1

Modern Acousto-Optics: Physics and Applications

Dr. Alexandre S. Shcherbakov

Image Science Group, Optics Department,

National Institute for Astrophysics, Optics & Electronics ( INAOE )

Part 1: Acousto-optical technique for spectrum analysis

of radio-wave and optical signals in astrophysics.

Part 2: Multi-wave coupled states in acousto-optics.

2

Part 1:

Acousto-optical technique

for multi-channel spectrum analysis

of ultra-high frequency radio-signals and

optical signals over all the visible range;

possible applications to the astrophysical needs.

In collaboration with: 1) Astrophysics Department of the INAOE,

2) Molecular Technology GmbH ( Berlin, Germany ),

3) Saint-Petersburg State Polytechnic University (St. Petersburg, Russia),

4) Institute for Applied Physics (Madrid, Spain).

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Outline

Introduction: Regimes of operation for the acousto-optic cells.

The Bragg acousto-optical interaction in anisotropic medium.

A one-phonon non-collinear acousto-optical interaction:

The number of resolvable spots. Estimations for a tellurium dioxide crystal.

A two-phonon non-collinear light scattering:

The number of resolvable spots. Estimations for a tellurium dioxide crystal.

Experimental data related to a multi-phonon light scattering in tellurium dioxide cell.

General schematic arrangement for prototype of an advanced acousto-optical spectrometer.

The first experimental data related to the spectrum analysis

of analogous high-frequency radio-wave signals within a one-phonon light scattering.

Collinear acousto-optical interaction and the opportunity of filtering analogous optical signals.

Potential spectral resolution of a calcium molybdate crystal based collinear filter.

Conclusive remarks.

4

Regimes of operating for the AO- cells

npsinsin 0p

1LQ2

1. Raman-Nath regime ( short L ),

in which the diffraction maxima

are given by

n2sin B

1LQ2

Scattering light by a thin dynamic

acoustic grating.

There are two limiting cases for light diffraction (scattering) regimes :

2. Bragg regime ( long L ),

in which the Bragg condition

is given by

where and

=V / f.

, ,

where .

Scattering light by a thick dynamic

acoustic grating.

The parameter Q is the Klein – Cook factor characterizing the regime.

5

12

1SPM2)cos(q

The Raman-Nath limit; Q << 1.

Conservation laws inherent in such interactions are

or

or 321 kkk

kp

321 E

Since in conventional acousto-optics, 1 , 3 10 14 Hz and 2 = 10 9 Hz, so

that / 10 –5 and Ephoton 10 -5 Ephonon.

Because c = 3 10 10 cm/s and V 3 10 5 cm/s, we yield

V / c 10 5, 1 / c / V, and kphoton Kphonon.

Plots for the light intensities in various orders of scattering

The Bragg limit; Q >> 1.

6

Realizing the parallel spectrum analysis by

a TeO2 acousto-optical cell in the Bragg regime

7

A one-phonon light scattering

1. Normal scattering 2. Optimized anomalous scattering

,Vn2

fsin 1

The comparison for TeO2 -crystal shows that generally: D f1 < D f0 .

,f

V1

Vn2

fsin

2

1

2

22

1,0

11,0

D

2

1

2

0

2nn D

D Vf0

2L0

)Lf(V 1

)Lf(V 0

1

2

1fL

Vn2f

D

L

n2V2f 0

0

D

Lx

Lx

2L0

8

A two-phonon acousto-optical interaction

General arrangement of optical beams and a two-phonon acousto-optical

interaction in a TeO2 single crystal

Kkk m1m

,nnnsin21

20

100

.nnV2f2

21

20

122

Conservation laws: m+1 = m + and ( m = 0,1,2 ).

A two-phonon Bragg scattering of light occurs when

9

9

General schematic arrangement of the advanced

prototype of acousto-optical spectrometer for the

Large Millimeter Telescope (GTM)

14

10

The light beam passing through glass prisms.

