# Elastic Waves

This is a continuation from the previous tutorial - guided-wave magneto-optic devices.

**Introduction**

Scattering of light by acoustic waves was first investigated by Brillouin. The acoustic frequencies involved in Brillouin scattering fall in the ultrasonic and hypersonic regions.

Hypersonic waves in a medium are caused by thermal excitation, whereas ultrasonic waves can be excited electronically using piezoelectric transducers. The acoustic waves used in acousto-optics are generally ultrasonic waves that have frequencies in the range between about 100 kHz and a few gigahertz.

The basic principles of acousto-optic devices are based on the scattering of light by the periodic index variations generated by an acoustic wave in the supporting medium.

These periodic index variations form a moving index grating, generated by a traveling acoustic wave, or a standing index grating, generated by a standing acoustic wave. Because the speed of sound in dielectric media that are used for device applications typically falls in the range between 1 and 10 km s^{-1}, the index gratings generated by ultrasonic acoustic waves have grating periods ranging from the order of 1 μm to a few centimeters.

A unique property of the index grating created by an acoustic wave is that its period and modulation depth can be varied by varying the frequency and amplitude, respectively, of the acoustic wave through variation in the electronic signal applied to the transducer. Therefore, the operating parameters of an acousto-optic device can be controlled electronically.

Practical acousto-optic devices include modulators, beam deflectors, frequency shifters, couplers, switches, and spectrum analyzers.

**Elastic Waves**

An acoustic wave in a medium is an elastic wave of space- and time-dependent periodic deformation in the medium. A ** traveling** plane acoustic wave can be expressed as

\[\tag{8-1}\mathbf{u}(\mathbf{r},t)=\boldsymbol{\mathcal{U}}\cos(\mathbf{K}\cdot\mathbf{r}-\Omega{t})\]

A ** standing** plane acoustic wave is a combination of two contrapropagating traveling waves of equal amplitude, wavelength, and frequency. It can be described by

\[\tag{8-2}\mathbf{u}(\mathbf{r},t)=\boldsymbol{\mathcal{U}}\cos(\mathbf{K}\cdot\mathbf{r})\cos\Omega{t}\]

In (8-1) and (8-2), \(\mathbf{u}(\mathbf{r},t)\) represents the time-dependent displacement of the point at \(\mathbf{r}\) in the medium subject to deformation, \(\boldsymbol{\mathcal{U}}\) is the amplitude of the elastic wave, and \(\mathbf{K}\) and \(\Omega=2\pi{f}\) are its wavevector and angular frequency, respectively.

The physical meaning of the displacement vector \(\mathbf{u}(\mathbf{r},t)\) is that in a fixed coordinate system, an atom, or an ion, in the medium at location \(\mathbf{r}\) before deformation is moved to location \(\mathbf{r}+\mathbf{u}\) under deformation caused by the elastic wave. Therefore, \(\mathbf{u}(\mathbf{r},t)\) describes the motion of the particles in the medium supporting the acoustic wave.

The direction of the amplitude vector \(\boldsymbol{\mathcal{U}}\) defines the polarization of the acoustic wave, while that of the wavevector \(\mathbf{K}\) describes the direction of propagation of the wave. The two contrapropagating traveling waves that form the standing wave expressed in (8-2) propagate in \(\mathbf{K}\) and \(-\mathbf{K}\) directions, respectively.

The amplitude of the wavevector is

\[\tag{8-3}K=\frac{2\pi}{\Lambda}=\frac{\Omega}{v_\text{a}}=\frac{2\pi{f}}{v_\text{a}}\]

where \(\Lambda\) and \(v_\text{a}\) are the wavelength and phase velocity, respectively, of the acoustic wave.

For any given direction of propagation of an acoustic wave in any medium, there are ** three** orthogonal normal modes of polarization. If one mode is polarized along the direction of \(\mathbf{K}\), the directions of polarization of the other two modes are perpendicular to \(\mathbf{K}\).

An acoustic wave polarized in the direction of \(\mathbf{K}\) is known as a ** longitudinal wave**, while one with a polarization perpendicular to \(\mathbf{K}\) is called a

**or a**

*transverse wave***.**

*shear wave*In isotropic media and cubic crystals, the three normal modes are always one purely longitudinal and two purely transverse for any direction of acoustic wave propagation.

