Linear electro-optic effect

FIGURE 7.1.5 Birefringence induced by the electro-optic effect and the electric field, (a) Linear electro-optic effect. Linear electro-optic coefficient gradient, (b) Quadratic electro-optic effect. Coefficient of is proportional to (Rn — Rn) value. 

Figure I. Comparative quantities for selected tensor components of second harmonic generation (left) and the linear electro-optic effect (right) (measured at 1.06 pm wavelength). (The strain free quantity, f, was measured at 0.633 pm wavelength except in the case of GaAs which was measured at - 0.9 pm). (Reproduced with permission from Ref. 8. Copyright 1982, Laser Focus.)...

The proportionality constants a and (> are the linear polarizability and the second-order polarizability (or first hyperpolarizability), and x(1) and x<2) are the first- and second-order susceptibility. The quadratic terms (> and x<2) are related by x(2) = (V/(P) and are responsible for second-order nonlinear optical (NLO) effects such as frequency doubling (or second-harmonic generation), frequency mixing, and the electro-optic effect (or Pockels effect). These effects are schematically illustrated in Figure 9.3. In the remainder of this chapter, we will primarily focus on the process of second-harmonic generation (SHG). 

Pockel cells use the linear electro optic effect in crystals to turn the plane of polarization when an external electric field is applied. A detailed description with references is given in... 

Where P is the polarisation and the others the linear (1) and non-linear, second (2) and third order (3) terms. Examples of important second order effects are frequency doubling and linear electro-optic effects (Pockles effect), third order effects are third-harmonic generation, four-wave mixing and the quadratic electro-optic effect (Ken-effect). 

Historically, the earliest nonlinear optical (NLO) effect discovered was the electro-optic effect. The linear electro-optic (EO) coefficient rij defines the Pockels effect, discovered in 1906, while the quadratic EO coefficient sijki relates to the Kerr effect, discovered even earlier (1875). True, all-optical NLO effects were not discovered until the advent of the laser. 

This is called the linear electro-optic effect, also called the Pockels effect . 

In non-polar, isotropic crystals or in glasses, there is no crystallographic direction distinguished and the linear electro-optic effect is absent. Nevertheless a static field may change the index by displacing ions with respect to their valence electrons. In this case the lowest non-vanishing coefficients are of the quadratic form, i.e. the refractive index changes proportionally to the square of the applied field Kerr effect . 

The linear electro-optic effect (see Chapter 4) transfers the periodically modulated space-charge field into a refractive index grating ... 

When investigating the polar structure by photo-induced light scattering we assume that the largest contribution to the initial optical noise is due to diffraction of the pump beam on optical inhomogeneities located at boundaries of ferroelectric domains [9], Figure 9.12 illustrates this concept schematically. Internal electric fields Ei (random fields) yield local perturbations 5n of the index of refraction via the linear electro-optic effect 5n = - n rssEi. 

The manifestation of the presence of polar nanodomains in strong rls in terms of the electro-optic effect was first demonstrated by Burns and Dacol [3] in measurements of the T dependence of the refractive index, n. For a normal ABO3 fe crystal, starting in the high-temperature PE phase, n decreases linearly with decreasing T down to Tc at which point n deviates from linearity. The deviation is proportional to the square of the polarization and... 

However, the linear response of a dielectric to an applied field is an approximation the actual response is non-linear and is of the form indicated in Fig. 8.6. The electro-optic effect has its origins in this non-linearity, and the very large electric fields associated with high-intensity laser light lead to the non-linear optics technology discussed briefly in Section 8.1.4. Clearly the permittivity measured for small increments in field depends on the biasing field E0, from which it follows that the refractive index also depends on E0. The dependence can be expressed by the following polynomial ... 

In 1875 John Kerr carried out experiments on glass and detected electric-field-induced optical anisotropy. A quadratic dependence of n on E0 is now known as the Kerr effect. In 1883 both Wilhelm Rontgen and August Kundt independently reported a linear electro-optic effect in quartz which was analysed by Pockels in 1893. The linear electro-optical effect is termed the Pockels effect. 

G. F. Lipscomb, A. F. Garito, and R. S. Narang, An exceptionally large linear electro-optic effect in the organic solid MNA, /. Chem. Phys. 75 1509-1516 (1981). 

The odd order susceptibilities are nonzero in all materials. However, owing to the fact that x is a third rank tensor, the second order susceptibility is nonzero only in noncentrosym-metric materials, that is, materials possessing no center of symmetry. The focus of this paper is on second order processes, and the relationships between the bulk susceptibility, second harmonic generation, and the linear electro-optic effect. For second harmonic generation, Xijl is symmetric in ij, leading to the relationship between the second harmonic coefficient dijk and the bulk second order susceptibility x 2)[i2l... 

It has been established experimentally that the origin of the electro-optic effect in organic materials is largely electronic. This implies that the linear electro-optic coefficient can be estimated from the second harmonic coefficient. By properly accounting for the dispersion (using a two level model), the electronic contribution to the electro-optic coefficient is calculated to be r5 3 - 2.4 0.6 x 10 m/V at X-O.S m. Measured values of the electro-optic coefficient are in agreement within experimental uncertainty. These values compare favorably with that of GaAs (r4i - 1.2 x 10 m/V). 

For certain macroscopic nonlinear parameters the tensor notation can be simplified due to the intrinsic symmetry of the experiment, e.g., second-harmonic generation and the linear electro-optic effect. Let us first consider SHG. The second-order contribution to the polarization is given by Eq. (9). 

The linear electro-optic effect arises from the ability of a material medium to change its refractive index under the action of an external electric field. This variation is proportional to the external field strength... 

As already mentioned, the only techniques sensitive to the polar order are even order nonlinear optical techniques such as the already-described second harmonic generation and linear electro-optic effect (cf. Chapter 2). The hrst technique offers a high sensitivity to the fast electronic contributions to susceptibility and is widely used. As already mentioned, it also gives the opportunity to study the kinetics of the poling by in situ measurements [152]. 

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