2 2 21

10

(n sin φ)[1 n sin (α δ)]dB ,

d n cos φ cos (α δ)

sin φδ arcsin .

n

A 4-prism beam expander

A multi-prism beam expander

.)B(Bm

1m

;mm1d0 ;mm35D

;5.1n .30o

11

The flatness can be estimated by about 25% ;

The experimentally estimated average transmittance is about 70%.

Experimental plot for a one-dimensionally expanded beam profile.

Transmittance of a 4-prism BK-7 glass beam expander

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Optical aperture of a cell: D = 3.5 cm ; Length of interaction: L = 1.0 cm ;

Slow shear elastic wave velocity: V = 0.660 .10 5 cm/s ;

Operating light wavelength: = 633 nm ; Refractive indices: no = 2.30 , ne = 2.46 ;

Central acoustic frequency: f0 = 75 MHz ; Frequency bandwidth: D f = 45 MHz ;

Ideal theoretical frequency resolution: f = V/D = 18.8 KHz ;

The estimated maximum number of resolvable spots, i.e.

the number of parallel frequency channels for spectrum analysis: N = 2360.

The Bragg tellurium dioxide

acousto-optic cell

Efficiency and frequency bandwidth of a TeO2-cell

13

Intensity of the scattered light versus

the applied electric power.

This diagram represents

the obtained degree of

linearity in a dependence

between the electric power

and the irradiance of light.

Experimental light scattering efficiency of a TeO2 - cell

The observed variations are

potentially conditioned by

the interplay between an

electronic impedance of the

matching circuits and an

acoustic impedance of the

crystalline piezo-electric

transducer.

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Experimental frequency bandwidth of a TeO2 - cell

MHz65f D

15

Thus, at the current step of our researches an advanced prototype of

the acousto-optic spectrometer can be characterized by:

1. Total frequency bandwidth: D f = 65 MHz ;

2. Frequency resolution: f = 42 KHz ;

3. The number of resolvable spots or

the number of parallel frequency channels for spectrum analysis: N = 1550.

An advanced prototype of the acousto-optic spectrometer.

The resulting resolution characterization

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Scheme of the collinear acousto-optical interaction

in a uniaxial crystal

The transmitted

light The incident

light

z

The acoustic (elastic) wave beam

x

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Collinear acousto-optical interaction

in terms of the wave vectors

,kKk 10

11 ,k

10 ,kp

,E

00 ,k

,K

Acoustic wave is slow v << c , so that , but10 ,10 kKk

In the special case of mismatched wave numbers, when detunings

and are connected with the slow wave and , ,

we yield and .

Thus, the total detuning can be described via only a mismatch related

to just the wave vectors

K

D11 ,k

00 ,k

KD

D KK D D

1,0kKK D 1,0D

,K

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Schematic arrangement for the experiments

with filtering optical signals by just collinear AO-cell

Collinearacousto-optical cell

U H Fgenerator

Oscilloscope Band-pass filter

Pulsegenerator

Polarizer 1 Polarizer 2 Continuous-wave laser

Photodetector

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Physical parameters of the collinear acousto-optical cell used

currently in experiments for filtering analogue optical signals:

.A7.5nm569.0Lf

V o

0

00

This cell is successfully exploited during our experiments in the INAOE.

Tetragonal single crystal CaMoO4 (along the x - axis):

L = 4.5 cm, T = 15 ms, V = 2.95 105 cm/s, D = 0.4 – 4.0 mm,

M2 = 6.16 10 -16 s3/ kg, and f0 = 61.3 MHz for 0 = 532 nm.

In this particular case of the CaMoO4 - crystal, one can expect

the following tuning dependence and spectral resolution:

,f

VnD

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Conclusive remarksThe current situation looks like:

1a. At the moment, an advanced prototype of an acousto-optic spectrometer has been created

in INAOE using a TeO2-crystal cell, which operates in the frequency range of 60 MHz

with the frequency bandwidth of 65 MHz and frequency resolution of 42 KHz.

