PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Light scattering observation of oscillation of a bubble in acoustic
cavitation cloud
Takanobu Kuroyama1
1 National Institute of Technology, Gifu College, Japan

ABSTRACT
High-intensity ultrasound in liquid generates oscillating fine bubbles called acoustic cavitation. Although
the behavior of the oscillation of the single bubbled driven by ultrasound is revealed, the behavior of the
bubbles in the multi-bubble system is not clear because of the difficulty of the measurement. To overcome
the problem, we propose an improved light scattering measurement technique employing confocal optics.
In this system, a laser beam irradiates the bubble cloud. The light scattered by a bubble in a tinymeasurement volume, which is formed by the confocal optics, is measured. The detected scattered lightintensity is proportional to the square of the bubble diameter. Thus, the oscillation of a bubble in the cloud
can be observed with the scattered light. The concept and the principle of measurement are described andthe characteristics are analyzed by numerical simulation. The verification experiment is performed and theoscillation of a bubble in acoustic cavitation formed under the ultrasonic horn is measured. The single
bubble oscillation-like behavior of a bubble in the acoustic cavitation is observed and it may indicate theability of the proposed method to measure the oscillation of a single bubble in the acoustic cavitation.
Keywords: Acoustic cavitation, Oscillation, Laser scattering

1. INTRODUCTION
Acoustic cavitation means a cavitation phenomenon induced by the high-intensity ultrasound in
the liquid. The negative pressure of the ultrasound causes the deposition of the dissolved gases in theliquid and the deposited gases form tiny bubbles. These bubbles volumetrically oscillate with thefluctuation of the ultrasound pressure. The bubbles gradually expand with the negative ultrasoundpressure and rapidly contract with the positive ultrasound pressure within a ultrasound period even if
the bubble is driven by the sinusoidal ultrasound pressure. The shrinkage stage of the bubble
accompanies the violent collapse and the semi-adiabatic compression of the inner gas. The collapseof the bubbles produces various chemical and physical phenomena, such as the sonoluminescence (1),sonochemiluminescence (2), production of radicals (3), and microjet (4).
The physics of the above mentioned nonlinear bubble oscillation has been revealed theoreticallyand experimentally for the single bubble case. Especially, the single bubble oscillation experiment
performed by Gaitan and Crum made further understanding of the bubble oscillation (5). In the
single bubble oscillation experiment, a bubble is trapped at the antinode of 101 kHz order ultrasonicstanding wave. The trapped bubble stably oscillates with an accurate period without translationalmotion. Thus, the bubble oscillation can be easily observed with the stroboscopic imaging (6) or thelaser scattering measurement (7, 8). However, the bubble oscillation in acoustic cavitation with multibubbles is difficult to be observed because the bubbles move randomly. The bubble oscillation in
ordinal acoustic cavitation is complicated because the bubbles mutually interact with each other andthe formation, coalescence, and collapse of the bubble occur. Thus, the oscillation characteristics ofthe bubble in the acoustic cavitation are still unclear.
In this paper, we propose a measurement method of the oscillation of a single bubble in acoustic
cavitation employing the laser scattering. The purpose of this paper is to establish the measurement
principle of the method. First, the concept and the theory of the measurement method are described.
The numerical simulation is performed to clarify the characteristics of the proposed method and theresult of the experimental observation of the bubble oscillation is discussed.

1kuroyama@gifu-nct.ac.jp

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2. Measurement Principle
2.1 Concept of Proposed Method
The optical system of the proposed method is shown in Figure 1. Acoustic cavitation bubbles inwater are irradiated by a Gaussian laser beam and the bubbles scatter the laser. An imaging lens is
positioned so that the image of the side scattered light from bubbles are formed on the pinhole. The
output of the photodetector is proportional to the power of the scattered light image on the pinhole
aperture. This lens system is called confocal optics. The scattered light only arises when the laser
beam crosses the bubbles and the bubble image is formed away from the optical axis of the lenssystem when the bubble is far from the optical axis. Thus, the photodetector only detects thescattered light when a bubble is positioned in the measurement volume, where the laser beam has
high light intensity near the optical axis of the lens system.
The size of the measurement volume, V, is approximated asV ≈ [π(d / M )2 / 4]D ,(1)

where d, M, D are the pinhole aperture diameter, the magnitude of the lens system, and the laser
beam diameter, respectively. Assuming that the bubble is spatially randomly distributed, the numberof bubble within the measurement volume k follows the Poisson distribution with a mean number of
bubbles in the measurement volume μ denoted as Pk , such that

