# Acousto-Optic Modulators

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

The acoustic wave of an acousto-optic modulator is ** amplitude modulated**. The operation of an acousto-optic modulator is based on the dependence of the acousto-optic diffraction efficiency on the intensity of the acoustic wave.

The acoustic intensity can be controlled by an electrical signal that generates the acoustic wave in a modulator. An acousto-optic modulator is an electronically addressed ** amplitude modulator** that accepts an electrical modulation signal to vary the intensity of an optical beam accordingly.

Acousto-optic modulators have been put to many different applications. The straightforward application is amplitude modulation of an optical beam, thus encoding a modulation signal on an optical carrier or providing loss modulation to an optical system such as a Q-switched or mode-locked laser. Sophisticated applications include time-domain convolution and correlation of wide-band RF signals in signal processing systems.

Generally speaking, an acousto-optic modulator can operate either in the Bragg regime or in the Raman-Nath regime. In the low-efficiency limit, the efficiency of the first diffraction order of a Raman-Nath type modulator is similar to the diffraction efficiency of a Bragg-type modulator. For low-efficiency diffraction from a traveling wave with \(|\kappa|l\ll1\), the efficiency of the first diffraction order of a Raman-Nath-type modulator is, according to (8-56) [refer to the acousto-optic diffraction tutorial],

\[\tag{8-103}\eta_1=J_1^2(2|\kappa|l)\approx|\kappa|^2l^2=\frac{\pi^2M_2l^2}{2\lambda^2}I_\text{a}\]

while the diffraction efficiency of a Bragg-type modulator with perfect phase matching is, according to (8-86) [refer to the acousto-optic diffraction tutorial],

\[\tag{8-104}\eta_\text{PM}=\sin^2|\kappa|l\approx|\kappa|^2l^2=\frac{\pi^2M_2l^2}{2\lambda^2}I_\text{a}\]

A similar comparison can be made in the case of diffraction from a standing wave using (8-94) and (8-97) [refer to the acousto-optic diffraction tutorial].

However, notwithstanding this similarity, an acousto-optic modulator operating in the Raman-Nath regime has a few disadvantages in comparison to one operating in the Bragg regime. For this reason, most acousto-optic modulators designed for practical applications are of Bragg type.

One obvious disadvantage of a Raman-Nath-type modulator easily seen from figure 8-3 [refer to the acousto-optic diffraction tutorial] is that none of the diffraction orders can reach a diffraction efficiency higher than 34%. In comparison, a phase-matched Bragg-type modulator has a maximum diffraction efficiency of 100%.

Other disadvantages stem from the condition in (8-49) [refer to the acousto-optic diffraction tutorial], which is required for Raman-Nath diffraction.

Under the constraint of this condition, the acceptable interaction length \(l\) decreases quadratically as a function of acoustic frequency and linearly as a function of optical wavelength. At high acoustic frequencies and/or for long optical wavelengths, the interaction length becomes impractically small so that a very large acoustic intensity is required in order to have a significant diffraction efficiency.

A modulator of Raman-Nath type is thus limited to applications with low acoustic frequencies and, consequently, small bandwidths. Such limitations do not apply to a modulator of Bragg type.

**Traveling-wave modulators**

Traveling acoustic waves are used in the majority of acousto-optic modulators. The most important performance characteristics to be considered for a traveling-wave modulator are its diffraction efficiency \(\eta\), its bandwidth, measured by a 3-dB modulation bandwidth \(f_\text{m}^\text{3dB}\), and its speed, measured by a modulation response risetime, \(t_\text{r}\).

A traveling acousto-optic modulator is normally operated in the Bragg regime with a focused optical beam of small spot size to reduce the acoustic transit time across the optical beam and thus increase its modulation bandwidth and speed.

Figure 8-8 below shows the diagram of a typical solid-state acousto-optic modulator operating with a traveling acoustic wave in the Bragg regime.

The acousto-optic cell, which can be a crystal or a noncrystalline glass, is attached to a piezoelectric transducer at one end and is terminated by an angled surface at the other end.

The piezoelectric transducer consists of a metallic electrode, a piezoelectric crystal such as LiNbO_{3}, and one or more metallic bonding layers for the attachment of the piezoelectric crystal to the acousto-optic cell.

It converts the applied RF electrical signal into an acoustic signal and couples the acoustic power to the acousto-optic cell to generate the traveling acoustic wave.

