunerk~al methods o o - 11 e : tomiz. a e applied [tos t... O,,.

1 local sdund pressure. ___ _-2___ __

23. 1~ lXRDC ..O _-_"_:"- ___--._ j! .__

14~ Numerical techniques using Fourier transforms are useful in 1'

_....._ ___ _

5 pressure field evaluation sinct an intermediate.stepyieldr f01' Fourier don, narrw bea with beha ang op t h q __r

1.7 waves comadigtp sound a e i

18 the phase treWidtardation propo a oto the oa %.19 possrble to__onstruct the sounid falit~et~ planof_ ...npi !_0_20 other word... n t of os f o.... . - Lo the, !d IM 2021 puted from a set of acousto noptic datea taken oVer a single 23.

22 transverse v ,. 2223 ed evopry. ,t f . e__ov YeLsou.d. suit•.e-a •J .__ not accounted foru.This total field concept is valid wthen the "_),.• sound field can be described by ;the el.mholtz equation, thus el- 226 iminating application to non-linear or highly attenuated sound ! 262-7 fields.28 . 28

2In a series of papers 1- 3 directed toward transducer ca'l1- _2.30 bration Cook and Berlinghieri have described one method of data ' 303-1 collection which we will call Metihod A. Acousto-optic data is32 collected at the terminus of the light paths as shown in Figure ; -3.. 1. These light paths are parallel to each other and are in a 333h plane parallel to the surface of !the transducer. Sufficient data __

•_can be acquired from interrogatio,,n of the file]d in one direction ,3.5.if the field is symmetric. If the field is not symmetric, Method

_3:,_A involves rotation of the transd ucer about an axis normal to the ...transducer surface, such as CC'.,

_ lHere, we present an alternative method of data collection

which we will call Method B. in Method B the transducer is ro-S.. tated about a line AA' parallel to the transducer surface and lo-

-cated in the plane of the light paths. The line AA' is also per-', pendicular to the light paths.lie will demonstrate how both methods allow the generation of

1,7

-4 Idata in the angular spectrum (plane wave dcconposition) domain"". !with both phase and amplitude information to allow evaluation of

-:L Ithe pressure field at the plane of measurement using Fourier• transforms. . 4 *.,

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Data a-'.pectrusdo-_4main in a polar format through a! series of one-dimensional Four-.

lier transforms. The pressure field at the plane of measurement

-! can then be obtained by a two- dimensional inverse Fourier trans-7form. DFFT algorithms, however, ;require data to bh in a rectan- ._7

8gular format. Two alternatives are to interpolate the rectangu- 8.9 lar data from the polar data or to perform a Hanke. __

10 ourier-Bessel transform.Th olar data becomes less dense 10

12 there. On theoth _d, dheedqpment of efficient.algorithms .1213 f_ iýHnkgI--Etn•'fks is now an ýctive area of research.A- - ,l4 :0at.4(15 'Method B exhibits three attractive features. The first is- .

6that throue f one-r xmien-s•_ 16IlT spectrum domain can be built in d format which closelyapo 1713 mates a recta__gu•r,rasýte..IOThe' second _feat:uL'rej..'that.the a.. :L19 collected methd.l._smid .JJ:] t ]..t pt .9.20 domain and-,'he1 -,Ai- -sz._dona in. 1a_.p..r21 field at the plane of measurementi, therefore, requires only a 2.22 ' e .os4ieen , lez nstdnal--txans~forms-...- &-tbird-f eatureJ 22

_21' to the direction of sound propagation bly a single inverse 2.425 one-dimensional transform. .5

26,

A 2.0 THEORY OF METHOD A 28

._0_ Consider a harmonic sound ffeld being produced by a planarz ._O3L transducer. Let the pressure at p distance z from the transducer ..:]_12 be expressed as -i

2% I 33314

-3.1 P(x,y,z,t) = DK,y,z)exp(-jw•t) (1) 3.'

