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Photoelastic effect

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This is a continuation from the previous tutorial - elastic waves.

Mechanical strain in a medium causes changes in the optical property of the medium due to the photoelastic effect.  The basis of acousto-optic interaction is the dynamic photoelastic effect in which the periodic time-dependent mechanical strain caused by an acoustic wave induces periodic time-dependent variations in the optical properties of the medium.

The photoelastic effect is traditionally defined in terms of changes in the elements of the relative impermeability tensor caused by strain:

\[\tag{8-7}\eta_{ij}(\mathbf{S})=\eta_{ij}+\Delta\eta_{ij}(\mathbf{S})=\eta_{ij}+\sum_{k,l}p_{ijkl}S_{kl}\]

where \(p_{ijkl}\) are dimensionless elasto-optic coefficients, also called strain-optic coefficients or photoelastic coefficients, and they form a fourth-rank tensor.

Because \(\eta_{ij}=\eta_{ji}\) and \(S_{kl}=S_{lk}\), the elasto-optic tensor \([p_{ijkl}]\) is symmetric in \(i\) and \(j\) and in \(k\) and \(l\). The rules of index contraction defined in (115) [refer to the propagation in an anisotropic medium tutorial] can be used to reduce the double indices \(ij\) and \(kl\) to single indices \(\alpha\) and \(\beta\):

\[\tag{8-8}p_{ijkl}=p_{jikl}=p_{ijlk}=p_{jilk}=p_{\alpha\beta},\qquad\text{where }\alpha,\beta=1,2,\ldots,6.\]

In general, \(p_{\alpha\beta}\ne{p_{\beta\alpha}}\). Then, (8-7) can be expressed as

\[\tag{8-9}\eta_\alpha(\mathbf{S})=\eta_\alpha+\Delta\eta_\alpha(\mathbf{S})=\eta_\alpha+\sum_\beta{p_{\alpha\beta}}S_\beta\]

where the elements \(S_\beta\) are defined by the following rules:

\[\tag{8-10}\begin{align}S_1&=S_{xx},\qquad S_2=S_{yy},\qquad S_3=S_{zz},\\S_4&=2S_{yz},\qquad S_5=2S_{zx},\qquad S_6=2S_{xy}\end{align}\]

Note that the factor 2 in the definitions of \(S_4\), \(S_5\), and \(S_6\) deviates from the standard rules used in index contraction.

Similarly to the electro-optic Kerr effect, the photoelastic effect exists in all matters, including centrosymmetric crystals and isotropic media. Acousto-optic interactions are not precluded by any symmetry property of a medium.

Table 8-1 below lists the matrix form of \(p_{\alpha\beta}\) for various point groups. Also included in the table is the \(p_{\alpha\beta}\) matrix for isotropic media, which has only two independent elements, \(p_{11}\) and \(p_{12}\).

The elasto-optic coefficients are dimensionless. The largest of them for a particular material typically has a value on the order of 0.1-0.5.

 

Table 8-1. Matrix form of elasto-optic coefficients for all point groups

Many crystals, for example LiNbO3 and TeO2, that are of interest in acousto-optic applications are piezoelectric. The elasto-optic coefficients of such crystals are modified by the piezoelectric effect due to the fact that the strain produced by an acoustic wave generates an electric field in a crystal, which also causes index changes in the crystal through the electro-optic effect.

The modifications due to this secondary effect can be quite significant in certain crystals. These modifications depend on the direction of propagation of the acoustic wave. Furthermore, they can rotate the index ellipsoid and induce birefringence, as can be imagined from our experience with the Pockels effect.

The modifications to \(p_{ijkl}\) due to this piezoelectric effect still maintain the symmetry described in (8-8), but they do not follow the matrix form listed in Table 8-1. Their matrix form depends on the combination of the wave propagation direction and crystal symmetry.

