Acousto-Optic Devices and Applications

This is a continuation from the previous tutorial - nonlinear compensation for digital coherent transmission.

1. Introduction

When an acoustic wave propagates in an optically transparent medium, it produces a periodic modulation of the index of refraction via the elasto-optical effect. This provides a moving phase grating which may diffract portions of an incident light into one or more directions.

This phenomenon, known as the acousto-optic (AO) diffraction, has led to a variety of optical devices that perform spatial, temporal, and spectral modulations of light. These devices have been used in optical systems for light-beam control and signal-processing applications.

Historically, the diffraction of light by acoustic waves was first predicted by Brillouin in 1922. Ten years later, Debye and Sears and Lucas and Biquard experimentally observed the effect.

In contrast to Brillouin’s prediction of a single diffraction order, a large number of diffraction orders were observed. This discrepancy was later explained by the theoretical work of Raman and Nath.

They derived a set of coupled wave equations that fully described the AO diffraction in unbounded isotropic media. The theory predicts two diffraction regimes; the Raman-Nath regime, characterized by the multiple of diffraction orders, and the Bragg regime, characterized by a single diffraction order.

Although the basic theory of AO diffraction in isotropic media was well understood, there had been relatively few practical applications prior to the invention of the laser. It was the need of optical devices for laser beam control that stimulated extensive research on the theory and practice of acousto-optics.

Significant progress of the AO devices has been made during the past two decades, due primarily to the development of superior AO materials and efficient broadband transducers. By now, acousto-optics has developed into a mature technology and is deployed in a wide range of optical system applications.

It is the purpose of this tutorial to review the theory and practice of bulkwave acousto-optic devices and their applications. The review emphasizes design and implementation of AO devices. It also reports the status of most recent developments.

In addition to bulkwave acousto-optics, there have also been studies on the interaction of optical guided waves and surface acoustic waves (SAW). However, the effort has remained primarily at the research stage and has not yet resulted in practical applications. As such, the subject of guided-wave acousto-optics will not be discussed here.

This tutorial is organized as follows:

The next section discusses the theory of acousto-optic interaction. It provides the necessary background for the design of acousto-optic devices. The important subject of acousto-optic materials is discussed in the section following. Then a detailed discussion on three basic types of acousto-optic devices is presented. Included in the discussion are the topics of deflectors, modulators, and tunable filters. The last section discusses the use of AO devices for optical beam control and signal processing applications.

2. Theory of Acousto-Optic Interaction

Elasto-optic Effect

The elasto-optic effect is the basic mechanism responsible for the AO interaction. It describes the change of refractive index of an optical medium due to the presence of an acoustic wave. To describe the effect in crystals, we need to introduce the elasto-optic tensor based on Pockels’ phenomenological theory.

An elastic wave propagating in a crystalline medium is generally described by the strain tensor \(\mathbf{S}\), which is defined as the symmetric part of the deformation gradient

\[\tag{1}S_{ij}=\left(\frac{\partial{u_i}}{\partial{x_j}}+\frac{\partial{u_j}}{\partial{x_i}}\right)/2\qquad{i,j=1\text{ to }3}\]

where \(u_i\) is the displacement.

Since the strain tensor is symmetric, there are only six independent components. It is customary to express the strain tensor in the contracted notation

\[\tag{2}S_1=S_{11},\quad{S_2}=S_{22},\quad{S_3}=S_{33},\quad{S_4}=S_{23},\quad{S_5}=S_{13},\quad{S_6}=S_{12}\]

The conventional elasto-optic effect introduced by Pockels states that the change of the impermeability tensor, \(\Delta{B}_{ij}\), is linearly proportional to the symmetric strain tensor.

\[\tag{3}\Delta{B_{ij}}=p_{ijkl}S_{kl}\]

where \(p_{ijkl}\) is the elasto-optic tensor. In the contracted notation,

\[\tag{4}\Delta{B_m}=p_{mn}S_n\qquad{m,n,=1}\text{ to }6\]

Most generally , there are 36 components. For the more common crystals of higher symmetry, only a few of the elasto-optic tensor components are non-zero.

In the above classical Pockels’ theory, the elasto-optic effect is defined in terms of the change of the impermeability tensor \(\Delta{B}_{ij}\). In the more recent theoretical work on AO interactions, analysis of the elasto-optic effect has been more convenient in terms of the nonlinear polarization resulting from the change of dielectric tensor \(\Delta\epsilon_{ij}\). We need to derive the proper relationship that connects the two formulations.

Given the inverse relationship of \(\epsilon_{ij}\) and \(B_{ij}\) in a principal axis system \(\Delta\epsilon_{ij}\) is:

\[\tag{5}\Delta\epsilon_{ij}=-\epsilon_{ii}\Delta{B}_{ij}\epsilon_{jj}=-n_i^2n_j^2\Delta{B}_{ij}\]

where \(n_i\) is the refractive index. Substituting Eq. (3) into Eq. (5), we can write:

\[\tag{6}\Delta\epsilon_{ij}=\chi_{ijkl}S_{kl}\]

where we have introduced the elasto-optic susceptibility tensor,

\[\tag{7}\chi_{ijkl}=-n_i^2n_j^2p_{ijkl}\]

For completeness, two additional modifications of the basic elasto-optic effect are discussed as follows.

Roto-optic Effect

Nelson and Lax discovered that the classical formulation of elasto-optic was inadequate for birefringent crystals. They pointed out that there exists an additional roto-optic susceptibility due to the antisymmetric rotation part of the deformation gradient.

\[\tag{8}\Delta{B}_{ij}'=p_{ijkl}'R_{kl}\]

where \(R_{ij}=(S_{ij}-S_{ji})/2\).

It turns out that the roto-optic tensor components can be predicted analytically. The coefficient of \(\mathbf{p}_{ijkl}\) is antisymmetric in \(kl\) and vanishes except for shear waves in birefringent crystals. In a uniaxial crystal the only nonvanishing components are \(p_{2323}=p_{2313}=(n_o^{-2}-n_e^{-2})/2\), where \(n_o\) and \(n_e\) are the principal refractive indices for the ordinary and extraordinary wave, respectively. Thus, the roto-optic effect can be ignored except when the birefringence is large.

Indirect Elasto-optic Effect

In the piezoelectric crystal, an indirect elasto-optic effect occurs as the result of the piezoelectric effect and electro-optic effect in succession. The effective elasto-optic tensor for the indirect elasto-optic effect is given by

\[\tag{9}p_{ij}^*=p_{ij}-\frac{r_{im}S_me_{jn}S_n}{\epsilon_{mn}S_mS_n}\]

where \(p_{ij}\) is the direct elasto-optic tensor, \(r_{im}\) is the electro-optic tensor, \(e_{jn}\) is the piezoelectric tensor, \(\epsilon_{mn}\) is the dielectric tensor, and \(S_m\) is the unit acoustic wave vector.

The effective elasto-optic tensor thus depends on the direction of the acoustic mode. In most crystals the indirect effect is negligible. A notable exception is LiNbO\(_3\). For instance, along the \(z\) axis, \(r_{33}=31\times10^{-12}\) m/v, \(e_{33}=1.3\) c/m\(^2\), \(E_{33}^s=29\), thus \(p^*=0.088\), which differs notably from the contribution \(p_{33}=0.248\).

Plane Wave Analysis of Acousto-optic Interaction

We now consider the diffraction of light by acoustic waves in an optically transparent medium in which the acoustic wave is excited. An optical beam is incident onto the cell and travels through the acoustic beam. Via the elasto-optical effect, the traveling acoustic wave sets up a spatial modulation of the refractive index which, under proper conditions, will diffract the incident beam into one or more directions.

In order to determine the detailed characteristics of AO devices, a theoretical analysis for the AO interaction is required. As pointed out before, in the early development, the AO diffraction in isotropic media was described by a set of coupled wave equations known as the Raman-Nath equations.

In this model the incident light is assumed to be a plane wave of infinite extent. It is diffracted by a rectangular sound column into a number of plane waves propagating along different directions. Solution of the Raman-Nath equations gives the amplitudes of these various orders of diffracted optical waves.

In general, the Raman-Nath equations can be solved only numerically and judicious approximations are required to obtain analytic solutions. Solutions of these equations can be classified into different regimes that are determined by the ratio of the interaction length \(L\) to a characteristic length \(L_o=n\Lambda^2/\lambda_o^{10}\) where \(n\) is the refractive index and \(\Lambda\) and \(\lambda_o\) are wavelengths of the acoustic and optical waves, respectively.

