Download Acoustooptic interaction of two light beams in a paratellurite crystal

ISSN 1064-2269, Journal of Communications Technology and Electronics, 2007, Vol. 52, No. 6, pp. 678–683. © Pleiades Publishing, Inc., 2007.
Original Russian Text © V.B. Voloshinov, K.B. Yushkov, 2007, published in Radiotekhnika i Elektronika, 2007, Vol. 52, No. 6, pp. 727–733.
RADIO PHENOMENA
IN SOLIDS AND PLASMA
Acoustooptic Interaction of Two Light Beams
in a Paratellurite Crystal
V. B. Voloshinov and K. B. Yushkov
Received October 17, 2005
Abstract—Double acoustooptic diffraction of light in a paratellurite crystal is investigated experimentally and
theoretically. The anisotropic Bragg interaction with three diffraction maxima is considered in the case when
two monochromatic light beams are incident on the crystal simultaneously. For each beam, double diffraction
is realized, so that the propagation direction of one of the beams coincides with the propagation direction of the
second-order diffracted ray for the other beam. It is shown that the diffraction pattern consists of three diffraction maxima: two of these are formed by the zero- and second-order diffracted rays, while the third maxima is
formed by the first-order diffracted rays. The possibility of summation of the radiation intensities at the diffraction maxima is analyzed.
PACS numbers: 43.35.Sx, 78.20.Hp
DOI: 10.1134/S1064226907060113
INTRODUCTION
Light diffraction by ultrasound in crystals is widely
employed in optoelectronics to control luminous fluxes
[1–4]. Paratellurite (TeO2) crystals, which exhibit an
extremely high acoustooptic (AO) quality are applied in
AO devices [1–5]. This material allows realization of
anisotropic diffraction characterized by the rotation of
the polarization plane of a diffraction wave with respect
to the polarization of incident light [1–4]. One of the
characteristic features of the AO interaction in birefringent crystals is multiple diffraction of light by ultrasound [1, 2, 6–14], in particular, the double Bragg interaction. In the case of double diffraction, the conditions
for the Bragg synchronism are fulfilled at a fixed ultrasound frequency for light scattering from the zero diffraction order into the first- and second-order diffraction maxima simultaneously [1, 2, 6, 7].
Diffraction modes with several light rays at the outputs of AO cells have recently attracted considerable
interest of researchers [10–20]. In order to realize such
modes, electric signals with specially chosen amplitudes, frequencies, and phases are applied to a piezoelectric transducer. In addition, an important problem
in acoustooptics is coherent summation of the amplitudes of light fields, which provides for combination of
several optical rays in a single total light beam [15–20].
In this study, we investigate a possible scheme of summation of the intensities of light rays. The scheme
involves double diffraction of light by ultrasound in
paratellurite [21]. In contrast to studies [15–18], here,
two monochromatic light rays with equal frequencies
and identically oriented polarization planes are incident
on a cell. Light diffraction by monochromatic sound is
realized. As a result of interaction, the light of six diffraction maxima is observed at the output of an AO
device. The maxima are combined in three light beams.
Each of the beams is formed by the rays of two diffraction maxima with identical polarization directions but
different frequencies and phases.
The propagation directions of incident light rays in
the plane of the AO interaction are chosen with allowance for the synchronism conditions. The interaction
develops in the ( 110 ) plane of a íÂé2 crystal when
the wave vector of a slow shear ultrasonic wave is oriented at the angle α = 10° to the [110] axis. In the considered case of double diffraction, the sound frequencies are much higher than the acoustic frequencies in
variants of the AO interaction that were investigated
earlier [15–20].
