# Acousto-Optic Deflectors

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

The acoustic wave of an acousto-optic deflector is ** frequency modulated**. Unlike an acousto-optic modulator, which is an amplitude modulator, an acousto-optic deflector is a

**, which allows its acoustic frequency to be varied electronically.**

*frequency modulator*Acousto-optic deflectors have many applications. A ** frequency shifter**, which has the sole purpose of generating a diffracted optical beam at an optical frequency shifted by the amount of the acoustic frequency from the input optical frequency, can be considered as the simplest form of an acousto-optic deflector. Acousto-optic deflectors are also used in such diverse applications as optical scanners, spatial light modulators, RF pulse compressors, and programmable optical interconnecters.

An acousto-optic deflector generally functions in the Bragg regime with a traveling wave. Its most important performance characteristics are its diffraction efficiency \(\eta\) and its number of resolvable spots \(N\). An acousto-optic deflector has a structure similar to that of a traveling-wave acousto-optic modulator shown in figure 8-8 [refer to the acousto-optic modulators tutorial], but it is always operated with a highly collimated optical beam with the parameter \(a\ll1\) to increase the value of \(N\).

The basic principle of acousto-optic deflectors is simple: the acousto-optic deflection angle, \(\theta_\text{def}=\theta_\text{d}-\theta_\text{i}\), which has an absolute value of \(2\theta_\text{B}\) in the case of nonbirefringent Bragg diffraction, is determined by the phase-matching condition, which can be varied by varying the value of the propagation constant \(K\) of the acoustic wave. Because \(K=2\pi{f}/v_\text{a}\), the deflection angle can be varied by varying the acoustic frequency.

The efficiency of a deflector is also given in (8-106), or (8-107) in the low-efficiency limit [refer to the acousto-optic modulators tutorial]. Because an acousto-optic deflector always uses the configuration of small-angle diffraction, its interaction length is defined by the length of its transducer: \(l=L\).

Certain requirements, such as the condition that \(Q\gg4\pi\) for the device to function in the Bragg regime, apply to both modulators and deflectors, but many key parameters of a deflector are subject to considerations different from those for determining the parameters of a modulator.

While the modulation signal applied to a modulator is the amplitude variations on a constant acoustic carrier frequency, the signal applied to a deflector has a constant amplitude but a varying acoustic frequency.

The overall bandwidth of an acousto-optic deflector is subject to both the Bragg bandwidth of the acousto-optic interaction and the transducer bandwidth determined by the frequency dependence of the conversion efficiency \(\eta_\text{t}\). The bandwidth considered in the following refers to the Bragg bandwidth alone.

In an acousto-optic deflector, both the propagation direction of the incident optical beam and that of the acoustic wave are usually fixed when the value of \(K\) is varied by varying the acoustic frequency.

In this situation, the angle of incident, \(\theta_\text{i}\), is fixed, and the variation in the deflection angle is simply determined by the change in the angle of diffraction: \(\delta\theta_\text{def}=\delta\theta_\text{d}\).

Because the value of \(\theta_\text{i}\) is fixed, perfect phase matching in all directions is not possible if the direction of vector \(\mathbf{K}\) is fixed when the value of \(K\) is varied. As shown in figure 8-11 below, the angle of diffraction, \(\theta_\text{d}\), is now determined by the requirement for phase matching in the \(x\) direction parallel to \(\mathbf{K}\).

For small variations of \(K\), the value of \(\theta_\text{d}\) varies linearly with \(K\). If the acoustic frequency is varied over a range \(\Delta{f}\) from \(f_0-\Delta{f}/2\) to \(f_0+\Delta{f}/2\), where \(f_0\) is the center frequency, the propagation constant of the acoustic wave varies over a range \(\Delta{K}=2\pi\Delta{f}/v_\text{a}\). Correspondingly, the deflection angle varies over a range of

\[\tag{8-130}\Delta\theta_\text{d}=\frac{\Delta{K}}{k_\text{d}}=\frac{\lambda}{n_\text{d}v_\text{a}}\Delta{f}\]

where \(n_\text{d}\) is the index of refraction seen by the diffracted optical beam.

