Acousto-optic collinear diffraction of arbitrary polarized light.

Abstract

Acousto-optic collinear diffraction of light is a type of light scattering interesting as a physical phenomenon and important from practical point of view. This type of diffraction is used in tunable acousto-optic filters. In this case, the incident light has to be linearly polarized along one of the acousto-optic crystal anisotropy axes. If the incident light is not primordially polarized, a polarizer should be disposed at the input of the acousto-optic cell. Unfortunately, this leads to loss of half the light power. The present work is devoted to theoretical and experimental investigation of collinear acousto-optic interaction peculiarities that arise when the incident optical beam is not polarized. It is shown that the output diffraction spectrum in the general case contains four components which have different polarization and frequency. Beating of these components leads to modulation of light intensity passed through the output analyzer. In this work, dependences of modulation components amplitudes on frequency and power of the acoustic wave are examined for different polarizer and analyzer orientations.

Full text

Page 1

Acousto-optic collinear diffraction of arbitrary polarized
light
S. Mantsevich and V. I. Balakshy
Dept. of Physics, M.V.Lomonosov Moscow State Univ., Vorobyevy Gory, Bldg. 1, 119991
Moscow, Russian Federation
manboxx@mail.ru
Acoustics 08 Paris
833

Page 2

Collinear acousto-optic diffraction of arbitrary polarized light is studied theoretically and experimentally. It is
shown that the diffraction spectrum at the output of a collinear acousto-optic cell contains in the general case
four optical components which have different polarization and frequency. The frequency shift is caused by the
Doppler effect. Beating of these four components leads to modulation of light intensity passed through the
output analyzer. In this work, the amplitudes of the modulation components are calculated as functions of
frequency and power of the acoustic wave for different polarizer and analyzer orientations. The dependences of
the optical beam intensity on the analyzer orientation are examined experimentally for different values of the
acoustic power.
1
Introduction
In practical use of collinear acousto-optic (AO) interaction
a reasonable question of choosing the incident light beam
polarization arises. The choice depends on the type of the
medium where the light beam and the acoustic wave
propagate. In other words, the type of the material which
the AO cell is fabricated from should be taken into account.
In modern acousto-optics, solid materials such as crystals
and glasses are mainly used. In these materials, the axes of
anisotropy which define the polarization of optical
eigenmodes either are predetermined by crystal symmetry
or appear under the action of ultrasound excited in the
medium. However, in any case the magnitude of the AO
effect strongly depends on the incident light polarization
[1,2]. Therefore, the correct choice of the input light
polarization is very important in any AO experiment. Usual
recommendations amount to the following: in an
anisotropic medium the incident radiation has to have the
polarization of one of the medium eigenmodes, whereas in
an isotropic medium the polarization vector has to be
directed along one of the sound-induced anisotropy axes.
However, AO interaction of an arbitrary polarized light is
undoubtedly of theoretical and practical interest as well.
Analyzing this problem, the authors [3,4] have shown that
in the Raman-Nath regime of diffraction the rotation of the
polarization plane can occur through an angle depending on
the acoustic power. A similar result has been obtained in [5]
with respect to the 1st order of the Bragg diffraction. As for
the intermediate regime of diffraction which corresponds
better to a real situation, the situation is much more
complicated. In this case the additional phase shift appears
in all diffraction orders [6-8] which has to be taken into
consideration at the analysis of the output light polarization.
Due to this effect, the linearly polarized optical beam
becomes elliptically polarized [9-11]. Changing the power
or the frequency of the acoustic wave, one can control the
light polarization.
In all papers mentioned only quasi-orthogonal geometry of
AO interaction was examined. The question about the
influence of light polarization on collinear diffraction
characteristics is open until now. The given work presents
results of such an investigation. The calculations have been
made in the plane-wave approximation. Preliminary
experiments are fulfilled with an AO collinear cell made of
a calcium molybdate (CaMoO4) single crystal.
2
Basic relationships
The distinguishing peculiarity of the collinear AO
interaction is that the incident and diffracted optical beams
and the acoustic wave propagate in an anisotropic medium
along the same direction. The principle scheme for
realization of collinear diffraction is shown in Fig. 1
[12,13]. In the case of a CaMoO4 cell, an acoustic wave
excited by a piezoelectric transducer propagates first along
the Z crystallographic axis and then, after reflection from
the input optical face of the cell, is transformed into a shear
mode propagating along the X axis. The regime of traveling
acoustic wave is provided by an acoustic absorber placed at
the output end of the cell. A laser beam passes through the
cell near the X axis and diffracts in the acoustic field,
changing its polarization to the orthogonal one. At the
conventional applications of the collinear cell as a spectral
filter, the polarizer is oriented in such a manner that the
input radiation has ordinary or extraordinary polarization,
while the analyzer is crossed with the polarizer. Due to this
geometry, the diffracted radiation is separated from the
incident one. However, if the incident light is not polarized,
half the light power is lost in this process. In this work, we
examine the case of arbitrary polarization of the incident
light.
Fig.1. Principle scheme of collinear AO interaction
Let us suppose that the incident radiation is linearly
polarized at the angle α with respect to the Y axis (Fig. 2).
Entering the crystal, the optical wave
Y
i E and
clear that in the general case these two waves are not equal
in amplitude. However, if the incident light is nor polarized
at all, these components are equal to each other as at
o
45
=α
. Thus, our consideration includes the cases of
linearly polarized and non-polarized light.
Since the phase matching condition is fulfilled for the both
components equally, they diffract in the acoustic field
independently from each other. In this case the ordinary
wave
i E is split into two
waves
Z
i E polarized along the Y and Z axes. It is
Y
i E diffracts into +1st order, forming the waves
Y
E0
Acoustics 08 Paris
834

