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Real time liquid surface acoustical holography Pille, Peter 1972

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C ' REAL TIME LIQUID SURFACE ACOUSTICAL HOLOGRAPHY by Peter P i l l e B. Eng. Careleton U n i v e r s i t y 1970 A THESIS SUMBITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We .accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1972 In presenting this thesis in pa rt i al fu I f i lrnen t of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver 8, Canada Date A B S T R A C T This thesis provides an analysis of an acoustic imaging tech-nique using holography and the i n t e r a c t i o n of acoustic and l i g h t waves at a liquid-gas i n t e r f a c e . Real time o p t i c a l images of objects that have been transilluminated with u l t r a s o n i c waves i n a l i q u i d medium can be obtained. The l i q u i d surface acts as a detector of the u l t r a s o n i c energy. When coherent l i g h t i s r e f l e c t e d o f f the l i q u i d surface an o p t i c a l image of the object i s obtained. An analysis i s presented of the mechanisms involved i n c l u d i n g an analysis of the transient motion of the l i q u i d surface. i TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF ILLUSTRATIONS • i i i ACKNOWLEDGEMENT v 1. INTRODUCTION 1 2. LINEAR SYSTEMS ANALYSIS OF THE IMAGING PROCESSES 2 2.1 The Basic System 2 2.2 Surface-Light I n t e r a c t i o n 6 2.3 Plane Wave Reference . . . . . . . . . . . . 7 2.4. Focused Image Hologram 11 2.5 Lensless Fourier Transform Hologram . 18 3. RADIATION PRESSURE AND THE FUNCTION g . . . . ' 23 3.1 Radiation Pressure of Sound Waves 23 3.2 E f f e c t c f the Minitank =cn the Radiation Pressure . . . 28 3.3 The Transfer Function g^ 31 4. LIQUID SURFACE RESPONSE TO RADIATION PRESSURE . 39 4.1 Steady State Response 39 4.2 Transient Response of the L i q u i d Surface 41 4.2.1 Formulation of the Problem 41 4.2.2 Results f o r Single S p a t i a l Frequencies . . . . 45 4.3 The Surface S p a t i a l Frequency Response to Pulsed Radiation Pressure 53 5. ACOUSTIC BEAM INTENSITIES, SOUND FIELD PARTICLE OSCILLATION, SURFACE BULGE 63 5.1 The U l t r a s o n i c Beam I n t e n s i t i e s 63 5.2 The Ultrasound F i e l d P a r t i c l e O s c i l l a t i o n 65 5.3 The Surface Bulge 66 6. CONCLUSIONS 70 APPENDIX A Fouri e r Transforms, D i f f r a c t i o n , Imaging . . . . . . 73 APPENDIX B Numerical Solution of L i q u i d Surface Motion . . . . 77 APPENDIX C L i q u i d Surface Response 82 REFERENCES . . 83 i i LIST OF ILLUSTRATIONS Figure Page 2.1 The b a s i c imaging system 2 2.2 Lig h t r e f l e c t i o n o f f the surface h(x) 7 2.3 Focused image hologram 12 2.4 Optics for focused image hologram 13 2.5 Transform plane 15 2.6 Lensless Fourier transform hologram . . . 19 3.1 Waves at a boundary 25 3.2 Minitank 28 3.3 Reflections i n the minitank 29 3.4 |g-^(n,0)| vs. r\/ky for values of d/A^ from 2 to 2 1/2 . . . 33 3.5 /g^Cn.O) vs. v j k ± f o r values of d/A2 from 2 to 2 1/2 . . 34 3.6 |g (n,0)| vs. n/ky for values of' d/A 2 from 9 to 9 1/2 . . 35 3.7 /g^Ql.O) vs. n / ^ f o r values of d/A 2 from 9 to 9 1/2 . . 36 4.1 L i q u i d surface deformation . . 41 4.2 Response of l i q u i d surface to step function r a d i a t i o n pressure 46 4.3 Response of l i q u i d surface to a pulse of r a d i a t i o n pressure 47 4.4 Overdamped l i q u i d surface response to step function r a d i a t i o n pressure 48 4.5 Response of l i q u i d surface to an impulse of r a d i a t i o n pressure 49 4.6 L i q u i d surface (freon E-5) s p a t i a l frequency response to step function r a d i a t i o n pressure 55 4.7 L i q u i d surface (water) s p a t i a l frequency response to step function r a d i a t i o n pressure 56' 4.8 L i q u i d surface (freon E-5) s p a t i a l frequency response to a pulse of r a d i a t i o n pressure 57 i i i Figure Page 4.9 L i q u i d surface (freon E-5) s p a t i a l frequency response. to a step function r a d i a t i o n pressure with a f i n i t e f l u i d depth 58 4.10 L i q u i d surface (freon E-5) s p a t i a l frequency response to a pulse of r a d i a t i o n pressure with a f i n i t e depth of f l u i d 59 5.1 Radiation pressure 68 5.2 Radiation pressure 69 A . l D i f f r a c t i o n 74 A. 2 Lens 74 A.3 Imaging with a lens 75 A. 4 Fourier transform with a lens 76 B. l F l u i d d ivided i n t o a rectangular mesh of c e l l s 77 B.2 P o s i t i o n of pressure and v e l o c i t i e s for c e l l i . j . . . . 78 B.3 Deformation of the l i q u i d surface . . . " 78 i v ACKNOWLEDGEMENT I wish, to express my gratitude to Dr. B.P. Hildebrand who was my o r i g i n a l supervisor and for suggesting the to p i c for the t h e s i s , and for h i s continuing i n t e r e s t i n my progress. Also, thanks to Dr. R.W. Donaldson for h i s support during the l a t t e r stages of the work as my supervisor. I also wish to acknowledge Miss Norma Duggan for typing the t h e s i s . Also, I am gr a t e f u l to the National Research Council f o r f i n a n c i a l assistance. V I. INTRODUCTION Since 1966 much research has taken place applying the p r i n c i p l e s of holography to a c o u s t i c a l imaging methods, with ultrasound x>?aves used to i l l u m i n a t e objects, and with images reconstructed using lightwaves. Much of t h i s work i s described i n references [1-3,], and a review of the important developments and r e s u l t s are described i n reference [4], Of i n t e r e s t for t h i s thesis i s the method of obtaining images of objects using a l i q u i d surface to detect the ultrasound energy that i s used to i l l u m i n a t e an object. E a r l y work i n t h i s method was done by Mueller and Sheridon [5], Green [6], and Brenden [7]. Applications of the technique are i n the nondestructive t e s t i n g of materials [8], and p o t e n t i a l l y for medical diagnosis i n imaging human tissue structure (e.g. detecting tumors). The purpose of t h i s thesis i s to provide an analysis of the form of l i q u i d surface a c o u s t i c a l holography that gives images i n r e a l time. The treatment i s an extention of the e a r l y work i n references [5-7] i n that the image obtained i s r e l a t e d to the object by t r a n s f e r func-tions that are c h a r a c t e r i s t i c of the l i q u i d surface and the geometry of the imaging method. An analysis i s made of the transient motion of the l i q u i d surface used and i s shown to be of c r i t i c a l importance i n the performance and understanding of the imaging system. Chapter 2 presents a t h e o r e t i c a l analysis of the imaging process i n terms of the t r a n s f e r functions that describe the system. In Chapters 3 and 4 the p a r t i c u l a r form of these t r a n s f e r functions are derived. 1 2 I I . LINEAR SYSTEMS ANALYSIS OF THE IMAGING PROCESS This chapter presents an analysis of an imaging technique [5, 6,7] which provides r e a l time o p t i c a l images of objects that have been transilluminated with u l t r a s o n i c waves. A hologram i s formed at a l i q u i d surface by u l t r a s o n i c waves, and then an image of the object i s obtained by r e f l e c t i n g l i g h t from the l i q u i d surface. In sections 2.1-2.3 the b a s i c r e l a t i o n s h i p s for the system are derived. In sections 2.4 and 2.5 two d i f f e r e n t imaging methods are analysed and the o p t i c a l images obtained are r e l a t e d to the a c o u s t i c a l f i e l d which describes the o r i g i n a l object. Since t h i s form of imaging i s space and time dependent, three dimensional Fourier transforms and convolution operations w i l l be used, with the notation defined i n Appendxx A. 2.1 The Basic System We s h a l l consider the formation of a c o u s t i c a l holograms as shown i n figure 2.1. Water filled. tank Figure 2.1 The b a s i c imaging system 3 I t i s desired to obtain an o p t i c a l image of the i n t e r i o r of an object which i s opaque to l i g h t . The object i s placed i n a water f i l l e d tank and transilluminated by u l t r a s o n i c waves from a transducer t Q . The acoustic x^aves transmitted through the object i n t e r f e r e with a reference x^ave of the same frequency produced by transducer t to form a hologram at the l i q u i d surface of the minitank. The f l u i d i n the minitank i s separated from the water of the main tank by a t h i n membrane which allows ultrasound waves to pass through but prevents any large scale disturbances i n the main tank from dis r u p t i n g the surface. Also, since the f l u i d properties are important i n the imaging system, the f l u i d of the minitank can be chosen x^ith properties that are more s u i t a b l e for imaging than i s x^ater. The acoustic waves are r e f l e c t e d o f f the surface and create a varying r a d i a t i o n pressure pattern on the surface. This pressure i s p a r t i a l l y determined by the i n t e n s i t y of the acoustic waves at the surface, which causes the surface to deform i n t o a pattern of r i p p l e s that represents a hologram. An image of the object i s then obtained by r e f l e c t i n g l i g h t from a l a s e r o f f the surface. The acoustic f i e l d d i s t r i b u t i o n i n a plane (x,y) i n the f l u i d xtfill be represented by the v e l o c i t y p o t e n t i a l [15] §(x,y,t). Since if) i s a s c a l a r and s a t i s f i e s the s c a l a r wave equation, the s c a l a r laws of d i f f r a c t i o n apply d i r e c t l y to <J>. Also, <j> w i l l be considered to be a complex quantity, and since the u l t r a s o n i c f i e l d w i l l be monochromatic, there x ^ i l l be an implied time f a c t o r of the form e (where j = / - l ) . However, t h i s time dependence w i l l not be e x p l i c i t l y shown, and the time dependence of <(>(x,y,t) w i l l r e f e r to time changes i n amplitude, f o r example a step change 4 <Kx,y,t) = <j>(x,y) u(t) where • , Cl t > 0 ^ ( t ) = ( o t < 0 Consider the v e l o c i t y p o t e n t i a l <t>^(x,y,t) i n c i d e n t at the sur-face (x,y plane) to be the sum of the v e l o c i t y p o t e n t i a l due to the object <{>^  (x,y, t) , and due to the reference cj> ( x , y , t ) . Then ^ ( x . y . t ) = ^ ( x . y . t ) + <|>r(x,y,t) (2-1) A r a d i a t i o n pressure i s exerted on the surface due to <j>^. However, t h i s pressure d i s t r i b u t i o n i s dependent on the angles at which the various plane wave components of C|K s t r i k e the surface. I t w i l l be shown i n Chapter 3 that we can consider an e f f e c t i v e value of the i n -cident f i e l d to be given by 4> f f.(x,y, t) = .^(x^y ,t) * g1(x.,.y.,'t) (2-2) The symbol * denotes a three dimensional convolution (see Ap-pendix A) operation i n the x,y, and t dimensions. The complex function g^(x,y,t) i s the e f f e c t i v e v e l o c i t y p o t e n t i a l at the surface when the inc i d e n t v e l o c i t y p o t e n t i a l i s an impulse 6 ( x , y , t ) . The time dependence of g-^(x,y,t) i s due to the i n t e r n a l r e f l e c t i o n s of the ultrasound waves in s i d e the minitank. The r a d i a t i o n pressure d i s t r i b u t i o n p^ on the surface (x,y) i s then given by P r(x,y,t) = c | * e f f ( x , y , t ) | 2 (2-3) This r e s u l t i s derived i n Chapter 3. The constant C depends on the density of the f l u i d i n the minitank and the wavelength of the u l t r a -sound used. The height of the deformation, h, of the l i q u i d surface from 5 the quiescent l e v e l (z = 0 plane) i s dependent on the properties of the f l u i d , the s p a t i a l frequencies inherent i n the function p^ . and time t. I t w i l l be shown i n chapter 4 that h(x,y,t) = p r(x,y,t) * g 2(x,y,t) (2-4) The function g 2(x,y,t) i s r e a l and represents the response of the l i q u i d surface to an impulse 6(x,y,t) of r a d i a t i o n pressure. From equations (2-1) to (2-4) the surface height i s given by h(x,y,t) = C|(cj>r +'<j,1) * gx\2 * g2 (2-5) We can write (2-5) as h(x,y,t) = h r(x,y,t) + h Q(x,y,t) + h ^ x . y . t ) + h*(x,y,t) (2-6) where hr(x.,y.,t) = C|(j>r *.g ri 2 *.g 2 (2-7) h o(x,y,t) = C l ^ * g x | 2 * g 2 (2-8) hiCx.y.t) = C{((|.r * g-J* (^ * § 1 ) } * g 2 - (2-9) and the symbol denotes the complex conjugate. Note that the height h^ i s proportional to the square of the amplitude of the reference beam, h Q i s pr o p o r t i o n a l to the square of the object beam amplitude, while h^ and h^ are proportional to the product of the reference and object amplitudes. In l a t e r sections i t w i l l be shown that the u s e f u l l i n f o r -mation from the object being imaged i s contained i n the functions i n -volvi n g h 1(x,y,t) . Experiments [7] have shown that considerably b e t t e r images can be obtained i f advantage i s taken of the transient motion of the l i q u i d surface. Thus the transducers t and t are used to produce ^ r o pulses of acoustic waves, and the f l u i d surface i s sampled at some 6 optimum time by r e f l e c t i n g pulsed coherent l i g h t o f f the surface to obtain a r e a l time image i n the o p t i c a l f i e l d a f t e r processing through sui t a b l e o p t i c s . 