# 3.2: Electro-Optics

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Electro-Optic Coefficients

Typically, the magnitude of material polarization in a dielectric is proportional to the strength of an applied electric field.

\[\overrightarrow{P} = \overrightarrow{D} - \epsilon_0\overrightarrow{E} = \epsilon_0\chi_e\overrightarrow{E} \nonumber \]

In this equation \(\chi_e\) is the **electric susceptibility**, and it is unitless. It is defined in Sec. 2.2.3 and related to permittivity by Equation 2.2.8. However in other materials, the material polarization depends nonlinearly on the applied electric field. Materials for which the material polarization depends linearly on the external electric field are called linear materials while others are called nonlinear or electro-optic materials. The electro-optic effect occurs when an applied external electric field induces a material polarization in a material where the amount of polarization depends nonlinearly on the external field. The name involves the word optic because the external field is often due to a visible laser beam. However, the external field can be from any type of source at any frequency, and a material polarization will occur even with a constant applied electric field. A large enough external electric field will cause a material to melt or to crystallize in a different phase, but this effect is not the electro-optic effect. Instead, the electro-optic effect only involves a change in the material polarization, not the crystal structure, and the change involved is not permanent.

We can write the magnitude of the material polarization as a function of powers of the applied external field.

\[ |\overrightarrow{P}| = \epsilon_0 \chi_e |\overrightarrow{E}| + \epsilon_0 \chi^{(2)} |\overrightarrow{E}|^2 + \epsilon_0 \chi^{(3)} |\overrightarrow{E}|^3 + ... \label{3.2.2} \]

The quantity \(\chi^{(2)}\) is called the chi-two coefficient, and it has units \(\frac{m}{V}\). The quantity \(\chi^{(3)}\) is called the chi-three coefficient, and it has units \(\frac{m^2}{V^2}\) [27] [42, ch. 1].

If an infinite number of terms are included on the right side of Equation \ref{3.2.2}, any arbitrary material can be described. In most materials, only the first term of Equation \ref{3.2.2} is needed while \(\chi^{(2)}\), \(\chi^{(3)}\), and all higher order coefficients are negligible, and these materials are not electro-optic. Materials with \(\chi^{(2)}\), \(\chi^{(3)}\) or other coefficients nonzero are called electro-optic. It is rare to need more coefficients than \(\chi_e\), \(\chi^{(2)}\), and \(\chi^{(3)}\) to describe a material.

The effect due to the \(\epsilon_0 \chi^{(2)} |\overrightarrow{E}|^2 \) term is called the **Pockels effect** or **linear electro-optic effect**. It was first observed by Friedrich Pockels in 1893 [3, p. 382] [10]. In this case the material polarization depends on the square of the external field. The effect due to the \(\epsilon_0 \chi^{(3)} |\overrightarrow{E}|^3\) term is called the Kerr effect or the quadratic electro-optic effect. In this case, the material polarization depends on the cube of the external electric field. John Kerr first described this effect in 1875 [3, p. 382] [10].

While some authors use the coefficients \(\chi_e\), \(\chi^{(2)}\), and \(\chi^{(3)}\), this effect is most often studied by optics scientists who instead prefer index of refraction n, a unitless measure introduced in Sec. 2.2.3. In electro-optic materials, the index of refraction is a nonlinear function of the strength of the external electric field. Instead of expanding the material polarization in a power series as a function of the external field strength as in Equation \ref{3.2.2}, the index of refraction is expanded. Pockels and Kerr coefficients are defined as terms of this expansion.

As described by Equation 2.2.2, material polarization is the difference in \(\frac{C}{m^2}\) between an external electric field in a material and the field in the absence of the material.

\[ |\overrightarrow{P}| = |\overrightarrow{D}| - \epsilon_0 |\overrightarrow{E}| \nonumber \]

With some algebra, we can identify the displacement flux density component and the overall index of refraction. Add two terms which sum to zero to Equation \ref{3.2.2}.

\[ |\overrightarrow{P}| = \epsilon_0 \chi_e |\overrightarrow{E}| + \epsilon_0 \chi^{(2)} |\overrightarrow{E}|^2 + \epsilon_0 \chi^{(3)} |\overrightarrow{E}|^3 - \epsilon_0 |\overrightarrow{E}| \nonumber \]

