Many optical techniques are used to study ultrasonicfields, such as the Schlieren method,1 Bragg scatter-ng,2 and Raman–Nath scattering.3 Raman–Nath
scattering is used in a technique called optical diffrac-tion tomography4 ~ODT! to obtain information on thespatial structure of ultrasonic fields. These tech-niques have in common that they utilize diffractioneffects. For this reason they are suitable for investi-gating ultrasonic fields of only a certain diffractivepower, either because of high acoustic intensities orshort wavelengths. The use of time-averaged holo-graphic interferometry to study ultrasonic fields hasalso been demonstrated,5 and holographic interferom-etry combined with tomography has been used to in-vestigate beam profiles.6
In this paper I describe how time-averaged TV ho-lography may be used to investigate ultrasonic fields,both measuring projections of a field, analogous toSchlieren measurements, and computing tomographicreconstructions of a field, analogous to ODT. It hasbeen known for some time that time-averaged TV ho-lography may be used to measure acoustic fields intransparent media,7–9 and it has also been demon-strated that it may be used to measure projections of
R. Rustad [email protected]! is with the Insti-tute of Physics, Norwegian University of Science and Technology,N-7034 Trondheim, Norway.
7368 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
acoustic fields in water. A mathematical descrip-tion of both projection measurements and tomographicreconstructions is presented, and the techniques aredemonstrated through measurements of an ultrasonicfield in water produced by an annular medical ultra-sound probe11 operated at 3.25 MHz in cw mode.
2. Theory
A light ray of wave number k traversing an acousticfield of frequency V along a path S is subject to amodulation of its optical phase equal to
c~t! 5 k *S
m~r!cos@Vt 1 F~r!#ds
5 kpop *S
P~r!cos@Vt 1 F~r!#ds. (1)
m~r! is the maximal change in the optical refractiveindex at a given location. It is related to the pres-sure amplitude P~r! through the piezo-optic constantpop 5 dnydP. F~r! is the phase of the acoustic field.It is possible to measure the modulation c~t! withgreat precision by use of TV holography.12 Eachpixel in the TV image is a measurement of the mod-ulation of the optical path length along the path alight ray traverses on its way from the source,through the acoustic field, to the target of the TVcamera. The two-dimensional distribution of suchmeasurements in a TV frame is a projection of theacoustic field’s amplitude and phase. In tomogra-phy, the objective is to recover the acoustic field’s
s
dmosgp
Fm
r
pressure amplitude P~r! and phase F~r! with highpatial resolution from a series of such projections.Since TV holography measurements of ultrasound
o not rely on diffraction effects, but measure theodulation of optical phase directly, the limitations
n acoustic intensity and wavelength are much lessevere than those pertaining to other optical investi-ation techniques. Raman and Nath introduced thearameter3
y 5 kmL, (2)
where k is the optical wave number, m is the maximalchange of optical refractive index, and L is the inter-action distance between the optical and the acousticwaves. If Raman–Nath or Bragg diffraction occurs,only the zeroth diffraction order, i.e., the undiffractedlight, is of importance in TV holography. The higherdiffraction orders are frequency shifted and do notinterfere with the reference wave in the interferom-eter. They simply contribute to the backgroundlight intensity, which is removed through filtering inthe signal processing in TV holography.12 However,Raman–Nath diffraction may lead to total extinctionof the zeroth diffraction order. Raman and Nathshowed3 that the intensity in the zeroth order is pro-portional to J0
2~y!, which has its first zero at y 52.405. This is equal to a modulation of the phase ofthe light outside the measuring range of current TV-holography systems. It is therefore assumed thatdiffraction does not represent a limiting factor in TV-holography measurements of ultrasonic fields at thistime.
It is assumed that the light propagates alongstraight lines, i.e., that ray bending is insignificant.Klein and Cook introduced a parameter13
Q 5K2L
k, (3)
Fig. 1. Pressure phasor P of magnitude P and phase F and itseal and imaginary parts PRe and PIm, respectively.
where K is the acoustic-wave number. It is commonto assume that ray bending does not occur when
Qy ,, 1. (4)
For the field studied here it would take a pressureamplitude of 100 kPa to bring Qy close to 1. Thisgives y ' 2.2, which again is outside the range of theTV-holography system.
A. Projections
Equation ~1! can be written in the form
c~t! 5 kpopHcos~Vt! *S
P~r!cos@F~r!#ds
2 sin~Vt! *S
P~r!sin@F~r!#dsJ; kpopFcos~Vt! *
S
PRe~r!ds 2 sin~Vt! *S
PIm~r!dsG .