2.2 Surface-Light Interaction The o p t i c a l image of the object i n the water tank can be ob-tained by r e f l e c t i n g coherent l i g h t o f f the l i q u i d surface. The r i p p l e pattern on the surface causes phase changes i n the r e f l e c t e d l i g h t which w i l l be determined i n this s e c t i o n . We s h a l l consider a beam of plane p a r a l l e l waves of l i g h t to be r e f l e c t e d o f f the surface h(x). The surface h(x) (figure 2.2) i s considered to be l o c a l l y f l a t , with the radius of curvature large com-pared to the wavelength of l i g h t . The r e f l e c t e d wave at the surface h(.a) i s U = B e ~ j k d e i k [ ( a + c ) s i n 6 z ] £ j k [ ( x - a ) s i n ( e z - 6 s ) + (z-h)cos (Q^S,,) ) s (2-10) where a,c and d are defined as shown i n f i g u r e 2.2. The f i r s t exponential f a c t o r accounts f or the phase l a g from the value of the f i e l d at (x,z) = (a+c,0). The second f a c t o r i s the value of the f i e l d at (a+c,0). The slope of the l i q u i d surface w i l l be quite small so that 8 << 1 and, by s e t t i n g z = 0 i n (2-10) then the s l i g h t d i s t r i b u t i o n U"s becomes, f o r a two dimensional surface h(x,y) , U (x,y) = B e - j 2 V e j ( u * X + V > (2-11) s where \ = k J l C O S 6x£ 7 A i s the wavelength of the l i g h t , and cos 0 , cos 8 and cos 6 are the d i r e c t i o n cosines of the. wave r e f l e c t e d o f f the surface z = 0. We assume that (2-11) represents the l i g h t f i e l d d i s t r i b u t i o n i n the plane z = 0. This i s an approximation which we can say i s v a l i d i f the r i p p l e pattern of the surface , or the space varying component of h i s not large compared to a wavelength of l i g h t . For a much more d e t a i l e d treatment of the r e f l e c t i o n of l i g h t o f f rough surfaces than than which w i l l be used i n t h i s thesis see references [9,10]. Figure 2.2 Light r e f l e c t i o n o f f the surface h(x) 2.3 Plane Wave Reference In t h i s s ection x^ e s h a l l determine the r e l a t i o n s h i p between the l i g h t d i s t r i b u t i o n at the l i q u i d surface and the acoustic d i s t r i -bution <j>, at the surface when plane waves of l i g h t U and acoustic J- s 8 reference d> are used, r The reference transducer t produces plane waves (j)^ , so that the surface z = 0 we have * r ( x , y , t ) = ^ ( x - y . t ) e j ( u r x + V ° (2-12) where u = k cos 6 r s xs v = k cos 6 r s ys . k = 2TT/A s s 2 2 2 cos 8 + cos 8 + cos 8 = 1 xs ys zs A g i s the wavelength of the u l t r a s o n i c wave i n the f l u i d i n the minitank and cos8 are the d i r e c t i o n cosines of the plane wave. T y p i c a l frequen-cies are 1 to 10 MHz. The amplitude R^ i s shown as a function of time i f the ultrasound i s pulsed and as a function of x and y to account f o r deviations from a perfect plane wave. Using (2-12), and with some manipulation we f i n d that the sur-face deformation h(x,y,t) given by (2-6) can be written as h(x,y,t) = h r + h Q + h 2 e " J ( u r X + V> + h* e ^ ( u r X + V > (2-13) where h 2 (x,y,t) = (R* «fr2) *'{g 2 ej(V + v f y ) } (2-14) R 2(x,y,t) = R x *' { g l e " j ( u r X + V 0 } (2-15) cb 2(x,y,t) = ^ * g x . (2-16) The function R 2 i s determined by the amplitude R^ of the reference wave i n (2-12), and i f R^ i s a constant then R 2 i s a constant, simply R 2 = R 1 ^ ( u r , v r , 0) (2-17) 9 where g1 (x,y,t) ^ > g (u,v,(o) (see Appendix A, Fourier Transform) If we write the function (x,y,t) i n (2-14) i n the form h 2 (x,y,t) = h a (x,y,t) e U ^ ' V > t ) (2-18) then (2-13) becomes h(x,y,t) = h + h + 2 h (x,y,t) cos (u x + v y - a(x,y,t)) r o 3. i j-(2-19) Thus i n the t h i r d term of (2-19), the information from the ob-j e c t f i e l d h^ (x,y,t) i s represented as amplitude and phase modulating n J a s p a t i a l c a r r i e r of frequency yu^ + , producing v a r i a t i o n s i n the height of the f l u i d . From (2-14), (2-15) and (2-16) •h2 Cx,y,t) = • { • R 2 x (^ * •g 1)} *"{g2 e J ( u r x + v r y V (2-20) * I f R 2 i s a constant, given by (2-17), then the function h 2 (x,y,t) represents a f i l t e r e d version of the f i e l d function <j>^  at the surface of the f l u i d . Then i f we take the Fourier transform of (2-20) with R 2 constant, h (u,v,a>) = R 2* ^ (u,v,w) g (U,V,CJ) g 2 ( u - u r > v-v , u>) (2-21) In (2-21) we can consider the function <j>^(x,y,t) to have been passed through a f i l t e r with tran s f e r function g± (u,v,w) g 2 (u-u r, v - v r, to) and amplified by R 2 . I f i s not a constant then there i s some ad-d i t i o n a l d i s t o r t i o n of <J>^  as given by (2-15) and (2-20). The more slowly the amplitude R^ i n (2-12) varies i n space and time, the l e s s 10 the d i s t o r t i o n of the object information cj> . When a coherent plane wave of l i g h t i s r e f l e c t e d o f f the sur-face we obtain a d i s t r i b u t i o n given by (2-11). In order to evaluate (2-11) we make use of the following r e l a t i o n . .-jacosb °? , ..n _ . . -jnb e = Z (-j) J n ( a ) e n=-°° where J i s a Bessel function of the f i r s t kind of order n. Then from n (2-11) and (2-19) the l i g h t at the surface becomes U (x,y,t) = ! U (x,y,t) e j ( u n x + V n y ) (2-22) n=-°° where TT / ,_\ / . \ H T, ~J2w n(h + h ) T l N ina TJ (x,y st) = (-j) B e £ v r o J (4w h ) e J n ^ n Z a u = u„ - n u n I r v = v. - n v n I r As mentioned e a r l i e r (2-22) i s v a l i d i f the space varying component of the surface deformation h(x,y,t) i s not large compared to a wavelength of l i g h t . Any constant component i n h i s of no concern. Equation (2-22) consists of a se r i e s of sums of three d i s t i n c t f a c t o r s . The fa c t o r J (4w h ) e?na contains the information of the n £ a' a c o u s t i c a l f i e l d d i s t r i b u t i o n <j>^ . A look at graphs of Bessel functions shows us that J i s of no value to us. However f o r nl > 0 we see that o 1 1 near the o r i g i n there i s approximately a l i n e a r r e l a t i o n s h i p between J (x) and x. Since n (0) = 1/2 then f o r 4w„h << 1 we have i a J l < 4 W 4 h a > ~~ 2 V a 11 For l a r g e r values of 4w^h we s t i l l have an approximately l i n e a r re-l a t i o n s h i p up to values of the argument of about 1. Thus f o r n = +1, from C2-18) J 1(4w J ih a) e j a ~ 2w^h2 (x,y,t) (2-23) The f a c t o r e j 2 w £ ^ r + o^"* causes phase d i s t o r t i o n s i n the l i g h t d i s t r i b u t i o n . The term h Q can be made small by reducing the power of the transducer t . I f h i s a constant ( i . e . R, a constant, (2-12)) o r 1 then i t introduces no d i s t o r t i o n , and then h^ represents a uniform height or bulge of the f l u i d surface due to the reference u l t r a s o n i c wave. A c t u a l l y , i n p r a c t i c e the f l u i d may not be uniformly r a i s e d but may be d i s t o r t e d . This can be minimized by pu l s i n g the transducers and the l i g h t source so that the surface i s sampled before d i s t o r t i o n e f f e c t s become i n t o l e r a b l e . Also, we can add a noise term h^ to h^ + h Q to account f o r any noise at the surface. I f we consider the i n t e n s i t y of the l i g h t at the surface the f a c t o r e ~ ^ W ?,^r + ^o + *nN^ drops out and introduces no d i s t o r t i o n as long as the space varying component of h(x,y,t) does not become large compared to a wavelength of l i g h t . In (2-22) each term represents l i g h t d i f f r a c t e d , i n t o d i f f e r e n t regions of space. The factor exp ( j [ u 0 - nu ) x + (v 0 - nv )y]) deter-mines the dire c t i o n s i n t o which the d i f f e r e n t orders or terms with d i f -ferent values of n are d i f f r a c t e d . R e c a l l that u 0 and v are determined by the d i r e c t i o n cosines of the l i g h t beam as in d i c a t e d i n (2-11). Simi-l a r l y , and v^ _ are determined by the acoustic reference beam as given i n (2-12). 2.4 Focused Image Hologram We can now make use of the previous r e s u l t s to carry out an 12 analysis of a s p e c i f i c form of imaging. Some of the best r e s u l t s [4] have been obtained with the method of the focused image hologram. This method i s due to Smith and Brenden [11]. An acoustic lens i s used to focus the f i e l d d i s t r i b u t i o n of the object <j>o(x,y) onto the surface d i s t r i b u t i o n <j>^ (x,y) as i n figure 2.3. The imaging could also be done with two acoustic lenses which perform a double Fourier transform of <J> . r o Image plane Laser Spatial filter Figure 2.3 Focused image hologram Ignoring the time delay i n propagating from the object to the surface, and using a s i n g l e lens, at the surface the image cf>^(x,y) i s r e l a t e d to the object <j>o(x,y) by equation (A-7) i f T T 2 z /2a 2 k « 1 o w where kw = 2IT/A w w 13 and A i s the wavelength of the u l t r a s o n i c wave i n water, w & <j>^(x,y,t) appears as the function '<)> (-mx,-my,t) passed through a bandlimited f i l t e r P(V(u + v ) z /ak. ) and with d i s t o r t e d phase. v J o w r The phase d i s t o r t i o n would not be present i f two acoustic lenses were used, however, the phase d i s t o r t i o n i s of no immediate concern since we are ultimately i n t e r e s t e d i n i n t e n s i t y only. The r e s o l u t i o n i s l i m i -ted by the s i z e of the lens and the distance to the lens, so the value of ak^/z should be made as large as conveniently p o s s i b l e . In order to obtain an image from equation (2-22) i t i s necessary to i s o l a t e one of the terms with n = + 1. This i s possible since each term i s d i f f r a c t e d i n t o a d i f f e r e n t region of space. I f a lens i s used to Fourier transform the l i g h t d i s t r i b u t i o n at the surface then a pinhole i n the transform plane can be used to allow only the desired order to pass through. Then a second Fourier trans-from performed by another lens w i l l restore the desired image. Surface Transform plane Image plane Figure 2.4 Optics f o r focused image hologram. In f i g u r e 2.4 the f l u i d surface i s shown v e r t i c a l f o r con-venience. The f o c a l lengths of the lenses are f^ and f^-A- s p a t i a l f i l t e r (pinhole) i s placed i n the transform plane. The f i e l d d i s t r i -bution U.(x,y,t) represents the image of the object <j> ( x , y , t ) . 14 An exact Fourier transform i s performed by lens only i f i s of i n f i n i t e s i z e , or the hologram area i s small and contains l i m i t e d s p a t i a l frequencies. In order to reduce the e f f e c t s of v i g n e t t i n g by the f i n i t e s i z e of the lens, the lens should be of approximately the same s i z e or l a r g e r than the hologram area. The s p a t i a l frequencies i n the d i s t r i b u t i o n u* s(x,y,t) w i l l not deviate greatly from the s p a t i a l frequency of the l i g h t used to r e f l e c t o f f the surface since X0 « X. . x, w This means that there w i l l be no high s p a t i a l frequencies and no l i g h t d eflected f a r o f f axis f o r the lens to i n t e r c e p t . In (2-22) the terms for high values of n are deflected f a r to e i t h e r side of the zero order term but these high order terms are of no i n t e r e s t . Taking the s p a t i a l Fourier transform of u"s i n (2-22) with lens i n f i g u r e 2.4 we obtain i n the transform plane (see (A-8)) U f(x,y,t) = d A ^ ) ? U n (u-u n, v - v n , t ) (2-24) n=-°° where u = ( k £ / f x ) x v = ( k ^ / ^ ) y u = u - nu n SL r v = v - nv n I r Thus we have a displacement of the various orders i n the transform . plane by an amount (u„ - nu ) f,/k„ i n the x d i r e c t i o n I x 1 I (v - nv ) f-i/k. i n the y d i r e c t i o n Jo TC -i- J6 / The transform plane with the various orders appears as i n figu r e 2.5. If there i s l i t t l e overlap of the d i s t r i b u t i o n U_^ or with the other orders then a pinhole can be used to allow the desired 15 A Figure 2.5 Transform plane order to pass through with the rest of the orders blocked. From (2-12) and (2-24) and using the r e l a t i o n k s i n 0 = k s i n 8 s zs w zw the distance between the centers of adjacent orders i s r = f a (\/\ 7) s i n 8 z w .(2-25) where the angle of the reference beam to the normal to the surface i s 8 i n the minitank and 8 i n the main (water) tank. The maximum spa-zs zw t i a l frequency i n the o r i g i n a l object, k w, represents a distance i n the transform plane r = f 1 X £ / A w (2-26) However, there may be higher s p a t i a l frequencies present due to the extra phase fa c t o r i n (2-22) containing h +h +h.T. r to r o N Thus from (2-25) and (2-26) we see that there w i l l be some overlap of the various terms i n (2-24) i n the transform plane. The terms of order |n| > 1 w i l l be dim and may not cause any serious overlap, but the zero order term i s the b r i g h t e s t . To reduce the overlap the angle 8 that the acoustic reference beam makes with the normal to the sur-zw face should be made s u f f i c i e n t l y l a r g e . T y p i c a l values f o r the wavelengths may be A - 0.15 mm and A = 0.6 ym, so that A /A = 4 x 10 . 