# Acousto-optic diffraction

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This is a continuation from the previous tutorial - photoelastic effect.

We see from the preceding two tutorials that the space- and time-dependent periodic permittivity changes induced by a traveling plane acoustic wave of the form given in (8-1) [refer to the elastic waves tutorial] can be generally expressed as

\[\tag{8-34}\Delta\boldsymbol{\epsilon}=\Delta\tilde{\boldsymbol{\epsilon}}\sin(\mathbf{K}\cdot\mathbf{r}-\Omega{t})\]

where \(\mathbf{K}\) depends on both the polarization and the propagation direction of the acoustic wave.

In general, \(\Delta\tilde{\boldsymbol{\epsilon}}\) is a function of the strain and the rotation generated by the acoustic wave in the medium, the elasto-optic coefficients of the medium, the mode and direction of the acoustic wave, and the frequency and polarization of the optical wave, but it is independent of the values of \(K\) and \(\Omega\).

When an optical wave at a frequency \(\omega\) is incident on this medium, the interaction between the optical wave and the periodic modulation described by (8-34) can generate diffracted optical waves at frequencies \(\omega\pm\Omega\).

The diffracted waves at \(\omega\pm\Omega\) can be diffracted once more to generate waves at frequencies \(\omega\pm2\Omega\). If this process is allowed to cascade, we will end up with a series of diffracted optical waves at frequencies \(\omega+q\Omega\), where \(q\) admits both positive and negative integers and is the ** order of acousto-optic diffraction**.

For acoustic-optic diffraction from a traveling acoustic wave, each diffraction order has a unique propagation direction. Therefore, there is a single wavevector \(\mathbf{k}_q\) associated with the optical wave component at the frequency \(\omega_q=\omega+q\Omega\). Following the formulation of the coupled-wave theory discussed in the coupled-wave theory tutorial, the total optical field consisting of all interacting components can then be expressed in the form of (5) [refer to the coupled-wave theory tutorial]:

\[\tag{8-35}\mathbf{E}(\mathbf{r},t)=\sum_q\boldsymbol{\mathcal{E}}_q(\mathbf{r})\text{e}^{\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega_qt}=\sum_q\hat{e}_q\mathcal{E}_q(\mathbf{r})\text{e}^{\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega_qt}\]

where \(\hat{e}_q\) is the unit vector defining the polarization of \(\boldsymbol{\mathcal{E}}_q\).

According to (8-14) [refer to the photoelastic effect tutorial], the polarization induced by the interaction of the acoustic wave with the optical field component of the frequency \(\omega_q\) is

\[\Delta\mathbf{P}_q(\mathbf{r})=\Delta\boldsymbol{\epsilon}(\omega_q)\cdot\mathbf{E}_q(\mathbf{r})=\Delta\boldsymbol{\epsilon}(\omega_q)\cdot\boldsymbol{\mathcal{E}}_q(\mathbf{r})\exp(\text{i}\mathbf{k}_q\cdot\mathbf{r})\]

Because \(\omega\gg\Omega\), dispersion of the medium within the frequency range of interaction can be ignored to take \(\Delta\boldsymbol{\epsilon}(\omega_q)=\Delta\boldsymbol{\epsilon}\). According to (6) [refer to the coupled-wave theory tutorial], the total induced polarization is

\[\tag{8-36}\begin{align}\Delta\mathbf{P}(\mathbf{r},t)&=\sum_q\Delta\boldsymbol{\epsilon}\cdot\boldsymbol{\mathcal{E}}_q(\mathbf{r})\text{e}^{\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega_qt}\\&=\sum_q\Delta\tilde{\boldsymbol{\epsilon}}\cdot\boldsymbol{\mathcal{E}}_q(\mathbf{r})\sin(\mathbf{K}\cdot\mathbf{r}-\Omega{t})\text{e}^{\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega_qt}\\&=\frac{1}{2\text{i}}\sum_q\Delta\tilde{\boldsymbol{\epsilon}}\cdot\boldsymbol{\mathcal{E}}_q(\mathbf{r})\left[\text{e}^{\text{i}(\mathbf{k}_q+\mathbf{K})\cdot\mathbf{r}-\text{i}\omega_{q+1}t}-\text{e}^{\text{i}(\mathbf{k}_q-\mathbf{K})\cdot\mathbf{r}-\text{i}\omega_{q-1}t}\right]\\&=\frac{1}{2\text{i}}\sum_q\Delta\tilde{\boldsymbol{\epsilon}}\cdot\left[\boldsymbol{\mathcal{E}}_{q-1}(\mathbf{r})\text{e}^{\text{i}(\mathbf{k}_{q-1}+\mathbf{K})\cdot\mathbf{r}}-\boldsymbol{\mathcal{E}}_{q+1}(\mathbf{r})\text{e}^{\text{i}(\mathbf{k}_{q+1}-\mathbf{K})\cdot\mathbf{r}}\right]\text{e}^{-\text{i}\omega_qt}\end{align}\]

Comparing (8-36) with (6) [refer to the coupled-wave theory tutorial], we find that

\[\tag{8-37}\Delta\mathbf{P}_q(\mathbf{r})=\frac{1}{2\text{i}}\Delta\tilde{\boldsymbol{\epsilon}}\cdot\left[\boldsymbol{\mathcal{E}}_{q-1}(\mathbf{r})\text{e}^{\text{i}(\mathbf{k}_{q-1}+\mathbf{K})\cdot\mathbf{r}}-\boldsymbol{\mathcal{E}}_{q+1}(\mathbf{r})\text{e}^{\text{i}(\mathbf{k}_{q+1}-\mathbf{K})\cdot\mathbf{r}}\right]\]

Using (11) [refer to the coupled-wave theory tutorial], we have the following equation for coupling of optical waves through their interaction with a traveling acoustic wave in an isotropic medium:

\[\tag{8-38}(\mathbf{k}_q\cdot\boldsymbol{\nabla})\boldsymbol{\mathcal{E}}_q=\frac{\omega_q^2\mu_0}{4}\Delta\tilde{\boldsymbol{\epsilon}}\cdot\left[\boldsymbol{\mathcal{E}}_{q-1}\text{e}^{\text{i}(\mathbf{k}_{q-1}+\mathbf{K}-\mathbf{k_q})\cdot\mathbf{r}}-\boldsymbol{\mathcal{E}}_{q+1}\text{e}^{\text{i}(\mathbf{k}_{q+1}-\mathbf{K}-\mathbf{k}_q)\cdot\mathbf{r}}\right]\]

In an anisotropic medium, the valid coupled-wave equation can be obtained by taking the transverse component on both sides of (8-38) according to (18) [refer to the coupled-wave theory tutorial]. From this coupled-wave equation, the following general observations can be made:

- Each optical frequency is directly coupled only to its neighboring frequencies shifted by \(\Omega\) or \(-\Omega\).
- Coupling between optical waves of different polarizations is possible in both isotropic and anisotropic media because \(\Delta\tilde{\boldsymbol{\epsilon}}\) is generally anisotropic. In an isotropic medium, this coupling is possible when \(p_{11}\ne{p_{12}}\), as can be seen from the demonstration in the photoelastic effect tutorial.
- The coupling efficiency depends on the polarization and the propagation direction of the optical waves being coupled, as well as on the polarization and the propagation direction of the acoustic wave.
- Coupling between \(\boldsymbol{\mathcal{E}}_q\) and \(\boldsymbol{\mathcal{E}}_{q-1}\) is phase matched when \(\mathbf{k}_{q-1}=\mathbf{k}_q-\mathbf{K}\), whereas the phase-matching condition for coupling between \(\boldsymbol{\mathcal{E}}_q\) and \(\boldsymbol{\mathcal{E}}_{q+1}\) is \(\mathbf{k}_{q+1}=\mathbf{k}_q+\mathbf{K}\). The efficiency for the coupling between two specific wave components of different frequencies depends critically on the amount of phase mismatch in the coupling process.

