# Acousto-optic tunable filters

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This is a continuation from the previous tutorial - acousto-optic deflectors.

An optical grating can be used for the separation or filtering of optical frequencies, as is seen in any grating spectrometer and in s distributed Bragg reflector as discussed in the grating waveguide couplers tutorial.

It is also possible to use the index grating generated by an acoustic wave in a medium for such purposes. One advantage of such an acousto-optic filter, or acousto-optic spectrometer, is that it is electronically tunable because the period of the acousto-optic grating can be varied by altering the acoustic frequency.

This electronic tunability allows an acousto-optic filter to have many sophisticated functions that are not possible with an ordinary spectroscopic device. For example, the acoustic frequency can be rapidly scanned or quickly switched from one to another, thus allowing very rapid optical spectrum analysis. The acoustic signal driving the filter can also be amplitude or frequency modulated, thus imposing a desired amplitude or frequency modulation on the optical signal at the selected optical frequency.

As a result, the applications of acousto-optic tunable filters cover a very wide range from wavelength tuning of a laser to wavelength-division multiplexing and demultiplexing in optical communication systems, as well as various spectroscopic analyses.

An acousto-optic tunable filter functions exclusively in the Bragg regime because its tunability and frequency selectivity rely entirely on the phase-matching condition. It generally uses a traveling acoustic wave because the acoustic frequency cannot be tuned continuously in a standing-wave device.

Practical acousto-optic filters, except for those in waveguide structures discussed in the next tutorial, function exclusively with birefringent Bragg diffraction in anisotropic crystals.

The most important characteristic parameters of an acousto-optic tunable filter are its chromatic resolving power, its angular aperture, and its efficiency. Because high spectral resolution is one of its merits, it generally operates with the parameter \(a\gg1\) with a large angular aperture.

The selectivity and resolution of the optical frequency in the interaction of an optical wave with a grating increase linearly with interaction length. To have a long interaction length with an acousto-optic grating while avoiding the need to expand the acoustic beam width and the consequential reduction of the acoustic intensity at a given acoustic power, it is necessary to use collinear, or nearly collinear, Bragg diffraction.

For collinear, ** codirectional** Bragg diffraction, the phase-matching condition is \(k_\text{d}=k_\text{i}\pm{K}\). The optical wavelength selected under this phase-matching scheme in an acousto-optic filter driven by an acoustic wave at frequency \(f\) is

\[\tag{8-149}\lambda=\frac{\Delta{n}v_\text{a}}{f}\]

where \(\Delta{n}=|n_\text{d}-n_\text{i}|\).

For collinear, ** contradirectional** Bragg diffraction, the phase-matching condition is \(k_\text{d}=-k_\text{i}\pm{K}\), and the relation between the selected optical wavelength and the acoustic frequency becomes \(\lambda=(n_\text{d}+n_\text{i})v_\text{a}/f\).

The contradirectional scheme is not practical for an ordinary acousto-optic tunable filter using a typical acousto-optic material because it requires an acoustic frequency on the order of 10 GHz for a typical optical wavelength, say \(\lambda\) = 1 μm.

In contrast, over the entire optical spectral range, the acoustic frequencies required for an acousto-optic tunable filter functioning in the codirectional scheme are, according to (8-149), generally in the practical range of 10 MHz to 1 GHz for typical acousto-optic materials, such as LiNbO_{3}, TeO_{2}, and CaMoO_{4}, that are used for this purpose.

Therefore, the ideal phase-matching scheme for a practical acousto-optic filter is collinear, codirectional Bragg interaction. In certain applications, exact collinear interaction is not possible due to the limitation of the parameters of the available acousto-optic materials. Then noncollinear birefringent Bragg interaction under a special tangential phase-matching condition can also be used to design practical noncollinear acousto-optic tunable filters.

** For an acousto-optic tunable filter that uses either collinear or noncollinear interaction**,

**.**

*an anisotropic crystal is needed and there is always a polarization change between the incident and diffracted optical waves*** This statement**,

**,**

*however***, as we shall see in the next tutorial. In this tutorial, we consider only collinear filters for simplicity, but the general concepts apply to noncollinear filters as well.**

*is not true for guided-wave acousto-optic tunable filters*Figure 8-16 below shows two configurations for collinear acousto-optic tunable filters.

The interaction length \(l\) is measured along the longitudinal direction, rather than t he transverse direction, of the acoustic beam. Therefore, the interaction length in an acousto-optic tunable filter is not determined by the transducer length: \(l\ne{L}\).

Because of birefringent Bragg interaction, the filtered optical beam at the selected wavelength has a polarization different from that of the unfiltered beam. In general, a very high extinction ratio can be easily achieved in separating orthogonally polarized optical waves. By using proper polarizing optical components to separate the filtered beam from the unfiltered beam, a very high signal-to-noise ratio can be obtained. This is another advantage of acousto-optic tunable filters.