In anisotropic crystals other than those in the triclinic system, the normal modes again consist of one purely longitudinal wave and two purely transverse waves if the acoustic wave propagates along a crystal axis of two-, three-, four-, or six-fold symmetry.

In general, however, the polarization directions of the normal modes of an acoustic wave in an anisotropic crystal are not necessarily parallel or perpendicular to the direction of wave propagation. Then, a mode that has its polarization close to the direction of \(\mathbf{K}\) is called ** quasi-longitudinal**, and one whose polarization is close to being perpendicular to \(\mathbf{K}\) is called

**. Figure 8-1 illustrates the characteristics of different modes of acoustic waves.**

*quasi-transverse*

At a given acoustic frequency in a given medium, the acoustic velocity \(v_\text{a}\), and consequently, the wavelength \(\Lambda\), and the value of \(K\) all depend on both the direction of propagation and that of the polarization of an acoustic wave.

In an isotropic medium, the two transverse modes are degenerate, meaning that they have the same acoustic velocity, but they are generally not degenerate with the longitudinal mode.

In a cubic crystal, the two transverse modes are degenerate only for waves propagating along certain directions, such as the [100] and [111] directions of the crystal.

In anisotropic crystals, all three normal modes are generally nondegenerate.

Deformation of a medium can be characterized by a second-rank ** displacement gradient tensor** defined by

\[\tag{8-4}\frac{\partial{u_i}}{\partial{x_j}}\]

where the indices \(i,j=1,2,3\) represent the coordinates \(x\), \(y\), \(z\).

The mechanical strains associated with deformation are described by a symmetric ** strain tensor**, \(\mathbf{S}=[S_{ij}]\), defined by

\[\tag{8-5}S_{ij}=\frac{1}{2}\left(\frac{\partial{u_i}}{\partial{x_j}}+\frac{\partial{u_j}}{\partial{x_i}}\right)\]

The three tensor elements \(S_{xx}\), \(S_{yy}\), and \(S_{zz}\) are ** tensile strains**, while the other elements \(S_{yz}=S_{zy}\), \(S_{zx}=S_{xz}\), and \(S_{xy}=S_{yx}\) are

**.**

*shear strains*In addition, there is an antisymmetric ** rotation tensor**, \(\mathbf{R}=[R_{ij}]\), defined by

\[\tag{8-6}R_{ij}=\frac{1}{2}\left(\frac{\partial{u_i}}{\partial{x_j}}-\frac{\partial{u_j}}{\partial{x_i}}\right)\]

Clearly, \(R_{xx}=R_{yy}=R_{yy}=0\), while \(R_{yz}=-R_{zy}\), \(R_{zx}=-R_{xz}\), and \(R_{xy}=-R_{yx}\).

Note that the elements of the strain and rotation tensors, as well as those of the displacement gradient tensor, are dimensionless. For elastic deformation caused by an acoustic wave such as that described by (8-1), all of these tensor elements are space- and time-dependent quantities.

If the reference coordinate system is chosen such that one of its axes lines up with \(\mathbf{K}\), a longitudinal acoustic wave generates only one tensile strain component and no rotation while a transverse acoustic wave generates only shear strains and rotation.

For example, if we take \(\mathbf{K}=K\hat{x}\), a longitudinal wave has \(\boldsymbol{\mathcal{U}}=\mathcal{U}\hat{x}\). Then, the only nonzero element of the strain tensor is \(S_{xx}\), and all elements of the rotation tensor vanish.

For a transverse wave, \(\boldsymbol{\mathcal{U}}=\mathcal{U}_y\hat{y}+\mathcal{U}_z\hat{z}\). If both \(\mathcal{U}_y\) and \(\mathcal{U}_z\) are nonzero, the only nonzero elements of the strain tensor are \(S_{xy}=S_{yx}\) and \(S_{zx}=S_{xz}\), while the nonzero elements of the rotation tensor are \(R_{xy}=-R_{yx}\) and \(R_{zx}=-R_{xz}\).

If an acoustic wave is neither purely longitudinal nor purely transverse, it can generate both tensile and shear strains as well as many elements of the rotation tensor.

The next part continues with the photoelastic effect tutorial.