Consequently, acousto-optic diffraction displays many different phenomena under different experimental conditions. Each phenomenon is useful for certain applications.

**Raman-Nath diffraction**

We consider the diffraction, in an isotropic medium, of a plane optical wave at a frequency \(\omega\) by a column of plane acoustic wave in a geometry shown in figure 8-2(a) below.

The acoustic wave propagates in the \(x\) direction so that \(\mathbf{K}=K\hat{x}\). The value of \(K\) depends on the polarization of the acoustic wave, as is demonstrated in the photoelastic effect tutorial.

The incident optical wave propagates in a direction close to normal to the acoustic column so that its wavevector \(\mathbf{k}_\text{i}\) makes a small angle \(\theta_\text{i}\) with respect to the \(z\)-coordinate axis, as shown in figure 8-2(a).

The acoustic wave column has a finite width in the \(z\) direction, but it extends in the \(x\) direction far beyond the region of interaction. We also assume that the interaction is two-dimensional so that there are no variations in the \(y\) direction for both the optical and the acoustic waves.

With these assumptions, any changes in the amplitude of the optical field caused by its interaction with the acoustic wave column occur only along the \(z\) direction even though the propagation direction of the optical wave might not be parallel to the \(z\)-coordinate axis. Consequently, we have \(\boldsymbol{\mathcal{E}}_q(\mathbf{r})=\boldsymbol{\mathcal{E}}_q(z)\) though \(\mathbf{k}_q=k_{q,x}\hat{x}+k_{q,z}\hat{z}\) and \(k_{q,x}\ne0\) in general.

Under the conditions discussed above, the coupled-wave equation in (8-38) can be written as

\[\tag{8-39}\begin{align}\frac{\text{d}\boldsymbol{\mathcal{E}}_q}{\text{d}z}=\frac{\omega_q^2\mu_0}{4k_{q,z}}\Delta\tilde{\boldsymbol{\epsilon}}\cdot&\left\{\boldsymbol{\mathcal{E}}_{q-1}\exp[\text{i}(k_{q-1,x}+K-k_{q,x})x+\text{i}(k_{q-1,z}-k_{q,z})z]\right.\\&\left.-\boldsymbol{\mathcal{E}}_{q+1}\exp[\text{i}(k_{q+1,x}-K-k_{q,x})x+\text{i}(k_{q+1,z}-k_{q,z})z]\right\}\end{align}\]

according to (12) [refer to the coupled-wave theory tutorial].

Because the field amplitudes in this equation vary with \(z\) only, the \(x\)-dependent phases on the right-hand side of this equation must vanish, resulting in the following phase-matching condition:

\[\tag{8-40}K=k_{q,x}-k_{q-1,x}=k_{q+1,x}-k_{q,x}\]

This phase-matching condition determines the propagation direction of each diffracted wave components.

Because \(\omega\gg\Omega\), we can take the approximation that \(k_q=n(\omega+q\Omega)/c\approx{n\omega/c}=k\). Then, (8-40) can be written as

\[\tag{8-41}K=k(\sin\theta_q-\sin\theta_{q-1})=k(\sin\theta_{q+1}-\sin\theta_{q})\]

where \(\theta_q\) is the directional angle of \(\mathbf{k}_q\) with respect to the \(z\) axis, as shown in figure 8-2(b) above.

The zeroth order, \(q=0\), represents the undiffracted component with \(\mathbf{k}_0=\mathbf{k}_\text{i}\) and \(\theta_0=\theta_\text{i}\) at the original frequency \(\omega_0=\omega\). The recursion relation in (8-41) can then be reduced to

\[\tag{8-42}\sin\theta_q=\sin\theta_\text{i}+\frac{qK}{k}\]

For small angles of incidence and diffraction, (8-42) can be written as

\[\tag{8-43}\theta_q\approx\theta_\text{i}+\frac{qK}{k}=\theta_\text{i}+q\frac{\lambda}{n\Lambda}=\theta_\text{i}+q\frac{\lambda{f}}{nv_\text{a}}\]

where \(n\) is the refractive index of the medium.

By expressing the phase mismatch in the \(z\) direction in terms of the angle of propagation for each wave component, (8-39) becomes

\[\tag{8-44}\frac{\text{d}\mathcal{E}_q}{\text{d}z}=\frac{\omega_q^2\mu_0}{4k\cos\theta_q}\left[\Delta\tilde{\epsilon}_{q,q-1}\mathcal{E}_{q-1}\text{e}^{\text{i}kz(\cos\theta_{q-1}-\cos\theta_q)}-\Delta\tilde{\epsilon}_{q,q+1}\mathcal{E}_{q+1}\text{e}^{\text{i}kz(\cos\theta_{q+1}-\cos\theta_q)}\right]\]

where \(\Delta\tilde{\epsilon}_{q,q-1}=\hat{e}^*_q\cdot\Delta\tilde{\boldsymbol{\epsilon}}\cdot\hat{e}_{q-1}\) and \(\Delta\tilde{\epsilon}_{q,q+1}=\hat{e}^*_q\cdot\Delta\tilde{\boldsymbol{\epsilon}}\cdot\hat{e}_{q+1}\). The coupled equations represented by (8-44) are known as the ** Raman-Nath equations**.

The solution of (8-44) depends on many experimental parameters. In the special case when the direction of propagation of the incident optical wave is normal to the direction of propagation of the acoustic wave so that \(\theta_\text{i}=0\), the phase-mismatch parameters in (8-44) can be approximated as

\[\tag{8-45}kz(\cos\theta_{q-1}-\cos\theta_q)\approx{kz}\left(q-\frac{1}{2}\right)\frac{K^2}{k^2}\]

\[\tag{8-46}kz(\cos\theta_{q+1}-\cos\theta_q)\approx{-kz}\left(q+\frac{1}{2}\right)\frac{K^2}{k^2}\]

using (8-43) to expand \(\cos\theta_q\).

If the interaction length \(l\) along the \(z\) direction is small so that

\[\tag{8-47}q\frac{K^2l}{k}\ll1\]

the cumulative phase mismatch over the interaction length can be neglected. Then (8-44) can be approximated as

\[\tag{8-48}\frac{\text{d}\mathcal{E}_q}{\text{d}z}\approx\frac{\omega_q^2\mu_0}{4k}(\Delta\tilde{\epsilon}_{q,q-1}\mathcal{E}_{q-1}-\Delta\tilde{\epsilon}_{q,q+1}\mathcal{E}_{q+1})\]

In this situation, acousto-optic coupling allows many diffraction orders to be observed. This is the regime of ** Raman-Nath diffraction**.

The condition for Raman-Nath diffraction is usually stated as

\[\tag{8-49}Q=\frac{K^2l}{k}=2\pi\frac{\lambda{l}}{n\Lambda^2}=2\pi\frac{\lambda{f}^2l}{nv_\text{a}^2}\ll1\]

although the condition in (8-47) is more precise when high diffraction orders are considered.

Typically, one chooses \(Q\le0.3\) for Raman-Nath diffraction. In addition to this condition, ** Raman-Nath diffraction occurs only when the optical wave propagates in a direction normal, or nearly normal, to the direction of propagation of the acoustic wave**.

