Pockels tensor

The susceptibility tensor Xpl is related to the Pockels tensor rjjK [Eq. (3-16)] [19],... 

Comments on NLO and Electrooptic Coefficients. Typically, the Pockels effect is observed at relatively low frequencies (up to gigahertz) so that slower nonlinear polarization mechanisms, such as vibrational polarizations, can effectively contribute to the "r" coefficients. The tensor used traditionally by theorists to characterize the second-order nonlinear optical response is xijk Experimentalists use the coefficient dijk to describe second-order NLO effects. Usually the two are simply related by equation 31 (16) ... 

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. 

Example 15.4-3 Both the piezoelectric effect and the Pockels effect involve coupling between a vector and a symmetric 7(2). The structure of K is therefore similar in the two cases, the only difference being that the 6 x 3 matrix [rqi is the transpose of the 3x6 matrix [diq where i 1, 2, 3 denote the vector components and q= l,. .., 6 denote the components of the symmetric 7(2) in the usual (Voigt) notation. Determine the structure of the piezoelectric tensor for a crystal of C3v symmetry. 

An alternative quantity is also traditionally used to characterize the Pockels non-linearity. Here an electro-optical tensor r is defined which is related to the Pockels susceptibility j (-a) a),0) by equation (31) (Singer et al, 1987),... 

The basis of NLO-effects arising from susceptibilities of second order, is the interaction of three electric fields with a material. The practical implementation of optical devices requires strong, coherent and monochromatic radiation and hence, laser technology. Not all of the interacting fields need to be optical fields, however. In devices that make use of the Pockels effect, an externally applied electric field is used to alter reversibly the refractive index of a material. In a second harmonic generation (SHG) process two photons of circular frequency w can be transformed into one photon of frequency Iw. SHG is the NLO effect used most for the evaluation of /3-tensor elements in solution. 

Summation over repeated indices is implied and is the th-order susceptibility tensor that describes the interaction between the electric fields and the material. The first two terms on the right-hand side of Equation 8.A1 give the spontaneous polarization and linear optics effects. The last two terms lead to various phenomena in nonlinear optics. They include SHG and EO Pockels and Kerr effects. The EO susceptibilities are obtained by combining optical and static fields therefore, the susceptibilities that describe the EO Pockels and Kerr effects are (-co, co, 0) and x% respectively. In a... 

From the two descriptions of the Pockels effect in the frameworks of electro-op-tics and nonlinear optics, one can show that the electro-optic tensor elements and the second-order nonlinear susceptibility elements are related by... 

The NLO materials used so far are mainly inorganic materials, such as quartz, KDP, zinc oxide, LiNbOs, etc. The ions in these inorganic materials are responsible for the NLO properties. The ions have a great mass so that the NLO coefficients of these NLO materials are small at high frequency. Table 6.11 lists elements of the NLO tensors of the Pockel effect and of the second harmonic generation for some widely used NLO materials, Xp and x(d ... 

In the same way as in the two previous sections, the first hyperpolarizabilities associated to three different nonlinear effects were calculated the static response and two dynamic effects, the dc-Pockels response characterized by P(-(b o),0) and the second harmonic generation response, p(-2 to, ). Again, in the dynamic case ffl=1.16 eV. The results are listed in Tables XIV, XV, and XVI for the ID clusters, and in Tables XVII, XVIII, and XIX for the 2D and 3D clusters. Similarly to the dipole moment case, only the major tensor component, Paaa> is considered. 

Quadratic Response (3rd rank tensors) (1) electric field effects Linear electrooptic (Pockels) effect, three-wave mixing, SHG. (2) radiation/magnetic field Faraday Effect. 

Eckart et have investigated the vibrational contributions to a and P for cyclopropenone and cyclopropenethione and for their saturated and non-cyclic counterparts. While in most cases the vibrational contributions are about 10% of the total value of a tensor component, it is suggested that the vibrational contribution to the Pockels effect tensor, P(—co oj, 0), might be as much as 50%. 

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. 

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... 

Dipolar chromophores must be assembled into a noncentrosymmetric lattice to translate molecular optical nonlinearity to maaoscopic electro-optic (EO) aaivity, which is one example of second-order NLO properties. The EO coefficient in units of pmV , rs3, is the principal element of the linear Pockel s EO effect tensor and denotes the magnitude of refractive index shift (Ai ) obtained for an applied low-frequency electric field. This... 

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