The photoelastic effect associated with the rotational deformation characterized by the rotation tensor was not included in the definition of the elasto-optic coefficients \(p_{ijkl}\) expressed in (8-7). A complete description of the photoelastic effect has to include the contributions from both strain and rotation as

\[\tag{8-11}\Delta\eta_{ij}(\mathbf{S},\mathbf{R})=\sum_{k,l}(p_{ijkl}S_{kl}+p'_{ijkl}R_{kl})\]

In this relation, \(p_{ijkl}\) can be considered to include modifications due to the piezoelectric effect, with the understanding that they are not exactly the same as those listed in Table 8-1 and that they depend on the direction of propagation of the acoustic wave. If the indices, \(i\), \(j\), \(k\), and \(l\) are referenced to the principal axes of a crystal, we have

\[\tag{8-12}p'_{ijkl}=-\frac{1}{2}\left(\frac{1}{n_i^2}-\frac{1}{n_j^2}\right)(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})\]

where \(n_i\) and \(n_j\) represent the principal indices of refraction of the crystal.

It can be seen that \(p'_{ijkl}\) is symmetric in \(i\) and \(j\) but is antisymmetric in \(k\) and \(l\). Therefore, index contraction cannot be applied to the indices \(k\) and \(l\) in (8-11) and (8-12).

Nevertheless, in (8-11) the indices \(i\) and \(j\) can still be contracted by following the rules in (115) [refer to the propagation in an anisotropic medium tutorial] into a single index \(\alpha\) to write \(\Delta\eta_{ij}=\Delta\eta_\alpha\) for it to be used to express the index ellipsoid in the form of (6-19) [refer to the electro-optic effects tutorial].

From (8-12), we find that \(p'_{ijkl}\) vanishes for isotropic media and cubic crystals and that the rotational effect is significant only in strongly birefringent crystals.

In the treatment of acousto-optic diffraction using coupled-wave theory, it is desirable to express the photoelastic effect caused by strain and rotation in a medium formally in terms of a change in the permittivity of the medium as

\[\tag{8-13}\boldsymbol{\epsilon}(\omega,\mathbf{S},\mathbf{R})=\boldsymbol{\epsilon}(\omega)+\Delta\boldsymbol{\epsilon}(\omega,\mathbf{S},\mathbf{R})=\boldsymbol{\epsilon}(\omega)+\epsilon_0\Delta\boldsymbol{\chi}(\omega,\mathbf{S},\mathbf{R})\]

where \(\boldsymbol{\epsilon}(\omega)\) is the dielectric permittivity tensor of the medium in the absence of strain and rotation fields. The effect of strain and rotation on an optical field \(\mathbf{E}(\omega)\) propagating in a medium is characterized by a polarization of

\[\tag{8-14}\Delta\mathbf{P}(\omega,\mathbf{S},\mathbf{R})=\Delta\boldsymbol{\epsilon}(\omega,\mathbf{S},\mathbf{R})\cdot\mathbf{E}(\omega)\]

Once the elements of \(\Delta\boldsymbol{\eta}\) caused by strain and rotation are found through the procedure described above, the elements of \(\Delta\boldsymbol{\epsilon}\) can be found using the relation in (6-17) or that in (6-18) [refer to the electro-optic effects tutorial]. 

 

Acousto-optic figure of merit

In the case when \(\Delta\eta_{ij}\) is independent of rotation tensor elements but is a function of strain tensor elements only, we can use (8-7) and (6-18) [refer to the electro-optic effects tutorial] to express the photoelastic changes in the permittivity tensor as

\[\tag{8-15}\Delta\epsilon_{ij}=-\epsilon_0n_i^2n_j^2\Delta\eta_{ij}=-\epsilon_0n_i^2n_j^2\sum_{k,l}p_{ijkl}S_{kl},\qquad{i,j,k,l}=1,2,3.\]

The strain tensor elements depend on the propagation direction and the polarization mode of the acoustic wave. For an acoustic wave that has a wavevector \(\mathbf{K}\) and an angular frequency \(\Omega\), the strain tensor elements vary with space and time as

\[\tag{8-16}S_{kl}=\mathcal{S}_{kl}\sin(\mathbf{K}\cdot\mathbf{r}-\Omega{t})\]

where \(\mathcal{S}_{kl}\) is the amplitude of the strain.

The magnitude of vector \(\mathbf{K}\) depends on the polarization of the acoustic wave because longitudinal and transverse acoustic modes propagating in the same direction generally have different velocities.

