Question

A material has relative permittivity \(\epsilon_{r} x\) when no external electric field is applied. The coefficient \(\chi^{(2)}\) is measured in the presence of an external electric field of strength \(|\vec{E}| .\) Assume that \(\chi^{(3)}\) and all higher order coefficients are zero. Find the Pockels coefficient \(\gamma\) as a function of the known quantities \(\epsilon_{r} x, \chi^{(2)},\) and \(|\vec{E}|\)

Short answer

\( \gamma = \frac{\chi^{(2)}}{2(\epsilon_r x - 1)} \).

Step-by-step solution

Understand the Electric Susceptibility Relation

Start by understanding that the electric susceptibility at zero electric field is given by the relative permittivity: \( \epsilon_r - 1 \). Therefore, for our system, \( \chi^{(1)} = \epsilon_r x - 1 \).

Formulating the Polarization Equation

The polarization \( P \) in the presence of an external electric field can be expressed using: \( P = \epsilon_0 (\chi^{(1)} + \chi^{(2)}|\vec{E}|) \vec{E} \). Given \( \chi^{(3)} \) and higher order terms are zero, the nonlinear polarization term 'only' involves \( \chi^{(2)} \).

Introducing the Pockels Coefficient

The Pockels effect describes a linear relation in polarization with the applied electric field, formally represented by the Pockels coefficient \( \gamma = \frac{\chi^{(2)}}{2(\epsilon_r - 1)} \) when expressing it as a function of the susceptibilities and the permittivity.

Solving for the Pockels Coefficient \( \gamma \)

Substitute the expression for \( \chi^{(1)} = \epsilon_r x - 1 \) into the formula derived from above, \( \gamma = \frac{\chi^{(2)}}{2(\epsilon_r x - 1)} \). Now you have an expression for \( \gamma \) in terms of the given parameters.

Key concepts

Relative Permittivity

Relative permittivity, often denoted as \( \epsilon_r \), is a measure of a material's ability to permit electric field lines to pass through it. It is also referred to as the dielectric constant. This quantity indicates how much electric potential energy is stored in a material when it is placed in an electric field.

• A higher relative permittivity means that the material can store more energy.
• It is a unitless measure, comparing the permittivity of a material to that of the vacuum.

When no external electric field is applied, if a material exhibits a relative permittivity given by \( \epsilon_r x \), it has a certain intrinsic ability to polarize. In this context, \( \epsilon_r \) influences the electric susceptibility and, important for the Pockels effect, helps describe how the material responds to applied electric fields.

Electric Susceptibility

Electric susceptibility, noted as \( \chi \), quantifies how easily a material can be polarized by an electric field. This parameter reflects the extent of the polarization within a dielectric material when an external electric field is applied.

• When there is no external field, the susceptibility is directly related to the relative permittivity: \( \chi^{(1)} = \epsilon_r x - 1 \).
• Susceptibility is indicative of the linear response of a material to an external field.

In the calculation for the Pockels coefficient, understanding electric susceptibility assists us in setting the baseline for how much more the material will react once an additional external field is applied. This foundational parameter shows how a dielectric material might stray, or polarize, from its equilibrium when no field is present.

Nonlinear Polarization

Nonlinear polarization refers to the behavior of a material in which its polarization response to an electric field is not directly proportional to the field itself.

• In our case, given that the nonlinear polarization is denoted principally by \( \chi^{(2)} \), and \( \chi^{(3)} \) and higher orders are zero, the response is primarily quadratic.
• The nonlinear polarization can particularly come into effect in strong fields which are common in photonic applications where light alters the electric field in the material.

The importance of nonlinear polarization in the context of the Pockels effect lies in its addition to the linear polarization, represented by \( \chi^{(1)} \), to produce the overall polarization \( P \). This signifies that while the linear term encodes the simple response of the material, the nonlinear term demonstrates more complex interactions between the material and the electric field.

Pockels Coefficient

The Pockels coefficient \( \gamma \) is a key parameter in describing the Pockels effect, which involves linear electro-optic phenomena where the refractive index of a material changes linearly with an applied electric field.

• Expressed as \( \gamma = \frac{\chi^{(2)}}{2(\epsilon_r x - 1)} \), it quantifies this linear relationship.
• This linear change in refractive index means that materials with a significant Pockels coefficient are highly applicable in optical modulation devices, such as waveguides and photonic circuits.

The calculation of \( \gamma \) illustrates how the initial susceptibility, altered by the presence of a field, gives a practical measure of a material’s usefulness in electro-optic applications. Understanding the Pockels coefficient aids in designing and predicting material behaviors in fields demanding high-speed optical data processing.