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0F VAREGUS STANEXNG WAVE RATIGSI

Thesis for the Em a! M. S.
MECHIGAN STATE UNWERSWY
Eifly D. Cook
1959

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DIFFRACTION OF LIGHT BY ULTRASONIC WAVES OF
VARIOUS STANDING WAVE RATIOS

Billy D. Cook

A Thesis
Submitted to the College of Science and Arts Michigan
State University of Agriculture and Applied
Science in partial fulfillment of

the requirement for the
degree of

MASTER OF SCIENCE
Department of Physics

1959

 

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ACKNOWLEDGEE‘IT

The author expresses his gratitude to Dr. E. A.
Hiedemann for his guidance in this study. The
author also expresses his thanks to Dr. W. G. Mayer
and Dr. K. L. Zankel for their suggestions and help.
The financial assistance from a U. 8. Army Ordnance
Contract and a fellowship grant of the National

Science Foundation is gratefully achnowledged.

DIFFRACTION OF LIGHT BY ULTRASONIC WAVES OF

VARIOUS STANDING WAVE RATIOS

by

Billy D. Cook

An Abstract
Submitted to the College of Science and Arts Michigan
State University of Agriculture and Applied
Science in partial fulfillment of

the requirement for the
degree of

MASTER OF SCIENCE
Department of Physics

1959

ABSTRACT

The theory for the diffraction of light by plane
ultrasonic waves of various standing wave ratio is derived.
The medium disturbed by the ultrasound is considered to
act as an optical phase grating. By evaluating the dif-
fraction integral for the light amplitude, expressions
for the time dependent and time average light intensities
are found for the diffraction spectrum. These expressions
reduce to those known for progressive and stationary waves.

Measurement of time dependent and time average light

intensities indicate that the theory is valid.

INTRODUCTION .
THEORY . .

TABLE OF CONTENTS

General Procedure.

Diffraction of Light by Standing Waves
Doppler Shift of the Diffracted Light

Time Dependence of Light Intensity

for Stationary Waves.

EXPERIMENTAL STUDY .

Apparatus and Procedure
Time Dependent Intensity Measurements

Average Light Intensity Measurements

Sm 1&de o o
BIBLIOGRAPHY .

ll

12

l4
14
16
19

23

24

-111

 

Fig.
Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

6.

FIGURES

Special Tank to Absorb Ultrasound . .

Schematic Diagram of Experimental
Apparatus . . . . . . . . .

Time Dependent Light Intensity
Observations . . . . . . . .

Theoretical Curves for Time Dependent
Light Intensities . . . . . .

EXperimental Results for Average Light
Intensity of the Zero Order for the
SJVR equal one o o o o o o o 0

Experimental Results for Average Light
Intensity of the Zero Order for the
SYJR equal tVVOo o o o o o o 0

Experimental Results for Average Light
Intensity of the Zero Order for the
SWR equal infinity . . . . . .

15

15

17

18

21

INTRODUCTION

Diffraction of light by ultrasonic waves has been
studied for many years. These studies have considered
two extreme cases: the progressive wave and the
stationary wave. The object of this investigation is
the study of diffraction of light by ultrasonic waves
of various standing wave ratios.

Standing waves may be considered as a superposi-
tion of two periodic wave trains of the same frequency
and travelling in opposite directionsl. If the wave
trains are continuous sinusoidal progressive waves, a
fixed distribution of nodes or partial nodes and anti-
nodes occur in Space. Stationary waves are standing
waves in which the energy flux is zero at all pointsz.
Since in the field of ultrasonics the usual method of
obtaining standing waves is by reflection, it is con-
venient to distinguish the different wave trains in
terms of an incident wave and a reflected wave. Define
the wave travelling in the direction of the net energy
flow as the incident wave and the other as the reflected
wave. The relative amplitudes of these waves are

given by the standing wave ratio which is defined as

the ratio of the incident wave amplitude to that of the

l\‘)

reflected wave. In the limiting cases, i.e., the
stationary wave and the progressive wave, the SWR*
is equal to one and infinity, respectively.

The medium disturbed by an ultrasonic wave was
considered by Raman and Nath3’4’5to act like an optical
phase grating. For both progressive and stationary
plane waves, they predict for normal incidence of the

light to the ultrasonic beam, that the light emerges

at angles 9,, given by

Sm 9h: hA/A' 11:01.1 :2

I j I

(l)

where /\ is the wavelength of the light and A"F the
ultrasonic wavelength. In a later papers, they predict
a Doppler shift in the various orders of diffracted
light. For progressive waves the incident light of
frequency P” emerges in the nth order at the frequency
of V+ h V’ ‘ where 1’“ is the frequency of the ultra-

sonic waves. The intensity of the light in the nth

order is given by

2
1:n =';Th (V, (2)

 

* O 9 O .
Hereinafter, the expreSSion standing wave ratio

will be abbreviated to SWR.