16 -3 Thus the dimensions of the various orders i n the transform plane as in d i c a t e d by (2-24) and (2-25) would normally be quite small. However, i f the f o c a l length f i s made large enough then i t may be possible to perform f i l t e r i n g with s u i t a b l e transparencies instead of simply a pinhole i n the transform plane. From (2-24) we have i n the transform plane f o r the term of order n = +1 U f(x,y,t) = (l/A^f-p \ ( u f , v f , t) i T ( u f , v f ) • P ( u f f j / k ^ P , v f f j / k ^ P ) (2-27) where u f = u - u x = ( k ^ / f p x - u £ + u r V f - v _ V ] L = ( k ^ / ^ ) y - v £ + v r p ( x y ) = / l 'for + y Z < 1 r < k X ' y ; { 0 otherwise P i s the radius of the p u p i l i n the transform plane, and g^ (u^, v^) represents the transmittance of the transparency used. I f the various terms i n (2-24) do not overlap s i g n i f i c a n t l y and p i s made large enough we can set P = 1. Also, i f no transparency i s used g = 1. Applying (A-8) to (2-27) and using (2-24) we obtain f o r the l i g h t d i s t r i b u t i o n i n the image plane, with P = 1 U ±(x,y,t) = ( V V [ U 1 (x^'V * h (x'y)] e J ( U l x + v i y ) (2T28) where - ( f x / f 2 ) x y - - ( f ^ ) y 17 I^Cx.y.t) i s defined i n (2-22) g T(x,yt) g T(u,v) s g and * and <—^ denote a s p a t i a l convolution and a s p a t i a l Fourier trans-frora r e s p e c t i v e l y , as in d i c a t e d i n Appendix A. The information from the object <j>o(x,y,t) i s contained i n (2-28). To see t h i s we set g" = 1. Then from (2-20), (2-22) and (2-23) the i n t e n s i t y i n the image plane becomes U ±(x,y,t) 2 = (B 2 w £ f 1 / f 2 ) 2 |{R*(x,y,t) [^(x-y-t) * g ; L (x,y,t)]} *'{g 2(x,y,t) e j ( u r * + V ° (2-29) where, as i n (A-7), section A.5 ^ ( x . y ) = m e j ( x 2 + y 2 ) m k w / 2 f [^(-n^-my) * q(m/(x 2+y 2)) ] In order to bring the quadratic phase f a c t o r caused by the use of a s i n g l e acoustic lens out of the convolution operations i n (2-29) we require that f o r some length dimension d^[12] § 1 (x,y,t) = 0 and g 2(x,y,t) e j ( u r X + V r y ) : 0 fo r \/x 2 + y 2 > d. and (mk ./2f) d 2 « 1 (2-30) W JL Then we can write | u . ( x , y , t ) | 2 = (m B 2 w £ f ^ f j ) 2 |{R2 (x,y,t) [[* o(-mx, -my, t) * q(m/(x+y ))] * g 1(x,y,t)]} M g 9 ( x , y , t ) e j ( u r X ' V r y ) } | 2 (2-31) 18 Thus the image we see appears as the scaled object function <f>o(x mf^/f^, y mf^/f^), passed through a f i l t e r w i t h t r a n s f e r function (1/m2) P (/(u 2+v 2) z./ak ) |. (u,v,co) then m u l t i p l i e d by the f a c t o r R^Cx.y.t) and passed through another f i l t e r I 2 ( u - u r » v " V r > co) In the preceeding analysis we set g = 1 (no transparency) and the phase d i s t o r t i o n s i n .U, due to the f a c t o r exp (-i 2 w„ (h + h + r 1 J I r o h^)) were of no consequence since we were i n t e r e s t e d i n the i n t e n s i t y of IL only. However i f use i s made of a transparency i n the transform plane then we have for the i n t e n s i t y i n (2-28) using (2-23) and (2-22) U^x.y.t) 2 = (B 2 w£ f x / f 2 ) 2 | { e - j 2 w J i ( h r + h Q + h N) ^ ( ^ i t ) } I ^ ( X ) 5 0 | 2 ( 2 _ 3 2 ) with h 2 (x,y,t) defined i n (2-20). The f a c t o r exp (-j 2 w ^ ( h r + h Q + h )) i n the convolution may cause unpredictable r e s u l t s , and i t may be a matter of experiment to see i f some form of g^ can be used to improve the image q u a l i t y . 2.5 Lensless Fourier Transform Hologram [26] We s h a l l consider one more method of image formation by f o r -ming a l e n s l e s s Fourier transform hologram as i n fi g u r e 2.6. An acoustic point source at (x^, y ) i n the f l u i d i s used as a reference and i s the same distance z^ from the surface as the object cj) o(x,y,t). The reference i s denoted by an impulse R-^t) <S(x-xr, y-y r) Thus at the surface, by Fresnel d i f f r a c t i o n (A-4) the 19 Plane wave light source Image plane 2 =0 0 F i g . 2.6 Lensless Fourier transform hologram refe-renoe wave i s 4 ( x,y , t ) = ( R . ( t ) A V e ^ V 2 * ! ^ * " * / + ^ r Y r J . w x and from the object 2 2 j / . s .., /, % j (k /2zY (x + y ) <f>, (x,y,t) = (1/X z n) e J V w r 3 X W X (2-33) • " ( x k w / z 1 , y \ J \ , t) P(x/d 2, y/d 2) where s • <Kx,y,t) <j> (u,v,t) 2 2 j./ _\ I t ^ \ j ( k /2z n) : ( X + y ) • (.x,y,t) = <j>o(x,y,t) e J V w V y T> / / J / i \ f l f o r ( x 2 + y 2 ) < d 2 P (x/d 2, y/d 2) = | Q o t h e r w i s e 2 P accounts for the f i n i t e s i z e of the hologram. From (2-2) and (2-3) the r a d i a t i o n pressure i s (2-34) 20 p r(x,y,t) = C | l * r ( x , y , t ) + ^ ( x ^ t ) ] . * g± ( x , y , t ) | (2-35) = ( C / z ^ ) 2 |R 2(t) e ~ j ( u r X + V 0 + <j>2(x,y,t)|2 (2-36) where ur = <>K/zl> x r v r = ( k w / z l > y r 4>9(x,y,t) = g (x,y,t) *'{*((k /z.)x , (k /z )y , t) Z X W X W X •p ( x/d 2, y/d 2)} R 2 ( t ) = R ^ t ) * [ g l ( x , y , t ) e " j ( u r X + V 0 ] We have dropped a constant phase f a c t o r i n R 2 as constant phase f a c t o r s here are of no importance. Also, we have brought a quadratic phase 2 2 f a c t o r exp [j (k w/2z^)(x + y )] out of the convoltuion operation i n (2-35) which we can do i f for some d 3[12] (see (3-33)) (k l v/2z 1) d 3 2 « 1 (2-37) and for g± (x,y,t) z 0 ( x 2 + y 2 ) < d 3 Thus we obtain f o r p r(x,y,t) i n (2-36) a form s i m i l a r to the case of a plane wave reference i n s e c t i o n 2.3 . The l i g h t d i s t r i b u t i o n at the surface i s then s i m i l a r to (2-22) and i s U g(x,y,t) = Z U n(x,y,t) e j ( U n X + V n y ) (2-38) n=-°° where TT / x.\ / -\ n -n -J2w n(h + h + h . J _ ,. , x jna U (x,y,t) = (-j) B e £ r o N J (4w h ) e J n II & a 21 u = u. + nu n I r v = v. + nv n it r and h r(x,y,t) = ( C / Z ; L A w ) 2 j R 2 ( t ) | 2 * g 2 (x,y,t) 2 2 h o(x,y,t) = (C/z 1A w) | <j) 2(x,y,t)| * g 2 (x,y,t) h a(x,y,t) e i a ( X i V f t ) = {R* (t) 4>2(x,y,t)} J • r / - i ( u x + v y ) , * {g 2(x,y,t) e J V r r' y} To obtain an image a lens i s used to Fourier transform the sur-face l i g h t d i s t r i b u t i o n . In the transform plane of the lens, or the image plane, we have d i f f e r e n t d i f f r a c t e d orders as i n figur e 2.5. As-suming no s i g n i f i c a n t overlap of the orders, then applying ( A - 8 ) to (2—3-8) and-with 4w h < 1 we obtain for the order n = +1 the l i g h t d i s -s. t r i b u t i o n U f(x,y,t) = A D (u,v,t) * { e j c ( Q + ^ ) / 2 [(4 (t) [^(u.v.t) * (<j,o (-cfl, -cv, t) * q 2 (/("u2 + v 2 ) ) ) ] ) i 2 (u + u r , v + v r , t ) ] J (2-39) with A = - j 2 w B e 2 u = ( k ^ ) x - u± v = ( k ^ ) y - v x q 2(u) = 2TT d 2 J 1 ( d 2 u ) / u f^ = lens f o c a l length 22 - j 2w„(h + h 4- h) s — , . i J & r o N <r^> D (u,v,t) 2 2 We have taken the f a c t o r exp (j c (u + v )/2) out of a convolution with i n (2-28) and thus we require [12] ( z J l k ) (7r/d 0) 2 « 1 1 w z r~ 2 2 since q^ (v(u + v )) i s n e g l i g i b l e f o r r 2 2 d 2V(u + v ) > TT In (2-39) the object function <j>o appears with a los s of r e s o l u -t i o n shoTm by the space convolution with q^ due to the f i n i t e hologram s i Then i t i s passed through a temporal f i l t e r g1 (u,v,t) that depends on the space coordinates (u,v). The m u l t i p l i e r (t) then changes the amplitude i n time. Then i t i s passed through another temporal f i l t e r g"2 which also depends on the space corrdinates (u,v) . The quadratic phase -factor then act-s as a m u l t i p l i e r over space. F i n a l l y there i s a d d i t i o n a l d i s t o r t i o n caused by the s p a t i a l convolution with D. The d i s t o r t i o n due to D did not appear i n the focused image method discussed i n section 2.4 (equation (2-31)). However i n equation (2-32) the f a c t o r exp (-j 2 w^  (h + h Q + h N ) ) does appear and i t seems that i n (2-32) and (2-39) this d i s t o r t i o n due to the phase f a c t o r w i l l be s l i g h t only i f the space varying part of the surface deformation h^ + h Q + h^ << A £. Another disadvantage of the present method arises from the demagnificat ion of the image by f,A /z A 1 x. 1 w Since A w >> A^ a microscope must be used to see the image. 23 I I I . RADIATION PRESSURE AND THE FUNCTION g 1(u,v,ti)) In t h i s chapter we s h a l l derive the function g^(u,v,co) which was used i n chapter 2 to determine the e f f e c t i v e value of the v e l o c i t y p o t e n t i a l at the l i q u i d surface of the minitank. The r a d i a t i o n pressure exerted by a sound f i e l d on the l i q u i d surface i s then given by the e f -f e c t i v e value of the v e l o c i t y p o t e n t i a l as in d i c a t e d by (2-3). In the f i r s t section the function g^ i s determined f o r a l i q u i d surface i n the absence of the minitank. Then i n the following sections the e f f e c t of the minitank on g^ i s determined. 3.1 Radiation Pressure of Sound Waves on a Free L i q u i d Surface In this s e ction we s h a l l determine the r a d i a t i o n pressure exer-ted by a sound f i e l d on the fi x e d surface of an object. The l i n e a r i z e d equations describing sound f i e l d s i n i d e a l f l u i d s are as follows [15] - c 2 A<j> = 0 (3-1) d t v = V <j> (3-2) P 1 = -P f£ (3-3) c 2 = | B - (3-A) P ' - ( J * - ) p' (3-5) o The symbols v, <j> , p and p denote the f l u i d v e l o c i t y , v e l o c i t y p o t e n t i a l , pressure and density r e s p e c t i v e l y . The v e l o c i t y of propaga-tio n of a sound wave i s given by c. The primes denote v a r i a t i o n s about 24 the static, value, v/here the s t a t i c value i s i n d i c a t e d by the subscript o. i . e . p --- p* + p o P = P' + p o The l i n e a r i z e d equations are v a l i d i f v << c or p 1 << p o In a sound f i e l d , the r a d i a t i o n pressure p r > exerted on the f i x e d surface of an i n t e r f a c e between d i f f e r e n t f l u i d s or on the sur-face of an o b s t i c l e i n the f l u i d i s given by the time averaged momentum f l u x per u n i t area of the surface. For an i d e a l f l u i d t h i s becomes [14] P r = p' n + p v (v'n) (3-6) The bar denotes a time average, and n i s a unit vector normal to the surface, p o i n t i n g out of the f l u i d . I f the sound f i e l d consists of waves that vary s i n u s o i d a l l y i n time then p' from l i n e a r theory i s zero. However, t h i s i s only a f i r s t order approximation, and to obtain a more accurate value for p' we must use the equations of non-linear acoustics. Then f o r an i d e a l f l u i d we have [14,15] (3-7) which includes terms up to second order. The values of p' and v i n (3-7) can be obtained from the l i n e a r i z e d equations. I f the v e l o c i t i e s at the surface i n question are normal to the surface (z=0) then the r a d i a t i o n pressure p^ i s also along the normal, and i s , from (3-6) and (3-7) (3 T8) 25 We s h a l l now determine the values of the density p' and normal v e l o c i t y v • When a t r a v e l l i n g plane wave <JK s t r i k e s a plane boundary (z=0) between two d i f f e r e n t f l u i d s we obtain the r e f l e c t e d and transmitted waves <j> and d>2 as shown i n fi g u r e ( 3 . 1 ) . Figure 3 .1 Waves at a boundary I f the inc i d e n t , r e f l e c t e d and transmitted waves are / ^ A j(u , x + w,z) 4>. (x,z) = A . e 1 1 ' l x • r (x,z) = A R e ^ u l x " W l z ) 0>2 (x,z) = A 2 eJ<V< + * 2 Z ) ( 3 -9 ) then we have the following r e l a t i o n s h i p s between the three waves [15] A Z . cos 9., - Z . cos6„ r _ _2 1 1 2 A^ Z 2 cos0^ + Zy cos0 2 ( 3 - 1 0 ) R 12 2 Z 2 cosS.^ P 2 Z 1 cos0 2 + Z 2 cos0 1 ( 3 - 1 1 ) 26 12 k. PC sin0. (3-12) (3-13) u. = k. sine, i i l w. = k. cos6. i i i k. = 2TT/A. I l i = 1, 2 (3-14) R.. i s the r e f l e c t i o n c o e f f i c i e n t of a plane wave t r a v e l l i n g i j i n medium ( i ) and r e f l e c t e d from a plane boundary with medium (j) as defined i n (3-10)!. S i m i l a r l y T. . i s the transmission c o e f f i c i e n t of i j a plane wave t r a v e l l i n g from medium ( i ) i n t o medium ( j ) as defined by (3-11). As a plane wave tr a v e l s through a boundary the s p a t i a l f r e -quency u^ along the x d i r e c t i o n does not change from one medium to another. We s h a l l write, u. = k. sin0. l l l (3-15) a l s o = n cose. = /{i - n 2/k. 2) (3-16) Then we can write R.., T.., w. i n terms of s p a t i a l frequency n. i j i j l I f medium (2) i s a gas and medium (1) i s a l i q u i d then >> p 2 and R ^ ~ -1- Then the v e l o c i t y p o t e n t i a l (j) i n the f l u i d (1) becomes the sum of the inc i d e n t and r e f l e c t e d waves as tj>(x,z) = T>i(x,z) + ^ ( x . z ) i 2 A. s i n w z e 1 J i 1 (3-17) 27 Then from (3-2) the v e l o c i t i e s v and v at the surface z=0 are x z v z=0 z=0 = 0 - j 2 w A. e J u l x  J 1 i = j 2 V7 <(,. 