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MICHIGAN STIATE UNIVERSITY LIBRARIES

3 1293 01020 5874

5 H): 0.55:.

This is to certify that the

dissertation entitled

THEORETICAL AND EXPERIMENTAL STUDIES OF
SECOND ORDER BRAGG DIFFRACTION OF AN
ACOUSTO-OPTIC INTERACTION

presented by

Bing—Chuan Hsu

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in EIectricaI Eng.

 

 

Date 8/ 29/ 94

 

 

LIBRARY
Michigan State
Unlverslty

 

 

 

PLACE ll RETURN BOXtomnwombcInckouImInyourncom.
TOAVOID FINESMunonorbdonddodm.

DATE DUE DATE DUE DATE DUE '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hsu IoAn Minna". WM Oppommhy IMMOI'I
W1

THEORETICAL AND EXPERIMENTAL STUDIES OF
SECOND ORDER BRAGG DIFFRACTION OF AN
ACOUSTO-OPTIC INTERACTION

By

Bing-Chuan Hsu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1994

ABSTRACT

THEORETICAL AND EXPERIMENTAL STUDIES OF
SECOND ORDER BRAGG DIFFRACTION OF AN
ACOUSTO-OPTIC INTERACTION

By

Bing-Chuan Hsu

Bragg diffraction of an acousto-optic interaction could be used to modulate light or
to obtain information about a number of characteristics of an acoustic wave. In this thesis,
the theoretical and experimental studies are focused on the second order Bragg diffraction.
A partial wave approach is used to investigate the acousto-optic interaction of the Bragg

diffraction.

The numerical simulations which provide a better understanding of the interaction of
Bragg diffraction were carried out for various acoustic frequencies, acoustic pressures, and
incident angle of the light beam. The criteria of the optimum system parameter sets which
provide a maximum second order Bragg diffraction were established. In addition, experi-
ments were designed and performed to verify the theory developed. These experiments

include the intensity distribution of the diffraction mode with various acoustic amplitudes,

acoustic frequencies, and incident angles of the light. The experimental results agreed

well with the theory developed.

Furthermore, acoustic Bragg imaging using first and second order modes were
obtained. The image resolution of the first and second order modes were compared, and
the images with various acoustic frequencies were analyzed. The image results indicate
that the second order Bragg image provides better image resolution than that of the first
order mode. It is also observed that the Bragg image resolution is directly proportional to
the acoustic frequency and acoustic beam width. The results of this investigation could be
applied to areas of nondestructive evaluation of materials as well as in noninvasive clinical

diagnostics.

ACKNOWLEDGMENTS

The author is sincerely grateful to his academic advisor Dr. Bong Ho who provided
much of the guidance, counsel and encouragement throughout the course of this research.
Gratitude must also be expressed to Dr. Timothy A. Grotjohn, Dr. Robert O. Barr and Dr.

Wayne W. Repko for their sincere interest and encouraging advice.

Furthermore, I would like to thank my parents, Mr. and Mrs. Hsing-Chang and

Chueh Kim Hsu, and my wife Chun-Mei Kuo for their encouragement and support.

TABLE OF CONTENTS

LIST OF FIGURES vii
1 INTRODUCTION 1
1.1 Introduction ......................................................................................................... 1
1.2 Outline of the Thesis ............................................................................................ 3

2 THEORY OF LIGHT DIF FRACTION BY SOUND WAVES 7

2.1 Introduction .......................................................................................................... 7
2.2 Review of Light Diffraction by Sound Wave ...................................................... 7
2.3 Theory of Light Diffraction by Sound Wave ....................................................... 9
2.3.1 Solution of the Optical Wave Equation ................................................... 12
2.3.2 Characterizations of Acousto—Optics System ......................................... 19
2.4 Diffraction in the Bragg’s Region ...................................................................... 22
2.4.1 First order Bragg Diffraction ................................................................... 23
2.4.2 The Second order Bragg Diffraction ....................................................... 28

3 NUMERICAL ANALYSIS OF THE ACOUSTO-OPTICS SYS-

TEM 35
3.1 Introduction ....................................................................................................... 35
3.2 Numerical Analysis of the First order Diffraction ............................................ 35
3.2.1 Diffraction in the Raman-Nath Region ................................................... 38
3.2.1 Diffraction in Bragg Region ................................................................... 40
3.3 Numerical Analysis of the Second Order Bragg Diffraction ............................ 45
3.3.1 Optimum set for the second order Bragg Diffraction ............................. 45
3.3.2 Second order mode Intensity Variation with Parameter a and V ............ 47
3.3.3 Second order mode Intensity Variation with Parameters Q and a: ........ 53
3.3.4 Deflection Resolution and Angular Resolution: ..................................... 57

4 CRITERIA FOR BRAGG DIFFRACTION EFFECTS IN

ACOUSTO-OPTICS SYSTEM 63
4.1 Introduction ....................................................................................................... 63
4.2 Basic Formulation of Bragg Diffraction ........................................................... 63

4.2.1 Criteria for Bragg Diffraction ................................................................ 66
4.3 Criteria For The Second Order Bragg Diffraction ............................................ 72

5 EXPERIMENTAL MEASUREMENTS FOR BRAGG DIFFRAC-

TION 78
5.1 Introduction ....................................................................................................... 78
5.2 Experimental Measurement System ................................................................. 79
5.2.1 Network for Acoustic Impedance Matching .......................................... 81

5.3 Experimental Results ........................................................................................ 83
5.3.1 Bragg diffraction with various V and Q value ........................................ 83
5.3.2 The Bragg diffraction with various incident angle ................................. 87

6 APPLICATION OF ACOUSTO-OPTIC INTERACTION 92
6.1 Introduction ....................................................................................................... 92
6.2 Acoustic Imaging base on the Bragg Diffraction ............................................. 93
6.2.1 Experiment Facility and Measurement System ...................................... 95
6.2.2 Experimental Result of Bragg Diffraction Imaging ............................... 98

7 CONCLUSION AND SUGGESTION FOR FUTURE STUDY 105

7.1 Conclusion ...................................................................................................... 105
7.2 Suggestion for Future Application .................................................................. 106
APPENDIX 108
A.1 The Solution for The First Bragg Diffraction ............................................... 108
A2 The Solution for The Second Bragg Diffraction ........................................... 111
A3 The Derivation of dlzl dQ ............................................................................ 117
BIBLIOGRAPHY 119

vi

LIST OF FIGURES

Figure 2.1 Diagram of the acoustic diffraction grating. ................................................... 10
Figure 2.2 Orientation of the partial wave vector. ........................................................... 14
Figure 2.3 Diagram for diffraction path. .......................................................................... 23
Figure 3.1 First order light intensities versus Q and V variations with (1:05. ................. 37
Figure 3.2 First order light intensities versus Q and 0t variations with V=3. ................... 41
Figure 3.3 First order light intensities versus Q with V=3 and 0:05. ............................. 43

Figure 3.4 Zeroth and first order light intensities versus incidence angle with Q

as a parameter. ................................................................................................. 44
Figure 3.5 Diffracted light intensities versus Q and V variations with 0t=1 ..................... 46
Figure 3.6 Diffracted light intensities versus V and 0t variations with Q=13. .................. 48
Figure 3.7 Diffracted light intensities versus V and 0t variations with Q=13. .................. 50
Figure 3.8 Light intensities versus V with a=1. ............................................................... 52
Figure 3.9 Second order light intensities versus Q and or variation. ................................ 54
Figure 3.10 Light intensities versus frequency f (MHz) with V=7.5 and (1:1. ................ 56
Figure 3.11 Deflected light by varying acoustic frequency. ............................................. 59
Figure 4.1 Bragg diffraction developed from path length consideration. ........................ 65
Figure 4.2 Schematic drawing of Bragg diffraction. ........................................................ 67
Figure 4.3 Second order light intensities versus f, N and a. ............................................ 70
Figure 4.4 Light intensities versus a with f = 17 MHz. .................................................... 71

vii

Figure 5.1 (a) Block diagram of the measurement system. (b) A plexiglass water
tank for observing acousto-optical interaction. ............................................... 80

Figure 5.2 (a) Electrical equivalent circuit of the acoustic transducer.
(b) T matching network. ................................................................................... 82

Figure 5.3 (a) First and Second order diffraction intensities versus V at double
Bragg angle with Q = 10.5. (b) First and Second order diffraction
intensities versus V at double Bragg angle with Q = 12.5. .............................. 85

Figure 5.4 (a) The First order diffraction versus Q with V = 3 and a = 0.5.
(b) The second order diffraction versus Q with V = 6.5 and 0t = 1.
(c) At each Q the acoustic amplitude was optimized to achieve maximum
power in the second order with a=1. ............................................................... 86

Figure 5.5 (a) The First order diffraction versus or with Q = 12.5.
(b) The second order diffraction versus on with Q = 12.5.
(c) The comparison of the angular sensitivity to the diffractions

with various (1. .................................................................................................. 89

Figure 5.6 The first order light intensities versus 0: with Q = 5.6, 12.5 and 19. .............. 90
Figure 6.1 Acoustic rays from source S interact with optical rays from source 0 to

form new light rays, which seem to come from 0’, a virtual image of S. ...... 94

Figure 6.2 Experimental setup for Bragg diffraction imaging. ........................................ 96

Figure 6.3 Experimental samples for imaging objects. .................................................... 99

Figure 6.4 The first and second Bragg diffraction imaging for four round washers
with acoustic frequency f = 17.5 MHz and beam size L = 2.54 cm.
(a) ~ ((1) show the imaging result by using the first Bragg diffraction.
(e) ~ (h) show the imaging result by using the second Bragg diffraction. ....100

Figure 6.5 The first Bragg imaging with MS U shape in different acoustic frequencies.
(a) Bragg imaging by using f = 17.5 MHz and L = 2.54 cm.
(b) Bragg imaging by using f = 20 MHz and L = 3.175 cm. .......................... 102

Figure 6.6 The second Bragg imaging with STAR shape in different acoustic frequencies.

(a) Bragg imaging by using f = 17.5 MHz and L = 2.54 cm.
(b) Bragg imaging by usingf= 20 MHz and L = 3.175 cm. .......................... 103

viii

Chapter 1

Introduction

1.1 Introduction

The interaction of light and acoustic waves gives rise to light diffraction patterns
whose intensity distribution in the various orders can be used to obtain information about
a number of characteristics of the acoustic wave. The interaction displays different char-
acteristics depending on the width of the interaction region, the light and acoustic wave-
lengths, and the amplitude of the acoustic wave. Roughly speaking, if the width and
amplitude of the acoustic beam are sufficiently small, and the operating frequency is low,
the diffraction phenomenon was explained by Raman and Nath and frequently referred to
as Raman-Nath diffraction. Conversely, Bragg diffraction is observed if the width, fre-

quency, and amplitude of the acoustic beam are sufficiently large.

In this thesis, effort has been focused in investigating the Bragg diffraction. The
analysis of Bragg diffraction closely parallels the earlier treatment by Klein and Cook but
is extended to the second order Bragg diffraction which includes a greater number of
experimental conditions. Moreover, the procedure described in this thesis leads directly to
equations which can be solved numerically. The interaction of light and acoustic waves is
analyzed for various experimental conditions. The optical wave equation is solved by

resolving the light traveling through the acoustic beam into a system of plane waves

whose amplitudes are described by a set of difference-differential equations. These equa-
tions provide a large extent of physical insight into the nature of light diffraction by acous-

tic waves even though analytic solutions are obtained for several special cases.

Numerical results are obtained by converting the set of difference-differential equa-
tions into a set of difference equations which can be solved by computer simulation. For
analyzing the acousto-optic system in various experimental conditions, it will be shown
that the solution of these difference-differential equations can depend only on a small
number of dimensionless combinations of experimental parameters. Since the actual
choice of these parameters is somewhat arbitrary, the equations are written in terms of
parameters corresponding most directly with typical experimental variables. It will be
shown that the intensities of the second order Bragg diffraction can be greatly enhanced if
certain system parameter sets are chosen, and that the second order Bragg diffraction gives

a finer angular resolution than the first order interaction.

The criteria of the Bragg diffraction and optimum second order Bragg diffraction
are also investigated. The results show that even if the criteria for the Bragg diffraction
are met, the mode intensities in the second order are Still comparatively weak. Therefore,
the requirement for optimum second order Bragg diffraction becomes a critical issue. In
addition, the criteria indicate that the second order Bragg diffraction requires only half the
acoustic frequency of the first order to provide the same degree of sharpness of diffraction
(i.e. light intensity versus incident angle) with first order. This makes the second order

Bragg diffraction much more feasible in practical acousto—optic systems.

The acoustic waves can modulate the amplitude and phase of a coherent light beam.
Furthermore, it can deflect, focus, and even shift the frequency of the light beam. As a
result, the light beam can reconstruct an acoustic imaging or provide detailed information
on the thermal vibration in solids and liquids. Diffraction of light waves by acoustic

waves takes several different forms, depending on their wavelengths and the dimensions

of the interaction region. In the industry, acousto-optic modulation has been widely used
in laser ranging, optical signal-processing systems, optical communication, high-speed
optical deflectors, acoustic traveling-wave lens devices, Q switching, amplitude and fre-
quency modulation, and mode-locking. Acousto-optic modulators can in general be used
for similar applications to electro-optic modulators, though they are not so fast as electro~
optic modulators. On the other hand, because the electro-optic effect usually requires
voltages in the kilovolt range, the drive circuitry for modulators based on this effect is
much more expensive than for acousto-optic modulators, which operate at low voltages.
The advantage by using the second order Bragg diffraction in acousto-optic system is that
this system provides not only larger deflection angle and frequency shift in light beam, but
high efficiency and double resolution in beam deflector and imaging reconstruction. In
recent years, acoustical imaging has been widely applied in the areas of clinical diagnos-
tics and non-destruction evaluation of materials. One major drawback of this type of
imaging system is that it requires either a mechanical or electronic scanning mechanism.
Such a system involves complex design and hardware implementation. As a result, the
system becomes very bulky and costly. Acousto-optical imaging could add some attrac-
tive features to the conventional acoustic imaging techniques. In view of the Bragg dif-
fraction, we see that there is a possibility of providing a real-time imaging system without
using a mechanical or electronic array scanner. In addition, high power solid-state laser
sources have become available in recent years due to the advancement of laser technology.
By using such diodes as laser beam sources, the proposed method of acousto-optical inter-
action could lead to a way of providing the possibility of low-cost, portable and real-time

imaging system.

1.2 Outline of the Thesis

Chapter 2 provides theoretical background to the interaction of the light and acoustic
wave. The problem of light diffraction by an acoustic field will be treated by solving the
optical wave equation which resolves the light into a system of plane waves. The ampli-
tudes of these waves passing through the acoustic field are given by a set of difference-dif-
ferential equations. The analytic solutions of the difference—differential equation are
found for cases when the frequencies of the acoustic wave is either low or very high. The
reason is that for low frequencies the acoustic wave produces an optical phase grating
effect, and at high frequencies light diffraction occurs when the light is incident on the
acoustic waves at angles associated with Bragg diffraction. With a double incident Bragg
angle between the light and acoustic wave, a closed-form expression of the second order
Bragg diffraction is derived by applying the Laplace transform technique to the difference
equations. Several experimental parameters which lead to a deeper understanding of the
Bragg diffraction are defined, and those parameters correspond most directly to the practi-

cal experimental variables.

The numerical analysis in acousto-optic system is carried out in chapter 3. Before
the experimental setup are designed to verify the theory developed, numerical simulation
can readily be implemented to see the mode amplitude variations for a given set of system
parameters (Q, V, a). The advance analysis for the second order Bragg diffraction is ana-
lyzed in details. After going through the general analysis of acousto-optic interaction, it
appears that we can adjust the system parameters to obtain a significant strength of the
second order Bragg diffraction. Furthermore, it appears that the second order Bragg dif-

fraction provides a sharper resolution than the first order.

Chapter 4 studies the criteria for Bragg diffraction effects in the acousto-optic sys-

tem. In this chapter, two types of criteria are discussed. One criterion analyzes the mini-

mum requirement for satisfying the Bragg diffraction. Another criterion relates to the
conditions for maximum second order Bragg diffraction. The result indicates that Bragg
diffraction is to be associated not only with high acoustic frequency F (unit in MHz) but

also with wide widths L of the acoustic beam. More precisely, it depends strongly on the
values of “LF2 “. The sharpness of the Bragg diffraction will increase with increasing the

L172 values. However, in order to maximize the second order Bragg diffraction other sys-
tem parameters such as acoustic pressure, dielectric constant of the medium, temperature,
etc. have to be considered as well. Trade off among parameters have to be made for

optimum operation.