Because \(\Delta\tilde{\epsilon}_{q,q-1}\) and \(\Delta\tilde{\epsilon}_{q,q+1}\) have different values and both vary with \(q\), there is no general solution to the coupled equations of (8-48).

Nevertheless, with the incident optical wave propagating in the \(z\) direction and the acoustic wave propagating in the \(x\) direction, we do have \(\Delta\tilde{\epsilon}_{q,q-1}=\Delta\tilde{\epsilon}_{q,q+1}=\Delta\tilde{\epsilon}_\text{id}\), which is independent of \(q\) in the following special situations.

- If the acoustic wave is longitudinal polarized and the incident optical wave is linearly polarized either in the \(x\) or \(y\) direction, then \(\hat{e}_q=\hat{e}_\text{i}\) for all \(q\).
- If the acoustic wave is a \(y\)-polarized transverse wave, then \(\hat{e}_q=\hat{e}_\text{i}\) for all even values of \(q\) and \(\hat{e}_q=\hat{e}_1\) for all odd values of \(q\). This statement is true irrespective of the polarization state of the incident optical wave, which can be linear, circular, or elliptical.

In these special cases, (8-48) can be written as

\[\tag{8-50}\frac{\text{d}\mathcal{E}_q}{\text{d}z}=\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{id}}{4k}(\mathcal{E}_{q-1}-\mathcal{E}_{q+1})\]

where the approximation of \(\omega_q\approx\omega\) is taken. We can cast (8-50) in the form

\[\tag{8-51}\frac{\text{d}\mathcal{E}_q}{\text{d}\xi}=\frac{1}{2}(\mathcal{E}_{q-1}-\mathcal{E}_{q+1})\]

by taking

\[\tag{8-52}\xi=\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{id}}{2k}z\]

The recursion relation in (8-51) is that of the Bessel functions given in (21) [refer to the step-index fibers tutorial]. Therefore, its solutions are the Bessel functions:

\[\tag{8-53}\mathcal{E}_q(z)=\mathcal{E}_0(0)J_q\left(\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{id}}{2k}z\right)\]

where \(\mathcal{E}_0(0)\) is the amplitude of the incident optical wave, which is the zeroth order with \(q=0\), at the input plane \(z=0\). We can define a coupling coefficient

\[\tag{8-54}|\kappa|=\frac{\pi}{\lambda}\left(\frac{M_2I_\text{a}}{2}\right)^{1/2}\]

where \(\lambda\) is the optical wavelength in free space and \(M_2\) is related to \(\Delta\tilde{\epsilon}_\text{id}\) according to the relation in (8-22) [refer to the photoelastic effect tutorial]. Then, for an interaction length \(l\), we have

\[\tag{8-55}\mathcal{E}_q(l)=\mathcal{E}_0(0)J_q(-2|\kappa|l)=\mathcal{E}_0(0)J_{-q}(2|\kappa|l)\]

where we have used the identity \(J_q(-x)=J_{-q}(x)\) for the Bessel functions. The leading orders of the Bessel functions have been plotted in figure 2 of the step-index fibers tutorial.

The ** Raman-Nath diffraction efficiency** for order \(q\) over an interaction length \(l\) is

\[\tag{8-56}\eta_q=\frac{I_q(l)}{I_0(0)}=J^2_{-q}(2|\kappa|l)=J_q^2(2|\kappa|l)=\eta_{-q}\]

Because \(J^2_{-q}=J^2_q\) according to (20) [refer to the step-index fibers tutorial], the diffraction orders \(q\) and \(-q\) have the same diffraction efficiency.

Figure 8-3 shows the Raman-Nath diffraction efficiencies of a few leading orders.

In the above, we have considered Raman-Nath diffraction in an isotropic medium. Raman-Nath diffraction in an anisotropic crystal that involves a polarization change between successive orders would require successive anisotropic phase matching and is generally not possible.

**Example 8-2**

A longitudinal acoustic wave propagates in a piece of fused silica glass in the \(x\) direction. An optical wave at 632.8 nm wavelength propagating in the \(z\) direction is diffracted by this acoustic wave in the Raman-Nath regime. (a) If the acoustic frequency is kept at \(f\) = 100 MHz, what is the limit on the interaction length \(l\)? (b) If the acousto-optic interaction length is \(l\) = 1 cm, what is the requirement on the acoustic frequency \(f\)? (c) Find the first-order diffraction efficiency for an interaction length of \(l\) = 1 cm and an acoustic intensity of \(I_\text{a}=1\text{ W cm}^{-2}\).

(a) For \(f\) = 100 MHz, we have the following limit for the interaction length from (8-49):

\[l\ll\frac{nv_\text{a}^2}{2\pi\lambda{f^2}}=1.3\text{ mm}\]

From these results, we see clearly that Raman-Nath diffraction takes place only at low acoustic frequencies or short interaction lengths.

(b) For \(l\) = 1 cm, the requirement from (8-49) that \(Q\ll1\) sets the following limit on the acoustic frequency:

\[f\ll\left(\frac{nv_\text{a}^2}{2\pi\lambda{l}}\right)^{1/2}=36\text{ MHz}\]

(c) From table 8-2 [refer to the photoelastic effect tutorial] we find that \(M_2=1.5\times10^{-15}\text{ m}^2\text{ W}^{-1}\), using (8-54), we find that \(|\kappa|=13.64\text{ m}^{-1}\) for \(I_\text{a}=1\text{ W cm}^{-2}\). Therefore, \(2|\kappa|l=0.2728\) for \(l\) = 1 cm, and the first-order diffraction efficiency for \(q=1\) and \(q=-1\) alike is

\[\eta_1=\eta_{-1}=J^2_1(0.2728)\approx0.018\]

**Bragg diffraction**

When the interaction length \(l\) is sufficiently large so that

\[\tag{8-57}Q=\frac{K^2l}{k}=2\pi\frac{\lambda{l}}{n\Lambda^2}=2\pi\frac{\lambda{f}^2l}{nv_\text{a}}\gg1\]

the cumulative phase mismatch for each pair of coupled wave components cannot be neglected when solving (8-44).

Consequently, it is necessary to have perfect, or nearly perfect, phase matching between two coupled wave components in order to have a significant diffraction efficiency from one of them to the other. This condition defines the regime of ** Bragg diffraction**. In practice, \(Q\ge4\pi\) is often chosen for Bragg diffraction.

The incident wave, being the zeroth order with a wavevector \(\mathbf{k}_\text{i}\) and a frequency \(\omega\), is directly coupled only to the diffraction orders \(q=1\) and \(q=-1\). It can be seen by taking \(q=0\) in (8-38) that the phase-matching condition for generation of the diffraction order \(q=1\) at the ** up-shifted frequency** \(\omega_1=\omega+\Omega\) is

\[\tag{8-58}\mathbf{k}_\text{d}=\mathbf{k}_1=\mathbf{k}_\text{i}+\mathbf{K}\]

whereas that for the generation of the diffraction order \(q=-1\) at the ** down-shifted frequency** \(\omega_{-1}=\omega-\Omega\) is

\[\tag{8-59}\mathbf{k}_\text{d}=\mathbf{k}_{-1}=\mathbf{k}_\text{i}-\mathbf{K}\]

For any diffraction order \(q\) to be generated, it is found by reduction that the phase-matching condition of \(\mathbf{k}_q=\mathbf{k}_\text{i}+q\mathbf{K}\) has to be satisfied for \(\omega_q=\omega+q\Omega\).

In addition, because each diffraction order is directly coupled only to its neighboring orders, Bragg diffraction at a high order requires successive generation of low diffraction orders, thus simultaneous satisfaction of corresponding phase-matching conditions. With the exception of some very special cases, these requirements cannot be fulfilled.