Then the photoelastic permittivity tensor is a function of space and time:

\[\tag{8-17}\Delta\boldsymbol{\epsilon}=\Delta\tilde{\boldsymbol{\epsilon}}\sin(\mathbf{K}\cdot\mathbf{r}-\Omega{t})\]

where \(\Delta\tilde{\boldsymbol{\epsilon}}\) is the amplitude of \(\Delta\boldsymbol{\epsilon}\), and its elements are

\[\tag{8-18}\Delta\tilde{\epsilon}_{ij}=-\epsilon_0n_i^2n_j^2\sum_{k,l}p_{ijkl}\mathcal{S}_{kl}\]

The intensity in watts per square meter of an acoustic wave that has a strain amplitude \(\mathcal{S}\) is given by

\[\tag{8-19}I_\text{a}=\frac{1}{2}\mathcal{S}^2\rho{v_\text{a}}^3\]

where \(\rho\) is the density of the medium and \(v_\text{a}\) is the velocity of the specific acoustic mode under consideration.

Therefore, the strain amplitude in (8-18) can be properly calculated from the acoustic intensity using the relation:

\[\tag{8-20}\mathcal{S}=\left(\frac{2I_\text{a}}{\rho{v_\text{a}}^3}\right)^{1/2}\]

In acousto-optic diffraction, as we shall see in the following section, the coupling coefficient between an incident optical wave of a unit polarization vector \(\hat{e}_\text{i}\) and a diffracted optical wave of a unit polarization vector \(\hat{e}_\text{d}\) is determined by the following effective permittivity:

\[\tag{8-21}\Delta\tilde{\epsilon}_\text{id}=\hat{e}^*_\text{i}\cdot\Delta\tilde{\boldsymbol{\epsilon}}\cdot\hat{e}_\text{d}\]

The diffraction efficiency, however, is proportional to the square of the coupling coefficient in the low-efficiency limit. In practical acousto-optic applications, it is usually convenient to use an acousto-optic figure of merit that is defined as

\[\tag{8-22}M_2=\frac{|\Delta\tilde{\epsilon}_\text{id}|^2}{2\epsilon_0^2n_\text{i}n_\text{d}I_\text{a}}=\frac{n_\text{i}^3n_\text{d}^3p^2}{\rho{v_\text{a}}^3}\]

where \(n_\text{i}\) and \(n_\text{d}\) are the refractive indices seen by the incident and diffracted optical waves, respectively, \(p\) is an effective elasto-optic coefficient properly characterizing the interaction, and \(v_\text{a}\) is the velocity of the acoustic mode involved in the interaction.

Note that the parameters in the definition of \(M_2\) have to be chosen properly according to the mode of operation. Therefore, the figure of merit is specific to the m ode of operation. It depends on both the propagation direction and the polarization mode of the acoustic wave, as well as on the polarizations of the optical waves.

The figure of merit \(M_2\) has the unit of cubic seconds per kilometer, which is equivalent to square meters per watt. The properties of some representative acousto-optic materials are listed in Table 8-2 below. The values of \(M_2\) and \(v_\text{a}\) listed in this table are measured under specific experimental conditions. They are subject to changes under different conditions.

Another factor to be considered in the practical applications of an acousto-optic material is the acoustic attenuation due to acoustic absorption of the medium, which generally increases with the square of the acoustic frequency.

As a consequence, popular materials such as silica glass, TeO2, PbMoO4, and Ge are limited to applications at acoustic frequencies well below 1 GHz and are often used in an acoustic frequency range between 10 and 500 MHz. GaP can be used in the acoustic frequency range between 500 MHz and 1 GHz. Because of its relatively low acoustic attenuation, LiNbO3 is suitable for applications at high acoustic frequencies up to 5 GHz.

 

Isotropic medium

We consider, for simplicity, the propagation of a plane acoustic wave in an isotropic medium. From the discussions in the preceding section, there are one longitudinal and two transverse modes for an acoustic wave in an isotropic medium. We can define the \(x\) direction of the coordinate system to be the direction of propagation of the acoustic wave so that \(\mathbf{K}=K\hat{x}\). Then a longitudinal wave can be expressed as

\[\tag{8-23}\mathbf{u}(x,t)=\hat{x}\mathcal{U}\cos(K_\text{L}x-\Omega{t})\]

The two orthogonal transverse modes are degenerate.  A transverse wave polarized in the \(y\) direction can be written as

\[\tag{8-24}\mathbf{u}(x,t)=\hat{y}\mathcal{U}\cos(K_\text{T}x-\Omega{t})\]

and that polarized in the \(z\) direction as

\[\tag{8-25}\mathbf{u}(x,t)=\hat{z}\mathcal{U}\cos(K_\text{T}x-\Omega{t})\]

Because \(K_\text{L}\ne{K_\text{T}}\) in general, the longitudinal and transverse waves have different acoustic velocities, \(v_\text{a,L}=\Omega/K_\text{L}\) and \(v_\text{a,T}=\Omega/K_\text{T}\), respectively.