The angled termination surface reflects the acoustic wave away from the incident direction so as to prevent the reflected acoustic wave from interacting with the optical beam. The back surface is also loaded with an acoustically absorbing material to reduce acoustic reflection.

The cross-sectional area of the acoustic beam in the modulator shown in figure 8-8 is \(HL\), which is defined by the length \(L\) and height \(H\) of the transducer. The acoustic intensity is

\[\tag{8-105}I_\text{a}=\frac{P_\text{a}}{HL}=\frac{\eta_\text{t}P_\text{e}}{HL}\]

where \(P_\text{a}\) is the acoustic power delivered by the transducer to the acoustic medium, \(P_\text{e}\) is the power of the electric signal driving the transducer, and \(\eta_\text{t}\) is the conversion efficiency of the transducer from electric to acoustic power.

Using (8-82) and (8-86) [refer to the acoustic-optic diffraction tutorial], we find that the diffraction efficiency of a Bragg-type traveling-wave modulator with perfect phase matching can be expressed as

\[\tag{8-106}\eta_\text{PM}=\sin^2\left[\frac{\pi}{\lambda}\left(\frac{M_2}{2HL}P_\text{a}\right)^{1/2}l\right]=\sin^2\left[\frac{\pi}{\lambda}\left(\frac{M_2}{2HL}\eta_\text{t}P_\text{e}\right)^{1/2}l\right]\]

In the low-efficiency limit, the diffraction efficiency is linearly proportional to the modulation power:

\[\tag{8-107}\eta_\text{PM}\approx\frac{\pi^2M_2l^2}{2\lambda^2HL}P_\text{a}=\frac{\pi^2M_2l^2}{2\lambda^2HL}\eta_\text{t}P_\text{e},\qquad\text{if }\eta_\text{PM}\ll1\]

For a modulator using a traveling acoustic wave, a time-dependent \(P_\text{e}(t)\) that carries the modulation signal is applied to the device so that the diffraction efficiency varies with time.

When an acousto-optic modulator is used as a loss modulator, the transmittance from the incident optical beam to the undiffracted optical beam is \(T=1-\eta_\text{PM}\) in the situation of perfect phase matching.

Most acousto-optic modulators take the configuration of small-angle Bragg interaction shown in figure 8-8. In this configuration, acousto-optic interaction takes place when the optical beam passes through the width of the acoustic beam. The interaction length is then defined by the length of the transducer: \(l=L\).

The amplitude modulation signal that is applied to a traveling-wave acousto-optic modulator can be either a continuously varying signal, as shown in figure 8-9(a) below, or a pulsed digital signal, as shown in figure 8-9(b) below.

In either case, the modulation signal appears as a modulation on the amplitude, thus on the intensity, of the acoustic wave at a carrier frequency of \(f_0=\Omega_0/2\pi\).

The frequency, \(f_\text{m}\), of a sinusoidal modulation signal that is imposed on an acoustic carrier wave of a frequency \(f_0\) must satisfy the condition that \(f_\text{m}\lt{f}_\text{0}\). This modulation generates two side-band acoustic frequencies at \(f_0-f_\text{m}\) and \(f_0+f_\text{m}\).

Through the dependence of the acousto-optic diffraction efficiency on the intensity of the acoustic wave, the intensities of the diffracted and undiffracted optical beams vary with the amplitude modulation signal accordingly.

In a practical modulator, both the incident optical beam and the acoustic beam have a certain degree of beam divergence because of the finite dimensions of their cross-sectional sizes.

The performance characteristics of a traveling-wave acousto-optic modulator are determined by three basic parameters: (1) the factor \(Q\); (2) the *beam divergence ratio*

\[\tag{8-108}a=\frac{\Delta\theta_\text{o}}{\Delta\theta_\text{a}}\]

of the optical beam divergence \(\Delta\theta_\text{o}\) to the acoustic beam divergence \(\Delta\theta_\text{a}\); and (3) the *acoustic transit time*

\[\tag{8-109}\tau_\text{a}=\frac{d}{v_\text{a}}\]

defined as the time it takes for an acoustic wavefront to travel across a light beam that has a beam diameter \(d\) at the interaction point.