In practical applications, an acousto-optic deflector is controlled by varying the acoustic frequency for the diffracted optical beam to address different spatial locations. In the ** random access** mode of operation, the acoustic frequency is changed discretely from one to another to access random positions. In the

**mode of operation, the acoustic frequency is varied continuously so that the deflection angle changes continuously.**

*continuous scan*An important parameter for an acousto-optic deflector is the number \(N\) of resolvable spots. Over a given deflection range \(\Delta\theta_\text{d}\), this parameter is determined by the divergence of the diffracted optical beam through the following relation:

\[\tag{8-131}N=\frac{\Delta\theta_\text{d}}{\Delta\theta_\text{o}}\]

as illustrated in figure 8-11 above.

If the optical beam has a Gaussian spatial profile, \(\Delta\theta_\text{o}\) has the form given in (8-111) [refer to the acousto-optic modulators tutorial]. Substituting (8-111) and (8-130) in (8-131), we have

\[\tag{8-132}N=\frac{\pi}{4}\tau_\text{a}\Delta{f}\]

where \(\tau_\text{a}\) is the acoustic transit time defined in (8-109) [refer to the acousto-optic modulators tutorial]. For a deflector, \(\tau_\text{a}\) is also called the ** time aperture** of the device.

The constant multiplication factor \(\pi/4\) in (8-132) is specific to a Gaussian beam. This factor is different for different optical beam profiles but is generally on the order of unity. Thus the number of resolvable spots is given by the time-bandwidth product \(\tau_\text{a}\Delta{f}\).

For a given value of \(\Delta{f}\), the value of \(N\) is solely determined by the value of \(\tau_\text{a}\). To increase the number of resolvable spots within a given frequency bandwidth, it is necessary to increase the acoustic time aperture by collimating the optical beam or by choosing an acousto-optic medium that has a low acoustic velocity.

However, the acoustic transit time determines the response time of a deflector. In a deflector operating in the random access mode, it takes a temporal delay of \(\tau_\text{a}\) for the deflector to address a new spatial position when the acoustic frequency is changed from one to another. Therefore, the ** scan rate**, defined as the number of spots addressed per second, is \(1/\tau_\text{a}\) in the random access mode. The scan rate in the continuous scan mode is generally much higher than \(1/\tau_\text{a}\).

Usually the speed requirement of a deflector is not very demanding and can be easily satisfied. Therefore, the choice of the parameter \(\tau_\text{a}\) for a deflector is primarily determined by the desired number of resolvable spots within a given frequency bandwidth.

We have seen in the preceding section that the minimum height \(H\) of the transducer for an acousto-optic modulator is determined by the optical spot size. This is because the optical beam is normally focused to a small round spot.

For a deflector, however, the optical spot size, \(w_0=v_\text{a}\tau_\text{a}/2\), in the acoustic wave propagation direction parallel to \(\mathbf{K}\) as determined by (8-109) [refer to the acousto-optic modulators tutorial] is usually quite large because a deflector normally requires a relative large value of \(\tau_\text{a}\) for a large value of \(N\).

In this situation, it is not necessary to maintain a round spot shape because the optical spot size in the direction perpendicular to \(\mathbf{K}\) is irrelevant to the transit time \(\tau_\text{a}\). The optical beam can then take an elliptic spot shape as seen in figure 8-11 above for a small value of \(H\) to increase the diffraction efficiency at a given acoustic power level.

In this case, the height of the transducer is limited by the divergence of the acoustic beam to the order of

\[\tag{8-133}H\approx{v_\text{a}}\left(\frac{\tau_\text{a}}{f_0}\right)^{1/2}\]

To prevent the second harmonics of the frequencies within the operating bandwidth from interfering with the function of a deflector, the highest frequency within the bandwidth has to be smaller than the second harmonic of the lowest frequency: \(f_0+\Delta{f}/2\lt2(f_0-\Delta{f}/2)\). For a given center frequency, this requirement sets the upper limit for the bandwidth:

\[\tag{8-134}\Delta{f}\le\frac{2}{3}f_0\]

The maximum value of the ** fractional bandwidth**, defined as \(\Delta{f}/f_0\), of an acousto-optic deflector is set by this condition at 0.67.

