# 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.

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 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.

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.

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

## 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.

### 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.

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.

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.

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.

### 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.

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.

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 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 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.

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.

### Acousto-optic Deflector

The early development of the AO deflector was aimed at laser beam scanning applications. The primary goal was to realize a large number of resolvable spots so that it may be used to replace mechanical scanners such as the rotating polygons.

In recent years, however, most of the effort was directed toward Bragg cells for optical signal processing applications. In addition to the high-resolution AO deflectors, the most intensive activity has been the development of wideband Bragg cells.

In the following we discuss in detail the various system applications of AO deflectors. First, we shall briefly discuss the present status of both types of AO deflectors (or Bragg cells). A summary of the representative devices is shown in Table 4.

**High-resolution Cell**

One of the earlier designs of high-resolution cells was first demonstrated by Warner et al. using the low-shear [110] mode in TeO\(_2\). Sizable reduction of drive power was achieved by using tangential phase matching of the birefringent diffraction. Due to the slow velocity along [110] axis, \(V=0.62\) mm/\(\mu\)sec, the optical aperture size of a 70 \(\mu\)sec device is only 4.3 cm.

The on-axis has two drawbacks.

First, the tangential frequency \(f_t\), determined by Eq. (58), is relatively low. At 633 nm, \(f_t\) is equal to 37 MHz. This limits the usable bandwidth to below 25 MHz.

Second, at the tangential phase-matching frequency, the second-order diffraction is also matched. The first-order diffracted light is rediffracted into the second order and results in a dip at the midband of the deflector.

By utilizing the acoustically rotated (AR)-type birefringent diffraction, Yano et al. demonstrated high efficient diffraction in a TeO\(_2\) cell without the midband dip. The primary drawbacks of the AR-type cell are the reduced aperture (and thus resolution) and significant optical aberration due to the large acoustic beam walkoff in TeO\(_2\).

Another technique of raising the tangential phase-matching frequency is to use the optically rotated (OR) configuration. Since the acoustic wave is along the principal axis, in this case, there is no acoustic beam walkoff, and high resolution is obtainable.

The upshift in \(f_t\) allows wider bandwidth to be realized. The optical aperture along the [110] axis is limited by the acoustic attenuation. The device shown in Table 4 is designed to operate at the tangential phase-matching frequency of 90 MHz with a 50-MHz bandwidth.

For an aperture size of 2.5 cm (or time aperture of 40 \(\mu\)sec), the OR-type TeO\(_2\) device shown in Table 4 has an overall resolution of about 2000 spots. A larger resolution can be realized with a longer optical aperture. Figure 8 shows a maximum resolution of 3500 spots for 5-cm aperture.

In recent years several new AO materials with exceptionally low acoustic velocities were developed. The acoustic velocities of slow shear waves in Hg\(_2\)Cl\(_2\) and Hg\(_2\)Br\(_2\) along the [110] direction are 0.347 mm/\(\mu\)sec and 0.282 mm/\(\mu\)sec, respectively.

Very large time apertures are obtainable with crystals of moderate size. As shown in Fig. 9, an AO deflector resolution of 5300 spots is projected for a 5-cm-long Hg\(_2\)Cl\(_2\) crystal along the [110] direction.

Recently, an experimental Hg\(_2\)Cl\(_2\) device was constructed that had a measured bandwidth of 30 MHz. Due to difficulties in the device fabrication and the large curvature of the slow acoustic mode, relatively small time aperture was realized. The potential of the exceedingly large deflector resolution as predicted in Fig. 9 is yet to be experimentally demonstrated.

**Wide Bandwidth Cell**

Two types of wideband Bragg cells have been developed. These include the phased array GaP cell and the birefringent LiNbO\(_3\) cell. The material GaP has very large figure of merit \(M_3\) and relatively low acoustic attenuation.

Two modes of particular interest are the L [111] and S [100] modes. Both have low acoustic slowness curvature and thus allow efficiency enhancement due to anisotropic beam confinement. The effective figure of merit \(M_3/\sqrt{B}\) for L [111] and S [100] are equal to 98 and 48, respectively.

The wideband GaP device also achieves an efficiency enhancement by using acoustic beam steering. To simplify fabrication of the transducer, planar phased-array configuration was used. The achievable efficiency and bandwidth of these phased array GaP devices are believed to be best state-of-the-art performance.

AO Bragg cells with even larger bandwidths have been demonstrated using the low attenuation materials. The best known design was a LiNbO\(_3\) device using a \(y-z\) propagating off-axis \(x\)-polarized shear wave. The device demonstrated an overall bandwidth of 2 GHz, a peak diffraction efficiency of 12 percent/watt and about 600 resolvable spots.

**Multichannel Cell**

As an extension to usual single-channel configuration there has been considerable activity in the development of multichannel Bragg cells (MCBC). The MCBC uses a pattern of multiple individually addressed transducer electrodes in the transverse plane.

The use of the MCBC allows the implementation of compact two-dimensional optical signal-processing systems such as optical page composers, direction-of-arrival processors, etc.