In chapter 5, the experimental system setup and the measurement procedures are
described. The experimental results for verifying the Bragg diffraction are presented. The
experimental verifications include the intensity distribution of the diffraction mode with
various acoustic amplitude, acoustic frequency and incident angle of the light beam. The
results show that when the incident angle equals to twice of the Bragg angle, the second
order diffraction can be driven to its peak intensity under a certain system parameter sets.
Comparing the angular sensitivity of the diffraction modes, the second order diffraction
provides a sharper angular resolution than first order mode. In general, the experimental

results agreed with the theory quantitatively.

An acoustic imaging application by using Bragg diffraction has been mentioned in
chapter 6. A brief description of Bragg imaging theory will be given here. The imaging
system setup and the measurement procedures will be described in detail. Imaging resolu-
tion using different acoustic frequencies and beam width will be examined. The experi-
mental result shows that the final Bragg image resolution is directly proportional to the
acoustic frequency. In addition, the imaging resolution for the first and second order are
compared, and the result indicates that the second Bragg diffraction imaging provides a

sharper image resolution than the first order. This could be very useful in medical diag-

nostic applications. Since biological materials have higher attenuation for higher acoustic
frequency, there will be a trade off between depth of penetration and image resolution.
However, from the results of this research we noticed that one can preserve image quantity

by using second order diffraction.

Chapter 2

THEORY OF LIGHT DIFFRACTION BY
SOUND WAVES

2.1 Introduction

In this chapter, a very brief historical survey of the field of light diffraction by sound
waves will be given. Then, the acousto—optical interaction will be analyzed by the partial
wave approach. The second order Bragg diffraction will be derived in detail and solved
for the special case that the incident angle of the light approaches two times of the Bragg
angle. Several experimental parameters which lead to a deeper understanding of the
Bragg diffraction are defined, and those parameters correspond most directly to the practi-

cal experimental variables.

2.2 Review of Light Diffraction by Sound Wave

Diffraction of light by high-frequency sound waves, often called Brillouin Scatter—
ing,[” was first observed by Debye and Searsm in the United States and by Lucas and

Biquardm in France. Brillouin’s original theorym predicted a phenomenon closely analo-
gous to X-ray diffraction in crystals: plane waves of light striking the acoustically induced

planes of compression and rarefaction at a certain critical angle would be partially

7

8

reflected. There would be only one critical angle(i.e. Bragg angle), since the spatial struc-
ture of acoustically induced density variations is essentially sinusoidal and hence contains
no space harmonic components. Brillouin predicted that the reflected light would be Dop—
pler shifted by an amount equal to the sound frequency because the reflecting layers them-

selves move with sound velocity.

A fundamental theory for light distribution in the diffraction orders was first pro-

posed by Raman and Nathm’m. They assumed that the sound column would act essen—
tially as a two-dimensional phase grating which was not too thick in the direction of light
propagation. Their theory correctly predicted the Bessel function dependence of the inten-

sity of the various orders as a function of sound pressure when the light beam is perpen-
dicular to the sound column. For the case of oblique incidence, it was treated later”). The

frequency change in the diffracted light was also investigated“. Next”), they departed
from the simple phase grating model and formulated a wave equation of the incident light
in the acoustically perturbed medium. Then they went on to obtain an infinite set of cou-
pled differential equations which described the spatial behavior of the various modes tra—

versing the sound beam. Finally, they extended their theory to arbitral angles of incidence

of the rigid“.

Although the fundamental theory of the acousto-optics diffraction was derived by

Raman and Nath, the results of that theory are not in a form which can provide a easy way

for the numerical evaluation of experimental situations. In 1969, Klien and CookllOm”
developed an unified approach for acousto-optical diffraction. In their papers, they
extended to include a greater number of experimental conditions in the infinite set of dif-

ferential equations, and those equations can be solved numerically. The results by using

those equations confirmed with those given by Parisean.“2" 13] However, their discussion

was restricted to only first order Bragg diffraction with different acoustic wave frequen-

[141417] who discussed the first-order

cies at various incidence angles. There were others
Bragg diffraction from alternate approaches and applications but the second order Bragg

diffraction has been generally ignored.

In 1981, Poon and Korpellls] give a brief discussion on the efficiency of second

[19] in the study of

order Bragg diffraction. A similar result has been given by Alfemess
holograms. These results described the interaction when the incident angle is precisely at
twice the Bragg angle. The physical setups for these two separate investigations are quite
different in nature. Consequently, there were some discrepancies from their analyses. In
this research, partial wave approach is used to analyze the acousto-optical interaction.

The condition of the interaction in terms of system parameters such as incident angle,

acoustic and laser frequencies, and the size of the interaction region are investigated.

2.3 Theory of Light Diffraction by Sound Wave

The interaction between the acoustic wave and the light can be treated by resolving
the diffracted light into a series of plane waves. Consider a monochromatic light beam

incident at an angle 9 upon an acoustic beam of width L as shows in Fig. 2.1.

The electric field intensity E is being polarized along the y-direction. For simplicity,
let us assume the medium is non-magnetic, non-conducting and source free. The scattered

field is described by the following set of Maxwell’s equations.

V-D = p (2-1)

VXE = 410% (M)

Vxn = .3. (2-3)
a:

V-F? = 0 (2-4)

10

I? Sound waves

 

 

 

 

 

 

 

 

 

 

X+ ‘
Light A.
..._>

e k n(x,t)

i

A

I
+—>

 

 

Figure 2.1 Diagram of the acoustic diffraction grating.

11

where D = eEy, and the permittivity 8 can be put in terms of the refraction index n as

_2
E—nEO.

For source free p = 0, Eq. (2- 1) becomes
V - (nzeoEy) = O
n2V . 15y + Vn2 - Ey = 0 (2-5)

where Vn2 = 2nVn = 2211-3-31; (n is only function ofx, see Fig. 2.1)

Since the electric field is uniform in the Y direction and Vn2 = 0 in Y direction, Eq. (2-5)

becomes V ~ E = 0. To obtain the scattered E field, we proceed to take curl of Eq. (2-2)

a

 

VxVxE = —lJ.0-a—;Vxn (2-6)
V (V - E) - V22 = —uO-§—t- r§a7n280£y] (2-7)
- 2323 DE 3 2 32 2
.. I I

Since the radian frequency of acoustic wave is small compared with the radian frequency
of the light wave, the last two terms in Eq. (2-8) can be ignored. Then, Eq. (2-8) becomes
2
a z?
V(V - E) — V22 = —p.0€0n2 —2-y (2-9)
3:
Because V ' E = 0, the optical wave equation which describes the propagation of

the electric field intensity can be written as

12

2
WE (x, z, r) = [g] §§E(x,z,t) (2—10)
I

l

Juoeo .

 

where C =

2.3.1 Solution of the Optical Wave Equation

The acoustic field is assumed to be composed of plane waves. All frequency compo-
nents of the sound field are assumed to be measurable. Further, the medium is assumed to
be nondispersive and nonabsorbable so that the frequency components retain a constant
phase relationship and remain constant in magnitude. It is also assumed that the medium
is not birefringent, i.e. the refractive index distribution is independent of the polarization
of the incident light beam. For acoustic waves with arbitrary periodic waveform, the

refractive index in the region of the sound field (0 < z < L) can be expressed as

n(x,t) = n0+ 2 njsin [non—km +51]
j=1

where (0* and k* are the radian frequency and wave number of the sound wave respec-
tively, nj is the amplitude of the jth Fourier component and 81- is its relative phase. The
coefficients nj are assumed to be directly proportional to the harmonic coefficients describ-
ing the density change and consequently may be considered to be directly proportional to

the various harmonic coefficients of the acoustic pressure. For the n2(x,t), we have

2 2 °° . . .
n (x, t) = no + 2n0[ 2 njsrn [j ((0*t — k*x) + 51.1] + higher order terms
1= 1

The quantity of the higher order terms can be neglected because compared to the

magnitude of the no terms the perturbation of n is small. Then it2 becomes

l3

2 2 °° . .
n (x,t) = n0+2n0 Z njsrn [j((!)*t—k*x) +81] (211)
i=1

In the region of the sound field the electric field intensity of the light beam can be

expressed as

jLwt—nOI-f)

E = A(x,z,t)e (2-12)

Since A(x,z,t) is a periodic function with the variable k*x, it can be expended into a com-

plex Fourier series as follows

A(x.z,:) = 2 ¢,,,<z)e""‘°°"""

m=—oo

Then, the electric field intensity becomes

i(m(o*t—Im - f)

¢m(2)e (2-13)

E(r, t) = em"

M

m

where 2,, .-. norm?

I = (xi+yy+22)
In the spatial coordinates, the phase term can be expanded as
If I” = nokxsine + nokzcosfl + mk*x . (2-14)

The vectorial relationship between the propagation vectors is shown in Fig. 2.2.

From Fig. 2.2, we have

nok sin 9 — mk*

 

nokcosO = tan (O—Bm) . (215)

Since both 6 and Bm are very small quantities Eq. (2-15) can be approximated as

l4

 

 

 

x

A Sound wave

Light
\fl l A
9
>
m k
nok mk*

 

 

 

Figure 2.2 Orientation of the partial wave vector.

 

15

mk“

9-...
nok

=9_p

m .
This implies that

mk*
B," = a; (2-16)

where [3m relate to the directions of the wave vectors. These directions are measured with

respect to the direction of the incident light beam. Notice that Eq. (2—16) resembles the

usual grating equation. As a matter of fact, one can consider the acoustic wave function

[20] where the grating spacing is replaced by the

constitutes a sinusoidal amplitude grating
wavelength of the acoustic wave. Eq. (2-13) is thus equivalent to assuming that within the
acoustic field the light distribution can be expanded into a series of partial waves labeled
by m which propagate in directions described by the usual grating, Eq. (2-16), and which

have amplitudes labeled by ¢m- Our problem is to determine the values of the cm for var-

ious values of the experimental parameters. Since the electric field intensity E is a func-

tion of x, z, and t, Eq. (2-10) can then be reduced to

82 82 282
_%+_§L=[g]_§ can
8): dz at
Substituting Eq. (2-13) into Eq. (2-17), we have
32 ' 0° 2 imam—I .r)
517E. = em” 2 —¢m (z)(n0ksin9+mk*) e m
m z _oo
325 1'0)! 0° i(mw*t—km-}) 32 i(mm*t—km.})
_2 = e 2 e —2¢m(z) +2 (—in0kcosO)e
82 m = _.,° 82 (2-18)

[(mw‘t—km-

§¢m(z)—(n0kcose) 26 })¢m(z) ]

Z

The LHS of Eq. (2-17) is then

l6

2 2 co
* .1," r a do
85+ age wmz ei(m(1)l ) _:m - (21H OkCOSG)-a—
3x 32 =_.,., 82 z (2-19)

2
—( ( nOk sinO + mid“) + (nokcosO) 2J¢m]
In order to simplify the expression in the following development, let us define the function

flx, z, t) as follows,

“marl—Em.”

E(x,z,t) = em” 2 (lime = elwtf(x, 2,1) .

(2-20)

where k =

QIS

The last two terms in Eq. (2—20) can be neglected, since the time variation off is of

the order of magnitude of (0"(variation in the refraction index), that is

21:23 2
'54 “ "‘ ’l

a 2
— €«lkfi

 

The RHS of Eq. (2-17) becomes

4» e ' (2-21)

Substituting the expressions of n2 and I”, ' I given by Eq. (2-11) and Eq. (2-14) respec-

tively into Eq. (2-21) and then comparing with Eq. (2-19), we obtain

 

 

 

2
3 ¢ 11(1) 2
—i-m— (2in0kcose)$m + [(noksine + mk*) + (nokcosO)2] (pm
52 (2-22)
2 2 . 2 °° 1'5- 45-
: —n0k (pm + znOk 2 ")(‘bm _je j—¢m+je 1)
j= 1
Since ¢m(z) is a slow varying function of 2, then
2
34>". a ¢,,,
nOk-a—E » _2
dz
We have then
M 2
ZnOkCOSO-é—zm + ill noksinO + mk*) + (nokcosO) 2:19,"
. 2 2 2 °° 1'5- . 45-
: (—r)n0k ¢m—n0k z nj(¢m_je ’-¢)m+je 1) .
j= 1
Rearranging, we have
30 °° kn. i8. —i5,
”1 I I J
E + 2 ZeosO[¢m-je —¢M+je ]
f = 1¢ (2-23)
I
m

. 2 2 2 2
= W[(noksrn9+mk*) + (nokcosO) —n0k]

l8

RHS of Eq.(23)
1'11

2
W [ (mk*) + 2n0mkk* sine + ,2?)sz sin29 + c0829) .- ngkz]

2
ik* 2 k .
2
ik* Lm
2Ln0kcosO

k .
[m + 2%]; Sine] ¢m
Conforming with the parameters used by Klein and Cook, let’s define the following
quantities,

k*2L
nokcose

 

Q:

(1 = -—n0-k— sin 0
k*

kn .L
__l_

i cosO

where L is the width of the sound beam.

Since the incident angle 9 is less than one degree, we can rewrite Q, V, and or in the

following simplified forms

 

*

Q = k L (2-24)
nok

or = —n -k—sin6 (2-25)

0k*
Vi = knjL . (2‘26)
In terms of these parameters, Eq. (2-23) becomes
BO," 1 W i5j 4'5}- iMQ
-a—Z- +§Z2 le:¢m__je —¢m+je :|= Y(m—20t) (pm . (2-27)

1 1

19

For a pure sinusoidal sound field, n (x, t) = no + nl (03*1‘ — k*x) . In other words,

under such excitation we have j = 1. In order to notation the subscript of V will be

dropped in the subsequent development. Finally we have

61¢

v imQ
33m+iz(¢m—l—¢m+l) = (m—Za)¢m (2‘28)

2L
Equation (2-28) is similar to the equation developed by Roman and Nathm, and by

Klein and Cook who developed a unified theory which is valid over a wide range of exper-

imental conditionsm]. The parameters in Eq. (2-28) such as the acoustical pressure, the
angle of incidence of the light on the sound field, and the sound wavelengths can easily be
varied in a given experimental system. From an analytical viewpoint, we would like to
use two dimensionless variables which are simply related to the two easily varied quanti-
ties. A third parameters is needed to complete the description of the interaction. For con-
venience, this third parameter should be independent of the acoustic pressure and angle of
incidence. The parameters Q, V, and or used here satisfy the above conditions while other
common choices of dimensionless parameters do not. It should be emphasized that exper-
imental systems characterized by the same set of values of Q, V, and or are equivalent

though k, k*, n], no, and L may differ. Choices of parameters which are combinations of
Q, V, and a have sometimes been taken for mathematical convenience. Also before exper-
imental setups are designed to verify the theory developed, numerical simulation can easy

be performed by using parameters (Q,V,0t) to see the mode amplitude variations.

2.3.2 Characterizations of Acousto-Optics System

Equation (22?) shows that the quantity Q is a measure of the differences in phase of
the various partial wave due to their different directions of propagation. When the phase

difference in these waves becomes large (say Q > 1), the diffracted light tends to remain in

20

the lower orders. In fact, for large Q the diffraction is not found to occur at normal inci-
dence. This phenomenon can be explained using elementary results of coupled mode the-
orym]. A coupled mode interpretation of difference-differential equations in the Eq. (2-
27) identifies the variable Vj as coupling coefficient between the light waves. The cou-
pling coefficient VJ- couples waves which differ in order number by j. The coefficient V1
couples adjacent waves, V2 couples alternate waves, etc. The amount of energy transfer
between the waves depends upon the coupling coefficient and the degree of synchroniza-
tion of the waves. In Eq. (2-27), the term mQ(m-20t) indicates the degree of synchronous,
i.e. the phase difference between the waves is nearly constant, energy transfer between the
various orders readily occurs. On the other hand, if the waves are highly nonsynchronous,

the energy transfer is extremely weak.

 

2
. . . . . k . k* L
F l f bl d , = — — = , E .
or ana ysrs o o rque rncr ence usrng or n0k* srnO and Q nokcosfi q
(2-28) can be rewritten as
my v 2
Em+fi(¢m-1_¢m+l) = i['£2ZQ_+mk*tan9]¢m (2-29)
For Q << 1 Eq. (2-29) becomes
do v .
a?“ + E. (¢m_ 1 — ¢m+ 1) = rmk* ran9¢m . (230)
Let ¢m = rymeimrmne , we have the difference-differential equation as follows
4W", V( -ik*ztan0 ik*ztan6)
32 +51 wm_1e —wm+1e = 0 (2-31)

When the light beam is at oblique incidence, the phase term, k*ztan9 , arises due to
the fact that the light progresses through the acoustical field it encounters a continuous

change of phase of the acoustical waves. If the light travels in such a manner that it enters

21

and leaves the sound field at points having a phase change of 2m (n being an integer).

Consequently, the diffraction effects are completely cancelled. Therefore, if the phase

term k*LtanO equal to 2nrt the diffraction effects will disappear at the angle of

212! nA

tanO = L .