This condition has the same form as that of (8-115) [refer to the acousto-optic modulators tutorial] if we identify the carrier frequency of a modulator with the center frequency of a deflector and \(\Delta{f}\) with \(2f_\text{m}^\text{3dB}\).

Note, however, that the bandwidth \(\Delta{f}\) and the modulation bandwidth \(f_\text{m}^\text{3dB}\) are completely different in terms of their physical originals and their implications for device performance.

As we have seen in the acousto-optic modulators tutorial, the modulation bandwidth \(f_\text{m}^\text{3dB}\) is determined by the acoustic transit time \(\tau_\text{a}\); it is related to the response speed of a device.

In contrast, the bandwidth \(\Delta{f}\) is determined by the maximum acceptable phase mismatch of a deflector; it is irrelevant to the speed of the device but determines the largest deflection angle and thus the number of resolvable spots.

For a deflector, the response speed is still related to its modulation bandwidth as \(f_\text{m}^\text{3dB}\approx0.75/\tau_\text{a}\) for \(a\ll1\), but \(\Delta{f}\ne2f_\text{m}^\text{3dB}\). Indeed, \(\Delta{f}\) and \(f_\text{m}^\text{3dB}\) are not directly related. Therefore, \(\tau_\text{a}\) and \(\Delta{f}\) can be simultaneously optimized to maximize the value of \(N\) in (8-132).

**Nonbirefringent deflectors**

When the value of \(K\) is varied, both the angle of the incidence and the angle of diffraction have to change accordingly in order to maintain perfect phase matching. When the incident angle \(\theta_\text{i}\) is fixed, a change in the value of \(K\) without a corresponding change in the direction of vector \(\mathbf{K}\) results in a phase mismatch.

Though phase matching in the \(x\) direction parallel to \(\mathbf{K}\) is maintained through a change in the value \(\theta_\text{d}\), a phase mismatch along the \(z\) direction perpendicular to vector \(\mathbf{K}\) cannot be avoided.

On either side of the acoustic frequency range, the maximum deviation from the center frequency is \(\Delta{f}/2\). The corresponding maximum deviation in the propagation constant is \(\Delta{K}/2\) from the center value \(K_0\), as illustrated in figure 8-12 below.

If \(\theta_\text{i}\) is chosen to be the angle \(\theta_\text{i0}=\pm\theta_\text{B0}\) for perfect phase matching at \(f_0\), the phase mismatch that appears at either edge of the bandwidth is

\[\tag{8-135}|\Delta{k}|=\frac{K_0}{4k}\Delta{K}=\frac{\pi\lambda{f_0}}{2nv_\text{a}^2}\Delta{f}\]

to first order.

Instead of ensuring perfect phase matching at \(f_0\), the incident angle can be chosen as

\[\tag{8-136}\theta_\text{i}=\theta_\text{i0}\left[1+\left(\frac{\Delta{f}}{2f_0}\right)^2\right]\]

to minimize the largest phase mismatch for a given bandwidth \(\Delta{f}\). Then the phase mismatch can be minimized to the following value:

\[\tag{8-137}|\Delta{k}|=\frac{\pi\lambda{f_0}}{2nv_\text{a}^2}\Delta{f}\left[1-\left(\frac{\Delta{f}}{2f_0}\right)^2\right]\]

This phase mismatch results in a reduction in the diffraction efficiency. The maximum acceptable phase mismatch depends on the interaction length \(l\) and the acceptable reduction of efficiency at the edges of the bandwidth. A convenient general criterion is to limit the value of \(|\Delta{k}|l\) to less than \(0.9\pi\):

\[\tag{8-138}|\Delta{k}|l=\frac{\pi\lambda{f_0}l}{2nv_\text{a}^2}\Delta{f}\left[1-\left(\frac{\Delta{f}}{2f_0}\right)^2\right]\approx\frac{\pi}{2}\frac{\Delta\theta_\text{d}}{\Delta\theta_\text{a}}\le0.9\pi\]

where \(\Delta\theta_\text{a}\) is the divergence of the acoustic beam defined in (8-112) [refer to the acousto-optic modulators tutorial].