The Raman-Nath equations admit analytic solutions in the two limiting cases. In the Raman-Nath regime, where \(L\ll{L_o}\), the AO diffractions appear as a large number of different orders. The diffraction is similar to that from a thin phase grating. The direction of the various diffraction orders are given by the familiar grating equation, \(\sin\theta_m=m\lambda_o/n\Lambda\), where \(m\) is the diffraction order.

Solution of the Raman-Nath equations shows that the amplitude of the \(m\)th-order diffracted light is proportional to the \(m\)th-order Bessel functions. The maximum intensity of the first-order diffracted light (relative to the incident light) is about 34 percent. Due to this relatively low efficiency, AO diffractions in the Raman-Nath regime are of little interest to device applications.

In the opposite limit, \(L\gt{L_o}\), the AO diffraction appears as a predominant first order and is said to be in the Bragg regime. The effect is called Bragg diffraction since it is similar to that of the x-ray diffraction in crystals.

An important feature of the Bragg diffraction is that the maximum first-order diffraction efficiency obtainable is 100 percent. Therefore, practically all of today’s AO devices are designed to operate in the Bragg regime.

In the intermediate case, \(L\le{L_o}\), the AO diffractions now appear as a few dominant orders. The relative intensities of these diffraction orders can be obtained by numerically solving the Raman-Nath equations.

This problem has been studied in detail by Klein and Cook, who calculated numerically the diffracted light intensities for the different regimes. Their analysis shows that when \(L=L_o\), the maximum first-order diffraction efficiency is about 90 percent. Thus \(L=L_o\) may be used as a criterion for insuring the AO device operated in the Bragg regime.

Many modern high-performance devices are based on the light diffraction in anisotropic media. In this case, the indices of the incident and diffracted light may be different. This is referred to as birefringent diffraction.

The classical Raman-Nath equations are no longer adequate and a new formulation is required. We have previously presented a plane wave analysis of AO interaction in anisotropic media. Results of the plane wave analysis for the Bragg diffraction are summarized as follows.

The AO interaction can be viewed as a parametric process where the incident optical wave mixes with the acoustic wave to generate a number of polarization waves. The polarization waves in turn generate new optical waves at various diffraction orders.

Let the angular frequency and wave vector of the incident optical wave be denoted by \(\omega_i\) and \(\mathbf{k}_i\), respectively, and those of the acoustic waves by \(\omega_a\) and \(\mathbf{k}_a\).

In the Bragg limit, only the first-order diffracted light grows to a finite amplitude. The polarization wave is then characterized by the angular frequency \(\omega_d=\omega_i+\omega_a\) and wavevector \(\mathbf{K}_d=\mathbf{k}_i+\mathbf{k}_a\). The total electric fields of the optical wave can be expressed as

\[\tag{10}\bar{E}(r,t)=\frac{1}{2}(\hat{\mathbf{e}}_iE_i(z)e^{j(\omega_it-\bar{k}_o\cdot\bar{r})}+\hat{\mathbf{e}}_dE_d(z)e^{j(\omega_dt-\mathbf{K}_d\cdot\bar{r})})+\text{c.c.}\]

where \(\hat{\mathbf{e}}_i\) and \(\hat{\mathbf{e}}_d\) are unit vectors in the directions of electric fields for the incident and diffracted optical waves, respectively, \(\mathbf{E}_i(z)\) and \(\mathbf{E}_d(z)\) are the corresponding slowly varying electric field amplitudes, and c.c. stands for complex conjugate.

The electric field of the optical wave satisfies the wave equation,

\[\tag{11}\nabla\times\nabla\times\mathbf{E}+\frac{1}{c^2}\left(\bar{\bar{\boldsymbol{\epsilon}}}\cdot\frac{\partial^2\bar{E}}{\partial{t^2}}\right)=-\mu_o\frac{\partial^2\mathbf{P}}{\partial{t^2}}\]

where \(\bar{\bar{\boldsymbol{\epsilon}}}\) is the relative dielectric tensor and \(\mathbf{P}\) is the acoustically induced polarization. Based on the Pockel’s theory of the elasto-optic effect,

\[\tag{12}\mathbf{P}(\mathbf{r},t)=\epsilon_o\bar{\bar{\chi}}\cdot\mathbf{S}(\mathbf{r},t)\cdot\mathbf{E}(\mathbf{r},t)\]

where \(\chi\) is the elasto-optical susceptibility tensor given in Eq. (7). \(\mathbf{S}(r,t)\) is the strain of the acoustic wave

\[\tag{13}S(\bar{r},t)=\frac{1}{2}\hat{\mathbf{s}}Se^{j(\omega_at-\bar{\mathbf{k}}_a\cdot\bar{r})}+\text{c.c.}\]

where \(\hat{\mathbf{s}}\) is a unit strain tensor of the acoustic wave and \(S\) is the acoustic wave amplitude. Substituting Eqs. (10), (12), and (13) into Eq. (11) and neglecting the second-order derivatives of electric field amplitudes, we obtain the coupled wave equations for AO Bragg diffraction.

\[\tag{14}\frac{dE_i}{dz}=j(Cn_d/n_i)S^*E_d\]

\[\tag{15}\frac{dE_d}{dz}-j\Delta{k}E_d=jCSE_i\]

where \(C=\pi{n}_i^2n_dp/2\lambda_o\), \(n_i\) and \(n_d\) are the refractive indices for the incident and diffracted light, \(p\) is the effective elasto-optic constant for the particular mode of AO interaction, and \(\Delta{k}\) is the magnitude of momentum mismatch between the polarization wave and free wave of the diffracted light.

\[\tag{16}\Delta{k}=|\mathbf{K}_d-\mathbf{k}_d|=|\mathbf{k}_i+\mathbf{k}_a-\mathbf{k}_d|\]

Eqs. (14) and (15) admit simple analytic solutions. At \(z=L\), the intensity of the first-order diffracted light (normalized to the incident light) is

\[\tag{17}I_1=\frac{I_d(L)}{I_i(0)}=\eta\;\text{sinc}^2\frac{1}{\pi}\left(\eta+\left(\frac{\Delta{k}L}{2}\right)^2\right)^{1/2}\]

where \(\text{sinc}(x)=(\sin{\pi}x)/\pi{x}\), and

\[\tag{18}\eta=\frac{\pi^2}{2\lambda_o^2}\left(\frac{n^6p^2}{2}\right)S^2L^2=\frac{\pi^2}{2\lambda_o^2}M_2P_a\left(\frac{L}{H}\right)\]

In the preceding equation, we have used the relation \(P_a=\frac{1}{2}\rho{V^3}S^2LH\) where \(P_a\) is the acoustic power, \(H\) is the acoustic beam height, \(\rho\) is the mass density, \(V\) is the acoustic wave velocity, and \(M_2=n^6p^2/\rho{V^3}\) is a material figure of merit.

Equation (17) shows that for sufficiently long interaction length the diffracted light builds up only when the momentum is nearly matched. For exact phase matching, (\(\Delta\mathbf{k}=0\)), the peak intensity of the diffracted light is given by

\[\tag{19}I_p=\sin^2\sqrt{\eta}\]

When \(\eta\ll1\), \(I_p\approx\eta\). The diffraction efficiency is thus linearly proportional to acoustic power. This is referred to as the weak interaction (or small signal) approximation which is valid when the peak efficiency is below 70 percent. As acoustic power increases, the diffraction efficiency saturates and approaches 100 percent. Thus in the Bragg regime, complete depletion of the incident light is obtainable. However, the acoustic power required is about 2.5 times that predicted by the small signal theory.

When the momentum is not matched, the fractional diffracted light \(I_1\), in Eq. (17) can be approximated by

\[\tag{20}I_1=I_p\;\text{sinc}^2\psi\]

where \(\psi=\Delta{k}L/2\pi\) is the phase mismatch (normalized to \(2\pi\)). We shall show later that \(\Psi\) determines the frequency and angular characteristics of AO interaction. In the next section we shall first consider the case \(\psi=0\); i. e., when the AO wavevectors are exactly momentum matched.

Phase Matching

It was shown in the preceding section that significant diffraction of light occurs only when the exact momentum matching is met.

\[\tag{21}\mathbf{k}_d=\mathbf{k}_i+\mathbf{k}_a\]

In the general case of AO interaction in an anisotropic medium, the magnitudes of the wave vectors are given by:

\[\tag{22}k_i=\frac{2\pi{n_i}}{\lambda_o}\qquad{k_d}=\frac{2\pi{n_d}}{\lambda_o}\qquad{k_a}=\frac{2\pi}{\Lambda}\]

where \(\Lambda=V/f\) is the acoustic wavelength, and \(V\) and \(f\) are the velocity and frequency of the acoustic wave. In Eq. (22), the small optical frequency difference (due to acoustic frequency shift) of the incident and diffracted light beams are neglected.