Pk = µ k exp(−µ) / k!
µ = NV(2),

where N is the number density of the bubbles (9). The probability of existing two or more bubbles in
the measurement volume can be reduced by decreasing the mean number of bubbles μ. The mean
number of bubbles μ can be set to small by reducing the measurement volume.
The side scattered light intensity is almost proportional to the square of the bubble diameter (10).
Thus, the radial oscillation of the bubble varies the scattered light intensity. When a single bubble islocated in the measurement volume, the radial oscillation can be measured with the fluctuation of thephotodetector output.
2.2 Scattering of Gaussian Laser Beam by Bubble in Water
Schematic diagram of coordinates for Gaussian beam scattering is shown in Figure 2. The bubble
at origin is irradiated with a Gaussian beam focused to the half-width w0 at the point (xf, yf zf). The
far-field intensity of the scattered light is expressed asI(r,θ,φ) ∝ ⎡⎣| S1 (θ,φ) |2 + | S2 (θ,φ) |2 ⎤⎦ / r 2 ,(3)where r, θ, and φ are the distance between the bubble center and the origin, the scattering angle, andthe angle between the x-axis and the scattering plane as shown in Figure 3, respectively. The
scattering function for x-polarized light S1(θ, φ) and for y-polarized light S2(θ, φ) are expressed as

Photo detectorx

Scattered light(Off-axis)

PinholeLens

Scattered
light

z
yf

BubblesFigure 1 – Concept of proposed method

Beam
waist

xf
Bubble

Laser
beam

4814

Laser
beam

zf
y

Figure 2 – Gaussian beam scattering coordinates∞

2l + 1⎡⎣ − jmα lm π l |m| (θ) + β lm τ l |m| (θ) ⎤⎦ exp( jmφ) ,
2l(l
+
1)
l=1 m=− ll

S1 (θ,φ) = ∑ ∑∞2l + 1⎡⎣ jmβ lm π l |m| (θ) + α lm τ l |m| (θ) ⎤⎦ exp( jmφ) ,
2l(l
+
1)
l=1 m=− l(4)lS2 (θ,φ) = ∑ ∑(5)where πlm (θ) and τlm (θ) are the angular functions. The partial-wave scattering amplitudes αlm and βlm
are the function of the beam waist position (xf, yf zf), half beam diameter w0 , and the bubble
properties, such as the diameter and the refractive index (11).
2.3 Imaging Characteristics of Lens System
The scattered light of the bubble is condensed on the pinhole at the entrance of the photodetector
by the imaging lens as shown in Figure 4. The front focal distance l1 and the back focal distance l2
satisfy the lens formula l1 −1 + l2 −1 = f −1 where f is the focal length of the lens. The magnitude of thelens system M is calculated as M = −l2 / l1 , where the negative sign means the inversion of the image.
The light amplitude on the image plane, which is on the pinhole, becomes the convolution between
the apparent light amplitude distribution of the scattered light on the object plane x-z and theamplitude point spread function. Thus, the x-polarized light amplitude on the image plane c1 (x, z; xb ,yb , zb ) for the scattered light from a bubble located at (xb , yb , zb ) is shown as

c1 (x, z; xb , yb , z b ) = c1 '(x / M , z / M; xb , yb , z b ) ,(6)

c1 '(x, z; xb , yb , z b ) = s1 (x, z; xb , yb , z b ) * h(x, z; yb ) ,(7)

where s1 (x, z; xb , yb , zb ) is the apparent x-directional light amplitude of the scattered light on theobject plane (12). The amplitude point spread function h(x, z; yb ) is given byh(x, z; yb ) = ∫ P(ρ, yb )J 0 kρNA x 2 + z 2 ρdρ ,)(8)P(ρ, yb ) = exp[ j2 yb knsin 2 (α / 2)ρ 2 ] ,(9)NA = nsin(α) ,(10)tan(α) = (DL / 2) / l1 ,

(11)