1b. Nevertheless, this prototype of an acousto-optic spectrometer is under construction and

testing in INAOE . This prototype is mainly oriented on applying multi-phonon regimes

of light scattering to improve the frequency resolution ( CONACyT project # 61237 ).

2. The second step can be done with involving a LiNbO3-crystal cell operating in the frequency

range of 1.0 - 1.5 GHz with the frequency resolution of ~ 200 KHz ( under consideration ).

3. The third step can be developed in the frequency range of 3 - 4 GHz with the frequency

resolution of ~ 1.0 MHz ( for the needs of cosmology; currently exists as a proposal ).

In parallel, one can say:

a. At the moment, the other prototype of an acousto-optic spectrometer is under construction and

testing in INAOE . This prototype is based on the specifically designed a TeO2-crystal cell

and is mainly oriented on applying multi-phonon regimes of light scattering to improve the frequency

resolution

b. Finally, the researches directed on creating a prototype of an optical filter based on

a CaMoO4-crystal collinear acousto-optic cell, which is potentially capable of overlapping

both visible and near infrared optical ranges, have been already initiated in the INAOE.

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Part 2:

Solitary multi-wave coupled states

in acousto-optics

Alexander S. Shcherbakov

Image Science Group, Department of Optics,

National Institute for Astrophysics, Optics & Electronics,

Puebla, Mexico.

In collaboration with: Molecular Technology GmbH, Berlin, Germany and

Saint-Petersburg State Polytechnic University, St.Petersburg, Russia.

Financial support: CONACyT, project # 61237, Mexico;

and Molecular Technology GmbH, Berlin, Germany.

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Abstract and general survey.

Physical background and formulation of theory.

Analytical model and its simplifications.

Effect of the phase mismatches.

A one-phonon light scattering.

A two-phonon light scattering.

A three-phonon light scattering.

Revealing and observing five-wave multi-component

non-collinear acousto-optical coupled states.

Revealing and observing three-wave dissipative

collinear acousto-optical coupled states.

Conclusion.

Outlines

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The physical principles of realizing the Bragg regime of one-, two-

and three-fold scattering of light by acoustic phonons in optically

anisotropic crystals in specially elaborated case are considered.

The exact and closed analytical models with slightly mismatched

wave numbers for describing such regimes are developed. Both the

analysis and the computer simulations are oriented to revealing the

specific acousto-optical solitary waves in the form of multi-component

coupled states, which can appear within a multi-phonon Bragg light

scattering in optically birefringent media.

These solitary waves have been observed and identified in the

core of various experiments with a multi-phonon Bragg light scattering

in uniaxial crystals.

Abstract

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Vector diagrams for the wave vectors of light

and acoustic wave in a TeO2 – single crystal

When these cross-points are lying equidistantly, the two- or three-fold

scattering of light can be provided by only one harmonic acoustic wave,

as shown here for a tellurium dioxide crystal. By this is meant that under

certain conditions, i.e. at a set of the angles of light incidence on

selected crystal cut and at the fixed frequency of the acoustic wave, one

can observe the Bragg scattering of light caused by sequential

participation of one, two or even three identical phonons in a crystal.

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A three-phonon light scattering:

The conservation laws and angles of scattering

,mp1p ,Kkk mp1p

.3and,2,1,0p

A two-phonon light scattering:

;nnV2f221

20

122

;nn2Vf221

20

133

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Dielectric permittivity and boundary conditions

Under action of the elastic wave, the dielectric permittivity takes the form

.tyKsint,y mm10

Generally, in crystalline materials, both 0 and 1 are tensors.

We assume that area of propagation for the elastic wave is bounded by

two planes x = 0 and x = L, and that a triplet or a quartet of the plane

electromagnetic waves with various states of polarization

pppppp

3or2

0p

pin tsinykcosxkiexpAE

strike the plane x = 0 at the angles P ( p = 0, 1, 2, 3 ) to the x-axis.