Isotropic Diffractions

Consider first the case of isotropic diffraction: \(n_i=n_d=n\). At exact phase matching, the acoustic and optical wavevectors form an isosceles triangle, as shown in Fig. 1a.

**Figure 1**. Wavevector diagram of acousto-optic interaction. (a) Isotropic diffraction, (b) birefringent diffraction.

The loci of the incident and diffracted optical wavevectors fall on a circle of radius \(n\). From the figure, it is seen that the incident and diffracted wave vectors make the same angle with respect to the acoustic wavefront

\[\tag{23}\sin\theta=\frac{\lambda_o}{2n\Lambda}=\frac{\lambda_of}{2nV}\]

For typical AO diffraction, small-angle approximation holds, i. e., \(\sin\theta_b\approx\theta_b\). The deflection angle outside the medium is

\[\tag{24}\Delta\theta=\frac{\lambda_of}{V}\]

The deflection angle is thus linearly proportional to the acoustic frequency. The linear dispersion relation forms the basis of the AO spectrum analyzer, an important signal processing application to be discussed later.

Referring to Fig. 1a, a change of incidence angle \(\delta\theta_i\) will introduce a change of acoustic frequency (or optical wavelength) for exact phase matching. Thus, this phase matching condition for isotropic diffraction is critical to the angle of the incident light.

Anisotropic Diffraction

Next, consider the Bragg diffraction in an optically anisotropic medium such as a birefringent crystal.

The refractive index now depends on the direction as well as the polarization of the light beam. In general, the refractive indices of the incident and diffracted light beams are different.

As an example, consider the Bragg diffraction in a positive uniaxial crystal. Figure 1b shows the wavevector diagram for the acoustic coupling from an incident extraordinary wave (polarized parallel to the \(c\)-axis) to a diffracted ordinary wave (polarized perpendicular to the \(c\)-axis).

From the wave vector diagram shown in the figure and using the law of cosines one obtains:

\[\tag{25}\sin\theta_i=\frac{\lambda_o}{2n_i\Lambda}\left[1+\frac{\Lambda^2}{\lambda_o^2}(n_i^2-n_d^2)\right]\]

\[\tag{26}\sin\theta_d=\frac{\lambda_o}{2n_d\Lambda}\left[1-\frac{\Lambda^2}{\lambda_o^2}(n_i^2-n_d^2)\right]\]

Notice that the first term on the right-hand side of the preceding equations is the same as Eq. (23) and thus represents the usual Bragg condition for isotropic diffraction, while the remaining terms denote the modification due to the effect of anisotropy.

Adding Eqs. (25) and (26) yields (in the small angle approximation) the same angle as the case of isotropic diffraction. However, addition of the second term in the above equations has significantly changed the angle-frequency characteristics of AO diffraction.

In order to show the distinct characteristics of birefringent diffraction, we shall consider AO diffraction in the constant azimuth plane of a uniaxial crystal. In general the refractive indices are functions and are dependent on the direction of the propagation.

For a uniaxial crystal the refraction indices for the ordinary and extraordinary waves are \(n_o\) and \(n_e(\theta_e)\), respectively .

\[\tag{27}n_e(\theta_a)=n_o\left(\cos^2\theta_e+\frac{\sin^2\theta_e}{e^2}\right)^{-1/2}\]

where \(\theta_e\) is the polar angle of the \(e\)-wave, \(e=n_E/n_o\), \(n_E\) is the refractive index for the \(e\)-wave polarized along the \(c\)-axis. Since the indices of refraction appear in both Eq. (25) and (26), the incident and diffraction angles are not separable. In the following analysis we shall decouple these equations and derive an explicit solution for the frequency angular relations.

Apply the momentum matching condition along the \(z\)-axis and use Eq. (22) to get

\[\tag{28}n_e(\theta_e)\cos(\theta_e-\theta_c)=n_o\cos(\theta_o-\theta_c)=n_z\]

where \(\theta_c\) is the polar angle for the \(z\)-axis, \(n_z\) is the \(z\)-component of the refractive indices.

For an \(e\)-wave input, Eq. (28) can be readily solved to give the polar angle \(\theta_o\) of the diffracted light.

\[\tag{29}\theta_o=\theta_c+\cos^{-1}\left(\frac{n_z}{n_o}\right)\]

The case of \(o\)-wave input is more complicated. After some tedious but straightforward algebra, the following formula for the diffracted \(e\)-wave are obtained. Introducing

\[\tag{30}\tan\phi_c=e\tan\theta_c\]

\[\tag{31}\phi_e=\phi_c+\cos^{-1}[n_z/n_e(\phi_c)]\]

yields the polar angle of the diffracted light,

\[\tag{32}\tan\theta_e=e\tan\phi_e\]

Once the directions of the incident and diffracted light are determined, the frequency-angular characteristics of AO diffraction can be determined by Eq. (25) or (26).

As an example, consider the AO diffraction in the polar plane of a shear wave TeO\(_2\) crystal. Figure 2 shows the dependence of the incident angle \(\theta_i\) and diffraction angle \(\theta_d\) as a function of \(n_a=\lambda_o/\Lambda\), the ratio of optical and acoustic wavelengths for a specific example where the polar angle \(\theta_a\) of the acoustic waves is equal to 100\(^\circ\).

The plots exhibit two operating conditions of particular interest. The point \(\theta_i=\theta_1\), referred to as the tangential phase matching (TPM), shows at this angle there exists a wide range of acoustic frequencies that satisfy the phase matching condition. This operating condition provides the optimized design for wideband AO deflectors (or Bragg cells).

The figure also shows that there exists two operating points, \(\theta_2\) and \(\theta_3\), where the phase matching is relatively insensitive to the changes of \(\theta_i\).

The operating condition is referred to as noncritical phase matching (NPM). AO diffraction at these points exhibit a large angular aperture characteristic that is essential to tunable filter applications.

**Figure 2**. Dependence of incident and diffraction angles vs. ratio of optical and acoustic wavelengths.

Figure 4 shows incident light for ordinary (\(\theta_o\)) polarizations. The plot for incident light with extraordinary polarization is similar.

Frequency Characteristics of AO Interaction

The plane-wave analysis can be used to determine the frequency and angular characteristics of AO interaction. In this approach the acoustic wave is approximated as a single plane wave propagating normal to the transducer. The frequency or angular dependence is obtained from the phase mismatch caused by the change of acoustic frequency or incident optical wave direction. This is referred to as the phase mismatch method.

**Figure 3**. Acousto-optic wavevector diagram showing the construction of momentum mismatch.

Referring to the wavevector diagram shown in Fig. 3, the momentum match \(\Delta\mathbf{k}\) is constrained to be normal to the boundary of the medium (i. e., along the \(z\)-axis).

An approximate expression of \(\Delta{k}\) is

\[\tag{33}\Delta{k}=((k_i^2+k_a^2-k_d^2)-2k_ik_a\sin\theta_i)/2k_d\]

where \(\theta_i\) is the angle of incidence shown in Fig. 3.

According to Eq. (20) the bandshape of the AO interaction is a function of the phase mismatch.

\[\tag{34}W(\psi)=\text{sinc}^2(\psi)\]

where \(\psi=\Delta\mathbf{k}L/2\pi\) is the phase mismatch (normalized to \(2\pi\)). Substituting Eq. (22) into Eq. (33), we obtain the following expression for the phase mismatch.

\[\tag{35}\psi=\left(\frac{L}{2\lambda_on_d}\right)\left(\left(\frac{\lambda_o}{\Lambda}\right)^2-2n_i\left(\frac{\lambda_o}{\Lambda}\right)\sin\theta_i+(n_i^2-n_d^2)\right)\]

It is convenient to normalize the acoustic frequency to a center frequency \(f_o\) (wavelength \(\Lambda_o\)). In terms of the normalized acoustic frequency \(F=f/f_o\), the phase mismatch function \(\psi\) can be written as,

\[\tag{36}\psi=\left(\frac{\mathscr{l}}{2}\right)(F^2-F_bF+F_c)\]

where

\[\tag{37}\mathscr{l}=\frac{L}{L_o},\quad{L_o}=\frac{n_d\Lambda_o^2}{\lambda_o},\quad{F_b}=\frac{n_i\Lambda_o}{\lambda_o}\sin\theta_i,\quad{F_c}=\left(\frac{\Lambda_o}{\lambda_o}\right)^2(n_i^2-n_d^2)\]

Isotropic Diffraction

In this case \(n_i=n_d=n_o\), \(F_c=0\). By choosing \(F_b=1+(\Delta{F}/2)^2\), the phase mismatch function can be written as

\[\tag{38}\psi=\frac{\mathscr{l}}{2}F\left(F-1-\frac{\Delta{F^2}}{2}\right)\]

where \(\Delta{F}\) is the fractional bandwidth of the AO interaction, the diffraction efficiency reduces to 0.5 when \(\psi=0.45\). This corresponds to a fractional bandwidth

\[\tag{39}\Delta{F}=\frac{\Delta{f}}{f_o}=\frac{1.8}{\mathscr{l}}\]

To realize octave bandwidth, (\(\Delta{F}=2/3\)) for instance, the normalized interaction length \(\mathscr{l}\) is equal to 2.7. Figure 4a shows the bandshape of isotropic AO diffraction.