(5)
The two integrals are the integrals over the real andthe imaginary parts of the phasor P along the path S.Figure 1 illustrates the relationship between the pha-sor P and its real and imaginary parts. I define theintegrals
PRe 5 *S
PRe~r!ds,
PIm 5 *S
PIm~r!ds. (6)
The modulation c~t! can now be characterized by aphasor ac of magnitude ac and phase f, which aregiven by
ac 5 pop~PRe2 1 PIm
2!1y2,
f 5 tan21SPIm
PReDmod 2p, (7)
so that
c~t! 5 kac cos~Vt 1 f!. (8)
igure 2 illustrates the relationship between theodulation phasor ac and the integrals )Re and )Im.The techniques presented here are applicable to
acoustic waves of any shape, but a case of specialinterest arises when the acoustic field is a harmonicplane wave and the path of integration is normal tothe direction of propagation of the acoustic wave.Assume that z is the direction of propagation of theacoustic wave and x the direction of propagation ofthe optical wave and that the acoustic field is of a
1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7369
t
T
t
so
T
ia
a
7
finite width L centered at x 5 0. Equation ~5! canhen be written as
c~t! 5 kpop *21y2
1y2
P~r!dx$cos~Vt!cos@F~z!#
2 sin~Vt!sin@F~z!#%. (9)
In this case
ac 5 pop *21y2
1y2
P~r!dx 5 popP~z!L,
f 5 F~z!, (10)
where
P~z! 51L *
21y2
1y2
P~r!dx (11)
is the average pressure amplitude. f, and hence F,may be measured directly. Sinusoidal phase mod-ulation of the reference arm of the interferometer ata frequency f 5 V 6 Df, where Df is in the range of1–10 Hz, allows for real-time observation of phasefronts and their propagation.14
B. TV-Holography Projections
c~t! is measured by phase-modulated time-averagedV holography, as described elsewhere.8,12 In the
TV image, the intensity at a given location I~x, y! isproportional to the square of the zero-order Besselfunction of the first kind15 J0
2~x!. The argument ofthis function is given by the vector difference betweenthe modulation phasor ac~x, y! and the phasor de-scribing the modulation of the reference arm of theinterferometer ar:
I~x, y! } J02@kuar 2 ac~x, y!u# 5 J0
2(k$ac2~x, y! 1 ar
2
2 2ac~x, y!ar cos@f~x, y! 2 u#%1y2). (12)
Fig. 2. Modulation phasor ac of magnitude ac and phase f, can itsreal and imaginary parts, the integrals pop)Re and pop)Im.
370 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
The term u is the phase of the modulation of thereference arm of the interferometer. A modulationamplitude ar is chosen so that I work on the linearpart of the first lobe of the Bessel function,14 and I dohe approximation
I~x, y! } J02(k$ac
2~x, y! 1 ar2 2 2ac~x, y!ar
3 cos@f~x, y! 2 u#%1y2)
< Ib~x, y! 2 c~x, y!ac~x, y!cos@f~x, y! 2 u#.
(13)
The terms Ib and c are pixel-dependent constants.To determine ac and f, four measurement frames arerecorded. The phase u is shifted by 90° between con-ecutive frames, and four intensity distributions arebtained:
The constant Ib~x, y! is eliminated in the subtrac-tions, and c~x, y! is determined from calibrationframes, as described in Ref. 12. The piezo-optic con-stant pop for water is given in Ref. 16 as pop 5 1.51 310210 Pa21.
C. Tomography
Tomography is well known from a large number ofapplications in medicine, research, and industry, andfor this reason is not discussed in detail here. Theaim of tomography is to reconstruct a function in twoor three dimensions from its projections. I want toreconstruct the pressure amplitude and the phase ofan ultrasonic field in a plane perpendicular to the TVtarget of the interferometer, i.e., a plane that isspanned by the set of rays that strike a horizontal lineof pixels on the TV target. Let x and y be coordinatesin this plane and t be the position along a line beingmaged onto the TV target. The TV target is at anngle u to the x axis, so that
t 5 x cos~u! 1 y sin~u!, (16)
s shown in Fig. 3.
itffi
crbt
The integrals in Eqs. ~6! obtained at an angle u cannow be written as
PRe,u ~t! 5 *2`
`
*2`
`
PRe~x, y!d@x cos~u!
