Pockels Effect
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This is a continuation from the previous tutorial - Electro-Optic Effects.
The majority of electro-optic devices are based on the Pockels effect. Structurally isotropic materials, including all gases, liquids, and amorphous solids such as glass, show no Pockels effect because they are centrosymmetric. Among the 32 point groups in the 7 crystal systems, 11 are centrosymmetric, and the remaining 21 are noncentrosymmetric.
It is important to note that the linear optical property of a crystal is determined only by its crystal system, as mentioned in the propagation in an anisotropic medium tutorial and summarized in Table 2 in that tutorial, but its nonlinear optical properties, including its Pockels coefficients, further depend on its point group.
An instructive example is that all cubic crystals have isotropic linear optical properties but not isotropic crystal structures. Two cubic crystals belonging to different point groups can have very different nonlinear optical properties. Among the cubic crystals, C, Si, and Ge are centrosymmetric materials of diamond structure that show no Pockels effect, whereas GaAs, InP, and other III-V semiconductors are noncentrosymmetric materials that have nonvanishing Pockels coefficients.
For the Pockels effect,
\[\tag{6-21}\Delta\eta_\alpha=\sum_kr_{\alpha k}E_{0k}\]
which can be written explicitly in matrix form:
\[\tag{6-22}\begin{bmatrix}\Delta\eta_1\\\Delta\eta_2\\\Delta\eta_3\\\Delta\eta_4\\\Delta\eta_5\\\Delta\eta_6\end{bmatrix}=\begin{bmatrix}r_{11}&r_{12}&r_{13}\\r_{21}&r_{22}&r_{23}\\r_{31}&r_{32}&r_{33}\\r_{41}&r_{42}&r_{43}\\r_{51}&r_{52}&r_{53}\\r_{61}&r_{62}&r_{63}\end{bmatrix}\begin{bmatrix}E_{0x}\\E_{0y}\\E_{0z}\end{bmatrix}\]
Even for a noncentrosymmetric material, the number of nonvanishing independent elements in its \(r_{\alpha k}\) matrix is generally reduced by its symmetry. Table 6-1 below shows the matrix form of the Pockels coefficients for the 21 noncentrosymmetric point groups.
Table 6-1. Matrix form of Pockels coefficients for noncentrosymmetric point groups
\[\text{Triclinic}\qquad\qquad\qquad1\;\;\begin{bmatrix}r_{11}&r_{12}&r_{13}\\r_{21}&r_{22}&r_{23}\\r_{31}&r_{32}&r_{33}\\r_{41}&r_{42}&r_{43}\\r_{51}&r_{52}&r_{53}\\r_{61}&r_{62}&r_{63}\end{bmatrix}\]
\[\text{Monoclinic}\qquad\begin{aligned}&2\\(2\;&\parallel\;\hat{y})\end{aligned}\;\begin{bmatrix}0&r_{21}&0\\0&r_{22}&0\\0&r_{23}&0\\r_{41}&0&r_{43}\\0&r_{52}&0\\r_{61}&0&r_{63}\end{bmatrix}\qquad\begin{aligned}&m\\(m&\perp\hat{y})\end{aligned}\;\begin{bmatrix}r_{11}&0&r_{13}\\r_{21}&0&r_{23}\\r_{31}&0&r_{33}\\0&r_{42}&0\\r_{51}&0&r_{53}\\0&r_{62}&0\end{bmatrix}\]