In addition to the design rules used for single-channel cells, other performance parameters such as crosstalk, amplitude, and phase tracking must be considered.

Another critical issue is power handling. Since the drive power is proportional to the number of channels, the excessive amount of heat dissipation at the transducer will introduce a thermal gradient that severely degrades the Bragg cell performance.

Early work on MCBC uses the \(y-z\) cut shear-wave wideband LiNbO\(_3\) cell design. Good amplitude and phase tracking were obtained over the operating bandwidth of 1 GHz. The channel-to-channel isolation, typically about 25 dB, was limited by RF crosstalk.

In a later development of GaP MCBC, reduction of electrical crosstalk (to -40 dB) was obtained by using stripline interconnection structures. The use of the anisotropic collimating modes in GaP also brings the advantage of lowering acoustic crosstalk.

**Laser Scanning**

There are a variety of laser applications that require laser beam scanning. Acousto-optic deflectors provide a simple solid state scanner which eliminates the inherent drawbacks of mechanical scanners due to moving parts such as facet errors and the requirement of realignment because of bearing wear.

For certain applications such as beam-addressed optical memory, the rapid random access capability of AO deflectors offers a distinct advantage.

One of the first practical AO deflector applications was a laser display system described by Korpel et al. The AO device used water as the interaction medium. Due to its high acoustic attenuation, the water cell had limited bandwidth.

By incorporating superior AO materials such as PbMoO\(_4\) and TeO\(_2\), a number of laser scanning systems were later developed that demonstrated significant performance improvements.

A notable example was the TV rate laser scanner described by Goreg et al. With a drive power as low as 50 mW, the TeO\(_2\) deflector operated as a horizontal scanner and achieved the specified TV rate and 500 resolvable spots.

We have shown (Fig. 9) that using optimized designs, the AO deflector can achieve 2000 to 4000 resolvable spots with good efficiency and acceptable access time. In the late seventies, such AO deflectors became commercially available. Using a 2000-spot AO deflector, Grossman and Redder demonstrated a high-speed laser facsimile scanner with 240 pixel/in resolution.

An intriguing approach to significantly increase the deflector resolution is the use of a traveling lens. The basic concept, proposed by Foster et al., utilizes the refractive effect of an acoustic pulse to form a traveling lens.

The laser beam from a primary deflector passes through a second cell that has an acoustic pulse traveling across the optical aperture in synchronicity with the deflected beam from the first cell. The additional focusing by the traveling acoustic lens reduces the final spot size and thus increases the overall number of resolvable spots.

The concept of the traveling wave acoustic lens was experimentally demonstrated with an enhancement factor from about 10 to 40. Combined with a primary deflector with moderate resolution, this technique should be capable of achieving more than 10,000 pixels/scan line resolution. One major drawback of this device is the relatively high drive power required.

In the design of the AO deflector for linear scanning, one must consider the effect due to finite RF sweep rate. An important characteristic of linear frequency modulation (LFM) operation is the cylindrical focusing effect.

When an LFM acoustic pulse is passing through the cell, the incoming beam is diffracted by an angle \(\theta=\lambda_of/V\), where \(f\) is the frequency of the local acoustic wave.

At a distance \(\Delta{x}\) away, the angle of diffraction will change by \(\Delta\theta=\lambda_o/V\Delta{f}\) where \(\Delta{f}\) is the corresponding change of acoustic frequency.

The diffracted rays will focus to a point located at a distance \(F\) from the center of the cell

\[\tag{88}F=\frac{\Delta{x}}{\Delta\theta}=V^2/\left(\lambda_o\frac{\Delta{f}}{T}\right)\]

range \(\Delta{f}\) where \(\Delta{f}/T\) is the linear scan rate and \(T\) is the time for sweeping over frequency. An additional lens may be used to compensate for the lens effect.

The finite scan rate also degrades the resolution of the AO deflector. When the acoustic transit time \(\tau\) (which is equal to flyback time) becomes an appreciable portion of the scan time \(T\), there is a reduction of the effective aperture by the factor \(1-\tau/T\). This results in a resolvable number of spots for the scanning mode,

\[\tag{89}N=\tau\Delta{f}(1-\tau/T)\]

To avoid loss of resolution the general practice is to choose \(\tau\ll{T}\). This limits the speed or resolution of the scanner.

Another method is the AO chirp scanner. The basic concept is to exploit the cylindrical lens effect described above.

An LFM or chirp RF with a duration less than the acoustic transit time is applied to the transducer. As the acoustic chirp travels through the cell, due to the cylindrical lens effect, it produces a focused spot which moves down in the image plane with the acoustic wave velocity.

To make efficient use of the laser power a prescanner is used to track the traveling chirp. For a chirp pulse of duration \(\tau_c\), the size of the focused spot is equal to the angular size of the incident beam multiplied by the focal length.