1 y sin~u! 2 t#dxdy,
PIm,u ~t! 5 *2`
`
*2`
`
PIm~x, y!d@x cos~u!
1 y sin~u! 2 t#dxdy. (17)
These are the Radon transforms17 of the real and themaginary parts of the pressure phasor. Severalechniques are available for reconstructing a functionrom its Radon transform. I use the technique ofltered backprojection as described in Ref. 18. The
Fig. 3. Recording geometry for tomography by TV holography.The projection of a function in the plane x–y is recorded by a TVamera at an angle u. The term t is the position along a lineunning through the origin at the same angle, and d is the distanceetween points on this line being imaged onto adjacent pixels onhe TV target.
Fig. 4. Experimental setup for TV-holography measurements of
algorithm is applied to the real and the imaginarycomponents separately. The output of the tomo-graphic reconstruction algorithm is a matrix with adiscrete representation of the reconstructed function.For the real and the imaginary parts of the pressurephasor, the two matrixes PRe and PIm are computed.Each matrix element represents a contribution to theintegrals in Eqs. ~6! and ~17!, so that
PRe~i, j! 5 P cos~F!d,
PIm~i, j! 5 P sin~F!d. (18)
The term d is the width of the matrix elements and isequal to the distance between points in the recon-struction plane imaged onto adjacent pixels in a lineon the TV target. The matrix elements are small, soP and F are assumed to be constant over each ele-ment. The pressure and the phase at any position inthe reconstruction plane can be computed from thecorresponding matrix elements by
P~x, y! 51d
@PRe2~i, j! 1 PIm
2~i, j!#1y2,
F~x, y! 5 tan21 PIm~i, j!PRe~i, j!
mod 2p. (19)
sonic fields in water. See text for explanation of notation used.
Fig. 5. Schematic drawing of the medical ultrasound probe.
ultra
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aw
t
7
The reconstruction process is repeated for all the hor-izontal lines in the TV image. This produces recon-structions in parallel, horizontal planes, one on top ofthe other. These planes are combined to form athree-dimensional matrix that represents a full-fieldreconstruction of the ultrasonic field.
3. Experimental
A. Setup
Figure 4 shows the experimental setup used to mea-sure ultrasonic fields in water with TV holography.A medical ultrasound probe radiates vertically into atank of water. The tank is made from glass, hasplane sides, and measures 60 mm 3 60 mm 3250 mm. The ultrasonic field is attenuated by a5-cm layer of foam rubber in the bottom of the tank toprevent interference from reflected waves. An im-aging interferometer is looking into the tank. Thelight source is a 5-mW He–Ne laser operated at632.8 nm. The light is split into an object arm and areference arm by beam splitter BS. The object arm
Fig. 6. Projection measurement of the ultrasonic field outside tmplitude, ~right! a gray-scale representation of the projection of thite to p. The cap of the probe is semitransparent, and the wa
Fig. 7. Plot of the amplitude projection ac in nanometers alonghe line A–A in Fig. 6.
372 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
traverses a tilted glass plate that may be rotated bya stepper motor. This is speckle averaging unit SA.Mirrors M3–M5 direct the object arm around thetank. On the far side of the tank the laser beamstrikes a series of diffusers that produce a speckledillumination of the tank. Rotation of the tilted glassplate between the acquisitions of TV frames shifts thephase of the speckles and speckle averaging isachieved.
The light from the diffusers traverses the ultra-sonic field in the tank, and the resulting speckle fieldpasses aperture-and-lens assembly A-L1 and inter-feres with the reference arm of the interferometer inbeam combiner BC. The reference arm of the inter-ferometer traverses electro-optic modulator EOM,which is used for sinusoidal phase modulation of thereference light. The object plane of the lens is at thecenter of the ultrasonic field.
The interferogram is recorded by a CCD camera.The TV signal is high-pass filtered and square recti-
of the probe: ~left! a gray-scale image of the projection of thease. The phase is wrapped so that black corresponds to 2p and
onts inside the cap are visible in the phase image.
Fig. 8. Close-up of the projection of the phase of the field at the tipof the probe. The measured data have been filtered once with a3 3 3 spatial median filter.
he tiphe phve fr
iatofia
fied19 in electronic processing unit EPU before it isdigitized by a frame grabber for further processing ona computer. The computer controls a signal gener-ator providing the drive signals for the probe and theelectro-optic modulator in the reference arm of theinterferometer. For the purpose of tomography, theprobe is mounted on a fixture that allows its angularposition to be determined with great accuracy. Thetank is placed on a vertical translator so that theultrasonic field can be studied at different distancesfrom the probe.