With Q << 1 the solution of Eq.(29) at z = L is

.mk"‘Llane

_ 2v (.k*Ltan9) “—2
¢m—Jm[k*LtanO 5‘" 2 ]e ' (2‘32)

 

 

The intensity of the mth order can be written as

 

lm=Jm V Qa (2-33)
7

Equation (2-33) shows that when Q is very small the intensity distribution follows the

manner of a sinc function. The diffraction pattern is symmetric for all angles.

For Q >> 1 the diffraction effects are found to be symmetric for oblique incidence.

As a matter of fact, the diffraction occurs predominantly around the angles associated with

Bragg diffraction given byml‘”

. _ m3. _
srnO — Z—Ano . (2 34)

This phenomenon can be explained by the use of coupled mode theory. From Eq.(29)

sz

the phase term is found to be 217 + mk*tan9. The phase term will be vanished when

m2Q

TL— + mk*tanO = 0. Hence we have

22

 

tan9 = —mQ . (2-35)
2k*L

Equation (2-35) can be reduced to the form of Eq.(34). The energy transfer between
the coupled modes is most effective between orders which have the same phase term.
Therefore, most diffracted light will lie into the order which satisfies the condition given

by Eq.(35). From the manner in which the parameter at is defined Bragg diffraction occurs

when a = , and the light is diffracted predominantly into the nth order. The sharpness

E
2
of the light intensity distribution in the vicinity of the Bragg angles as a function of angle

depends on the Q value. This will be shown by numerical analysis in next chapter.

2.4 Diffraction in the Bragg’s Region

Although the Q is related to k“, L, no, and k, however, in a typical experimental
setup, the no and k are fixed. The width and the frequency of the sound beam are the major

factors to vary the Q value. Experiments for the low frequency and the shorter width of
sound beam are said to be in the Debye-Searsm or Raman-Nath14] region (Q << 1). The

solution for the Roman-Nath interaction has well been srudiedlznflnom”[161. In this dis-
sertation, we will concentrate on the solution of the second order Bragg Diffraction.
When the acoustic beam is wide enough and acoustic wavelength is short, the interaction
can be observed in the Bragg region. The diffracted light reaches a maximum when the
incident acoustic wave is at a particular angle. Figure 2.3 depicts the geometrical relation-
ship between the light and acoustic beams. Equation (236) states the constructive inter-

ference condition for the interaction.

I

ml = 21 —1 (2-36)

I!

23

 

 

   
 

n=1 n0 Diffracted beam

Light beam

 

 

 

 

Air Water

Acoustic beam

 

 

Figure 2.3 Diagram for diffraction path.

 

24

where An = k/n0 is the wavelength of the light beam inside the interaction region. The

l is the wavelength of the incident light. Under constructive interference, the acoustic

wavelength A, length of interaction L and optical path I, are related to the optical path 1

and incident angle 9 in the following way,

A lsinO
L = 21cosO (2-37)

1’ = LcosO = 2100829
Substituting these relations into Eq. (2-36), we have
min = 2Asin9 (2-38)

or
ml = 2An0 sin 9 (2-39)

Using Snell’s law, nsinOi = nosinO, we have sin 9i = nosinO for n =1 in air, and Eq.
(2-39) becomes
. _ m?»
srnOi —- 2 A (2-40)
Equation (240) describes the Bragg relation for constructive interference at some specific

incident angles.

2.4.1 First order Bragg Diffraction

For the first order mode m=1, Eq. (2-40) becomes

. __ 7L
srnOi - 2A (2-41)
Since 9i has a value of less then one degree, one can approximate sin 9i E 91.. The angle

becomes,

25

-2: ..
68.2/\ (242)

where GB is the well-known Bragg anglem. When the incident angle is at the Bragg

angle, the diffraction mode intensity is at its maximum.

In the case of m = 1, Eq. (2-28) becomes

“’0 V

a ”WW-1‘11) = 0 (2'43)
“’1 V -Q

E +i(¢0-¢2) =1i(1—2a)¢1 (2‘44)

These equations describe the possible coupling between various modes. In Eq. (2-
43) the zeroth mode is coupled to the i1 modes. The quantity 5% relates to the coupling

coefficient. Equation (2-44) describes the coupling between the first order mode and its

adjacent mode. In order for modes to be coupled, the imaginary term in Eq. (2-44) should
be vanished. The condition for the first order mode to be coupled is when or = 0.5.
Under this condition, Eq. (2-25) becomes

nosrnO = 27 = fl

Since sin 9i = no sinO , the incident angle before the light passes through the sound
beam will be sin 91' :- 9i = 21A . Therefore, for or = 0.5, the incident angle is equal to the

Bragg angle, and this result confirms the geometrical relation in Fig. 2.3.

26

To evaluate the intensity of the first order mode, let us consider the coupling between

(1)0 and $1 modes. Let’s take a simple case of or = 0.5, Eq. (2-43) and Eq. (2-44) reduce

to
61¢
_0__‘:¢ = 0
dz 2L1 (245)
(“1’1 V .Q
21? +55% = rim-200(1)]

From an analytical viewpoint, we ignore (1),] and (1)2 because in this case, ¢-1 and (p2
are very small compared with $0 and (1)]. Equation (2-45) can be solved by using Laplace

transformation. Putting this set of equations in matrix form, Eq. (2-45) becomes

90_ 0C1¢0

. — (2-46)
(11 —C1C2 ‘1’1

V
where C1 - 27:

C2 — 12—1:(1—2(X) .

[20]

For a Linear system, one can solve a set of difference-differential equation by

using the Laplace transformation technique. For a system [X] = [A] [X , the solution is

X (z) = eA’XO , where X0 is a initial condition, and the 8A2 can be presented as

6’” =3’1[<sI—A>-‘l

Applying this approach to Eq. (246) we have

90(2) ____ 311312 41,-,“ = ell¢inc (2-47)
q’19?) 621622 0 e2l¢inc

27
where (trim is the mode amplitude of incident light at z = 0, and

e =3’1P 3—C2 q
I] sz-sC2+C%J

 

— —C 1
e =g'1 1
21 32—5C2+C%J

 

 

 

h

Then from Eq. (2-47), ¢1(z) can be put in the following form (The detailed steps are

shown in Appendix A.1)

-Q
_ —2V r4—I:(1—2a)z . (0’ )

01(2) — 6 e srn 4L2 (2-48)
where o = (4V2+ [Q(1-2a)]2)1/2.
Finally, the first order mode intensity 11 at z = L becomes

4v2 . 2(0)
= . * = _ — , -4

I1 (3)] ‘1’] 02 srn 4 (2 9)

Similarly, one can obtain ¢0(z), and the zeroth mode intensity 10 at z = L is
4V2 . 2 o
[0 =1—3?81n(2):1—Il . (2-50)

The solutions given in Eq. (2-49) and Eq. (2-50) are consistent with those given by
Phariseanlmml. From these equations, we can see that the distributions of the first order
mode not only depend on the acoustical pressure and frequency of the acoustical wave but

also the incident angle.

For the case of or = 0.5 (the incident angle equal the Bragg angle) the first order

mode intensity is expected to have maximum intensity. The mode intensities are

28

1V)

COS '2'
2

Il sin (‘5)

The numerical solution of mode intensity variation will be given in the following

(2-51)

chapter. It is noticed that the intensities predicted by Eq. (2-49) and (2-50) are quite accu-
rate under the condition Q >> 1. In Eq. (25]) the V value was chosen to be (2n-1)rt, n
being integer number. Under this condition, all light shall lie in the first order favored by

Bragg diffraction.

2.4.2 The Second order Bragg Diffraction

For the second order mode m = 2, Eq. (2-38) becomes

(2-52)

>|>’

sin 9i =

Since 9‘. also has a value of less then one degree, we can assume sin 9i E Di. Com-

paring with Eq. (2-42), we have

923 = 298 =

>|>’

where 023 is equal to twice of the Bragg angle. When the incident angle of the light

approaches to the double Bragg angle, the light is in phase and interferes constructively.

As a result, the second order mode intensity reaches a maximum.

For m = 2, Eq. (2-37) becomes

“’2 v .2Q
23 +-2—,:(¢,-¢3) = 70—00% (2-53)

29

This indicates that $2 couples to (131 and 93- Consider the coupling between (1)0, (1)1 and (1)2
(ignore ¢_1 and (p3 for the same reason as in the first order case.) We have a set of differ-

ence differential equation given below,

 

'd¢0_l -0
IE 21.!-

d¢

< 1}: _ _.Q _ (2-54)
23 +2L(¢0 $2) "21.“ 2004)]

“’2 v 2Q

‘ 3'2- +2E¢1 =1T(1-01)¢2

This set of equations can be solved by using Laplace transformation. In matrix

form, the set of equations can be expressed as

 

 

 

 

 

 

- . P _ i' _

(1)0 0 C1 0 (1)0

‘11 = ‘C1 C2 C1 91 (2-55)
V

where C1 = 2L

Since this is a 3x3 matrix, and the coefficients in the matrix could be complex quan-
tities, finding the mode solutions becomes rather involved. A brief procedure is shown
below, the detail steps will be given in appendix A2. The coefficient matrix of the system

equation is

in C1 01
A = —C1 C2 C1 . (2-56)
0 —C1 C3

 

 

30

Following the similar procedure in solving the first order mode, we have

r- q

A err 312 813
Z _ .1 _

e “’3 [(sl-A)"‘] " 621622 323
_e3r 332 e3;

 

 

 

 

 

 

¢(z) = eAZMO) (2-57)
rr (0)1
O (pine
where (13(0)= ¢1(0) = 0
45(0) 0
and
F—C S—C CZC —C C2 _
(s 2)( 3)+1 l(3 3) 1
-1 1
[SI—A] =3 —C](s-C3) s(s—C3) sC1
_ C12 —sC1 s(s-C2+C%)J
where A = 53— (C2+C3)32+LC2C3+2C12)s—C3C%
The set of equations is then in the following form.
i ‘ ' ‘ i ‘
4’0”) 311812913 ¢inc err
(2-58)

4’1“) = 821822 823 0 ‘ e21 ¢inc °

92(2)‘ _e3r e32 e33_

 

 

 

 

 

 

Next, we can find em, em and e31 by the use of Laplace transform. The results are as

follows

sz-s(C2+C3) +C2C3+Cf
err = 3 2 2 2 (2’59)
5 —s (C2+C3) +s(C2C3+2C1)+C3C1

31

 

 

- —C (s—C)
e21 =34 3 2 1 3 2 2 (2'60)
3 —s (C2+C3) +s(C2C3+2C1)+C3C1
2
C
_ -1 1
e31 -3 3 2 (2-61)

2 2
s -s (C2 + C3) + s(C2C3 + 2Cl) + C3Cl
Since we are interested in second order Bragg diffraction for maximum excitation of

. . . 7t .
the second order mode, the rncrdent angle rs 923 = A , and thrs means 0: equals to one

[see Eq. (2-25)]. Therefore, with or = 1 the C’s become

v
Cl—Z
__-_Q_
C2‘(’)2L
C3=0

For solving the mode intensities with or = 1, one substitutes above C’s to Eq. (2-59) ~

Eq. (2-61), and the results are given as follows
(a). The intensity of the first order mode with or = 1.

Using or = l in Eq. (2-60), it yields

 

_ —C
e21 =3 1 2 1 (2-62)
3 —C2s+2C12

The detailed derivation of 82] will be shown in Appendix A.2, then we have

1'
e21 = (—2)ge sin(g) , atz=L (2-63)

where O’ = (8V2 +Q2)1/2

32

From Eq. (2-58) we have

(bl = e21 - (trim: , so the intensity of the first order mode is

2 . 2 o
11 = ¢1'¢1* = 4—srn (2)11.“ (2-64)

The value of the parameter 0 in Eq. (2-64) for the first order mode is different than
the one used in Eq. (2-49). The reason is that when a = 0.5, the light is in phase and inter-
feres constructively at the first order diffraction, thus (1)2 is small and can be ignored.
However, in Eq. (2-64), or = l, the light is in phase and interferes constructively at the sec-
ond order diffraction. As a result, (1)2 becomes large, and can no longer be neglected. In
the case of or = l, the first order mode is strongly depend upon the parameter V and much
more so than the case of or = 0.5. Later, more specify information will be provided in the

numerical analysis.

(b). The intensity of the second order mode with or = 1.

From Eq. (2-61), e31 becomes

 

2C2
1 -1 1
e := — ’ ‘ . (2'65)
3‘ 2 g [slsz— C23 + 2C1)]
From Appendix A.2 we can derive e31 sa follows
e - l 1 —e-i%(igsin(9) + cos(9)) at z - L (2-66)
31 - 2 o 4 4 _

where o = (8v2 +Q2)1/2

33

Then the amplitude of the second order mode (1)2 becomes

(1)2 = 631 ' ¢inc (2'67)
And the intensity of the second order 12 is
1 0 Q 2 Q . o . Q 2
=02 02* = 2'1 C084 — cosz) + (Esrnz — srn?) [inc (2-68)

It is interesting to note that for proper Q and o values, 12 can ideally be excited to

100% from Imc This indicates that if the incident angle of the light is twice of the Bragg

angle the line can transfer most of its energy to [2.

(c). The intensity of the zero order mode with or = 1.

For the (1)0 part whenOt= l,we have
2 2
5 —2C +C
ell = 1[ ]=31[']'e31= 1‘631 (2’69)

33 — C2s2 + 2C2s S

and
(2-70)

4’0 = ell¢inc

Therefore, the intensity of the second order mode at z = L becomes

greaserwse» examiner]...

I0 = ¢0'¢0* =

Finally, we have

1 0 Q 2 Q. o . Q 2
Z[(cos—+cos-) +(Esrnz+srnz [inc

’0 = 4 4
2V . o 2
11 = [B-srnz] 1W (2-72)

-1 . gr (9° a]
12—4[(cos4 cos4 + csrn4—srn4 [inc

34

 

where
l
2
0' = (8V2+Q2)
0t - (—n )LsinO -l
_ 0 k* _
2
Q = LL.
nokcosfl
knlL
V =
0039

The equations of (1)0, $1 and 02 derived above satisfy the solution of the difference

differential equation given by Eq. (2-54) with a equals to one, and these confirms that the
result derived in Eq. (2-72) are consistent. Also, the equation set, Eq. (2-72), is com
pletely general for a pure sinusoidal excitation case. An interesting property of the second
order mode results from pure sinusoidal grating is that its maximum value is periodic with
Q/4 and 6/4. With proper Q and o values, the diffraction efficiency of the second order
will ideally be 100%. In chapter 4 we will discuss the conditions for high efficient diffrac-

tion in second order mode diffraction. From Eq. (272), it is easy to show that

IO +l1 +125 inc‘ This implies that only a small diffraction appears in the high order

modes and that the total energy of the system is practically conserved.

In the case when a = l, the conditions for maximum intensity for the second order
mode are quite different from those for maximizing the first order mode intensity when
or = 0.5. For at = 0.5 when the Q values are large, the first order mode is heavily depen-
dent on the value of V. However, when at = 1 the second order mode intensity becomes
function of both Q and V. For maximizing the second order Bragg diffraction, we need to
adjust Q and V to certain values which match the criterion for maximum second order dif-
fraction. This means that only proper choices of acoustic frequency, amplitude, and width

of acoustic beam can provide a high efficiency second order Bragg diffraction.

Chapter 3

Numerical Analysis of the Acousto-optics System

3.1 Introduction

In the previous chapter, we derived the conditions under which the second order
Bragg diffraction occurs. Before the experimental setup is designed to verify the theory,
numerical simulation can be readily performed to see the mode amplitude variations for a
given set of system parameters (Q, V, or). Also, from the results of the numerical analysis,
it is helpful to verify the criterion for determining an optimal set of parameter values for
the acoustic optical interaction. In this chapter, the simulation for the first order and sec-

ond order Bragg diffraction will be carried out in detail.

3.2 Numerical Analysis of the First order Diffraction

Klein and Cooklm used numerical simulation to show the Roman and Nath condi-
tion and the first order Bragg diffraction. In the Bragg diffraction case, they only simu-
lated certain Q and V values. However, a through analysis of the interaction was not

given. For the sake of comparison, a detail mode variation under all parameter changes

35

36

will be very useful in understanding the interaction as well as comparing with the second

order mode variations.

First, we use the difference-differential equation set in Eq. (2-54) to perform the
numerical simulation. Since Eq. (2-54) involves (1)0, (p, and (b2, the numerical result for the
first order mode will be more accurate than the result obtained from the difference-differ-
ential equation set Eq. (2-45), which was used by Klein and Cook considering only the 90
and (1)] modes. Since the magnitudes of the higher order modes drop out rapidly, accurate

result can be obtained by using the first three order modes. As incident angle of the light
beam approaches the Bragg angle (say or = 0.5), the second order diffraction will be very
weak when a large Q value and a small V value is used. Then, the first order mode inten-

sity can either be obtained by Eq. (2-54) or Eq. (2-45). However, for small Q, the diffrac-

tions analyzed by Eq. (2-45) did not work precisely, and we need Eq. (2-54) to analyze the

diffractions in this region.