Birefringent Diffraction

Consider next the case of anisotropic AO diffraction in a birefringent crystal. We consider the case when \(n_i\gt{n_d}\). We put \(F_t=\sqrt{F_c}=(\Lambda_o/\lambda_o)\sqrt{n_i^2-n_d^2}\).

By choosing \(F_b=2\), the phase-mismatch function now takes the form

\[\tag{40}\psi=\frac{\mathscr{l}}{2}(F^2-2F+F_t^2)\]

Equation (40) shows that there are two frequencies where the mismatch is zero. If we choose \(F_t=1\) the two frequencies coincide. This is referred to as the tangential phase matching, (TPM) since the acoustic wavevector is tangential to the locus of the diffracted light wavevector. At the tangential frequency the AO interaction exhibits wide bandwidth characteristics.

If the center frequency is chosen to be slightly off from the TFM frequency, the AO bandpass becomes a double peak response with a dip at the center frequency. It has an even larger bandwidth.

If we choose \(F_t^2=1-\Delta{F}^2/8\), the phase mismatch at the center frequencies and the band edges will be equal. The 3 dB bandwidth is obtained by letting \(\psi=0.45\) and the fractional bandwidth of birefringent diffraction is then,

\[\tag{41}\Delta{F}=\frac{\Delta{f}}{f}=\left(\frac{7.2}{\mathscr{l}}\right)^{1/2}\]

Figure 4b shows the AO bandshape of birefringent diffraction with octave bandwidth (\(\Delta{F}=2/3\)). Notice that for octave bandwidth, the normalized interaction length is equal to 16.2. This represents an efficiency advantage factor of 6 compared to isotropic diffraction.

Acousto-optic Interaction of Finite Geometry

The plane wave analysis based on phase mismatch just presented provides an approximate theoretical description of AO interaction in the Bragg regime.

However, applicability of this method may appear to be inadequate for real devices since the interaction geometry usually involves optical and acoustic beams of finite sizes with nonuniform amplitude distributions, e. g., a Gaussian optical beam and a divergent acoustic beam.

One approach for taking into account the finite AO interaction geometry is to decompose the optical and acoustic beams into angular spectrum of plane waves and apply the plane wave solution as in a standard Fourier analysis.

To make the approach analytically tractable, a practical solution is to impose the simplifying assumption of weak interaction. Since the basic AO interaction is modeled as a filtering process in the spatial frequency domain, the approach is referred to as the frequency domain analysis.

In the case of weak interaction, there is negligible depletion of the incident light. Using the approximation of constant amplitude of the incident optical wave, Eq. (15) can be integrated to yield the diffracted optical wave.

At the far field, (\(z\rightarrow\infty\)), the plane wave amplitude of diffracted light becomes

\[\tag{42}E_d(\mathbf{k}_d)=jCS(\mathbf{k}_a)E_i(\mathbf{k}_i)\delta(\mathbf{k}_d-\mathbf{k}_i-\mathbf{k}_a)\]

The preceding equation shows that in the far field the diffracted optical wave will be nonzero only when the exact momentum-matching condition is satisfied. Thus, assuming the incident light is a wide, collimated beam, the diffracted light intensity is proportional to the power spectra of the acoustic wave components which satisfies the exact phase matching condition.

As an example we use the frequency domain approach to determine the bandpass characteristics of the birefringent AO deflector. Suppose the AO device has a single uniform transducer of length \(L\) in the \(z\) direction and height \(H\) in the \(y\) direction.

The incident optical beam is assumed to be a plane wave propagating near the \(z\) axis in the \(x\;z\) plane (referred to as the interaction plane). To take into account the finite size of the transducer, the acoustic beam is now modeled as an angular spectrum of plane waves propagating near the \(x\) axis.

\[\tag{43}S(\bar{\sigma})=S_o\cdot\text{sinc}(\sigma_xL)\;\text{sinc}(\bar{\sigma}_yH)\delta\left(\sigma_z-\frac{1}{\Lambda}-\frac{\Lambda}{2}(B\sigma_y^2+C\sigma_z^2)\right)\]

where \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are the spatial frequency components of the acoustic wavevector, \(B\) and \(C\) are the curvatures of acoustic slowness surface for the \(y\) and \(z\) directions.

We consider the AO diffraction only in the interaction plane where the phase-matching condition is satisfied. The diffracted light intensity distribution is proportional to the acoustic power spectra; i. e.,

\[\tag{44}I_d=\eta\;\text{sinc}^2\left(\frac{L}{\Lambda}\theta_a\right)=\eta\;\text{sinc}^2\psi\]

Referring to Fig. 3, we have

\[\tag{45}\psi=\frac{L}{\Lambda}(\sin\theta_i-\sin\bar{\theta}_i)\]

where the angles \(\theta_i\) and \(\bar{\theta}_i\) are the optical incidence angle and Bragg angle (at exact phase matching), respectively. Substituting the Bragg angle \(\theta_i\) from Eq. (25) into Eq. (45), we obtain the same result as Eq. (35). Thus, the frequency-domain analysis and the phase-mismatch method are equivalent.

In the above analysis the weak field approximation is used. The case of strong interaction has also been studied. However, due to the mathematical complexities involved, only numerical solutions are obtainable.

3. Acousto-Optic Materials

The significant progress of AO devices in recent years has been largely due to the development of superior materials such as TeO\(_2\) and GaP. In this section we shall review the material issues related to AO device applications. A comprehensive list of tables summarizing the properties of AO materials is presented at the end of this section.

The selection of AO materials depends on the specific device application. An AO material suited for one type of device may not even be applicable for another. For example, GaP is perhaps the best choice for making wideband AO deflectors or modulators. However, since GaP is optically isotropic, it cannot be used for tunable filter purposes.

Some of the requirements for materials’ properties apply to the more general cases of optical device applications, e. g. high optical transparency over wavelength range of interest, availability in large single crystals, etc. We shall restrict the discussion to material properties that are particularly required for AO device applications.

Acousto-optic Figures of Merit

A large AO figure of merit is desired for device applications. There are several AO figures of merit that have been used for judging the usefulness of an AO material. The relevant one to be used depends on the specific applications.

Several AO figures of merit are defined in the literature. These include:

\[\tag{46}M_1=\frac{n^7p^2}{\rho{V}}\quad{M_2}=\frac{n^6p^2}{\rho{V^3}}\quad{M_3}=\frac{n^7p^2}{\rho{V^2}}\quad{M_4}=\frac{n^8p^2V}{\rho}\quad{M_5}=\frac{n^8p^2}{\rho{V^3}}\]

where \(n\) is the index of refraction, \(p\) is the relevant elasto-optic coefficient, \(\rho\) is the density, and \(V\) is the acoustic wave velocity. These figures of merit are generally listed as the normalized quantities \(M\) (normalized to values for fused silica).

Based on Eq. (18), the figure of merit \(M_2\) relates the diffraction efficiency \(\eta\), to the acoustic power \(P_a\) for a given device aspect ratio \(L/H\).

\(M_2\) is the AO figure of merit most often referred to in the literature and is widely used for the comparison of AO materials. This is a misconception, since from the viewpoint of device applications, \(M_2\) is usually not appropriate. Comparison of AO materials (or modes) based on \(M_2\) can lead to erroneous conclusions.

