Electro-Optic Effects

This is a continuation from the previous tutorial - Surface Input and Output Couplers.

The optical property of a dielectric material can be changed through an electro-optic effect in the presence of a static or low-frequency electric field \(\pmb{E}_0\). The result is a field-dependent susceptibility and thus a field-dependent electric permittivity.

\[\tag{6-1}\mathbf{P}(\omega,\pmb{E}_0)=\epsilon_0\boldsymbol{\chi}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)=\epsilon_0\boldsymbol{\chi}(\omega)\cdot\mathbf{E}(\omega)+\epsilon_0\Delta\boldsymbol{\chi}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)\]

and

\[\tag{6-2}\mathbf{D}(\omega,\pmb{E}_0)=\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)=\boldsymbol{\epsilon}(\omega)\cdot\mathbf{E}(\omega)+\Delta\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)\]

where field-independent \(\boldsymbol{\chi}(\omega)=\boldsymbol{\chi}(\omega,\pmb{E}_0=0)\) and \(\boldsymbol{\epsilon}(\omega)=\boldsymbol{\epsilon}(\omega,\pmb{E}_0=0)\) represent the intrinsic linear response of the material at the optical frequency \(\omega\), while \(\Delta\boldsymbol{\chi}\) and \(\Delta\boldsymbol{\epsilon}\) represent changes induced by the low-frequency field \(\pmb{E}_0\).

We can write \(\mathbf{D}(\omega,\pmb{E}_0)=\mathbf{D}(\omega)+\Delta\mathbf{P}(\omega,\pmb{E}_0)\), where \(\Delta\mathbf{P}(\omega,\pmb{E}_0)=\Delta\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\cdot\mathbf{E}(\omega)\). The total permittivity of the material in the presence of the applied field is then

\[\tag{6-3}\boldsymbol{\epsilon}(\omega,\pmb{E}_0)=\boldsymbol{\epsilon}(\omega)+\Delta\boldsymbol{\epsilon}(\omega,\pmb{E}_0)=\boldsymbol{\epsilon}(\omega)+\epsilon_0\Delta\boldsymbol{\chi}(\omega,\pmb{E}_0)\]

The dielectric permittivity tensor \(\boldsymbol{\epsilon}(\omega)\) in the absence of an applied electric field is diagonal in the coordinate system defined by the intrinsic principal dielectric axes, \(\hat{x},\hat{y},\) and \(\hat{z}\), of the dielectric material. The electro-optically induced changes usually generate off-diagonal elements in addition to changing the diagonal elements:

\[\tag{6-4}\boldsymbol{\epsilon}(\omega)=\begin{bmatrix}\epsilon_x & 0 & 0 \\0 & \epsilon_y & 0 \\0 & 0 & \epsilon_z\end{bmatrix},\qquad\text{while}\qquad\boldsymbol{\epsilon}(\omega,\pmb{E}_0)=\begin{bmatrix}\epsilon_x+\Delta\epsilon_{xx} & \Delta\epsilon_{xy} & \Delta\epsilon_{xz}\\\Delta\epsilon_{yx} & \epsilon_y+\Delta\epsilon_{yy} & \Delta\epsilon_{yz}\\\Delta\epsilon_{zx} & \Delta\epsilon_{zy} & \epsilon_z+\Delta\epsilon_{zz}\\\end{bmatrix}\]

in the coordinate system of \(\hat{x},\hat{y},\) and \(\hat{z}\) axes.

As discussed in the propagation in an anisotropic medium tutorial, \(\boldsymbol{\epsilon}\) for a dielectric material is a symmetric tensor. This remains true for an electro-optic material subject to an applied electric field. Therefore

\[\tag{6-5}\epsilon_{ij}=\epsilon_{ji}\qquad\text{and}\qquad\Delta\epsilon_{ij}=\Delta\epsilon_{ji}\]

for field-dependent permittivity tensors.