Page 3

(zero order) and
Z
E1 (+1st order), while the extraordinary
i E diffracts into –1st order, originating the waves
E0 (zero order) and
E1
Z
E0 have the same frequency ω as the incident light,
whereas the waves
E1 and
E1
have shifted frequencies
Ω+ω
wave
Z
Z
Y
− (–1st order). The waves
Y
E0 and
ZY
−, due to the Doppler effect,
and
Ω−ω
respectively.

Fig. 2. Polarization of incident and diffracted waves
The analyzer oriented at the angle β let pass only a part of
every component. Thus, at the system output the four
components can be written as
⎥⎦
⎤
⎢⎣
⎡
⎟⎠
⎞
⎜⎝
⎛
+−
⎟⎠
⎞
⎜⎝
⎛
Κ
2
π
−
Κ
2
==
2
expsinc
2
coscoscoscos
01
R
lktj
R
jEEE
oi
Y
d
ωβαβ
, (1)
(
ω
)
⎥⎦
⎤
⎢⎣
⎡
⎟⎠
⎞
⎜⎝
⎛
−−Ω+
Κ
2
π
−==
2
expsinc
β
sin cos
2
sin
12
R
lktj
A
EEE
ei
Z
d
αβ
, (2)
⎥⎦
⎤
⎢⎣
⎡
⎟⎠
⎞
⎜⎝
⎛
−−ω
⎟⎠
⎞
⎜⎝
⎛
π
Κ
2
+
Κ
2
βα=β=
2
expsinc
2
cossinsinsin
03
R
lktj
R
jEEE
ei
Z
d
, (3)
()
⎥⎦
⎤
⎢⎣
⎡
⎟⎠
⎞
⎜⎝
⎛
+−Ω−ω
π
Κ
2
βα=β=
−
2
expsinccossin
2
cos
14
R
lktj
A
EEE
oi
Y
d
, (4)
where A is the Raman-Nath parameter proportional to the
acoustic wave amplitude, R is the dimensionless phase
A +=Κ
propagation constants for the ordinary and extraordinary
optical waves accordingly, l is the AO interaction length.
The phase mismatch R depends on the optical wavelength
λ and the acoustic frequency f:
−
−π=
mismatch [1],
22
R
,
o k and
e k are the