For the longitudinal wave given in (8-23), the only nonvanishing strain tensor element is the tensile strain

\[\tag{8-26}S_1=S_{xx}=\mathcal{S}_1\sin(K_\text{L}x-\Omega{t})\]

where \(\mathcal{S}_1=\mathcal{S}_{xx}=-K_\text{L}\mathcal{U}\) is the amplitude of the space- and time-dependent periodic strain wave.

Using (8-15) and the matrix form of \(p_{\alpha\beta}\) for isotropic media in Table 8-1, we find that

\[\tag{8-27}\Delta\boldsymbol{\epsilon}=\Delta\tilde{\boldsymbol{\epsilon}}\sin(K_\text{L}x-\Omega{t})=-\epsilon_0n^4\begin{bmatrix}p_{11}&0&0\\0&p_{12}&0\\0&0&p_{12}\end{bmatrix}\mathcal{S}_1\sin(K_\text{L}x-\Omega{t})\]

For the \(y\)-polarized transverse wave given in (8-24), the only nonvanishing strain tensor elements are the shear strains \(S_{xy}=S_{yx}\); thus

\[\tag{8-28}S_6=2S_{xy}=\mathcal{S}_6\sin(K_\text{T}x-\Omega{t})\]

where \(\mathcal{S}_6=2\mathcal{S}_{xy}=-K_\text{T}\mathcal{U}\).

Because \(p_{66}=(p_{11}-p_{12})/2\) in an isotropic medium, we have

\[\tag{8-29}\Delta\boldsymbol{\epsilon}=\Delta\hat{\boldsymbol{\epsilon}}\sin(K_\text{T}x-\Omega{t})=-\epsilon_0n^4\begin{bmatrix}0&\frac{1}{2}(p_{11}-p_{12})&0\\\frac{1}{2}(p_{11}-p_{12})&0&0\\0&0&0\end{bmatrix}\mathcal{S}_6\sin({K_\text{T}x-\Omega{t}})\]

Similarly, for the \(z\)-polarized transverse wave given in (8-25), we have

\[\tag{8-30}S_5=2S_{zx}=\mathcal{S}_5\sin(K_\text{T}x-\Omega{t})\]

and

\[\tag{8-31}\Delta\boldsymbol{\epsilon}=\Delta\hat{\boldsymbol{\epsilon}}\sin(K_\text{T}x-\Omega{t})=-\epsilon_0n^4\begin{bmatrix}0&0&\frac{1}{2}(p_{11}-p_{12})\\0&0&0\\\frac{1}{2}(p_{11}-p_{12})&0&0\end{bmatrix}\mathcal{S}_5\sin(K_\text{T}x-\Omega{t})\]

where \(\mathcal{S}_5=2\mathcal{S}_{zx}=-K_\text{T}\mathcal{U}\).

We see from the above examples that the elements of the \(\Delta\hat{\boldsymbol{\epsilon}}\) tensor have the form

\[\tag{8-32}\Delta\hat{\epsilon}_{ij}=-\epsilon_0n^4p\mathcal{S}\]

where \(p\) is the appropriate elasto-optic coefficient and \(\mathcal{S}\) is the amplitude of the appropriate strain tensor element representing the traveling acoustic wave under consideration.

In an isotropic medium, the acousto-optic figure of merit defined in (8-22) is simplified as

\[\tag{8-33}M_2=\frac{n^6p^2}{\rho{v_\text{a}^3}}\]

Note that even in an isotropic medium, \(M_2\) still depends on the mode of the acoustic wave and the polarization of the optical waves involved in the interaction.