In the previous chapter, it has been shown that the first order mode can be synchro-

nous with zeroth order mode and gives maximum mode intensity if the incident angle of
light field is exactly equal to the Bragg angle 08 = 221' The differential equations given

in Eq. (2-54) are used to simulate the acousto-optic system. Figure 3.1(a) shows the distri-

bution of It with Q values from 1 to 40 and V values from 1 to 20 when the incident angle

of the light field is equal to the Bragg angle (or = 0.5). Figure3.l(b) shows the result

solved by Eq. (2-45) with or = 0.5. It can be readily seen that Fig. 3.1(a) gives more
detailed information in the mode distribution of the first order than Fig.3.1(b), especially
when Q < 15 and the value of V is large. This will be useful fact for experimental verifica-

tion and practical applications. The first order diffraction has a rapid change in the small

   

   
  

    
    
     

"""" W’lll‘ 00/” "271' (”I

1,,
III 111/“111

W i111.) 2
Ill/Ill” ”’0 will!" I" ’\\\\\’//l/I
[Ii/[Ml WI .5

l

Intensity

.0 .° .° .0
b or an d
l l l l

 

    

 

o
K I l
x

o
(b)

Figure 3.1 First orerd lightin veurs saQ nd V variations with 0t=0.5.
(aen)Th uerrm CEIII’CS UIICWIIOIIhC SICI'Id ring¢0,¢1and¢2.

(b) Con Siefd only 1110, 4’1-

38

Q region, and Q depends quadratically on the acoustic frequency. Consequently, one can
use acoustic frequency to vary the diffraction intensities in this region. For large Q and
small V, figures 3.1 (a) and (b) show the same character of diffracted light. Though we use
Eq. (2-54) to analyze the interaction, comparing Fig.3.1(a) with (b), we see that for maxi-

mum 11 = (2n — 1) rt (n being integer) can be applied for the case of a = 0.5 as well.

The result shown in Fig.3.1(a) is appeared to be correct, therefore, in future analysis we

will refer to those numerical data.

3.2.1 Diffraction in the Raman-Nath Region

In order to compare the characteristics of the Bragg region with the Raman-Nath
Region, a brief review of the properties of the Raman and N ath diffraction is given below.

Mth Q <<1, the intensity of the nth diffraction order is given by

 

- 22
I =flifln2 an
n n QB '
2

For normal incident of the light beam, the intensity can be reduced

“=Jfiw. om
Raman and Nath first obtained this result and indeed suggested the correct angular

dependence from geometric argumentsm. Occasionally, one finds in the literature the
designation of Raman and Nath diffraction as all diffraction at normal incidence. How-
ever, in the interests of historical accuracy and a consistent description of the problem,
Klein and Cook suggested the designation of Raman and Nath diffraction for all diffrac-

tion under condition Q << 1.

39

The acoustic fields described by the condition Q << 1 are equivalent to optical grat-
ings which produce only a modulation of the phase of light passing through them. The
light is modulated in phase but not in amplitude. This agrees with the early speculation of

Raman and Nath that a sound beam of narrow width and low frequencies could be consid-

ered as an optical phase gratingm. The parameter V, known as the Raman and Nath
parameter, describes completely the diffraction in this limiting case. It measures the
degree of phase modulation of the light produced by the acoustic wave. Examining the

intensity, the following properties in the Raman and Nath region are observed.

0 The diffraction pattern is symmetric at all angles of incidence since 1n = Ln. Many
diffraction orders can be observed. When the angle is normal incidence, the inten-

sities of the first few diffraction orders is a function of V.

o The angular dependence of the arguments of the Bessel functions is such that the

effect of oblique incidence is to alter the effective value of V by a factor of

(sinQTa)/Qz£. Maximum diffraction will occur at normal as well as on both

sides with respect to the normal. When Qor = 2m: (m being a nonzero integer), all

diffraction effects will disappear.

0 The light intensities in In and I,n will vanish together when the argument of Eq. (2-

33) is equal to any root of the nth order Bessel function. This can be made to occur
by changing either V or or. The intensity of a particular order as a function of the

incidence angle will be symmetrical with respect to the normal.

The above observations are restricted for Q << 1. When the magnitude of Q

approaches to unity the observed light intensity distribution begins to deviate from those

predicted by Eq. (2-33). The excess amount of light is found in the lower orders. Also,

the maxima and minima of the light intensities as functions of V are shifted, and the total

40

diffraction effect does not vanish at Qa = Zmrt. These effects become more pronounce for

larger Q value.

3.2.2 Diffraction in Bragg Region

For Q << 1 the diffraction is described by Eq. (2-33) while for Q >> 1 Eq. (2-51)

should be used. However, when the Q value is in the range from 1 to 10, there is no close

form solution for the difference-differential equation Eq. (2-54). Numerical methods have

to be used to analyze the interaction.

The mode intensity variation of the zeroth and first diffraction orders for normal

incidence was given by Klein and Cookm]. They mentioned that for Q < 2, the depen-
dence on acoustical pressure is similar in nature to Raman-Nath diffraction but differs in
essential detail. It is seen that more light appears in the lower diffraction order than is pre-
dicted by Raman and Nath. By summing the intensities of the zeroth and the first orders,
it is found that essentially no light appears in the second or higher orders with Q = 4. For
Q = 7, only a small fraction appears in the first order. When Q = 10, diffraction effects
essentially vanished at normal incidence for small acoustic pressures. However, in the
case of oblique incidence the diffractions have quite different characteristics than these
from normal incidence when Q is large. From previous development, we have established
that V = (2n-l)rt is the condition for providing a high efficiency diffraction in the first
order. In order to investigate the mode intensity variation as a function of the incident
angle of the light field and the frequency of the acoustic wave in the first order diffraction,

various 0t and Q values are used while keeping V = 3. The mode distribution of I, is

shown in Fig. 3.2. Figure 3.2 depicts the rapid change in the angular dependence with Q

for the first order diffraction. The first order mode is excited to its maximum when the

 

41

1(1). V=3. L=3.175cm

    
         

‘
\' é

  

§

\

 

§

 
     

 

 

         

 

   
  

I,” [I z 11

Ill ”if \i\\

[Ill/”11111111 111111111“ 11
llllllllljlll,111111111 \\\\\\\\\\\111lllllli11.t\-4

‘ ””llllllllfljl/\\\\\\\\\\\\\\\\\\\\\1\“.“‘“1“n

Variable O 0 0

Variable Alpha

Figure 3.2 First order light intensities versus Q and or variations with

V=3.

42

incident angle approaches the Bragg angle (or = 0.5) with large Q. This is referred to as
the First Bragg Diffraction. It is observed that for small Q, even though the incident angle

is the Bragg angle and V = (2n-1)rt, the magnitude of ’1 remains small.

Figure 3.3 shows the mode intensities versus various Q values with V = 3 and or =
0.5. When Q = l, a considerable amount of light is found to be appeared in the orders
other than the zeroth and first. Most of this light will be in the negative first order or pos-
itive second order. For Q < 3, the diffracted light is very similar in nature to Raman and
Nath diffraction but differs in detail. As Q is increased, less light appears in those orders.
It is found that essentially no light appears in the second or higher order while Q > 12, and
most light is found to appear in first order with only a small fraction in zeroth order. When
Q >> 12, all light lies in the first order, and this phenomenon was considered as Bragg dif-
fraction. It is seen that for Q = 6, about 92 percent of light lies in the first order, and when
Q = 18, it contains almost 99 percent. These results are seen to be in agreement with the

results given by Phariseau.

Mth V = 3, Fig. 3.4 shows the angular variation with Q for the zeroth and first
orders. It is interesting to note that when Q < 2, the dependence on the incident angle is
similar to those of the Raman and Nath diffraction. A slight difference is that the zeroth
order is symmetrical with respect to normal incidence, whereas the first order are symmet-
ric about their respective Bragg angle (or = i 1 / 2 ). This is not very apparent with small Q
values, however, at larger Q values it becomes noticeable since the range of or decreases.
Figure 3.4 also illustrates that when Q is increased further, Bragg diffraction predominates
over other effects. Then, in order to observe the Bragg diffraction phenomenon, high
acoustic frequencies are required. Comparing Figs. 3.4 (b), (c) and (d), the sharpness of

the intensity distribution is seen to increase with Q.

Intensity

 

43

V=3. Alpha=0.5. L.—.3.175cm

 

 

 

 

 

 

Variable 0

Figure 3.3 First order light intensities versus Q with V=3 and 0t=0.5.

 

 

O-t. V-3
I v v v v v v t 1
0.9 1
0.8 1
0. .
0. 4

Intonotty
O
In

 

 

 

 

 

 

 

1(1)
0.4 F J
0.3 t <
0.2 ’
°~1 ’ I(0) ______________ 1
v0 0.1 0.2 0.3 0.4 05 0.5 0.7 0.0 0.9 1 v0 0.1 02 0.3 0.4 0.5 0.6 0. 7 0.0 0.9 I
Am m

(a) (b)

 

 

 

 

 

 

 

 

 

0-10, V-J
1 V Y T fir Y T r r 1
\
\
0. ‘\ - -:
\ I t
\ I
0 \ ’I “
I
~‘ 1) ,’
0 \ I .
‘ I
‘ I
o. “ I q -
I
i ‘ , f
C o. -1 C _
\
0.4 ’ \ [I 4
\ I
\
0.3L \ ,' <
‘ 1(0) I
‘ I
02> \ I j
‘ I
‘ I
\
0.1 > \\ [I .
\ ‘ I I
A L 1 1 r - 1 1 r l 1 n 1
v0 01 0 2 0 3 0 4 0.5 0 6 0 7 0 0 0 9 1 V0 0.1 02 0.3 0.4 0.5 0.0 0.7 0.0 0.9 I

(C) ((0

Figure 3.4 Zeroth and first order light intensities versus incidence angle with
Q as a parameter.

45

3.3 Numerical Analysis of the Second Order Bragg Diffraction

In this section we will concentrate on the analysis of the second order Bragg diffrac-
tion. When the incident angle equals twice of the Bragg angle 023 = [5‘ (say or = 1), the

second order mode can be synchronous and give maximum intensity. With or = 1, Eq. (2-

72) describes the diffractions with pure sinusoidal grating, and this interaction can be
referred to as a double diffraction or the Second order Bragg Diffraction. The expression
given by Eq. (2-72) is rather complex, in order to obtain more insight of the interaction, a
numerical simulation has been performed. Since Eq. (2-72) only provides the special case

a = 1, if we want to observe the mode distribution with different incident angle, we still

need to simulate the whole system by using difference-differential equation Eq. (2—54).

3.3.1 Optimum set for the second order Bragg Diffraction

Mth or = l and L = 3.1756m, the numerical simulation is shown in Fig. 3.5. Figure

3.5 shows the distribution of the diffracted intensities with Q values (0 to 40) and V values

(0 to 15). As can be seen from Fig. 3.5, when the incident light make a double Bragg
angle with respect to the acoustic beam, most energies are coupled between the zeroth,

Fig. 35(3), and the second order, Fig. 3.5(c), and only a small fraction appears in the first

order diffractions, Fig. 3.5(b). In the first Bragg diffraction case (0t = 0.5 see Fig. 3.1(a))
when Q values are large, all energies are coupled to the zeroth and the first order diffrac-
tions and both maximum diffractions are only a function of V. However, in the second
Bragg diffraction (or = 1), all diffracted light are not only a function of V but also a func-
tion of Q. Figure 3.5(c) indicates that the maximum second order diffraction is periodic

with Q and V. When an optimum set of Q and V values is used, the intensity [2 reaches a

46

1(0). Alpha-1. Ld.t75cm

 

(a)

 

      
     

 
 
 

 

505‘ 9”’ .. -. -_ ..

20.4 ....... I I)" ’l/iI/ll/I

I ’ ’ ’A’ 2’” ‘7”; I” l/l/

I ’I/ ”110

oooooo,’, /’ ’ ,"” I’ll ’ I “\ . 3

.3» """ " ii”"”"5;i””7///{i\l\\
’ \§I'£/ A \\\\\{”/ 15

(b)

Variabbo ° 0 Va.“ v

 

 

(0)

“Mb” 0 0 Variable v

Figure 3.5 Diffracted light intensities versus Q and V variations with 0t=1

47

maximum, and almost 99% energy couples from line to 12. In the first order Bragg diffrac-

tion, there is a wide range of system parameters to be used to obtain the maximum first
order diffraction. In the second order Bragg diffraction, the maximum intensity can only
be achieved if a certain set of parameter is selected. In Fig. 3.5(c), there are three sets of
Q and V values for obtaining the maximum 12, they are: (Q=13, V=7.5), (Q=25, V=10),
and (Q=38, V=11.5). These parameter sets show that for small Q value (say Q = 13) the
minimum required V value to excite the maximum 12 is smaller than the V value for large
Q value (say Q = 38). The physical interpretation of this is as follows. For a given width
of an acoustic beam with low Q values, the required acoustic frequency as well as acoustic

power could be lower. This property is also same to the first order Bragg diffraction.

From the results of the above simulation, we learn how the second order Bragg dif-
fraction can be excited to its maximum intensity with certain or, V and Q parameters. Next

chapter, experimental setup will be described.

3.3.2 Second order mode Intensity Variation with Parameter a and V

To analyze the distribution of the second order Bragg diffraction with V and a as

parameters, we choose the optimum set (Q=13) for detail analysis. Figure 3.6 shows the
distribution of diffracted light intensities with Q = 13, while V and on are variables. For a
system setup with the incident light, medium and the width of acoustic beam are fixed, the
Q value can be directly related to the acoustic frequency. Furthermore, V can be related to

the acoustic pressure and a in term of the angle between the light and the acoustic beam.

For a given acoustic frequency such as to give Q = 13, Fig. 3.6 shows the distribu-
tion of the diffracted light with various intensities of the acoustic beam and angles of inci-

dence. It is observed that from Figs. 3.6 (a) and (b) when the incident angle approaches

48
Ito). 0.13, L-SJ75an

. Av Will/4”
' nWfigfif?’

 
  
  

  

 
 
 

’ ’
y.
" ".
’

 

.
‘~
‘o
‘~
‘.

 

 

      
 

    
   

 
 
 

 

   
       

 
  
    

  

 

       

(a)
-- -mllllwl/I   .
\W/ /M\ \‘N “411' (0;... ;
. SEW/0‘ ”‘3'“; 2
(b) mm. v o 0 WWW “m
1:... tall/{i’\1\ ’
(C)

v i V 0 Variable Alpha

Figure 3.6 Diffracted light intensities versus V and a variations with Q=13.

49

the Bragg angle (CL = 0.5), most of the energies are coupled from the zeroth to the first
order modes. With a = 0.5 since the intensity of the first order diffraction can be maxi-
mized by adjusting the acoustic beam intensity, under such excitation the second order dif-
fraction is too weak to be observed experimentally. However, when the incident angle
equals to twice of the Bragg angle (on = 1, see Fig. 3.6(c)) it is noted that the most energy
is transferred from the first order to the second order mode. In addition, the second order
mode intensity can be maximized by adjusting the acoustic beam intensity. This indicates
that the second order diffraction can be strongly excited with a pure sinusoidal grating
when the incidence angle of the light beam approaches to twice of the Bragg angle with an
optimum set of Q and V values. Under such excitation, the first order mode intensity 11
can only reach about 45% of 1,01 (see Fig. 3.5(b)). When the incident angle greater than
the double Bragg angle (say on > 1), only a small fraction of the incident light appears in
the first or higher order. Practically all diffraction effects vanish for a >> 1 under small

acoustic pressures.

With Q = 13, Fig. 3.7 displays more detail variation of the angular distribution when
the V value changes. Mth V = 1, some diffractions appear on the first order at a = 0.5, but
most of the energies are still contained in the zeroth order. When V increases to 3, the first
order reaches its maximum at a = 0.5, and a noticeable second order diffraction was

observed at a = 1. When V = 5 (see Fig. 3.7(c)), the first order dips down to 40% of 1,01.

