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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

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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

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(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

cos coscoscos

01

R

lktj

R

jEEE

oi

Y

d

ωβαβ

, (1)

(

ω

)

⎥⎦

⎤

⎢⎣

⎡

⎟⎠

⎞

⎜⎝

⎛

−−Ω+

Κ

2

π

−==

2

expsinc

β

sincos

2

sin

12

R

lktj

A

EEE

ei

Z

d

αβ

, (2)

⎥⎦

⎤

⎢⎣

⎡

⎟⎠

⎞

⎜⎝

⎛

−−ω

⎟⎠

⎞

⎜⎝

⎛

π

Κ

2

+

Κ

2

βα=β=

2

expsinc

2

cossinsin sin

03

R

lktj

R

jEEE

ei

Z

d

, (3)

()

⎥⎦

⎤

⎢⎣

⎡

⎟⎠

⎞

⎜⎝

⎛

+−Ω−ω

π

Κ

2

βα=β=

−

2

expsinccos sin

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

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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.

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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.

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