\[\text{Orthorhombic}\qquad\qquad222\;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\r_{41}&0&0\\0&r_{52}&0\\0&0&r_{63}\end{bmatrix}\qquad{mm2}\;\begin{bmatrix}0&0&r_{13}\\0&0&r_{23}\\0&0&r_{33}\\0&r_{42}&0\\r_{51}&0&0\\0&0&0\end{bmatrix}\]
\[\begin{align}\text{Tetragonal}\qquad\qquad\qquad4\;\begin{bmatrix}0&0&r_{13}\\0&0&r_{13}\\0&0&r_{33}\\r_{41}&r_{42}&0\\r_{42}&-r_{41}&0\\0&0&0\end{bmatrix}\qquad\bar{4}\;\begin{bmatrix}0&0&r_{13}\\0&0&-r_{13}\\0&0&0\\r_{41}&r_{42}&0\\-r_{42}&r_{41}&0\\0&0&r_{63}\end{bmatrix}\\422\;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\r_{41}&0&0\\0&-r_{41}&0\\0&0&0\end{bmatrix}\qquad{4mm}\;\begin{bmatrix}0&0&r_{13}\\0&0&r_{13}\\0&0&r_{33}\\0&r_{42}&0\\r_{42}&0&0\\0&0&0\end{bmatrix}\qquad\bar{4}2m\;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\r_{41}&0&0\\0&r_{41}&0\\0&0&r_{63}\end{bmatrix}\end{align}\]
\[\begin{align}\text{Trigonal}\qquad\qquad\qquad\qquad3\;\begin{bmatrix}r_{11}&-r_{22}&r_{13}\\-r_{11}&r_{22}&r_{13}\\0&0&r_{33}\\r_{41}&r_{42}&0\\r_{42}&-r_{41}&0\\-r_{22}&-r_{11}&0\end{bmatrix}\\32\;\begin{bmatrix}r_{11}&0&0\\-r_{11}&0&0\\0&0&0\\r_{41}&0&0\\0&-r_{41}&0\\0&-r_{11}&0\end{bmatrix}\qquad3m\;\begin{bmatrix}0&-r_{22}&r_{13}\\0&r_{22}&r_{13}\\0&0&r_{33}\\0&r_{42}&0\\r_{42}&0&0\\-r_{22}&0&0\end{bmatrix}\end{align}\]
\[\begin{align}\text{Hexagonal}\qquad\qquad\qquad\qquad6\;\begin{bmatrix}0&0&r_{13}\\0&0&r_{13}\\0&0&r_{33}\\r_{41}&r_{42}&0\\r_{42}&-r_{41}&0\\0&0&0\end{bmatrix}\qquad\bar6\;\begin{bmatrix}r_{11}&-r_{22}&0\\-r_{11}&r_{22}&0\\0&0&0\\0&0&0\\0&0&0\\-r_{22}&-r_{11}&0\end{bmatrix}\\622\;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\r_{41}&0&0\\0&-r_{41}&0\\0&0&0\end{bmatrix}\qquad6mm\;\begin{bmatrix}0&0&r_{13}\\0&0&r_{13}\\0&0&r_{33}\\0&r_{42}&0\\r_{42}&0&0\\0&0&0\end{bmatrix}\qquad\begin{aligned}\bar6&m2\\(m&\perp\hat{x})\end{aligned}\;\begin{bmatrix}0&-r_{22}&0\\0&r_{22}&0\\0&0&0\\0&0&0\\0&0&0\\-r_{22}&0&0\end{bmatrix}\end{align}\]
\[\text{Cubic}\qquad\qquad432\;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\0&0&0\\0&0&0\\0&0&0\end{bmatrix}\qquad\begin{aligned}&23\\&\text{and}\\&\bar43m\end{aligned}\;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\\r_{41}&0&0\\0&r_{41}&0\\0&0&r_{41}\end{bmatrix}\]
Some crystal point groups are of particular interest.
- Cubic \(\bar{4}3m\). Most III-V semiconductors, such as GaAs, InP, AlAs, and GaP, and many II-VI compounds, such as ZnTe, ZnSe, CdTe, and HgSe, are cubic crystals of \(\bar43m\) symmetry. They have isotropic linear optical properties with \(n_x=n_y=n_z=n_o\). The \(r_{\alpha k}\) matrix has only three nonvanishing elements with the same value: \(r_{41}=r_{52}=r_{63}\).