To ensure that the modulator operates in the Bragg regime, the value of \(Q\) has to satisfy (8-57) [refer to the acousto-optic diffraction tutorial]. More precisely, to eliminate parasitic coupling of optical intensity to high diffraction orders sufficiently, it is necessary to have

\[\tag{8-110}Q=2\pi\frac{\lambda{l}}{n\Lambda^2}\ge4\pi\]

The divergence of an optical beam is inversely proportional to its beam waist with a proportionality factor determined by the shape of the beam profile. For an optical beam of the fundamental Gaussian spatial profile discussed in the Gaussian beam tutorial, the divergence is given by (137) [refer to the Gaussian beam tutorial] as

\[\tag{8-111}\Delta\theta_\text{o}=\frac{2\lambda}{\pi{n}w_0}\]

where \(w_0\) is the minimum Gaussian beam spot size in the acoustic medium.

The acoustic beam can be considered to have a rectangular transverse spatial distribution with a beam width \(L\) determined by the length of the transducer, as shown in figure 8-8. Its divergence is given by

\[\tag{8-112}\Delta\theta_\text{a}=\frac{\Lambda}{L}\]

Assuming that acousto-optic interaction takes place at the optical beam waist, as in the most favorable situation, the acoustic transit time is given by \(\tau_\text{a}=d_0/v_\text{a}=2w_0/v_\text{a}\).

** The most important characteristics to be considered for a traveling-wave acousto-optic modulator are the diffraction efficiency and the modulator response**. Optimization of these characteristics dictates the choice of the values of \(a\) and \(\tau_\text{a}\) in designing a practical acousto-optic modulator.

In the analysis of acousto-optic diffraction presented in the acousto-optic diffraction tutorial, we have considered only the interaction between plane optical and plane acoustic waves, both of which were assumed to have zero divergence. In this ideal situation, perfect phase matching can be accomplished over the entire optical wavefront when the conditions in (8-60) [refer to the acousto-optic diffraction tutorial] are satisfied. Then the formulas in (8-106) and (8-107) above are accurate for calculation of the diffraction efficiency.

In a realistic situation where the optical beam has a nonzero divergence angle \(\Delta\theta_\text{o}\), the optical wavefront covers a range of directions from \(-\Delta\theta_\text{o}/2\) to \(\Delta\theta_\text{o}/2\) with respect to its central direction of propagation.

If the acoustic beam has an infinite plane wavefront with a single, well-defined direction of propagation, the phase-matching conditions in (8-60) [refer to the acousto-optic diffraction tutorial] cannot be simultaneously satisfied for the entire range of optical wavefront directions. Consequently, the overall diffraction efficiency of the optical beam is degraded. In this situation, the formulas for the diffraction efficiency given in (8-106) and (8-107) become inaccurate but tend to overestimate the real efficiency of the device.

In reality, however, the acoustic beam also has a nonzero divergence, and its wavefront covers a range of directions from \(-\Delta\theta_\text{a}/2\) to \(\Delta\theta_\text{a}/2\) with respect to its central direction of propagation.

If \(\Delta\theta_\text{a}\ge\Delta\theta_\text{o}\), it is possible for different optical wavefront directions to be phase matched by different acoustic wavefront directions. Therefore, to lessen the degradation in diffraction efficiency caused by divergence of the optical beam, a small value of \(a\) is required, meaning that ** the optical beam has to be more collimated than the acoustic beam for maximizing the diffraction efficiency**.

The response of a traveling-wave acousto-optic modulator is primarily determined by the value of \(\tau_\text{a}\). If the modulation signal varies substantially within a time interval of \(\tau_\text{a}\), the acoustic wave intensity varies spatially across the width of the optical beam.

This spatial nonuniformity of acoustic intensity across the cross section of the optical beam leads to nonuniform diffraction of the optical beam. As a result, the modulation signal carried by the acoustic wave is not faithfully converted to the modulation of the optical beam.

For a given value of \(\tau_\text{a}\), this effect becomes more significant at higher modulation frequencies, thus degrading the response of a modulator to high-speed or high-frequency modulation signals.

Quantitatively, the modulator response is measured by the modulation bandwidth in the case of a continuously varying modulation signal, or by the modulation speed in the case of a pulsed modulation signal.

The modulation bandwidth is characterized by a 3-dB modulation frequency, \(f_\text{m}^\text{3dB}\), at which point the frequency response of a modulator rolls off to 50% of its maximum response.

The modulation speed is characterized by a ** risetime**, \(t_\text{r}\), which is defined as the time interval needed for the modulated optical intensity of the diffracted beam to rise from 10 to 90% of its steady-state value in response to a step modulation signal.

Detailed analysis of the response of a traveling-wave acousto-optic modulator involves convolution of the spatial intensity profile of the optical beam with the propagation of the acoustic wave carrying the modulation signal across the optical beam. The results depend on the spatial profile of the optical beam as well as on the value of \(a\).