1 T l (3-18) z=0 Thus the v e l o c i t y at the surface i s r e l a t e d to the in c i d e n t v e l o c i t y p o t e n t i a l by (3-18), and the v e l o c i t y at the surface due to a s i n g l e plane wave of s p a t i a l frequency n becomes z=0 j 2 k± ]J1 - n 2 / ^ 2 (3-19) z=0 For an a r b i t r a r y v e l o c i t y p o t a n t i a l d i s t r i b u t i o n i n 2 dimen-sions we can then associate a tr a n s f e r function r e l a t i n g the v e l o c i t y and the incident v e l o c i t y p o t e n t i a l at the surface as follows v z (u,v) = j 2 k x g± (u,v) J± (u,v) (3-20) or where v z (x,y) = j 2 k± g± (x,y): * ^ (x,y) g 1(x,y) Jl> gy (u,v) = y l - n 2 / ^ 2 2 2 , 2 n = u + v Note that g^(x,y) i s not a function of time, also, v here i s a s p a t i a l frequency and not v e l o c i t y . We can now determine the r a d i a t i o n pressure on the l i q u i d surface due to an a r b i t r a r y sound f i e l d . From (3-17) <j> n =' 0 and-z=U thus from (3-3) and (3-5) p'l . = 0. Then from (3-8) and (3-20) z=u 28 with v a complex function, the r a d i a t i o n pressure on the surface z i s p r(x,y) = p ^ 2 | g l ( x , y ) * <J). ( x , y ) | 2 (3-2.1) 2 This i s the r e s u l t that was used i n (2-2) with C = p k 1 . Note that o 1 since p^ _ i s a time average quantity the time average should be taken over several periods of the sound f i e l d o s c i l l a t i o n . 3.2 E f f e c t of the Minitank on the Radiation Pressure In the formation of a c o u s t i c a l holograms using a l i q u i d sur-face a minitank i s used as shown i n figure 3.2 . A th i n membrane separates the l i q u i d i n the minitank from the water i n the main tank, to prevent the surface from being disrupted by any disturbances i n the water that are not associated with the sound f i e l d . Air ^ Liquid surface hologram Water Minitank -Thin membrane F i g . 3.2 Minitank Pulses of ultrasound waves that form the hologram pass from the water through the membrane and bounce around i n the minitank. In t h i s section we s h a l l determine the e f f e c t of the waves that bounce around and include these e f f e c t s i n the function g^(u,v). I t w i l l be shown that a time dependency i s introduced into g^. We s h a l l assume that the presence of the membrane i n f i g u r e 3.2 has n e g l i g i b l e e f f e c t on the passage of sound waves, which requires 29 that the membrane be much thinner than a wavelength of sound. Con-s i d e r the s i t u a t i o n i n figu r e 3.3 where a plane wave <j)^  i n medium (1) passes into medium (2) (the minitank) and s t r i k e s the surface at time t = 0 and i s r e f l e c t e d at the surface and i n t e r f a c e (2) - (1) a number of times. j ( u x +w(z+d)) ^ ( x . z . t ) = e J V r y(t) y(t) = 0 for t < 0 1 for t > 0 Main tank (3-2 2) (2) i s Fi g . 3.3 Reflections i n the minitank The v e l o c i t y p o t e n t i a l incident on the surface i n medium z=0 = T 1 2 e J ( u 2 ^2d)\(t) (3-23) = T12 * i y(t) z=0 and the v e l o c i t y at the surface i s then, taking into account the multiple r e f l e c t i o n s = j 2 k 2 /U- n 2/k 2 2) T 1 2 E ( - l ) n R ^ e ^ n W 2 Q y ( t - t n ) 2 = 0 n = 0 (3-24) n^n j2nw d 30 2 2 The factor j 2 / ( l - n /k^ ) <(K i s s i m i l a r to the r e s u l t i n (3-19) when no minitank was present while T^ 2 i s the transmission c o e f f i c i e n t (see (3-11)) i n passing from medium (1) into medium (2). Each time the wave r e f l e c t s o f f the surface, the i n t e r f a c e (2) - (1), and reaches the surface again, i t changes i n phase by (-1) R ^ exp (j 2 w^d) along z, where (-1) and R are the r e f l e c t i o n c o e f f i c i e n t s (see (3-10)) at the surface and i n t e r f a c e ( 2 ) - ( l ) r e s p e c t i v e l y , and d i s the t h i c k -ness of the f l u i d l a y e r (2). The number of r e f l e c t i o n s o f f the sur-face i s given by n + 1. Also, t n = n 2 d/c /("l - n 2/k 2 2) (3-25) 2 2 where c i s the v e l o c i t y of propagation and 2d/c / ( l - n /k 2 ) i s the time between r e f l e c t i o n s o f f the surface. S i m i l a r l y j i f the wave <j>^  i s a short pulse or impulse ^ ( x . z . t ) = e J ( u l x + w l ( z + d ) ) 6(t) (3-26) then the v e l o c i t y at the surface w i l l be a sum of impulses delayed i n time of d i f f e r e n t phase and diminishing amplitude as follows v J - J 2 k 2 /(W/k/) T 1 2 * t ! (-1)" R 2 1 " e ^ V 5 ( t - t n ) I Z=0 n = - c o (3-27) We can r e l a t e the v e l o c i t y at the surface to the i n c i d e n t v e l o c i t y p o t e n t i a l i n terms of the three dimensional Fourier transform gj^ (u,v,w) as v z(u,v,to) = j 2 k 2 g 1 (n,to) ^ ( u . V j O ) ) (3-28) or v z ( x , y , t ) = j 2 k 2 g l ( x , y , t ) * <j>..(x,y,t) (3-29) 31 where 2 2 , 2 n = u + v and g^(x,y,t) i s the e f f e c t i v e value of the v e l o c i t y p o t e n t i a l (2-2) when the i n c i d e n t v e l o c i t y p o t e n t i a l i s an impulse i n space and time 6(x,y,t). Then with g 1(x,y,t) ^ » g ; L ( u , v , w ) = g 1 (n,w) we have l± (n.ui) = A i - n 2/k 2 2) T 1 2 ( n ) •Z ( - D n R 9 i n ( n ) e j 2 n d w 2 ( n ) e j U t n ( n ) (3-30) n=0 where e ^ U t n i s the Fourier transform i n time of 6 ( t - t ). n One f i n a l item that should be mentioned i s that (3-10) and (3-11) are v a l i d for 0 < n < (smaller of k , k ) 1 We therefore desire the properties of the f l u i d media (1) and (2) to be such that k 2 > k x (or c 2.s c^) otherwise no s p a t i a l frequencies from the object higher than k 2 w i l l pass through the i n t e r f a c e , from the water, i n t o the f l u i d i n the minitank. In other words, i f k 2 < k^, a l l plane waves s t r i k i n g the i n t e r f a c e (1) - (2) with angles 6 greater than the c r i t i c a l angle 0^ 0 = arc s i n (k„/k.) c 2 1 w i l l be t o t a l l y r e f l e c t e d at the i n t e r f a c e . 3.3 The Transfer Function (ri,w) In t h i s s ection we s h a l l consider the t r a n s f e r function 32 g^(u,v,w) i n more d e t a i l when a) = 0, i . e . when there i s no time v a r i a -t i o n of the inc i d e n t ultrasound f i e l d . This i s equivalent to looking at the time step response a f t e r i t has reached a steady state ( n = c o ) . The simplest case occurs when l i q u i d s (1) and (2) are the same. Then T.^ = 1, R^^ = 0 and g± (n,oj) = \Jl - n 2 / k 2 2 as there would be no i n t e r n a l r e f l e c t i o n s i n the minitank, but the high s p a t i a l frequencies would be attenuated (see figure 3,4). I f the two l i q u i d s are d i f f e r e n t then T ^ ^ 1» ^ ^ a n < ^ w e m u s t s u m t n e e f f e c t s of the i n t e r n a l r e f l e c t i o n s as given by (3-30). Some values of g^(n>0) are p l o t t e d i n figures 3.4 to 3.7 fo r various depths of f l u i d d. In figures 3-4 and 3.6 the curve r - 2 2 = f o r V.(l - n / . k . 0 " ) .is .the value of , g . (.n.,.o) i f the f l u i d s (1) and (2) are the same. I t i s apparent i n figu r e 3-4 that f o r some depths d shown the minitank has the e f f e c t of f l a t t e n i n g the curves f o r g^(n>0) for some depths, but not for others, and that the curves are f a i r l y sen-s i t i v e to depth. For the greater depths d i n figure 3.6 the curves become o s c i l l a t o r y f o r the higher s p a t i a l frequencies. The phases of g-^ruO) are p l o t t e d i n figures 3.5 and 3.7 corresponding to the absolute values i n figures 3.4 and 3.6 . In figur e 3.5 the phase d i s t o r t i o n does not appear s i g n i f i c a n t except f o r the high s p a t i a l frequencies. However, these high s p a t i a l f r e -quencies might not be present at the surface anyway because of the f i n i t e appertures used i n the imaging system. For the greater depths d i n figure 3-7 the phase d i s t o r t i o n becomes o s c i l l a t o r y f o r the higher s p a t i a l frequencies. Again, the curves i n figures 3.5 and 33 34 35 36 9 to 9 1/2 37 3.7 are f a i r l y s e n s i t i v e to the depth d. Although the curves i n figures 3.4 to 3.7 were for n = 0 0, the values of g^(n-O) for n ~ 2 or 3 are almost the same, except for the very high s p a t i a l frequencies. That i s , i t takes only a few r e f l e c t i o n s of the sound waves i n the minitank i n order to reach a steady state f o r the densities and wavelengths i n d i c a t e d . The smaller the r e f l e c t i o n c o e f f i c i e n t I^ C^u) the more quickly g^(n,0) reaches a steady state and the le s s e f f e c t the minitank has on g^. The reason why the higher s p a t i a l freqeuncies take more • r e f l e c t i o n s , n, to reach a steady state i s because increases with s p a t i a l frequency. Thus we see that the e f f e c t of the f l u i d l a y e r i n the mini-tank on the imaging processes i s to cause amplitude and phase d i s t o r -tions i n the image obtained, e s p e c i a l l y at high s p a t i a l frequencies. Although there may be some depths of the minitank f o r which the ampli-tude versus s p a t i a l frequency of g^ i s quite uniform, ( f i g u r e 3.4), there i s a phase d i s t o r t i o n associated with t h i s at high s p a t i a l f r e -quenices. Again, however, the high frequencies may be cut o f f by the f i n i t e apertures of the imaging system. Also, we can see that f or the lower s p a t i a l frequencies the resonance e f f e c t caused by constructive interference of the i n t e r n a l r e f l e c t i o n s increases the amplitude of g^. However, t h i s p o s i t i v e aspect does not appear s i g n i f i c a n t , as the more important consideration i s the uniformity of g^ with s p a t i a l frequency. I t does not appear obvious that there i s any optimum depth of the f l u i d i n the minitank. However, we can at l e a s t say that the curves i n figures 3.4 and 3.5 are f a i r l y uniform over a wider range of s p a t i a l frequencies than i n figures 3.6 and 3.7. Of the curves shown 38 the best may be for d/A^ = 2 3 / 8 . I t should also be mentioned that i n de r i v i n g g^ the l i q u i d l ayer was assumed to be of i n f i n i t e l a t e r a l extent. That i s , we assumed that there were no r e f l e c t i o n s o f f the sides of the f i n i t e minitank. I t may be desirable to have a sound absorbing material along the inner sides of the minitank to absorb the u l t r a s o n i c waves. In i n e q u a l i t y (2-30) and ( 2 -27 ) we assumed that there was some d^ or d^ such that g^(x,y,t) ~ 0 for \/fx 2 + y 2 ) > d^. This condition i s more e a s i l y met the more uniformly g^(n,t) i s extended i n s p a t i a l frequency. For the case ^ ( n ) = vfl - n 2/k 2) (3 -31) we can obtain g^(x,y) by taking the inverse Hankie transform of (3 -31) and with [ 2 2 ] , then g- L(r) = — — £ ( S i " k r k - cos rk} ( 3 -32 ) (rk) r f~2 2 2 where r = yx + y . At r = 0 we have g-^(O) = k / 3 , and f o r l a r g e r values of r, g-^(r) decreases and o s c i l l a t e s . The f i r s t zero i s at rk - 4 . 5 , so i f we take g^(r) to be n e g l i g i b l e beyond r = 4 .5/k, we can take dv d 3 = 4 .5/k ( 3 -33 ) F i n a l l y , i n the next chapter we s h a l l determine the response of the l i q u i d surface to a pulse of r a d i a t i o n pressure, i . e . , the function g 2 ( x , y , t ) . 39 IV. LIQUID SURFACE RESPONSE TO RADIATION PRESSURE The l i q u i d surface acts as a detector of the u l t r a s o n i c energy that i s produced by the transducers t and t Q of. f i g u r e 2.1. When sound waves are r e f l e c t e d o f f of or absorbed by an object, a r a d i a t i o n pressure [14] i s exerted on the object. In the case of the l i q u i d sur-face a deformation of the surface r e s u l t s which represents the hologram. The deformation of the surface i s time dependent. However, we s h a l l i n i t i a l l y consider the steady state case and look at the transient motion l a t e r . 