B. Projection Measurements
The ultrasonic field I have investigated is producedby an annular medical ultrasound probe11 operatedin cw mode at 3.25 MHz. A schematic drawing ofthis probe is provided in Fig. 5. Four concentric,annular transducer elements are attached to a con-cave support. The outer diameter of the active areais 14.7 mm. All transducer elements are driven inphase with each other. A flexible, semitransparent
Fig. 9. Plot of the phase along a vertical line in Fig. 8. Theunfiltered data are shown.
Fig. 10. Three light rays passing through an ultrasonic field of plaof the field is shown to the left. The lens system of the interferomwill illuminate the same pixel in the TV camera. They all undergfield, except when they pass through the maxima or the minima o
rubber cap filled with a liquid of matching acousticrefractive index covers the transducer elements.During normal use the cap and the liquid ensure goodacoustic contact between the transducer and the pa-tient’s body. They do not contribute to the focusingof the beam. The tip of the cap is 6 mm away fromthe transducer surface. The probe produces an ul-trasonic field focused at a distance of approximately72 mm from the tip of the probe. The half-width ofthe beam at the focus is estimated to be20 3.4 mm.
First, projections of the amplitude and the phase ofthe near field of the probe were measured. Theframes have a format of 256 3 256 pixels and cover16 3 12 mm. One of these projections is presentedn Fig. 6. A plot of the projection of the amplitudelong the horizontal line A–A immediately below theip of the probe is presented in Fig. 7. At the centerf the projection, the amplitude is ac 5 28 nm. If theeld is assumed to have a diameter of 11 mm, theverage pressure is P 5 17 kPa.A lens of greater magnification was used to record a
close-up of the projection of the phase of the field at thetip of the probe. This measurement is presented inFig. 8, and a plot of the phase along a line in thedirection of propagation is presented in Fig. 9. Eachpixel in Fig. 8 images an area in the object plane mea-suring approximately 12 mm 3 9 mm, but, because ofthe large entrance pupil of the magnifying lens system,the corresponding ray bundle has a diameter of nearly400 mm at the edge of the ultrasonic field. The mea-surement still seems to have very good spatial resolu-tion. This may in part be explained by Fig. 10.Light rays focused onto the same pixel may take dif-ferent paths through the acoustic field, but all thesepaths intersect at the focus in the middle of the field,and they all undergo approximately the same modu-lation, except when passing through the maxima andthe minima of the acoustic wave.
aves. One of the quadrature components of the amplitude phasoris focused at the center of the ultrasonic field, and all three rays
proximately the same phase modulation as they pass through theacoustic wave.
ne weter
o apf the
1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7373
pesjo4mwst
p
7
C. Tomographic Reconstructions
A series of 60 projections like the one presented inFig. 6 was recorded. The angular separation be-tween consecutive measurements was 3°. A recon-struction of the amplitude and the phase of theacoustic field was computed for a volume outside thetip of the probe.
In the tomographic reconstruction process it wasassumed that the light propagates along straight, par-allel paths. In reality, the light that illuminates onepixel of the TV target of the interferometer consists ofmany rays covering a volume shaped like a double conewith the apex in the object plane of the lens system.The opening of the cone is determined by the diameterof the entrance pupil of the lens system and the dis-tance to the object plane. In addition, the cones arenot parallel, but diverge slightly. The straight, par-allel beam model is still useful as a first approxima-tion, but elaborate models should replace it later.
Fig. 11. Reconstruction of the ultrasonic field at 3.25 MHz in a pof the transducer. The amplitude reconstruction is presented inphase is presented in a wrapped gray-scale representation, so tha
Fig. 12. Same reconstruction as in Fig. 11 in a plane parallel to thregion above the line A–A the probe intersects some of the projectthe phase image points at a disturbance in the phase.
374 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
In tomography, the spatial resolution in the recon-structions is determined by the spatial sampling fre-quency in the projections and the number ofprojections. Each projection should fulfill the sam-pling theorem, and the number of projections shouldbe somewhat larger than the number of samples ineach projection ~Ref. 18, p. 186!. If the number of
rojections is smaller than the number of samples inach projection, the number of projections limits thepatial resolution in the reconstructions. Sixty pro-ections were used here. This limits the spatial res-lution in each reconstruction to approximately00 mm. These reconstructions are in planes nor-al to the direction of propagation of the acousticave, and these planes are only 50 mm apart, so the
patial resolution in the direction of propagation is ofhe order of 50 mm.