For an optical beam of the fundamental Gaussian spatial profile, we have

\[\tag{8-113}f_\text{m}^\text{3dB}\approx\begin{cases}\frac{0.75}{\tau_\text{a}},\qquad\qquad{a\ll1}\\\frac{0.86-0.13a}{\tau_\text{a}},\qquad{a\gg1}\end{cases}\]

and

\[\tag{8-114}t_\text{r}\approx\begin{cases}0.65\tau_\text{a},\qquad\qquad\qquad{a\ll1}\\(0.45+0.25a)\tau_\text{a},\qquad{a\gg1}\end{cases}\]

It can be seen that, besides degrading the diffraction efficiency, a large value of \(a\) also degrades the modulator response.

From the discussions presented so far, it seems that for the best performance of a modulator, the value of \(a\) should be made as small as possible. This is not true, however, because \(a\) and \(\tau_\text{a}\) are both functions of the optical beam waist diameter.

For a given modulator with a fixed acoustic beam divergence, the value of \(a\) can be reduced only by reducing the value of \(\Delta\theta_\text{o}\) through collimation of the optical beam. The consequence is an increase in the optical beam waist diameter and a corresponding increase in the value of \(\tau_\text{a}\), thus degrading the bandwidth and the speed of the modulator.

From (8-113) and (8-114) above, it is clear that a small value of \(\tau_\text{a}\) is required for a large modulation bandwidth and, correspondingly, a high modulation speed. Indeed, to obtain a large modulation bandwidth and a high modulation speed, the optical beam has to be focused to a small beam waist located in the inaction region.

These conflicting requirements lead to the need for properly choosing an optimum value of \(a\) depending on the requirements of a particular application. Once this choice is made, the value of \(\tau_\text{a}\) and, consequently, the characteristics of the modulator response are basically determined.

Two additional issues regarding the functional reality of a traveling-wave acousto-optic modulator have to be considered.

First, the amplitude modulation signal carried by the acoustic wave generates side-band frequencies on both high- and low-frequency sides of the carrier frequency \(f_0\). For a modulator with a modulation bandwidth of \(f_\text{m}^\text{3dB}\), the side-band frequencies cover the range from \(f_0-f_\text{m}^\text{3dB}\) to \(f_0+f_\text{m}^\text{3dB}\). Clearly, the lowest side-band frequency has to be a positive frequency in order for the modulation signal not to be distorted.

In practical situations, it is often necessary to avoid nonlinear distortion of the modulation signal by requiring that the highest side-band frequency be smaller than the second harmonic of the lowest side-band frequency: \(f_0+f_\text{m}^\text{3dB}\lt2(f_0-f_\text{m}^\text{3dB})\). This requirement leads to the following condition for the carrier frequency:

\[\tag{8-115}f_0\ge3f_\text{m}^\text{3dB}\]

Another realistic issue is the need to separate the diffracted and the undiffracted optical beams cleanly at the output of the modulator. The clean separation between these two beams can be ensured by requiring that the deflection angle be larger than twice the beam divergence:

\[\tag{8-116}|\theta_\text{def}|=|\theta_\text{d}-\theta_\text{i}|\gt2\Delta\theta_\text{o}\]

In a modulator where the acousto-optic diffraction is nonbirefringent, \(|\theta_\text{def}|=2\theta_\text{B}\), and the above condition becomes

\[\tag{8-117}\theta_\text{B}\gt\Delta\theta_\text{o}\]

The conditions discussed above set some constraints on the physical parameters of a traveling-wave acousto-optic modulator:

1. **Condition for Bragg diffraction**.

The condition in (8-110) that \(Q\ge4\pi\) for Bragg diffraction sets the following minimum interaction length for a given acoustic carrier frequency:

\[\tag{8-118}L=l\ge\frac{2n\Lambda_0^2}{\lambda}=\frac{2nv_\text{a}^2}{\lambda{f_0}^2}\]

where \(\Lambda_0\) is the acoustic carrier wavelength corresponding to the acoustic carrier frequency \(f_0\) and \(n\) is the refractive index of the medium.