4.1 Steady State Response In the steady state there i s a balance between the l i q u i d surface height, the r a d i a t i o n pressure, gravity and surface tension as follows [ 6 ] P r(x,y) - Pg h (x-y) + y A h (x,y) = 0 (4-1) p^ i s the r a d i a t i o n pressure, p the f l u i d density, g the a c c e l e r a t i o n due to g r a v i t y , h the height of the surface above the quiesent l e v e l , and Y the c o e f f i c i e n t of surface tension. I f we take the Fourier transform of the terms i n (4-1) then h (u,v) = p r(u,v) (u,v) where g 2(u,v) = g 2(n) = l/(pg + yn 2) (4-3) 2 2 ^ 2 n = u + v The t r a n s f e r function g 2(u,v) was used i n (2-4) 40 Thus, as ind i c a t e d by (4-2) and (4-3) the height h of the surface deformation i s r e l a t e d to the r a d i a t i o n pressure P r by the low-pass t r a n s f e r function g^i^). The amplitude of the function g2(n) i s down by a fac t o r of 1/2 when the s p a t i a l frequency n i s given by n = /("pg/Y) (4-4) For freon E-5 (y = 16 dynes/cm, p = 1.8 gm/cm ) the h a l f amplitude frequency i s 1.7 cycles/cm. It appears then that the res o l u t i o n we can expect to obtain i n an image w i l l be l i m i t e d by the low pass response of the l i q u i d surface. For freon E-5 we may be severly l i m i t e d i n re s o l u t i o n by the 1.7 cycles/cm h a l f amplitude, yet we may be using u l t r a s o n i c waves with wavelengths of around 0.03 cm which i s 33 cycles/cm (5 MHz i n water). Also, we can see i n (4-3) that there i s a zero frequency •component to the surface deformation. In (4-3) t h i s represents a -uni-form l e v e t a t i o n or bulge of the surface, but i t may a c t u a l l y be a source of d i s t o r t i o n i n the image. I f the u l t r a s o n i c beams from transducers t and t of figu r e 2.1 cover only part of the surface of the minitank r D b then there w i l l be a d i s c o n t i n u i t y of r a d i a t i o n pressure at the edges of the area covered. Thus, rather than being uniform, the bulge may be curved, e s p e c i a l l y near the edges, and d i s t o r t the surface pat-tern containing the information of the hologram or v i o l a t e the condition that the space varying component of h should not be large compared to a wavelength of l i g h t . I t should be poss i b l e to avoid the bulge i f the u l t r a s o n i c beams cover the en t i r e surface, since then the zero frequency compo-nent of h cannot r a i s e up the surface. In t h i s case, the average pres-sure i n the f l u i d w i l l simply decrease [20]. However, there may s t i l l 41 be a d i s t o r t i o n or curvature of the surface at the sides of the mini-tank due to surface tension. I t remains a matter of experiment to see i f b e t t e r images can be obtained by i r r a d i a t i n g the e n t i r e surface. 4.2 Transient Response of the L i q u i d Surface 4.2.1 Formulation of the Problem It was suggested by B.B. Brendon to pulse the u l t r a s o n i c waves rather than to use continuous waves. I t was then found that much b e t t e r images were obtained. This improvement was a t t r i b u t e d to a transient increase i n the s p a t i a l frequency bandwidth of the l i q u i d surface response to the r a d i a t i o n pressure. An analysis of the transient motion of the l i q u i d surface response to the r a d i a t i o n pressure i s considerably more involved than for 'the steady state case. In the r e s t of t h i s s ection we w i l l solve t h i s problem by observing the response of the surface to a pulse of r a d i a t i o n pressure of s i n u s o i d a l space v a r i a t i o n . The f i n a l s o l u -t i o n w i l l be obtained numerically, by a f i n i t e d i f f e r e n c e scheme, and . the r e s u l t s w i l l be shown to f i t an a n a l y t i c expression. Z I. F i g . 4.1 L i q u i d surface deformation. 42 As shown i n figure 4.1 we have a l i q u i d l a y e r of depth d which i s assumed to be i n f i n i t e i n extent i n the x d i r e c t i o n . The sur-face i s deformed i n t o a s i n u s o i d a l pattern h(x,t) with s p a t i a l frequency n by the s i n u s o i d a l r a d i a t i o n pressure p^(x). We wish to determine the amplitude of t h i s s i n u s o i d a l deformation h(x,t) which v a r i e s i n time. I t i s assumed that the a i r above the l i q u i d has n e g l i g i b l e e f f e c t on the l i q u i d motion and thus the a i r w i l l be replaced by a vacuum. The problem i s described by the following equations (4-5) to (4-12). Throughout the f l u i d : |J = v A v - i Vp + g (4-5) dt p Vv = 0 (4-6) Boundary conditions: At the surface (z=h) 8 2 h 9 V z P =-p_ "Y — K + 2 pv-rf; (4-7) r 3x 2 9 2 9v 9v X + T — 2 = 0 (4-8) 3z 3x At the bottom (z = -d) v| , = 0 (4-9) 1 z= -d I n i t i a l conditions ( t = 0): h ( x ) | t = Q = 0 (4-10) v ( x , z ) | t = ( ) = 0 (4-11) The r a d i a t i o n pressure on the l i q u i d surface (z=h) w i l l take the form P r(x,t) = P r cos nx [y(t) - y ( t - A t ) ] (4-12) 43 P^ i s the constant r a d i a t i o n pressure amplitude, n = 2ir/X, At i s the pulse duration and 1 for t > 0 I 0 otherwise Also, we have the condition that h << A. The symbols v, p, v, t and g are the f l u i d v e l o c i t y , pressure, kinematic v i s c o s i t y , time and a c c e l e r a t i o n due to gravity, r e s p e c t i v e l y . Equation (4-5) i s the l i n e a r i z e d Navier-Stokes [16] equation f o r the motion of an incompressible viscous f l u i d , and must be s a t i s -f i e d by a l l f l u i d p a r t i c l e s . The l i n e a r i z a t i o n i s v a l i d i f h << A and h << d. Equation (4-6) i s the i n c o m p r e s s i b i l t i y condition. The boundary conditions are given by equations (4-7) to (4-9), Equation (4-7) i s obtained [18] from considerations of the balance of the r a d i a t i o n pressure, the viscous forces, and the surface tension at the liquid-vacuum i n t e r f a c e when h << A. Equation (4-8) states that the shear s t r e s s at the l i q u i d surface i s zero. Equation (4-9) i s the condition that a viscous f l u i d has zero v e l o c i t y at a f i x e d w a l l . The i n i t i a l conditions are given by (4-10) and (4-11). We w i l l solve (4-5) numerically using a s i m p l i f i e d form of the marker and c e l l f i n i t e d i f f e r e n c e method of Harlow and Welch [21] f o r incompressible f l u i d s . E f f e c t s due to acoustic streaming i n the ultrasound f i e l d w i l l not be accounted f o r . Only the v e l o c i t i e s w i l l be determined numerically, while the pressure can be determined as follows. I f we take the divergence of (4-5) and use (4-6) we f i n d that Ap = 0 (4-13) where A i s the Laplacian operator. From (4-5) and (4-9) we obtain the 44 boundary condition 3 2v ( f f " P v f ) = -Pg (4-14) 3z Now, we s h a l l assume that since the d r i v i n g function p^, the r a d i a t i o n pressure, varies s i n u s o i d a l l y along x, then the pressure p and the v e l o c i t y v^ also vary s i n u s o i d a l l y along x. Also, since h <<X and h << d-we assume that we can set h = 0 i n determining the pressure d i s t r i b u t i o n p ( x , z , t ) . Thus at some time t we set p(x,z,t) = P(z,t) cos nx - Pgz (4-15) v (x,z,t) = V (z,t) cos nx (4-16) z z where P(z,t) and V ( z , t \ are the amplitudes of the pressure and z component of v e l o c i t y r e s p e c t i v e l y at some time t . Equations (4-13) to (4-16) describe a boundary value problem fo r the pressure p i n the f l u i d l a y e r z = -d to z = 0. The s o l u t i o n at the time t i s p(x,z,t) = -pgz 3 2V (-d,t) + C ° S 1 n x fP(0,t) cosh n(z+d) + ^ sinh nz} (4-17) coshnd 1 v > ' n .2 dZ Equation (4-17) represents the f l u i d pressure p i n terms of the pressure amplitude P(0,t) at the f l u i d surface, and the second d e r i v a t i v e of the z component of the v e l o c i t y at the bottom. Now we can solve for the v e l o c i t y d i s t r i b u t i o n i n the f l u i d since we know the i n i t i a l conditions and the form of the pressure d i s t r i b u t i o n . I f (4-5) i s written i n f i n i t e d i f f e r e n c e form, and using (4-17), the v e l o c i t i e s can be solved for a sequence of times using a d i g i t a l computer. The method i s described i n Appendix B. 45 4 . 2 . 2 Results for Single S p a t i a l Frequencies A general s o l u t i o n of the problem described i n 4 . 2 . 1 may be quite complicated. However, for the case of i n t e r e s t where the d i s -s i p a t i o n of energy i n the f l u i d due to v i s c o s i t y i s small, we can f i t an a n a l y t i c expression to the r e s u l t s that were obtained using the numer-i c a l method described i n Appendix B. We s h a l l comment on these numerical r e s u l t s by making use of a n l a y t i c a l r e s u l t s that are a v a i l a b l e f or the propagation of gravity c a p i l l a r y waves on the surface of a f l u i d . I t i s known that when a l i q u i d surface i s subject to r a d i a t i o n pressure of some s p a t i a l frequency n> that the surface w i l l be deformed and may o s c i l l a t e and decay i n time [ 2 2 ] . This i s i n agreement with the r e s u l t s obtained using the numerical method described i n Appendix B. Figures 4 .2 ~to 4 . 5 are examples -of the v a r i a t i o n i n time of the amplitude H(t) of the s i n u s o i d a l surface deformation h(x,t) = H(t) cos nx when subject to r a d i a t i o n pressure of the form P^ cos nx for some duration At. A number of computer runs were made as i n d i c a t e d by table B - l , f o r various values of the l i q u i d density p, surface tension y> kinematic v i s c o s i t y v, s p a t i a l frequency n (n = 2TT/A) and f l u i d depth given by the r a t i o of depth to wavelength, d/X. The r e s u l t s obtained by the numerical method as shown by figures 4 . 2 to 4 . 5 are s i m i l a r to the f a m i l i a r t r a n s i e n t response of a second order system as described i n any book on l i n e a r systems. These curves are characterized by two parameters £ and u n- The damping r a t i o z, determines the overshoot of the response H(t) above the steady state value, and co^, the n a t u r a l frequency of o s c i l l a t i o n , determines the time scale. For a r a d i a t i o n pressure of duration At 2.0 1.5 3 ? 1.0 0.5 -0.0 0 <JOnAt - co X= 0.10 cm d/X = 0.258 V = 15.9 dyn/cm p=1.79 g/crr>3 V = 0.039 cm2/s S = 0.207 C0n = 1427 rad/s EQUATION (4-18) NUMERICAL ANAL YSIS 2 (sOnt 7 F i g . 4.2 L i q u i d surface response height H/H^ , vs. time w^t, to a step function of r a d i a t i o n pressure. 2.0 1.5 1.0 0.5 0.0 -0.5 1.0 •1.5 •Pr = 0 aOn&t = 3.86 x = 0.20 cm d/\ = 0.258 y = 15.9 dyn/cm P = 1.79 9/cm3 V = 0.039 cm2/s 0.147 GOn= Y 505 rad/s © © 15 EQUATION (4-18) NUMERICAL ANALYSIS 4.3 L i q u i d s u r f a c e r e s p o n s e h e i g h t H/H^, v s . t i m e w ^ t , t o a p u l s e o f r a d i a t i o n p r e s s u r e o f d u r a t i o n w n At = 3.86. 48 0.6 0.5 0.4 -0.2 O.J A - 0.02 cm d/X = 0.25 (jQnAt=co y = 5.0 dyn/cm — EQUATION p = 3.0 g/cm3 0.60 cm2/s (4-18) V = ' . 0,4 NUMERICAL 1.65 / ANALYSIS 6884 rad/s / / ^ A = 1.0 cm d/\=0.08 y = 20.0 dyn/cm P = 2.0 g/cm3 V = 0.25 cm2/s - £ =2.32 i [ (x)n = 33.9 rad/s i i i 0.0 2 Fig. 4.4 Overdamped liquid surface response height H/H ,^ vs. time w^t, to a step function of radiation pressure. 0.8 0.6 -0.4 -.0.2 -0.0 0.2 -0.4 F i g . 4.5 Liquid surface response height H/K^, vs. time w^t to an impulse of radiation pressure. = P /{k1 + B 2)/ Yn^ (see (C-l)) 50 the response H(t) can be written as p H(t) = X— Ul - j e ^ V sin(a) dt + 6)] y(t) pg+ny -[1 - j e " C u n ( t _ A t ) s i n (u>d(t-A.t) + 0)] y(t-At)} (4-18) where 3 = / ( i - c 2 ) to = gw d n 0 = tan B/C Equation (4-18) can be written i n other forms i f g i s imaginary or i f t > At. These are given i n Appendix C. It was found that close agreement with the numerical r e s u l t s .are obtained i f C = 2 npv /Tn/(p(Pg + Yn 2))} [tanh (nd)]E (4-19) <on2 = (Pg + yn2) j tanh ( nd) . (4-20) / -2.3 i f d/X < 1/4 ( 0 i f d/A > 1/4 The s o l i d curves i n figures 4.2 to .4.5 are obtained from equations (4-18), (4-19) and (4-20), while the numerical r e s u l t s are i n d i c a t e d by the c i r c l e s . The differences between the numerical r e -s u l t s and (4-18) to (4-20) are shown i n table B - l for the peak values of the curves H(t) and the times at which the peak occurs. The d i f f e r -ences are quite small f or the values of the parameters used. Figure 4.2 shows the response H(t) to a step function r a d i a t i o n pressure. Figure 4.3 i s s i m i l a r except that the r a d i a t i o n pressure i s of duration to^At = 3.86. Two cases of the response H(t) 51 when £ i s greater than one are shown i n f i g u r e 4.4 . The response H(t) to an impulse of r a d i a t i o n pressure ( a c t u a l l y a short pulse, w At = 0.0121) i s shown i n figu r e 4.5 . The response of the l i q u i d surface to r a d i a t i o n pressure i s s i m i l a r to the propagation of gra v i t y c a p i l l a r y waves on a l i q u i d surface. The s i m i l a r i t y comes from the fact that i n both s i t u a t i o n s there i s a si n u s o i d a l deformation of the surface, the amplitude of which decays i n time. In (4-20), the n a t u r a l frequency of o s c i l l a t i o n i s the same as the r e s u l t that can be obtained a n a l y t i c a l l y [17] for the frequency of o s c i l l a t i o n of c a p i l l a r y g r a v i t y waves propagating on the surface of an i d e a l f l u i d of depth d. Also, except f o r the dependence on depth ^d, the damping c o e f f i c i e n t £ 1 0 as given by (4-19) and (4-20) i s the same as that •which i s derived i n .