A stronger second order mode intensity is visible when V increases from 3 to 5. It is rather
surprising that the first order mode reappears at a = 1. This explains why in an experi-
mental setup the first order mode can easily be observed since there exists a wide range of
system parameters to excite the first order mode. When V is further increased to 7.5, Fig.
3.7 (d), the diffracted light energy is almost completely transferred into the second order
mode at a = 1. It is important that a strong second order mode excitation can provide

attractive features in engineering applications. The second order diffraction provides not

O:13,V:1. l=3175cm

 

 

 

 

 

 

 

 

50

0:13, V-J. [13.175071

 

 

 

 

 

 

 

 

 

 

 

 

1» ‘ I -------------------- 1’ _ "_' “‘,"‘~--
[I I") I, \‘ ’I
’ I ' \ I
II “0) l: \ ' I “0)
0", \‘ ’ 4 OBI ' ‘
I
. I
‘ I
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0.5L 4 ,05’ r. -..
Z‘ - I
Te 3 t 3‘
9 S j ‘.
E 5 '1 I'.
0.4» i 0“ I 1
W)
02. ' ' 02 ’ i
_--" l?)
o h. 1 0h "
L 1 L 1 I 1 l 1
0 0‘5 1 1.5 25 0 05 l 15 2 2.5
vm m Vii”. mu
(3) V = 1 (b) v = 3
Q=13. V:5,l£17san 0:13. V375. P41175011
‘ I‘ I‘ I‘\ I\\ ‘
I \ \ \
,’ . I0) , t I
l ‘ ' \/
0M I 1 ' <
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g , ‘ I E
0.4- ,I‘ ’ ' .
02'
0
0 05 1 1.5 2 2.5 25
Variaiema
(C)V=5 (d)V=7.5

Figure 3.7 Diffracted light intensities versus V and 0t variations with Q=13.

5]

only a larger deflection angle and frequency shift in the light beam, but also gives a better
resolution in beam deflector and optical imaging. Another interesting result has also been
noted as shown in Fig. 3.7(d) that when Q = 13 and V = 7.5 the maximum diffraction
appeared at or = 0.5 is not the first but the zeroth order, and the maximum of the first order

is shifted to a = 0.75 instead.

In previous section, we mentioned that using the Optimum parameter sets, for small

Q values the minimum required V values to excite maximum 12 is smaller than the V val-
ues for large Q values. On the other hand, the large Q provides a wider range of V values
which drives 12 to greater than 70% of the maximum 12. Comparing Fig. 3.8 (a) and Fig.
3.9 (c), when Q equal to 38, the magnitude of 12 is in excess of 70% of the maximum 12
with various V values from 9.5 to 13.8. However, when Q equal to 13, the ranges of the V
which have 12 greater than 70% of the maximum [2 are only from 6.2 to 9. This indicates
that if high Q is used for optimum set, there exists a broader range of V values for exciting

the second order Bragg order.

Physically, the V value is proportional to the amplitude of the Fourier component of
the acoustic wave and is directly related to the intensity of the acoustic grating. Then, the
V value is a function of the acoustical pressure.[23"24] In acousto-optical application, the

transfer function of amplitude modulation can be related to the result in Fig. 3.8.

52

0.13. AlphI-‘l . L—3.1 75cm

 

V I V Y ‘I V r I I

0.8 ‘-

manslty

0.4 '-

0.2-

 

 

 

 

(a) o 2 4 6 a 10 12 14 16 1s 20
Variable V

0-25. AIN‘I-1 , L-3. 1 75cm

7 *7 V V 1' '71' FTf——— ""'Y“ ’ .7—

 

0.8_

0.6 '-

0.21-

 

 

 

 

0 2 4 6 8 10 12 14 16 18 20
VlrllblOV

0.33. Alpha-1, L-3.1 75cm

 

0.8 '-

0.8»

Intensity

0.4 '-

02'-

 

 

 

 

10 12 14 16 18 20
Variablov

o
to
b
an
0

Figure 3.8 Light intensities versus V with (1:1.

53

3.3.3 Second order mode Intensity Variation with Parameters Q and a:

For observing the distribution of the second order diffraction with Q and or as vari-
ables, we choose optimum sets (V=7.5, V=10 and V=11.5) for in-depth analysis. Figure
3.9 shows the distributions of the second order mode with Q and 0t as parameters while
using the three optimum V values. From the mode distributions, one can see that with
optimum V values, the second order mode is very sensitive to Q and on. Only a narrow
range of a values can efficiently drive the second order mode. With or =1, the maximum

second order diffractions are located in Q = 13 for V = 7.5, Q = 25, for V = 10 and Q =38

for V = 11.5. Figure 3.9 indicates that the mode distributions with varying Q values are
quite different for each optimum V value. For small V value, say V = 7.5 in Fig. 3.9(a),
the Q value required to achieve the maximum second mode is small (Q =13), and the
range of a which can efficiently drive the second order mode is large. Under such excita-
tion, when Q is increasing, the intensity of the second order mode is decreasingly and the
range of a which can efficiently drive the second order mode becomes narrower. How-
ever, for case V = 11.5 in Fig. 3.9(c), the Q value required to achieve the maximum second
order is change to large (Q =38), and the mode distribution with Q various is wholly dif-

ferent to the case V = 7.5.

From Fig. 3.1(a) and Fig. 3.2, we see that when the conditions V = (2n — 1)1t and
a = 0.5 are met, the first order Bragg diffraction is strongly excited at large Q values. In
addition, when Q is large, the first order mode intensity is saturated and does not change
too much with increasing Q. Quite different from the first Bragg diffraction, the maxi-
mum second Bragg diffraction shown in Fig. 3.9 is heavily dependent on the Q value, and
for different Optimum V value, the mode distributions with various Q become versatile.

For a fixed acoustic beam, light wavelength and medium no, the Q value is only function

of the acoustic frequency. Then, an acousto-optical frequency modulation can be applied

by using the Bragg diffraction, and the second order Bragg diffraction provides a versatile

54

l(2). V-7.5. L-3.1 750m

1"“

 

 

 

Variable O Vartabla Alpha

1(2). v-1 1 .s. L-3. 175cm

   
 
 
 

  
 

 

   

1‘
0.8~4
03‘ - "
3.....- f
0.2-~ “ \ \‘ ,
.1 . Itt‘éi’:
4° . . _\\\\\\\\\:~__ ...... a 2

(C)

. 0 0
Variable 0 Variable Alpha

Figure 3.9 Second order light intensities versus Q and or variation.

55

transfer function than the first order one. Since only a short range of frequency can pro-
vide the maximum diffraction in the second order mode, it is useful that using the second

Bragg diffraction to implement a frequency filter to acousto-optical signal processing.

Equation (2-25) indicates that the variable or is a measure of the incidence angle of

the light on the sound field (see Fig. 2.3). In the Bragg diffraction, the diffraction angle is

6 = 331% , where m is the diffraction order. The diffraction angle 9 is thus proportional to

the acoustic frequency. Therefore, for a higher Q value the diffraction angle 9 is large.
Using this result, we can apply the Bragg diffraction to an acousto-optic(AO) deflector,
and using acoustic frequency controls the deflected light beam. Since the second Bragg
diffraction has a double Bragg angle, the A0 deflector using the second Bragg diffraction
would have a larger deflected light effect than that using the first Bragg diffraction. In the
next section we will also show that using the second Bragg diffraction would have a dou-
ble resolution on the acousto-optic deflectors. In addition, the second Bragg diffraction

provides a double frequency shifting to the diffracted light. Since the frequency shifting is

extremely important for heterodyning applicationlzsl'lm, the frequency modulation by
using the second Bragg diffraction is a good study t0pic for future application. Another
useful application is that since the second order intensity is also a function of the Q value,
using this result we can apply to acousto-optic devices and optical signal processing which
prefer using acoustic frequency to control the amplitude of the diffraction not by using

acoustic pressure.

For the purpose of demonstrating how the diffraction intensity be affected by the

acoustic frequency, let’s consider the incident light be a Helium-Neon Laser beam 3.0 =

6.328 x 10'5 cm, and the acoustic system has an acoustic beam width of L = 3.175 cm.

Figure 3.10 shows the mode distribution as a function of the acoustic frequency with V =

7.5. As can be seen, when the acoustic frequency approaches 17 MHz, the second order

56

V=7.5, Alpha-1, L-3.175cm

 

09*

08*

0.7 ~

Intensity
.0
U!
I

0.3 '-

0.2"

0.1 -

 

 

 

 

Frequency

Figure 3.10 Light intensities versus frequency f (MHz) with V=7.5 and (1:1.

57

diffraction reaches a maximum intensity. As a result, one can control the mode intensities

by varying the acoustic frequency.

3.3.4 Deflection Resolution and Angular Resolution:

One of the advantages of using the second Bragg diffraction in acousto—optical sys-

tem is due to the fact that it provides a better deflection resolution. Light deflec-

tion[28]"[30] by ultrasound is based on the relationship between the acoustic frequency and
the Bragg angle. This relationship can be obtained by substituting the quotient of sound

velocity v and acoustic frequency f for the acoustic wavelength A into the Bragg diffrac-

tion condition. For the first Bragg diffraction, the condition is 23in98 = 7);. With

A = , the condition becomes

K1.,t<

<I>’

23in93 = f (3-3)

Typically, the Bragg angles are rather small, 23in 98 can be approximated by 29 B .

Hence, 20 B represents the angle by which the diffracted light departs from the path of the

incident light. As indicated in Eq. (3-3), the direction of the diffracted light ca be changed
by varying the acoustic frequency f.

A (293) = (if? . (3-4)

[31] and for memory or

Light deflection is potentially useful for television projection
switching devices. The major factor in a light deflection devices is the number of angular

positions that can be clearly distinguished from each other, usually called the number of

58
resolvable spots. A light beam, no matter how well collimated, eventually spreads
through diffraction: far away from a source with uniform phase and amplitude across an
aperture D, radiation of wavelength It spread out over an angle il/ D . Figure 3.11 shows
that to determine the number of resolvable spots, we divide the angular displacement

A (29 B) by the unavoidable diffraction spread B = X/ D of a light beam projection

min
from an aperture D. This spread determines how small a spot one can make. Now for the

first Bragg diffraction the number of resolvable spots can be defined as follows:

)
=__B_ =M=Af€ 2A“ (3-5)

where D is the width of light beam that the acoustic wave traverse at its velocity v. Hence,

the transit time T of the sound across the optical aperture is D/v. The number of resolv-

able spots equals this transit time multiplied by the total acoustic frequency shift Af.

For the second Bragg diffraction, the condition is

. It
ZSrnflzB = 2—

I.

To vary the direction of the diffracted light, an incremental change of acoustical fre-

quency Af is required,

“2923) = 2Af§ . (34)

Hence, the number of resolvable spots for the second Bragg diffraction can be writ-

ten as

 

29
= A[ 23) = 2M? = 2Aft (3-8)

59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T Acoustic wave with
Variable frequency

 

Figure 3.11 Deflected light by varying acoustic frequency.

 

60

This clearly shows that the second order Bragg diffraction provides twice as many
resolvable spots when comparing with the first order diffraction. Even though, the second
order gives a better deflected resolution, it is however that there exists a limitation on the
number of resolvable spots. Since the diffraction efficiency of the second order diffraction
is a function of the acoustic frequency and the Optimal mode intensity which can only

occur over a range Of the acoustic frequencies, as a result the number of the resolvable

spots is limited. For example, Fig. 3.10 shows that the range of acoustic frequencies

which drive [2 above 70% of ’2max are fmin = 13.8 MHz and fmax = 21 MH, the band limit

is then Af = 7.2 MHz. With one inch diameter light beam. the transit time is ‘t = 17.1us,

thus the maximum number of resolvable spots is limited to N2 = 246.

Angular resolution of the diffracted beam is another important feature of the interac-

[32' probed the column of sound by varying the entrance angle of

tion. Cohen and Gordon
the incident light, and recorded the intensity of the diffracted light as a function of angular
position. The result shows that the angular distribution of the diffracted light intensity is
very similar to the distribution of power density in far-field diffraction pattern of the sound
beam. Consequently, one can explore acoustic radiation patterns by the use of the Bragg

diffraction. Our analysis shows that the second Bragg diffraction provides a better angular

resolution than the first Bragg diffraction.

In the case of Q = 13, for example, Fig. 3.12(a) shows the mode distribution of I,
and 12 as a function of on. When comparing 1, with 12, both intensities are being optimized.
The result shows that the 3 dB bandwidth of 12 is much narrower than that of the 11. In

other words, the response of the second order mode is more sensitive to the angular varia-
tion of the incident light. When the interaction is used for imaging purpose, the second

order mode should give a superior angular resolution.

61

In Figures 3.12( b), it is also observed that the narrowness of the 12 response is pro-

portional to the Q value. This can be explained as follows. Since the Q quantity is propor-
tional to the frequency of the sound wave, when the frequency of the sound wave is raised,
the wavelength becomes shorter, which in term will give a shorter equivalent slit spacing.

Shorter slit spacing in light diffraction always give narrower interference pattern.

62

Q=13. L=3.17501T|

 

I \
0.8 ~

1(2). V=7.5

Intensity

 

 

 

 

0 2 0.4 o 6 0.8 1 1.2 1.4 1 6 1.8 2
Variable Abha

Figure 3.12(a) Light intensities versus or with Q=13.

 

1(2). L.3.175cm
1» .
I \
--:o-13.v.7.s I; '-.‘
I: 2‘
' n I -"
o.8~ ....o-25,v 10 ’ _, ‘ .
l -_ \
.' '. 1
- :0:38.V=11.5 ' ' ‘

Intensity

 

 

 

 

 

 

0 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Varidrlo Abba

Figure 3.12(b) Second order light intensities versus Q, V and or.

 

Chapter 4

Criteria for Bragg Diffraction Effects in
Acousto-Optic System

4.1 Introduction

Acoustic-light diffraction occurs when the acoustic waves diffract the light beam in

a manner satisfying the Bragg diffraction law. [311435 1 Such phenomenon has been utilized
in the study of crystal structures by using X-rays. Strong intensity in a given portion of the

diffraction is Obtained only when the glancing angles of incidence 1p and reflection 9 are

equal, as in mirror reflection, and when It, A, and 9 satisfy the Bragg law m?» = 2Asin9
0» and A are light and acoustic wave length, respectively, and m is the order of diffrac—
tion). In this chapter two types criteria for acoustO-optic system will be discussed. One
criterion analyzes the minimum requirement in acoustic frequency and acoustic beam size
for satisfying the Bragg diffraction condition. Another criterion anatomizes the conditions

for maximum second order Bragg diffraction.

4.2 Basic Formulation of Bragg Diffraction

Let us consider the destructive and constructive interferences of light rays passing

through a plane acoustic beam in the same manner as has long been used in establishing

63

64

the angular relations for Bragg diffraction of X-rays in crystals. In Fig. 2.3, it shows that
Eq. (2-38) describes the condition for Bragg diffraction in the case of constructive inter-

ference, which can be expressed as follows:

m?» = 2Asin9 (4-1)

In order to understand the phenomenon of constructive interference, we will go
through the theoretical analysis. In Fig. 4.1, let the horizontal lines represent the acoustic
wave fronts, space A apart, the solid line indicates the wave crest. The width of the acous-
tic field is L. Let the distance from A] to C 1 represents the wave front of the incident light
beam, whose perpendicular rays A1 A2 , B] 82 , etc., make the glance angle 4) with the
acoustic wave fronts. As the light rays pass through the acoustic beam, a small portion of
the ray is assumed to be scattered at each acoustic wave front and the remainder pass
undeviated to the next wave front. Next, let us consider only the light rays scattered in the

specific direction A2 A3 , which makes the glancing angle 9 with the acoustic wave fronts,
and arrive later at a plane 83 33' which is normal to all the rays. By considering light path
lengths from the plane A1 C1 to the plane B3 83' we can find the conditions under which

two or more rays will arrive in phase and thus give constructive interference of the light

intensity in the 9 direction. These conditions will be examined below.

First, let us consider any two scattering points A2 and 82 in the first wave front. The
path length A1 A2 A3 will be equal to the path length 31 32 83 if 6 = ¢- For this condition,

then, there will be constructive interference of all scattered rays. Next, consider the single

incident light ray B] 82 which may be scattered from one or more acoustic wave fronts
besides the first, at 82, 82', and 82'. Thus, the scattered rays 82 B3, 82' 83', etc., will
arrive in phase at the B3 B3” plane if the path length 81 82 B3 equals the path length B]

82' B3', or if the difference in path length is m7». When 9 = (1), it satisfies the condition.

65

 

 

 

 

 

acoustic Waves

 

Figure 4.1 Bragg diffraction developed from path length consideration.

 

66

Reinforcement will occur in the direction 6 defined by the relation. The more acoustic
waves that a single light ray passes through, the more will be the reinforcement. As a
result, the diffraction spectra will be more intense and sharper. Further, if a single light
ray passes through fewer than two acoustic wave fronts, no reinforcement will occur, and

hence no Bragg diffraction can exist.

4.2.1 Criteria for Bragg Diffraction

The Bragg diffraction occurs when incident the angle 11 = 9 and sinO = (min) / (2A) .

It requires that the acoustic beam should be wide enough such that a single incident light

ray may cross one wave length of the acoustic beam. Figure 4.2 shows that

tan¢ = 1% (4—2)

where N is the number of acoustic wave fronts crossed by a light ray in traversing the
acoustic beam. In order to obtain Bragg diffraction in a liquid medium, the incident angle

should satisfy the following relationship,

. ml" 4
Sln¢ — '2—/-\— ( '3)
where A.“ is light wave length within the liquid medium, An = l/no, and no is the refrac-

tive index in the medium.