Setting the sweep time equal to \(\tau_c\) in Eq. (88) yields

\[\tag{90}d=\frac{F\lambda_o}{V\tau_c}=\frac{V}{\Delta{f}}\]

The total number of resolvable spots is approximately equal to \(\Delta{f}\tau\) if \(\tau_c\ll\tau\). The chirp scanner thus avoids the degradation resolution by reducing the duration of the chirp over the bandwidth \(\Delta{f}\). The technical difficulty of the chirp scanner is the requirement of a stable fast chirp with extremely linear frequency scanning characteristics.

Despite these promising results, the AO deflector has not been able to compete with mechanical scanners such as rotating polygons for general laser scanning applications.

Limited resolution, improved reliability, and, in particular, high cost have been the primary factors. As a result, in more recent developments, the AO deflectors are designed to be used in laser scanning systems for acquiring extremely wide bandwidth, random access capability, or to provide auxiliary functions to prime scanners such as wobble correction and facet tracking.

### Acousto-optic Modulator

Among the three basic types of AO devices, the AO modulator has been most well developed in the marketplace. A variety of AO modulators is commercially available that is suited to external or intracavity applications.

Compared to the competing electro-optic (EO) modulators, the AO modulator has many advantages that include low drive power, high extinction ratio, insensitivity to temperature change, simple drive electronics, and high safety factors.

Since the AO effect occurs in all crystal types as well as amorphous solids, and high optical quality, low-loss AO materials are readily available for intracavity applications.

Except in certain applications where very large bandwidths are required, AO modulators are generally preferred to their EO counterparts.

The AO modulators can be broadly classified into three categories: simple intensity modulators, intracavity modulators, and AO frequency shifters.

**Intensity Modulators**

Simple intensity AO modulators are used for general-purpose applications. The lower-cost types, usually made of glass (dense flint or Tellurite glass), are useful for bandwidths of up to about 10 MHz.

The use of superior materials such as PbMoO\(_4\) and TeO\(_2\) has extended modulation bandwidth to about 50 MHz. Chang and Hecht demonstrated the first GaP modulator with a risetime of 5 nsec. Later development has reduced the risetime to about 2 nsec, which corresponds to a modulation bandwidth of about 250 MHz. Table 5 lists a few selected AO modulators and some of the typical performances.

Historically, the most important market for AO modulators is for use in laser printers. Before the development of organic photoreceptors with spectral response extended to the near infrared, gas lasers such as HeCd or HeNe were used in laser printers. The requirement of an external modulator can be best met by the simple, low-cost AO modulator. The laser printer generally uses a flying-spot scanner configuration with the AO modulator operated in the focused beam mode in order to achieve the desired rise time.

Since the development of laser printers that use laser diodes as the optical source, the internal modulation capability of laser diodes has eliminated the need for external modulators. Therefore, for future laser printer applications, the use of AO modulators will probably be limited to cases where gas lasers are employed, or where the AO device offers a unique advantage. For example, the use of an AO device to perform combined deflector and modulation functions in the multibeam fast, dither scanners.

The AO modulators have also been used for laser modulation in the infrared. The development of efficient infrared AO materials has resulted in various laser modulation uses, ranging from 1.06 \(\mu\)m to 10.6 \(\mu\)m. One important application of AO modulators is to serve as an external modulator in laser communication systems.

For lower modulation bandwidth requirements, AO modulators are preferred to their EO counterparts due to lower optical insertion loss or greater contrast ratio. Efficient and fast miniature AO modulators suited to 1.06 \(\mu\)m can be constructed using GaP, GaAs, or GeAsSe.

Among these, GeAsSe glass is particularly suited to wideband uses due to its exceptionally low acoustic attenuations (\(\alpha_o=1.7\) dB/nsec GHz\(^2\)). Although it is slightly less efficient than both GaP and GaAs, it offers the important advantage of lower cost.

AO modulators have also been used to modulate CO\(_2\) lasers for various applications such as precision machining, range finding, and communications. At 10.6 microns, the most popular material has been single-crystal Ge. Early work of Ge modulators using longitudinal mode along [111] axes gave a high, large figure of merit (\(M_2=540\)) with respect to fused silica. Later, more accurate measurements have shown a smaller figure of merit (\(\mathbf{M}_2=120\)). Typical performance of the Ge modulator shows a risetime of 30 nsec and a diffraction efficiency of 5 percent/watt.

**Intracavity AO Modulators**

Due to the high optical quality of AO materials such as fused silica, AO modulators have been used exclusively inside a laser cavity. These intracavity applications include \(Q\)-switching, mode locking and cavity dumping.

\(Q\)-switching of YAG lasers has been an important requirement for industrial laser applications such as cutting, scribing, resistor trimming, and other material processing processes.

For \(Q\)-switching applications, an intracavity AO modulator is used to introduce an optical loss to keep the laser below threshold. When the loss is suddenly removed by switching off the acoustic pulse, the laser bursts into a short pulse with extremely high intensity. During the \(Q\)-switch period, the AO modulator should ideally add no additional loss to the laser cavity.