\(M_2\) is used only when efficiency is the only parameter of concern. In most practical cases, other parameters such as bandwidth and resolution must also be considered. To optimize the bandwidth product, the relevant figure of merit is \(M_1\).

\[\tag{47}\eta=\frac{\pi^2}{2\lambda_o^3f^2}M_1\mathscr{l}\frac{P_a}{H}\]

where \(\mathscr{l}\) is the normalized interaction and is determined by the specified (fractional) bandwidth.

In the design of AO deflectors, or Bragg cells, besides efficiency and bandwidth, a third parameter of interest is the aperture time \(\tau\). A minimum acoustic beam height \(H\) must be chosen to insure that the aperture is within the near field of the acoustic radiation. Let \(H=hH_o\) with \(H_o=V(\tau/f)^{1/2}\).

\[\tag{48}\eta=\frac{\pi^2M_3}{2\lambda_o^3f^{3/2}\tau^{1/2}}P_a\frac{\mathscr{l}}{h}\]

For wideband AO modulators, the acoustic power density \(P_d\) is often the limiting factor. The appropriate AO figure of merit is then \(M_4\), i. e.,

\[\tag{49}\eta=\frac{\pi^2}{2\lambda_o^4f^4}M_4P_d\mathscr{l}^2\]

In the design of AO tunable filters, the parameters to be optimized are the product of efficiency \(\eta\), the resolving power \(\lambda_o/\Delta\lambda\), and the solid angular aperture \(\Delta\Omega\). In this case the appropriate AO figure of merit is \(M_5=n^2M_2\).

Acoustic Attenuation

The performance of AO devices also depends on the acoustic properties of the interaction medium. Low acoustic attenuation is desired for increased resolution of deflectors or aperture of tunable filters.

The theory of acoustic attenuation in crystals has been a subject of extensive study. Generally, at room temperature \(\omega\tau_\text{th}\ll1\), where \(\omega\) is the angular frequency of the acoustic wave and \(\tau_\text{th}\) is the thermal phonon relaxation, the dominant contribution to acoustic attenuation is due to Akhieser loss caused by relaxation of the thermal phonon distribution toward equilibrium.

A widely used result of this theory is the relation derived by Woodruff and Erhenrich. It states that acoustic attenuation measured in nepers per unit time is given by

\[\tag{50}\alpha=\frac{\gamma^2f^2\kappa{T}}{\rho{V^4}}\]

where \(\gamma\) is the Gruneisen constant, \(T\) is the temperature, and \(\kappa\) is the thermal conductivity.

Equation (50) shows that the acoustic attenuation has a quadratic frequency-dependence near to that observed in most crystals.

In practice, in some crystals such as GaP, it has been found that the frequency dependence of attenuation \(\alpha\sim{f^n}\), where \(n\) varies in different frequency ranges and has an average value between 1 and 2.

The deviation from a quadratic dependence may be attributed to the additional extrinsic attenuation caused by scattering from lattice imperfections.

Optical Birefringence

Optical birefringence is a requirement for materials used in AO tunable filters. The requirement is met by optically birefringent crystals in order that the phase matching for the AO filter interaction can be satisfied over a large angular distribution of incident light.

For AO deflectors and modulators, optical birefringence is not necessary. AO devices with high efficiency, wide bandwidth, and large resolution are realizable with superior isotropic materials such as GaP.

However, in an optically birefringent crystal it is possible to achieve tangential phase matching, which provides an enhancement of (normalized) interaction length \(\mathscr{l}\) for a given fractional bandwidth.

This represents an increased interaction length advantage, by a factor of five or more, as compared to isotropic diffraction. The advantage is particularly significant when power density is the limiting factor, since the reduction of power density is proportional to \(\mathscr{l}^2\). The optical birefringence in LiNbO\(_3\) and TeO\(_2\) have been largely responsible for the superior performance of these two AO materials.

The driving acoustic frequency corresponding to the passband wavelength in an AO tunable filter is proportional to the crystal birefringence. It is desirable to lower the acoustic frequency for simpler construction and improved performance. Therefore, for AO tunable filter applications, the birefringence is preferably small.

For wideband AO Bragg cells using tangential phase matching in birefringent crystals, a larger value of birefringence is desirable since it limits the maximum frequency for wideband operations.

Tabulation of Acousto-optic Material Properties

To aid the selection of AO materials, the relevant properties of some promising materials are listed. Table 1 lists the values of elasto-optical tensor components. Table 2 lists the relevant properties of selected AO materials. The listed figures of merit \(M_1\), \(M_2\) and \(M_3\) are normalized relative to that of fused silica, which has the following absolute values.

\[\begin{align}M_1&=7.83\times10^{-7}\quad&&[\text{cm}^2\;\text{sg}^{-1}]\\M_2&=1.51\times10^{-18}\quad&&[\text{s}^3\text{g}^{-1}]\\M_3&=1.3\times10^{-12}\quad&&[\text{cm}^2\text{s}^2\text{g}^{-1}]\end{align}\]

**Table 1. Elasto-optic Coefficients of Materials**

4. Basic Acousto-Optic Devices

In this section, we present in detail the theory and practices of AO devices. Following our previous classification, three basic types of devices will be defined by the relative divergence of the optical and acoustic beams. Let

\[\tag{51}a=\frac{\delta\theta_o}{\delta\theta_a}\]

be the divergence ratio characterizing the AO interaction geometry. In the limit \(a\ll1\), the device acts as a deflector or spatial modulator. For the intermediate value \(a\approx1\), the device serves as a (temporal) modulator. In the other limit \(a\gg1\), the device provides a spectral modulation, i. e., a tunable optical filter.

Table 3 summarizes the interaction geometry and appropriate figures of merit for the three basic AO devices.

**Table 3. Figures of Merit for Acousto-Optic Devices**

Acousto-optic Deflector

Acousto-optic interaction provides a simple means to deflect an optical beam in a sequential or random-access manner. As the driving frequency of the acoustic wave is changed, the direction of the diffracted beam can also be varied. The angle between the first-order diffracted beam and the undiffracted beam for a frequency range \(\Delta{f}\) is approximately given by (outside the medium)

\[\tag{52}\Delta\theta_d=\frac{\lambda_o\Delta{f}}{V}\]

In a deflector, the most important performance parameters are resolution and speed. Resolution, or the maximum number of resolvable spots, is defined as the ratio of the range of deflection angle divided by the angular spread of the diffracted beam, i. e.,

\[\tag{53}N=\frac{\Delta\theta}{\delta\theta_o}\]

where

\[\tag{54}\delta\theta_o=\xi\lambda_o/D\]

where \(D\) is the width of the incident beam and \(\xi\) is the a factor (near unity) that depends on the incident beam’s amplitude distribution. For a nontruncated gaussian beam \(\xi=4/\pi\). From Eqs. (52), (53), and (54) it follows that

\[\tag{55}N\approx\tau\Delta{f}\]

Where \(\tau=D/V\cos\theta_o\) is the acoustic-transit time across the optical aperture.

Notice that the acoustic-transit time also represents the (random) access time and is a measure of the speed of the deflector. Equation (55) shows that the resolution is equal to time (aperture) bandwidth product.

This is the basic tradeoff relation between resolution and speed (or bandwidth) of AO deflectors. In the design of AO deflectors, the primary goal is to obtain the highest diffraction efficiency for the specified bandwidth and resolution (or time aperture).

In the following we consider the design of AO deflectors. Figure 5 shows the geometry of an AO deflector.

**Figure 5**. Acousto-optic deflector configuration.

A piezoelectric transducer is bonded to the appropriate crystal face, oriented for efficient AO interaction. A top electrode deposited on the transducer defines the active area with interaction length \(L\) and acoustic beam height, \(H\).

An acoustic wave is launched from the transducer into the interaction medium and produces a traveling phase grating. An optical beam is incident at a proper Bragg angle with respect to the acoustic wavefront.

The incident beam is generally modeled as a gaussian profile in the interaction plane with a beam waist \(2\omega_1\) at \(1/e^2\) of the intensity. The optical beam in the interaction plane is truncated to an aperture width \(D\).

For sufficiently long interaction length \(L\), the acoustic wave diffracts a portion of the incident light into the first order. The angular spectrum of the diffracted light is proportional to the acoustic power spectrum, weighted by the truncated optical beam profile.

The acoustic beam profile in the transverse plane is determined by the transducer height \(H\), and the acoustic diffraction in the medium. The optical intensity distribution can be taken as Gaussian with waist \(2\omega_2\) at \(1/e^2\) intensity. The diffracted light intensity is then proportional to the overlapping integral of the optical beam and the acoustic diffraction profile.