1. GENERAL SCHEME OF DIFFRACTION
OF TWO LIGHT RAYS INCIDENT
ON A CRYSTAL
The general schematic of diffraction and the propagation directions of incident and diffracted light beams
are displayed in Fig. 1. As is shown in the figure,
extraordinarily polarized light beam 1 is incident on a
crystal and transmitted through paratellurite. After
leaving the cell, the beam has amplitude C01. Simultaneously with the first beam, extraordinarily polarized
light beam 2 with amplitude C02 is incident on the crystal. The vector diagrams illustrating the photon–photon
interaction [1–4] are used to describe diffraction of two
beams. It is known that the length of the wave vector of
acoustic waves K = 2πf/V is determined by control-signal frequency f and sound velocity V, while the wave
number of light k = 2πn/ λ depends on optical wavelength λ and a crystal’s refractive index n.
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ACOUSTOOPTIC INTERACTION OF TWO LIGHT BEAMS
1
679
[110]
2πno/λ [001]
2
α
θB2
θB1
f*
C21, C02
C11, C–12
C01, C–22
k3
θd
K
k2
I3
I2
k1
I1
Fig. 1. Directions of rays in a paratellurite-crystal AO cell.
Figure 2 displays the vector diagram for the studied
case of double AO interaction in a paratellurite crystal.
K
2πno/λ
2πne/λ
It is seen from the figure that wave vector k 1 of light
beam 1 incident on ultrasound is oriented at Bragg
angle θB1 relative to the acoustic-wave front, while
Fig. 2. Vector diagram of the double anisotropic Bragg diffraction in paratellurite.
wave vector k 2 corresponding to the light scattered into
the first diffraction order is oriented at angle θd. For the
the length of the wave vector of light beam 2 coincides
first-order diffraction, the vector relationship k 1 + K =
k 2 is fulfilled and the beam with amplitude C11 corresponding to the positive first diffraction order is
observed at the exit from the crystal. Moreover, it is
seen from the vector diagram in Fig. 2 that, for the firstorder diffracted light beam, the synchronism condition
k 2 + K = k 3 (where k 3 is the wave vector of the second-order diffracted beam) is satisfied. Thus, in a crystal, light diffraction from the first into the second order
is possible [1, 6, 7]. As is seen from Fig. 1, secondorder diffracted light waves have amplitude C21. Since,
during each anisotropic diffraction transition, the polarization plane is rotated through an angle of 90°, the second- and zero-order diffracted radiation is extraordinarily polarized and the first-order diffracted light is characterized by the ordinary polarization. Note that, owing
to the Doppler shift, the frequencies of the first-and second-order diffracted light beams differ from the frequency of the incident light beams by quantities f and
2f, respectively.
In the considered variant of double diffraction, both
light beam 2 and 1 are guided into the crystal at the
Bragg angle. However, Bragg incidence angle θB2 of
the second beam is chosen such that the incident radiation propagates toward the second diffraction maximum, which is formed as a result of double diffraction
of the first beam. Thus, the wave vector of beam 2 has
the same orientation as wave vector k 3 . It is evident that
with the length of vector k 3 , because the ultrasonic frequency is much lower than the light frequency. Therefore, in the diagram, vector k 3 is common to both incident light beam 2 and the second-order diffracted beam
formed from beam 1.
It is obvious that, for the chosen propagation direction of light beam 2, the Bragg condition for scattering
into the negative first diffraction order, which is
described by the relationship k3 – k = k2, is fulfilled in
the crystal. Thus, when beam 2 is diffracted, a beam
that corresponds to the negative order of diffraction and
that has amplitude C–12 is formed at the exit from the
cell. This effect is illustrated in Fig. 1. The radiation of
the negative diffraction order can be scattered repeatedly into the minus second diffraction maximum. This
scattering process produces light with amplitude C–22
and is described by the vector relationship k 2 – K = k 1 .
The light frequencies at the maxima of the minus first
and minus second diffraction orders differ from the frequency of the incident beam by the quantities –f and
−2f, respectively.