The electro-optically induced nondiagonal permittivity tensor, \(\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\) given in (6-4), can be diagonalized. Its orthonormalized eigenvectors, \(\hat{X},\hat{Y},\) and \(\hat{Z}\), are the new principal dielectric axes of the material in the presence of an applied electric field \(\pmb{E}_0\). In general, they depend on the direction of \(\pmb{E}_0\). If the unit vectors \(\hat{X},\hat{Y},\) and \(\hat{Z}\) are expressed in terms of \(\hat{x},\hat{y},\) and \(\hat{z}\) as

\[\tag{6-6}\hat{X}=a_1\hat{x}+b_1\hat{y}+c_1\hat{z},\qquad\hat{Y}=a_2\hat{x}+b_2\hat{y}+c_2\hat{z},\qquad\hat{Z}=a_3\hat{x}+b_3\hat{y}+c_3\hat{z},\]

then transformation between the old coordinate system defined by \(\hat{x},\hat{y},\) and \(\hat{z}\) and the new coordinate system defined by \(\hat{X},\hat{Y},\) and \(\hat{Z}\) can be carried out using the following transformation matrix:

\[\tag{6-7}\mathbf{T}=\begin{bmatrix}a_1 & b_1 & c_1 \\a_2 & b_2 & c_2 \\a_3 & b_3 & c_3 \\\end{bmatrix}\]

Because both sets of vectors, {\(\hat{x},\hat{y},\hat{z}\)} and {\(\hat{X},\hat{Y},\hat{Z}\)}, that define the transformation matrix \(\mathbf{T}\) are orthonormal unit vectors, the transformation characterized by the matrix \(\mathbf{T}\) is an orthogonal transformation with the convenient characteristic that \(\mathbf{T}^{-1}=\tilde{\mathbf{T}}\), where \(\tilde{\mathbf{T}}\) is the transpose of \(\mathbf{T}\).

The relation in (6-6) between old and new principal axes can be written

\[\tag{6-8}\begin{bmatrix}\hat{X}\\\hat{Y}\\\hat{Z}\end{bmatrix}=\mathbf{T}\begin{bmatrix}\hat{x}\\\hat{y}\\\hat{z}\end{bmatrix}\qquad\text{or}\qquad\begin{bmatrix}\hat{x}\\\hat{y}\\\hat{z}\end{bmatrix}=\tilde{\mathbf{T}}\begin{bmatrix}\hat{X}\\\hat{Y}\\\hat{Z}\end{bmatrix}\]

The transformation of the coordinates of any vector \(\mathbf{r}=x\hat{x}+y\hat{y}+z\hat{z}=X\hat{X}+Y\hat{Y}+Z\hat{Z}\) in space is given by

\[\tag{6-9}\begin{bmatrix}X\\Y\\Z\end{bmatrix}=\mathbf{T}\begin{bmatrix}x\\y\\z\end{bmatrix}\]

or

\[\tag{6-10}\begin{bmatrix}x\\y\\z\end{bmatrix}=\mathbf{T}^{-1}\begin{bmatrix}X\\Y\\Z\end{bmatrix}=\tilde{\mathbf{T}}\begin{bmatrix}X\\Y\\Z\end{bmatrix}=\begin{bmatrix}a_1X+a_2Y+a_3Z\\b_1X+b_2Y+b_3Z\\c_1X+c_2Y+c_3Z\end{bmatrix}\]

Accordingly, the field components in the two coordinate systems are related through

\[\tag{6-11}\begin{bmatrix}E_X\\E_Y\\E_Z\end{bmatrix}=\mathbf{T}\begin{bmatrix}E_x\\E_y\\E_z\end{bmatrix},\qquad\begin{bmatrix}D_X\\D_Y\\D_Z\end{bmatrix}=\mathbf{T}\begin{bmatrix}D_x\\D_y\\D_z\end{bmatrix},\]

and so on. Diagonalization of \(\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\) to obtain its eigenvalues can be carried out using \(\mathbf{T}\) as

\[\tag{6-12}\mathbf{T}\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\mathbf{T}^{-1}=\mathbf{T}\boldsymbol{\epsilon}(\omega,\pmb{E}_0)\tilde{\mathbf{T}}=\begin{bmatrix}\epsilon_X & 0 & 0\\0 & \epsilon_Y & 0\\0 & 0 & \epsilon_Z\end{bmatrix}\]

