The electro-optic effect

In the earlier discussion on dielectrics (see Section 2.7.1) a linear relationship between P and E was assumed. The justification for this assumption rests on 

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  

It is evident that if the material has a centre of symmetry or a random structure as in the case of a normal glass, reversal of E0 will have no effect on n. This requires a to be zero, so that there is only a quadratic dependence of n on E0 (and, possibly, a dependence on higher even powers). If, however, the crystal is non-centrosymmetric, reversal of E0 may well affect n and so the linear term has to be retained. 

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. 

The small changes in refractive index caused by the application of an electric field can be described by small changes in the shape, size and orientation of the 

If we try to understand the transmission of light waves in biaxial crystals, we start from the concept of the indicatrix, and to attempt to visualize what shape this must have to show the variation of refractive index with vibration direction for such crystals. From our previous knowledge of the indicatrix for uniaxial crystals, an ellipsoid of revolution with two principal refractive indices, n0 and ne, it is a simple step to see that the indicatrix for biaxial crystals will be a triaxial ellipsoid with three principal refractive indices, n7, np and na. 

Certain transparent materials change their index of refraction, if an electric field is applied to them. This may be used to control the propagation of light. We will first give the basic explanation of the phenomenon. 

On the other hand, the application of a static or slowly varying electric field will be able to displace ions and electrons away from their equilibrium positions and, as a consequence, the polarizability of the electrons will be modified. In the description of H. A. Lorentz s electronic oscillators, the small shifts in the ionic positions modify the spring constants and restoring forces of the electronic oscillators. 

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 . 

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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). 

Changes in the refractive index by the electro-optic effect lead to phase encoding of the incident light distribution... 

As a method to control wavepackets, alternative to the use of ultra-short pulses, I would like to propose use of frequency-modulated light. Since it is very difficult to obtain a well-controlled pulse shape without any chirp, it is even easier to control the frequency by the electro-optic effect and also by appropriate superposition of several continuous-wave tunable laser light beams. 

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. 

For both electro-optic and NLO effects we would like the electric field (associated with an external applied voltage for the electro-optic effect and associated with the input light for NLO effects) to have a large effect on properties. 

Figure 9.12 Seed scattering at refractive index modulations induced by localized internal random fields via the electro-optic effect. The internal fields are also responsible for the formation of a rich ferroelectric domain structure. Here, a periodic sequence of domains with lengths A d is shown. Note, that the grating period of the refractive index modulation As is equal to the lengths of the ferroelectric domains.

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... 

A complete description of the electro-optic effect for single crystals necessitates full account being taken of the tensorial character of the electro-optic coefficients. The complexity is reduced with increasing symmetry of the crystal structure when an increasing number of tensor components are zero and others are simply interrelated. The main interest here is confined to polycrystalline ceramics with a bias field applied, when the symmetry is high and equivalent to oomm (6 mm) and so the number of tensor components is a minimum. However, the approach to the description of their electro-optic properties is formally identical with that for the more complex lower-symmetry crystals where up to a maximum of 36 independent tensor components may be required to describe their electro-optic properties fully. The methods are illustrated below with reference to single-crystal BaTi03 and a polycrystalline electro-optic ceramic. 

Despite these shortcomings it will become clear that in the one-dimensional NLO-phores treated in this section, which display a wide range of seemingly disparate chemical structures, the crude model works surprisingly well. Thus, as a consequence of the validity of the two-state model, their second-order polarizabilities in principle reduce to p-nitroaniline . The reader may even gain the impression that the efforts to improve on the hyperpolarizabilities of even the simplest and most easily accessible -n systems (like p-nitroaniline) have been futile. It is true that an efficiency-transparency trade-off exists At a given wavelength of absorption (related to A ) a maximum value for the second-order molecular polarizability per volume element exists which is not tremendously different from that of very basic unoptimized rr systems. However, for applications like the electro-optical effect, a bathochromic shift of the UV-visible absorption is tolerable so that to strive for maximum hyperpolarizabilities is a viable quest. Furthermore, molecular structures with the same intrinsic second-order polarizabilities may differ substantially in their chemical stabilities and their abilities to be incorporated into ordered bulk structures. 

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). 

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