[17.] f o r the damping of g r a v i t y waves p.rop.agating on the surface of a viscous f l u i d of i n f i n i t e depth. It i s shown i n [17] that f o r gravity waves t h i s value of damping c o e f f i c i e n t i s v a l i d only i f i t i s small, so that the f l u i d motion i s approximately that of an i d e a l f l u i d . Then f or f l u i d s of i n f i n i t e depth or of depth greater than about A/4 th i s condition i s v n << t^gn (4-21) 2 If we include the e f f e c t of surface tension by repla c i n g g by g + n y/p then (4-21) becomes v n 2 « V T g T ! + n 3 Y/P) (4-22) We would then expect that the equations (4-18) to (4-20) to apply i f (4-22) i s s a t i s f i e d . For freon E-5 or water, condition (4-22) i s e a s i l y s a t i s f i e d f o r s p a t i a l frequencies n from zero up to values of i n t e r e s t , for example 1000 rad/cm. At 10 MHz i n water the wavelength 52 i s 0.015 cm, for which the wave number k = 420 rad/cm. A c t u a l l y , f o r the curves i n f i g u r e 4.4 the condition (4-22) i s not s a t i s f i e d , as both sides of the i n e q u a l i t y are approximately the same. This seems to i n d i c a t e that the condition (4-22) i s too r e s t r i c t i v e when equations (4-18) to (4-20) are used. For f l u i d s of depth le s s than about A/4 a f a c t o r (tanh nd) applies as i n d i c a t e d i n (4-19). This was found to f i t the numerical r e s u l t s w e l l for values of d/A. at l e a s t as low as 0.02, but no smaller depths were used i n the numerical a n a l y s i s . I f the depth d i s held constant with the s p a t i a l frequency n varying, the dependence on depth becomes more s i g n i f i c a n t at the lower s p a t i a l frequencies and the v a l i d i t y of equations (4-18) to (4-20) may become doubtful for very low values of n. However no studies were made to determine at what point t h i s may be reached. Also, the damping c o e f f i c i e n t f or f l u i d s where the d i s -s i p a t i o n of energy i s very large may be quite d i f f e r e n t from that given by (4-19) and (4-20). This i s shown by the r e s u l t s i n reference [14,17] fo r the decay of s i n u s o i d a l deformation on a f l u i d surface. However, the equations (4-18), (4-19) and (4-20) seem to be s u f f i c i e n t f o r most f l u i d s and s p a t i a l frequencies we may use i n r e a l time l i q u i d surface holography. The numerical method i n Appendix B can be used to determine the surface motion f o r any p a r t i c u l a r f l u i d and s p a t i a l frequency that i s of i n t e r e s t . In the numerical analysis described i n Appendix B, the ac-c e l e r a t i o n due to gravity, g, was set to zero. This was done because the numerical method used was hot s u i t a b l e f o r accounting for the small v a r i a t i o n s i n pressure that occur at the surface due to g r a v i t y . 53 However, the e f f e c t of gravity i s n e g l i g i b l e f o r a l l but very low 1 s p a t i a l frequencies, and i n (4-19) and (4-20) g can be set to zero f o r most s p a t i a l frequencies of i n t e r e s t . F i n a l l y , f o r a s i n u s o i d a l deformation of s p a t i a l frequency n we can write the response of the surface amplitude H(t) to an impulse of r a d i a t i o n pressure of amplitude P^ . 6(t) as P UJ H(t) = e ? V sin(Bo) t) y(t) (4-23) , £ p n pg+yn I f a surface deformation h(x,y,t) i s r e l a t e d to an a r b i t r a r y r a d i a t i o n pressure p r(x,y,t) by (2-4), that i s h(x,y,t) = p r(x,y,t) * g 2 (x,y,t) or h (u,v,w) = p (u,v,w) g 0 (u,v,w) (4-24) r t~ where g2(x,y,t) i s the response of the surface to an impulse 6(x,y,t) of r a d i a t i o n pressure, then by taking the Fourier transform i n time of (4-23) we see that 2 = 1 W n g 2(u,v,w) = — ^ 2 (4-25) pg + YH W + 2j t,bi 03 - w - n n 2 2 2 R e c a l l that n = u + v , also, i t i s evident that = A g 2(u,v,to) = g 2 (-u,-v,-u) 4.3 The Surface S p a t i a l Frequency Response to Pulsed Radiation Pressure In l i q u i d surface holography the transducers t and t i n fi g u r e 2.1 are used to produce pulses of ultrasound. In t h i s s e ction we s h a l l look at the response of the l i q u i d surface f o r various s p a t i a l 54 frequencies when the r a d i a t i o n pressure i s a pulse or step i n time. Then i f P r(x,y,t) i s given by p (x,y,t) = P^ 6 (x ,y ) [u(t) - y( t - A t ) ] (4-26) r r i i.e.,'p i s unifjP^m i n s p a t i a l frequency P r ( x,y,t) <4> P r [y(t) - y( t - A t ) ] Then from (4-18) the Fourier transform i n (x,y) of the response of the surface h(x,y,t) f o r t h i s case i s _ _ t _ h(u,v,t) = p r (u,v,t) * g 2 (u,v,t) = H ( n , t ) (4-27) Figures (4-6) and (4-7) show the response H''(t) when At =°° (step response) f o r freon E-5 and water, r e s p e c t i v e l y , at various times t over a f r e -quency range from 1 to 2000 rad/cm. These curves are for an i n f i n i t e f l u i d depth d, or at l e a s t greater than 1/4 of the la r g e s t wavelength A = 2TT cm, so d > 1.5 cm approximately. In a l l figures of t h i s s e c t i o n H i s normalized by = P^/pg. We can see i n figures 4.6 and 4.7 that the high s p a t i a l f r e -quencies reach a steady state much sooner than the low frequencies. The s p a t i a l frequency response r i s e s at the rate of 10 db per decade of s p a t i a l frequency (db = 10 l o g H/FL^) and 20 db per time decade u n t i l i t i s close to the steady state value. Then the steady state drops at the rate of 20 db per decade of frequency at the higher frequencies. The o s c i l l a t i o n s of the curves near the steady state e s p e c i a l l y at the lower s p a t i a l frequencies correspond to the o s c i l l a t i o n s i n the response once the steady state value i s passed (overshoot) as i n fi g u r e 4,2. As mentioned i n section 4.1 the steady state response i s seen to be very poor, with the higher s p a t i a l frequencies dropping o f f 55 10 0 -10 -•20 h 3 ? 2r -30 -4 0 f -50 60 h 70 h 80 F i g . 4.6 L i q u i d surface (freon E-5) s p a t i a l frequency response H(n,t) to r a d i a t i o n pressure of a step function i n time. H„ = P /pg N r 56 10 QQ I F i g . 4.7 L i q u i d surface (water) s p a t i a l frequency response H(n,t) to r a d i a t i o n pressure of a step function i n time. H N " V P S 5 7 10 0 -10 -20 -30 4 0 50 60 •70 80 H 1—H-+-H-H- H 1 I I I -1 1 I I I Ij 10 100 7] rad/cm 1000 \0 ' y P V d At = 16. dyn/cm = 1.8 g/cm3 = 0.039 cm2/s = CD - IO-4 s F i g . 4.8 L i q u i d surface (freon E-5) s p a t i a l frequency response H(n-t) to r a d i a t i o n pressure of a pulse i n time. 58 70 -SO L- -F i g . 4.9 L i q u i d surface (freon E-5) s p a t i a l frequency response H(n,t) to r a d i a t i o n pressure of a step function i n time, with f i n i t e f l u i d depth. 59 10 0 10 -20 3 '30 -50 60 -70 -80 H— i — i i i 1111 1—i—i M i n i 1 1—h-i i i n 10 100 T) rad/cm 1000 y P V d = 16. dyn/cm - 1.8 g/cm3 = 0.039 cm2/s = 0.015 cm = 10~4 s F i g . 4.10 Li q u i d surface (freon E-5) s p a t i a l frequency response H(n,t) to r a d i a t i o n pressure of a pulse i n time, with f i n i t e f l u i d depth. H N = P r/pg. 60 at 20 db per decade. However, the transient response at the higher frequencies i s r e l a t i v e l y f l a t over a r e s t r i c t e d frequency range where the curves peak near the steady state curve. In (2-31) the information from the object <j>Q i s passed through the f i l t e r g 2 ( u ~ u r » v ~ v r ' u ) > where u and v depend on the o r i e n t a t i o n of the acoustic reference r r beam as in d i c a t e d i n (2-12). Thus by adjusting u^ and v^, advantage can be taken of the transient motion by s h i f t i n g the object information i n t o a region where the s p a t i a l frequency response i s momentarily f l a t and sampling the surface with the l a s e r at t h i s moment. Evidently, the greater the angle of incidence of the reference beam to the surface, the e a r l i e r i s the sample time. Consider that we have an imaging system i n which the s p a t i a l frequencies at the surface from the object range from zero to a maximum of n . The maximum amount by which the object information can "be s h i f t e d m i n s p a t i a l frequency i s k , where k = 2TT/X and X i s the wavelength W W w w of the ultrasound i n water. Thus the sampling time i s determined by any of the curves that has i t s f l a t region between n and r\ + k . m m w For example, for ultrasound at 10 MHz i n water k = 420 rad/cm, w but say that the apertures of the system l i m i t n m to 250 rad/cm. In f i g u r e 4.6 with freon E-5 i n the minitank we see that the curve for -4 / t = 0.5 x 10 s has a f l a t region from about 275 to 525 rad/cm. Thus r 2 2 -4 i f y ( u r + v r ) = 275 rad/cm then the sample time would be 0.5 x 10 s. The angle 0 of the reference beam i n water ( i . e . i n the main tank) zw to the normal to the surface i s then 2 2 0 = arc s i n v(u + v )/k zw r r w = 41.5° (4-28) 61 Another possible sample time i s 0.1 x 10 s with /(u + v r ) = 150 rad/cm and 9 = 2 1 ° . zw With the transducers t and t producing pulses of u l t r a -r o ° sound, the pulse length can be set so that the pulse terminates a f t e r the surface i s sampled. Then a f t e r s u f f i c i e n t time to allow the surface deformation to decay, another image can be obtained with another pulse of ultrasound. Figure 4.8 shows the response of the surface to a pulse -4 of r a d i a t i o n pressure of duration At = 10 s. The curves f o r the various times are terminated at those values of n where the response H(n-t) becomes negative, as the response decays and o s c i l l a t e s as shown i n f i g u r e 4.3. The high frequencies begin to decay r a p i d l y , but the -4 lower frequencies to the l e f t of the peak region for t = 10 s continue to r i s e , and then eventually to decay. I t i s -apparent 'in f i g u r e 4.8 -that we can s t i l l - s a m p l e "the surface for times t > At. For example, at 5 MHz i n water k = 210 w rad/cm. I f n = 105 rad/cm, pulse length At = 10 ^s, with the curve -3 i n f i g u r e 4.8 for t = 0.2 x 10 s (sample time), and the frequency 2 2 s h i f t /(u + ) = 105 rad/cm, the angle of the reference beam 6 = 3 0 ° . zw The time between the pulses of ultrasound i s determined by the rate at which the surface decays a f t e r the pulse ends. I f t = 0 at the beginning of the pulse, then a s u f f i c i e n t condition f o r the time t-^  at which the surface at a given s p a t i a l freqeuncy has decayed to w i t h i n a f r a c t i o n f of i t s height at the sample time t g where t ^ At, i s , from ( C - l ) . e ' ^ ^ l j? f e~ C a )n ts s i n (u,t +6 + 6) (4-28) 62 Then the time at which the next pulse can begin can be taken as t^ - t . Using the values of the previous example and applying (4-28) to the lowest s p a t i a l frequency within the bandwidth of i n t e r e s t , i . e . n = 100 _3 rad/cm, we obtain f o r f = 0.05, t ^ 5 4.5 x 10 s, or the period of the o pulses t, - t >. 4.3 x 10 s. r 1 s In the figures 4.6 to 4.8 the depth d of the minitank was as-sumed to be d > 1.5 cm so that there was no e f f e c t of the depth on the surface motion f o r the s p a t i a l frequencies shown. Figures 4.9 and 4.10 are s i m i l a r to figures 4.6 and 4.8 r e s p e c t i v e l y except that d = 0.015 cm. This means that f o r those s p a t i a l frequencies n < 2ir/4d = 100 rad/cm the surface motion i s more he a v i l y damped i n figures 4.9 and 4.10. The f a c t that a thin f l u i d l ayer damps the motion of low s p a t i a l f r e -quencies i s a desirable feature since t h i s contributes to the s t a b i l i t y of the f l u i d surface. 63 V. ACOUSTIC BEAM INTENSITIES, SOUND FIELD PARTICLE OSCILLATION, SURFACE BULGE In t h i s chapter we s h a l l consider the i n t e n s i t i e s of the acoustic beams required, the amplitude of the o s c i l l a t i o n of f l u i d p a r t i c l e s i n the u l t r a s o n i c f i e l d , and the motion of the surface bulge. 5.1 The U l t r a s o n i c Beam I n t e n s i t i e s In order to gain some idea of the i n t e n s i t i e s required of the -4 u l t r a s o n i c beams we s h a l l consider pulses of duration At = 10 s as i n figur e 4.8. With the reference <j> and object <j>^  v e l o c i t y p o t e n t i a l s both plane waves, then at the surface 4>r = R e J ( u r X + V r y ) [u(t) - y ( t - A t ) ] (5-1) ^ = A e j ( ' u l x + V i y ) [u(t) - y( t - A t ) ] (5-2) In f i g u r e s 3.4 and 3.5 we see that at l e a s t f o r a t h i n f l u i d l a y e r of freon E-5 on water the e f f e c t of multiple r e f l e c t i o n s i n the minitank f o r the lower s p a t i a l frequencies i s not s i g n i f i c a n t . Then the e f f e c t of the t r a n s f e r function g-^  (2-2) on the two beams §^ and <j>^  i s approximately • r * 8 l = a * r *1 * g l * a *1 where from fi g u r e 3.4 a s 1/2. From (2-3) and (3-21) the r a d i a t i o n pressure on the surface i s then p = a 2 p 0k 2 {R2 + A 2 + 2RA cos[(u -u.)x + (v -v.)y]} r r 2 2 r 1 r 1 •[y(t) - y ( t - A t ) ] (5-3) 64 The three pressure terms i n (5-3) correspond to the three height terms i n (2-19) and we can use (4-18) or figu r e 4.8 to deter-mine the amplitude of the surface deformation. In order that the l i n e a r i z a t i o n (2-23) be v a l i d we must have the surface height h due to the reference and object beams such that 3. 