3.i '' _3,

In the following discussion the t'ime variance will be dropped for 38)

,39- convenience. . ---.8ho:1 Line integrals across the pressure field at: z+zo can be. t ritten ' j j

.1- 3 1.3

-(xzo) f- (x,YZ 0 )dy (2),'., I p ,

w'here the limits of this integral, and others are taken from minus I-infinity to plus infin'ity. P(x,zo) can be seen as a "projection" I --

!of the pre3sure field P(x,yz 0 ). .5 1-

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3 In the -.6Etr i 6se r n .z sotained from"l- measurement of the Raman-Nath' parameter'V defined as _- . I .__6

.- V(x,zo) = [27-r ] A (x,zo) (3) _73 ~8-9 _.9..-O- where X is the optical wavelength in vacuum and k is the medium 1O

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12 chan in acoustic preKss!u.._...hi.s..p_.a .ri _L__ieýq.sixjL 2-3 •'oD _ • als!D aouru-_etardation ifrduceg h ay_ t f.ip

a common paranter used in most theories and can be inferred from 3._!15 acoustQ-2pti----- ..•.- - i5

_ 6 be measured\Iaý' tipBus t ehi er.Jra q rn the.- _1627. necessar h c ihe b.ie f1Q7tn?.nLi*_h _

1 terature ...............

e219 '1_9.ý0 We wil.. _li•.T•re~tpAn_ .t~w.j e C gi t. e.da..p r.•s sue. -nud• 20._21 the Fourier domain assuming p(x•zo) to be a measurable quantity 2 1

22 _%-_ ýs-tit-u.t QrLof-.-tw.o-_dimes ion a' .trans forxi ex BePrssion -_into..tthe .r2_22~3 integKr.pfL v _24 - . -24

26 .(x,zo) fff(kx,ky;zo)exp[j(lg)x'rky)ldydlcxdky (4) 26

23 128Swher& kx and ky are components o4 the acoustic wave vector k32. The integration of the y-variable can be completed yielding the 39:LDirac-S function 2w& (k). The sifting properties of the 3.

_2.,. 6 function upon integration over lk produce the desired result .32

13 1 33

ý(X~o) l/2wf(kxO.z)exCkx)dky (5)1 336 ,z J36

31,8 This result, sometimes terefrred "to as the "Fourier_ projection-slice theorem,, states that the Fourier transform of a

projection is a slice of the Fouiier transform of the projectedfunction. In other words, the one-dimensional transform of .. _

:-.. p(x,zo) produces a single line in the Fourier domain. This lineL- ilies perpendicular to the direction of the light paths, that is :43

I' the line of interrogation. Consequently, if one rotates the ...

!-5 1transducer as specified for Method A in equal angular increments --" i (essentially around the sound field axis), the projections obta-

l- ined are related to values in ;the Fourier domain located along:

"!- radial lines. In other words, taking a discrete one-dimensionalL transform of the projections p (X,Zo) yields values shown as cir- 7-:-10 eles in Figure 2. This result is not restricted to sound fields 0

la and-is --a--general--result of- projectio.n-,theory. - ..V. . . . . ... ... . . . . .. _ _ _ _ __ O'" - , '• L .',.. . • 1 "

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4" Fig. 2. Fourier domnain, Method A. !i[

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in. t................ .. ..... - -sa d.ma . ...romt ata_l hin the "Fourier "domain to. th'e •tiine'-spTacdomain from the polar -'•

""5 format. In addition to interpolation to a rectangular format, _-- Mersereau and Oppenheim suggesL 'other schemes to circumvent this _.,1. problem.9 -(

8 '8

__ ..O THEORY OF METHOD B ".0 • _O11 FIRST LI':Er 0? TITLE _i12 Consider the ogiri the X coorgdina:.% axes to be the 1213 51 fD bf:Efi7ee?5ion of the acoustic axis a dth .a3X

l.4 light slice. Again, z is the acoustic axis. The light pat. jl.

15 slice wilp_ _- _ a.-16 the project.A7e- . _ --- 117 account for the g 2118 11819 J ..:9

20 21...... 6) 2121 -22 __ _ _ _ _ _ _ ~ * ~ ~ _ _ _ _ _ 22

23

. The Fourier domain pressure ican be expressed a. 2526' 26

"28 .(kx,ky,z) = Po(kx,ky)exp(jkzz) (7) .•.

3_1_ where Po(kx,k ) is the Fourier comtnponent at z - 0 and k. is the I3

32 z-component of the acoustic wave vector. We again substitute the 32-

.13• Fourier description of the pressutre field in Equation (6) and in- 333 corporate Equation (7) to give 3--

35 _336 ' 363037 k(x,¢,z) = Ifff•o(kx,ky')exýtj(lcx-Fkyy"'-kzz))dy'dItxdky (8)

39_I 39

40- The light path slice and the x-axis defines a new coordinate"I'_ system (x,y',z'). This is related to the (x,y,z) coordinate sys-"L_ tem by the transformation43 h4y y'cos'" + z'siný -1-i

z -ylsin., + z'cosý) )

'Substituting this expression into Equation (8) and collecting L-7

terms of integration of dy', we find the integral I -9

501SfexpI-j(kcos --- kzsiný)y' 1dy!.-,20(kycos kjsinY(.l0j 1

-- F

- oI

1 -EV-.i flt.,:.. 1,,..-: .0D!Y "00 8

j3L If we df-...i. _M_

4".56 ky ='kztan*, (11) i

w we can write the Dirac-6 functioA of Equation (10) as 2,-6(k .y

9 ky'), Equation (8) can now be written as *i0,_ " 101L 2yf f ____l-X-- -__

13 M .. _ __ *

4-" exp[j(kxx+kyz'sin4 + kzz'cosý)ldkxd] y (12).