In the limit of low diffraction efficiency where \(|\Delta{k}|\gt\kappa\), the constraint in (8-138) ensures that the diffraction efficiency at the edges of the bandwidth is not reduced by more than 3 dB below that at the center frequency. Other criteria can be chosen based on the bandwidth and efficiency specifications of a device.

The condition in (8-138) can be appreciated from two different, but equivalent, viewpoints.

From one viewpoint, the relation \(|\Delta{k}|l\lt0.9\pi\) indicates that in order to limit the degradation of the diffraction efficiency caused by phase mismatch to an acceptable level, the interaction length has to be kept below an upper limit set by the amount of phase mismatch at the edges of a given bandwidth.

From another viewpoint, the relation \(\Delta\theta_\text{d}\le1.8\Delta\theta_\text{a}\) implied by (8-138) leads to a picture similar to that used in the acousto-optic modulators tutorial in the discussion of the relation between \(\Delta\theta_\text{a}\) and \(\Delta\theta_\text{o}\) for a traveling-wave acousto-optic modulator. For efficient interaction over the entire deflection range, the divergence \(\Delta\theta_\text{a}\) of the acoustic beam has to be large enough that the acoustic wavefront covers a sufficiently wide range of directions for all of the different diffraction directions within the range of \(\Delta\theta_\text{d}\) to be phase matched by different acoustic wavefront directions.

These two viewpoints are equivalent because the interaction length is defined by the width of the acoustic beam and a small acoustic beam width results in a large acoustic beam divergence.

Combining (8-131) and (8-138), we find that

\[\tag{8-139}a=\frac{\Delta\theta_\text{o}}{\Delta\theta_\text{a}}\le\frac{1.8}{N}\ll1\]

where \(N\) for a practical deflector is generally a large number.

The small value of \(a\) required by (8-139) can be appreciated by considering the fact that for an acousto-optic deflector, \(\Delta\theta_\text{o}\) has to be much smaller than \(\Delta\theta_\text{d}\) in order to have a large number of resolvable spots but \(\Delta\theta_\text{a}\) has to be on the order of \(\Delta\theta_\text{d}\) in order to prevent severe degradation in the diffraction efficiency due to phase mismatch at the edges of the range of deflection.

The lower limit for the length \(L\) of the piezoelectric transducer given by (8-118) [refer to the acousto-optic modulators tutorial] is still valid due to the requirement for a deflector to operate in the Bragg regime throughout the entire bandwidth. Because a lower frequency sets a more stringent lower limit for \(L\), we have to use the lowest frequency \(f_0-\Delta{f}/2\) for the Bragg condition.

In addition, the relation in (8-138) and the fact that \(l=L\) together set an upper limit for \(L\).

Combining these two limits, we find the following constraints for the length of the transducer in a nonbirefringent acousto-optic deflector:

\[\tag{8-140}\frac{1.8nv_\text{a}^2}{\lambda{f_0}\Delta{f}}\left[1-\left(\frac{\Delta{f}}{2f_0}\right)^2\right]^{-1}\ge{L}\ge\frac{2nv_\text{a}^2}{\lambda{f_0}^2}\left(1-\frac{\Delta{f}}{2f_0}\right)^{-2}\]