The propagation characteristics of an optical wave in the presence of an electro-optic effect are then determined by \(\epsilon_X,\epsilon_Y,\) and \(\epsilon_Z\) with the following new principal indices of refraction

\[\tag{6-13}n_X=\sqrt{\frac{\epsilon_X}{\epsilon_0}},\qquad n_Y=\sqrt{\frac{\epsilon_Y}{\epsilon_0}},\qquad n_Z=\sqrt{\frac{\epsilon_Z}{\epsilon_0}}\]

The discussions above describe a formal and systematic approach to treating an electro-optic effect in terms of changes in the permittivity tensor. However, electro-optic effects are traditionally defined in terms of the changes in the elements of the relative impermeability tensors as \(\boldsymbol{\eta}(\pmb{E}_0)=\boldsymbol{\eta}+\Delta\boldsymbol{\eta}(\pmb{E}_0)\), which is expanded in the following form:

\[\tag{6-14}\eta_{ij}(\pmb{E}_0)=\eta_{ij}+\Delta\eta_{ij}(\pmb{E}_0)=\eta_{ij}+\sum_kr_{ijk}E_{0k}+\sum_{k,l}s_{ijkl}E_{0k}E_{0l}+\cdots,\]

where the first term \(\eta_{ij}=\eta_{ij}(0)\) is the field-independent component, the elements of the \(r_{ijk}\) tensor are the linear electro-optic coefficients known as the Pockels coefficients, and those of the \(s_{ijkl}\) tensor are the quadratic electro-optic coefficients known as the Kerr coefficients. 

The first-order electro-optic effect characterized by the linear dependence of \(\eta_{ij}(\pmb{E}_0)\) on \(\pmb{E}_0\) through the coefficients \(r_{ijk}\) is called the linear electro-optic effect, also known as the Pockels effect.

The second-order electro-optic effect characterized by the quadratic field dependence through the coefficients \(s_{ijkl}\) is called the quadratic electro-optic effect, also known as the Kerr effect.

Both linear and quadratic electro-optic effects are nonlinear optical effects, as discussed above.

The Pockels effect does not exist in a centrosymmetric material, which is a material that possesses inversion symmetry. The structure and properties of such a material remain unchanged under the transformation of space inversion, which changes the signs of all rectangular spatial coordinates from \((x,y,z)\) to \((-x,-y,-z)\), and those of all polar vectors. As discussed in the optical fields and Maxwell's equations tutorial, an electric field vector is a polar vector that changes sign under the transformation of space inversion.

By simply considering the effect of space inversion, it is clear that the electro-optically induced changes in the optical property of a centrosymmetric material are not affected by the sign change in the applied field from \(\pmb{E}_0\) to \(-\pmb{E}_0\), meaning that \(\eta_{ij}(\pmb{E}_0)=\eta_{ij}(-\pmb{E}_0)\). As can be seen from (6-14) above, this condition requires that the Pockels coefficients \(r_{ijk}\) vanish. It can also be seen that the condition does not require vanishing of the Kerr coefficients \(s_{ijkl}\). Consequently, the Pockels effect exists only in noncentrosymmetric materials, while the Kerr effect exists in all materials, including centrosymmetric ones.

In (6-14), indices \(i\) and \(j\) are associated with optical fields, while indices \(k\) and \(l\) are associated with the low-frequency applied field. Because \(\eta_{ij}=\eta_{ji}\) and \(\Delta\eta_{ij}=\Delta\eta_{ji}\), indices \(i\) and \(j\) can be contracted using the index contraction rule of (115) [refer to the propagation in an anisotropic medium tutorial], thus reducing (6-14) to

\[\tag{6-15}\eta_\alpha(\pmb{E}_0)=\eta_\alpha+\Delta\eta_\alpha(\pmb{E}_0)=\eta_\alpha+\sum_kr_{\alpha k}E_{0k}+\sum_{k,l}s_{\alpha kl}E_{0k}E_{0l}+\cdots,\]

where \(\alpha=1,2,...,6\) with the meaning defined in (115) [refer to the propagation in an anisotropic medium tutorial].