Consequently, only one diffraction order, either \(q=1\) or \(q=-1\), is usually generated in Bragg diffraction from a traveling acoustic wave.

Bragg diffraction occurs in both isotropic and anisotropic media when the phase-matching condition in (8-58) or that in (8-59) is satisfied.

The polarization of the diffracted wave is determined by the property of the \(\Delta\tilde{\boldsymbol{\epsilon}}\) tensor and the polarization of the incident optical wave in much the same manner as that discussed above for Raman-Nath diffraction.

Therefore, in both isotropic and anisotropic media, acousto-optic Bragg diffraction can be accompanied by a change of polarization between the incident and diffracted waves.

In an isotropic medium, \(k_\text{d}\approx{k}_\text{i}=k\) no matter whether there is a change of polarization in the process or not. In an anisotropic medium, \(k_\text{d}=k_\text{i}\) when the incident and the diffracted waves have the same polarization, but \(k_\text{d}\ne{k}_\text{i}\) in general when they have different polarizations.

The type of acousto-optic diffraction in which \(k_\text{d}\ne{k}_\text{i}\) is called ** birefringent diffraction**, whereas that in which \(k_\text{d}=k_\text{i}\) is called

**.**

*nonbirefringent diffraction*With the incident wave propagating at a directional angle \(\theta_\text{i}\) and the diffracted wave propagating at a directional angle \(\theta_\text{d}\), the phase-matching conditions in (8-58) and (8-59) can be expressed as

\[\tag{8-60}k_\text{d}\cos\theta_\text{d}=k_\text{i}\cos\theta_\text{i},\qquad{k}_\text{d}\sin\theta_\text{d}=k_\text{i}\sin\theta_\text{i}\pm{K}\]

where the plus sign is for ** up-shifted diffraction** and the minus sign is for

**.**

*down-shifted diffraction*Therefore, for phase-matched, up-shifted Bragg diffraction, the angles of incidence and diffraction are

\[\tag{8-61}\theta_\text{i}=-\sin^{-1}\frac{K^2+k_\text{i}^2-k_\text{d}^2}{2k_\text{i}K}=-\sin^{-1}\frac{\lambda{f}}{2n_\text{i}v_\text{a}}\left[1+\frac{v_\text{a}^2}{\lambda^2f^2}(n_\text{i}^2-n_\text{d}^2)\right]\]

\[\tag{8-62}\theta_\text{d}=\sin^{-1}\frac{K^2+k_\text{d}^2-k_\text{i}^2}{2k_\text{d}K}=\sin^{-1}\frac{\lambda{f}}{2n_\text{d}v_\text{a}}\left[1+\frac{v_\text{a}^2}{\lambda^2f^2}(n_\text{d}^2-n_\text{i}^2)\right]\]

respectively.

For phase-matched, down-shifted Bragg diffraction, the signs of both \(\theta_\text{i}\) and \(\theta_\text{d}\) change from those given above for up-shifted diffraction.

Note that when \(k_\text{i}\ne{k}_\text{d}\), either the incident or the diffracted wave has to be an extraordinary wave; sometimes both of them are.

Therefore, in the above equations, the value of \(k_\text{i}\) depends on \(\theta_\text{i}\) and that of \(k_\text{d}\) depends on \(\theta_\text{d}\), in general.

In Bragg diffraction, the angles of incidence and diffraction are not limited to small values as is the case in Raman-Nath diffraction. As can be seen from (8-61) and (8-62), depending on the values of \(k_\text{i}\), \(k_\text{d}\), and \(K\) involved in the process, the values of \(\theta_\text{i}\) and \(\theta_\text{d}\) dictated by the phase-matching conditions can be anywhere between \(-\pi/2\) and \(\pi/2\), if they exist.

For certain combinations of \(k_\text{i}\), \(k_\text{d}\), and \(K\), there are no solutions for \(\theta_\text{i}\) and \(\theta_\text{d}\). In such cases, Bragg diffraction cannot occur because the required phase-matching condition cannot be satisfied.

Figure 8-4 above shows various phase-matching configurations for up-shifted and down-shifted Bragg diffraction from a traveling acoustic wave of a given wavevector \(\mathbf{K}\).

Several remarks on phase-matched Bragg diffraction can be made:

**(1)**. Bragg diffraction in an isotropic medium, with or without a change of polarization, and that in an anisotropic medium without a change of polarization between the incident and the diffracted waves are nonbirefringent because \(k_\text{d}\approx{k}_\text{i}=k\) in such cases. The angles of incidence and diffraction in nonbirefringent Bragg diffraction are of equal magnitude but opposite signs:

\[\tag{8-63}\theta_\text{i}=-\theta_\text{B},\qquad\theta_\text{d}=\theta_\text{B}\]

for up-shifted diffraction, and

\[\tag{8-64}\theta_\text{i}=\theta_\text{B},\qquad\theta_\text{d}=-\theta_\text{B}\]

for down-shifted diffraction, where

\[\tag{8-65}\theta_\text{B}=\sin^{-1}\frac{K}{2k}=\sin^{-1}\frac{\lambda{f}}{2nv_\text{a}}\]

is known as the ** Bragg angle**. For a monochromatic optical wave incident at any \(\theta_\text{i}\), there is always one and only one value of \(K\) that can satisfy the phase-matching condition for either up-shifted or down-shifted Bragg diffraction, depending on the sign of \(\theta_\text{i}\), as demonstrated in figures 8-4(e) and (f) above.

**(2)**. For Bragg diffraction in an isotropic medium, the values of \(K\) and \(f\) that allow phase-matched interaction fall in the following range:

\[\tag{8-66}0\le{K}\le2k,\qquad\text{or}\qquad0\le{f}\le\frac{2nv_\text{a}}{\lambda}\]

For birefringent Bragg diffraction in an anisotropic medium in general, the range of \(K\) and \(f\) for phase-matched interaction is:

\[\tag{8-67}|k_\text{i}-k_\text{d}|\le{K}\le|k_\text{i}+k_\text{d}|,\qquad\text{or}\qquad\frac{|n_\text{i}-n_\text{d}|v_\text{a}}{\lambda}\le{f}\le\frac{|n_\text{i}+n_\text{d}|v_\text{a}}{\lambda}\]

**(3)**. For birefringent Bragg direction in an anisotropic medium, there is a change of polarization between the incident and the diffracted waves, and \(k_\text{d}\) can be either smaller or larger than \(k_\text{i}\).

In the case when \(k_\text{d}\lt{k}_\text{i}\), Bragg diffraction occurs only if the incident angle satisfies \(\pi/2\ge|\theta_\text{i}|\gt\cos^{-1}(k_\text{d}/k_\text{i})\). For each acceptable value of \(\theta_\text{i}\), there may exist two values or only one value of \(K\) for phase matching if the diffracted wave is extraordinary so that the value of \(k_\text{d}\) depends on \(\theta_\text{d}\), but two values of \(K\) always exist for each acceptable \(\theta_\text{i}\) if the diffracted wave is ordinary. Up-shifted diffraction occurs when \(\theta_\text{i}\) has a negative value, as demonstrated in figure 8-4(a) and (c). Down-shifted diffraction occurs when \(\theta_\text{i}\) has a positive value, as demonstrated in figures 8-4(b) and (d).

In the case when \(k_\text{d}\gt{k}_\text{i}\), for any incident angle except \(\theta_\text{i}=0\), one value of \(K\) exists for up-shifted diffraction and another for down-shifted diffraction, if the values of both \(k_\text{i}\) and \(k_\text{d}\) are fixed. For given \(\mathbf{k}_\text{i}\) and \(\mathbf{K}\), this implies that \(\mathbf{k}_\text{d}\) for up-shifted diffraction and that for down-shifted diffraction have different directions and different magnitudes, as demonstrated in figures 8-4(g)-(j).