4 w. h < 1 £ a or h a < 0.04 \ I cos 6 (see (2-11)) (5-4) From (5-3) we take the r a d i a t i o n pressure from the t h i r d term P r = a 2 p 2 k 2 2 M ( 5 _ 5 ) and i f the s p a t i a l frequency of the object wave i s s h i f t e d to / T ( u r - U l ) 2 + (v - v 1 ) 2 } = 200 rad/cm (5-6) —3 then at t = 0.2 x 10 s from figure 4.8 the normalized surface height h i s 0.275 x 10~ 2, and n h a = h n P r / p g = h n a 2 k 2 2 RA/g (5-7) In order f o r (5-4) to be s a t i s f i e d the product of the beam amplitudes RA must then be given by RA < A„g/8TTh a \ 2 cos 6 (5-8) Jo n 2 Z £ The r e l a t i o n s h i p between the i n t e n s i t y I and the v e l o c i t y p o t e n t i a l amplitude <j> i n a plane wave i s given by [24] I = pto2 <}>2/2c (5-9) where w i s the frequency of o s c i l l a t i o n of the u l t r a s o n i c wave and c the v e l o c i t y of propagation. I f the reference beam amplitude i s l a r g e r 65 than the object beam amplitude so that say R = bA, then the reference beam i n t e n s i t y becomes, with 0 = 0 I_ < p 0 c bA„g / l 6 f T a 2 h (5-10) K. / x, n If we take b = 10, a = 1/2, h = 0.275 x 10" 2, c = 0.7 x 10 5 cm/s and n 3 -4 p = 1.8 gm/cm (freon E-5) with A^=0.6xl0 cm then T < 200 mW/cm2 (5-11) R and I < 20 mW/cm2 (5-12) However, i f an actual object i s used then the v e l o c i t y p o t e n t i a l <j)^  w i l l consist of a d i s t r i b u t i o n of frequencies, and the i n t e n s i t y given by (5-12) may have to be higher. Also, some energy i n the u l t r a -sound beams would be absorbed by the object and by the water, so that (5-11) and (5-12) may be too small-In water the absorption c o e f f i c i e n t a causes an attenuation of the amplitude of a propagating wave by the factor e . At [25] -3 -1 -2 -1 1 MHz a = 10 cm and at 10 MHz a = 4 x 10 cm , so that the i n t e n s i t y 2 i s down to 1/e = 0.14 at 1000 cm and 25 cm f o r 1 MHz and 10 MHz res-p e c t i v e l y . 5.2 The Ultrasound F i e l d P a r t i c l e O s c i l l a t i o n So f ar no mention has been made of the amplitude E, of the o s c i l l a t i o n of the f l u i d p a r t i c l e s at the frequency of the u l t r a s o n i c beams. This motion of the f l u i d p a r t i c l e s at the surface, as opposed the motion of the surface h due to the r a d i a t i o n pressure, can be con-sidered as a d d i t i o n a l noise. In a plane wave the amplitude of the f l u i d p a r t i c l e o s c i l l a t i o n 66 E i s r e l a t e d to the i n t e n s i t y I by [24] E 2 = 21/pew2 (5-13) At the f l u i d surface the wave i s r e f l e c t e d and E would be twice that given by (5-13) i f the wave i s approximately at normal incidence to the surface. Note that E i s i n v e r s l y p roportional to the frequency to of the u l t r a s o n i c wave. I f we consider the case with to = 1 MHz, I = '0.2 2 3 5 W/cm , and f o r the f l u i d properties p = 1.8 gm/cm and c = 0.7 x 10 cm/s (freon E-5) then from (5-13) E = 0.9 x 10" 6 cm (5-14) -4 If the l i g h t wavelength ^ = 0.6 x 10 c m then E/X £ = 0.015 (5-15) which i s comparable to h i n (5-4) Thus f or high i n t e n s i t y , low frequency u l t r a s o n i c beams the amplitude of the o s c i l l a t i o n of the f l u i d p a r t i c l e s at the surface may be comparable with the surface r i p p l e pattern containing the i n f o r -mation of the image. However, i f the l i g h t from the l a s e r that i s r e f l e c t e d o f f the surface has a pulse length which i s much longer than the period of o s c i l l a t i o n of the u l t r a s o n i c beams, then we would expect that the average e f f e c t of the f l u i d p a r t i c l e o s c i l l a t i o n s on the image obtained would be minimized. 5.3 The Surface Bulge In t h i s s ection we w i l l consider the e f f e c t of a large ampli-tude low s p a t i a l frequency r a d i a t i o n pressure term on the f l u i d surface. We can see i n (5-3) for the r a d i a t i o n pressure that the low s p a t i a l 2 2 frequency terms are proportional to R and A . I f i n the reference 67 beam the amplitude R i s s u f f i c i e n t l y l a r g e r than A i n the object beam 2 then since we have R i n the pressure, the reference beam may cause a large deformation of the surface at low s p a t i a l frequencies i n com-parison with the high s p a t i a l frequency term proportional to RA. I n i t i a l l y , t h i s would not seem to present any problems since the transient response to a pulse i n figure 4.10 ind i c a t e s that the low s p a t i a l frequencies respond slowly and are h i g h l y damped for a th i n f l u i d l ayer. Also, the surface r i p p l e height as given by (5-4) i s much less than a wavelength of l i g h t , so that i t would appear that the condition mentioned i n section 2-«2, that the varying part of the surface height should not be much l a r g e r than a wavelength of l i g h t , i s not v i o l a t e d . But we must note that f i g u r e 4.10 applies f o r a s i n g l e pulse, and i f a rapid succession of pulses are used, and since the low -spatial frequencies decay -slower than the high frequencies, the low frequencies may b u i l d up more than i n d i c a t e d by f i g u r e 4.10. Also, a large low s p a t i a l frequency term i n the r a d i a t i o n pressure may conceivably cause an undesirable d i s r u p t i o n of the sur-face, i f only part of the surface i s i r r a d i a t e d with u l t r a s o n i c waves. Then a bulge i s formed on the surface as mentioned i n s e c t i o n 4.1. At zero s p a t i a l frequency i n (4-18), the surface response has zero height, yet i n (4-1) we obtain f o r the steady state the height h = P^/pg. This d i f f e r e n c e i s due to the fac t that (4-18) applies f o r the case of a f l u i d of i n i f n i t e surface area, a l l of which i s subject to r a d i a t i o n pressure. The problem of the surface response to r a d i a t i o n pressure covering only a f i n i t e area of the surface can be approached i n the following way. Consider the simplest case with the r a d i a t i o n pressure 68 varying i n the x d i r e c t i o n and constant i n the y d i r e c t i o n along the surface. I f the hologram area i s small compared to the area of the sur-face of the minitank and p^ _ i s simply a pulse along the surface x and i n time t then p = P G (x) [y(t) - u(t-At)] J . i d (5-16) where the gate function G (x) = f 1 for -a < t < a I 0 otherwise and 2a i s the length of the hologram on the surface ( f i g u r e 5.1 ) Prfx) -3- X - a Figure 5.1 Radiation pressure I f the length of the minitank 2b, i s comparable to the length of the hologram 2a, then there are extra constraints on the f l u i d motion since the v e l o c i t i e s must be zero at the v e r t i c a l walls of the minitank. We can account for the zero h o r i z o n t a l v e l o c i t y at the walls by considering p^ to be a p e r i o d i c , even function along x as i n fi g u r e 5.2 . Then with p^ simply pulses i n x and t p (x,t) = P [y(t) - y ( t - A t ) ] Z G (x-2nb) i JL C l 11=-oo (5-17) Thus from symmetry, the h o r i z o n t a l v e l o c i t i e s i n the f l u i d at x = + b 69 •2b •b -a a b 2b Figure 5.2 Radiation pressure are zero. The v e r t i c a l v e l o c i t i e s at x = +b, however, are not n e c e s s a r i l y zero, but t h i s J o e s not appear to b e a serious drawback unless the f l u i d i s very viscous. Equation (5-17) f or p^ can be wr i t t e n i n terms of i t s Fourier s e r i e s i n x s o that P\ CO P r(x,t) = P r [y(t) - y ( t - At)] {^+ E cos rye} (5-18) n=l 2a s i n ( m r a / b ) (5-19) n b n i T a / b ri = nnx/b n Then, since each frequency component i n (5-18) would be attenuated as given by (4-18), the surface response h(x,t) can be written as h(x,t) = E a cos n x H(n ,t) (5-20) . n n n n=l (-b < x ^ b) where i n (4-18) we replace n by n . Thus the shape of the surface, the s p a t i a l frequency components present and the rate at which these s p a t i a l frequency components o s c i l l a t e i n time are given by (4-18), (5-19) and (5-20). 70 VI. . CONCLUSIONS In chapter 2 we have presented an analysis of a form of r e a l time imaging of objects using holography with ultrasound waves i l l u m i n a -t i n g the object i n water, and with the ultrasound energy detected by a l i q u i d surface. The images obtained i n the o p t i c a l domain by r e f l e c -t i n g coherent l i g h t o f f the surface were r e l a t e d to the object by the functions g^ and g^ which represent the impulse response of the l i q u i d l a y e r i n the minitank. I t was shown that the l i g h t d i s t r i b u t i o n at the surface has a form s i m i l a r to the acoustic d i s t r i b u t i o n at the surface. However, s u i t a b l e o p t i c a l methods are required to extract the useful/!' information to form an o p t i c a l image of the object. The focused image method was shown to be i n s e n s i t i v e to noise on the l i q u i d surface since the noise merely causes phase changes i n the l i g h t that forms the image. This i s true as long as the noise does not cause surface v a r i a t i o n s much l a r g e r than a wavelength of l i g h t and i f the various orders of l i g h t produced do not overlap i n the trans-form plane. Hie separation of the various orders i n the transform plane i s increased by i n c r e a s i n g the angle of incidence of the acoustic reference beam to the normal to the surface. Also, the focused image method avoids the problem of the image being demagnified by the r a t i o of the u l t r a s o n i c wavelength to the l i g h t wavelength. The phase d i s t o r t i o n of the l i g h t at the surface i s caused by the deformation of the surface due to the r a d i a t i o n pressure from the reference and object acoustic beams act i n g alone on the surface, plus any surface deformation caused by any external disturbances reaching the minitank. Thus any method of imaging that requires the l i g h t to • 71 propagate over a distance before an image can be formed may be subject to severe d i s t o r t i o n . This d i s t o r t i o n would be minimized i f the sur-face v a r i a t i o n s (h + h + h T i n (2-22)) that cause the phase d i s t o r t i o n r o N were much smaller than a wavelength of l i g h t . Since we expect h + h Q + h^ to be generally of low s p a t i a l frequency, and since the response of the l i q u i d surface to low s p a t i a l frequencies i s slow, then the phase d i s t o r t i o n due to h^ + h Q + h^ may be minimized by using pulses of ultrasound. The surface should be sampled before the response to the low s p a t i a l frequencies becomes large, and the time between pulses should be s u f f i c i e n t l y long to allow the surface to decay before the next pulse. Figures 4.6 to 4.10 show that the pulsed method of imaging i s necessary for any imaging system that uses the l i q u i d surface since the s p a t i a l frequency response of 'the l i q u i d surface 'to r a d i a t i o n pres-sure i n the steady state i s very poor f o r the s p a t i a l frequencies of i n t e r e s t , with ultrasound frequencies above 1 MHz. It i s the l i q u i d surface response given by the function g^ (which acts as a bandpass f i l -ter) as derived i n chapter 4, that i s the most important determinant i n the method of operation of the imaging system. Graphs of the l i q u i d surface response such as figures 4.6 to 4.10 and equation (4.18) can be used to determine the angle of incidence of the acoustic r e f e r -ence, the pulse length of the acoustic beams, the sampling time of the l i g h t beam and the time between the pulses of the acoustic beams. The function g^, derived i n chapter 3, which accounts f o r the angle of incidence of the acoustic waves to the l i q u i d surface and the presence of the minitank i s of l e s s importance than the function g^. However, g.. shows that at large angles of incidence of the ultrasound 72 a phase d i s t o r t i o n i s introduced i n t o the acoustic f i e l d at the surface i f the f l u i d i n the minitank i s d i f f e r e n t from the f l u i d i n the main tank. For thinner f l u i d s this e f f e c t i s l e s s pronounced. Also, a thi n f l u i d l a y e r i n the minitank i s desirable i n order to damp the motion of surface r i p p l e s of low s p a t i a l frequency. Further-more, as i n d i c a t e d i n section 3.2 the v e l o c i t y of propagation of the ultrasound i n the minitank should be less than or not much greater than the v e l o c i t y i n the main tank. F i n a l l y , a study of th i s thesis can give the reader an under-standing of the mechanisms involved i n using a l i q u i d surface i n an acoustic imaging system. 7 3 A P P E N D I X A F O U R I E R T R A N S F O R M S , D I F F R A C T I O N , I M A G I N G A . l Fourier Transform The three dimensional Fourier transform f(u,v,w) of a function f ( x , y , t ) w i l l be indicated by f(x,y,t) <-» f(u,v,u) where f(u,v,a>) = / / / f(x,y,t) e " j ( u x + ^ + W t ) dxdydt ( A - l ) —oo A two dimensional transform i n the s p a t i a l dimensions x,y w i l l be denoted by f(x,y,t) < A » f(u,v,t) A . 