Since the angle (1) is very small, both tan¢ and sin¢ can be approximated by (1).

Then, we have

 

NA m?»
T 2An0 ' (4'4)

Ill

67

 

 

 

 

 

 

 

 

 

 

 

Light acoustic waves V
w
I """ /
4,,” """" /’J/w
\Q / ’2' ””””” ‘ ,m MW
\/ ”If!” ’
\J”’/'" NA JV
\J’flf "
a“ "V.
J """""
A \
L \
7» 7m
Air Liquid

 

 

 

Figure 4.2 Schematic drawing of Bragg diffraction.

 

68

The acoustic wave length A is equal to t) / f, where t) is the velocity of acoustic beam

within the liquid, and f is the acoustic frequency. Substitute t) / f into Eq. (4-4), we have

sz- 2_N__t)2’2 (4_5)

In Eq. (4-5), N must 2 1 in order to give Bragg diffraction. For acoustic beam in

water, the refractive index no = 1.33, and the velocity u = 1.483 x 105 cm / sec (velocity in

water at 20 0C). By using Helium-Neom light, A = 6.328 x 10’5 cm and the acoustic beam

width L in cm, Eq. (4-5) becomes

Lf2= 9. 2447me10
The condition can be simplified by putting frequency in MHz. We have then

(4-6)

L172 = 9245/
m

where F is MHz.

Equation (4-6) is the criterion for Bragg diffraction in water, when the light source is

a Helium-Neom laser. The integer m describes different order of diffractions.

For m = 1, the criterion for the first order Bragg diffraction becomes

L172 = 924N. (4-7)

For m = 2 the criterion for the second order Bragg diffraction becomes

LF2 = 462N. (4-8)

It is clear that LF2 must have the largest value for the first order Bragg diffraction
and progressively lower value for the higher order modes. The first appearance of Bragg

diffraction occurs when N = l. The intensity of higher order mode can be normalized with

69

respect to the first order mode (N=1) for comparison purpose. Geometrically, the factor N
describes the number of acoustic wavelengths passed through by an undiffracted light ray
in traversing the acoustic beam. Therefore, a greater sharpness is expected in all orders
for larger N value. For various N values, Fig. 4.3 shows the distribution of the second
order mode intensities by varying the incident angle. It can be seen that the 3dB band-
width resolution with N = 4 is twice better than that with N = 2, while the resolution with
N = 6 is thrice as good as that with N = 2. This result could be very useful to the acousto-
optic Bragg imaging applications. It is estimated that the imaging resolution can be
enhanced to twice as good when the acoustic frequency is raised from 17 MHz to 24 MHz.
Judging from the results, the N value acts pretty much like a figure Of merit. However,
when N becomes too large, the further increase in N will not give marked improvement in
sharpness of diffraction. This is because that there is little intensity left in the traversing

light ray as it reaches the far side of the acoustic beam.

Another important factor for the interaction is m. From Eq. (4-6), different m value

indicates different order of diffraction for a given L172 value. For given Ll”2 value, for
example f= 17 MHz and L = 3.175 cm, the second order has N = 2 while the first order has
N = 1. Using the system setup mentioned above, Fig. 4.4 shows that the second order
Bragg diffraction provides a sharper resolution than the first order. This implies that if we
produce the same sharpness of diffraction in both the first and second orders, the acoustic
frequency for exciting the first order diffraction is twice that for the second order. In
experimental setup, lower acoustic operating frequency is preferred, since low frequency
transducers are easier to be fabricated as well as low cost. However, good resolution still
can be achieved by Operating at low frequency, provided that second order diffraction is

used.

70

l2. L=3.175<:m

 

N=2 with 1:17MHZ

N=4 with 1:24MHZ

0-8 N=6 with 1:29.5MH2

.0
or

Intensity

.0
A

0.2

 

 

    
 

 

 

Variable Alpha

Figure 4.3 Second order light intensities versus with N as a parameter.

71

1:17 MHz. L=3.175cm

 

 

 

 

‘ “ i
I(1). V=3 1(2). V=7.5
0.8 . .
E06 . .
(D
C
a:
E
0.4 - ,
0.2 - .
0 1 l 1 1 l 1 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4

Variable Alpha

Figure 4.4 Light intensities versus or with f = 17 MHz.

1.6

72

4.3 Criteria For The Second Order Bragg Diffraction

Although the incident angle of the light beam satisfies the double Bragg angle

9 = A/ A, the intensity of the second order is usually rather weak. The second order

28

diffraction intensity can be maximized by choosing an optimal set of system parameters.

In this section, we will analyze the condition for optimum second order Bragg diffraction.

VVrth incident angle 9 = 928 = K/A, the diffraction intensities are

 

 

2V . O 2
I1 = [KSInZ] [MC
2 2 (4'9)
-1 9_ Q) (Q-L-QH
l2 — 4 (c034 cos4 + O_srn4 srn4 [inc
1
2 2
where O’ = 8V2+Q
2
Q = k* L
nOkCOSG
knlL
v =
c050
F ° I (”2 0
or maxrmum 2, d—Q- — .
d12
The detailed derivation of E—Q will be given in Appendix A.3. Taking the derivative

of 12 with respect to Q, we have

5in = [gsing— sing][(-}1)cos§+ésing+(%)2(icosg—C—lysing)] (4-10)

73

For 12 to be maximum, [1 must be minimum. Ideally, for complete mode energy conver-

sion, the first order mode should be vanished. In Eq. (4-9) if I 1 = 0 we need 3 = nrt.

Substitute % = M: into Eq. (4-10), it becomes
cl! 2
2 - .1. - Q[ 9 (Q) 9]
d—Q- - 4srn4 cos4 O cos4 (411)
(”2 Q Q
For -d-—Q- = 0, sin-£1- must equal to zero, which requires that 4 = mrt . As aresult,

= nrt and g = mrt. Since O is related to Q and V

AIQ

the conditions for 12 maximum are

we need to consider the following cases.

- mrt , and m = even.

With m = even and = Irrt, Eq. (4-9) becomes

«BIO

(cosnrt— 1)2 (4-12)

Alt-

,2 =

For 12 maximum, cosnrt must be (-1), therefore, we need It to be odd, or let

= (2n—l)7t (4-13)

AIQ

where n = 1, 2, 3,....

. 2 V2 0 .
Since G = 8V2 + Q and 4 = (2n— 1) rt, we can obtain the corresponding V

value as follows:

74

4(2n— 1):: = (8V2+Q2)1/2

With Q = 4mrt, we have

1/2
V = J21t[(2n—1)2—m2] (4-14)

m+l

 

For V> 0, we need It 2

For example, When m = 2 and n = 2

4m7t = 25.13

Q

V 510

J21:[(2n—l)2—m2]1/2

This result confirms with the numerical solution of the optimum parameter set

(Q=25, V=10) which excited the second order Bragg diffraction to maximum diffraction.

Case2. (4, = mug = mrt,andm=odd.

With m = odd and g = nrt , Eq. (4-9) becomes

12 =2-11(cosnrt+1)2

For 12 to be maximum, cosnrr must be 1, therefore, we need It to be even, or let

= 2nrt

#10

where n = 1, 2, 3,....

75

O 2 2 V2 .
If we substitute 4 = 2m: into O = 8V + Q , we can obtain V as follows:

With Q

4mrt, and O = 8nrt, we have

V (4-15)

./§1t[(2n)2-m?‘]1/2

. m
For V > 0, the requrrement becomes n 2 2 .

For example, when m = 1 and n = 1

Q = 4mrts 13
and

v = J21E[(2n)2—m2]1/257.7

Also, form = 3 and n = 2

Q = 4m: 5 38
and

1/2
v = J2R[(2n)2—m2] a 11.75

Again, these results confirm with the Optimum parameter sets of (Q=13, V=7.5) and

(Q=38, V=11.5).

Finally, we can conclude that for the second order Bragg diffraction to occur, the Q

and V values must satisfy the following conditions:

Q=4mn

2 2 “2 m+l
_ _ = >
J21:[(2n l) m :1 form even and n _. 2 (4-16)

 

1/2
J21t[(2n)2—m2] form=oddandn2tg

76

Both m and n are integers.

In terms of physical quantities the Q value can be expressed as
2
*
Q = .__k__£‘__ = 4m1t. (4_17)
nokcosezB
The wave numbers of acoustic wave and light beam are k* = (21c)/A and

k = (210/1 respectively. Substituting k* and k into Eq. (4-17), we have

m .. L7» . (4-18)

— 2
2A nOcosem

 

The wavelength Of the acoustic wave is A = u/ f , where f is the frequency Of the acoustic
wave and t) is the wave velocity in liquid. The condition in Eq. (4-17) can be alternately
stated as

m - [J21 . (4-19)

2
2n01) cos 928

In addition to the necessary second order Bragg condition, an auxiliary condition, Eq. (4-
19) has to be satisfied for an effective excitation of the second order mode with a pure
sinusoidal grating. Notice that this auxiliary condition is completely independent of index

modulation. For a given L, A, and no, the above condition implies that the second order

mode can be effective excited only for discrete grating frequencies.

The auxiliary condition, Eq. (4-19), is by no mean contradictory tO the Bragg dif-
fraction condition given by Eq. (4-5). This can be shown as follows. Rewriting the Eq.

(4-19), we have

77

Zn v 2mcosG
Lf2= 0 23 . (4-20)

 

In the experimental setup, the light source is a He-Ne laser and water is used as interacting

medium. The wavelength of the laser beam is 632.8 nm, the velocity of the acoustic wave

in water is 1.483 x 105 cm / sec, and the refractive index in water is 1.33. For m =1, which

gives Q = 41: = 13, Eq. (4-20) becomes

1.722924.” 1012 cmtz2 (421)
The criterion for the second order Bragg diffraction in Eq. (4-8) is then LF2(cm x M2Hz) =

462 N. Therefore, for Q = 13 and the LF2 product above, it has N = 2 which is larger than

the minimum requirement of N 2 1 in Eq. (48). This means that in an acoustic-optic sys-
tem, if the criterion for optimizing the second order Bragg diffraction is satisfied, then the

criterion for Bragg diffraction is automatically satisfied.

knlL
050

 

From the definition of V = , for a given m value, one can choose n1 to sat-

28

isfy Eq. (4-16). Since n1 relates to the pressure of the transducer, one can adjust the acous-

tic power output tO furnish the required pressure.

Chapter 5

Experimental Measurements for Bragg Diffraction

5.1 Introduction

In this chapter, the experimental setup and the measurement procedures are
described, and the experimental findings for verifying the Bragg diffraction are presented.
The experimental verifications include the intensity distribution of the diffraction mode
with various acoustic amplitudes, acoustic frequencies and incident angles of the light
beam. The results show that when the incident angle equals to twice of the Bragg angle,
the second order diffraction can be exited to its maximum intensity at several particular
sets of system parameters. Comparing the angular sensitivity of the diffraction modes, the
second order diffraction provides a sharper angular resolution than the first order mode. In

general, the experimental results agreed with the theory qualitatively.

78

79

5.2 Experimental Measurement System

A protO-type system for observing the acousto-optical interaction has been designed
and constructed at the MSU Ultrasonic Laboratory. The acoustic wave is generated in
water which diffracts the incident laser beam. The full range of Bragg diffraction can eas-
ily be observed at a range of acoustical frequencies (6 to 30 MHz). The coherent light
source provides a well collimated beam for the interaction. A block diagram of the mea-
surement setup is depicted in Fig. 5.1 (a). The acoustic cell shown in Fig. 5.1(b) is made of
a plexiglass measuring 27 x 10 x 13 cm. A quartz transducer is mounted on one side of
the acoustic cell. Four different transducers are used in this experiment. The transducers
have a fundamental frequency of 5.8 MHz, 8.8 MHz, 17 MHz and 20 MHz respectively,

but they can also be excited at their odd harmonics.

The acoustic signal is generated from a RF generator (Model 191 constant ampli-
tude signal generator TEKTRON IX, Inc.), and then amplified by a RF. power amplifier
(Model 310L EN I Inc.) to attain a maximum power of 15 watts before triggering the trans-
ducer. The diffraction output is detected by a photo detector (Model 115-9) and power
meter (Model 40 Optometer made by United Detector Technology). A rotator which sup-
ports the acoustic cell is used to vary the angle between the incident light and acoustic
beam. The laser source is a 20 mW He-Ne laser. An acoustic absorber is placed at the end
of the acoustic cell, therefore, all of the transmitted acoustic signal after the interaction
region will be completely absorbed such that no standing wave exists in the interaction
region. The transducer is placed on one end of the cell. Since the transducer is submerged
in water, the heat generated by the transducer during high power Operation will be diffused

by the water without overheating the transducer.

80

 

Acoustic Absorber

 

 

 

Photo
Detector

 

 

. .... . Diffracted Power
_I light ‘ I Meter
[HO-NC Laser ! ‘41 V .

Razor

Blades
W‘

ACOUS C C011 Acoustic

Transducer

Matching RF. Power
Network Amplifier

 

 

Generator

 

Figure 5.1(a) Block diagram of the measurement system.

 

Acoustic Absorber

   
 
  

Acoustic Transducer

Optical window
for laser beam passage

 

 

 

Figure 5.1(b) A plexiglass water tank for observing acoustO-optical interaction.

 

81

5.2.1 Network for Acoustic Impedance Matching

Since the impedance of acoustic transducer and the output impedance of the r.f.
power amplifier vary as function of operating frequency, to maximize the acoustic output
of a acoustic beam it is necessary to match the impedance between the transducer and the
r.f. power amplifier by using a matching network. Figure 5.2(a) shows an electrical repre-

sentation of the acoustic transducer. The capacitance C0 is the transducer capacitance and
Ra is the real part of the impedance which represents acoustic power flow into the

medium. The technique of impedance matching of acoustic transducer is well reported
[37]~[39]. A T-matching network shown in Fig. 5.2(b) is chosen to put between the trans-
ducer and the power amplifier. There are many possible ways to match the acoustic sys—

tem, however the reason for choosing the T—matching network is as follows. The theory of

[40] states that only two reactive elements are required to match the

matching networks
input and output resistances Of two interfacing system. But, one needs to add a third ele-
ment. The reason is that one element of the T—matching network can be assigned a value
arbitrarily, and the other two components can be determined accordingly. From such a T-
matching network, it is rather easy to accomplish the matching at hand. Since we have
several transducers that need to be matched to the r.f. power amplifier over a wide range of
operating frequencies a simpler and more flexible matching network should be used. The
T-matching network can provide the functions required for the acousto—optics system. A
T—matching network (Made T-1000, Tucker Electronics Inc.) is used in our experiment.
By using the T-lOOO matching network, the power transferred to the acoustic transducer
can reach 75% ~ 85% of the input power. On the other hand, only 25% ~ 35% of input
power is transferred to the acoustic transducer, if the matching network is not used. For

such a poor transmitting efficiency, the eventual interaction would be too weak to be

Observed.

82

 

 

 

Ro((D)

 

 

Figure 5.2(a) Electrical equivalent circuit of the acoustic transducer.

 

 

 

 

_W
:1
\I
7|
\l
I
2"
8
o.

 

 

 

 

Figure 5.2(b) T matching network.

 

 

83

5.3 Experimental Results

Several experiments were performed to verify the simulation results obtained in
chapter 3. The pure sinusoidal grating was established by an acoustic transducer, while
the incident light source is from a He-Ne laser. For the sake of comparison of various
mode intensities, the measured irradiance of each mode is normalized with respect to the

total transmitted irradiance.

Experiments were performed in a way that the results can be compared with the the-
ory developed. To make quantitative comparison, the gratings were established by differ-

ent acoustic transducers. In the experimental setup, the wavelength Of the laser beam is K

= 6.328 x 10'5 cm, the acoustic velocity in water is v = 1.483 x 105 cm / sec and the refrac-

tive index in water is n = 1.33. With these parameters, the Q value is found to be

Q= 1.359x 1044-12-1. (5-1)

where f is the acoustic frequency and L is the width of the acoustic beam.

5.3.1 Bragg diffraction with various V and Q value

The first experiment was performed to observe the maximum diffraction as a func-
tion of V value. An acoustic transducer, which operates at a fundamental frequency of 5.8
MHz with 2.54 cm beam width, was driven at its third harmonic of 17.4 MHz. With f =
17.4 MHz and L = 2.54 cm, the Q value becomes 10.5. Using this setup with the double
Bragg incident angle 01 = 1, Fig. 5.3(a) shows the diffraction efficiency with various inten-
sities of the acoustic beam. In Fig. 5.3 the solid curve represents the theoretical result,
while the “0” curve indicates the experimental result. A separate experiment was per-
formed by using an acoustic transducer having a fundamental frequency of 17 MHz and

3.175 cm beam width. With an acoustic frequency of 17 MHz and beam width of 3.175

84

cm, the Q value becomes 12.5. Included in Fig. 5.3(b) are the theoretical and experimen-
tal results for Q = 12.5. It can readily be seen that the experiment result of the second
order mode is a function of V. For Q = 12.5 and V approaching to 6.5 the second order dif-

fraction reaches a maximum.