- Tetragonal \(\bar42m\). Crystals possessing tetragonal \(\bar42m\) symmetry include many commonly used nonlinear optical crystals, such as KH_{2}PO_{4} (KDP), KD_{2}PO_{4} (KD*P), NH_{4}H_{2}PO_{4} (ADP), ND_{4}D_{2}PO_{4} (AD*P), CsH_{2}AsO_{4} (CDA), AgGaS_{2}, and AgGaSe_{2}. These are uniaxial crystals with \(n_x=n_y=n_o\) and \(n_z=n_e\). The \(r_{\alpha k}\) matrix has only three nonvanishing elements with two independent values: \(r_{41}=r_{52}\ne{r_{63}}\).
- Trigonal \(3m\). The very useful nonlinear optical crystals LiNbO_{3}, LiTaO_{3}, and \(\beta\)-BaB_{2}O_{4} (BBO) of trigonal \(3m\) symmetry are uniaxial with \(n_x=n_y=n_o\) and \(n_z=n_e\). The \(r_{\alpha k}\) matrix has eight nonvanishing elements with four independent values: \(r_{13}=r_{23},r_{12}=r_{61}=-r_{22},r_{33},\) and \(r_{42}=r_{51}\).
- Orthorhombic \(mm2\). The nonlinear crystals KTiOPO_{4} (KTP), KTiOAsO_{4} (KTA), LiB_{3}O_{5} (LBO), KNbO_{3}, and Ba_{2}NaNb_{5}O_{15} have orthorhombic \(mm2\) symmetry. They are biaxial crystals with \(n_x\ne{n_y}\ne{n_z}\), and they have five independent nonvanishing Pockels coefficients in the \(r_{\alpha{k}}\) matrix: \(r_{13},r_{23},r_{33},r_{42},\) and \(r_{51}\).
A secondary effect due to the existence of piezoelectricity causes complexity in the determination of the Pockels coefficients of a crystal. A stress applied to a noncentrosymmetric polar crystal can induce an electric polarization in the crystal. This effect is called the direct piezoelectric effect.
In the converse piezoelectric effect, an electric field applied to the same crystal can induce a strain in the crystal.
The piezoelectric effect and the Pockels effect have similar symmetry properties: both vanish in centrosymmetric materials, and both are restricted by crystal symmetry in similar manners. Consequently, the piezoelectric effect exists in a crystal that shows the Pockels effect.
The strain generated in a crystal by an applied electric field can induce index changes through the photoelastic effect. In a free crystal, which is allowed to strain in response to the applied electric field, this secondary effect is comparable in magnitude to the primary effect that accounts for the index changes directly caused by the applied electric field.
Pockels coefficients measured at constant strain (indicated by S) with a crystal clamped reflect only the primary effect, whereas those measured at constant stress (indicated by T) with a crystal free and unclamped reflect the sum of the primary and secondary effects.
Table 6-2 below lists the properties of some representative electro-optic materials.
In practical device applications, an electro-optic crystal is not clamped, but its electro-optic coefficient is a function of the modulation frequency. At low modulation frequencies, the electro-optic response of the crystal is that of a free crystal at constant stress because the photoelastic response can follow the low-frequency modulation signal.
At high modulation frequencies, however, the photoelastic effect vanishes because the strain in the crystal cannot respond quickly enough to follow the modulation signal.
Consequently, the Pockels coefficients measured at constant stress have to be used for low-frequency modulation, but those measured at constant strain have to be used for high-frequency modulation.
Besides their dependence on the modulation frequency, the Pockels coefficients are also a function of temperature and optical wavelength. Because of these complications, only typical values of the Pockels coefficients measured at constant strain are listed in Table 6-2 except for those of KDP and ADP crystals, for which the \(r_{41}\) coefficient at constant stain is not available.
Table 6-2 Properties of representative electro-optic crystals
The technologically most important electro-optic materials are the III-V semiconductors, particularly GaAs and InP and related compounds, and the \(3m\) crystals, such as LiNbO_{3} and LiTaO_{3}. Electro-optic devices based on LiNbO_{3} are the most extensively studied and most well developed.
Those based on the III-V semiconductors are also intensively studied because they can be monolithically integrated with other optoelectronic devices, including semiconductor lasers, amplifiers, and detectors.