Another important requirement for the AO Q-switch is due to special characteristics of the YAG laser; it achieves maximum gain when operated in the unpolarized state. Thus, it is essential that the AO \(Q\)-switch does not introduce any polarizing effect.

Presently all of the AO \(Q\)-switches use UV-grade fused silica as the interaction medium. The low optical absorption, good homogeneity, and near strain-free material make it the only viable choice.

In a longitudinal mode device, the diffraction efficiency for light polarized perpendicular to the acoustic wave is five times greater than that of parallel polarization. Thus, the YAG laser will tend to operate in the polarization state with the low diffraction efficiency.

Because of this, today’s AO \(Q\)-switches are designed to use acoustic shear waves since they are insensitive to the state of polarization. Due to the extremely low AO figure of merit of this mode, substantial RF power (~ 50 W) is required to hold off a laser with a moderate gain.

In the mode-locking applications, a standing wave AO modulator is used inside the cavity to introduce a loss modulation with a frequency that is equal to longitudinal mode spacing. \(f_o=c/2L\) where \(L\) is the optical length of the laser cavity. The loss modulation provides a phase locking of the longitudinal modes and produces a train of short optical pulses.

In the \(Q\)-switching operation the maximum repetition rate is limited by the finite buildup time of population inversion, to about 50 KHz. To obtain higher repetition rates, the technique of cavity dumping is used.

In the cavity dumping mode of operation, short acoustic pulses are fed into an intracavity AO modulator to couple the laser energy out of the cavity. The laser is always kept above the threshold, thus the switching rate is limited by the switching speed of the AO modulator. Most cavity dumpers are fused silica as the interaction medium. To reduce drive power, the TeO\(_2\) modulator has been used for cavity dumping of YAG lasers.

There has been considerable activity in the development of intracavity AO devices that perform simultaneous \(Q\)-switching mode locking or cavity dumping.

### Acousto-optic Tunable Filter

Since the early work on AOTFs, there has been considerable effort to improve its performance for meeting system requirements.

The key performance parameters include tuning range, spectral resolution, angular aperture, out-of-band rejection, drive power, and image degradation.

Before a discussion of the applications, we shall first review the progress of these performance parameters. The performance characteristics of several typical AOTFs is given in Table 6.

**Spectral Tuning Range**

Early operation of the collinear AOTF was reported in the visible spectrum region. Later development extended the operating wavelength into the ultraviolet using crystal quartz and into the infrared using Tl\(_3\)AsSe\(_3\).

The extension of spectral range is limited primarily by the availability of efficient AOTF materials that are transparent in the desired wavelength regions. Since the crystal classes applicable to collinear AOTF are more restricted, most of the effort for extending the spectral range has been focused on noncollinear AOTFs.

Operation of noncollinear AOTF has been demonstrated in the UV region using crystal quartz, sapphire, and MgF\(_2\). Noncollinear AOTFs operated in the infrared up to about 11 microns have also been reported. The filter materials include TeO\(_2\), Tl\(_3\)AsSe\(_3\) and, most recently, HgCl\(_2\).

**Drive Power**

One of the major limitations of the AOTF is the relatively large drive power required. The problem is particularly severe in the infrared since the drive power is proportional to the square of the optical wave length.

As an example, the acoustic power density of an infrared TeO\(_2\) AOTF is estimated using a \(\theta_i\approx20^\circ\) design (\(\mathbf{M}_2\approx10^{-12}\) m\(^2\)/W) and an interaction length of 1 cm. At \(\lambda_2=4\;\mu\)m, the required acoustic power density is estimated to be about 8 W/cm\(^2\).

This relatively high power requirement is probably the most serious disadvantage of the AOTF and limits its use for certain important system applications (such as focal plane sensors). One technique to reduce the drive power is to utilize acoustic resonance enhancement.

In an acoustic resonator structure, the peak acoustic field at resonance can be orders of magnitude greater than that of a traveling acoustic wave. It is thus possible to decrease the drive power by operating the AOTF in the acoustic resonance mode.

Naturally, the resonant AOTF must be operated at the discrete resonant acoustic frequencies. For collinear AOTFs, the passband response (in acoustic frequency) consists of two to three modes.

For noncollinear AOTFS, the passband response consists of many resonant modes. The passband response between successive resonant peaks will be sufficiently overlapped. Thus, the continuous tuning of the resonant AOTF is obtainable. The bandpass is essentially the same as the conventional traveling wave case.

The drive power of the resonant AOTF is reduced by the enhancement of diffraction efficiency due to acoustic resonance. At resonance the enhancement factor can be shown to be equal to the reciprocal of one-way acoustic loss. The total acoustic loss includes losses due to acoustic attenuation in the filter medium and reflection loss at the boundaries.

The reduction of drive power due to acoustic resonance was demonstrated in a collinear CaMoO\(_4\) AOTF. Operating the filter at the peak of the acoustic resonance, a reduction in drive power of 22 dB was achieved.