Under momentum matching conditions, the peak diffraction efficiency of a Bragg cell is given by Eq. (18a)

\[\eta_o=\frac{\pi^2}{2\lambda_o^2}M_2\left(\frac{L}{H}\right)P_a\]

The diffraction efficiency can be increased by the choice of large \(L\) and small \(H\). However, the acoustic beam width (i. e., interaction length) \(L\) defines the angular spread of the acoustic power spectrum and is thus limited by the required frequency bandwidth.

The acoustic beam height \(H\) determines the transverse acoustic diffraction. A smaller value of \(H\), however, will increase the divergence of the acoustic beam in the transverse plane. An optimum acoustic beam height in the Bragg cell design is chosen so that the AO diffraction occurs within the acoustic near field.

\[\tag{56}H_o=\sqrt{BD}=V\sqrt{\frac{\tau}{f}B}\]

where \(D\) is the total optical aperture and \(B\) is the curvature of the acoustic slowness surface. Substituting Eq. (48) into Eq. (18), we obtain:

\[\tag{57}\eta_o=\frac{\pi^2P_a}{2\lambda_o^3f^{3/2}\tau^{1/2}}\left(\frac{M_3}{\sqrt{B}}\right)\cdot\mathscr{l}\]

where \(\mathscr{l}=L/L_o\) is the normalized acoustic beamwidth, and is determined by the specified fractional bandwidth. We have derived the frequency response of AO interaction. For the isotropic diffraction bandshape it was shown that, is related to the fractional bandwidth \(\Delta{F}=f/f_o\) by Eq. (39)

\[l\approx1.8/\Delta{F}\]

Equation (57) shows increased diffraction efficiency can be obtained by the selection of AO materials and modes with large effective figure of merit \(M_3^*=M_3/\sqrt{B}\) and applying techniques for increasing \(\mathscr{l}\).

The use of acoustic modes with minimum curvature is referred to as anisotropic acoustic beam collimation. A well-known example is a shear mode in GaP propagating along [110] direction. In this case \(B=0.026\).

Compared to an acoustically isotropic direction, the transducer height can be reduced by a factor of 6.2. Theoretically it is also possible to reduce the acoustic beam height by using acoustic focusing with cylindrical transducers. Since a deposited ZnO transducer is required, implementation of cylindrical acoustics is more complicated.

Another performance enhancement technique is to increase, by using AO diffraction in birefringent crystals. In this case it is possible to choose the acoustic wave vector to be approximately tangential to the locus of the diffracted light vector.

As a result, a large band of acoustic frequencies will simultaneously satisfy the momentum-matching condition. Equivalently, for a given bandwidth, a larger interaction length can be used, thus yielding an enhancement of diffraction efficiency. The normalized interaction length is related to the fractional bandwidth by Eq. (33).

\[\mathscr{l}=\frac{7.2}{\Delta{F^2}}\]

Compared to isotropic diffraction, the birefringent phase matching achieves an efficiency advantage factor of \(4/\Delta{F}\), which becomes particularly significant for smaller fractional bandwidths.

Two types of birefringent phase matching are possible; these include the acoustically-rotated (AR) and optically-rotated (OR) phase matching.

Figure 6a shows the wave vector diagram for AR tangential phase matching where the constant azimuth plane is chosen as the interaction plane and the acoustic wave vector is rotated in the plane to be tangential to the locus of the diffracted light wave vector.

The wave vector diagram shown in Fig. 6b describes the OR type of tangential phase matching. In this case the acoustic wave vector is chosen to be perpendicular to the optic axis. The incident light wave vector is allowed to rotate in different polar angles to achieve tangential phase matching at desired frequencies.

For both types of phase-matching schemes the tangential matching frequency, \(f_t=F_tf_o\), is given by

\[\tag{58}f_t=\frac{V(\theta_a)}{\lambda_o}\sqrt{n_i^2(\theta_i)-n_d^2(\theta_d)}\]

where \(\theta_i\), \(\theta_d\) and \(\theta_a\) denote the directions of the incident optical wave, diffracted optical wave, and acoustic wave, respectively.

**Figure 6**. Wave vector diagram for tangential phase matching. (a) Acoustically rotated geometry, (b) optically rotated geometry.

Another technique for increasing, while maintaining the bandwidth is to use acoustic beam steering. In this approach a phase array of transducers is used so that the composite acoustic wavefront will effectively track the Bragg angle.

The simplest phase array employs fixed inter relevant phase difference that corresponds to an acoustic delay of \(P\Lambda/2\), where \(P\) is an integer. This is referred to as first-order beam steering and can be realized in either a stepped array or a planar configuration.

The two types of phased array configurations are shown in Fig. 7. The stepped array configuration is more efficient; but less practical due to the fabrication difficulty. The following analysis addresses only the planar configuration of first-order beam steering.

**Figure 7**. Acoustic beam steering using phased array. (a) Stepped phased array, (b) planar phased array.

Consider the simplest geometry of a planar first-order beam-steered transducer array where each element is driven with an interelement phase difference of 180\(^\circ\). The acousto-optic bandpass response of this transmitter configuration is equal to the single-element bandshape multiplied by the interference (array) function; i. e.,

\[\tag{59}W(F)=\left(\frac{\sin\pi{X}}{\pi{X}}\right)^2\left(\frac{\sin{N\pi}Y}{N\sin\pi{Y}}\right)^2\]

\[\tag{60}X=\frac{L_eF}{2L_o}(F_b-F)\]

\[\tag{61}Y=\frac{1}{2}\left\{\frac{D}{L_o}F(F_b-F)+1\right\}\]

and \(L_e\) is the length of one element, \(N\) is the number of elements, where \(D\) is the center-to-center distance between adjacent elements.

For large \(N\), the radiation pattern for a single element is broad, the bandpass function is primarily determined by the grating (array) functions. The array function can be approximately given by Eq. (34) except that the phase mismatch is given by:

\[\tag{62}\psi=\frac{Nd}{2}\left\{F(F_b-F)+\frac{1}{d}\right\}\]

where \(d=D/L_o\).

The bandpass characteristics of the grating loss are the same as the birefringent diffraction case with an equivalent interaction length \(\mathscr{l}=Nd\) and tangential matching frequency \(F_1=1/\sqrt{d}\). The discussion on the interaction length-bandwidth relation of birefringent diffraction is thus directly applicable.

Referring to Eq. (59), notice that at the peak of the grating lobe for the phase-array radiation, the value of the single element radiation \(\text{sinc}^2x\) is approximately equal to 0.5. There is thus an additional 3-dB loss due to the planar phase array.

An interesting design is to combine the preceding techniques of tangential phase matching and acoustic beam steering in a birefringent phased-array Bragg cell. The approach allows a higher degree of freedom in the choice of acoustic and optic modes for optimized \(\mathbf{M}_3^*\), shifting of center frequency, and suppression of multiple AO diffractions.

Both techniques discussed here, decreasing acoustic beam height and increasing interaction length, allow the reduction of drive power required for obtaining a given diffraction efficiency.

The increase of \(\mathscr{l}\) is particularly significant since the power density is proportional to \(\mathscr{l}^2\). For instance, an increase of \(\mathscr{l}\) by a factor of 7 will reduce the power density by 50 times!

In most wideband AO cells high power density has been the dominant factor limiting the device performance. The deterious effect due to high power density includes thermal, gradient nonlinear acoustics and possible transducer failure.

Besides bandwidth, there are other factors that limit the usable time aperture of AO deflectors: maximum available crystal size, requirement of large optics, and, most basically, the acoustic attenuation across the aperture.

For most crystalline solids, the acoustic attenuation is proportional to \(f^2\). If we allow a change of average attenuation of \(\mathscr{L}\) (dB) across the band, the maximum deflector resolution is given by

\[\tag{63}N_\text{max}=\frac{\mathscr{L}}{\alpha_of_o}\]

where \(\alpha_o\) (dB/\(\mu\)sec GHz\(^2\)) is the acoustic attenuation coefficient.

It can be shown that the acoustic loss has negligible effects on the resolvable spot of the deflector. For most practical cases, a more severe problem associated with the acoustic loss is the thermal distortion resulting from heating of the deflector. The allowable acoustic loss thus depends upon other factors, such as the acoustic power level, thermal conductivity of the deflector material, etc.

It is instructive to estimate the maximum resolution achievable of AO deflectors. In the calculation, the following assumptions are made: octave bandwidth: \(f_o=1.5\Delta{f}\), maximum acoustic loss: \(\mathscr{L}=4\) dB, maximum aperture size: \(D=5\) cm.

