 ## Nonlinear Optics

For low electric field strengths, the optical properties of a material are well described by its linear susceptibility $\chi_{ij}$.

$$P_i= \chi_{ij}E_j,$$

which describes the linear response of the polarization $P_i$ to an applied electric field $E_j$. However, higher field strength ($10^5 - 10^{10}$ V/m) causes a bigger displacement of the charge density, and the relation between the polarization and electric field (described by the susceptibility) remains no longer only linear. For a mathematical treatment of nonlinearity, the polarization is expanded as a Taylor series.

$$P_i= \chi_{ij} E_j + \chi_{ijk} E_j E_k + \chi_{ijkl} E_j E_k E_l + \dots$$

The rank of the susceptibility tensors is given by the number of indices.

### Electric susceptibility as a function of Gibbs free energy

Under typical experimental conditions, a system in equilibrium will go to minimum of the Gibbs free energy $G(T,E_i,H_i,\sigma_{ij})$ where the temperature $T$, the electric field $E_i$, the magnetic field intensity $H_i$, and the stress tensor $\sigma_{ij}$ are held constant during the experiment. The polarization can be expressed as a derivative of the Gibbs free energy,

$$P_i = -\frac{\partial G}{\partial E_i}.$$

This can be further differentiated to obtain expressions for the linear and nonlinear susceptibilities

\begin{equation} \chi_{ij} = \frac{\partial P_i}{\partial E_j} = -\frac{\partial^2 G}{\partial E_i \partial E_j} \\ \chi_{ijk} = \frac{\partial^2 P_i}{\partial E_j \partial E_k} = -\frac{\partial^3 G}{\partial E_i \partial E_j\partial E_k} \\ \chi_{ijkl} = \frac{\partial^3 P_i}{\partial E_j \partial E_k \partial E_l} = -\frac{\partial^4 G}{\partial E_i \partial E_j \partial E_k \partial E_l}. \end{equation}

Besides the electric field, other static (or only slowly varying in comparison to the light oscillations) fields can affect the polarization as well. If the polarization is caused by:

• an electric field, it is called electro-optic effect,
• a magnetic field, it is called magneto-optic effect,
• a stress field, it is called elasto-optic (photoelastic) effect.

To quantify those effects mathematically, the polarization is expanded in the intensity $H_i$ and strain fields $\varepsilon_{ij}$:

\begin{equation} P_i = \underbrace{ P_i^0}_{\text{Spontaneous } \\ \text{polarization}} + \underbrace{\chi_{ij} E_j}_{\text{Dielectric} \\ \text{polarization}} + \underbrace{\chi_{ijk} E_j E_k}_{\text{Pockels effect} \\ \text{and SHG}} + \underbrace{\chi_{ijkl} E_j E_k E_l}_{\text{Kerr} \\ \text{effect}} + \underbrace{\chi_{ijk} E_j H_k}_{\text{Faraday} \\ \text{effect}} + \underbrace{\chi_{ijkl} E_j H_k H_l}_{\text{Cotton-Mouton} \\ \text{effect}} + \underbrace{\chi_{ijkl} E_j \varepsilon_{kl}}_{\text{Linear elasto-optic and} \\ \text{Linear acousto-optic effect}} + \dots \end{equation}

#### Spontaneous polarization

This effect is also referred to as Ferroelectricity. Some crystals exhibit a spontaneous electric polarization without an externally applied field. Above a critical temperature, this effect vanishes, and the materials only show normal polarizability. This effect is analogous to ferromagnetism.

#### Dielectric polarization

All materials show dielectric polarizability, which is described by the second-rank tensor $\chi_{ij}$. This susceptibility is closely linked to the dielectric constant by $\epsilon_{ij} = 1 + \chi_{ij}$. For optical frequencies, the index of refraction $n$ and the index of extinction $K$ are linked to the susceptibility: \begin{equation} \sqrt{1+\chi} = \sqrt{\epsilon} = n + i K. \end{equation}

#### Pockels effect

The Pockels effect is also called the linear electro-optic effect because it describes the linear change of the polarization when an external electric field is applied to the crystal. By the external electrical field, an induced birefringence of the crystal can be controlled.

