Linear Electro-Optic Pockels Effect

The application of an external electric field deforms the optical indicatrix (Fresnel ellipsoid) of crystals lacking a center of symmetry in such a way that its birefringence is changed. The dependence of the birefringence on the electric field E is linear, and can be analytically described by a change of the impermeability tensor a = (e ) by the electric field E [see Eq. (8.4a)] and the polarization P, respectively  

Since the electro-optic tensor has the same symmetry as the tensor of the inverse piezoelectric effect, the linear electro-optic (Pockels) effect is confined to the symmetry groups in which piezoelectricity occurs (see Table 8.3). The electro-optic coefficients of most dielectric materials are small (of the order of 10 m V ), with the notable exception of ferroelectrics such as potassium dihydrogen phosphate (KDP KH2PO4), lithium niobate (liNbOs), lithium tantalate (LiTaOs), barium sodium niobate (Ba2NaNb50i5), or strontium barium niobate (Sro.75Bao.25Nb206) (Zheludev, 1990). For example, the tensorial matrix of KDP with symmetry group 42m has the form 

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The T(3) rtjk gives the linear electro-optic (Pockels) effect, while the 7(4) ptjkl is responsible for the quadratic electro-optic (Kerr) effect qtjkl is the photoelastic tensor. 

Several techniques have been developed for determining the second-order susceptibility [24]. Of practical importance are methods that may be employed for aligned polymeric systems containing polar moieties [4, 8]. Methods making use of the Pockels or linear electro-optic (EO) effect are based on the measurement of the variation in the refractive index of thin polymer films induced by an external electric field. In this way, values of the electro-optic coefficients rss and in are obtained, which are related to the corresponding values through Eq. (3.16). 

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... 

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 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. 

In the literature however, other related parameters, besides x are often used to describe the macroscopic second-order NLO properties of materials. The SHG nonlinear coefficient d and the linear electro-optic coefficient r are the parameters commonly used for second-harmonic generation and the Pockels effect respectively [3, 5]. They are related to x according to Eqs. (4) and (5). 

Photonics is playing an ever-increasing role in our modern information society. Photon is gradually replacing the electron, the elementary particle in electronics. Several hooks and reviews have appeared dealing with the theory of nonlinear optics and the structural characteristics and applications of nonlinear optical molecules and materials [1—18]. Tlie earliest nonlinear optical (NLO) effect discovered was the electro-optic (EO) effect. The linear EO coefficient defines the Pockel effect, discovered in 1906, while the quadratic (nonlinear) EO coefficient s,i relates to the Kerr effect, discovered 31 years later (1875). Truly, all-optical NLO effects were not discovered until the discovery of lasers. Second harmonic generation (SHG) was first observed in a single crystal of quartz by Franken et al. [1] in 1961. They frequency doubled the output of a ruby laser (694.3 nm) into the 383... 

Electro-optic (EO) phenomena are related to the interaction of an electric field with an optical process. The classical electro-optic effects, the Pockels and the Kerr effect, discovered in 1893 and 1875 with quartz and carbon disulfide, respectively, refer to the induction of birefringence in certain materials under the influence of an external electric field. Application of an electric field to the sample causes a change in the refractive index. In the case of the Pockels effect. An is linearly proportional to E, the strength of the applied electric field [see Eq. (3-1)]. Hence, it is also called the linear electro-optic effect In contrast. An is proportional to E in the case of the Kerr effect [see Eq. (3-2)]. 

Linear electro-optical effect — Pockels effect An = rE Quadratic electro-optical effect — Kerr effect An = q2E ... 

Ceramic PLZT has a number of structures, depending upon composition, and can show both the Pockels (linear) electro-optic effect in the ferroelectric rhombohedral and tetragonal phases and the Kerr (quadratic) effect in the cubic paraelectric state. Because of the ceramic nature of the material, the non-cubic phases show no birefringence in the as-prepared state and must be poled to become useful electro-optically (Section 6.4.1). PMN-PT and PZN-PT are relaxor ferroelectrics. These have an isotropic structure in the absence of an electric field, but this is easily altered in an applied electric field to give a birefringent electro-optic material. All of these phases, with optimised compositions, have much higher electro-optic coefficients than LiNb03 and are actively studied for device application. 

X0>( linear electro-optical effect (Pockels-effect) Light-modulation... 

The first term in Equation (14.6) is related to initial refractive indices of the medium at three primary directions, n, Uy, n. The second term refers to the linear electro-optic effect, which is known as the Pockels effect, and the third term refers to the quadratic electro-optic effect, known as the Kerr effect. Here, and Sjj are electro-optic tensors for the linear and quadratic electro-optic effects, respectively. The second-order Kerr effect is small as compared to the first-order linear effect, so it is usually neglected in the presence of linear effect. However, in crystals with centro-symmetric point groups, the linear effect vanishes and then the Kerr effect becomes dominant. 

This effect is even less pronounced than the linear Pockels effect in crystals. In contrast to the linear electro-optic effect that is confined to crystals lacking a center... 

So, d22 coefficient corresponds to and d to d a . The linear electro-optic effect or Pockels effect (Fig. 2.1) can be defined as a rotation or deformation of... 

When the frequency (S2) of one of the applied electric fields is much lower than the other (S2 co), one often speaks of linear electro-optic (EO) or Pockels effect. An EO coefficient can be introduced by means of the expression ... 

The linear electro-optical (EO) behavior, i.e., the Pockels effect, constitutes a manifestation of nonlinear optical features of anisotropic and non-centro-symmetric media. Functional architectures based on host polymer matrixes and guest SiC nanoparticles (nc-SiC) as active chromophores were realized. The intrinsic dipole moments of the chromophore combined with the eventual polarization at the interfaces with the host matrix constitute the physical origin of the electro-optical responses. The experiments were carried out in hybrid materials based on SiC nanocrystals and matrixes such as PVK, PMMA, or PC. 

Noncentrosymmetric crystals show other properties in addition to frequency conversion, for instance the linear electro-optic or Pockels effect the linear change of the refractive index induced by an applied DC electric field. Furthermore, the point groups and allow for the existence of a permanent electric dipole moment. Indeed, crystals... 

Linear Electro-Optic Effect (Pockels Effect) and Electro-Optic Modulators [19,20]... 

Linear electro-optic effect (Pockels effect) refers to the modification of the refractive indices of certain NLO materials by a low-frequency or (DC) electric field. The amplitude of modulation field is assumed to be much larger than the light field Ex Ej. The resulting small refractive index change is linearly proportional to the electric field ... 

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). 

Whether or not the dependence is expressed in terms of E or P is a matter of choice it seems customary in the literature relating to single crystals to use the r coefficient for the linear Pockels effect and g for the quadratic Kerr effect. In the case of electro-optic ceramics r and R are most commonly used. 

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