The parameter V’ is given by

ZIrLM
A

 

V:
(3}
where l* is the amplitude of the periodically changing
index of refraction, L_ is the path length of the light
in the ultrasonic beam, and er” is the nth order Bessel
function.
For stationary waves, the Raman and Nath theory

predicts that for the nth order, the emerging radiation

consists of frequencies

1r+(Zr-+n)v* r=o t1. t2,--~
(4)

The intensity14".of each sub-component of frequency

74 (av +h)v' is

.’ 2 a
.2-2 (5)
for each of the progressive waves composing the sta-

tionary wave. The total intensity in the nth order is

given by

° 2 (6)
In = Z JPZ (V) Jr-H(V)

The theory of Raman and Nath considers the medium
to act only as a phase grating and assumes that there
is no amplitude modulation of the light by the ultra-
sonic waves. More general theoretical considerations

7

by others have shown that the Raman and Nath theory

is valid for

nLAV/M.X' <1
(7)

whereichis the index of refraction of the undisturbed
medium.
Experimental confirmation in liquids of the Raman

8 and F. Sandersg.

and Nath theory has been made by R. Bar
An asymmetry of the light intensity about the zero order
in the diffraction pattern was detected for both pro-
gressive and stationary waves. Zankel and Hiedemannl
have explained this asymmetry for progressive waves to
be caused by finite amplitude distortion.

ll have shown that

Pande, Pancholy, and Parthasarathy
one may obtain optically a phase grating corresponding
to that of a stationary wave without having an actual
ultrasonic stationary wave. Using two piezoelectric

tranducers driven from the same high frequency oscillator,

they produced identical ultrasonic progressive waves

travelling in opposite directions in two different
vessels, the sides of which were close together and
containing the same liquid. For normal incidence of
the light to both ultrasonic beams the diffraction
pattern was the same as one caused by a stationary wave.
A variation of the method described above may be
used to obtain, in an optical sense, standing waves of
various standing wave ratios. With separate control of
the sound output of the transducers, one can produce a

full range of SWR from one to infinity.

THEORY

General Procedure

 

In considering the ultrasonic diffraction of light
caused by an optical phase grating, an integral method
may be used to determine the diffraction phenomena.

For collimated monochromatic light at normal incidence
to a transmission phase grating, the amplitude distri-
bution of the diffracted light is given by

. D {[1}?! sine + V(X,tfl
A(9)=Celwt 6 dx

~O

where 20 is the width of the light beam. A is the

wavelength of light, to is the angular frequency of

the light, and 9 is the angle at which the light is

diffracted. The term.\l(¥,t)is a dimensionless parameter

describing the optical phase grating in Space and time.
The value of A11)ifisin.general a complex quantity.

The time dependent light intensity I, is given by

I =AA’31A12
(9)

and the time average light intensity I is given by

- T
I = -,': / [A12 dt.
(10)
The parameter V(¥, t) is related to the instan-
taneous sound pressure p(x,t) through the periodic
change of the index of refraction 1409?). For a given

Sound pressure pfnfi.

v(x,t) = Eli-E note) = 2.1521 P(x,t)

(ll)

where X.is the piezo-optic constant for the medium.

Diffraction of Light by Standing_Waves

The instantaneous pressure of an ultrasonic

sinusoidal standing wave may be written as

p: P sin (w’t-k'x)+-£- $1}: (0)"! MR”)
(13)
where w’and K‘are the acoustic angular frequency and

wave constant, respectively, and a.is the standing wave

ratio (8'1". )0

The optical grating corresponding to the ultrasonic

standing wave is described by

V: V Sl'fllw’f—K’x)+{- 3n} (‘9'! +k'x).