For a given value of \(a\), we find, using (8-111) and (8-112), that this condition requires the Gaussian beam spot size located at the interaction region to be subject to the following condition:

\[\tag{8-119}d_0=2w_0\ge\frac{8}{a\pi}\Lambda_0=\frac{8v_\text{a}}{a\pi{f_0}}\]

For a given acoustic beam width \(L\) and a given optical beam spot size \(w_0\), the condition for Bragg diffraction sets the limit for the lowest acceptable carrier frequency:

\[\tag{8-120}f_0\ge\left(\frac{2nv_\text{a}^2}{\lambda{L}}\right)^{1/2}=\frac{4v_\text{a}}{a\pi{w_0}}=\frac{8}{a\pi\tau_\text{a}}\]

where we have used the relation \(\tau_\text{a}=d_0/v_\text{a}=2w_0/v_\text{a}\) by taking \(d\) in (8-109) to be the beam waist diameter \(d_0\).

2. **Condition for beam separation and side-band limitation**.

Using the definitions for \(\theta_\text{B}\), \(\Delta\theta_\text{o}\), and \(\Delta\theta_\text{a}\) in (8-65) [refer to the acousto-optic diffraction tutorial], (8-111), and (8-112), respectively, it can be shown that the condition \(\theta_\text{B}\gt\Delta\theta_\text{o}=a\Delta\theta_\text{a}\) given in (8-117) for clean separation between the diffracted and the undiffracted beams sets the following lower limits for the acoustic beam width:

\[\tag{8-121}L=\frac{a\pi{n}w_0\Lambda}{2\lambda}\ge\frac{2anv_\text{a}^2}{\lambda{f_0}^2}\]

and the following lower limit for the optical beam spot size:

\[\tag{8-122}d_0=2w_0\ge\frac{8}{\pi}\Lambda_0=\frac{8v_\text{a}}{\pi{f_0}}\]

This latter constraint sets the following lower limit for the acoustic carrier frequency:

\[\tag{8-123}f_0\ge\frac{4v_\text{a}}{\pi{w_0}}=\frac{8}{\pi\tau_\text{a}}\]

This condition guarantees that \(f_0\ge3.4f_\text{m}^\text{3dB}\) because \(f_\text{m}^\text{3dB}\le0.75/\tau_\text{a}\) for any value of parameter \(a\), according to (8-113).

Therefore, the condition that \(f_0\ge3f_\text{m}^\text{3dB}\) given in (8-115) imposed by side-band consideration is automatically satisfied as long as the condition in (8-122) for clean beam separation is satisfied.

From these discussions, we see that the value of the parameter \(a\) determines whether the physical parameters of a traveling-wave acousto-optic modulator are dictated by the condition for Bragg diffraction or by the condition for clean beam separation.

In the case when \(a\lt1\), the limits set by the condition for Bragg diffraction determine the physical parameters of the device because they are more stringent than those required by the condition for clean beam separation.

In the case when \(a\gt1\), the limits set by the condition for clean beam separation are more stringent and thus define the physical parameters of the device.

In applications where the modulation bandwidth and speed have to be maximized, the optimum choice for the value of \(a\) is

\[\tag{8-124}a=1.5\]

for \(f_\text{m}^\text{3dB}=0.65/\tau_\text{a}\) and \(t_\text{r}=0.85\tau_\text{a}\) with a small value of \(\tau_\text{a}\) due to a focused beam waist in this situation. Then the acoustic beam width and the optical beam spot size are limited by the constraints set by (8-121) and (8-122), respectively, while the acoustic carrier frequency is subject to the condition in (8-123).

In applications where the diffraction efficiency and the collimation of the optical beam have to be maximized at the expense of modulation speed, \(a\ll1\) is chosen so that \(f_\text{m}^\text{3dB}=0.75/\tau_\text{a}\) and \(t_\text{r}=0.65\tau_\text{a}\) with a large value of \(\tau_\text{a}\) due to an unfocused beam waist. Then the acoustic beam width and the optical beam spot size are limited by the constraints set by (8-118) and (8-119), respectively, while the acoustic carrier frequency is subject to the condition in (8-120).

In all applications, the height of the transducer, however, only has to be \(H\le\sqrt{2}d_0=2\sqrt{2}w_0\) to cover the spot size of the optical beam at the interaction point.

The modulation bandwidth and modulation speed discussed above take into consideration only the interaction of the acoustic wave with the optical beam. Clearly, the overall response of a modulator to an electrical modulation signal is also subject to the bandwidth of the piezoelectric transducer and its supporting electronic circuitry. This ** transducer bandwidth** is characterized by the frequency dependence of the conversion efficiency \(\eta_\text{t}\) defined in (8-105).