Two cross sections through the reconstruction areresented here. Figure 11 shows a gray-scale rep-
normal to the direction of propagation, immediately below the tipscale so that lighter gray corresponds to higher pressure. The
ck corresponds to 2p and white to p.
ection of propagation intersecting the center of the probe. In theThe reconstruction is not reliable in this region. The arrow in
lanegrayt bla
e dirions.
Fig. 13. Projection measurements of the field at different distances from the probe. All the amplitude plots use the same gray scale. Inthe topmost projection, the probe is visible. The second projection shows the focus of the field, at the white markers, 72 mm below thetip of the probe, while the last projection shows the field as it hits a layer of foam rubber 150 mm from the tip of the probe. The horizontallines are distance markers, 10 mm apart.
1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7375
7
resentation of the pressure amplitude and the phaseof the ultrasonic field in a horizontal cross sectionimmediately outside the tip of the probe. The recon-struction shows that the field has a peak pressureamplitude of approximately 32 kPa, while the aver-age pressure amplitude along a line running throughthe center of the field is 17 kPa, in agreement withthe plot in Fig. 7. The phase plot shows that thewave is concave. The corrugations seen around thecenter of the phase plot are of the order of 1 rad.
Figure 12 shows a gray-scale representation of thephase and the amplitude of the ultrasonic field in aplane parallel to the direction of propagation. Theplane intersects the center of the probe. The annu-lar structure of the near field is apparent in both crosssections. In the vertical cross section in Fig. 12, onlythe part of the field that is below the tip of the probeis reliably reconstructed. The phase image of thiscross section shows the good spatial resolution of thisreconstruction. Also in Fig. 12, a disturbance in thephase is visible at the center of the field approxi-
Fig. 14. Plot of the amplitude projection ac in nanometers at thefocus. The bar shows the estimated half-width of the beam at thefocus.
Fig. 15. Projection of the field as it strikes
376 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
mately four wavelengths below the tip. This point,which is indicated by an arrow in the figure, coincideswith a minimum in the amplitude.
Figure 13 shows projection measurements of theultrasound beam at three different distances fromthe probe. All three measurements are obtained un-der equal conditions with the same excitation of theprobe. The amplitude plots use the same gray scale,and the change in acoustic intensity and beam diam-eter along the beam is evident. The phase plotsshow how the shape of the phase fronts changes fromconcave near the probe to plane at the focus, andfurther to convex after the focus.
A plot of the amplitude projection at the focus isshown in Fig. 14. When compared with the plot inFig. 7 it shows that the acoustic intensity nearly dou-bles from the probe tip to the focus. The beam di-ameter is nearly halved.
Finally, another projection of the field as it strikesthe foam rubber at the bottom of the tank is pre-sented in Fig. 15. The beam has a higher intensitythan it had when the measurements in Fig. 13 wereacquired. Interference with the reflected wave andscattering on inhomogeneities in the foam rubber aremore evident in this measurement than in the one inFig. 13.
4. Conclusion
The theory of measuring ultrasonic fields in water byTV holography has been presented. The experimen-tal procedure and the processing of the measureddata have been explained. Both projection measure-ments and tomographic reconstructions of a 3.25-MHz ultrasonic field produced by a medicalultrasound probe have been presented. The mea-surements have a spatial resolution of better than100 mm.
TV holography combined with tomography pro-vides a powerful tool for investigating ultrasonicfields. The projection measurements are of a quality
er of foam rubber at the bottom of the tank.
a lay
7. O. J. Løkberg, “Sound in flight: measurement of sound fields
equal to Schlieren photography, but the optical in-strumentation is simpler since there is no need for acollimated light beam or accurately positioned stopsand mirrors. In addition, the phase is measured si-multaneously with the amplitude. Since the TV-holography measurements do not rely on diffractionof light, this technique is also applicable to acousticfields of low intensities and long wavelengths.
The tomographic reconstructions give quantitativemeasurements in three dimensions of both the am-plitude and the phase of ultrasonic fields, as doesODT. The technique is self-calibrating, meaningthat, if the interferometric measurements are cali-brated, no further calibration is needed to obtain cor-rect quantitative measurements of the amplitude.The spatial resolution is at least as good as what hasbeen reported for ODT, and the experimental proce-dure is simpler and applicable to acoustic fields overa wider range of frequencies and intensities.
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