The first two terms can be combined, and \(\epsilon_0 |\overrightarrow{E}| \) can be distributed out.

\[ |\overrightarrow{P}| = [ (\chi_e + 1) + \chi^{(2)} |\overrightarrow{E}| + \chi^{(3)} |\overrightarrow{E}|^2 + ... ] \epsilon_0 |\overrightarrow{E}| - \epsilon_0 |\overrightarrow{E}| \nonumber \]

The first term is the displacement flux density.

\[ \overrightarrow{D} = \epsilon_{r\;eo} \overrightarrow{E} = \left[ (\chi_e + 1) + \chi^{(2)} |\overrightarrow{E}| + \chi^{(3)} |\overrightarrow{E}|^2 + ... \right] \epsilon_0 |\overrightarrow{E}| \label{3.2.6} \]

The quantity in brackets in Equation \ref{3.2.6} is the relative permittivity, \(\epsilon_{r\;eo}\). Since we are considering electro-optic materials, it depends nonlinearly on the applied external field. Assuming the material is a perfect dielectric with \(\mu = \mu_0\), the index of refraction is the square root of this quantity. It represents the ratio of the speed of light in free space to the speed of light in this material, and it also depends nonlinearly on the applied external field.

\[n_{eo} = \sqrt{\epsilon_{r\;eo}} \label{3.2.7} \]

The index of refraction must be larger than one because electromagnetic waves in materials cannot go faster than the speed of light, so the quantity \(\frac{1}{\epsilon_{r\;eo}}\) must be less than one.

Some authors expand the term \(\frac{1}{\epsilon_{r\;eo}}\) in a Taylor expansion instead of the material polarization, and electro-optic coefficients are defined with respect to this expansion [42].

\[\frac{1}{\epsilon_{r\;eo}} = \frac{1}{\epsilon_{r\;x}} + \gamma |\overrightarrow{E}| +s|\overrightarrow{E}|^2 + ... \label{3.2.8} \]

The coefficient \(\gamma\) is called the Pockels coefficient, and it has units \(\frac{m}{V}\). The coefficient s is called the Kerr coefficient, and it has units \(\frac{m^2}{V^2}\). In the absence of nonlinear electro-optic contributions, we can denote the relative permittivity as \(\epsilon_{r\;x}\) and the index of refraction as \(n_x\) where

\[\epsilon_{r\;x} = {n_x}^2 = \chi_e + 1. \nonumber \]

The expansion of Equation \ref{3.2.8} is guaranteed to converge because \(\frac{1}{\epsilon_{r\;eo}} < 1\). Example values of the Pockels electro-optic coefficient are listed in Table 3.1.1.

With some algebra, the overall index of refraction neo can be written in terms of the Pockels and Kerr coefficients. Equations \ref{3.2.7} and \ref{3.2.8} can be combined.

\[n_{eo} = \left( \frac{1}{\epsilon_{r\;x}} + \gamma|\overrightarrow{E}| + s|\overrightarrow{E}|^2 + ... \right)^{-1/2} \nonumber \]

\[n_{eo} = \left[ \frac{1}{\epsilon_{r\;x}} \left( 1 + \gamma \epsilon_{r\;x} |\overrightarrow{E}| + s \epsilon_{r\;x} |\overrightarrow{E}|^2 + ... \right)\right]^{-1/2} \nonumber \]

\[n_{eo} = n_x \left[ 1 + \gamma {n_x}^2 |\overrightarrow{E}| + s {n_x}^2 |\overrightarrow{E}|^2 + ... \right]^{-1/2} \label{3.2.12} \]

The quantity of Equation \ref{3.2.12} in brackets can be approximated using the binomial expansion and keeping only the first terms.

\[\left(1+\gamma \mathrm{n}_{x}^{2}|\overrightarrow{E}|+s \mathrm{n}_{x}^{2}|\overrightarrow{E}|^{2}+ ... \right)^{-1 / 2} \approx\left(1-\frac{1}{2} \gamma \mathrm{n}_{x}^{2}|\overrightarrow{E}|-\frac{1}{2} s \mathrm{n}_{x}^{2}|\overrightarrow{E}|^{2}\right) \nonumber \]

Finally, the overall index of refraction can be written as a polynomial expansion of the strength of the external electric field [10, p. 698].

\[\mathrm{n}_{e o} \approx \mathrm{n}_{x}\left(1-\frac{1}{2} \gamma \mathrm{n}_{x}^{2}|\overrightarrow{E}|-\frac{1}{2} s \mathrm{n}_{x}^{2}|\overrightarrow{E}|^{2}\right) \nonumber \]

The Pockels electro-optic effect is called the linear electro-optic effect while the Kerr effect is called the quadratic effect due to the form of the equation above.