(l3)
The diffraction integral, Equation (8), becomes
0
14(0): Cetwt/ ch41)!
-D
([v “31 (K'x-uth- .‘a’; slHk‘u-a'flj
X C d

X,

(14)

n Q
where LL= 2 /). and I: smO . Using the identity

. . o .
ecb sin-F : z J... (b) elr.f
’3', (15)

Equation (14) reduces to an integrable form

D
{at v " v [(nshft ulr-mx
A=Ce f2 2 J’Lfv) J.(Z)e X

-D 88-. 's-. d '

(l6)
Integrating, one obtains

A = zce‘”t[ f; :2) 1,0) 74%)

83-. D's-Q

x e£(r*3)‘°'t SI." [u14-(r-x) K']D . (1'7)
[u1+0'-S)K"]

To normalize, it is assumed that the light amplitude

at 9= 0 equals unity for V=O . Thus one finds
C=I/2D
(18)

Let n=r-$. and then Equation (17) becomes

a co 4'[ (2 r-niw'4w1t

A": Z LOO «7....."8 W»

n. -m V‘s-N (19)

where

sin [(u1+nk*)D]
[(u£+nk") o] '

 

w, =

(20)

For ideal diffraction,D-bco. W11 then has non—zero
values only for
LL.I " JG k’* := 0
(21)
which is the same condition given in Equation (1).
Thus the light is diffracted into discrete orders of

amplitude A" given by

" ' ( r-h)w'+w]t m
A»: 2 \TM") Jig) ed; (‘4)

D's-ow

10

since W" equals unity when the condition in Equation
(21) is satisfied.
The time dependent intensity in the nth order is
a. a
_ v v
In - 2 E LM Lb!) If?) J-(’i)
P‘s-b P‘s-C h P n

X a £2 (P-PJw't

(23)
and the average light intensity is
Q 2 V
_. 2 _
In - Z J; (V) Jr-" (a)
Pz-Q . (24)

Equation (24) reduces to the Raman and Nath results
for both progressive and stationary waves. For progres-

sive waves a—vm, and Equation (24) reduces to

v“ 2
In = J" (V)
(25)
as
Inn“) 2'» 0 V35"
:: .1. r- :n , (26)

For stationary waves a: 1 , and Equation (24) becomes

4° z
in = E “Tr-21V) JV—n(V)

rz—m (‘37)

11

Doppler Shift of the Diffracted Light

The motion of the optical phase grating produces
a Doppler shift in the diffracted light. Equation (25)
shows that for a standing wave the light in the nth order

contains components of the angular frequency

(28)
The average intensity of these non-coherent components

is

STEM Lid-‘5.) .

However, for a progressive wave the only component
having a non-zero intensity is the one whereren .

Thus the angular frequency of light in the nth order for

a progressive wave is

w+nw*
(30)
with the average intensity given in Equation (25).
The terms in the summation in Equation (24) are the
intensities of the non-coherent components. However,

the actual frequency shift is so small that it is not

easily observable.

12

Time Dependence of Light Intensity fog Stationary Waves
For stationary waves the optical grating may be

described by

V(X,f) = V cos co't sin k'x.
(31)

The diffraction integral then becomes

 

D
A - C Ciwt/ ecu-(x 4-4.. Vans (0'? 6"! 5"!
-D

dx'

(32)
Using Equation (15), integrating, and normalizing as

before, one obtains

A: E: Jn(Vcos w't)€‘.wt Wu

has-W

(53)

For [>400 , the light intensity in the nth order is

In -.—. (1’va m we),
(34}
Raman and Nath obtained Equation (34) which is equiva-
lent to Equation (23) when 431-. Equation (34) is
more suitable for physical interpretation and for time

dependent calculations than Equation (23). Comparing

15

the light intensity for the progressive wave, Equa-
tion (25) and the light intensity for a stationary
wave, Equation (34), one sees that they are similar.
The argument of the Bessel function is modulated in
time for the stationary wave. This modulation is ex-
pected since the stationary wave may be considered

sinusoidal in space with a time varying amplitude.

 

14

EXPERIMENTAL STUDY

Apparatus and Procedure

 

.An optical phase grating corresponding to that of
an ultrasonic standing wave is produced with two ad-
jacent progressive waves travelling in opposite direc-
tions. The optical axis intercepts both sound beams.
The two progressive waves are obtained by eliminating

reflections in a specially designed tank. This tank

“Tn-3.. pr:- Tr

is a modification of one described by Hargrove, Zankel,
and Hiedemannlz. The sound travelling in each direction
is absorbed in a Castor oil termination. Figure 1 shows
the tank with the castor oil terminations separated
from the water by a thin membrane. As the specific
impedance of castor oil and water are very nearly the
same, there are essentially no reflections at the
interfaces.

A schematic diagram of the experimental apparatus
is shown in Figure 2. The mercury light source illumi-
nates the source SlitSLq . The collimated beam pro-
duced by lens L,is normal to both sound beams. The
light intensity of the diffracted orders is measured by
a photomultiplier.

Two air backed quartz transducers are driven by a

500 watt crystal controlled radio frequency oscillator.

16

 

 

WINDOW ‘
(#43 SOUND-«-
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"
MEflBRANE

CASTOR OIL—-r'

J_J .q

 

 

 

 

115. 1. Spacial Tank to Absorb Ultrasound.