1

0

(

where J0 , n, α, NA, and DL are the first-kind zeroth-order Bessel function, the refractive index of themedium surrounding the lens, the angle of aperture, the numerical aperture, and the aperture
diameter of lens, respectively (13). The optical wavenumber k is given by k = 2π / λ where λ is the
optical wavelength.
2.4 Photodetector Output
The apparent x-polarized light amplitude s1 (x, z; xb , yb , zb ) in Eq. 7 can be calculated from thescattering function shown in Eq. 4. The spatial Fourier transformation S1 ’(νx, νz) of the apparent
x-polarized light amplitude for a bubble at origin s1 ’(x, z) = s1 (x, z; 0, 0, 0) is given by

S1 '(ν x , ν z ) = ∫∫ s1 '(x, z)exp[2π(ν x x +ν z z)]dxdz ,(12)where νx and νz are the x- and z-directional spatial frequency, respectively. The spatial frequencies
are related with the propagation angle of the scattered light as (14)cos(θ) = λν z ,(13)cos(φ) = λν x .

(14)

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x

x
Scattering
plane
l2

z

Bubblezxb

φLaser Bubble
beam

zb

Lens

yb

Image
plane
(Pinhole)
Pinholeaperture

l1
Scatteringy angle, θFigure 3 – Scattered light coordinatesyBubble image
Figure 4 – Imaging coordinates

Therefore, the apparent light amplitude s1 ’(x, z) can be obtained by the inverse Fourie transformationof the scattering function by replacing the spatial frequency with the converted propagation angle.
When the bubble is located at (xb , yb , zb ), the light amplitude distribution displaces from the origin as
s1 (x, z; xb , yb , zb ) = s1 ’(x − xb , z − zb ). The y-directional displacement of the bubble yb does not affectthe apparent light amplitude but affects the amplitude point spread function of the lens system. From
the definition of the coordinates shown in Figures 2 and 4, the bubble displacement and the beam
waist position have a relationship as xf = −xb , yf = −yb , and zf = − zb − zbw where zbw is the distance
between the lens axis and the beam waist position. The power of the x-polarized scattered light
passing through the pinhole aperture becomes the integration of the light intensity on the aperture as

O1 (xb , yb , z b ) = ∫∫ 2 2( x +z )<( d /2)2

| c1 (x, z; xb , yb , zb ) |2 dxdz .

(15)

Similarly, the power of the y-polarized scattered light passing through the pinhole aperture isexpressed as

O2 (xb , yb , z b ) = ∫∫ 2 2( x +z )<( d /2)2

| c2 (x, z; xb , yb , zb ) |2 dxdz ,(16)

where c2 (x, z; xb , yb , zb ) is the y-polarized light amplitude on the image plane. The output of thephotodetector, O(xb , yb , zb ), is proportional to the total power as

O(xb , yb , z b ) ∝ O1 (xb , yb , z b ) + O1 (xb , yb , z b ) .

(17)

3. Simulation of Measurement Property of Optical System
3.1 Experimental Setup of Measurement System
The schematic diagram of the experimental setup is shown in Figure 5. A bolt-clamped
Langevin-type transducer (BLT) with a horn radiates ultrasound into the water in a glass vessel withan inner dimension of 40×40×68 (mm3 ). The diameter of the horn output surface is 30 mm. The BLT
is driven by the sinusoidal voltage with frequency of 19.29 kHz. The acoustic cavitation bubbles as
the measurement targets mainly distribute just below the horn surface.
The acoustic cavitation bubbles below the horn are irradiated by a Gaussian laser beam generated
by a He-Ne laser source. The beam waist is located at the exit aperture of the laser source and has 0.4
mm of half beam diameter. The distance between the exit aperture and the horn axis is 600 mm. The
scattered light from the bubbles enters a lens and the condensed scattered light is measured with anavalanche photodiode module (APD). The front focal length of the lens is 80 mm and the diameter ofthe lens aperture is 48 mm. The diameter of the active area, which corresponds to the diameter of thepinhole aperture, is 1.5 mm. The magnification of the lens system is 3 and the numerical aperture is0.287. Thus, the object plane equivalent pinhole diameter d / |M| is 0.5 mm.