It can generally be said that the experimental results agree with the theory well. The
experimental result of the first order diffraction is very close to the theoretical prediction
which shows that the intensity starts decreasing when the incident angle is approaching
twice of the Bragg angle. The intensity of the second order is 30% lower than we
expected. This could due to surface reflection on the imperfection of the experimental

setup. However, it is strong enough for practical application.

The next experiment was performed to Observe the maximum diffraction of the sec-
ond order interaction as a function of Q. It is more difficult to observe the diffraction in
various Q values, since this experiment needs a wide range of acoustic frequencies to pro-
vide different Q values and we have only a limit of transducers available. However, each
transducer can still be driven over a narrow range frequencies around its central frequency,
enough data were collected to show the diffraction effects. However for each frequency
we need to compensate the diffraction intensity by power spectrum distribution of the
individual transducer. Figure 5.4(a) shows the experiment result of the first order diffrac-
tion with various Q values. Since the Bragg angle is a function of the acoustic wave-
length, for each acoustic frequency the Bragg angle will be different. Therefore, at each Q
value, the incident angle was optimized to achieve maximum diffraction in the first order
interaction. The result indicates that when Q is large, the first order diffraction becomes
very efficiency. This means that higher acoustic frequency and wider acoustic beam pro-

vide higher diffraction efficiency in the first order Bragg diffraction.

85

 

 

lntonaity
.0
4

0.4’

0.21-

 

- : Theorelcd

o : Experimental

 

 

 

 

 

 

8
Variable V

 

 

 

08>

Intensity

 

. :Theaellcal

o : Experimental

 

 

 

 

 

€06" 1
E
1 0.4
0.2» i
G
16 0 16
VanableV
Figure 5.3(a) First and Second order diffraction intensities versus V
at double Bragg angle with Q = 10.5.
11-
1 0.8- .
30.6
S
1 0.4»
o.2~ .
G
16 0 16

 

8
Variable V

Figure 5.3(b) First and Second order diffraction intensities versus V at

double Bragg angle with Q = 12.5.

 

8
Variable V

86

 

 

 

 

 

 

 

1 i .
0.9». It ............... fi .,
.1“ U
0.8- [I ..
0.7»- .1
g_O.6v- ..
C
o . a
:5- 0'5 - Theoretical
0.4r- ..
.' E xpen mental
0.3- .4
0.2» ..
0.1 ~ " .
a o 4 1 1 1 J 1
( ) O S 10 15 2O 25 30
Variable O
1 f" -1
0,9 >- - : Theoretical .

.‘ . : Experimental

 

 

 

 

0.2“ 7'. .4
"I
o 1)- -r .
(b) o 1 l A 1 1 L
O 5 1O 15 20 25 30
Variable O

 

 

 

 

 

0.61-
? - -
s 0.5“ . - -.. .l
O.‘ l' I .4
0‘3" - : Theoretical -
O-2l‘ ..'.. :Expenmontal .4
0.1 r-
C 0 A L ‘ l L A
( ) 5 10 15 20 25 30
Variable 0

Figure 5.4 (a) The First order diffraction versus Q with V = 3 and or = 0.5.
(b) The second order diffraction versus Q with V = 6.5 and 01 = 1.
(c) At each Q the acoustic amplitude was optimized to achieve
maximum power in the second order with 01:1.

87

For the second order diffraction, the experiment result shown in Fig 5.4(b) illustrates
that the maximum diffraction efficiency is very sensitive to the Q values. In this experi-
ment the V value is set to 6.5 and the incident angle is optimized to on = 1. The diffraction

[2 is driven to maximum when Q approaches 12.5.

From the theoretical prediction in Fig. 3.5(c), the maximum diffraction 12 is also
related to the intensity of acoustic beam V. Therefore, we repeated the experiment with
optimizing the acoustic amplitude to achieve maximum power in the second order at each
Q, and the result is shown in Fig. 54(0). The experiment result indicates that when inci-
dent angles are at double of the Bragg angle, the second order diffraction becomes very
strong. Unlike the first order, the maximum second order mode only appears in certain Q
values which can be Obtained by adjusting the acoustic frequency and beam width. A

rough check confirmed that the criterion of the Q value for the maximum second order is

very close to Q = 4m: (m is integer).

5.3.2 The Bragg diffraction with various incident angle

In this section, the experiments describe the diffraction modes with different incident
angles. An acoustic transducer with Q =12.5 (17 MHz acoustic frequency and 3.175 cm
acoustic beam) was used in the experiments. The acoustic amplitude was optimized to

obtain the maximum diffraction in the first and second order mode. With the frequency f =
17 MHz and the velocity of acoustic wave in the water v = 1.483 x 105 cm / sec, the wave-
length Of the acoustic wave is 8.72 x 10'3 cm. With these physical parameters, the theoret-

ical Bragg angle GB is predicted to be 0.208 0, and the double Bragg angle 023 equals to

0.416 0. From the definition or = 1% - sine, the or equals to 0.5 for 93 = 0.208 0, and

equals to 1 for 623 = 0.416 0. Figure 5.5(a) shows the intensity distribution of the first

88

order with various a. From the experiment result, when the incident angle approaches

0.185 0 (equivalent to a = 0.44), the first order diffraction is driven to maximum intensity.

The Bragg angle GB used in the experiment is very close to the theoretical Bragg angle.

Figure 5.5(b) indicates that the incident angle which drives the second order diffrac-

tion to maximum power is 0.371 0 (equivalent to or = 0.88). The value of the double
Bragg angle found in the experiment also closely confirmed with the theoretical predic-
tion. Observing the experiment result of the first and second order shown in Fig. 5.5(c),
the response of the second order diffraction is more sensitive to the angular variation of
the incident light. For imaging application purpose, the second order diffraction should

provide a better angular resolution than the first order diffraction.

From the experimental results, we can conclude that when the incident angle equals
to twice of the Bragg angle, with proper system setup, the second order diffraction can be
strongly excited and a better angular resolution can be achieved. The limitation of the sec-
ond Bragg diffraction is that only certain optimum parameter sets in the system setup can
provide the maximum diffraction. But for the first Bragg diffraction when Q is large

enough, the first order diffraction can always be excited to its maximum intensity.

Another interest property of diffraction observed is that the angular sensitivity of the
diffraction is proportional to the Q values. Three Q values, 5.8, 12.5 and 19 were used to
check this property. Figure 5.6 shows the intensity distribution of the first order mode
with various or values. In order to compare the mode intensity distribution, the maximum
diffraction of each data set is normalized to unity and the location of the peak value is

shifted to or = 0.5. The result shows that the 3 dB bandwidth of 11 for Q = 19 is only one
third of the bandwidth of I 1 for Q = 5.8. With this property, the Bragg diffraction can pro-

vide a higher angular resolution in large Q values. A rough check confirmed that the sec-

ond order Bragg diffraction also have the same property.

89

 

0.9

0.8

0.7

0.6

0.5

1m»;

0.4 ’-
0.3 c

0.21-

(a) 0.1 1}.

 

1(1)

- : Theoretical

: Experimental

 

 

_lL .1.
0.6 0.8
Vanable Alpha

1,6

 

0.9 '-

0.8'-

0.7

0.8 *-

11mm

0.4 1
° 0.3L

0.2"

(b)

 

0.5»

- : Theorohcal

..‘.. : Experimental

1(2)

 

 

 

lnlens'ty
_o
b

 

1

 

 

 

0.2

0.4

0.8 0.8
Variable Alpha

1

Figure 5.5 (a) The First order diffraction versus or with Q
(b) The second order defloration versus 01 with Q
(c) The comparison of the angular sensitivity to the dif-
fractions with various or.

1.8

= 12.5.
= 12.5.

9O

 

 

 

 

1 ’- . ‘0 1 ~ 0 _,
V x ..o.. :O=5.6
..O / \ 0.
0’ I '-
I ‘1 Q. 20:12.5
b- ." I \ '. -l
0.8 .0 I \ .V
x l x 0, -x- :O=19
O )1 \\ 10
PO 6 I 1
.2 . O 0 II \ '0
.9 I ‘1 ’
S \ O
i‘ t .
I ‘
0.4 ~ I x _ 1
I ‘ o-
\ 1
I .\ o
I .
0.2 ~ I ‘ ~ 1
0.
/§ 0 ‘X- *\ '_
X X 2 o
o 1 1 1 1 1 1 ”X‘—
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Variable Alpha

Figure 5.6 The first order light intensities versus 01 with Q = 5.6, 12.5 and 19.

91

All experiments we performed were up shifted interaction. This means that the inci-
dent angle between light beam and the normal of the acoustic beam has positive value, and
the diffractions appear in plus order diffraction. For the incident angle is negative, the dif-
fractions are down shifted interaction, and minus order diffraction will appear strong. All

the down shifted interactions will be symmetrical to the up shifted interactions.

Chapter 6

Application of Acousto-optic Interaction

6.1 Introduction

From previous discussion it is clear that an acoustic cell may be used to modulate
light by applying Bragg diffraction theory. In recent years, many Bragg diffraction appli—
cations have been investigated, including acousto—optic modulation, Bragg diffraction
imaging, Optical signal processing, television projection, light deflection, frequency shift-
ing, the guided wave effect, acoustic velocity measurement, Q switching, and mode lock-
ing in lasers.[4”’[59] Some products are already on the market. Newport ElectrO—Optics
Systems Inc. Offers acoustO-optic (A0) modulators, A0 beam deflectors, A0 mode lock-
ers, a Q-switchin g system, and a 2-D beam projector all based on Bragg diffraction theory.
However, most of the applications utilize the first Bragg diffraction. By applying the sec-

ond Bragg diffraction, greater resolution can be achieved.

In this chapter a Bragg diffraction imaging system was designed and performed.
Imaging resolution for the first and second orders are observed. Imaging quality for dif-

ferent acoustic frequencies and beam widths are compared.

92

93

6.2 Acoustic Imaging base on the Bragg Diffraction

One useful application of Bragg diffraction is the acoustic imaging. The technique

of acoustic visualization has potential applications in nondestructive testing Of materials

and medical diagnostics.[601 Korpellml'lml was the first to show that acoustic Bragg dif-

fraction possesses image-translation properties. The basic diffraction imaging method

was also independently developed by Tsail65]'[67], and Wadel681'l69] for different applica-
tions. It is possible to use pulsed sound and pulsed light sources to improve the usefulness
of Bragg diffraction imaging by providing range gating to achieve the depth discrimina-

tion.[7O]

A brief description of Bragg imaging theory is given below. Figure 6.1 shows that
the interaction between the waves emanating from a point source S of single frequency
sound and a point source 0 of monochromatic light produces diffracted light that appears
to come from a point 0’; a virtual image of the sound sources but one that is visible to the

eye. For a number of sound sources S 1, S 2, S3, etc., the virtual image points 01’, 02’, 03’,

etc., form a corresponding pattern. In this translation process, amplitude ratios and phase

angles are preserved and an acoustic field is transformed into an equivalent Optical field.

For the first order Bragg diffraction imaging, the angle between incident light ray

and acoustic beam equals the Bragg angle 03, and it may also be shown that 050’ is an
isosceles triangle with apex angle equal to 293. This affords a convenient construction for
locating virtual images which has been used in Fig. 6.1 in order to illustrate the image

transformation properties. As 03 is usually a small angle, we can see that the imaging

involves a rotation of close to 90°. This is the reason why with this technique it is very

convenient to obtain an image of a cross section perpendicular to the propagation direction

94

 

 

 

 

 

 

 

Figure 6.1 Acoustic rays from source S interact with optical rays from
source 0 to form new light rays, which seem to come from 0’,
a virtual image of S.

 

95

of the acoustic wave. The imaging process involves a demagnificated factor M which

1

equals the ratio oflight wavelength to sound wavelength, i.e. M = 25in98 = A'

In optical microscopy, resolution is limited by the numerical aperture (the sine of
one half the angle formed by the most divergent rays that the Optical system accepts from
Object) and the wavelength of the light. In acousto-optical imaging, the same rule applies.
We must substitute the wavelength of the acoustic wave for that of the light and consider
the angle formed by acoustic and optical rays. Therefore, the minimum resolvable detail
is limited to NM times the size of the point source of light, and this cannot be smaller than
half a wavelength of acoustic wave divided by the numerical aperture of the incident light
beam. For higher acoustic frequency, the wavelength of the acoustic wave is shorter, and

this provides a better diffraction imaging resolution.

For the second order Bragg diffraction imaging, the imaging process is the same as

above except that the Bragg angle GB is changed to the double Bragg angle 023_ There-

fore, the demagnificated factor M equals to 2% , and the minimum resolvable detail is one

half of the first order. This means that the imaging resolution by the second order Bragg

diffraction is twice that Of the first order Bragg diffraction.

6.2.1 Experiment Facility and Measurement System

A diagram of the measurement system used in our experiment is shown in Fig. 6.2.
Figure 6.2 shows that a He-Ne laser beam is expanded in diameter by the telescope

formed by spherical lenses S 1 and 52, then the beam size is controlled by the aperture. A
cylindrical converging lens, C1 forms a vertical wedge Of light which passes through the

acoustic field and comes to a line focus at P which is perpendicular to the plane of the

96

 

 

  
   
 

  

Laser
Exciter

C1 Cylindrical
Converging Lens

Aperture

.0

 

 

 

 

 

 

 

    

 

Mirror + I
Telescope

 

Acoustic
Cell

Matching
Network

Acoustic
Absorber
Screen
C2 Cylindrical or
Vrdrcon

Projecting Lens

    

Mask C3 Aspect-Ratio

Correcting Lens

Object

Acoustic
Transducer

7 RF Power - - rgna
Amplifier Generator

 

Figure 6.2 Experimental setup for Bragg diffraction imaging.

 

97

paper. When the light beam within the acoustic cell interacts with an acoustic wave, the

Bragg diffracted images which include the information Of the acoustic pattern appear at

0+ and 0' respectively, where the ‘+’ corresponds to the Doppler up shifted light and the

‘-’ corresponds to the Doppler downshifted light.

For imaging in the direction vertical to the paper there is no demagnificated

involved. A cylindrical projecting lens C2 and a cylindrical aspect-ratio correcting lens C3

compensate for the horizontal demagnification inherent in this imaging process. While
the imaging in the horizontal direction (and thus the resolution in that direction) depends
on the angular selectivity of Bragg diffraction, the imaging in the vertical direction is
essentially a shadow of the acoustic wave cross section in the region of the acoustO-optic
interaction. Therefore, the vertical resolution improves as Object is moved away from the

transducer and toward the light beam, while the horizontal resolution is unchanged.

Since we need the diffracted image to be focused on the screen, the projection lens

C2 may have a shorter focal length than lens C1. Also the positions Of C2, C3 and the
screen distance must be properly chosen. By changing the position of C2, different acous-

tic cross sections can be brought to a focus on the screen. The undiffracted beam and one
of the images are removed by a masking stop. The other diffracted image is projected to
the screen. A video camcorder, RCA pro 807, is used to line in the imaging signal, and the

signal is captured by a video capture card added on the PC computer.

Two transducers with different beam size and acoustic frequencies are used to com-
pare the imaging results. The first transducer has a fundamental frequency of 5.8 MHz
with 2.54 cm diameter for beam size but is driven at its third harmonic at 17.5 MHz. The
other one has a fundamental frequency of 20 MHz with 3.175 cm diameter for beam size.
For each transducer, the incident angle and the acoustic amplitude were Optimized to

achieve maximum power in Bragg interaction.

98

6.2.2 Experimental Result of Bragg Diffraction Imaging

In this section, six experimental samples shown in Fig. 6.3 were used as imaging
Objects for Bragg imaging interaction. Four of the samples are round aluminum washers.
The biggest one has a 18 mm outside diameter and an 8 mm inside diameter. The smallest
one has a 5 mm outside diameter and 3 mm inside diameter. The width of the aluminum
washer is 5 mm for the biggest one and 1mm for the smallest one. Another object made
by steer material has a star shape and has a 1.2 mm width for each star side. The last
Object is cut from a thin copper sheet with the characters “MS U” and has a 2 mm width for

its line shape.

Three acoustic imaging experiments are performed. One experiment compares the
imaging resolution Of the first order Bragg imaging with that of the second order mode.

The others analyze the imaging resolution with various acoustic frequencies.