For optimum performance of a deflector, both a large \(\Delta{f}\), for a large number of resolvable spots, and a large \(L\), for a high efficiency, are desired. The optimum values of \(\Delta{f}\) and \(L\) for a nonbirefringent deflector can be found by solving (8-140) with equals sign for both places of the \(\ge\) sign to be

\[\tag{8-141}\Delta{f}=0.525f_0\qquad\text{and}\qquad{L}=3.68\frac{nv_\text{a}^2}{\lambda{f_0^2}}\]

*Example 8-7*

A LiNbO_{3} acousto-optic deflector for 1.3 μm optical wavelength is desired to have a 1 GHz bandwidth with 100 resolvable spots. The optical axis \(\hat{z}\) of the crystal is parallel to the [001] direction, and the \(y\)-coordinate axis is taken to be in the [010] direction. The acoustic wave is a transverse mode propagating in the [001] \(z\) direction and polarized in the \(y\) direction. Its acoustic velocity is \(3.59\text{ km s}^{-1}\). Optical deflection takes place in the \(yz\) plane, and the incident optical wave is a Gaussian beam of an elliptical spot shape polarized in the \(x\) direction. The ordinary and extraordinary indices of refraction for LiNbO_{3} at 1.3 μm are \(n_\text{o}=2.222\) and \(n_\text{e}=2.145\), respectively.

(a) Find the required acoustic time aperture \(\tau_\text{a}\) and the optical spot size \(w_0\) in the \(z\) direction.

(b) Find the optimum center frequency \(f_0\).

(c) Find the optimum dimensions \(L\) and \(H\) for the transducer to maximize the efficiency.

(d) What is the peak diffraction efficiency for 1 W of acoustic power?

**(a)** With \(\Delta{f}\) = 1 GHz and \(N\) = 100, we find by using (8-132) that \(\tau_\text{a}=4N/\pi\Delta{f}=127.3\) ns. Then the spot size in the \(z\) direction is \(w_0=v_\text{a}\tau_\text{a}/2=228.5\) μm. This is not the required spot size in the \(x\) direction, however, as the beam is elliptical.

**(b)** For interaction of the \(x\)-polarized optical wave with the \(z\)-propagating, \(y\)-polarized transverse acoustic mode in LiNbO_{3}, the only coupling is through the \(\Delta\epsilon_\text{xx}\) element of the acousto-optic permittivity tensor with a figure of merit of \(M_2=3.15\times10^{-15}\text{ m}^2\text{ W}^{-1}\) at \(\lambda\) = 1.3 μm. Therefore, this deflector is nonbirefringent with both incident and diffracted waves polarized in the \(x\) direction. We can then use (8-141) to find that the optimum center acoustic frequency is \(f_0=\Delta{f}/0.525=1.9\) GHz for \(\Delta{f}\) = 1 GHz.

**(c)** From (8-141), the optimum length of the transducer is

\[L=3.68\frac{nv_\text{a}^2}{\lambda{f_0^2}}=\frac{3.68\times2.222\times(3.59\times10^3)^2}{1.3\times10^{-6}\times(1.9\times10^9)^2}\text{ m}=22.5\text{ μm}\]

and, from (8-133), the optimum height is

\[H=v_\text{a}\left(\frac{\tau_\text{a}}{f_0}\right)^{1/2}=3.59\times10^3\times\left(\frac{127.3\times10^{-9}}{1.9\times10^9}\right)^{1/2}\text{ m}=29.4\text{ μm}\]

**(d)** The peak diffraction efficiency at \(f_0\) = 1.9 GHz for \(P_\text{a}\) = 1 W is, for \(l=L\),

\[\eta_\text{PM}=\frac{\pi^2M_2L}{2\lambda^2H}P_\text{a}=\frac{\pi^2\times3.15\times10^{-15}\times22.5\times10^{-6}}{2\times(1.3\times10^{-6})^2\times29.4\times10^{-6}}\times1=0.7\%\]

**Birefringent deflectors**

For a nonbirefringent deflector, the constraints in (8-140) dictate that increasing the bandwidth \(\Delta{f}\) leads to a reduction in the length \(L\), and vice versa. It is therefore not possible to increase both \(\Delta{f}\) and \(L\) simultaneously above their respective optimum values given in (8-141).

