Quadratic electro-optic effect

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). 

The Kerr quadratic electro-optic effect Single-crystal BaTiO3 (T>TC)... 

In passive mode-locking, an additional element in the cavity can be a saturable absorber (e.g., an organic dye), which absorbs and thus attenuates low-intensity modes but transmits strong pulses. Kerr lens mode-locking exploits the optical Kerr63 or DC quadratic electro-optic effect here the refractive index is changed by An = (c/v) K E2, where E is the electric field and K is the Kerr constant. 

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

It was recently shown that doping poly(pPIN) with iodine produces a semi-conductive material, with conductivities of about 8 X 10 S cm typical of non-conjugated doped polymers [99]. This material also showed a large quadratic electro-optic effect which would make it useful for non-linear optics applications [100]. 

Rajagopalan H., Vippa R, Thakur M., Quadratic electro-optic effect in a nano-optical material based on the nonconjugated conductive polymer, poly(beta-pinene), App/. Phys. Lett., 88(3), 2006. 

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. 

In Equation (14.7), the linear term vanishes and only the Kerr effect term survives. The electrooptic tensor for the Kerr effect varies with different molecular stmcture. For an isotropic Uquid, its quadratic electro-optic effect coefficients can be represented by the following matrix ... 

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. 

Narasimhamurty, T. S. (1981). Kerr quadratic electro-optic effect Pockels phenomenological theory, in Photoelastic and Electro-Optic Properties of CrystalsAnonymous, pp. 359-362, Plenum Press, New York. 

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). 

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. 

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. 

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 . 

This electro-optical effect, commonly observed as transient changes in optical birefringence of a solution following application, removal, or reversal of a biasing electric field E(t), has been used extensively as a probe of dynamics of blopolymer solutions, notably by O Konski, and is a valuable tool because it gives information different in form, but related to, results from conventional dielectric relaxation measurements. The state of the subject to 1975 has been comprehensively presented in two review volumes edited by O Konski (25). The discussion here is confined to an outline of a response theory treatment, to be published in more detail elsewhere, of the quadratic effect. The results are more general than earlier ones obtained from rotational diffusion models and should be a useful basis for further theoretical and experimental developments. 

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. 

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. 

Potassium tantalate-niobate [K(Ta Nbi jc)03, KTN] is one of the ferroelectric materials with the perovskite structure, and is a sohd solution of potassium tantalate (KTaOs) and potassium niobate (KNbOs). The Ciuie temperature of KTN for the cubic to tetragonal transition varies with Ta/Nb ratio, and is lowered with increasing Ta substitution (Triebwasser, 1959). The ferroelectric properties of KTN, therefore, can be controlled by the Ta/Nb ratio. The nonferroelectric cubic phase of KTN atx= 0.65 is known to show photorefractive effect based upon a large quadratic electro-optic coefficient at room temperature (Gausic, 1964 Orlowski, 1980). 

Particular nonlinear optical phenomena arise also when static electric or magnetic fields are applied. The molecular states and selection rules are thereby modified, leading, for instance, to higher-order, nonlinear-optical variants of the linear (Pockels) and quadratic (Kerr) electro-optical effect, or of the linear (Faraday) and quadratic (Cotton-Mouton) magneto-optical effect. 

Quadratic NLO effects arising from (3 and x(2) include SHG, the electro-optic (EO, Pockels) effect and frequency mixing (parametric amplification). SHG is actually just a special case of a three-wave... 

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