It can be seen from table 6-2 that the Pockels coefficients of the III-V semiconductors are relatively small compared with those of other important electro-optic materials. However, this disadvantage is generally compensated by using advanced semiconductor processing technologies. For example, small waveguide structures with optimized overlap of the applied electric field and the optical field can be made in a III-V semiconductor to maximize the electro-optic modulation efficiency. The intrinsic electro-optic effect in a III-V material can also be substantially enhanced by incorporating artificially tailored structures, such as quantum-well structures, in the material.
Index changes and rotation of principal axes
Depending on the symmetry of a specific material being used and the direction of the electric field being applied to the material, the index changes induced by the Pockels effect may or may not be accompanied by a rotation of principal axes. This fact is best illustrated through real examples.
We first consider LiNbO_{3}, which is a negative uniaxial crystal of \(3m\) symmetry. The following analysis applies equally to other \(3m\) crystals although some of them, such as LiTaO_{3}, are positive uniaxial crystals.
1. The electric field is applied along the optical axis: \(E_{0x}=E_{0y}=0,E_{0z}\ne0\). In this case, the changes induced by the Pockels effect are \(\Delta\eta_1=r_{13}E_{0z},\Delta\eta_2=r_{13}E_{0z},\) and \(\Delta\eta_3=r_{33}E_{0z}\). The index ellipsoid becomes
\[\tag{6-23}\left(\frac{1}{n_o^2}+r_{13}E_{0z}\right)x^2+\left(\frac{1}{n_o^2}+r_{13}E_{0z}\right)y^2+\left(\frac{1}{n_e^2}+r_{33}E_{0z}\right)z^2=1\]
Equivalently, by using (6-4) and (6-18) [refer to the electro-optic effects tutorial], the field-dependent dielectric permittivity tensor can be found:
\[\tag{6-24}\boldsymbol{\epsilon}(\pmb{E}_0)=\epsilon_0\begin{bmatrix}n_o^2-n_o^4r_{13}E_{0z}&0&0\\0&n_o^2-n_o^4r_{13}E_{0z}&0\\0&0&n_e^2-n_e^4r_{33}E_{0z}\end{bmatrix}\]
The principal axes are not rotated: \(\hat{X}=\hat{x},\hat{Y}=\hat{y},\) and \(\hat{Z}=\hat{z}\). The crystal remains uniaxial with the same optical axis, but the indices of refraction are changed. Since the induced changes are generally so small that \(|r_{13}E_{0z}|\ll{n_o^{-2}}\) and \(|r_{33}E_{0z}|\ll{n_e^{-2}}\), the new principal indices of refraction are
\[\tag{6-25}n_X=n_Y\approx n_o-\frac{n_o^3}{2}r_{13}E_{0z},\qquad n_Z\approx n_e-\frac{n_e^3}{2}r_{33}E_{0z}\]
2. The electric field is applied along the \(y\) axis: \(E_{0x}=E_{0z}=0,E_{0y}\ne0\). Then the induced changes are \(\Delta\eta_1=-r_{22}E_{0y},\Delta\eta_2=r_{22}E_{0y},\) and \(\Delta\eta_4=r_{42}E_{0y}\). The index ellipsoid becomes
\[\tag{6-26}\left(\frac{1}{n_o^2}-r_{22}E_{0y}\right)x^2+\left(\frac{1}{n_o^2}+r_{22}E_{0y}\right)y^2+\frac{1}{n_e^2}z^2+2r_{42}E_{0y}yz=1\]
The corresponding dielectric permittivity tensor is