The analysis of the general diffraction scheme and
of the vector diagram shows that six diffraction maxima
are formed at the exit from the crystal. These are combined into three light beams. The two extreme light
beams consist of the zero- and second-order diffracted
beams with amplitudes C01 and C–22 and amplitudes C21
and C02. As is seen from the vector diagram, all of these
beams are extraordinarily polarized. The ordinarily
polarized light beam at the center of the diffraction pat-
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VOLOSHINOV, YUSHKOV
θB, deg
25
ters operating in the presence of nonpolarized optical
radiation [23–25]. Anisotropic diffraction corresponding to point B is not employed in AO devices in the case
when a high-frequency acoustic wave propagates in
paratellurite far from the [110] axis [21]. The calculations have shown that the angle–frequency characteristics intersect at point B when λ = 633 nm and the ultrasonic frequency is f * = 188.4 MHz. In this situation,
when the incident radiation is ordinarily polarized, the
two branches of angle–frequency characteristics intersect.
Ä
15
2
5
1
Ç
–5
0
100
1
2
200
300
400
f, MHz
Fig. 3. Frequency dependences of the Bragg angles of light
incidence on an acoustic wave front for (1) ordinarily and
(2) extraordinarily polarized light beams. Experimental
data are shown as triangles and squares.
tern is formed by the first- and minus-first-order diffracted beams with amplitudes C11 and C–12.
2. DEPENDENCES OF THE BRAGG INCIDENCE
ANGLES ON THE ACOUSTIC FREQUENCY
Having analyzed the vector diagram in Fig. 2, we
can represent the acoustic frequency as a function of
light incidence angle f in the case of the extraordinary
optical polarization:
V
2
2
2
f ( θ B ) = --- ( n i sin θ B – n o – n i cos θ B ),
λ
(1)
where no is the medium’s refractive index for an ordinary wave and ni is the medium’s refractive index for an
extraordinary wave [1, 2, 7, 12–14]. Refractive index ni
depends on the direction of light propagation. In a similar manner, the Bragg frequency can be represented as
a function of the incidence angle of ordinarily polarized
light. Typical angle–frequency characteristics in paratellurite are depicted in Fig. 3. In the calculations based
on formula (1), the principal refractive indexes
observed in paratellurite at no = 2.26 and ne = 2.41 and
a typical of λ = 633 nm were used. The velocity of
shear acoustic waves was chosen to be V(10°) = 7.09 ×
104 cm/s.
It is seen from Fig. 3 that, for the considered geometry of interaction in a íÂé2 crystal, angle–frequency
dependences θB(f) intersect at points A and B. All possible intersection points in uniaxial crystals and the corresponding geometries of the AO interaction are classified in studies [14, 22]. The Bragg diffraction mode that
is observed in paratellurite and that corresponds to
point A has been investigated comprehensively and is
used in modulators [16, 17, 19, 20] and tunable AO fil-
3. INTENSITIES OF DIFFRACTED LIGHT
IN THE CASE OF DOUBLE SCATTERING
In order to analyze double scattering, we have found
the light intensities at the three diffraction maxima
shown in Fig. 1 for both positive and negative diffraction orders. In this study, we consider the case of phase
synchronism, which ensures the highest efficiency of
diffraction [1].
Let optical beam 1 be incident on a crystal. With
allowance for phase ϕ1 of the acoustic wave, the system
of differential equations for the complex amplitudes of
the zero C01(x), positive first (C11(x)), and positive second
(C21(x)) diffraction orders takes the following form [1, 7]:
dC 21
dC 01
q
q
- = --- C 11 exp ( iϕ 1 );
----------- = – --- C 11 exp ( – iϕ 1 ); ---------2
dx
2
dx
(2)
dC 11
q
----------- = --- [ C 01 exp ( iϕ 1 ) – C 21 exp ( – iϕ 1 ) ].