The first imaging experiment used a 17.5 MHz acoustic transducer with a diameter
of 2.54 cm. Four washers were used for comparing the imaging resolution to the first and
second Bragg diffraction imaging. The imaging results are shown in Fig. 6.4. Figures 6.4
(a) ~ (d) are the first order Bragg images of the four washers and figures 6.4 (e) ~ (h) are
the second order Bragg images. For the biggest washer, there is a little difference between
the first and second Bragg images. However, when the Object (washer) size is reduced, the
resolution difference between the first and second Bragg images becomes evident. For the
smallest washer, the inside circle of the washer in the second order Bragg image, Fig.
6.4(h), can be distinguished while only a small spot shows up in the first order Bragg
image, Fig. 6.4(d). It is clear that the second Bragg diffraction provides a better imaging

resolution than the first Bragg diffraction.

99

 

“we ”I(NV‘.:I'§.;.

.._ . _ i .
. .. . .
~ \ ..' ' ....3 '
.._.,._.:u . . .: .. u .. .
, .. . . . .
. . . .
' ._.__ 3... e 'n .. . r ..
I. t I . e

 

 

 

 

 

Figure 6.3 Experimental samples for imaging objects.

 

 

 

 

 

Figure 6.4 The first and second Bragg diffraction imaging for four round wash-
ers with acoustic frequency f = 17.5 MHz and beam size L = 2.54
cm. (a) ~ (d) show the imaging result by using the first Bragg dif-
fraction, and (e) ~ (h) show the imaging result by using the second
Bragg diffraction.

l01

The next experiment shows that the imaging resolution is proportional to the acous-
tic frequency and the width of the acoustic beam. The “MS U ” shaped object was used to
demonstrate the first Bragg imaging interaction with two different acoustic frequencies.
The imaging results in Fig. 6.5(a) were produced by using acoustic frequency f = 17.5
MHz and beam size L = 2.54 cm (which gives Q = 10.5). The images shown in Fig. 6.5(b)
were produced by usingf= 20 MHz and L = 3.175 cm (Q = 19.2). Comparing Fig. 6.5(a)
and (b), it is obvious that the image in Fig. 6.5(b) has a better imaging resolution than Fig.
6.5(b). This means that higher acoustic frequency and wider acoustic interaction area pro-

vide better Bragg imaging quality.

Figure 6.6 shows the star object imaging results using the second order Bragg dif-
fraction. In this experiment, the system setup is the same as the previous experiment but
the acoustic amplitude and incident angle of light were optimized to produce a higher dif-
fraction intensity of the second order mode. From Fig. 6.6(a) to 6.6(b), Operating fre-
quency and interaction region length are both raised. The outside of the star shapes from
the Bragg images in Fig. 6.6 (a) and (b) are both clear and sharp. However, the inside of
the star is less clear in Fig. 6.6(a) than in Fig. 6.6(b). This means that with a higher Q
value, the second order Bragg diffraction imaging method also provides better imaging

resolution.

From above results, one can conclude that the final Bragg image resolution is
directly proportional to the acoustic frequency. Therefore, it is desirable to use the highest
possible operating frequency. However, for medical diagnostic applications, biological
materials generally have higher attenuation at higher frequencies, so it is not practical in
medical applications with high acoustic frequency. Using the same acoustic frequency,
second order Bragg imaging should provide a better imaging resolution than the first order
Bragg imaging. Based on this result, the second Bragg diffraction has high potential in

medical diagnostic applications than the first order mode. Another useful application of

 

 

 

 

 

Figure 6.5 The first Bragg imaging with MSU shape in different acoustic frequen-
cies. (a) Bragg imaging by usingf= 17.5 MHz and L = 2.54 cm. (b)
Bragg imaging by usingf= 20 MHz and L = 3.175 cm.

103

 

 

 

 

 

Figure 6.6 The second Bragg imaging with STAR shape in different acoustic frequen-
cies. (a) Bragg imaging by usingf= 17.5 MHz and L = 2.54 cm. (b) Bragg
imaging by usingf= 20 MHz and L = 3.175 cm.

104

Bragg imaging is in the area of nondestructive material testing. Since acousto-Optic inter-
action can provide real-time capability and high resolution properties, it should be a pow-

erful technique for nondestructive evaluation.

Chapter 7

Conclusion and Suggestion for Future Study

7.1 Conclusion

In this dissertation, the second order Bragg diffraction of an acousto-Optic interac-
tion has been investigated in detail by using the partial wave approach. The intensity of
light diffraction waves passing through the acoustic field are governed by a set Of differ-
ence-differential equation. The numerical simulations are obtained for various acoustic
frequencies, acoustic pressures, and incident angles of the light beam. The criteria of the
Optimum system parameter sets (Q, V) which provide a maximum second order Bragg dif-
fraction are established. These results indicated that when the incident angle equals to the
Bragg angle, with V = (2n - 1) It , n being integer number, the first order Bragg diffrac-
tion can be strongly excited at large Q values (high acoustic frequency and wide acoustic
beam width). However, for Optimal excitation of the second order Bragg diffraction, the
incident angle must equal to twice of the Bragg angle, and only a discrete range of Q and

V values can provide the maximum second order Bragg diffraction.

The experiments were designed and performed to verify the theory developed.

These experiments include the intensity distribution of the diffraction mode with various

105

106

acoustic amplitudes, acoustic frequencies, and incident angles of the light. The experi-
mental results agreed well with the theory developed. Finally, acoustic Bragg imaging
experiments are performed. The image resolution of the first and second order mode are
compared, and the images with various acoustic frequencies are analyzed. The image
results indicate that the second order Bragg image provides better image resolution than
that of the first order mode. In addition, the Bragg image resolution is directly propor-

tional to the acoustic frequency and acoustic beam width.

The advantage of the second Bragg diffraction is that the second order Bragg diffrac-
tion provides a double Bragg angle to deflect diffraction light, and a double frequency
shifting to the diffraction light. In addition, the angular sensitivity of the second order dif-
fraction light is better than the first order mode. Therefore the acoustic Bragg imaging of
the second order mode can provide a superior imaging resolution than that of the first
order mode. Also, in light deflection system, the number of resolvable spots of the second
order Bragg diffraction are better than that of the first order mode. The drawback of the
second Bragg diffraction is that the system parameters required to excite the second order
mode are more critical than that of the first order mode, and this makes a little complicate

requirement in application of the second Bragg diffraction.

7.2 Suggestion for Future Application

Acousto-optic interactions are useful in A0 modulation, light deflection, frequency
shifting, and Bragg diffraction imaging. Therefore, there are many topics to pursue in
future applications. For example, in the imaging application, the acoustic Bragg image
could be a powerful tool to the nondestructive evaluation of materials. For real-time
imaging, a pulsed Bragg diffraction system by using the second order Bragg diffraction

would be a good topic to purpose in evaluating the internal damage structure of composite

107

material. In addition, high power solid-state laser sources have become available in recent
years due to the advancement of laser technology. By using such diodes as laser beam
sources, the proposed method of acousto-Optical interaction could lead to a way of provid-

ing the possibility of low-cost, portable and real-time imaging systems.

APPENDIX

Appendix A

A.l The Solution for The First Bragg Diffraction

TO evaluate the intensity of the first order mode, let us consider the coupling between

(1)0 and 411 modes. From difference-differential equation Eq.(2-45), we have

(180 v

E -§Z¢1 = 0 (A1)
d¢l+ V - ' l 201)

21—2 2L¢0 ’ ’2L( ’ 4’1

Equation (A.1-1) can be solved by using Laplace transformation. Putting this set of

equations in matrix form, Eq. (A-l) becomes

' 0 C
90 = 1 00 (A-2)
in ‘C1 C2 ¢1
V
where C] = i

__ .2 _
€2-12L(1 201) .

108

109

[20]

For a Linear system, one can solve a set of difference-differential equation by

using the Laplace transformation technique. For a system [X] = [A] [X] , the solution is

X (z) = eAZXO , where X0 is a initial condition, and the 6’” can be presented as

8A2 = .g.1:(sl_A)—l] .

Applying this approach to Eq. (A-2) we have

00(2) ___ 311312 (1),-”c = ell¢inc (A-3)
4’1“) 621322 0 e2l¢inc

where em is the mode amplitude of incident light at z = 0, and

 

e = " A-4
11 a l:.12---.1C2 +Cf] ( )

 

1 —C1 5
e = " . (A- )
21 g [sz-sCerCd

[71]

From the table of the Laplace transformation , we have

_ gw
2 2:2 213:7.I_zlsin(thI—§2) . (A-6)
s + ws+w 1_§

C
2
Let w = C 1 and i = -2_C , the Laplace transform pair becomes as follows
I

 

 

110

2 7 2
_1. C1

 

C1 sz—Czs+Cf

 

 

4C v 4v2

_ 2

C
letO = (4V2+(1—201)2Q2) ,then 1-——2 = —.
4C2 2V

1

Finally, Eq. (A-5) becomes

1E(1-20)Z

e __e sin(£—z)
21— O 4L '

2V

 

Then,

ig(l—2a)zv

. C
¢1 = €210¢inc = ~28 8810(Z)¢inc atz=L.

where e = (4v2+ [Q(1-20t)]2)1/2 .

Finally, the first order mode intensity I] at z = L becomes

4V2 . 2 O
r, = 1,1,1: = ___... (Z),

(A-7)

(A-8)

(A-9)

111
thrc [inc _ ¢inc (Dino

Similarly, one can obtain 00(2), and the zeroth mode intensity IO at z = L is

N
g
I
IN
m
0‘.
D
N
A
tso

‘)¢mc = 1—Il . (A-10)

A.2 The Solution for The Second Bragg Diffraction

To evaluate the intensity Of the second order mode, let us consider the coupling

between 1110, 01 and (1)2 modes. From difference-differential equation Eq.(2—53), we have

 

(100 v

[x ‘52‘1’1 =0

.6141] v .Q A-ll
a heme-1’2) = “er.“‘za’q’i ‘ )
(“’2 v .2Q

‘21:» 7311-7114112

Following the similar procedure in solving the first order mode, we have

s2-5(C2+C3) +C2C3+C?

 

 

 

-1
1911 = (A-12)
a s3-52(C2+C3) +s(C2C3+2C?)+C3Cf
- -—C (s—C)
821:3.1 3 2 I 3 2 2 (A-13)
s -s (C2+C3) +s(C2C3+2C1)+C3C1
2
C
_ -1 1
e31 -3 3 (A-l4)

s - .92 (C2 + C3) + 5(C2C3 + 2C?) + C3Cf

112

V
where C] - i,
C2 = 1274(1 -2(1)

C3 =i-2—LQ(1-0t) .

Since we are interested in second order Bragg diffraction for maximum excitation of

. . . )t .
the second order mode, the 1nc1dent angle 18 928 = — , and thls means 0t equals to one

A

[see Eq.(2-25)]. Therefore, with a = 1 the C’s become

then

V
Cl-Z—L
__~_Q
C2—(z)2L
C3=0

For solving 621 for (1 = 1, Eq. (A-13) can be reduced as follows

_ —C
321 =3 1 2 1 - (A-15)
s —C2s+2cf

From the Eq. (A-6), we have the transform pair

 

2 -1§wl

2 w zwwe sin(th1-§2) . (A-16)
s +2§ws+w ,/1_§2

 

 

Let W2 = 2C? and 2§w = —C2,

V
w=./§Cl=./§2—L

113

 

 

g = __2 ___ [_Q_
2w zfiv
l
f 2 _ _1_ l( 2 2))?
1—§ — 2V(2 8V +Q
then, substituting w, 1:, and J1 - 5,2 to Eq. (A-15) and Eq. (A—16), we have
2
e 1 .1 4‘31
21 ‘ ‘—
2Cl sz—C25+2C%
. Q v
f _12Jivfi2—l'z 1
- 1 2Cle . fiv 1(1(8V2 2))?
"2C 1/25‘“ 2717i ”"2 z
I_L(l(8v2+Q2))
2V 2
Q

l
where o = (8V2+Q2)2.

Then, the (1)] at z = L is

-Q

_1_—

V 4 . 0’
$1 = €21.¢inc = (-356 sm(Z)'¢inc'

Therefore, the intensity of the first order mode is

V2 . 2 o
11 = ¢,-¢,* = 43551n (2)1,“

(A-l7)

(A-18)

(A-19)

114

For solving e31 for at = 1, Eq. (A-l4) can be reduced as follows

2
1 201 ,

e31 : i 3.11502 — C23 + 2Cl )]

(A-ZO)

From the table of the Laplace transform, we have the transform pair

W

2 e—gw’sin(wx/1——?t—w)

 

 

42 ”:14,
sLs +2§ws+w J

1:?)

 

where w = atan[

JT“

Let w2 = 2C? and 2§w = —C2,then

 

w = JiCl = 7:23:
C . Q
i‘ ‘27?» z 372?
1

.f""2'-_1_ 2 2§-_1_

F: — 2,/§v( Q) zfivc
‘1’ = atan[ 11:53] = atan(zé)

=> tanw = (lg)

=> smw = (5%,)

=>cosw = (4355—5;

where(Q2—<52)1/2 = 12,/5V

Jugz

(A-21)

(Q2_02)1/2

io

115

Substituting w, 11!, Q, and J1 — £2 to Eq. (A-20) and Eq. (A-21), we have

0
—z—z
e31 = % 1+g‘16-gl/e 4L sin(fzz—w) . (A-22)

Since sin (ai B) = sina- cosBi sinB' cosa, then

 

       

sin(‘-1—(;:z— 111i)— — 2fV8181n(40 z)— zjiVCOS(-4—OZ z.) (A-23)

Substituting Eq. (A-23) into Eq. (A-22), 63] becomes

 

.Q
-l-—Z
1+e 41‘ HAT—z ' Q sin(%z)——£—cos(—O—z):| . (A-24)

1
631—5 0’ lzfiv

Then, the $2 at z = L is

_,-Q
(1)31 = 6314me =% l—e 4(i%sin(§)+cos(g)) «pine. (A-25)

Therefore, the intensity of the second order mode becomes as follows

_ , * _ . *.
2 ' ¢31 4’31 ' 63] 931 line

= 211 1—e_lz(i%sin(§) + cos(g)) 1 —ei%( (-i)%31n(g)+ 905042)) .(Ac26)

2 2
_1( _(Q Q Q “(9)2” °D
_41— chin4sinz Zeos4cos4+ o 51" 4+C°S4 [inc

(0

116
Since

2 y . . 2
(cosx — cosy) + (; (smx— smy))

2 2 2
= 1-2£sinxsiny-2cosxcosy+(i) sin x+cos x

12 can be represented as follows

._ 1 0 Q 2 Q - G - Q 2
I2 — 3(cosZ—cosz) +(EsinZ—smz linc. (A-27)

For solving intensity of the zero order mode with a = 1, Eq. (A-12) can be reduced

as follows

 

 

 

e“ :34. 332—2i24rC122] {31 (sZ—sczncf ng
_5 “C23 +2C13 5(52 —Czs+2C1) (A-28)
=g-13]—e31 = 1—1331
and
q’0 = e 11%“: ' (A39)

Therefore, the intensity of the'second order mode at z = L becomes

’0 = q’0'1’0’“ = iKCOS‘Gj‘iCOSeDZ (QS‘"(Z)+ We» ]’inc' (A30)

Finally, we have

_1 9 Q)2(2-9 cf]
10 — 4[(cos4+cos4 + os1n4+s1n4 [inc
2
6
I1 = [B—Sinz] [inc (A-31)
-1 9 Q)2 (Q 9- 9f]
I2 - 4 (0084 cos4 + osin4 sini- [inc

' d

117

where

 

 

1
2
o = (8V2+Q2)
k .
a = (—n0);;sm9 = 1
2
Q _ k* L
— nokcose
knlL
v =
c059

A.3 The Derivation of —

dz2
d Q

From Eq. (A-3l), we have

_1_ <2- 2)2 (29,2)?
[2-—4[(cos4 cos4 + osm4 sm4 [inc

2 1 /2 dlz
where o = 8V2 + Q . Let line = 1, therefore — can be represented as follows

dQ

Z-l-QZ = %[(COS§_ cosgxf1 + ism?) + (gsing— singxésing + sz —}1cos%)]

 

118

Since 6 includes parameter Q, we need take a derivative of o with Q. Then

_1
3% = (mg) 2g = 2

. do
W1th Zi—Q- , f1 and f2 become

f - icosg - (-1)sin(Egg - (—1
I'dQ 4' 4 4dQ‘ 4

_ d( _1. <3)- 412 9_LQ- 9-2(1 gal-9)
fz—EO s1n4 —o 4Ccos4 (52081“4—0'2 cos osm

dl
Substituting f1 and f2 to T5 , we have

:fl—Z 5F (cosg— cosQX sing-95in?) + (Qsing— sing)

dQ 4 4 4 4 o o
2
0 Q 1 o 1 o 1 Q
{ San+62(4C084_osm4 —4cosz]]

_12-9_-2)(_l) 91-9 Q_(l gal-9)
-2(Gsm4 81114 I: 4 COS4+GSln4+ 2 40084 081114

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