Acousto-optic devices are based on the periodic modulation of the optical index of refraction caused by an acoustic wave propagating in a transparent material.  For an acoustic wave with wave vector k_s, propagating in a material with the right acoustical and optical properties, the change in the index of refraction Dn(r,t) is given by

Dn(r,t) » Dn exp(i(k_s×r - w_st),

where w_s/k_s = v_s, the speed of sound in the material. 

An acousto-optic device is constructed by bonding an acousto-electric transducer onto a photo-elastic medium, so that acoustic waves can be launched into the medium.  The transducer is usually a piezoelectric crystal, metalized on both faces so that an electric field can be applied, which induces a strain throughout the piezoelectric crystal.  Through appropriate choice of transducer crystal cut and orientation the time dependent strain within the transducer is coupled into the photo-elastic medium with a frequency-dependent efficiency dictated by the acoustic impedance matching of the transducer and bonding layers.  By applying a sinusoidally varying electric field to the transducer, that is within its acoustic resonant bandwidth, we can launch a propagating acoustic wave into the photo-elastic medium.

Assume an acoustic wave is traveling in a transparent material with a wave vector pointing into the x-direction.

Assume an electromagnetic wave is traveling in the material with a wave vector making an angle q_B with the z-direction.  We may consider the acoustic wave as a beam of phonons, with frequency w_s and wave vector k_s, with w_s/k_s = v_s, where v_s is the speed of sound in the material.  We may consider the electromagnetic wave as a beam of photons with frequency w_i and wave vector k_i, with w_i/k_i = c/n, where n is the index of refraction of the undisturbed material at frequency w_i.

The phonons have energy hw_s and momentum hk_s, while the photons have momentum hk_i and energy hw_i.  If there exists some coupling between the acoustic wave and the electromagnetic wave, then a phonon may be annihilated or created as a result of this interaction.  However momentum and energy have to be conserved.  If a phonon is annihilated, a photon has to take up the energy and the momentum.

Ultrasonic acoustic waves and electromagnetic waves both can have wavelengths on the order of a micrometer, but the associated frequencies differ greatly.  The speed of sound in a material is typically on the order of 10³ m/s.  The frequency of an acoustic wave with wavelength in the micrometer range is on the order of 10⁹ Hz (FM radio frequency).  The frequency of the electromagnetic wave with wavelength in the micrometer range (the near infrared region) is on the order of 10¹⁴ Hz.  We have w_i ~ 10⁵ w_s.

So when a phonon is annihilated, the momentum of a photon changes appreciably, while its energy changes negligibly.  This is only possible when the electromagnetic wave is incident at the Bragg angle q_B with respect to the acoustic wave.  We need

sinq_B » k_s/(2k_i).

If the Bragg condition is satisfied for the incident wave, then the scattered photons form a diffracted wave with wave vector k_d = k_i + k_s and frequency w_d = w_i + w_s » w_i.

The diffracted wave forms a beam that is deflected from the incident beam.  This beam can also interact with the acoustic wave.  If a photon in the diffracted wave emits a phonon, than it is deflected back into the incident wave.

k_d – k_s = k_i

The coupling between the between the acoustic wave and the electromagnetic wave is a result of the modulation of the optical index of refraction caused by an acoustic wave.

Maxwell’s equations in a macroscopic medium are

,

.

In regions with no free charge and current densities in a non-magnetic lih medium we may write

.

Now assume that the index of refraction as a function of space and time is given by

n = n₀ + Dn(r,t),

where Dn(r,t) << n₀.  Then

n² » n₀² +2n₀Dn(r,t),

and we approximately have

.

But

.

We have Ñ×D = 0,  Ñ×eE = eÑ×E + E×Ñe = 0.

Assume the incident wave is polarized in the y-direction E = Ej.  The index of refraction and therefore e are functions of x and t.  The gradient of e points into the ±x-direction and E×Ñe = 0.  Similarly, Ñ×E = 0.  (Since r_f = 0, Ñ×E = -Ñ×P/e₀, and P = P_y(x)j, Ñ×P = 0.)  Therefore

.

We obtain the inhomogeneous wave equation with a source term.  We expect the incident wave

E_i = jA_i(z)exp(i(k_i×r - w_it),

to be the source of the deflected wave

E_d = jA_d(z)exp(i(k_d×r - w_dt),

with k_d = k_i + k_s and w_d = w_i + w_s » w_i.