Results of the calculation are summarized in Fig. 8 where the deflector resolution is plotted for a number of selected AO materials. The figure clearly shows the basic tradeoff relation between the speed and the resolution of AO deflectors. A large number of resolvable spots is obtainable for low-bandwidth detectors using materials with slow acoustic velocities such as TeO\(_2\) and Hg\(_2\)Cl\(_2\). The maximum resolution of AO deflectors is limited to a few thousand.

For certain applications such as laser scanning, a few thousand resolvable spots are insufficient, and further increase of deflector resolution is desirable. One early design for increased resolution involves the use of cascade deflectors.

Another technique is to use higher-order AO diffractions. One interesting design was to utilize the second-order diffraction in an OR-type birefringent cell. Since both the first- and second-order diffraction are degenerately phase matched, efficient rediffraction of the first order into the second order was obtained. However, the use of the second-order diffraction allows the deflector resolution to be doubled for a given bandwidth and time aperture.

**Figure 8**. Resolution vs. access time for acousto-optic deflectors.

Acousto-optic Modulator

The acousto-optic interaction has also been used to modulate light. In order to match the Bragg condition over the modulator bandwidth, the acoustic beam should be made narrow, as in the case of deflectors.

Unlike the case of deflectors, however, the optical beam should also have a divergence approximately equal to that of the acoustic beam, so that the carrier and the sidebands in the diffracted light will mix collinearly at the detector to give the intensity modulation.

Roughly speaking, the optical beam should be about equal to that of the acoustic beam, i. e., \(a=\delta\theta_o/\delta\theta_a\approx1\). The actual value of the divergence ratio depends on the tradeoff between desired efficiency and modulation bandwidth.

The divergent optical beam can be obtained by using focusing optics. Figure 9 shows the diffraction geometry of a focused-beam AO modulator.

**Figure 9**. Diffraction geometry of acousto-optic modulator.

In most practical cases the incident laser beam is a focused gaussian beam with a beam waist of diameter \(d\). The corresponding optical beam divergence is

\[\tag{64}\delta\theta_o=\frac{4\lambda_o}{\pi{nd}}\]

The acoustic wave generated from a flat transducer is assumed to have a uniform amplitude distribution of width \(L\) and height \(H\). The corresponding acoustic beam divergence in the interaction plane is

\[\tag{65}\delta\theta_a=\frac{\Lambda}{L}\]

The ratio of optical divergence to acoustic divergence becomes

\[\tag{66}a=\frac{\delta\theta_o}{\delta\theta_a}=\frac{4\mathscr{l}}{\pi{f_o}\tau}\]

where \(\tau=d/v\) is the acoustic transit time across the optical aperture. The characteristics of the AO modulator depend on \(a\) and \(\mathscr{l}\), and thus the primary task in the design of the modulator is to choose these two parameters so that they will best meet the device specifications.

In the following we consider the design of a focused beam AO modulator for analog modulation. We assume the incident laser beam profile is Gaussian, and calculate its diffraction by an amplitude-modulated (AM) acoustic wave.

Three acoustic waves, the carrier, the upper, and the lower sidebands will generate three correspondingly diffracted light waves traveling in separate directions. The modulated light intensity is determined by the overlapping collinear heterodyning of the diffracted optical carrier beam and the two sidebands.

Using the frequency domain analysis, the diffracted light amplitudes can be calculated. Figure 10 shows the calculated AO modulation bandwidth as a function of optical-to-acoustic divergence ratio.

**Figure 10**. Modulation bandwidth \(\times\) acoustic transit time vs. ratio of optical and acoustic beam divergence.

In the limit \(a\ll1\), the modulation bandwidth approaches the value

\[\tag{67}f_m\approx0.75/\tau\]

where \(\tau=d/V\) is the acoustic transit time.

Equation (60) shows that the modulation bandwidth may be increased by reducing the size of the optical beam. Further reduction of the beam size, however, will increase the angular spread of the optical beam \(\delta\theta_o\) to become greater than \(\delta\theta_a\); i. e., \(a\gt1\).

The divergence angle of the diffracted beam is now determined by the acoustic divergence \(\delta\theta_a\) through the momentum-matching condition. Thus, the modulation bandwidth starts to decrease as a increases. On the other hand, a larger value of a increases the interaction length \(l\) and thus the efficiency. A plot of the product of bandwidth and efficiency shows a broad maximum near \(a=15\).

In some laser systems it is desirable that the AO modulation does not introduce noticeable change to the guassian distribution of the incident beam. To satisfy this more restricted requirement an even smaller value for \(a\) should be chosen.

The effect of the parameter \(a\) on the eccentricity of the diffracted beam was analyzed based on numerical calculation. The result shows that to limit the eccentricity to less than 10 percent the divergence ratio value for a must be about 0.67. In this region, \(0.67\lt{a}\lt1.5\), the modulation bandwidth is approximately given by

\[\tag{68}f_m=0.7/\tau\]

Another important case is the digital, or pulse modulation. Maydan calculated the rise time and efficiency of pulsed AO modulators. His results show that an optimized choice of \(a\) is equal to 1.5, and that the corresponding rise time (10 to 90%) is

\[\tag{69}t_r=0.85\tau\]

An additional constraint in the design of AO modulators is that the diffracted beam must be separated from the incident beam. To obtain an adequate extinction ratio, the angle of separation is chosen to be equal to twice that of the optical beam divergence. It follows that the acoustic frequency must be greater than

\[\tag{70}f_o=\frac{8}{\pi\tau}\]

Comparing to Eq. (68), the center frequency of the AO modulator should be about 4 times that of the modulation frequency.

The focused-beam-type AO modulator has certain disadvantages. The diffraction spread associated with the narrow optical beam tends to lower the diffraction efficiency. More importantly, the focusing of the incident beam results in a high peak intensity that can cause optical damage for even relatively low laser power levels.

For these reasons, it is desirable to open up the optical aperture. Due to the basic issue of acoustic transit time, the temporal bandwidth of the wide-beam AO modulator will be severely degraded.

In certain applications, such as the laser display system, it is possible to use a much broader optical beam in the modulator than that which would be allowed by the transit time limitation.

The operation of the wide-beam modulator is based on the ingenious technique of scophony light modulation. A brief description of the scophony light modulator is presented as follows.

The basic idea, applicable to any system which scans a line at a uniform scan velocity, is to illuminate a number of picture elements, or pixels, in the modulator (window) onto the output line, such that the moving video signal in the modulator produces a corresponding image of the pixels which travels across the beam at sound velocity.

The image can be made stationary by directing the image through a deflector that scans with equal and opposite velocity. Now, if the window contains \(N\) spots, then \(N\) picture elements can be simultaneously exposed in the image at any instant, and each picture element will be built up over the access time of the window which is equal to \(N\) times the spot time.

Since the spots are immobilized, there is no loss of resolution in the image, provided that the modulator bandwidth is sufficient to produce the required resolution. The design of the wide-beam AO modulator is thus the same as that of the Bragg cell.

Acousto-optic Tunable Filter

The acousto-optic tunable filter (AOTF) is an all-solid-state optical filter that operates on the principle of acousto-optic diffraction in an anisotropic medium.

The center wavelength of the filter passband can be rapidly tuned across a wide spectral range by changing the frequency of the applied radio frequency (RF) signal.

In addition to the electronic tunability, other outstanding features of the AOTF include: large angular aperture while maintaining high spectral resolution, inherent intensity, and wavelength modulation capability.

The first AOTF, proposed by Harris and Wallace, used a configuration in which the interacting optical and acoustic waves were collinear. Later, the AOTF concept was enlarged by Chang in a noncollinear configuration.

**Figure 11**. Collinear acousto-optic tunable filter. (a) Transmissive configuration, (b) wave vector diagram.

Figure 11a shows the schematic of a transmissive-type collinear AOTF. It consists of a birefringent crystal onto which a piezoelectric transducer is bonded. When an RF signal is applied, the transducer launches an acoustic wave which is reflected by the acoustic prism and travels along a principle axis of the crystal.

The incident optical beam, passing through the input polarizer, propagates along the crystal axis and interacts collinearly with the acoustic waves. At a fixed RF frequency, only a narrow band of optical waves is diffracted into the orthogonal polarization and is selected by the output analyzer.

The center wavelength of the passband is determined by the momentum matching condition. Figure 11b shows the wave-vector diagram for the collinear AO interaction in a uniaxial crystal.