: ,uuusomc TRANSDUCER

WWW—TM
SI LI 9: L2 p"

V

 

 

 

 

1'13. 8. Bohantio Diagram of hpariaantal Apparatus.

We're.

16

The transducers are one inch square. The ultrasonic
output of each transducer is controlled with a variable
inductance used as a transformer to provide an impedance
.match between the transmitter and the transducer.

For time dependent measuiements the output signal
of the photomultiplier is amplified by two wide band
amplifiers in cascade and displayed on an oscilloscope.
The average light intensity is measured with a micro-
photometer photomultiplier.

The measurements are made at 1.0 me in water at

room temperature.

Time Dependent Intensity Measurements

From Equation (34) one sees that for stationary
waves, the light intensity of the diffracted orders varies
periodically at the frequency of the sound. Figure 3
shows the oscilloscope displays of the time dependent
intensity of the zero order and the first order. The
approximate value of \l as given in Equation (34) is 2.5
for the displays. Theoretical curves calculated for
Equation (54) are shown in Figure 4. Comparison of
Figures 3 and 4 shows the fair agreement of theory and

experiment.

17

 

Figure 3 - Time dependent light intensity of zero and firs:
orders of diffraction for stationary waves.

18

 

 

 

 

 

1r 31:72
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1'13. 4. Theoretical Curyas for Time Dspandant Light
Intensities or Zero and First Ordars.
Stationary lavas of Amplituda v33; .

19

Average Light Intensitqueasurements

The average light intensity of the zero dif-
fraction orders are shown for standing wave ratios
of one, two, and infinity in Figures 5, 6, and 7.
For progressive waves, SWR::OD , the theoretical
curves predict zero light intensity in the orders
for certain values of V . These zero values are not
obtained experimentally. This discrepancy may be
attributed to the Fresnel field intensity variation
of the sound beam and to finite amplitude effects.
Since the standing wave is a composition of two
progressive waves, the deviations from the theory
may be attributed to the same effects. Other sources
of error are the non-linear response of the micro—
photometer and the mechanical vibration of the
optical system. Small vibrations of the components
of the optical system move the images of the dif-
fraction orders about the entrance slit of the
photomultiplier. Within experimental accuracy the
intensity of the orders is symmetric about the zero

order.

20

 

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25

SUMMARY

EXpressions for the diffraction of light by plane
ultrasonic waves of various standing wave ratios are
derived using the method of Raman and Nath. This
method consists of evaluating a diffraction integral
for an optical phase grating. The expressions of
the light intensity of the diffraction spectrum re-
duce to those given by Raman and Nath for progressive
and stationary waves.

The light intensity in the diffracted orders is
found to be time dependent for standing waves with
finite standing wave ratio (SWR). In application to
diffraction type stroboscopes, the amount of light
modulation in the diffracted orders can now be deter-
mined for various SWR.

It is theoretically shown that the magnitudes of
non-coherent components resulting from Doppler shift
depend on the SWR. Further, for finite SWR, the
frequency shift is found to be independent of SWR.

Measurements of time dependent and average

light intensities indicate that the theoretical re-

sults are valid.

c _ .__.._-.-A ....

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8.
9.

10.

ll.

12.

24

BIBLIOGRAPHY

Lord Rayleigh, The Theory 2: Sound, Vol. II,
page 75, Dover, New York, 1945.

 

American Institute 9: Physics Handbook 5 - 57,
McGraw-Hill, New York, 1957.

 

 

C. V. Raman and N. S. Nath, Proc. Ind. Acad. _

C. Raman and N. Nath, Proc. Ind. Acad. Sci. A2, T a
4-6 - 412, (1955). f

C. Raman and N. Nath, Proc. Ind. Acad. Sci. A2,
415 - 420, (1955).

 

C. Raman and N. Nath, Proc. Ind. Acad. Sci. Ag
75 - 84, (1956).

G. Wannier and R. Extermann, Helv. Phys. Acta.
9; 520 - 532 (1956).

R. Bar, Helv. Phys. Acta., 3, 265 — 284 (1956).

F. Sanders, Canad. Journ. Res., 13, 158 - 171
(1956).

K. Zankel and E. Hiedemann, J. Acoust. Soc. Am.,
pg, 44 - 54, (1959).

A. Pande, M. Pancholy, and Parthasarathy, J. Sci.
Industr. Res. g, 2, (1944).

L. Hargrove, K. Zankel, and E. Hiedemann,
J. Acoust. Soc. Am. 51, 1566 - 1571, (1959).

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