,p0p .ckk121

0ppp

Here:

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A one-fold or one-phonon regime of light

scattering in an acousto-optical modulator

The other regime of light diffraction occurs with a large L. In this case,

the dynamic acoustic grating is rather thick, so during the analysis of

diffraction one has to take account of the phase relations between waves

in different orders. Such a regime can be realized only when the angle

of light incidence on a thick acoustic grating meets the Bragg angular

condition and Q = L/L2 >> 1. Usually, the Bragg regime

includes the incident and scattered light modes and the acoustic mode.

B

B

L n2sin B

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A one-phonon light scattering

With p = ( 0, 1 ) , the set of Eqs.(8) gives

,xi2expxCq

dx

xdC011

0

.xi2expxCqdx

xdC000

1

Using the boundary conditions C0 ( x = 0 ) = 1, C1 ( x = 0 ) = 0 and the

notation q2 = q0 q1, one can find

20

22

20

2

22

1 qxsinq

qС

1

2

1fL

Vn2f

9

A two-fold or two-phonon light scattering

With p = ( 0, 1, 2 ) , equations can be rewritten as

,xi2expxCq

dx

xdC01

0

.xi2expxCqxi2expxCq

dx

xdC1200

1

,xi2expxCq

dx

xdC11

2

There are two different frequency mismatches 0 and 1. We assume

precise angular alignment and extend 0 and 1 into a series in powers

of only the frequency detuning f – f2 for the current frequency f relative

to the central frequency f2 for a two-phonon light scattering.

In the first order approximation, we obtain from the vector diagram:

.00

10

A two-phonon light scattering;

the frequency bandwidth

The frequency detuning is

The first unity-level maximum for |C2|2 can be reached at

One can estimate the bandwidth of a two-phonon light

scattering as:

.2xq

.f2Vnff1

22

02

.f4

1f 12

11

A three-phonon light scattering

With p = ( 0, 1, 2, 3 ) , equations can be rewritten as

,xi2expCqdx

dC01a

0

,xi2expCqxi2expCqdx

dC12n00a

1

There are three different frequency mismatches 0 , 1, and 2 . We

assume a precise angular alignment and extend these mismatches into

a series in powers of only the frequency detuning f – f3 for the current

frequency f relative to the central frequency f3 for a three-phonon light

scattering. In the first approximation, we find from vector diagram that

3321

00 fffVn2

,xi2expCqxi2expCqxd

Cd23a11n

2

.xi2expCqxd

Cd22a

3

12

A three-phonon light scattering; simulations

and estimations of the frequency bandwidth

The frequency detuning is

The first unity-level maximum for |C3|2 can be reached at

Taking 2 0L = / 6 from the figure at x = L , one can estimate the

bandwidth of a two-phonon light scattering as:

.xqn

.fVn2ff1

32

003

.f6

1f 13

13

Originating five-wave Bragg non-collinear

weakly-coupled acousto-optical states.

The solutions to the equations governing the three-phonon interaction:

,2

xqPcosSq2

2

xqScosPq2

q412

IxqC n22n22

2n0

,2

xqPsinPq2S

2

xqSsinSq2P

)q41(2q2

I

xqC

n22n22

2

n1

,2

xqPcos

2

xqScos

q41

IqxqC

nn

2n2

.2

xqPsinS

2

xqSsinP

)q41(2

IxqC nn

2n3

14

Having two sets of the localization conditions, which are given by

where k and m are the whole numbers; m > k, because P > S.

Putting, k ≠ 0 and using the parameters P and S, one may obtain

Pm22Sk22Lqn 1) ,

correlating k and m and the parameter q.

2222q41q21kq41q21m ,

Graphical interpretation of the 1-st set of localization conditions.

The localization conditions

(19)

15

The last two figures show various interplays between the whole number m

and the parameter q for a set of the whole numbers k.

Graphical interpretation of the 2-nd set of localization conditions.

,P/)m2(2S/)k2(2Lqn 2)

In this case, the inequality m > k as valid, but now k = 0 is acceptable,

and the interralation between k and m and the parameter q is given by:

2222q41q21k21q41q21m21 .