\[\tag{6-27}\boldsymbol{\epsilon}(\pmb{E}_0)=\epsilon_0\begin{bmatrix}n_o^2+n_o^4r_{22}E_{0y}&0&0\\0&n_o^2-n_o^4r_{22}E_{0y}&-n_o^2n_e^2r_{42}E_{0y}\\0&-n_o^2n_e^2r_{42}E_{0y}&n_e^2\end{bmatrix}\]
Because of the existence of the \(yz\) term in (6-26), which corresponds to the existence of the off-diagonal terms in \(\boldsymbol{\epsilon}(\pmb{E}_0)\) in (6-27), the new principal axes \(\hat{Y}\) and \(\hat{Z}\) are rotated away from \(\hat{y}\) and \(\hat{z}\) while \(\hat{X}\) remains the same as \(\hat{x}\):
\[\tag{6-28}\hat{X}=\hat{x},\qquad\hat{Y}=\hat{y}\cos\theta+\hat{z}\sin\theta,\qquad\hat{Z}=-\hat{y}\sin\theta+\hat{z}\cos\theta\]
The angle of rotation \(\theta\) and the new principal indices of refraction can be found by eliminating the \(yz\) term in (6-26) or, equivalently, by diagonalizing \(\boldsymbol{\epsilon}(\pmb{E}_0)\) in (6-27) through a transformation matrix \(\mathbf{T}\) defined by (6-6) and (6-7). For LiNbO_{3}, since \(n_o\gt{n_e}\) and \(n_o^2-n_e^2\gg|n_o^2n_e^2r_{42}E_{0y}|\gt|n_o^4r_{22}E_{0y}|\) for any \(E_{0y}\) below the material breakdown field of the order of 100 MVm^{-1}, it can be shown that
\[\tag{6-29}\theta\approx-\tan^{-1}\frac{n_o^2n_e^2r_{42}E_{0y}}{n_o^2-n_e^2}\]
and
\[\tag{6-30}\begin{aligned}&n_X\approx{n_o}+\frac{n_o^3}{2}r_{22}E_{0y}\\&n_Y\approx{n_o}-\frac{n_o^3}{2}r_{22}E_{0y}+\frac{1}{2}\frac{n_o^3n_e^4}{n_o^2-n_e^2}(r_{42}E_{0y})^2\\&n_Z\approx{n_e}-\frac{1}{2}\frac{n_e^3n_o^4}{n_o^2-n_e^2}(r_{42}E_{0y})^2\end{aligned}\]
The crystal biaxial in the presence of an electric field applied in the \(y\) direction. Note that not only do the index changes depend on the applied field, but the angle of rotation of the principal axes is a function of \(E_{0y}\) as well.
3. The electrical field is applied along the \(x\) axis: \(E_{0x}\ne0,E_{0y}=E_{0z}=0\). The induced changes are \(\Delta\eta_5=r_{51}E_{0x}=r_{42}E_{0x}\) and \(\Delta\eta_6=-r_{22}E_{0x}\). Then, we have
\[\tag{6-31}\frac{1}{n_o^2}x^2+\frac{1}{n_o^2}y^2+\frac{1}{n_e^2}z^2+2r_{42}E_{0x}zx-2r_{22}E_{0x}xy=1\]
and
\[\tag{6-32}\boldsymbol{\epsilon}(\pmb{E}_0)=\epsilon_0\begin{bmatrix}n_o^2&n_o^4r_{22}E_{0x}&-n_o^2n_e^2r_{42}E_{0x}\\n_o^4r_{22}E_{0x}&n_o^2&0\\-n_o^2n_e^2r_{42}E_{0x}&0&n_e^2\end{bmatrix}\]
In this case, all three new principal axes \(\hat{X},\hat{Y},\) and \(\hat{Z}\) are rotated away from the original principal axes. The crystal also becomes biaxial. Because \(n_o\ne{n_e}\), the angles of rotation depend on the magnitude of \(E_{0x}\). Again, the new principal axes and their corresponding principal indices of refraction in the presence of \(E_{0x}\) can be found by eliminating the \(zx\) and \(xy\) terms in (6-31) or by diagonalizing \(\boldsymbol{\epsilon}(\pmb{E}_0)\) in (6-32).
Because \(r_{33}\) is the largest electro-optic coefficient of LiNbO_{3}, the largest index change is obtained in \(n_Z\) when the electric field is applied along the \(z\) axis.