()
0
2
2ff
V
l
nn
V
f
lR
oe
−
π
=
⎟
⎠
⎞
⎜
⎝
⎛
λ
(5)
where V is the ultrasound velocity,
πλ=
2
ee
kn
are the refractive indices,
frequency of collinear phase matching.
At the system output, these four waves interfere with each
other. Beatings of shifted (2),(4) and unshifted (1),(3)
components result in appearing intensity light modulation
with the acoustic frequency Ω, while the beatings of
differently shifted components (2) and (4) give intensity
modulation with the frequency Ω
πλ
0f is the
=
2
oo
kn
and
2
. Therefore, the output
light intensity contains three components and can be written
in the form
(
110
cos
ϕ+Ω+=
tIIEI
i
These components can be separated from each other and
measured in the experiment.

)()
[]
22
2
2cos
ϕ+Ω+
tI
(6)
3
Results of calculations
Below results of computations of the normalized intensities
0 I , 1I and
The calculations have been fulfilled for a fixed polarization
of the incident light (
45
=α
) and a discrete set of analyzer
orientations (
,0o
=β
2 .11
,
5 .22
2 I are presented for different values A and R.
o
o
o
,
o
45 ).
Fig. 3 demonstrates the dependence of
the Raman-Nath parameter (in fact, on the acoustic wave
amplitude) at the frequency of phase matching
0
=
R
). Firm curves correspond to the case when the
analyzer lets pass the radiation polarized along the Y axis. It
is seen that the constant component is always equal to 0.5,
the second harmonic is absent completely and the first
harmonic changes sinusoidally, reaching maximum value
0.5 at the points
, 2
π=
A23π
points the light intensity varies harmonically with the
frequency Ω from zero to the incident light intensity
The AO cell produces 100% modulation of the optical
beam without any light losses. However, the acoustic power
required in this case is 4 times less than in the conventional
variant of collinear diffraction, when the incident light
polarization is chosen along the proper axes of the crystal.
Another interesting variant takes place at
lines). In this case, the constant component changes from 1
to 0.5, the first harmonic is absent and the second harmonic
attains extremum at the point
geometry one can also obtain 100% modulation without
light losses, but at the frequency Ω
Fig. 4 displays mismatch characteristics obtained at fixed
values
2
π=
A
. As mentioned above, the mismatch R can
be varied in experiments by means of changing either the
acoustic frequency or the optical wavelength. In the latter
case, the obtained curves may be considered as
transmission functions of the collinear AO filter [1,2].
The case
0
=β
is of most interest. Here the constant
component does not depend on the mismatch R. The second
harmonic is absent at all. Thus, only registering the first
harmonic, one can perform the spectral analysis of optical
radiation at the given disposition of the polarizers. The
spectral transmission function differs from the sinc2-
function that is typical for the conventional collinear filter:
in its central part there is rather a broad area with a
practically constant transmission coefficient. At the point of
phase matching (R = 0) the output beam is 100%-modulated
without any optical losses. As mentioned above, the
required acoustic power is 4 times less; this peculiarity
should be considered as an important advantage. However,
the variant discussed has also a disadvantage – too large
side lobes of the transmission function in comparison to the
situation when analyzer and polarizer are crossed
(conventional filter). The plots in Fig. 4c are calculated for
0 I , 1I and
2 I on
0f (when
, … This means that at these
2
i E .
o
45
=β
(chain
π=
A
. Thus, at this
2
.
o
Acoustics 08 Paris
835

Page 4

2
π=
A
higher. In the case
; in the case
π
o
=
45
A
=
the values
, the first harmonic is absent,
2I are two times
β