#### SHG = Second Harmonic Generation

The SHG effect is only observable for very strong light intensities (laser). An outgoing wave with double the frequency of the impinging light is generated in the crystal by nonlinear effects. The intensity of the frequency-doubled wave has a quadratic dependency on the incident beam intensity. More information about SHG can be found in this Student Project.

#### Kerr effect

Similar to the Pockels effect, also the Kerr effect induces a birefringence by changing the refractive indices. This finally leads to polarization of the transmitted light. The change shows a quadratic dependency from the applied external electric field, so it is called the quadratic electro-optic effect.

When a magnetic field is applied along the axis of the light passing through a certain crystal the plane of polarization of the light beam is rotated. The rotation angle depends on the length of the medium and the strength of the magnetic field.

#### Cotton-Mouton effect

The Cotton-Mouton effect is also called the quadratic electro-magnetic effect because the change in the polarization state of the light passing through the crystal depends on the square of the applied magnetic field.

#### Linear photoelastic effect

When a static stress field is applied to an elasto-optic crystal, the crystal structure is distorted and this can induce a birefringence. This was used to observe stress fields in mechanical parts before the time of numerical simulations.

#### Linear acousto-optic effect

The stress field is present in the form of sound waves passing through the crystal. Photons of the light beam can scatter with the acoustic phonons of the sound wave. Thereby the refractive index of the crystal is changed.

### Nonlinear susceptibility data for crystals

In Chapter 1.1 in Boyd, Nonlinear optics (see References at botton) an order-of-magnitude estimation was made for the nonlinear susceptibilities of crystals. This gives the following results: \begin{equation} \chi^{(2)} = 1.9 \cdot 10^{-12} \frac{\text{m}}{\text{V}} \\ \chi^{(3)} = 3.8 \cdot 10^{-24} \frac{\text{m}^2}{\text{V}^2} \end{equation}

Data for real crystals is shown in Table 1.

Table 1: Nonlinear susceptibility data for selected crystals. Data is taken from Boyd, Nonlinear optics (Table 1.5.3 and 4.3.1)
Crystal $\chi_{ij}$ in $\frac{\text{m}}{\text{V}}$ $\chi_{ijk}$ in $\frac{\text{m}^2}{\text{V}^2}$
order-of-magnitude estimation$~~~$ $~~~1.9 \cdot 10^{-12}$ $~~~3.8 \cdot 10^{-24}~~~~~~~~~~~~~~~$
CdS $~~~ 2.78 \cdot 10^{-12}$ $~~~ 9.8 \cdot 10^{-20}$
GaAs $~~~ 740 \cdot 10^{-12}$ $~~~ 1.4 \cdot 10^{-18}$
BBO $~~~ 4.4 \cdot 10^{-12}$ $~~~ -$
KDP $~~~ 0.86 \cdot 10^{-12}$ $~~~ -$
Aceton $~~~ -$ $~~~ 1.5 \cdot 10^{-21}$

Whether nonlinear effects can be observed or not, depends therefore on the strengths of the electric field of the impinging em-wave, which is related to the intensity $I$ by the following equation: \begin{equation} |E| = \sqrt{\frac{2I}{c \epsilon_0}}. \end{equation} Three examples show the impact of the intensity:

• Sunlight on the earth surface has an Intensity about $1000$ W/m², resulting in an electric filed strength of $|E| = 868$ V/m .
• A laser pointer with continuous wave output of 1 mW focused on 0.1 mm² has an intensity (Intensity = Power/Area) of $10^4$ W/m², yielding a electric field strength of $|E| = 2.74 \cdot 10^3$ V/m .
• A pulsed laser source emitting a 25 fs pulse with 4 mJ pulse energy hat a momentary power of 0.16 TW, and an intensity of $1.6 \cdot 10^{18}$ W/m² if focused on 0.1 mm². This gives an electric field strength of $|E| = 3.4 \cdot 10^{10}$ V/m .