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Bolt-clamped Langevin-typetransducer with hornx
He-Ne laser source

z
Lens

y

Glass vessel
Avalanche photo diodemoduleFigure 5 – Experimental setup
The output signal of the APD is recorded via an analog to digital converter (ADC). The sampling
frequency of the ADC is 25 MHz. The recorded signal of the APD is band-limited below 10 MHz by
the frequency response of the APD. The driving voltage and the current fed to the BLT are
simultaneously recorded by the ADC and the driving power is calculated.
3.2 Spatial Sensitivity Characteristics to Bubble Position
Numerical simulation of the photodetector output was performed to reveal the characteristics ofthe proposed method. The bubble position dependence of the photodetector output for the bubble
diameter of 10 μm is shown in Figure 6. The photodetector output is normalized by the maximum
value observed in the simulation for the bubble diameter of 10 μm. photodetector output has a large
value in the circular region centered at (xb , zb ) = (0, 0) on yb = 0 plane as shown in Figure 6(a). The
diameter of the circular region is almost the same as the object plane equivalent pinhole diameter d /|M|. Although the boundary of the region is blurred by the diffraction limit of the lens system and thedefocus, the shape of the measurement volume in xb -zb cross section is dominated by the shape of thepinhole for any yb . The output variation along the xb -axis is caused by the radial light intensity
variation of the incident Gaussian laser beam. On the zb = 0 plane, as shown in Figure 6(b), thephotodetector output decreases with increasing |yb | because of not only the intensity variation of theincident laser beam but also the defocusing of the lens system. As a result, the output variation on
this plane is larger than that on the xb -zb plane as shown in Figure 6(c). Figure 7 shows the simulation
results for the bubble diameter of 100 μm. The output characteristics are almost the same as the caseof 10 μm except for the absolute value of the output. The photodetector output is normalized by the
maximum value in each simulation. Thus, the maximum values are both unity but the absolute
maximum value for the 100 μm case is about 102 times larger than that of 10 μm case.
Let we define the measurement volume V as the region where the photodetector output has a value
larger than 1/e2 of its half maximum value as
1

(b)

0

−500
−500

0
zb (μm)500 −5000yb (μm)500

01Norm. photodetector output(a)Norm. photodetector output

xb (μm)

500

(c)

0.5

0

zb

yb

−500

0
yb or zb (μm)

500

Figure 6 – Bubble position dependence of photodetector output for bubble diameter of 10 μm. (a) Output
on xb-zb plane. (b) Output on xb-yb plane. (c) Output along xb and zb-axis.

4817

1

(b)

0

−500
−500

0
zb (μm)500 −5000yb (μm)

(c)

zb

0.5

0

500

1

Norm. photodetector output(a)Norm. photodetector output

xb (μm)

500

0

yb

−500

0
yb or zb (μm)

500

Figure 7 – Bubble position dependence of photodetector output for bubble diameter of 100 μm. (a) Output

0.6

N = 5 mm−3
10 mm−3
20 mm−3

Probability, Pk

0.5
0.4
0.3
0.2
0.1
0.0

0 1 2 3 4 5 6 7 8 9 10
Number of bubbles inmeasurement volume, k

Photodetector output (arb. unit)

on x-y plane. (b) Output on x-z plane. (c) Output along xb and zb-axis.
105
Squared (0, 300, 0) (0, 0, 0)
104 bubble diam.
103
102
101
100
10-1
(300, 0, 0)
10-2(xb, yb, zb) = (300, 300, 0) (μm)
10-3

100

101
Bubble diameter (μm)

102

Figure 8 – Probability of existing k bubbles

Figure 9 – Relationship between bubble diameterin measurement volume

and photodetector output

V = ∫∫∫O( xb ,zb ,zb )>max[O( xb ,zb ,zb )]exp(−2)

dx dy dz .

(18)

The measurement volumes calculated from the simulation result for the bubble diameter of 10 and
100 μm are 142.6×10−3 and 138.4×10−3 mm3 , respectively. These values are almost the same as thesimple approximation value 157.0 ×10−3 mm3 calculated with Eq. 1. The probability of existing kbubbles, whose diameter is 10 μm, in the measurement volume calculated with Eq. 2 is shown in
Figure 9 for the bubble number density N of 5, 10, and 20 mm−3 . For the bubble number density of 5
mm−3 , the probabilities for k = 0 and 1 are high. Thus, when the relatively low number density
condition, the photodetector output represent the signal from a single bubble.
3.3 Bubble Diameter Dependence of Photodetector Output
The relationship between the photodetector output and the bubble diameter is shown in Figure 9.
The plots are for four bubble position and the dashed line showing the slope corresponding to thesquare of the diameter is also plotted. Although the absolute value decreases with increasing the
distance between the origin of the optical system and the bubble position, the photodetector output
keeps monotonical increase with the bubble diameter and the square relation with the bubble
diameter. Therefore, the radial oscillation of the bubble can be monitored from the output signal.