**(4)**. When \(\theta_\text{i}=\pi/2\) or \(-\pi/2\), the phase-matching configurations are collinear. In these configurations, also shown in figure 8-4, \(\mathbf{k}_\text{i}\) and \(\mathbf{k}_\text{d}\) are both collinear with \(\mathbf{K}\) but can be either parallel or antiparallel to each other.

**Example 8-3**

The angles of incidence and diffraction for Bragg diffraction are determined by the parameters of both the optical and the acoustic waves in the medium, including \(\lambda\), \(n_\text{i}\), and \(n_\text{d}\) for the optical waves and \(f\) and \(v_\text{a}\) for the acoustic wave. To see the dependences of \(\theta_\text{i}\) and \(\theta_\text{d}\) on these parameters clearly, it is convenient to define a dimensionless normalized acoustic frequency:

\[\tag{8-68}\hat{f}=\frac{\lambda{f}}{(n_\text{i}+n_\text{d})v_\text{a}}\]

Then \(\theta_\text{i}\) and \(\theta_\text{d}\) given in (8-61) and (8-62) for up-shifted Bragg diffraction can be expressed as

\[\tag{8-69}\theta_\text{i}=-\sin^{-1}\frac{\hat{f}+\hat{f}_\text{min}/\hat{f}}{1+\hat{f}_\text{min}},\qquad\theta_\text{d}=\sin^{-1}\frac{\hat{f}-\hat{f}_\text{min}/\hat{f}}{1-\hat{f}_\text{min}}\]

where

\[\tag{8-70}\hat{f}_\text{min}=\frac{n_\text{i}-n_\text{d}}{n_\text{i}+n_\text{d}}\]

Note that \(\hat{f}_\text{min}\) can be either positive or negative depending on whether \(n_\text{i}\gt{n}_\text{d}\) or \(n_\text{i}\lt{n}_\text{d}\).

For down-shifted Bragg diffraction, the signs of both \(\theta_\text{i}\) and \(\theta_\text{d}\) change from those seen in (8-69).

For nonbirefringent Bragg diffraction, we simply set \(n_\text{i}=n_\text{d}=n\) so that \(\hat{f}_\text{min}=0\) and \(\theta_\text{i}=-\theta_\text{d}=-\theta_\text{B}\) with

\[\tag{8-71}\theta_\text{B}=\sin^{-1}\hat{f}\]

We find from (8-69) that \(\theta_\text{i}\) and \(\theta_\text{d}\) have solutions only if the frequency \(\hat{f}\) falls within the range:

\[\tag{8-72}|\hat{f}_\text{min}|\le{f}\le1\]

This condition is the same as that in (8-67) for birefringent Bragg diffraction and that in (8-66) for nonbirefringent Bragg diffraction. Therefore, \(|\hat{f}_\text{min}|\) is the minimum normalized acoustic frequency that allows a phase-matched birefringent Bragg interaction, whereas there is no minimum acoustic frequency for phase-matched nonbirefringent Bragg diffraction.

In the case when \(n_\text{i}\gt{n}_\text{d}\) so that \(\hat{f}_\text{min}\gt0\), we also find from (8-69) that \(\theta_\text{d}=0\) while \(|\theta_\text{i}|\) has a minimum value of \(|\theta_\text{i}^\text{min}|\) for an incident angle of

\[\tag{8-73}\theta_\text{i}^\text{min}=-\sin^{-1}\frac{2|\hat{f}_\text{min}|^{1/2}}{1+|\hat{f}_\text{min}|}=-\cos^{-1}\frac{n_\text{d}}{n_\text{i}}\]

at the frequency of

\[\tag{8-74}\hat{f}_\text{t}=|\hat{f}_\text{min}|^{1/2}\]

In the case when \(n_\text{i}\lt{n}_\text{d}\), we find \(\theta_\text{i}=0\) and \(\theta_\text{d}=\theta_\text{d}^\text{min}=\cos^{-1}n_\text{i}/n_\text{d}\) at the frequency \(\hat{f}_\text{t}\).

Phase matching for birefringent Bragg diffraction at \(\hat{f}_\text{t}\) so that one angle is zero and another has a minimum absolute value is known as ** tangential phase matching** or

**because either \(\mathbf{k}_\text{d}\) (in the case when \(n_\text{i}\gt{n}_\text{d}\)) or \(\mathbf{k}_\text{i}\) (in the case when \(n_\text{i}\lt{n}_\text{d}\)) is perpendicular to \(\mathbf{K}\) in this situation.**

*90° phase matching*To illustrate the dependencies of \(\theta_\text{i}\) and \(\theta_\text{d}\) on the value of \(\hat{f}\) numerically and to compare birefringent and nonbirefringent interactions, we consider a birefringent case with \(n_\text{i}=2.2\) and \(n_\text{d}=1.8\) and a nonbirefringent case with \(n=2\) so that \(n_\text{i}\gt{n}_\text{d}\) and \(n_\text{i}+n_\text{d}=2n=4\).

For birefringent diffraction, we find that \(\hat{f}_\text{min}=0.1\) and that \(\theta_\text{i}=\theta_\text{i}^\text{min}=-35.1^\circ\) and \(\theta_\text{d}=0\) at \(\hat{f}_\text{t}=0.316\).

The values of \(\theta_\text{i}\) and \(\theta_\text{d}\) as a function of \(\hat{f}\) are shown in figure 8-5(a) for birefringent Bragg diffraction and in figure 8-5(b) for nonbirefringent Bragg diffraction.

The ** deflection angle** \(\theta_\text{def}=\theta_\text{d}-\theta_\text{i}\) measured from the direction of the incident wave to that of the diffracted wave is shown in figure 8-5(c) for both cases.

As can be seen, the dependences of \(\theta_\text{i}\) and \(\theta_\text{d}\) on \(\hat{f}\) differ significantly between birefringent and nonbirefringent cases, particularly at frequencies near and below \(\hat{f}_\text{t}\).

However, there is very little difference in the deflection angle \(\theta_\text{def}\) between the two cases except at frequencies near and below \(|\hat{f}_\text{min}|\) where birefringent diffraction has a cutoff but nonbirefringent diffraction does not.

Taking the approximation that \(\omega_1\approx\omega\approx\omega_{-1}\) and considering the general situation that the phase-matching condition may not be perfectly satisfied, we find from (8-38) the following coupled equations for Bragg diffraction:

\[\tag{8-75}\cos\theta_\text{i}\frac{\partial\mathcal{E}_\text{i}}{\partial{z}}+\sin\theta_\text{i}\frac{\partial\mathcal{E}_\text{i}}{\partial{x}}=\mp\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{id}}{4k_\text{i}}\mathcal{E}_\text{d}\text{e}^{\text{i}\Delta\mathbf{k}\cdot\mathbf{r}}\]

\[\tag{8-76}\cos\theta_\text{d}\frac{\partial\mathcal{E}_\text{d}}{\partial{z}}+\sin\theta_\text{d}\frac{\partial\mathcal{E}_\text{d}}{\partial{x}}=\pm\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{di}}{4k_\text{d}}\mathcal{E}_\text{i}\text{e}^{-\text{i}\Delta\mathbf{k}\cdot\mathbf{r}}\]

where \(\Delta\tilde{\epsilon}_\text{id}=\hat{e}_\text{i}^*\cdot\Delta\tilde{\boldsymbol{\epsilon}}\cdot\hat{e}_\text{d}=\Delta\tilde{\epsilon}_\text{di}^*\).