2 Convolution The convolution of two functions f^(x,y,t) * f 2 ( x , y , t ) i n the x,y,t dimensions i s defined as f ^ C x . y . t ) * f 2 ( x , y , t ) OO jjj ^ ( e . n . x ) f 2 ( x - c , y-n, t-x) dedndT ( A-2) s A convolution i n x,y only w i l l be denoted f^(x,y) * f 2 ( x , y ) , while i n t the convolution w i l l be denoted f ^ ( t ) & f 2 ( t ) . A . 3 D i f f r a c t i o n Freshel D i f f r a c t i o n (ignoring constant phase factors) The f i e l d f^(x,y) at a distance Z q from f Q ( x , y ) i s given by [13] 74 ' 0 M Fig . A . l D i f f r a c t i o n f ^ x . y ) = (1/XZ q) f 0(x,y) * e j ( k 2 / z o ) ( x 2 + y 2 ) " 2 2 / i /-v -\ 1/ \ j(k/2z )(x + y ) = (1/Xz ) f(u,v) e J o J o (A-3) (A-4) where t, N 4: / x i(k/2z ) ( x 2 + y 2 ) f(x,y) = f Q ( x , y ) e J oJ u = (k/z )x o v = (k/z )y k = 2TT/A X i s the wavelength of the r a d i a t i o n . A.4 Thin Lens F i g . A.2 Lens A th i n lens introduces a quadratic phase change i n the f i e l d I 75 f (x,y) such that [13] f l (x,y) - f o ( x , y ) e - J ( k / 2 f ) ( x 2 + y 2 ) p ( x / a > y / a ) (A-5) where PU,y) = {I I 2 2 1 for x + y < 1 ttherwise a i s the lens radius f i s the lens f o c a l length A.5 Imaging With a Lens y A F i g . A.3 Imaging with a lens 4>,(x,y) Applying (A-4) to $ ; (A-5) to <j> ; and (A-4) to 4>, then the O 3L D image <f)1 of the object function § i s 2 2 s ^ ( x . y ) = m e J a r [(^(-mx.-my) e J m r ) * q (mr) ] (A-6) where m = z /z, o 1 r = /("x2 + y 2) a = k/2z q(r) = (1/2TT) 2aa J 1 ( 2 a a r ) / r ^ P(u/2aa, v/2aa) 1/z + 1/z. = 1/f o 1 a i s the lens radius, i s a Bessel function of the f i r s t 76 kind or order 1, f i s the lens f o c a l length. Equation (A-6) can be s i m p l i f i e d since J^(x)/x i s n e g l i g i b l e f o r x > TT. I f the geometry of the imaging system i s such that [12] then a ( iT/2aa) << 1 ^ ( x . y ) = m e j ( m k / 2 f ) r [<j> (~ m x, - my) * q (mr) ] (A-7) A.6 Fourier Transform With a Lens [13] y A A V •v0(*,y) 'i F i g . A.4 Fourier transform with a lens The l i g h t d i s t r i b u t i o n V Q(x,y) from the front f o c a l plane of the lens becomes the d i s t r i b u t i o n V^(x,y) i n the back f o c a l plane. With V o(x,y) A V q(U,V) then ignoring a constant phase f a c t o r and the f i n i t e s i z e of the lens V ; L(x,y) = (1/Af) V o ( x k / f , yk/f) (A-8) 77 APPENDIX B NUMERICAL SOLUTION OF LIQUID SURFACE MOTION The numerical method that was used to determine the l i q u i d surface motion when subject to r a d i a t i o n pressure i s based on a method by Harlow and Welch. A d e t a i l e d d e s c r i p t i o n of the method i s given i n [ 2 1 ] . With t h i s method the Navier-Stokes equation i s written i n f i n i t e d i f f e r e n c e form, and the v e l o c i t i e s of the f l u i d p a r t i c l e s throughout the f l u i d are determined numerically f or a sequence of time steps and the subsequent motion of the surface i s followed by updating the p o s i t i o n of marker p a r t i c l e s at the f l u i d surface a f t e r each time step. The l i q u i d i s divided up i n t o rectangular c e l l s as i n figure B . l with the f l u i d pressure defined i n s i d e the center of each c e l l and the h o r i z o n t a l v e l o c i t i e s u and the v e r t i c a l v e l o c i t i e s v defined on the c e l l boundaries as i n figure B.2. The marker p a r t i c l e s are i n i -t i a l l y set at the center of the second from the top row of c e l l s to mark the p o s i t i o n of the surface. The surface motion i s constrained to stay w i t h i n the second from the top row of c e l l s . \/4 Fluid 1 / surface J Bottom of minitank J [ J L F i g . B . l F l u i d divided i n t o a rectangular mesh of c e l l s . 78 V. . P; • F i g . B.2 P o s i t i o n of pressure and v e l o c i t i e s f o r c e l l i , j . The deformation of the surface h i s assumed to' take the form H(t) cos nx when the r a d i a t i o n pressure takes the form P^ . cos nx [u(t) -u ( t - A t ) ] . Since the surface deformation h i s assumed to be much l e s s than the wavelength A or the depth of the f l u i d d, and the ac c e l e r a t i o n due to gravity i s set to zero, then from symmetry considerations i t i s only necessary to apply the numerical method to a length of f l u i d (along x) of A/4 (figu r e B.3) of depth d. J _ F i g . B.3 Deformation of the l i q u i d surface As in d i c a t e d i n section 4.2.1 the l i n e a r i z e d Navier-Stokes equation can be used, so that the non-linear terms i n the f i n i t e d i f -ference, equation i n [21] can be dropped. Also, since we have an anal-y t i c expression (4-17) f o r the pressure, the i t e r a t i o n required i n [21] to determine the f l u i d pressure d i s t r i b u t i o n can be 79 replaced by using (4-17). With the pressure at the surface, the surface deformation and the v e l o c i t y d i s t r i b u t i o n varying along x as cos nx, (4-7) can be used to determine the pressure at the surface f o r each time step of the numerical method. With the heavy l i n e bounding the f l u i d on the l e f t i n figure B . l considered to be a f r e e - s l i p w a l l , and the heavy h o r i z o n t a l l i n e a n o - s l i p w a l l , the v e l o c i t i e s requried by the numerical method outside the w a l l are given by [21]. At the r i g h t 'wall' the h o r i z o n t a l velo-c i t i e s on e i t h e r side of the wall are set equal, while the v e r t i c a l ve-l o c i t i e s and pressures on e i t h e r side of the r i g h t w a l l are set to be the negative of each other. Note that the r i g h t 'wall' i s neither a f r e e - s l i p w a l l nor a n o - s l i p w a l l as discussed i n [21] , since f l u i d i s allowed to pass through i t . The h o r i z o n t a l v e l o c i t i e s on the r i g h t w a l l are determined by the numerical method. The v e l o c i t i e s at the surface are set as i n [21], with the added condition that the shear stress at the surface be zero, which determines the h o r i z o n t a l v e l o c i t i e s outside the surface i n the top row of c e l l s . Thus, with the modifications that have been discussed, the motion of the l i q u i d surface can be determined by using a s i m p l i f i e d form of Harlow and Welch's method. Since the shape of the surface i s known to have the form cos nx, surface tension forces can be accounted f o r , and since h << X shear and normal stresses at the surface can be accounted f o r since the surface can f o r t h i s purpose be assumed to be f l a t . A number of computer runs were made f o r various parameter values as l i s t e d i n table B-1. The r e s u l t s were t y p i c a l l y those of the res-ponse of a second order system to a pulsed or step f o r c i n g function, as shown by figures 4.2 to 4.5. The r e s u l t s were seen to f i t c l o s e l y 80 the expressions (4-18), and (4-19) and (4-20) and the percentage overshoot P.O. and the damped frequency of o s c i l l a t i o n taken from the numerical r e s u l t s are l i s t e d i n table B-1. The peak value of the response H(t) as determined by (4-19) was compared with the numerical r e s u l t s , and the di f f e r e n c e i s shown i n table B-1 under column A as a percentage of the numerical r e s u l t s . A p o s i t i v e percentage indicates that (4-19) gave a higher value. Also, the per-centage difference i n the damped frequency OJ^ obtained i s l i s t e d under column B, where the damped frequency i s determined by the time at which H(t) reaches i t s peak value. The damping r a t i o £ l i s t e d was determined from (4-19). 81 TABLE B - l [ A cm d/A P g./cm Y dyn/cm . v x l O 2 cm Is P.O. W d rad/s A % B % 1 0.20 0.258 1.79 15.9 3.90 65.3 491 -1.5 +1.6 0.147 2 0.10 0.258 1.79 15.9 3.90 53.3 1372 -1.2 +1.8 0.207 3 0.05 0.258 1.79 15.9 3.90 39.4 3880 -0.9 -0.5 0.293 4 0.40 0.258 1.79 15.9 3.90 76.0 178 -2.3 +1.7 0.104 5 0.05 0.258 1.79 15.9 1.95 64.5 3910 -1 +2 0.147 6 0.10 0.258 7.16 15.9 3.90 75.2 697 -1.8 +1.9 0.104 . 7 0.20 0.258 1.79 3.98 3.90 43 236 -3.5 +2.1 0.293 8 0.05 0.50 1.79 15.9 1.95 73 4110 -6.2 +0.9 0.147 9 0.05 0.129 1.79 15.9 1.95 30.9 3090 -1.6 +3.4 0.369 10 0.20 0.50 1.79 15.9 3.90 73.5 511 -6.2 +1.4 0.147 11 0.05 0.04 1.79 15.9 •0.0 103.5 2065 -1.7 +0-9 0.0 12 0.05 0.129 1.70 15.9 0.0 105.6 3270 -2.7 +4.9 0.0 .13 0.20 0..04 1.7.9 15.. 9 .0..3.90 . .30.4 224 -5 .4 +8.0 0.369 14 0.20 0.02 1.79 15.9 0.9075 19.6 151 +1.7 +10.6 0.438 15 1.0 0.07 1.00 72.0 1.00 85.9 82.6 +0.5 +4.0 0.045 16 0.05 0.25 2.00 10.0 10.0 - 2870 - +1.7 0.25 17 0.02 0.25 3.00 5.00 6.00 0.0 - - - 1.65 18 1.0 0.08 2.00 20.0 25.0 0.0 - - - 2.32 19 0.05 1.00 1.79 15.9 1.95 73.4 4110 -6.1 +0.9 0.147 Computer runs to determine the l i q u i d surface response H(t) to a step function r a d i a t i o n pressure were made with the parameters i n d i c a t e d i n the f i r s t f i v e columns. The r e s u l t i n g percentage overshoot (P.O.) of H(t) and damped frequency of o s c i l l a t i o n oo^are shown; also, the values of P.O. and UJ^ obtained from (4-19) and (4-20) are given i n columns A and B re s p e c t i v e l y as a percentage of the values i n the columns P.O. and to^. Numbers 1, 2 and 16 are shown i n figures 4.3, 4.2 and 4.5 r e s p e c t i v e l y , and 17 and 18 i n figu r e 4.4. 82 APPENDIX C LIQUID SURFACE RESPONSE The equation f o r the motion of a l i q u i d surface when subject to r a d i a t i o n pressure, (4-18) can also be wr i t t e n as P r- 9 -Cu) t H(t) = r _ V ( A + B ) — n sin(3u t + G + 6) (C-l) ( p g + Yn ) for t ^ At, C < 1, and where 3 =yj{\.- C 2) . Cu At . . A = -e n s i n ($„ At) n B = e C W n A t cos (B_ At) - 1 n " tan 0 = B/C tan 6 = A/B Equation (C-l) s t i l l holds i f 3 i s imaginary (c > 1) as we can use the following r e l a t i o n s cos jx = cosh x s i n j x = j sinh x tan j x = j tanh x 83 REFERENCES 1. A.F. Metherell, H.M.A. El-Sum and Lewis Larmore, e d i t o r s , A c o u s t i c a l  Kologfaphy, Plenum Press, Inc., New York, 1969, V o l . 1. 2. A.F. Metherell and-Lewis Larmore, e d i t o r s , A c o u s t i c a l Holography, Plenum Press, Inc., New York, 1970 , V o l . 2. 3. A.F. Metherell, e d i t o r , A c o u s t i c a l Holography, Plenum Press, Inc., New York 1971, V o l . 3. 4. R.K. Mueller, "Acoustical Holography", Proceedings of the IEEE, Vol. 59, No. 9, p. 1319, Sept. 1971. 5. R.K. Mueller and N.K. Sheridan, Appl. Phys. L e t t e r s 9: 328-329, 1966. 6. P.S. Green, "Acoustical Holography with the L i q u i d Surface R e l i e f Conversion Method", Lockheel I n t e r v a l Report No. 6-77-67-42, Sept. 1972. 7. B.B. Brenden, "A Comparison of A c o u s t i c a l Holography Methods", A c o u s t i c a l Holography, Vol. 1, Ch. 4, pp. 57-71. 8. A c o u s t i c a l Holography, Plenum Press, V o l . 3, pp. 129-171. 9. Beckmann and Spizzichino, The Scattering of Electro-Magnetic Waves  from Rough Surfaces, MacMillan Co., 1963 10. B.P. Hildebrand, "An Analysis of the L i q u i d Surface as a Li g h t Scatterer", Optics and Device Development Technical Report 68-15, B a t t e l l e Memorial I n s t i t u t e , Richland, Wash., 1968. 11. R.B. Smith, and B.B. Brenden, IEEE Symposium on Sonics and U l t r a -sonics, Sept. 25-27, New York, 1968. 12. A. Popoulis, Systems and Transforms with Applications i n Optics, McGraw H i l l , 1968, pp. 205, 418. 13. J.W. Goodman, Introduction to Fourier Optics, McGraw H i l l , 1968, pages 60, 80. 14. Z.A. Gol'dberg, "Acoustic Radiation Pressure", i n High-Intensity  U l t r a s o n i c F i e l d s , edited by L. D. Rozenberg, Plenum Press, New York, 1971. 15. L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course of Th e o r e t i c a l  Physics, V o l . 6, Pergamon Press, London, 1959, Chapter 8. 16. L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course of T h e o r e t i c a l  Physics, V o l . 6, Pergamon Press, London, sections 12, 15. 84 17. L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course of T h e o r e t i c a l  Physics, VoL 6, Pergamon Press, London, 1959, sections 12, 25, 61. 18. L.D. Landau and E.M. L i f s h i t z , F l u i d Mechanics Course of Th e o r e t i c a l  Physics, Vol. 6, Pergamon Press, London, 1959, sections 15, 60. 19. J.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Pro- ducts , Academic Press, New York 1965, page 688, formula 6.5671. 20. R.F. Beyer, "Radiation Pressure i n a Sound Wave", Am. Jour. Phys., Vol. 18, No. 1, Jan. 1950, page 25. 21. F.H. Harlow and J.E. Welch, "Numerical C a l c u l a t i o n of Time-Dependent Viscous Incompressible Flow of F l u i d with Free Surface", The Physics of F l u i d s , Vol. 8, No. 12, Dec. 1965, page 2182. 22. B.P. Hildebrand and B.B. Brenden, An Introduction to A c o u s t i c a l  Holography, Plenum Press, New York, 1972, page 157. 23. H.F. Budd, "Dynamical Theory of Thermoplastic Deformation", J-. Ap. Phys., V o l . 36, No. 5, May 1965, page 1613. 24. J. B l i t z , Fundamentals of U l t r a s o n i c s , Butterworths, 1963, page 16. 25. E.G. Richardson, U l t r a s o n i c Physics, E l s e v i e r Publishing Co., Amster-dam, New York. 1962. page 188. 26. J.N. Goodman, Introduction to Fourier Optics, McGraw-Hill 1968, page 227. 

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