**Example 8-5**

A fused silica traveling-wave acousto-optic modulator using the longitudinal acoustic mode at an acoustic carrier frequency of \(f_0\) = 100 MHz is designed for the optical wavelength at \(\lambda\) = 1.064 μm. The optical wave is polarized in a direction perpendicular to the propagation directions of both the optical and the acoustic waves. The physical length \(L\) and height \(H\) of the transducer are chosen so that the device can be used for both high-speed application with a focused optical beam and low-speed application with a collimated optical beam.

(a) Find the optimum optical beam spot size and the values of \(f_\text{m}^\text{3dB}\) and \(t_\text{r}\) for the high-speed application.

(b) Find the values of \(f_\text{m}^\text{3dB}\) and \(t_\text{r}\) for the low-speed application with a collimated optical beam waist diameter of \(d_0\) = 1 mm.

(c) If the transducer efficiency is \(\eta_\text{t}\) = 60%, what is the electrical modulation power needed to obtain a modulation loss of 10% for the device?

From table 8-2 [refer to the photoelastic effect tutorial], we find that \(v_\text{a}=v_\text{a,L}=5.97\text{ km s}^{-1}\) for a longitudinal acoustic wave in silica glass. At \(\lambda\) = 1.064 μm, \(n\) = 1.45 for pure silica, as can be calculated by using (96) [refer to the dispersion in fibers tutorial].

(a) For the high-speed application, we choose \(a\) = 1.5. Then the beam spot size is limited by (8-122) to be

\[d_0=2w_0\ge\frac{8v_\text{a}}{\pi{f_0}}=\frac{8\times5.97\times10^3}{\pi\times100\times10^6}\text{ m}=152\text{ μm}\]

Therefore, the beam waist can be focused to a minimum of \(w_0\) = 76 μm to obtain a minimum acoustic transit time of \(\tau_\text{a}=8/\pi{f_0}=25.5\) ns using (8-123). We then have \(f_\text{m}^\text{3dB}=0.65/\tau_\text{a}=25.5\) MHz and \(t_\text{r}=0.85\tau_\text{a}=21.7\) ns. The width of the acoustic beam is required by (8-121) to be

\[L\ge\frac{2anv_\text{a}^2}{\lambda{f_0}^2}=\frac{2\times1.5\times1.45\times(5.97\times10^3)^2}{1.064\times10^{-6}\times(100\times10^6)^2}\text{ m}=1.46\text{ cm}\]

We can then choose a transducer length of \(L\) = 1.5 cm.

(b) With \(L\) = 1.5 cm and \(w_0=d_0/2=500\) μm for the low-speed application, we then find that

\[a=\frac{\Delta\theta_\text{o}}{\Delta\theta_\text{a}}=\frac{2\lambda{f_0}L}{\pi{n}w_0v_\text{a}}=\frac{2\times1.064\times10^{-6}\times100\times10^6\times1.5\times10^{-2}}{\pi\times1.45\times500\times10^{-6}\times5.97\times10^3}=0.235\]

Because \(a=0.235\ll1\), we have to use (8-120) to find that \(\tau_\text{a}\ge8/a\pi{f_0}=108\) ns. We then have \(f_\text{m}^\text{3dB}=0.75/\tau_\text{a}\le6.9\) MHz and \(t_\text{r}=0.65\tau_\text{a}\ge70\) ns for the low-speed application with a collimated beam spot size of \(d_0\) = 1 mm.

Because this device is to be used for both high-speed and low-speed applications, the height of the transducer is dictated by the larger beam spot size in the low-speed application to be \(H\ge\sqrt{2}d_0=1.4\) mm. Therefore, we can choose a transducer height of \(H\) = 1.5 mm.

(c) Because \(M_2\propto{n^6}\), we can use the value of \(M_2=1.5\times10^{-15}\text{ m}^2\text{ W}^{-1}\) for \(n=1.457\) at 632.8 nm to find that \(M_2=(1.45/1.457)^6\times1.5\times10^{-15}\text{ m}^2\text{ W}^{-1}=1.46\times10^{-15}\text{ m}^2\text{ W}^{-1}\) for \(n=1.45\) at 1.064 μm. Because both \(\theta_\text{i}\) and \(\theta_\text{d}\) are very small in the operation of this device, we have \(l=L\). For a modulation loss of 10%, we need \(\eta_\text{PM}=0.1\). With \(\eta_\text{t}\) = 60%, the required electrical power can be found by using (8-107) to be

\[P_\text{e}=\frac{2\lambda^2H}{\pi^2M_2L}\frac{\eta_\text{PM}}{\eta_\text{t}}=\frac{2\times(1.064\times10^{-6})^2\times1.5\times10^{-3}\times0.1}{\pi^2\times1.46\times10^{-15}\times1.5\times10^{-2}\times0.6}\text{ W}=2.6\text{ W}\]

This is the required power for low-speed modulation. For high-speed modulation, the required power would be somewhat higher for the same modulation loss of 10% because of the degradation in efficiency at a high modulation speed with a focused optical beam.