In collinear, codirectional Bragg diffraction, if both the frequency and the propagation direction of the acoustic wave are fixed, the phase mismatch caused by a wavelength deviation of \(\delta\lambda\) can an angular deviation of \(\delta\theta_\text{i}\) in the incident optical wave is

\[\tag{8-150}\Delta{k}\approx\frac{2\pi(n_\text{d}-n_\text{i})}{\lambda}\left[-\frac{\delta\lambda}{\lambda}+\frac{(\delta\theta_\text{i})^2}{2} \right]\]

A very important characteristic of an acousto-optic tunable filter is its optical spectral resolution. Over a spectral width of \(\Delta\lambda\) centered at the wavelength \(\lambda\) that is selected by a given acoustic frequency \(f\) according to (8-149), the largest wavelength deviations away from \(\lambda\) are \(\delta\lambda=\pm\Delta\lambda/2\) at the two edges of the spectral width. For a perfectly collimated optical beam with \(\delta\theta_\text{i}=0\), the general criterion that \(|\Delta{k}|l\le0.9\pi\) leads to the following ** spectral width passed by the filter**:

\[\tag{8-151}\Delta\lambda=\frac{0.9\lambda^2}{\Delta{n}l}\]

The ** chromatic resolving power** of the acousto-optic filter is then given by

\[\tag{8-152}R=\frac{\lambda}{\Delta\lambda}=\frac{\Delta{n}l}{0.9\lambda}=\frac{fl}{0.9v_\text{a}}=\frac{l}{0.9\Lambda}\]

Hence the resolving power is simply given by the number of acoustic wavelengths covered by the interaction length and is linearly proportional to the interaction length.

Another important characteristic of an acousto-optic tunable filter is its ** angular aperture**. As is the case in the application of any spectroscopic instrument, the input optical beam incident on the acousto-optic tunable filter is usually not well collimated. A large angular aperture is thus needed for a high optical collection efficiency at the input, which then leads to a high overall efficiency for the device.

The angular aperture of a given acousto-optic tunable filter is determined by the maximum angular divergence \(\Delta\theta_\text{o}\) allowed for the incident optical beam. An angular divergence of \(\Delta\theta_\text{o}\) in the incident optical beam corresponds to maximum angular deviations of \(\delta\theta_\text{i}=\pm\Delta\theta_\text{o}/2\) with respect to the center direction of the beam. For any given optical spectral component, we find that

\[\tag{8-153}\Delta\theta_\text{o}=2\left(\frac{0.9\lambda}{\Delta{n}l}\right)^{1/2}=\frac{2}{R^{1/2}}\]

by applying the criterion \(|\Delta{k}|l\le0.9\pi\).

Note that \(\Delta\theta_\text{o}\) is the maximum allowable optical beam divergence ** inside** the acousto-optic medium. The difference in the refractive index between the medium and the free space causes a change in the divergence of an optical beam between free space and the acousto-optic medium. Therefore, the angular apertures for the incident and the diffracted waves in free space outside the medium are, respectively,

\[\tag{8-154}\Delta\theta_\text{i}=n_\text{i}\Delta\theta_\text{o}=\frac{2n_\text{i}}{R^{1/2}}\qquad\text{and}\qquad\Delta\theta_\text{d}=n_\text{d}\Delta\theta_\text{o}=\frac{2n_\text{d}}{R^{1/2}}\]

For birefringent interaction in a collinear filter, one of the two optical waves has to be an extraordinary wave but both optical waves have to be polarized in a plane that is perpendicular to the acoustic \(\mathbf{K}\) direction because \(\mathbf{k}_\text{i}\parallel\mathbf{k}_\text{d}\parallel\mathbf{K}\). Therefore, interaction is generally restricted to the plane that is perpendicular to the optical axis of the crystal with the extraordinary optical wave polarized in the direction along the optical axis.

Because the interaction length is not determined by the transducer length in a collinear configuration, it is possible to increase the diffraction efficiency and the resolving power of a collinear acousto-optic tunable filters simultaneously by choosing \(l\gt{L}\). The length \(l\) is normally limited by the attenuation of the acoustic wave and by the availability of long acousto-optic crystals.

**Example 8-9**

It is possible to make a collinear LiNbO_{3} acousto-optic tunable filter utilizing a transverse acoustic mode that propagates in the [010] \(y\) direction and is polarized in the [100] \(x\) direction. The acoustic velocity of this mode is \(v_\text{a}=4.08\text{ km s}^{-1}\). The birefringent collinear interaction with \(\mathbf{k}_\text{i}\parallel\mathbf{k}_\text{d}\parallel\mathbf{K}\parallel\hat{y}\) then couples the ordinary wave polarized with \(\hat{e}_\text{o}=\hat{x}\) and the extraordinary wave polarized with \(\hat{e}_\text{e}=\hat{z}\) through the acousto-optic tensor elements \(\Delta\epsilon_\text{xz}=\Delta\epsilon_\text{zx}\). This filter is used to select an ordinary incident wave spectrally at \(\lambda\) = 1.3 μm with a chromatic resolving power of \(R=10^3\). The ordinary and extraordinary refractive indices of LiNbO_{3} at 1.3 μm are \(n_\text{o}=2.222\) and \(n_\text{e}=2.145\), respectively. Find the required acoustic frequency \(f\) and the required interaction length \(l\). What are the spectral linewidth \(\Delta\lambda\) and the angular apertures \(\Delta\theta_\text{i}\) and \(\Delta\theta_\text{d}\)?

The required acoustic frequency is

\[f=\frac{\Delta{n}v_\text{a}}{\lambda}=\frac{|2.222-2.145|\times4.08\times10^3}{1.3\times10^{-6}}\text{ Hz}=241.66\text{ MHz}\]

For \(R=10^3\), the needed interaction length is

\[l=\frac{0.9\lambda{R}}{\Delta{n}}=\frac{0.9\times1.3\times10^{-6}\times10^3}{|2.222-2.145|}\text{ m}=1.52\text{ cm}\]

The spectral linewidth is simply \(\Delta\lambda=\lambda/R=1.3\text{ nm}\). The angular apertures are \(\Delta\theta_\text{i}=2n_\text{o}/R^{1/2}=0.141\text{ rad}=8.05^\circ\) and \(\Delta\theta_\text{d}=2n_\text{e}/R^{1/2}=0.136\text{ rad}=7.77^\circ\).

The next part continues with the guided-wave acousto-optic devices tutorial.