from the relation that \(\boldsymbol{\eta}=(\boldsymbol{\epsilon}/\epsilon_0)^{-1}\) defined in (111) [refer to the propagation in an anisotropic medium tutorial], it can be seen that \(\boldsymbol{\eta}\) in the absence of \(\pmb{E}_0\) is a diagonal tensor in the coordinate system defined by \(\hat{x},\hat{y},\) and \(\hat{z}\) with the following eigenvalues:

\[\tag{6-16}\eta_x=\frac{\epsilon_0}{\epsilon_x}=\frac{1}{n_x^2},\qquad\eta_y=\frac{\epsilon_0}{\epsilon_y}=\frac{1}{n_y^2},\qquad\eta_z=\frac{\epsilon_0}{\epsilon_z}=\frac{1}{n_z^2},\]

where \(n_x,n_y,\) and \(n_z\) are the principal indices of refraction of the material in the absence of an applied electric field.

In the presence of an applied field, \(\boldsymbol{\eta}(\pmb{E}_0)\) is generally not diagonal in this coordinate system. Using the relation \(\boldsymbol{\eta}\cdot\boldsymbol{\epsilon}/\epsilon_0=1\), the relation between \(\Delta\boldsymbol{\epsilon}\) and \(\Delta\boldsymbol{\eta}\) can be found:

\[\tag{6-17}\Delta\boldsymbol{\epsilon}=-\frac{1}{\epsilon_0}\boldsymbol{\epsilon}\cdot\Delta\boldsymbol{\eta}\cdot\boldsymbol{\epsilon}\qquad\text{and}\qquad\Delta\boldsymbol{\eta}=-\frac{1}{\epsilon_0}\boldsymbol{\eta}\cdot\Delta\boldsymbol{\epsilon}\cdot\boldsymbol{\eta}\]

When \(\boldsymbol{\eta}\) and \(\boldsymbol{\epsilon}\) in the absence of \(\pmb{E}_0\) are diagonalized, the relations in (6-17) can be written explicitly as

\[\tag{6-18}\Delta\epsilon_{ij}=-\epsilon_0\frac{\Delta\eta_{ij}}{\eta_i\eta_j}=-\epsilon_0n_i^2n_j^2\Delta\eta_{ij}\qquad\text{and}\qquad\Delta\eta_{ij}=-\epsilon_0\frac{\Delta\epsilon_{ij}}{\epsilon_i\epsilon_j}=-\frac{\Delta\epsilon_{ij}}{\epsilon_0n_i^2n_j^2}\]

In the absence of an electric field, the index ellipsoid of a material is that given by (117) [refer to the propagation in an anisotropic medium tutorial] with its principal axes aligned with \(\hat{x},\hat{y},\) and \(\hat{z}\). Changes in the optical property of the material induced by an electro-optic effect deform the index ellipsoid into a new one described by

\[\tag{6-19}(\eta_1+\Delta\eta_1)x^2+(\eta_2+\Delta\eta_2)y^2+(\eta_3+\Delta\eta_3)z^2+2\Delta\eta_4yz+2\Delta\eta_5zx+2\Delta\eta_6xy=1\]

whose principal axes no longer line up with \(\hat{x},\hat{y},\) and \(\hat{z}\) unless \(\Delta\eta_4=\Delta\eta_5=\Delta\eta_6=0\). To find the principal axes of this new ellipsoid and their corresponding principal indices of refraction, we can perform a coordinate rotation in space to eliminate the cross-product term containing \(yz,zx,\) and \(xy\). From the discussions above, it can be seen that this procedure is the same as the coordinate rotation used to diagonalize \(\boldsymbol{\epsilon}\). Thus, we can use (6-9) to transform (6-19) into

\[\tag{6-20}\frac{X^2}{n_X^2}+\frac{Y^2}{n_Y^2}+\frac{Z^2}{n_Z^2}=1\]

where \(n_X,n_Y,\) and \(n_Z\) are the same as those given in (6-13). The principal axes of this ellipsoid are simply the same \(\hat{X},\hat{Y}\), and \(\hat{Z}\) as those found from the eigenvectors of \(\boldsymbol{\epsilon}\) and given in (6-6). Figure 6-1 illustrates the concept described here. 

 The next part continues with the Pockels Effect tutorial.