Acoustic resonance enhancement was also demonstrated using a noncollinear TeO\(_2\) AOTF operated in the infrared region of 2 to 5 \(\mu\)m. About nine resonant modes were observed within one filter passband. Due to the relatively high acoustic attenuation in TeO\(_2\), a power reduction factor of 16 was obtained.

**Out-of-Band Rejection and Sidelobe Suppression**

The ability of spectral discrimination is determined by the ratio of peak transmission and out-of-band rejection of the AOTF. The out-of-band rejection for an AOTF is determined by two factors.

Overall out-of-band rejection is determined by the contrast ratio; i. e., the fraction of undiffracted light leakage through crossed polarizers. In practice the contrast can also be limited by residual strain of the filter medium. A large contrast ratio is obtainable in the noncollinear AOTF, where the incident and dif fracted light are spatially separated.

Near the filter band, out-of-band rejection is determined by the sidelobe structure of the filter passband response. The AOTF bandpass response is proportional to the power spectra of the acoustic field. For uniform acoustic excitation, the AOTF exhibits a sinc\(^2\)-type bandpass characteristic with the nearest sidelobe about 13 dB below the main lob .

Suppression of the high sidelobes can be obtained by techniques of amplitude apodization. In the collinear AOTF, an acoustic pulse apodized in time is launched into the filter medium. By utilizing a triangular window, the first sidelobe was reduced by \(-22\) dB with a collinear quartz AOTF.

For the noncollinear AOTF, the apodization can be realized by weighted acoustic excitation at a transducer array. This was demonstrated in a noncollinear TeO\(_2\) AOTF. A maximum sidelobe of \(-26\) dB was achieved using a Hamming-type window.

**Spectral Resolution**

For certain applications such as lidar receiver systems it is desirable to significantly narrow the passband of the AOTF. In principle, increased spectral resolution can be obtained by using long interaction length. However, there exists a basic tradeoff relation between the angular and spectral bandwidth. For instance, the angular aperture of a TeO\(_2\) AOTF with 0.1 percent bandwidth (e. g., 5 Å at 0.5 \(\mu\)m) the angular is only \(\pm3^\circ\).

One approach for realizing narrow passband while maintaining a large angular aperture is to use the wavelength dispersion of birefringence.

In the spectral range near the absorption band edge, certain uniaxial semiconductors such as CdS exhibit anomalous birefringence dispersion. The dispersion constant \(b\) in Eq. (74) can become orders of magnitude larger than the birefringence.

Near the dispersive region, the AOTF would exhibit a large enhancement of resolution without degradation of angular aperture (which is not affected by the dispersion).

Experimentally the concept of the dispersive AOTF was demonstrated using CdS as the filter medium. Over the wavelength 500 to 545 nm, a FWHM of 2 nm was obtained over an angular aperture of 38\(^\circ\).

The AOTF possesses many outstanding features that make it attractive for a variety of optical system applications. To proceed from experimental research to practical system deployment, the merits of the AOTF technology must be compared against other competing and usually more matured, technologies.

Similar to the case of AO deflectors, the AOTF is most attractive to certain special applications that utilize its unique characteristics. The significant features of the AOTF include: rapid and random-access tuning, large optical throughput while maintaining high spectral resolution, and good image resolution, inherent amplitude, frequency modulation and wavelength multiplexing capability, and small rugged construction with no moving parts. In the next section we shall review the various applications of AOTFs.

**Rapid-scan Spectrometers**

The rapid , random-access tuning of the AOTF makes it well-suited to rapid-scan spectrometer applications, such as time-resolved spectra analysis. The random access speed of the AOTF is limited by the acoustic transit time and is typically on the order of a few seconds.

Compared to the conventional grating spectrometer, the AOTF approach offers the advantages of higher resolution, larger throughput, and amplitude/wavelength modulation capability.

Currently, inexpensive photodetector arrays with high performance are not yet available at wavelengths longer than 1.1 \(\mu\)m. Thus, the AOTF-based rapid-scan spectrometer will have clear advantages in the infrared.

Other practical advantages, including easy computer interface, small size, and rugged construction, which make it suitable to field applications. These advantages were demonstrated in several experimental models of AOTF-based spectrometers.

As an example, we briefly describe an infrared spectrometer operating over the tuning range from 2 to 5 micrometers. The system used a noncollinear TeO\(_2\) AOTF with a full width at half maximum (FWHM) of 7 cm\(^{-1}\) over an f/3 angular aperture.

The AOTF device is capable of random accessing any wavelength within the tuning range in less than 10 microseconds. The actual scan speed is limited by the sweep of the RF synthesizer.

The system has been used as a radiometer to measure the radiation spectrum of a rocket engine placed in a high-altitude simulation chamber. The measurement of the entire scan was completed in 20 milliseconds. Compared to conventional grating spectrometers, this represents a significant increase in the data acquisition speed.

Another experimental type of AOTF spectrometer, also operated in the infrared, was developed for use as a commercial stack analyzer. Current approaches based on in-situ spectrometers generally require absorption measurement at two selected wavelengths (differential spectroscopy) or the change of absorption with respect to wavelength change (derivative spectroscopy).