2
dx
The equalities C01(0) = 1 and C11(0) = C21(0) = 0 express
the initial conditions. In (2), q is the factor of AO coupling and x is the coordinate oriented perpendicularly to
the boundaries of the acoustic column. In a similar
manner, the system of equations describing the double
AO interaction for light beam 2 can be obtained:
dC 02
q
----------- = --- C –12 exp ( iϕ 2 );
2
dx
dC –22
q
------------ = – --- C –12 exp ( – iϕ 2 );
2
dx
(3)
dC –12
q
------------ = --- [ C –22 exp ( iϕ 2 ) – C 02 exp ( – iϕ 2 ) ].
2
dx
This system should be combined with the following initial conditions: C02(0) = 1 and C–12(0) = C–22(0) = 0. The
analysis has shown that a solution to system (2) at the
exit from the acoustic column at x = l can be represented in the form
ql
1
C 11 = ------- sin ------- exp ( iϕ i );
2
2
2 ql
= sin ---------- exp ( 2iϕ 1 );
2 2
2 ql
C 01 = cos ---------- ;
2 2
C 21
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ACOUSTOOPTIC INTERACTION OF TWO LIGHT BEAMS
while system (3) yields the solution
2 ql
C 02 = cos ---------- ;
2 2
C –22
1
ql
C –12 = – ------- sin ------- exp ( – iϕ 2 );
2
2
(5)
2 ql
= sin ---------- exp ( – 2iϕ 2 ) ,
2 2
which differs from (4) in phase factors.
The calculations have confirmed that, when only one
light beam is incident on an AO cell, the total energy of
incident light at the exit from the sound column can be
collected in the second diffraction maximum at ql =
2 π; i.e., C21C *21 = I21 = 1.0 and C–22 C *–22 = I–22 = 1.0.
No more than 50% of the light power incident on a
crystal can be concentrated in the first diffraction order.
When ql = π/ 2 , the intensity of the first diffraction
maximum is I11 = I–12 = 0.5, 25% of the incident light
flux is concentrated in the zero diffraction order, and
25% of the incident light flux is concentrated in the second diffraction order.
The analysis of solutions (4) and (5) has shown that
the light amplitudes at diffraction maxima depend on
sound-wave parameters q and ϕ and on length l of the
region of the light–sound interaction. The light amplitudes in the zero, first, and second diffraction orders
prove to be identical; however, the wave phases are different for the maxima with positive and negative numbers. When two light beams are simultaneously incident on an AO cell, the interaction in the acoustic column develops in the same spatial region. Therefore, in
relationships (2)–(5), the phases of the acoustic wave
can be assumed equal: ϕ1 = ϕ2 = ϕ. Generally, the optical waves of each of the two partial diffraction maxima
that, at the exit from the cell, form the first, second, and
third light beams have different amplitudes, phases, and
frequencies. This circumstance should be taken into
account in the calculation of the resulting amplitudes of
electromagnetic waves in the three beams leaving the
crystal. Evidently, the light amplitudes in all of these
beams are modulated with sound frequency f and the
intensities in these beams are modulated with the frequency 2f. If a low-frequency receiver records only the
energy characteristics of beams, beats with the frequency 2f are averaged. Then, the intensities of the
three beams leaving the crystal are described by the
expressions
2 ql
I 1 = I 3 = 0.5 ⎛ 1 + cos -------⎞ ;
⎝
2⎠
2 ql
I 2 = sin ------- .
2
(6)
Thus, when the sound amplitude has the optimum
value, i.e., when ql = π/ 2 , the light intensities at the
side maxima take their minimum values: I1 = I3 = 0.5. In
this situation, at the center maximum, which combines
the rays of the first diffraction orders, the summation of
the light-field amplitudes yields the intensity I2 = 1. The
following main conclusion can be drawn: The light
681
power equal to the sum of the powers of initial optical
beams cannot be collected at the center maximum,
because a half of the light power incident on the cell is
concentrated in two side beams.