## Electro-Optic Effect in Crystalline Materials

As with the piezoelectric effect, we can determine which crystalline insulating materials will exhibit the Pockels effect by looking at the symmetries of the material. To determine if a crystal can show the Pockels effect, determine the crystal structure, identify the symmetries, and determine its crystal point group. The Pockels effect occurs in noncentrosymmetric materials, materials with a crystal structure with no inversion symmetry. Of the 32 crystal point groups, 21 of these groups may exhibit the Pockels electro-optic effect. For materials in these crystal point groups, \(\chi^{(2)}\) and the Pockels coefficient (\(\gamma\)) are nonzero. These 21 groups are also the piezoelectric crystal point groups [10, ch. 18], and they are listed in Table 2.3.1. In some crystalline materials which belong to these crystal point groups, the Pockels effect is nonzero but too small to be measurable.

From Table 2.3.1 we can see that all materials that are piezoelectric are also Pockels electro-optic and vice versa. Also, all materials that are pyroelectric are piezoelectric but not the other way around. Thus, if a device is used as an electro-optic device, and the device is accidentally mechanically stressed or vibrated, the material polarization will be induced by piezoelectricity. In many devices, these effects simultaneously occur, and it can be difficult to identify the primary cause of a material polarization when multiple effects simultaneously occur.

Tables of Pockels electro-optic coefficients for crystals can be found in reference [27] and [42].

The Kerr electro-optic effect can occur in crystals whether or not they belong to a crystal point group which has a center of symmetry, so some materials exhibit the Kerr effect but not the Pockels effect. In many materials, the Kerr effect is quite small.

## Electro-Optic Effect in Amorphous and Polycrystalline Materials

Table 2.3.1 only applies to crystalline materials because only crystalline materials have a specific crystal structure and can be classified into to a crystal point group. However, crystalline, polycrystalline, and amorphous materials can all be electro-optic. In amorphous and polycrystalline materials, the electro-optic effect is necessarily nonlinear. When an external electric field, for example from a laser, is applied, a material polarization develops. The charge separation in that region induces a material polarization in nearby atoms. Just as materials can be ferroelectric piezoelectric and ferroelectric pyroelectric, amorphous and polycrystalline materials can be ferroelectric electro-optic.

## Applications of Electro-Optics

Some controllable optical devices are made from electro-optic materials. Examples of such devices include controllable lenses, prisms, phase modulators, switches, and couplers [10]. Operation of these devices typically involves two laser beams. One of these beams controls the material polarization of the device. The intensity, phase, or electromagnetic polarization of the second optical beam is altered as it travels through the device [10, p. 698-700]. Combinations of these electro-optic devices are used to make controllable optical logic gates and interconnects for optical computing applications [10, ch. 21] [31, ch. 20].

Most memory devices are not made from electro-optic materials, but some creative memory device designs involve electro-optic materials. For example, electro-optic materials are used for some rewritable memory [10, p. 712] [27, p. 534] and for hologram storage [10, ch. 21] [27, ch. 20].

Also, electro-optic materials are used in liquid crystal displays [10, ch. 18]. Liquid crystals are electro-optic materials because an external voltage alters their material polarization [10, ch. 18].

Electro-optic materials are also used to convert an optical beam at one frequency to an optical beam at a different frequency. Second harmonic generation involves converting an optical beam with photons of energy \(E\) to a beam with photons at energy \(\frac{1}{2}E\) [10, ch. 19] [27, ch. 18] [31, ch. 16]. Electro-optic materials are used in the second harmonic generation process as well as in the related processes of third harmonic generation, three wave mixing, four wave mixing, optical parametric oscillation, and stimulated Raman scattering [10, ch. 19].