For up-shifted diffraction, \(\Delta\mathbf{k}=\mathbf{k}_\text{d}-\mathbf{k}_\text{i}-\mathbf{K}\), and the top signs are used on the right-hand side of both (8-75) and (8-76).

For down-shifted diffraction, \(\Delta\mathbf{k}=\mathbf{k}_\text{d}-\mathbf{k}_\text{i}+\mathbf{K}\), and the bottom signs are used.

According to the discussions in the coupled-wave theory tutorial, it is also understood that in the case of diffraction in an anisotropic medium, (8-75) and (8-76) represent the transverse components of the fields being coupled. We can define the following normalized amplitude for an optical field:

\[\tag{8-77}A=\left(\frac{2k}{\omega\mu_0} \right)^{1/2}\mathcal{E}\]

so that, according to (98) [refer to the propagation in an isotropic medium tutorial], the intensity of the field is simply

\[\tag{8-78}I=|A|^2\]

Then, the coupled equations for Bragg diffraction can be written as

\[\tag{8-79}\cos\theta_\text{i}\frac{\partial{A}_\text{i}}{\partial{z}}+\sin\theta_\text{i}\frac{\partial{A}_\text{i}}{\partial{x}}=\text{i}\kappa{A}_\text{d}\text{e}^{\text{i}\Delta\mathbf{k}\cdot\mathbf{r}}\]

\[\tag{8-80}\cos\theta_\text{d}\frac{\partial{A}_\text{d}}{\partial{z}}+\sin\theta_\text{d}\frac{\partial{A}_\text{d}}{\partial{x}}=\text{i}\kappa^*{A}_\text{i}\text{e}^{-\text{i}\Delta\mathbf{k}\cdot\mathbf{r}}\]

where, for up-shifted or down-shifted diffraction, respectively,

\[\tag{8-81}\kappa=\text{i}\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{id}}{4k_\text{i}^{1/2}k_\text{d}^{1/2}}\qquad\text{or}\qquad\kappa=-\text{i}\frac{\omega^2\mu_0\Delta\tilde{\epsilon}_\text{id}}{4k_\text{i}^{1/2}k_\text{d}^{1/2}}\]

Using (8-22) [refer to the photoelastic effect tutorial], we find that the Bragg coupling coefficient can be expressed in terms of the acousto-optic figure of merit as

\[\tag{8-82}|\kappa|=\frac{\pi}{\lambda}\left(\frac{M_2I_\text{a}}{2}\right)^{1/2}\]

which has the same form as that of the coupling coefficient for Raman-Nath diffraction defined in (8-54) above.

In general, \(\theta_\text{i}\) and \(\theta_\text{d}\) can have any values between \(-\pi/2\) and \(\pi/2\), and \(A_\text{i}\) and \(A_\text{d}\) can vary with both \(x\) and \(z\).

However, two extreme cases that have much simplified solutions for the coupled equations in (8-79) and (8-80) are of particular interest and are discussed in the following.

**Small-angle Bragg diffraction:** \(\theta_\text{i}\approx0\)

In case the angles of incidence and diffraction are both very small, \(\cos\theta_\text{i}\) and \(\cos\theta_\text{d}\) can both be approximated by unity, and \(\sin\theta_\text{i}\) and \(\sin\theta_\text{d}\) are both approximately zero.

This is the situation in which the optical waves propagate almost perpendicularly to the acoustic wave. It normally occurs when the acoustic wavelength is much larger than the optical wavelength but the interaction length is large so that (8-57) is satisfied.

In this case, the field amplitudes, \(A_\text{i}\) and \(A_\text{d}\), vary primarily with \(z\) only, and the phase mismatch is \(\Delta\mathbf{k}=\Delta{k}\hat{z}\). The coupled equations in (8-79) and (8-80) then reduce to

\[\tag{8-83}\frac{\text{d}A_\text{i}}{\text{d}z}=\text{i}\kappa{A}_\text{d}\text{e}^{\text{i}\Delta{k}z}\]

\[\tag{8-84}\frac{\text{d}A_\text{d}}{\text{d}z}=\text{i}\kappa^*{A}_\text{i}\text{e}^{-\text{i}\Delta{k}z}\]

The boundary conditions are \(A_\text{i}(0)\ne0\) and \(A_\text{d}(0)=0\).

These coupled equations describe codirectional coupling of the incident and the diffracted waves with symmetric coupling coefficients. They have the solutions obtained in the two-mode coupling tutorial for codirectionally coupled modes when we identify \(\Delta{k}\) with \(2\delta\).

Therefore, the ** codirectional Bragg diffraction efficiency** over an interaction length \(l\) is

\[\tag{8-85}\eta=\frac{I_\text{d}(l)}{I_\text{i}(0)}=\frac{|A_\text{d}(l)|^2}{|A_\text{i}(0)|^2}=\frac{1}{1+\Delta{k}^2/4|\kappa|^2}\sin^2\left(|\kappa|l\sqrt{1+\Delta{k}^2/4|\kappa|^2}\right)\]

The diffraction efficiency in the case of perfect phase matching is

\[\tag{8-86}\eta_\text{PM}=\sin^2|\kappa|l\]

Using (8-82), it is found that the acoustic intensity needed for 100% Bragg diffraction efficiency is

\[\tag{8-87}I_\text{a}=\frac{\lambda^2}{2M_2l^2}\]

**Collinear Bragg diffraction**: \(\theta_\text{i}=\pm\pi/2\)

Another special case of particular interest is collinear Bragg diffraction, which occurs when the incident optical wave propagates collinearly with the acoustic wave.

The wavevector \(\mathbf{k}_\text{i}\) can be either parallel or antiparallel to \(\mathbf{K}\), corresponding to \(\theta_\text{i}=\pi/2\) and \(-\pi/2\), respectively.

The phase-matching condition in (8-60) then requires that \(\theta_\text{d}\) be either \(\pi/2\) or \(-\pi/2\). Therefore, \(\mathbf{k}_\text{d}\) is collinear with both \(\mathbf{k}_\text{i}\) and \(\mathbf{K}\).

As can be seen from the collinear phase-matching configurations shown in figure 8-4 above, the diffracted optical wave can propagate either codirectionally or contradirectionally with respect to the incident optical wave.

** Collinear Bragg diffraction in an isotropic medium is always contradirectional** because \(k_\text{i}=k_\text{d}=K/2\) in this situation. However,

**because two values of \(K\) exist for phase-matching in each case when \(k_\text{i}\ne{k}_\text{d}\).**

*collinear Bragg diffraction in an anisotropic medium can be codirectional or contradirectional*Codirectional Bragg diffraction occurs in an anisotropic medium with an acoustic wave at a low frequency corresponding to a small \(K\) value, whereas contradirectional Bragg diffraction occurs with an acoustic wave at a high frequency corresponding to a large \(K\) value.

From the above discussions, we find that the coupled equations describing collinear Bragg diffraction can be either those of codirectional coupling or those of contradirectional coupling, depending on whether the propagation directions of the incident and the diffracted waves are codirectional or contradirectional.