The localization conditions

(20)

16

Spatial-frequency distributions for a quartet of optical

components in a multi-pulse five-wave weakly coupled

state with q = 4.5 and

The distributions are completely locked with 0 = 0.

an eight-pulse component in

the zero order of scattering

a nine-pulse component in

the first order of scattering

a nine-pulse component in

the second order of scattering

an eight-pulse component in

the third order of scattering

.2Lqn

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Experimental results:

the tellurium dioxide acousto-optical cell

1. A Te02 -crystal has rather dispersive refractive index: n0 = 2.26 at

633 nm, n0 = 2.33 at 488 nm, and n0 = 2.35 at 442 nm. The

ultrasound velocity equals v = 0.6 105 cm/s for the slow shear acoustic

mode running exactly along the [110]-axis with the displacement vector

directed along [110]-axis. The figure of acousto-optical merit for this

shear mode wave in a TeO2-crystal is M2 = 1.2 10-15 s3/g , which is the

highest one for solid-state acousto-optical materials in the visible range

known up to now.

2. At first, one has to check the realization of just Bragg regime for light

scattering in the chosen cell. In such a regime, the Klein-Cook

parameter Q should exceed unity. Operating at the light-blue optical

wavelength 488 nm and at the lowest expected acoustic wave

frequency 40 MHz with L = 1 cm one can calculate Q = 10, which

confirms the Bragg character of light scattering in the regime selected

within the visible range of light spectrum.

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Scheme of the collinear acousto-optical interaction

in a uniaxial crystal

The transmitted

lightThe incident

light

z

The acoustic (elastic) wave

x

en

on

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A triplet of nonlinear differential equations describing the regime of

a weak coupling with linear acoustic losses is given by

,Ut

U

V

1

x

U

,xi2expUCq

t

C

c

1

x

C11

00

,xi2expUCqt

C

c

1

x

C00

11

where describes losses of the non-optical wave.

Co-directional collinear acousto-optical interaction

in the presence of linear acoustic losses.

The exact solutions to this equations in the case of a weak coupling are given by

;0GxGcosxС2

22

2

22

22

0

;0GxGsinq

qxС

2

22

2

1

02

1

(21)

(22)

(23).Uqq2

102

20

the localization process may has 3 stages:

1) The localizing non-optical pulse is incoming through the facet x = 0 ;

2) The pulse is passing along the crystal ;

3) The pulse is issuing through the output facet x = L .

Such a process can be illustrated for the scattered light waves under conditions:

.)Vc()Vc(xTor0,0

;)Vc()Vc(xT)Vc()Vc(x,xxGLG

;)Vc()Vc(xT,xxGxG

;T0,0GxG

0

0

If the spatial length x 0 of the non-optical pulse is much shorter than L , i.e. whenVLTVxT 00 .

The localization condition can be written as

N)0(G)x(G , where: N = 1, 2, …

The localization conditions

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a) b)

c) d)Intensity of the scattered light components vs. and L with = 2 and = 0.05.

a) = 1.5, the beginning of one-pulse dissipative coupled state; b) = 2.4, a one-pulse dissipative

coupled state; c) = 3.5, an intermediate stage; and d) = 6.0, a two-pulse disspative coupled state.

An example: Numerical simulations of the light wave 21C

22

Schematic arrangement for the experiments

with the collinear AO-cell

Collinearacousto-optical cell

U H Fgenerator

Oscilloscope Band-pass filter

Pulsegenerator

Polarizer 1 Polarizer 2 Continuous-wave laser

Photodetector

23

We have developed a special approach to Bragg scattering of light in

optically uniaxial crystals marked by the inclusion of multi-fold

processes. In particular, the configurations related to one-, two-, and

three-phonon scattering processes have been analyzed in details to

highlight both the angular-frequency conditions peculiar for their

realization and the characteristics that can be optimized from the

viewpoint of potential applications.

The analysis performed has given us the opportunities for revealing

and observing various multi-wave collinear and non-collinear multi-

component acousto-optical coupled states.

Thank you

Conclusion