Using birefringent Bragg diffraction under the special condition of ** tangential phase matching**, also known as

**, as discussed in Example 8-3 [refer to the acousto-optic diffraction tutorial], it is possible to increase the values of both \(\Delta{f}\) and \(L\) for a birefringent deflector beyond their optimum values for a nonbirefringent deflector, thus increasing the number of resolvable spots and the diffraction efficiency simultaneously.**

*90° phase matching*In the application of a deflector, it is required that \(\theta_\text{d}\) varies sensitively to the acoustic frequency \(f\) while \(\theta_\text{i}\) is fixed. Therefore, the conditions for birefringent tangential phase matching in a deflector are (1) \(n_\text{i}\gt{n}_\text{d}\) so that \(k_\text{i}\gt{k}_\text{d}\) and (2) \(K_0^2=k_\text{i}^2-k_\text{d}^2\) so that perfect phase matching occurs at the center frequency \(f_0\) with \(\theta_\text{d}=0\), which are shown in figure 8-13 below.

This frequency \(f_0\) is determined by the normalized frequency \(\hat{f}_\text{t}\) found in (8-74) [refer to the acousto-optic diffraction tutorial] as \(f_0=(n_\text{i}+n_\text{d})v_\text{a}\hat{f}_\text{t}/\lambda\). Under this phase-matching condition, the maximum phase mismatch that appears at the edges of the bandwidth is

\[\tag{8-142}|\Delta{k}|\approx\frac{(\Delta{K})^2}{8k_\text{d}}=\frac{\pi\lambda}{4n_\text{d}v_\text{a}^2}(\Delta{f})^2\]

Applying the criterion of \(|\Delta{k}|L=|\Delta{k}|l\le0.9\pi\), we find that the bandwidth and the transducer length of a birefringent deflector under the tangential phase-matching condition are subject to the following constraints:

\[\tag{8-143}\frac{3.6n_\text{d}v_\text{a}^2}{\lambda(\Delta{f})^2}\ge{L}\ge\frac{2n_\text{d}v_\text{a}^2}{\lambda{f_0}^2}\left(1-\frac{\Delta{f}}{2f_0}\right)^{-2}\]

This condition can be satisfied for the largest bandwidth allowed by the condition in (8-134). Therefore, it leads to the following optimum values for the bandwidth and the interaction length, respectively:

\[\tag{8-144}\Delta{f}=\frac{2}{3}f_0\qquad\text{and}\qquad{L}=8.1\frac{n_\text{d}v_\text{a}^2}{\lambda{f_0}^2}\]

In comparison to (8-141), we see that tangential phase matching for a birefringent deflector allows an interaction length that is more than twice that of the optimum length for a nonbirefringent deflector while having a 27% increase in its bandwidth to reach its allowable maximum.

The upper limit of length \(L\) can be further doubled if we move the phase-matching point slightly away from the tangential point by choosing \(K_0^2=k_\text{i}^2-k_\text{d}^2+(\Delta{K})^2/8\), as shown in figure 8-14 below.

Under this arrangement, perfect phase matching occurs at the two frequencies of \(f_0\pm\Delta{f}/2\sqrt{2}\), and the maximum phase mismatch appears at the center frequency as well as at both edges of the bandwidth. This maximum phase mismatch is

\[\tag{8-145}|\Delta{k}|\approx\frac{(\Delta{K})^2}{16k_\text{d}}=\frac{\pi\lambda}{8n_\text{d}v_\text{a}^2}(\Delta{f})^2\]

which is only half the value of that in (8-142). Consequently, instead of (8-143), we have

\[\tag{8-146}\frac{7.2n_\text{d}v_\text{a}^2}{\lambda(\Delta{f})^2}\ge{L}\ge\frac{2n_\text{d}v_\text{a}^2}{\lambda{f_0^2}}\left(1-\frac{\Delta{f}}{2f_0}\right)^{-2}\]