In either case, with \(\theta_\text{i}=\pm\pi/2\), \(\theta_\text{d}=\pm\pi/2\), and \(\mathbf{K}=K\hat{x}\), both \(A_\text{i}\) and \(A_\text{d}\) vary only with \(x\) and \(\Delta\mathbf{k}=\Delta{k}\hat{x}\). The coupled equations in (8-79) and (8-80) then reduce to

\[\tag{8-88}\pm\frac{\text{d}A_\text{i}}{\text{d}x}=\text{i}\kappa{A}_\text{d}\text{e}^{\text{i}\Delta{k}x}\]

\[\tag{8-89}\pm\frac{\text{d}A_\text{d}}{\text{d}x}=\text{i}\kappa^*{A}_\text{i}\text{e}^{-\text{i}\Delta{k}x}\]

In (8-88), the plus or minus sign on the left-hand side is chosen respectively according to whether \(\mathbf{k}_\text{i}\) points in the direction of \(\hat{x}\) or \(-\hat{x}\).

Similarly, in (8-89), the plus or minus sign is chosen respectively according to whether \(\mathbf{k}_\text{d}\) points in the direction of \(\hat{x}\) or \(-\hat{x}\).

**(1)** When \(\mathbf{k}_\text{i}\) and \(\mathbf{k}_\text{d}\) point in the same direction, the same sign is chosen in both (8-88) and (8-89) for codirectional coupling between the incident and the diffracted waves. The boundary conditions for codirectional Bragg diffraction are \(A_\text{i}(0)\ne0\) and \(A_\text{d}(0)=0\), and the diffraction efficiency is that given in (8-85) in general and that in (8-86) for perfect phase matching.

** Collinear, codirectional Bragg diffraction is birefringent and is possible only in anisotropic media. It is always accompanied by a change of polarization between the incident and the diffracted waves**.

**(2)** When \(\mathbf{k}_\text{i}\) and \(\mathbf{k}_\text{d}\) point in the opposite directions, different signs are chosen in (8-88) and (8-89) for contradirectional coupling between the incident and the diffracted waves. For contradirectional Bragg diffraction over an interaction length \(l\), the boundary conditions are \(A_\text{i}(0)\ne0\) and \(A_\text{d}(l)=0\) if \(\mathbf{k}_\text{d}=-k_\text{d}\hat{x}\) and are \(A_\text{i}(0)\ne0\) and \(A_\text{d}(-l)=0\) if \(\mathbf{k}_\text{d}=k_\text{d}\hat{x}\). The solutions obtained in the two-mode coupling tutorial for contradirectionally coupled modes can be used when we identify \(\Delta{k}\) with \(2\delta\). Therefore, the ** contradirectional Bragg diffraction efficiency** is

\[\tag{8-90}\eta=\frac{I_\text{d}(0)}{I_\text{i}(0)}=\frac{|A_\text{d}(0)|^2}{|A_\text{i}(0)|^2}=\frac{\sinh^2\left(|\kappa|l\sqrt{1-\Delta{k}^2/4|\kappa|^2}\right)}{\cosh^2\left(|\kappa|l\sqrt{1-\Delta{k}^2/4|\kappa|^2}\right)-\Delta{k}^2/4|\kappa|^2}\]

The diffraction efficiency in the case of perfect phase matching is

\[\tag{8-91}\eta_\text{PM}=\tanh^2|\kappa|l\]

** Collinear, contradirectional Bragg diffraction is possible in both isotropic and anisotropic media. A change of polarization between the incident and the diffracted waves may or may not occur in this process**.

**Example 8-4**

An optical wave at 632.8 nm wavelength interacts with a longitudinal acoustic wave of a frequency \(f\) = 100 MHz in a piece of fused silica glass over an interaction length of \(l\) = 4 cm. The incident optical wave is polarized in a direction \(\hat{e}_\text{i}\) that is perpendicular to vector \(\mathbf{K}\) of the acoustic wave. Is this interaction in the Bragg regime? What incident angle \(\theta_\text{i}\) should be chosen for phase matching? What is the deflection angle \(\theta_\text{def}\)? Find the acoustic intensity that is required for a 100% diffraction efficiency if that is possible.

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 and \(n=1.457\) at 632.8 nm. At the acoustic frequency of \(f=100\text{ MHz}\), we find that

\[Q=2\pi\frac{\lambda{f}^2l}{nv_\text{a}^2}=2\pi\times\frac{632.8\times10^{-9}\times(100\times10^6)^2\times4\times10^{-2}}{1.457\times(5.97\times10^3)^2}=30.6\gg1\]

Therefore, the interaction is in the Bragg regime, and phase matching is required. Because this is nonbirefringent phase matching, the angles of incidence and diffraction are both defined by the following Bragg angle:

\[\theta_\text{B}=\sin^{-1}\frac{\lambda{f}}{2nv_\text{a}}=\sin^{-1}\frac{632.8\times10^{-9}\times100\times10^6}{2\times1.457\times5.97\times10^3}=0.21^\circ\]

We then have \(\theta_\text{i}=-\theta_\text{B}=-0.21^\circ\) and \(\theta_\text{def}=2\theta_\text{B}=0.42^\circ\) for up-shifted diffraction and \(\theta_\text{i}=\theta_\text{B}=0.21^\circ\) and \(\theta_\text{def}=-2\theta_\text{B}=-0.42^\circ\) for down-shifted diffraction.

Because \(\theta_\text{i}\approx0\) in this problem, this is a case of small-angle Bragg diffraction. It is therefore possible to accomplish a 100% diffraction efficiency. Because \(\hat{e}_\text{i}\perp\mathbf{K}\), the relevant figure of merit for this interaction is \(M_2^\perp=1.5\times10^{-15}\text{ m}^2\text{ W}^{-1}\) found in example 8-1 [refer to the photoelastic effect tutorial]. We then find, using (8-87), that the required acoustic intensity is

\[I_\text{a}=\frac{\lambda^2}{2M_2l^2}=\frac{(632.8\times10^{-9})^2}{2\times1.5\times10^{-15}\times(4\times10^{-2})^2}\text{W m}^{-2}=8.29\text{ W cm}^{-2}\]

**Diffraction from a standing acoustic wave**

So far only diffraction from a traveling acoustic wave has been considered. We have seen that each spatial diffraction order defined by a wavevector \(\mathbf{k}_q\) at a diffraction angle \(\theta_q\) contains a single, uniquely defined frequency of \(\omega_q=\omega+q\Omega\). This is not the case, however, for diffraction from a standing acoustic wave.

A standing acoustic wave can be considered as a linear superposition of two contrapropagating traveling waves with both \(\mathbf{K}\) and \(-\mathbf{K}\) existing simultaneously for phase matching.

The implication of this situation is two-fold:

(1) Both up-shifted and down-shifted frequencies are simultaneously generated in each phase-matched direction of diffraction.

(2) Each shifted optical frequency generated by diffraction can be diffracted back to the direction of the incident wave with a further shift in frequency.

This process cascades. Figure 8-6 below shows the cascading process in the case of Raman-Nath diffraction from a standing acoustic wave.

At the output, each of the even spatial orders, including the undiffracted zeroth order, consists of all of the frequencies up- or down-shifted by even multiples of \(\Omega\), whereas each of the odd spatial orders consists of all of the frequencies up- or down-shifted by odd multiples of \(\Omega\).

For Bragg diffraction from a standing acoustic wave, the incident angle can be either \(\theta_\text{i}\) or \(-\theta_\text{i}\) of the form given in (8-61) because \(\mathbf{K}\) and \(-\mathbf{K}\) exist simultaneously.

In either case, both up-shifted and down-shifted frequencies are generated in the direction of the corresponding \(\mathbf{k}_\text{d}\).

This process cascades. Consequently, the undiffracted beam in the \(\mathbf{k}_\text{i}\) direction contains a series of even side bands at \(\omega\pm{2m}\Omega\), and the diffracted beam in the \(\mathbf{k}_\text{d}\) direction contains the odd side bands at \(\omega\pm(2m+1)\Omega\).

