acousto optic
Acousto optic basics
The acousto optic effect is generally used to control the intensity or the position of a laser beam. For instance, it can be used inside a laser cavity working as a Qswitch (see "laser tutorial") enabling the laser to emit pulses or used out of the laser cavity for beam modulation or beam deflection. The acousto optic effect is induced by the propagation of an acoustic wave in an optical material. Indeed, it generates periodic dilatations and compressions of the material and therefore periodic variations of its refraction index around a medium value. The material can be assumed to a succession of low and high index layers behaving as a phase grating diffracting light. The acoustic wave is generated by a piezoelectric transducer attached to the optical material and submitted to a periodic voltage.
Let consider an istropic material.
The variation of deformation of the material regarding the direction of propagation at the contact with the transducer is 
where Ia is the intensity of the acoustic wave, va is its velocity and ρ is the density of the material. At a distance z, the amplitude of deformation is S(z,t) and the index variation is :
where Pp is the photoelastic constant of the material and n its refraction index.
A sinusoidal acoustic wave produces then an index variation :
.
Ka and Ωa are respectively the wave number and the pulsation of the acoustic wave.
The index is :
.
where M is the figure of merit caracterizing the efficiency of the material expressed by the following formula :
.
When propagating in the volume of the material, the acoustic wave has a divergence depending on the size of the transducer :
.
d is the width of the acoustic wave and Λa its wavelength.
There are two configurations that can be modellized easily. The first concerns the case of thin acoustic waves and the second the case of thick acoustic waves. For intermediate configurations, analytical calculations are more complicated.
Raman Nath effect
The Raman-Nath effect applies for a thin acoustic wave. The diffracted wave is then very similar to the one diffracted by a diffraction grating (see "diffraction" tutorial).
Let consider a transmissive grating with a refraction index 
and a thickness e which is small enough to neglect any propagation effect in the grating. An incident plane wave, is diffracted in several orders (several direction) as shows the diffracted amplitude :
.
,
,
.
λ0 is the wavelength of the optical wave in the air and θi is the incidence angle (in the material) and Λ is the grating pitch (see the scheme on the left side).
The diffracted wave is the superposition of plane waves propagating in the directions θp verifying :
.
p is the diffration order.
The intensity diffracted in the order p is proportionnal to 
.
The normalized intensity for the different orders 0 (blue), 1 (red), 2 (green) and 3 (yellow) depending on Φ are represented on the left-side curves. The maximum diffraction efficiency is obtained for the order 1 and can not exceed 33%.
In an acousto optic material, the acoustic wave is travelling. Therefore, the grating is travelling with the same velocity.
The refraction index is then :
.
The amplitude of the diffracted wave is :
.
Because of the conservation of the cinematic moment, the pulsations and thus the wavelengths of the diffracted orders are all different. The pulsation is given by :
.
If the thickness of the acoustic wave increase, the diffracted wave is the result of intereference from diffracted waves at different positions in the thickness of the material.
Bragg conditions
Let consider a plane wave illuminating a grating with an incident angle θi = 0. The optical wavelength is considered much smaller than the grating pitch and the diffraction angle in the order p is considered small as well as the grating thickness d. The phase shift between the wave diffracted in the order p at the grating entrance and the one diffracted at the grating output is :
.
.
Consequently, the interference become destructive in the order p for a thickness :
.
For a thickness larger than Lc, the efficiency in the order p is significantly decreasing. The formula shows that Lc decreases with the order. Thus, increasing the grating thickness makes the diffraction orders disappear starting from the higher order.
Let us take advantage of this paragraph to mention that for acousto optics, the thickness 
is generally considered as the critical interaction length. It corresponds to the distance covered by a wave propagating perpendiculary to the acoustic wave for running through half of an acoustic wavelength in the transverse plane ( plane perpendicular to the light wave propagation ) because of the diffraction. Indeed, at a given time, the acousto optic can be schematized as a succession of low index and high index zones of thickness 
. The optical wave can be therefore considered as diffracted by FORMULA thick slits. The Raman-Nath regime can only occure if the interaction length is far below Lc.
Back to gratings, for an incidence angle θi, the amplitude of the wave diffracted in the order p is :
.
.
A0 ( θp ) is the amplitude of the wave diffracted in the order p per unit length.
This shows that if 
, the grating is considered thin and several orders can be diffracted (see "Raman-Nath" section).
If the thickness increases, the diffraction efficiency decreases for all orders except for the one verifying the following equation :
.
This applies for the order 0 and the order p where 
. This condition is called the Bragg condition. The absolute value of the incident ( and diffracted ) wave is then :
*
.
In an acousto optic, the Bragg configuration applies for orders -1 and therefore :
*
.
As the grating is moving at the acoustic wave velocity, there is a frequency shift of the diffracted wave given by : 
.
The efficiency in the bragg conditions is :
.
A maximum efficiency of almost 100% is obtained for 
.
The acoustic intensity is then :
.
For an incidence angle close to the Bragg angle, the efficiency of the diffracted wave is :
.
The angular acceptance Δ θ of the acousto optic is defined as the angle variation referred to the Bragg angle for which the efficiency drops to 50% (3 dB).
It is :
*
Also, the frequency bandwidth Δ Fa is defined as the frequency variation referred to the exact frequency in Bragg configuration inducing an efficiency drop of 50% (3 dB).
Δ Fa is given by the formula :
.
(*) the angles are calculated in the acousto optic material. The Fresnel law must be applied to them to calculate the corresponding values in the air.
Quality factor
For an acousto optic, the Quality factor is defined as follows :
.
The acousto optic can be considered as thin and its optical behavior can be modellized according to the Raman-Nath regime if :
.
Q is then much smaller than 1.
The Bragg regime model is applicable when Q > 1.
In the Bragg regime, the efficiencies in the order 0 and 1 are then respectively :
and
.
Acousto optic modulator
When used as a light modulator, an acousto optic material is crossed by a modulated acoustic wave.
At Bragg incidence angle, the beam is only deflected in the Bragg direction when the acoustic power is switched on. Obviously, the modulation frequency has to be much smaller than the acoustic wave frequency.
The modulation bandwidth is limited by the rise time tr of the acoustic wave along the beam, tr being defined in this case as the time for the acoustic wave to propagate along the beam.
For a top hat beam, 
while for a gaussian beam (TEM00 mode - see "laser tutorial"),
( D is the beam diameter ).
The modulation frequency bandwidth is calculated from the modulation transfer function ( MTF ) which gives the contrast regarding the modulation frequency. For a gaussian beam, the MTF is :
.
FM is the modulation frequency.
fc is the cut-off frequency ( 
).
The MTF drops of 50 % for 
.
References
"Cours effets electro, acousto et magneto-optiques application à la modulation et la déflexion de la lumière" -
Ecole Nationale Supérieure des Télécommunications - 1988 - author : P. Gallion.
"Effets et composants Electro & acousto optique" - Institut dOptique Graduate School - 2016 - author : Ivan Sortais, Jean Michel Jonathan.
"Cours études expérimentales" - Ecole Nationale Supérieure des Télécommunications - 1988 - author : P. Chabran, C. Debarge.
"Acousto-optic theory - Application notes" - AA OPTOELECTRONIC.