Therefore, while the bandwidth remains at its allowable maximum, the interaction length is doubled:

\[\tag{8-147}\Delta{f}=\frac{2}{3}f_0\qquad\text{and}\qquad{L}=16.2\frac{n_\text{d}v_\text{a}^2}{\lambda{f_0^2}}\]

In the above discussions, we have assumed that \(n_\text{d}\) is a constant that does not vary with acoustic frequency. In the situation when the diffracted beam is an extraordinary wave, however, \(n_\text{d}\) can be a function of the diffraction angle and thus a function of the acoustic frequency. Then the optimization process is more complicated than what is discussed above, but the general concepts are still valid.

**Example 8-8**

A LiNbO_{3} acousto-optic deflector for 1.3 μm optical wavelength is desired to have a 1 GHz bandwidth with 100 resolvable spots such as the one described in Example 8-7 above. The acoustic wave is still a transverse mode propagating in the [001] \(z\) direction, but is now polarized in the \(x\) direction. Its acoustic velocity is also \(3.59\text{ km s}^{-1}\). Optical deflection still takes place in the \(yz\) plane, and the incident optical wave is still a Gaussian beam of an elliptical spot shape polarized in the \(x\) direction.

(a) Find the required acoustic time aperture \(\tau_\text{a}\) and the optical spot size \(w_0\) in the \(z\) direction.

(b) Find the optimum center acoustic frequency \(f_0\).

(c) Find the optimum dimensions \(L\) and \(H\) for the transducer to maximize the efficiency.

(d) What is the peak diffraction efficiency for 1 W of acoustic power?

**(a)** With \(\Delta{f}\) = 1 GHz and \(N\) = 100, we still find, using (8-132), that \(\tau_\text{a}=4N/\pi\Delta{f}=127.3\) ns. Then the spot size in the \(z\) direction is \(w_0=v_\text{a}\tau_\text{a}/2=228.5\) μm. These two parameters are the same as those found in Example 8-7.

**(b)** For interaction of the \(x\)-polarized optical wave with the \(z\)-propagating, \(x\)-polarized transverse acoustic mode in LiNbO_{3}, the only coupling is through the \(\Delta\epsilon_{xy}=\Delta\epsilon_{yx}\) and \(\Delta\epsilon_{zx}=\Delta\epsilon_{xz}\) elements of the acousto-optic permittivity tensor with a figure of merit of \(M_2=1.075\times10^{-14}\text{ m}^2\text{ W}^{-1}\) at \(\lambda\) = 1.3 μm. Therefore, this deflector is birefringent with both an ordinary incident wave polarized in the \(x\) direction and an extraordinary diffracted wave polarized in the \(yz\) plane. We find that \(n_\text{i}\gt{n}_\text{d}\) because LiNbO_{3} is negative uniaxial. Therefore, tangential phase matching so that \(\theta_\text{d}=0\) for the optimum performance of the deflector is possible. This occurs at \(\theta_\text{i}=-15.13^\circ\) for up-shifted diffraction, or \(\theta_\text{i}=15.13^\circ\) for down-shifted diffraction, at an acoustic frequency of \(f_\text{t}\) = 1.6 GHz for tangential phase matching. Because \(\theta_\text{d}=0\), the diffracted wave is polarized in the \(z\) direction with \(n_\text{d}=n_\text{e}=2.145\). For this device, the center acoustic frequency is dictated by the tangential phase-matching condition to be \(f_0=f_\text{t}=1.6\text{ GHz}\) rather than by the desired \(\Delta{f}\) and the optimum condition in (8-144). With \(\Delta{f}\) = 1 GHz, we find that \(\Delta{f}/f_0=0.625\lt2/3\).