4. EXPERIMENTAL INVESTIGATION
OF DIFFRACTION IN PARATELLURITE
We performed an experimental investigation on a
paratellurite AO cell where a slow shear acoustic wave
was excited by a piezoelectric transducer made from xcut lithium niobate. The calculation has shown that,
when the length of the piezoelectric transducer is l =
1.4 cm and the ultrasonic frequency is f = 180–
190 MHz, the sound divergence is 4 mrad. The acoustic-wave velocity is oriented at the angle α = 10° to the
[110] axis in the ( 110 ) plane. A helium–neon laser providing for linearly polarized emission at the wavelength
λ = 633 nm served as a light source. Before being incident on the crystal, the emission had been divided
into two identically polarized beams of equal intensities. The divergence of each of the incident beams
was equal to 3–4 mrad, a value that is comparable to
the divergence of an ultrasonic beam.
In the experiments, when each of the two incident
light beams was diffracted separately, the efficiency in
the second diffraction order reached 70–80%. Less than
20% of the incident radiation remained in the first diffraction order. The experiment showed that the most
effective double scattering was observed at the ultrasonic frequency f = 187.9 MHz. The experimental value
of the acoustic frequency proved to be close to the value
f* = 188.4 MHz, which had been predicted theoretically. The transducer’s electric voltage determined by
the condition ql = 2 π reached the value U = 6 V,
which corresponded to an ultrasonic power of 0.8 W.
The dots in Fig. 4 show the experimental values of double-diffraction efficiency I21 as a function of the ultrasonic frequency. It is seen that diffraction into the second order is observed over a narrow frequency range:
∆f = 0.4 MHz. This circumstance indicates the high
selectivity of the employed Bragg interaction mode.
Similar results were obtained for beam 2 incident on the
crystal.
In addition, we have experimentally studied the
multiple Bragg light scattering in the case when two
extraordinarily polarized light rays are incident on the
crystal. These rays were guided to the acoustic wave
front at the Bragg angles, and the ultrasonic frequency
was equal to the frequency of double scattering. As had
been predicted theoretically, when two light rays were
incident on the crystal, the diffraction pattern contained
two maxima of the zero and second orders and a combined diffraction maximum of the first order. At the
control-signal frequency f = 187.9 MHz, the diffractedlight intensities were measured at each of the three diffraction maxima as functions of the electric-signal voltage at the piezoelectric transducer (Fig. 5). Curve 4
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VOLOSHINOV, YUSHKOV
I21, arb. units
0.6
I, arb. units
2.1
4
2.0
1.9~
~
1.2
1
0.4
0.8
0.2
3
0.4
2
0
0
0
186
188
2
4
6
U, V
190 f, MHz
Fig. 4. Frequency dependence of the light intensity in the
second diffraction order.
shows the total intensity of all of the three beams at the
exit from the cell, i.e., illustrates the conservation of the
optical power in the AO device. The data presented in
Fig. 5 confirm that all the light at the exit from the AO
cell was actually concentrated only in three light beams
and that the diffraction pattern contained no other diffraction maxima.
The following conclusion can be drawn from the
measurement results presented in Fig. 5. There exist
two diffraction modes with different behavior of diffraction intensities I1, I2, and I3 (as functions of control
electric voltage U) in diffraction orders. These regions
correspond to low and high intensities of double diffraction. The experiment has shown that, at a low control voltage at the transducer (U < 2 V across a load of
50 Ω), the probability of a photon’s transition from the
first into the second diffraction order is small. In this
case, at the combined first-order diffraction maximum,
the energies of initial beams 1 and 2 are summed, I2 =
I11 + I21, and light-ray intensity I2 increases with the
sound power.
The optimum control voltage is U = 3 V. This value
corresponds to the condition ql ≅ 0.7π and agrees with
the theoretical prediction. At this control voltage, the
beam intensities in two extreme diffraction orders are
equal and take the minimum value: I1 ≅ I3 = 0.65 ± 0.03.