As a first-order approximation to the full solution of the problem we insert E_i on the right and E_d on the left side of the inhomogeneous wave equation and obtain

.

Here we assume that terms proportional to ѲA_d are small and can be neglected.

[E_d has only a y-component.  Write E_d = fg, where f = A_d(z) and g = exp(i(k_d×r - w_dt).
ѲE_d = Ñ×ÑE_d =  Ñ×Ñfg.   Ñfd = fÑg + gÑf. 
Ñ×Ñfg =  Ñ×(fÑg + gÑf) = Ñg×Ñf +fѲg + Ñf×Ñg +gѲf = 2Ñg×Ñf + fѲg + gѲf.
Ñg = ik_dexp(i(k_d×r - w_dt),  Ñf = ÑA_d(z), Ѳg = -k_d²exp(i(k_d×r - w_dt), Ѳf = ѲA_d(z).]

We have k_d = w_d(m₀e₀)^½n₀ and k_d×ÑA_d = cosq_B k_d (dA_d/dz).

(A_d is a function of z only, the gradient of A_d points into the z-direction, k_d makes an angle q_B with the z-direction.)

We therefore have

,

where we used w_d/k_d = c/n₀.

Considering the deflected wave as the source term for the incident wave we similarly obtain

.

We have two coupled equation, with the coupling constant equal to

.

The coupling constants are nearly, but not exactly equal to each other.  Electromagnetic energy is not exactly conserved.  The electromagnetic wave can exchange energy with the acoustic wave.

The system behaves approximately like a lossless coupled system.
dA_d/dz = kA_i, dA_i/dz = kA_d.
Our analysis of coupling of modes in space (with b₁ = b₂ = Db = 0 ) applies.

We therefore already know that if A_d = 0 at z = 0

A_i(z) = A_i(0) cos|k|z,    A_d(z) = -iA_i(0) sin|k|z,

with

.

The incident wave feeds the diffracted wave, and the diffracted wave feeds back into the incident wave, as long as the two waves do not separate spatially, i.e. as long as the beam cross section is much larger than Lsinq_B, where L is the width of the region supporting the acoustic wave.  The diffracted wave is shifted up in frequency by approximately one part in 10⁵.

The preceding analysis can also be applied to a bulk grating, if we let v_s = w_s = 0.  Then we have "frozen" index variations and w_d = w_i.  The incident and diffracted wave are coupled and energy is fed from one to the other.

For a material with a fixed acoustic velocity, the acoustic wavelength is a function of the frequency of the drive signal.  The acoustic wavelength controls the angle of deflection.  The amplitude of the acoustic wave determines Dn and therefore controls the fraction of the electromagnetic radiation that is deflected.

Keeping the transducer frequency constant and modulating the power to the transducer modulates the amplitude of the deflected beam.  If we only monitor the deflected beam, we have no intensity when there is no power to the transducer.  If we choose the length L appropriately we can achieve maximum power in the deflected beam for maximum power in the acoustic beam.  In this way we can use the acousto-optic device as a modulator.

We can also use acousto-optic device as a deflector.  We keep the acoustic power constant but vary the acoustic wavelength to change the Bragg angle.

If we set up a standing acoustic wave instead of a traveling acoustic wave in the interaction region, then we have phonons with frequency w_s and wave vectors k_s and –k_s.

Assume a photon incident at the Bragg angle absorbs a phonon with wave vector k_s.  Its wave vector changes from k_i to k_d.  The frequency of the photon increases, w_d = w_i +w_s.  For k_d to change back to ~k_i, the deflected photon now can emit a phonon with wave vector k_s or absorb a phonon with wave vector -k_s.  Its frequency therefore can change to w_i or to w_i + 2w_s.

The incident photon also may first emit a phonon with wave vector -k_s to change its wave vector from k_i to k_d.  The frequency of the photon then decreases, w_d = w_i - w_s.  The deflected photon again can emit a phonon with wave vector k_s or absorb a phonon with wave vector -k_s.  Its frequency therefore can change to w_i or to w_i - 2w_s. Since w_s << w_i, waves with frequencies w_i ± m w_s remain phase matched over a large distance for many values of m.  We obtain many sidebands, the sidebands cascade.