Pockels susceptibility

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

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 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). 

The applied voltage in effect changes the linear susceptibility and thus the refractive index of the material. This effect, known as the linear electrooptic (LEO) or Pockels effect, modulates light as a function of applied voltage. At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different... 

The linear susceptibility yy1 1 is related to optical refraction and absorption. The most common effects due to second-order susceptibility x(2) are frequency doubling x (-2co co, co) and the EO (Pockels) effect x(2)(- 0, co). The third-order susceptibility y 3) is responsible for such phenomena as frequency tripling and the Kerr effect. 

Thus, the applied field, E2, changes the effective linear susceptibility (i.e. the dependence of the polarization on the light field, Eft. Since the linear susceptibility is related to the refractive index, the refractive index of the material is also changed by the applied field. This is known as the linear electrooptic (EO) or Pockels effect and can be used to modulate the polarization or phase of light by changing the applied voltage. 

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

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). 

The second-order NLO properties are of interest for a variety of NLO processes [1-3]. One of the most relevant is the SHG, originated by the mixing of three waves two incident waves with frequency co interact with the molecule or the bulk material with NLO properties, defined by a given value of the quadratic hyperpolarizability, fi, or of the second-order electrical susceptibility, respectively, to produce a new electrical wave, named SH, of frequency 2co. Another important second-order NLO process is the electrooptic Pockels effect which requires the presence of an external d.c. electric field, E(0), in addition to the optical E co) electrical field. This effect produces a change in the refractive index of a material proportional to the applied electric field, and can be exploited in devices such as optical switches and modulators [1-3]. 

Here ai is the frequency of the first strongly absorbing electronic transition in the molecule, and la and at are the fundamental wavelengths in second harmonic generation and for electrooptic coefficient measurements, respectively. The electrooptic effect (Pockels effect) is related to the corresponding second-order NLO susceptibility and by knowing the SHG coefficients, one can also estimate the electrooptic coefficients. 

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