1_7 The integral oy._dy.. can be eva.1aatged _x.Apng •4.g px.pe .4 1__........._'""__"_"__"__"_"_' •1 18

1 8 t i e s o f t h • . -.t u ; _- t_.._L_.. ..19 . .-

20D AiDADY.PESS _ ____20

21 P(x42z) = /. - fo(kx,ky') X 2 2122 y2223_ -7 ' -'23.

exp[j(kxx+ky'z'siný+kz'zcos0)]dkx (13) _PL

26 26

27 Since the origin of the coordinate system is in t:he plane of the?88 light path slice, we have z' = z= 0. Equation (13) now becomes AR

_30 3031 ý(x,ý,zo) = /21rfo(kx,ky')exp(jkxx)dk. (14) .A

13133

IL Equation (11) can be restated as, 3;

36 36I2_ 19_ ,Cb (k i-k 2 )sin4- (15a'. -

40 If kx is small compared to k, th4n the foillowlng approximation .

holds . , I -i

I..51I13Syl kn4 (15b)

' When this approximation is sibstituted into Lquation (14),"" ip(x, ,zo) becomes p(xlky' ,z0 )

I I | .....

-"[... ~(~y)z"~~ 2" f )o(kk 1 )exp(-jkyx~dx (16) '•• ".. ... ... ..... .. ..... . .. .. :.',- . ..-. ..-_: : . ... .-.-- . .- ..... ----. --

22 Wes.t 17t stre1.New York, ;:.Y. J.O012 V X

.1 EV_ - RmIai:J .HAFD _'G_ 9 Y 9EVE?. __ __ _ __ __ _

2 J3 which is the'fifll YU 1 , jh ..

It is important to note that g(xky ,zo) lies midway between-6 the Fourier domain and the time-space domain. If we take the in- 1

verse Fourier transform of P(x,kf ,zo) -with respect to the vari-__ able ky', we obtain 8

3. /7f jjy mQ2 52f:Jy zo cP ip y s± ys h ~ y~l 10~I92 _9 _ _

20o_. y- c "i "" Azyo22 ,uf. .---'-- _ __ __ _i

12 12

2732

J,.33 JMM.l•~, 2. f p>(k x.ky_•. Ego) _[l.xX+k y~y).J..yjkdky- 11-.. 1_3.

28zk•k',o , zo[xCjxx1 1) .__.'

15.. . .. .. 116 The term orýt-'f h -,-and.~de.cn.h_ gi.lg . -4,-,-_pii.e. _L1__

17w Thus fromthae setsofmefasurledmnt taken atFa given elevation 3.!x 17

._9transform _.t.p_b.k~ain_ e._pjes.s; •a~_lraisfo_ __au. 1_9_2•0 of x. The j _; .• Z _e _a.speqi4..• _ed e_-_.p a. _ _alro _t•__ Q j.ag =._ 2__0

1_ined by a series of such transý-orm s taken at equally spdced va- 2]1.

23 23"pIf we take the Fourier transform of a(x,kl Zo). with respect 24

to the variable x t we obtain di a ba *_26 2 6

28 P(kxtky, zo) i i(x1:ky',zo)[exp(-jkxx)dkx (18) _?.

3t_0 s3.1 hich is the pressure field in the Fourier domain. 3q o

3. Implementation of Method B ursing DFFT algorithms requires t.3o

.b the approximation that if' does oot depend on kx. This approxi..marion is valid when the ultrasoand is confined to a narrow beam _3 .

]j6 as with sound field produced by maost medical and NDE transducers. .. 6-i I ...

wer To illustrate this approximation we show in Figure o th'e 1 33-2.. nearly rectangular for'mat for 7data obtained from Equation (16) -A-!h l using an DFFT. The error incurrtd by assuming this foroat to be enc u

rectangular will be small if most of the radiated energy in near*''ithe origin in the Fourier domain.; The illustrotion in Figure 3 -'1.3- iis for a sound field produced by a circular transducer of radius -}

... a = I0X where X is the acoustic O4avelength. Each of the larger 7"!5concentric circles has associat~ed with it a percentage of total --

were calculated using an Airy !pattern to approximate the sound ...."