**Standing-wave modulators**

Acousto-optic modulators that utilize standing acoustic waves are used in some special applications such as laser mode locking. A standing-wave modulator provides sinusoidal amplitude modulation at a very high frequency. It differs from a traveling-wave modulator in many important aspects, from the device structure to the performance characteristics.

The most important performance characteristics of a standing-wave acousto-optic modulator are its diffraction efficiency \(\eta\) and its loss modulation frequency, \(f_\text{m}=2f\), at twice the acoustic frequency in the low-efficiency limit. It is always operated in the Bragg regime with a well-collimated optical beam.

In order to create a standing acoustic wave, the acousto-optic cell is made to be a resonant acoustic cavity. Instead of the angled surface of the acousto-optic cell of a traveling-wave device, the surface at the far end across the cell width is made parallel to the near end that is attached to the piezoelectric transducer, as shown in figure 8-10 below.

With a given cell width \(W\) measured in the direction of the acoustic wave, a standing acoustic wave is formed only when the acoustic wavelength satisfies the condition:

\[\tag{8-125}W=m\frac{\Lambda}{2},\qquad{m}=\text{integer}\]

Therefore, the device functions only at the following discrete acoustic resonance frequencies:

\[\tag{8-126}f=m\frac{v_\text{a}}{2W},\qquad{m}=\text{integer}\]

which are determined by the cell width and the acoustic velocity.

The resonance frequencies are sensitive to variations in the value of \(v_\text{a}\) caused by temperature fluctuations. In many applications of a standing-wave modulator, the temperature of the acousto-optic cell has to be carefully stabilized to maintain stable and efficient operation.

The acoustic power that h as to be delivered by the transducer to the acoustic resonator is equal to the product of the acoustic energy stored in the resonator and the decay rate, \(\gamma_\text{a}\), of that energy:

\[\tag{8-127}P_\text{a}=\left(\frac{I_\text{a}^\text{f}+I_\text{a}^\text{b}}{v_\text{a}}HLW\right)\gamma_\text{a}\approx\frac{2I_\text{a}^\text{f}}{v_\text{a}}HLW\gamma_\text{a}\]

where we have taken \(I_\text{a}^\text{f}\approx{I}_\text{a}^\text{b}\) for an efficient resonator.

The acoustic power, \(P_\text{a}\), is converted from an electrical power, \(P_\text{e}\), by a transducer, which has a conversion efficiency of \(\eta_\text{t}:P_\text{a}=\eta_\text{t}P_\text{e}\).

Using (8-97) and (8-102) [refer to the acoustic-optic diffraction tutorial], we find that the diffraction efficiency of a standing-wave acousto-optic modulator with perfect phase matching can be expressed as

\[\tag{8-128}\eta_\text{PM}=\sin^2\left[\frac{\pi}{\lambda}\left(\frac{M_2v_\text{a}}{HLW\gamma_\text{a}}P_\text{a}\right)^{1/2}l\cos\Omega{t}\right]=\sin^2\left[\frac{\pi}{\lambda}\left(\frac{M_2v_\text{a}}{HLW\gamma_\text{a}}\eta_\text{t}P_\text{e}\right)^{1/2}l\cos\Omega{t}\right]\]

In the low efficiency limit, we have

\[\tag{8-129}\eta_\text{PM}\approx\frac{\pi^2M_2l^2v_\text{a}}{\lambda^2HLW\gamma_\text{a}}\eta_\text{t}P_\text{e}\cos^2\Omega{t}=\frac{\pi^2M_2l^2v_\text{a}}{2\lambda^2HLW\gamma_\text{a}}\eta_\text{t}P_\text{e}(1+\cos2\Omega{t}),\qquad\text{if }\eta_\text{PM}\ll1\]

Again, \(l=L\) in the configuration of small-angle Bragg diffraction.