Another important example is the Pockels effect in a III-V semiconductor of \(\bar43m\) symmetry, such as GaAs or InP. A similar effect is seen when an electric field is applied along any of the original principal axes because \(n_x=n_y=n_z=n_o\) and the only nonvanishing Pockels coefficients are \(r_{41}=r_{52}=r_{63}\) for such a crystal. We therefore consider only the case when the field is applied along the \(z\) axes: \(E_{0x}=E_{0y}=0,E_{0z}\ne0\). Then, we only have \(\Delta\eta_6=r_{41}E_{0z}\). The index ellipsoid becomes
\[\tag{6-33}\frac{1}{n_o^2}x^2+\frac{1}{n_o^2}y^2+\frac{1}{n_o^2}z^2+2r_{41}E_{0z}xy=1\]
and the dielectric permittivity tensor becomes
\[\tag{6-34}\boldsymbol{\epsilon}(\pmb{E}_0)=\epsilon_0\begin{bmatrix}n_o^2&-n_o^4r_{41}E_{0z}&0\\-n_o^4r_{41}E_{0z}&n_o^2&0\\0&0&n_o^2\end{bmatrix}\]
The crystal has the following new principal axes:
\[\tag{6-35}\hat{X}=\frac{1}{\sqrt2}(\hat{x}+\hat{y}),\qquad\hat{Y}=\frac{1}{\sqrt2}(-\hat{x}+\hat{y}),\qquad\hat{Z}=\hat{z}\]
with the following new principal indices:
\[\tag{6-36}n_X\approx{n_0}-\frac{n_o^3}{2}r_{41}E_{0z},\qquad{n_Y}\approx{n_o}+\frac{n_o^3}{2}r_{41}E_{0z},\qquad{n_Z}=n_o\]
In the above, we have considered the simple cases where the electric field is applied only along one of the principal axes of the crystal. In certain practical situations, however, the applied electric field may not line up with any of the principal axes. The index changes and the rotation of principal axes can be found by following the same general procedure as illustrated above although the mathematics may be somewhat more complicated.
Example 6-1
Find the index changes and the birefringence at \(\lambda=1\) μm caused by an electric field of \(E_0=1\) MVm^{-1} applied to LiNbO_{3} and GaAs, respectively, in a direction along the z principal axis of the crystal.
The values of the Pockels coefficients and the refractive indices for both LiNbO_{3} and GaAs are listed in table 6-2. For LiNbO_{3}, an electric field applied along its \(z\) axis does not rotate its principal axes but only causes changes in its refractive indices. The crystal remains uniaxial. From (6-25), we find that the change in the ordinary index is
\[\Delta{n_o}=-\frac{n_o^3}{2}r_{13}E_{0z}=-\frac{2.238^3}{2}\times8.6\times10^{-12}\times1\times10^6=-4.82\times10^{-5}\]
while the change in the extraordinary index is
\[\Delta{n_e}=-\frac{n_e^3}{2}r_{33}E_{0z}=-\frac{2.159^3}{2}\times30.8\times10^{-12}\times1\times10^6=-1.55\times10^{-4}\]
The electro-optically induced birefringence is \(\Delta{n_o}-\Delta{n_e}\approx1\times10^{-4}\), which is almost three orders of magnitude smaller than the intrinsic birefringence of \(n_o-n_e=0.08\) for LiNbO_{3}. In normal device applications, the applied electric field typically falls in the range between 0.1 and 10 MVm^{-1}. Because the index changes are linearly proportional to the applied electric field, the electro-optically induced birefringence is typically two to three orders of magnitude smaller than the intrinsic birefringence of LiNbO_{3}.
For GaAs, which is originally non-birefringent, an electric field applied along its \(z\) axis causes a rotation of its \(x\) and \(y\) principal axes and a birefringence between them. From (6-36), we find that the electro-optically induced index changes are
\[\Delta{n_Y}=-\Delta{n_X}=\frac{n_o^3}{2}r_{41}E_{0z}=\frac{3.5^3}{2}\times1.2\times10^{-12}\times1\times10^6=2.57\times10^{-5}\]
The electro-optically induced birefringence is \(n_Y-n_X=\Delta{n_Y}-\Delta{n_X}=5.15\times10^{-5}\). Although this birefringence is smaller than that in the case of LiNbO_{3}, it is significant because GaAs is originally non-birefringent.
The next part continues with the Electro-Optic Modulators tutorial.