In this case, Eq. (21) yields a relation between the center of the passband and the acoustic frequency

\[\tag{71}\lambda_o=\frac{V\Delta{n}}{f}\]

where \(\Delta{n}\) is the birefringence. Equation (71) shows that the passband wavelength can be tuned simply by changing the frequency of the RF signal.

Experimental demonstrations of the collinear AOTF were reported by Harris and coworkers in a series of papers. For instance, a collinear AOTF was operated in the visible using CaMoO\(_4\) as the interaction medium.

The full width at half-maximum (FWHM) of the filter passband was measured to be 8 Å with an input light cone angle of \(\pm4.8^\circ\) (F/6). This angular aperture is more than one order of magnitude larger than that of a grating for the same spectral resolution.

The collinearity requirement limits the AOTF materials to rather restricted classes of crystals. Some of the most efficient AO materials (e. g., TeO\(_2\)) are not applicable for the collinear configuration. To utilize such materials, a new AOTF configuration was proposed in which the acoustic and optical waves are noncollinear.

**Figure 12**. Schematic of noncollinear acousto-optic tunable filter.

Figure 12 shows the schematic of a noncollinear AOTF. The use of the noncollinear geometry has the significant advantage of simpler fabrication procedures. In addition, when the filtered narrowband beam is spatially separated from the incident broadband light, the noncollinear AOTF can be operated without the use of polarizers.

The basic concept of the noncollinear AOTF is shown by the wave vector diagram in Fig. 13 for an acoustically rotated AO interaction in a uniaxial crystal. The acoustic wave vector is so chosen that the tangents to the incident and diffracted light wave vector loci are parallel.

When the parallel tangents condition is met, the phase mismatch due to the change of angle incidence is compensated by the angular change of birefringence. The AO diffraction thus becomes relatively insensitive to the angle of light incidence, i. e., a process referred to as the noncritical phase matching (NPM) condition. Figure 13 also shows the NPM scheme for the special case of the collinear AOTF.

**Figure 13**. Wave vector diagram for noncollinear AOTF showing noncritical phase matching (NPM).

The first experimental demonstration of the noncollinear AOTF was reported for the visible spectral region using TeO\(_2\) as the filter medium. The filter had a FWHM of 4 nm at an F/6 aperture. The center wavelength is tunable from 700 to 450 nm as the RF frequency is changed from 100 to 180 MHz. Nearly 100 percent of the incident light is diffracted with a drive power of 120 mW. The filtered beam is separated from the incident beam with an angle of about 6\(^\circ\).

The plane wave analysis of birefringent AO diffraction presented earlier can be used to determine the wavelength and angular bandpass characteristics of the AOTF. For proper operation, the requirement of NPM must be satisfied, i. e., the tangents to the optical wave vector surfaces are parallel. The parallel tangents condition is the equivalent of collinearity of ordinary and extraordinary rays, i. e.,

\[\tag{72}\tan\theta_e=e^2\tan\theta_o\]

where \(e=n_E/n_o\), \(n_E\) is the refractive index of the extraordinary wave polarized along the optic \(c\)-axis. The momentum matching condition is

\[\tag{73}\tan\theta_a=(n_e\sin\theta_e-n_o\sin\theta_o)/(n_e\cos\theta_e-n_o\cos\theta_o)\]

\[\tag{74}f_a=(V/\lambda_o)(n_e^2+n_o^2-2n_en_o\cos(\theta_e-\theta_o))^{1/2}\]

where \(n_e(\theta_e)\) is the refractive index for the extraordinary wave with polar angle \(\theta_e\), i. e.,

\[\tag{75}n_e(\theta_e)=n_o(\cos^2\theta_e+(\sin^2\theta_e)/e^2)^{-1/2}\]

Substituting Eqs. (72) and (75) into Eq. (73) the acoustic wave angle, \(\theta_a\), can be written explicitly as a function of the optical wave angle, \(\theta_o\).

\[\tag{76}\tan\theta_a=\frac{-(\cos\theta_o+\sqrt{1+e^2\tan^2\theta_o})}{\sin\theta_o}\]

For small birefringence \(\Delta{n}=|n_c-n_o|\ll{n_o}\), an approximate solution for the acoustic frequency is

\[\tag{77}f_a=(V\Delta{n}/\lambda_o)(\sin^4\theta_e+\sin^22\theta_e)^{1/2}\]

where \(\theta_e=90^\circ\). Equation (77) reduces to the simple expression for the collinear AOTF (Eq. (71)).

The bandpass characteristics of the AOTF are determined by the growth of optical waves that nearly satisfy the phase-matching condition over the finite interaction length. For an acoustic column of uniform amplitude, the bandpass response is given by

\[\tag{78}T(\lambda_o)=T_o\;\text{Sinc}^2\sigma{L}\]

where \(T_o\) is the peak transmission at exact momentum matching, \(L\) is the interaction length, and \(\sigma\) is the momentum mismatch. It can be shown that

\[\tag{79}\sigma=-b\sin^2\theta_i(\Delta\lambda/\lambda_o^2)+(\Delta{n}/2\lambda_o)[F_1(\Delta\theta)^2+F_2(\sin\theta_i\Delta\phi)^2]\]

where \(\Delta\lambda\), \(\Delta\theta\), and \(\Delta\phi\) are deviations in wavelength, polar, and azimuth angles of the incident light beam, \(b\) is the dispersion constant defined by

\[\tag{80}b=\Delta{n}-\lambda_o\delta/\delta\lambda_o(\Delta{n})\]

and \(F_1=2\cos^2\theta_i-\sin^2\theta_i\), \(F_2=2\cos^2\theta_i+\sin^2\theta_i\).

Resolution. Equation (78) shows that half peak transmission occurs when \(\sigma{L}\approx0.44\). The full width at half-maximum (FWHM) of the AOTF is

\[\tag{81}\Delta\lambda=0.9\lambda_o^2/bL\sin^2\theta_i\]

Angular Aperture. The acceptance (half) angles in the polar and azimuth planes are given by:

\[\tag{82}\Delta\theta=\pm{n}(\lambda_o/\Delta{n}LF_1)^{1/2}\]

\[\tag{83}\Delta\phi=\pm{n}(\lambda_o/\Delta{n}LF_2)^{1/2}\]

The frequency domain method described previously can now be used to determine the AOTF bandpass characteristics for an incident cone of light. The filter transmission for normalized input light distribution \(I(\theta_i,\phi_i)\) is obtained by integrating the plane-wave transmission [Eq. (78)] over the solid angle aperture

\[\tag{84}I_d(\Delta\lambda)=\int_{\alpha_i,\;\beta_i}T(\Delta\lambda,\theta_i,\phi_i)I(\theta_i,\phi_i)\sin\theta_i\;d\theta_i,\;d\phi_i\]

Angle of Deflection. For noncollinear AOTF the diffracted light is spatially separate from the incident light. The angle of deflection can be determined from Eq. (72). For small birefringence, an approximate expression for the deflection angle is given by

\[\tag{85}\Delta\theta_d=\Delta{n}\sin2\theta_o\]

It reduces maximum when \(\theta_o=45^\circ\). For instance, in a TeO\(_2\) AOTF operated at 633 nm, \(\Delta{n}\approx0.15\), the maximum deflection angle for TeO\(_2\) AOTF is about 8.6 degrees.

As long as the angular aperture \(\Delta\theta\) is smaller than the deflection angle \(\Delta\theta_d\), the noncollinear AOTF can realize the angular aperture without the use of polarizers. Notice that if the small birefringent dispersion is ignored, \(\Delta\theta_d\) is independent of wavelength. This feature is important to spectral imaging applications.

Transmission and Drive Power. An important parameter in the design of an AOTF is the required drive power. The peak transmission of an AOTF is given by

\[\tag{86}T_o=\sin^2\left(\frac{\pi^2}{2\lambda_o^2}M_2P_dL^2\right)^{1/2}\]

where \(P_d\) is the acoustic power density. Maximum transmission occurs when the drive power reaches the value

\[\tag{87}P_a=\frac{\lambda_o^2A}{2M_2L^2}\]

where \(A\) is the optical aperture. The high drive power required for infrared AOTFs (particularly large-aperture imaging types) is perhaps the most severe limitation of this important device.

5. Applications

Acousto-optic devices provide spatial, temporal, and spectral modulation of light. As such, they can be used in a variety of optical systems for optical beam control and signal processing applications.

In this section we shall discuss the application of AO devices. The discussion will be limited to these cases where the AO devices have been successfully deployed and appear most promising for future development.