The primary advantage of such a wavelength-modulation technique is that the degradation of system accuracy due to variations of the light source, intensity, optical misalignment, etc. can be self-compensated. Since the AOTF is inherently capable of performing wavelength modulation, it is ideally suited to the gas stack analyzer applications.

In spite of the encouraging results demonstrated by these early experimental systems, to date the AOTF technology has not been employed in any commercial spectrometers. This may be due to many factors that include reliability, cost, and performance deficiencies such as high sidelobes, fixed resolution, and high drive power. In view of some of the more recent technological progress, certain special-purpose AOTF spectrometers are expected to appear in the commercial market.

**Multispectral Imaging**

An interesting AOTF imaging application was demonstrated by Watson et al. in the investigation of planetary atmospheres. A collinear CaMoO\(_4\) AOTF operated in the near-infrared was used to obtain spectral images of Jupiter and Saturn.

The filtered image was detected with a cooled silicon CID sensor array. The CID signals were read out into a microcomputer memory, processed pixel-by-pixel for subtraction of CID dark currents, and displayed on a color TV monitor.

Spectral images of Saturn and Jupiter were obtained with this system in the near-infrared at wavelengths characteristic of CH\(_4\) and NH\(_3\) absorption bands.

When the first noncollinear TeO\(_2\) AOTF was demonstrated, it was recognized that because of its larger aperture and simpler optical geometry, this new type of AOTF would be well-suited to spectral imaging applications.

A multispectral spectral experiment using a TeO\(_2\) AOTF was performed in the visible. The white light beam was spatially separated from the filtered light and blocked by an aperture stop in the immediate frequency plane. A resolution target was imaged through the AOTF and relayed onto the camera. Over the tuning range in the visible, good image resolution in excess of 100 lines/mm was obtained.

As an aid to the design of the AOTF-based imaging spectrometer, a ray-tracing program for optical imaging through AOTF has been described.

Recently, the use of noncollinear AOTF imaging in astronomy has been demonstrated by spectropolarimetry of stars and planets. To perform precise polarimetric imaging adds additional requirement on the design of the AOTFs.

Since the angles for noncritical phase matching (NPM) are different for ordinary and extraordinary rays, AOTFs based on standard designs cannot be, in general, optimized for both polarizations.

In another operation, AOTF spectral imaging has been shown to be a promising approach in the studies of biological materials. Treado et al. demonstrated the use of a visible AOTF to obtain the absorption spectral images of human epithelial cells. The same authors have also described the same techniques for obtaining high fidelity Raman spectral images.

**Fiber Optic Communication**

One important application where the AOTF has shown great potential is in the area of fiber-optic communication. Due to its capability of selecting a narrow optical band over a wide spectral range within a time duration of microseconds, the AOTF appears to be well-suited to perform wavelength division multiplexing (WDM) networks.

Both the bulkwave and guided wave type AOTF have been demonstrated for this application. Compared to the guided wave version, the bulkwave device has the advantage of lower optical insertion loss; however, the drive power is orders of magnitude higher.

For single-mode fiber applications, the laser beams are well-collimated, thus the stringent condition of NPM can be relaxed. A new type of noncollinear TeO\(_2\) AOTF using a collinear beam geometry was described. Because of the long interaction length of the new device geometry, narrower passband and lower drive power are obtainable.

Other applications of AOTFs include laser detection and tuning of dye lasers.

### Acousto-optic Signal Processor

The inherent wide temporal bandwidth and parallel processing capability of optics makes it one of the most promising analog signal-processing techniques. It is particularly suited for performing special integral transforms such as correlation, convolution, and Fourier transforms.

Implementation of the optical signal processing in real time requires a spatial light modulator (SLM) that impresses the electronic signal onto the optical beams. Up to now, the AO Bragg cell was by far the best developed and possibly the only practical one-dimensional SLM to use as the input electrical to optical transducer.

In the past decade there was significant progress on the key relevant components that include laser diodes, photodetectors, and AO Bragg cells. A number of AO signal processors have been developed that demonstrated great potential for various system applications.

The frequency-dispersion characteristics of the AO Bragg cell leads to the obvious application of signal spectrum analysis. Referring to Fig. 6b, the momentum-matching condition yields the deflection angle for the diffracted light beam.

\[\tag{91}\Delta\theta=2\sin^{-1}\left(\frac{\lambda_of}{2V}\right)=\frac{\lambda_o{f}}{V}\]

for \(\Delta\theta\le0.1\) radian, where \(\lambda_o\) is the optical wavelength, \(V\) is the acoustic velocity, and \(f\) is the acoustic frequency.

If a lens is placed behind the Bragg cell, it will transform the angular deviation into a displacement in the back focal plane (referred to as the frequency plane). The displacement of the first-order deflected spot from the undeflected spot in the frequency plane is thus proportional to the frequency of the input signal, i. e.,

\[\tag{92}X=F\lambda_o{f}/V\]

where \(F\) is the transform lens focal length.