We see that, when a standing-wave acousto-optic modulator is operated in the low-efficiency limit, the intensity of the diffracted beam at its output is sinusoidally modulated at ** twice** the acoustic carrier frequency with a

**that is linearly proportional to the driving power.**

*modulation depth*Unlike the situation in a traveling-wave modulator, there is no need to impose an additional modulation signal on the carrier. Therefore, \(P_\text{e}\) in the above equations is a constant. The transducer is driven by an unmodulated RF electrical signal at the desired acoustic frequency.

A standing-wave modulator is capable of modulating an optical beam at very high frequencies, but the allowed modulation frequencies cannot be tuned continuously because they are discretely defined by the resonance frequencies of the acousto-optic cell.

A standing-wave acousto-optic modulator is often used as a loss modulator, such as in its use as a mode locker for a mode-locked laser. The transmittance from the incident optical beam to the undiffracted optical beam in a loss modulator is \(T=1-\eta_\text{PM}\) in the case of perfect phase matching.

In the consideration of the performance of a standing-wave modulator, the only relevant parameters are \(Q\) and \(a\). The acoustic transit time \(\tau_\text{a}\) is irrelevant because the two contrapropagating acoustic waves that form the standing wave are not amplitude modulated.

To ensure operation in the Bragg regime, the requirement that \(Q\ge4\pi\) given in (8-110) still has to be satisfied, leading to the same minimum length given in (8-118) for the transducer.

Because the transit time is no longer relevant, the value of \(a\) is not subject to the conflicting requirements faced by a traveling-wave device. Therefore, the optical beam in a standing-wave modulator can be well collimated so that \(a\ll1\) to avoid degradation of the diffraction efficiency caused by divergence of the optical beam.

**Example 8-6**

A fused silica standing-wave acousto-optic modulator using the longitudinal acoustic mode at the acoustic frequency of \(f\) = 100 MHz is designed for the optical wavelength at \(\lambda\) = 1.064 μm with specifications similar to those of the traveling-wave modulator described in Example 8-5 above. The optical wave is polarized in a direction perpendicular to the propagation directions of both the optical and the acoustic waves. It has a collimated optical beam waist diameter of \(d_0\) = 1 mm. The transducer efficiency is also \(\eta_\text{t}\) = 60%, and the length \(L\) and height \(H\) of the transducer are to be chosen properly for this device. The resonant acousto-optic cell of this standing-wave modulator has a width of \(W\) = 3 cm, resulting in a decay rate of \(\gamma_\text{a}=4\times10^4\text{ s}^{-1}\). Find the modulation frequency and the electrical modulation power needed to obtain a peak modulation loss of 10% for the device.

Because the optical beam is collimated to have a beam waist diameter of \(d_0\) = 1 mm, we know from Example 8-5 that \(a\ll1\) in this situation. The length L is thus subject to the condition in (8-118):

\[L\ge\frac{2nv_\text{a}^2}{\lambda{f^2}}=\frac{2\times1.45\times(5.97\times10^3)^2}{1.064\times10^{-6}\times(100\times10^6)^2}\text{ m}=9.7\text{ mm}\]

which can be smaller than that chosen in Example 8-5. For easy comparison to Example 8-5, however, we choose the same transducer length of \(L\) = 1.5 cm and the same transducer height of \(H\) = 1.5 mm.

With an acoustic frequency of \(f\) = 100 MHz, the loss modulation frequency is \(f_\text{m}=2f=200\) MHz according to the discussions following (8-129). The operation of this device is small-angle Bragg diffraction with \(l=L\). From (8-129) the required electrical power for a peak modulation loss of \(\eta_\text{PM}^\text{max}\) = 10% can be found to be

\[\begin{align}P_\text{e}&=\frac{\lambda^2HW\gamma_\text{a}}{\pi^2M_2Lv_\text{a}}\frac{\eta_\text{PM}^\text{max}}{\eta_\text{t}}\\&=\frac{(1.064\times10^{-6})^2\times1.5\times10^{-3}\times3\times10^{-2}\times4\times10^4\times0.1}{\pi^2\times1.46\times10^{-15}\times1.5\times10^{-2}\times5.97\times10^3\times0.6}\text{ W}=263\text{ mW}\end{align}\]

In comparison to the traveling-wave modulation described in Example 8-5, we find that the standing-wave modulator has a much higher modulation frequency at a much reduced modulation power for a given modulation loss. However, a standing-wave modulator is not as versatile as a traveling-wave modulator because it only allows periodic sinusoidal modulation at twice its resonance frequencies.

The next part continues with the acousto-optic deflectors tutorial.