•.-field. Figure 3 show§ the format to be essentially rectangular •-:for 98% of the radiated energy in this case. Figure 4 shows the .- ,"

S[general curve for the fraction of total energy radiated as a", function of ka(sin4A)-..........-.. . "

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-DATA3 4.0__DATA ACQUISITIOU•:USIN•:yUETtI0D __K.

5 Equations (15a) and (15b) indicate that the angle * should -5"6 change by equal increments of sin €, or that-

sin " +1- n (19) !9

-12 where n = 1 2 ,...N with 2N+1 number of experimental points .213 E W-.4j=-L c'ifie:.Ihcrement for in _ .3.

SI.041.5 The maximum value of k observed in the data is 1_

. _-__7 '116

20 2021 From Equation (15b), we also know .2122 _P__

24 (ky)max ,k(sin4max) (21) 2 h

26 26.?_. Combining Equations (19), (20) and (21) yields 21

28

30 Aky I= kc (22) 30

32 _3-,_ The increments of y in the eransformed data returned by the 331L DFFT are y = mAy where m 0,1,2,1...,M and M is the number of po- 311

.Ž.ints used in the DFFT. From the :periodicity of the DFFT, we findI 35

IlAk~Ay = v(23)1 3

39

or ~3II

I:y , f" 3.2

"1.>-where c is the sound speed, and t~he acoustic frequency f c-- /X.

MAy c/(00f) . (25) . 9.

K... ............................................... . i .

2'T e .t. l'i L %*tr . •.%:t 6 ½ .

"e, Y±. 100.LL

1 EV. ,,-.,, ) . ...... ,., 1 13

3 5.0 EXPERIHENTAI RESULTS - -.5 Cook and Berlinghieri have eýperimentally demonstrated the _ ._5

6 validity of Method A. 2 They ge'nerated a rectangular array ofpoints in the Fourier domain by rptating the transducer through

8the proper angles and sampling .the linear scan at proper inter- 89 Ivals. They then used a two-dimensional DFFT to reconstruct the

10 acoustic field in the..._plane. of interrogation and in nearby ep _p _j ]2 I

13 ýh]f_8 as-% s.4 t--_o __clculae.___t.he__p.r._ssure ___d2s.t.i.u.tion 1

T, over a transverse plane for a 1;.25 cm. diameter PZT transducer _.4

15 Isubmersed in..w.erooThetansduce¢x[~as-operating•at_2.2_1hz4(A_ i_.* 16 .9.3 14 ad- 12 M"7 .a16__!- 9. A . -•a#!•o'fi ofRl_x_A1..".datapoists.-.using-.a-value-of_•-ý- .•-:

217 sin( 1degre)_ .were_used.o__jneas.u.ete_ ptep..urerin,_a_pla ae-5_cm..18 from the 'Tf"Mu55 7c• a i._sdaawa.used..to.- construct_'the] .1-9 pressure field-o.vyex..an-ar-ea 3..9__3..9_cm.2.The.xresuits.are-..shown._ 12.20 in Figure A5._AT.ei@e ects.of._ the.apptoximation -made,.in..Equation' 2021 (15) are noticeable in the reconstruction. The reconstructed.' 2lL22_ fieldis .slightly_.ellUL cr.j4 rather..than- circular-- due-to-the -out:---! P223 &di~d.spla~cement [_;,dae- e- mopoe tsh~..the_....recon-J 2?•21_4 struction. t

26, 26~

?7 6.0 REFERENCES -27-28 28

9. 1. J. Berlinghieri and B. Cook, J. Acoust. Soc. Am. 2?29.30q 58, 823-827 (1975). " .3231 31_

2. B. Cook and J. Berlinkhieri, Proc. IEEE Ultrason. "V_33 Symp. 75, CHO 944-SUI 133-135 (1975). 133.3-LI ].J

.L 3. B. Cook and J. Berling ,ieri, 3. Acoust. Soc. An.36 61, 147-1480 (1977).

4. J. P. Clero and C. H.; Durney, Proc. IEEE 67, 14-63 32.. __(1979). ! ..3i9 .1;0

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}-." B. g. Cook, J. Acoust. Soc. Am. 60, 95-99 (1976). [ "

9. R. Nersereau and A. Oppenhein, Proc. IEEE 62,/_' U . .... . . .13 19- 1"338 (1974 ). ' -- ' ' ,;;. . . .. . . . . - --- -

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