Figure 8-7 below shows the cascading process in Bragg diffraction.

A formal analysis of diffraction from a standing acoustic wave using coupled-wave theory can be carried out, but it is very complicated because each spatial order consists of many different frequency components. Closed-form analytical solutions can be obtained, however, without solving coupled-wave equations but by extending the results obtained from the analysis of diffraction from a traveling wave.

For a standing acoustic wave of the form given in (8-2) [refer to the elastic waves tutorial], we have

\[\tag{8-92}\Delta\boldsymbol{\epsilon}=\Delta\tilde{\boldsymbol{\epsilon}}\sin(\mathbf{K}\cdot\mathbf{r})\cos\Omega{t}=\Delta\tilde{\boldsymbol{\epsilon}}(t)\sin(\mathbf{K}\cdot\mathbf{r})\]

Because \(\omega\gg\Omega\), the temporal variation in \(\Delta\tilde{\boldsymbol{\epsilon}}(t)=\Delta\tilde{\boldsymbol{\epsilon}}\cos\Omega{t}\) is very slow compared to the optical cycles. Consequently, in place of (8-35), we can expand the field as

\[\tag{8-93}\mathbf{E}(\mathbf{r},t)=\sum_q\boldsymbol{\mathcal{E}}_q(\mathbf{r},t)\text{e}^{\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega{t}}=\sum_q\hat{e}_q\mathcal{E}_q(\mathbf{r},t)\text{e}^{\text{i}\mathbf{k}_q\cdot\mathbf{r}-\text{i}\omega{t}}\]

where \(q\) represents the spatial diffraction order and \(\mathbf{k}_q=\mathbf{k}_\text{i}+q\mathbf{K}\).

For Raman-Nath diffraction, it follows from the analysis leading to (8-55) that for the spatial diffraction order \(q\ge0\), we have

\[\tag{8-94}\begin{align}\frac{\mathcal{E}_q(l,t)}{\mathcal{E}_0(0,0)}&=J_q(-2|\kappa|l\cos\Omega{t})\\&=\sum_{n=0}\sum_{m=0}^\infty\sum_{p=-m}^m\frac{(-1)^{q+p}q!(2m)!(|\kappa|l)^m}{2^{q+2m}m!n!(q-n)!(m-p)!(m+p)!}J_{q+m}(2|\kappa|l)\text{e}^{\text{i}(q-2n-2p)\Omega{t}}\end{align}\]

and \(\mathcal{E}_{-q}(l,t)=(-1)^q\mathcal{E}_q(l,t)\) for the negative spatial orders.

In the expansion of (8-94), we have used the following multiplication theorem for the Bessel functions:

\[\tag{8-95}J_q(x\cos\phi)=\cos^q\phi\sum_{m=0}^\infty\frac{\sin^{2m}\phi}{m!}\left(\frac{x}{2}\right)^mJ_{q+m}(x),\qquad\text{for }q\ge0\]

As can be seen from (8-94), the temporal dependence of \(\mathcal{E}_q(l,t)\) contains all of the positive and negative even harmonics of \(\Omega\) if \(q\) is an even integer, and it contains all of the positive and negative odd harmonics of \(\Omega\) if \(q\) is an odd integer.

Therefore, each even spatial order consists of a series of frequency components at \(\omega\pm2m\Omega\), while each odd spatial order consists of a series of frequency components at \(\omega\pm(2m+1)\Omega\).

A similar analysis can be carried out for Bragg diffraction in different situations. For small-angle Bragg diffraction with perfect phase matching, we have

\[\tag{8-96}\begin{align}A_\text{i}(l,t)&=A_\text{i}(0,0)\cos(|\kappa|l\cos\Omega{t})\\&=A_\text{i}(0,0)\sum_{m=-\infty}^\infty(-1)^mJ_{2m}(|\kappa|l)\text{e}^{\text{i}2m\Omega{t}}\end{align}\]

\[\tag{8-97}\begin{align}A_\text{d}(l,t)&=\text{i}A_\text{i}(0,0)\sin(|\kappa|l\cos\Omega{t})\\&=\text{i}A_\text{i}(0,0)\sum_{m=-\infty}^\infty(-1)^mJ_{2m+1}(|\kappa|l)\text{e}^{\text{i}(2m+1)\Omega{t}}\end{align}\]

where, for the expansion, we have used the following identities:

\[\tag{8-98}\cos(x\cos\phi)=\sum_{m=-\infty}^\infty(-1)^mJ_{2m}(x)\text{e}^{\text{i}2m\phi}\]

\[\tag{8-99}\sin(x\cos\phi)=\sum_{m=-\infty}^\infty(-1)^mJ_{2m+1}(x)\text{e}^{\text{i}(2m+1)\phi}\]

It can be seen clearly from (8-96) and (8-97) that the undiffracted beam consists of the frequency component at \(\omega\) and all of the even side bands at \(\omega\pm2m\Omega\), while the diffracted beam consists of all of the odd side bands at \(\omega\pm(2m+1)\Omega\), as discussed above.

The concept of intensity refers to the flow of energy through a unit area. It can be clearly defined for a traveling wave but is not applicable to a standing wave. However, a standing wave can be considered to be the linear superposition of two contrapropagating traveling waves of equal amplitude and, therefore, of equal intensity.

For a standing acoustic wave described by (8-2) [refer to the elastic waves tutorial], we find the amplitude of the strain tensor element representing the two contrapropagating acoustic waves to be \(\mathcal{S}^\text{f}=\mathcal{S}^\text{b}=\mathcal{S}/2\), where \(\mathcal{S}\) is the same as the tensor element \(\mathcal{S}_{kl}\) defined in (8-18) [refer to the photoelastic effect tutorial].

Therefore, the intensities of the two contrapropagating traveling acoustic waves are given by

\[\tag{8-100}I_\text{a}^\text{f}=I_\text{a}^\text{b}=\frac{1}{2}\left(\frac{\mathcal{S}}{2}\right)^2\rho{v}_\text{a}^3\]

Using (8-20) [refer to the photoelastic effect tutorial] for the relation between \(I_\text{a}\) and \(\mathcal{S}\) used in defining \(M_2\) in (8-22) [refer to the photoelastic effect tutorial], we find from the above that \(I_\text{a}=4I_\text{a}^\text{f}=4I_\text{a}^\text{b}\). We then find, using (8-22), that

\[\tag{8-101}|\Delta\tilde{\epsilon}_\text{id}|^2=8\epsilon_0^2n_\text{i}n_\text{d}M_2I_\text{a}^\text{f}=8\epsilon_0^2n_\text{i}n_\text{d}M_2I_\text{a}^\text{b}\]

Using (8-101) in (8-53) and (8-81) for the cases of Raman-Nath and Bragg diffraction, respectively, we find that the coupling coefficient for both cases is

\[\tag{8-102}|\kappa|=\frac{2\pi}{\lambda}\left(\frac{M_2I_\text{a}^\text{f}}{2}\right)^{1/2}=\frac{2\pi}{\lambda}\left(\frac{M_2I_\text{a}^\text{b}}{2}\right)^{1/2}\]

This is the coupling coefficient that appear in (8-94) and in (8-96) and (8-97) for diffraction from a standing acoustic wave. It has a form different from that of (8-54) and (8-82) for diffraction from a traveling acoustic wave.

In both Raman-Nath and Bragg diffraction from a standing acoustic wave, the number of frequencies that appear in each spatially separated, diffracted or undiffracted, beam is determined by the dispersion and the bandwidth of the medium, as well as by the total length and the strength of interaction.

The next part continues with the acoustic-optic modulators tutorial.