(a)
(b)
(b)
Fig. 3. Normalized intensities
as functions of Raman-Nath parameter at
0 I (a), 1I (b) and
2I (c)
0
.
=
R
(a)
(b)
(c)
Fig. 4. Normalized intensities
as functions of mismatch parameter at
0 I (a), 1I (b) and
2I (c)
2
.
π=
A
however the second harmonic can be used effectively for
optical signal filtration. The transmission function is the
same as for the conventional filter, but one can obtain a
gain of 2 times in output optical intensity when the incident
radiation is not polarized because of no losses in light.
Acoustics 08 Paris
836

Page 5

4
Experimental results
An experiment was carried out to verify the theoretical
results. In the experiment we used an AO cell fabricated of
a CaMoO4 single crystal. An optical radiation of the He-Ne
laser with the wavelength
λ
shear acoustic wave propagates along the X axis of the
crystal. The AO interaction length l was about 4 cm. The
experiment was executed at the acoustic frequency
6 .46f
MHz which is the frequency of the collinear
phase matching for calcium molybdate. The input
polarization was chosen at the angle
axes. In the experiment, we measured the dependence of
0 I and
the angle β in the range from zero to 180 degrees. The
measurements were carried out for two values of acoustic
power which corresponded to the Raman-Nath parameter
magnitudes
4 . 1
=
A
and
=
A
dependences for the constant component
633
=
nm together with a
0=
o
45
=α
to the Y and Z
1I components of the diffracted optical beam on
8 . 1
. Fig. 5a presents these
0 I . It is seen that
0 I changes with the angle β according to the cosine law
and reaches zero at the points
...4 3 , 4 ππ=β
. The




Fig. 5. The constant component
as functions of the analyzer orientation
0 I and the 1st harmonic 1I

maximum value
experimental curves, one can mark a good agreement
between theory and experiment. Analogous curves for the
first harmonic amplitude 1I are represented in Fig. 5b.
0 I is 0.5. Comparing the theoretical and
5 Conclusion
The paper presents the first analysis of the polarization
effects that appear at the collinear AO interaction in the
case when the incident optical beam has an arbitrary
polarization. Entering the AO cell, the optical wave is split
into two components which diffract independently, forming
four components of the zero, +1st and –1st orders. Because
of the Doppler effect the components of the +1st and –1st
orders have shifted frequencies. Beatings of all the
components lead to intensity modulation of the light beam
at the output of the analyzer. It should be noticed that this is
the only case of diffracted light modulation when the light
is scattered by the traveling acoustic wave. Depending on
the analyzer orientation, one can get 100% modulation at
the frequency of ultrasound or at the doubled frequency.
Acknowledgments
This work has been supported in part by the Russian
Foundation for Basic Research, grants № 06-07-89309 and
№ 08-07-00498.
References
[1] V.I. Balakshy, V.N. Parygin, L.E. Chirkov, “Physical
principles of acousto-optics”, Moscow: Radio i Sviaz,
1985
[2] J.P. Xu, R. Stroud, “Acousto-optic devices: Principles,
design and applications”, N.Y.: Wiley & Sons Inc.,
1992
[3] A. Alippi, “Half-wave plate behaviour of ultrasonic
waves light modulators”, Opt. Commun., 8, 397-400
(1973)
[4] H. Eklund, A. Roos, S.T. Eng, “Rotation of laser beam
polarization in acousto-optic devices”, Opt. Quant. El.,
7, 73-79 (1975)
[5] S.V. Bogdanov, “Polarization of light diffracted on
elastic oscillations of the grating, Optics &
Spectroscopy, 49, 146-150 (1980)
[6] V.I. Balakshy, “Additional phase shift effect in
acoustooptic interaction and its applications”, Proc.
SPIE, 1844, 178-186 (1992)
[7] V.I. Balakshy, T.G. Kulish, “High orders of light
diffraction by ultrasound in the intermediate regime of
acoustooptic interaction. I. Theoretical consideration”,
Acustica - Acta Acustica, 84, 830-836 (1998)
[8] V.I. Balakshy,
V.Y. Molchanov, “High orders of light diffraction by
ultrasound in the intermediate regime of acoustooptic
I.V. Krylov, T.G. Kulish,
Acoustics 08 Paris
837