This value exceeds the theoretical quantity I1 = I3 = 0.5
only slightly. At the optimum signal voltage, the intensity I2 = 0.70 ± 0.03 has been recorded in the center
beam, while the theoretical prediction is I2 = 1. Thus,
we can conclude that the theory and experiment are in
qualitative agreement.
Evidently, one of the causes of the discrepancy
between the experimental and theoretical results is the
fact that optical and acoustic beams participating in diffraction are not plane waves. Therefore, all the incident
light could not be concentrated in the second diffraction
Fig. 5. Relative intensities I of diffracted light as functions
of electric-signal voltage U at a piezoelectric transducer.
Curves 1 and 3 correspond to the side diffraction maxima
with intensities I1 and I3, respectively; curve 2 corresponds
to the center maximum with intensity I2; and curve 4 corresponds to the sum of intensities I1 + I2 + I3. Experimental
data are shown as squares and circles.
order even when a single light beam was incident on
ultrasound. The experiment has additionally confirmed
that double diffraction of two light beams by monochromatic sound should be analyzed with allowance for
the amplitudes and phases of light waves. Obviously,
precisely because of the phase relationships, the light
fluxes of initial light beams could not be concentrated
experimentally in a single diffraction order with the
maximum efficiency. This conclusion fits the results
from [17, 19], where the effect of phase relationships
on the intensities of diffracted beams has been predicted.
It is seen from Fig. 5 that, over a wide range of control voltages (2 < U < 6 V), the diffraction efficiency in
all of the three diffraction orders varies from 60 to 70%.
This means that the optical power of incident light is
characterized by an approximately uniform distribution
among the three diffraction maxima. The experiment
has confirmed that, at a high control voltage (U > 4.5 V)
and ql > π, the intensities of light beams change only
slightly, while the ultrasound amplitude grows. This
circumstance means that the considered model of double AO interaction is not complete, because it disregards possible additional light-energy exchange
between the rays of all of the six partial rays at the exit
from an AO cell.
It has been found during the investigations that,
when incidence angles of light beams are fixed and the
acoustic frequency varies around the value f = f *, the
interaction between initial light beams 1 and 2 vanishes.
Thus, at the ultrasonic frequency f = 189.0 MHz, the
efficiency of double scattering is a small bounded quantity (I21 ≅ I–22 < 0.02) and the intensities of the first dif-
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ACOUSTOOPTIC INTERACTION OF TWO LIGHT BEAMS
fraction orders are substantial (I11 ≅ I–12 > 0.9). In this
situation, the light fluxes of incident beams are effectively summed in the first diffraction order. The experiment has shown that, when synchronism is lost and the
mode of double diffraction is disturbed, the first-order
diffracted light beams leaving the crystal are spaced by
an angle of approximately 4 mrad. This angle is comparable with the divergence angles of the light beams incident on the crystal. Therefore, the structure of the light
spot at the center of the diffraction pattern is complex
and composed of two spots that overlap only partly. Our
investigation has confirmed that the use of double diffraction makes it possible to concentrate a substantial
portion of the energy of two beams incident on an AO
cell in the center diffraction order.
CONCLUSIONS
When two light rays incident on a paratellurite-crystal AO cell are doubly diffracted in the crystal, there
exist interaction modes differing in the behavior of the
diffracted-light intensity as a function of the ultrasound
amplitude. At a small control voltage, the intensity of
the diffracted beam increases with the amplitude of the
control signal. The intensities of two light rays incident
on the crystal are summed at the common diffraction
maximum. At a substantial ultrasound power, the
energy is distributed approximately uniformly among
three light beams at the exit from the AO cell and the
intensities of the light rays are dependent only slightly
on the acoustic-wave power. In the considered case of
double AO interaction, two linearly polarized monochromatic light beams can be combined into a single
beam. However, the analysis has confirmed that this
combination of light beams is accompanied by a loss of
light power.
ACKNOWLEDGMENTS
The authors are grateful to V.V. Proklov for his interest in the problem and for helpful discussions of the
results.
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