**(c)** Because the value of \(\Delta{f}/f_0\) is smaller than its optimum value allowed in (8-144), the value of \(L\) can be larger than that given in (8-144). It is determined by its upper bound in (8-143) to be

\[L=\frac{3.6n_\text{d}v_\text{a}^2}{\lambda(\Delta{f})^2}=\frac{3.6\times2.145\times(3.59\times10^3)^2}{1.3\times10^{-6}\times(1\times10^9)^2}\text{ m}=76.6\text{ μm}\]

The height of the transducer can be chosen to be approximately

\[H=v_\text{a}\left(\frac{\tau_\text{a}}{f_0}\right)^{1/2}=3.59\times10^3\times\left(\frac{127.3\times10^{-9}}{1.6\times10^9}\right)^{1/2}\text{ m}=32\text{ μm}\]

**(d)** The peak diffraction efficiency at \(f_0=1.6\text{ GHz}\) for \(P_\text{a}=1\text{ W}\) is, for \(l=L\),

\[\eta_\text{PM}=\frac{\pi^2M_2L}{2\lambda^2H}P_\text{a}=\frac{\pi^2\times1.075\times10^{-14}\times76.6\times10^{-6}}{2\times(1.3\times10^{-6})^2\times32\times10^{-6}}\times1=7.5\%\]

which is more than ten times that of the nonbirefringent deflector described in Example 8-7 above. By further using the optimum phase-matching scheme illustrated in figure 8-14, the length \(L\) can be doubled to \(L=153.2\text{ μm}\), thus doubling the peak efficiency to \(15\%\). However, with this optimum phase-matching scheme, the peak efficiency does not occur at \(f_0=1.6\text{ GHz}\), but occurs at two frequencies: \(f_0+\Delta{f}/2\sqrt{2}=1.954\text{ GHz}\) and \(f_0-\Delta{f}/2\sqrt{2}=1.246\text{ GHz}\).

**Deflectors using phased-array transducers**

The phase mismatch that exists over the bandwidth of a deflector is caused by the fact that eh propagation directions of both the incident optical wave and the acoustic wave are fixed while the acoustic frequency is varied.

In the operation of an acousto-optic deflector, it is clearly not practical to vary the direction of the incident optical wave in response to variations in the acoustic frequency in order to maintain perfect phase matching. However, using a multiple-element ** phased-array transducer** shown in figure 8-15 below, the direction of the acoustic \(\mathbf{K}\) vector can be steered to satisfy the phase-matching condition better as the acoustic frequency is varied.

The simplest phased-array transducer consists of equally spaced elements of the same width, but a phase shift of \(\pi\) between each pair of adjacent elements is introduced. The interference among the acoustic waves generated by the phase-shifted elements causes the acoustic power to concentrate in the direction of constructive interference. This direction shifts from the normal to the transducer to an angle \(\theta\) given by

\[\tag{8-148}\theta\approx\sin\theta=\frac{\pi}{d_\text{t}K}=\frac{\Lambda}{2d_\text{t}}=\frac{v_\text{a}}{2d_\text{t}f}\]

where \(d_\text{t}\) is the center-to-center distance between adjacent elements in the transducer array, as shown in figure 8-15.

Therefore, as the acoustic frequency varies, the direction of the acoustic beam is steered accordingly.

In a nonbirefringent deflector using this simple phased-array transducer, the maximum phase mismatch over a given bandwidth can be reduced to the value given in (8-145) when the parameters of the transducer and the incident direction of the optical beam are chosen properly.

Therefore, the bandwidth \(\Delta{f}\) and the interaction length \(L\) of a nonbirefringent deflector using this phased-array transducer under optimum conditions are subject to the same constraints as given in (8-146). Their optimum values are thus those given in (8-147).

The simple phased-array transducer with a constant phase shift of \(\pi\) between adjacent elements does not completely eliminate phase mismatch within a given bandwidth, but it allows the performance of a nonbirefringent deflector to match the best performance of a birefringent deflector.

Further improvement to obtain closer phase matching over a large bandwidth is possible with a more sophisticated phased-array transducer to fine tune the acoustic beam direction. For most applications, however, a complicated transducer is not practical because the effort is not justified by the benefit gained.

The next part continues with the acousto-optic tunable filters tutorial.