The frequency dispersion of AO diffraction described above can be used to perform RF channelization. The simplest system architecture (shown in Fig. 14) is the power spectra analyzer (PSA) that measures the magnitude square of the Fourier transform.

It consists of a laser, collimating/beamforming optics, AO Bragg cell, Fourier transform lens, and linear photodetector array for real-time electronic readout. The laser beam amplitude profile is tailored by the beamforming optics so as to match the interaction aperture of the Bragg cell.

The Bragg cell diffracts a portion of the laser light into an angular distribution of intensities that are proportional to the input RF power spectra. The Fourier transform lens converts the angular diffracted beam into a linear density distribution coincident with the photodetector array. The output of the photodetector thus yields the real-time power spectra of the input RF signal.

Mathematically, the preceding operation yields a Fourier transform of the light distribution at the optical aperture of the Bragg cell

\[\tag{93}F(s)=\int{w(x)}\exp(-j2\pi{sx})dx\]

where \(w(x)\) is the weighting function, \(s=f\tau\) is the normalized frequency (\(\tau\) is the time aperture of the Bragg cell), and \(F(s)\) is the desired Fourier transform. The photodetector array in the transform plane measured the magnitude square of \(F(s)\), i. e., the instantaneous power spectra of the signal read into the Bragg cell.

The frequency resolution and sidelobes inherent to the instantaneous power spectra are determined by the weighting function. In practice, the incident optical beam has a gaussian distribution modified by the finite aperture.

The corresponding power spectra distribution can be obtained from a numerical calculation of Eq. (93). A typical spectrum is shown in Fig. 15 with a truncation ratio of 1.5. It shows a 3-dB width of \(\delta{f}=1.2/\tau\). This is the frequency resolution or minimum separation for resolving two equal signals.

In spite of its extreme simplicity, the AO channelized receiver described above is equivalent to a large number of contiguous narrowband receivers realized in small size. Unity POI is ensured for analysis of a large number of simultaneous signals over a wide instantaneous bandwidth.

An alternative approach for RF spectrum analysis is to implement a compressive receiver using acousto-optics. Functionally, the AO compressive receiver performs the Fourier transform via the chirp \(Z\) transform.

Using the identity \(2ft=(f-t)^2-f^2-t^2\), the Fourier transform can be written as:

\[\tag{94}F(f)=e^{-j\pi{f^2}}\int{f(t)}[e^{-j\pi{t^2}}\cdot{e}^{j\pi(f-t)^2}]dt=e^{-j\pi{f^2}}\left\{[e^{-j\pi^2}f(t)]\ast{e}^{j\pi^2}\right\}\]

where \(\ast\) denotes convolution.

The Fourier transform is accomplished by premultiplying the signal with a forward chirp, convolving with a reverse chirp, and then post multiplying with another chirp. Since the algorithm involves premultiplication, correlation, and post multiplication, it is referred to as the MCM scheme. For power spectrum analysis, the second multiplication can be neglected.

The desired convolution can be implemented using either a space-integrating, or a time-integrating architecture. The space-integrating compressive receiver is illustrated in Fig. 16a.

The signal to be analyzed, \(x(t)\), is multiplied by the forward chirp and input into the first Bragg cell. The optical amplitude distribution is imaged onto the second Bragg cell, which is driven by the reverse chirp. The product is spatially integrated by lens \(L3\) to produce the final output, \(\tilde{x}(f)\), which is (except for a phase factor) the Fourier transform of \(x(t)\).

The single photodetector measures the desired power spectrum and displays it as a function of time. The frequency resolution of the space-integrating compressive receiver is set by the finite aperture of the Bragg cell.

An alternative method of implementing the compressive receiver is the time-integrating architecture shown in Fig. 16b. Like the space-integrating system, the product of the input and the forward chirp from the first Bragg cell is imaged onto the second Bragg cell (driven by the reverse chirp). However, the product light distribution leaving the second cell is now time-integrated by the photodetector array (instead of the spatial integration due to a lens).

The time-integrating architecture has significant advantages. Since the integration time is now limited by the photodetector and is orders of magnitude larger than that obtainable from spatial-integration, very long integration time is realizable, thereby achieving fine frequency resolution and large time bandwidth products. In addition, the frequency resolution can be varied by changing the integration time and the chirp rate.

Theoretically, the AO compressive receiver using the space- or time-integrating architecture appears to have great potential for advanced receiver applications. In practice, since the achievable dynamic range and bandwidth are limited by the performance of photodetectors, the AO implementation of compressive receiver offers little performance advantage compared to that based on the more mature SAW technology. As a result, most of the previous efforts have been directed toward RF signal channelization. Therefore, in the remainder of the tutorial, the discussion will be limited to AO channelized receivers.

It has been pointed out that the major advantage of analog AO signal processor is its high throughput realized with small volume and low power consumption. A figure of merit for comparing different signal processors is \(B\log_2N\) where \(B\) is the bandwidth and \(N\) is the time-bandwidth product. Thus, the AO signal processor is most promising for wideband applications with the size and power constraint.