 

Example 8-1

Fused silica glass is an isotropic material that has only two independent elasto-optic coefficients, \(p_{11}=0.121\) and \(p_{12}=0.271\). A longitudinal acoustic wave at a frequency of 500 MHz is generated to propagate in the \(x\) direction. Use the data listed in Table 8-2 to find the wavelength of the acoustic wave and the figure of merit at 632.8 nm optical wavelength for optical waves of different polarizations. If the acoustic wave has an intensity of \(10\text{ W cm}^{-2}\), what are the photoelastic index changes?

From Table 8-2, we find that \(v_\text{a,L}=5.97\text{ km s}^{-1}\) for a longitudinal acoustic wave in silica glass. At an acoustic frequency of \(f\) = 500 MHz, the wavelength of the longitudinal acoustic wave is found using (8-3) [refer to the elastic waves tutorial] to be

\[\Lambda_\text{L}=\frac{v_\text{a,L}}{f}=11.92\text{ μm}\]

The longitudinal acoustic wave has a wavevector \(\mathbf{K}=K_\text{L}\hat{x}\). Therefore, the acousto-optic permittivity change \(\Delta\boldsymbol{\epsilon}\) has the form of that given in (8-27) for isotropic silica glass. For an optical wave that is linearly polarized in the \(x\) direction, parallel to the acoustic wavevector \(\mathbf{K}\), we have \(p=p_{11}\) and the following figure of merit:

\[M_2^\parallel=\frac{n^6p_{11}^2}{\rho{v_\text{a,L}^3}}=\frac{1.457^6\times0.121^2}{2.2\times10^3\times(5.97\times10^3)^3}\text{ s}^3\text{ kg}^{-1}=3.0\times10^{-16}\text{ s}^3\text{ kg}^{-1}=3.0\times10^{-16}\text{ m}^2\text{ W}^{-1}\]

For an optical wave that is polarized in any direction in the \(yz\) plane, perpendicular to the acoustic wavevector \(\mathbf{K}\), we have \(p=p_{12}\) and the following figure of merit:

\[M_2^\perp=\frac{n^6p_{12}^2}{\rho{v_\text{a,L}^3}}=\frac{1.457^6\times0.271^2}{2.2\times10^3\times(5.97\times10^3)^3}\text{ s}^3\text{ kg}^{-1}=1.5\times10^{-15}\text{ s}^3\text{ kg}^{-1}=1.5\times10^{-15}\text{ m}^2\text{ W}^{-1}\]

With \(I_\text{a}=10\text{ W cm}^{-2}=1\times10^5\text{ W m}^{-2}\), we have the following strain amplitude:

\[\mathcal{S}=\left(\frac{2I_\text{a}}{\rho{v_\text{a,L}^3}}\right)^{1/2}=\left[\frac{2\times10^5}{2.2\times10^3\times(5.97\times10^3)^3}\right]^{1/2}=2.07\times10^{-5}\]

Therefore, \(\Delta\tilde{\epsilon}_{xx}=-\epsilon_0n^4p_{11}\mathcal{S}=-1.13\times10^{-5}\epsilon_0\) and \(\Delta\tilde{\epsilon}_{yy}=\Delta\tilde{\epsilon}_{zz}=-\epsilon_0n^4p_{12}\mathcal{S}=-2.53\times10^{-5}\epsilon_0\). Because \(\Delta\boldsymbol{\epsilon}\) is diagonal and because \(|\Delta\epsilon_{ij}/\epsilon_0|\ll{n^2}\), we have

\[\Delta{n}_x(x,t)=\frac{\Delta\tilde{\epsilon}_{xx}}{2n\epsilon_0}\sin(K_\text{L}x-\Omega{t})=-3.38\times10^{-6}\sin(K_\text{L}x-\Omega{t})\]

\[\Delta{n}_y(x,t)=\frac{\Delta\tilde{\epsilon}_{yy}}{2n\epsilon_0}\sin(K_\text{L}x-\Omega{t})=-8.68\times10^{-6}\sin(K_\text{L}x-\Omega{t})\]

\[\Delta{n}_z(x,t)=\frac{\Delta\tilde{\epsilon}_{zz}}{2n\epsilon_0}\sin(K_\text{L}x-\Omega{t})=-8.68\times10^{-6}\sin(K_\text{L}x-\Omega{t})\]

where \(K_\text{L}=2\pi/\Lambda_\text{L}\) and \(\Omega=2\pi{f}\).

 

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


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