4. Experimental Results and Discussion
Figures 10(a-i) – 10(c-i) show the typical waveforms of the photodetector output obtained withthe above-mentioned setup for the BLT driving power of 6.2, 13.6, and 21.1 W, respectively. The
waveforms are lowpass-filtered with cutoff frequency of 1MHz. The several peaks of the waveform
observed in Figure 10(a-i) corresponds to the passing of a bubble in the measurement volume. Figure

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Photodetector output (a. u.)

0.0
0

Square root ofphotodetector output (a. u.)

0.5

0.6
(a-ii)
0.5
0.4
0.3
0.2
0.1
0.0
1209 1210

0.4

(a-i)

(b-i)

(c-i)

0.3
0.2
0.1
500

1000

1500

0

500
1000
1500
Time × Ultrasound frequency

0(b-ii)

1211

500

1000

1500

(c-ii)

1212 1213 1252 1253 1254 1255 1256 732
Time × Ultrasound frequency

733

734

735

736

Figure 10 – Experimentally obtained photodetector output. (a) – (c) correspond to BLT driving power of
6.2, 13.6, and 21.1 W, respectively. (ii) is magnified plot of square root of (i).
10(a-ii) shows the enlarged view of the peak around 1200th ultrasound period shown in Figure10(a-i). The vertical axis shows the square root of the photodetector output to interpret the output tothe bubble diameter because of the square relation between the output and the diameter. The bubble
diameter vibrates in almost the ultrasound period. The rapid contraction subsequent to the gradualexpansion and the bounce after the rapid contraction, which are often observed in the single bubble
oscillation (6, 7), are observed. This result may indicate that the waveform is derived from theoscillation of a single bubble in the measurement volume.
The number of peaks and the percentage of the time when the photodetector output has arelatively high value around 0.05 are increasing with increasing the BLT driving power. This isbecause that the number density of the bubble increases with the BLT driving power and theprobability of existing bubbles in the measurement volume increases. In the same manner, theincrease of the region with relatively high photodetector output without a clear peak is caused by theincreasing of the number of bubbles around the measurement volume, where has relatively low lightintensity of the incident beam. Thus, the waveforms without the clear peak may be derived from thescattered light from the many bubbles not from a single bubble. The clear peak is also derived fromthe sum of the scattered light from the measurement volume and peripheral region, but the
waveforms in the clear peak may be dominated by a few bubbles in the measurement volume because
the intensity of the incident laser beam in the measurement volume is much higher than that of theperipheral region. Supposing that the output waveform shown in Figures 10(b-ii) and 10(c-ii) is
derived from a single bubble, the source of the scattered light may be a relatively large bubble
because the rebound has not occurred and these have rectified sinusoidal-like waveform, which is the
typical waveform of the large bubble in single bubble oscillation. In the case shown in Figure10(b-ii), the maximum value of the waveform varied period by period. It may indicate thesub-harmonic oscillation of the bubbles.

5. CONCLUSIONS
A measurement method of oscillation of a bubble in acoustic cavitation employing the laser
scattering and the lens system is proposed. The concept and the principle are described and thenumerical simulation shows that the method can form a tiny measurement volume and the scattered
light from a single bubble, whose intensity is proportional to the bubble diameter, can be obtainedwith the method. A verification experiment measuring the acoustic cavitation formed under the hornof the BLT shows that the measurement system can obtain the single bubble oscillation-likewaveform and the waveform of the bubble varies with the BLT driving power. Thus, we conclude4819

that the proposed method can measure the oscillation of a single bubble in acoustic cavitation. We
will try to the further development of the output signal processing technique and the optimization of
the optical system for the quantitative investigation of the bubble oscillation.

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant Number 19K15000.

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