At present the major deficiency of AO signal processors such as the channelized receiver is the relatively low dynamic range. For channelized receivers, the dynamic range is the capability of analyzing a weak signal in the presence of strong signals.

For signals that are close in frequency, the instantaneous dynamic range is limited by the sidelobes of the AO diffraction, which can be effectively suppressed by the choice of the weighting window for the illuminated optical beam (apodization).

For signals far apart in frequency, the dynamic range is limited either by optical scattering or by the inmodulation products (IMPs) generated in the Bragg cell. The most dominant inband intermodulation products for two simultaneous signals at \(f_1\) and \(f_2\) are the third-order terms at \(2f_1-f_2\) and \(2f_2-f_1\).

The main contributions of IMPs are due to multiple AO diffractions occurring at high diffraction efficiencies. For two equal intensity signals the two-tone third-order IMP is equal to:

\[\tag{95}I_3=\frac{I_1^3}{36}\]

Where \(I_1\) is the first-order diffracted light efficiency.

The above theory assumes that exact momentum matching is met at each step of the multiple AO diffraction process. In an AO signal processor such as the PSA, the IMP in the Bragg cell becomes the dominant factor limiting the dynamic range only when the two signals are relatively far apart in frequency.

A refined theory of IMPs in Bragg cells was developed for the case of when the two signals are separated. It was shown that significant suppression of the IMPs can be obtained by using birefringent or phased-array Bragg cells.

In our recent experimental work on wideband Bragg cells, we have found in many cases that the dominant contribution of IMPs are not due to multiple AO diffractions, but are instead due to acoustic nonlinearities.

Further theoretical and experimental investigation on these acoustically generated IMPs were reported. It was shown that the process could involve second- or third-order acoustic nonlinearities. The nonlinear acoustically-generated IMP in turn diffracts the incident optical beam to an angular position that appears as the optical spurious mode with the corresponding frequency shift.

This type of IMP becomes increasingly severe in wideband Bragg cells where the acoustic power densities are high. The dynamic range degradation due to acoustic nonlinearity thus sets a fundamental limit on the largest bandwidth obtainable. Most of the recent Bragg cells have employed birefringent or phased-array designs to increase the interaction length. These techniques have further extended the high frequency limit of wideband Bragg cells.

Besides the generation of IMPs in Bragg cells, the PSA also has limited dynamic range due to the use of direct detection. Since the photodetector current is proportional to the RF power, the input dynamic range is only one half that of the output dynamic range.

One approach to overcome this deficiency is the use of an Interferometric Spectra Analyzer (ISA) configuration. The photodetector current is now proportional to the square root of the product of the signal and a constant reference. Therefore, theoretically the dynamic range of an ISA is twice that of a PSA.

VanderLugt described a special ISA configuration which used two Bragg cells, a signal cell, and a reference cell. The reference cell is used to provide a spatially modulated reference beam.

With proper arrangement, the interference term of the signal and reference beams will produce a heterodyning detector output with almost constant temporal IF (within one resolution) over the entire spatial frequency plane. The constant IF implementation scheme has the advantage of significantly lowering the required bandwidth of the detector arrays.

Based on the fixed offset IF scheme, a number of ISAs have been built and demonstrated increased dynamic range. Experimental results obtained from these ISAs have shown that the dynamic range is limited primarily by the spurious modulation associated with the wideband RF reference.

Figure 17 shows the schematic of an ISA using a Mach-Zehnder type of interferometer. The optical bench is referred to as the AO interferometer since it uses two Bragg cells to modulate the optical path difference of the two equal path arms.

The input signal is applied to one Bragg cell, the signal cell, through an RF amplifier. A wideband RF reference is fed to the second Bragg cell, the reference cell. The incident laser light is divided by the beam splitter and results in two diffracted laser beams out of the signal and reference cells. By appropriately adjusting the tilt between the two Bragg cells, the two diffracted beams are made to overlap at the Fourier plane and to mix collinearly at the desired intermediate frequencies (IF).

The signal cell is operated in the linear range at relatively low efficiency (about 5 percent or less) to avoid spurious signals due to intermodulation products (IMPs) and to achieve a high two-tone dynamic range. The reference is usually driven at high efficiency in order to improve receiver sensitivity. The IMPs generated from the reference cell will not severely degrade the receiver performance.

Theoretically, the ISA can significantly improve the dynamic range of an AO channelized receiver. Measured results from an experimental model reveals that the improvement is realizable only for purely CW signals.

For pulsed signals, the sidelobe rejection of the IF bandpass is severely degraded due to the inherent frequency gradient in the leading and trailing pulse edges. This introduces false signals in the sidelobe region and greatly increases the difficulty of processing the optical processor outputs.

Other critical problems include optical stability, complexity of wideband reference and IF subsystem, and most of all the implementation of wideband detector arrays.

The next tutorial introduces ** metal-coated fibers**.