Second-order electro-optic effects

Here, the first term is referred to as the first-order electro-optic effect (Pochels effect), and the second term is referred to as the second-order electro-optic effect (Kerr effect). The coefficients rij/, and Ry/, are ternary and quaternary tensor quantities known as the Pochels constant (first-order electro-optic constant) and Kerr constant (second-order electro-optic constant), respectively. As Table 7.1.3 shows, a second-order electro-optic effect is present in materials, including isotropic materials such as glass, whereas first-order electro-optic effects are only observed in piezoelectric crystals. In Table 7.1.3, electro-optic effects are present in crystals belonging to point groups. 

Figure 9.3 Schematic illustration of second-order nonlinear optical effects, (a) Second-harmonic generation. Two light fields at frequency go are incident on medium with nonvanishing / 2. Nonlinear interaction with medium creates new field at frequency 2 go. (b) Frequency mixing. One light field at frequency GO and one at frequency go2 is incident on nonlinear medium. Nonlinear interaction with medium creates new field at frequency goi + go2. (c) electro-optic effect. Static electric field E (0) applied over nonlinear medium changes phase of an incoming light field.

The summation runs over repeated indices, /r, is the i-th component of the induced electric dipole moment and , are components of the applied electro-magnetic field. The coefficients aij, Pijic and Yijki are components of the linear polarizability, the first hyperpolarizability, and the second hyperpolarizability tensor, respectively. The first term on the right hand side of eq. (12) describes the linear response of the incident electric field, whereas the other terms describe the nonhnear response. The ft tensor is responsible for second order nonlinear optical effects such as second harmonic generation (SHG, frequency AotAAin, frequency mixing, optical rectification and the electro-optic effect. The ft tensor vanishes in a centrosymmetric envirorunent, so that most second-order nonlinear optical materials that have been studied so far consists of non-centrosyrmnetric, one-dimensional charge-transfer molecules. At the macroscopic level, observation of the nonlinear optical susceptibility requires that the molecular non-symmetry is preserved over the physical dimensions of the bulk stmcture. 

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 electro-optic property of EO polymers comes from the NLO chromophores. When these chromophores are preferentially aligned to break the centrosymmetry of the material, the molecular level microscopic NLO effect of the molecules translates to the macroscopic second-order NLO effect of the polymer material. The poled material exhibits a strong macroscopic 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 EO effect is a second-order nonlinear optical (NLO) effect. Only non-centrosymmetrical materials exhibit second-order NLO effects. This non-centrosymmetry is a condition, both at the macroscopic level of the bulk arrangement of the material and at the microscopic level of the individual molecule. All electro-optic modulators that are presently used by telecom operators are ferro-electric inorganic crystals. The optical nonlinearity in these materials is to a large fraction caused by the nuclear displacement in the applied electric field, and to a smaller fraction by the movement of the electrons. This limits the bandwidth of the modulator. The nonlinear response of organic materials is purely electronic and, therefore, inherently faster. 

In this review, the focus is on electro-optic materials. Such materials are members of the more general class of second-order nonlinear optical materials, which also includes materials used for second harmonic generation (frequency doubling). The term second-order derives from the fact that the magnitude of these effects is defined by the second term of the power series expansion of optical polarization as a function of applied electric fields. The power series expansion of polarization with electric field can be expressed either in terms of molecular polarization (p Eq. 1) or macroscopic polarization (P Eq. 2)... 

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. 

The odd order susceptibilities are nonzero in all materials. However, owing to the fact that x is a third rank tensor, the second order susceptibility is nonzero only in noncentrosym-metric materials, that is, materials possessing no center of symmetry. The focus of this paper is on second order processes, and the relationships between the bulk susceptibility, second harmonic generation, and the linear electro-optic effect. For second harmonic generation, Xijl is symmetric in ij, leading to the relationship between the second harmonic coefficient dijk and the bulk second order susceptibility x 2)[i2l... 

For certain macroscopic nonlinear parameters the tensor notation can be simplified due to the intrinsic symmetry of the experiment, e.g., second-harmonic generation and the linear electro-optic effect. Let us first consider SHG. The second-order contribution to the polarization is given by Eq. (9). 

As already mentioned, the only techniques sensitive to the polar order are even order nonlinear optical techniques such as the already-described second harmonic generation and linear electro-optic effect (cf. Chapter 2). The hrst technique offers a high sensitivity to the fast electronic contributions to susceptibility and is widely used. As already mentioned, it also gives the opportunity to study the kinetics of the poling by in situ measurements [152]. 

X quantifies all second-order NLO effects such as SHG, electro-optic effect (Pockel) and frequency mixing, x is representative of third-order NLO effects such as THG, optical Kerr effect and two-photon absorption (TEA). The real part of 7 describes the nonlinear refractive index and its imaginary part the two-photon cross section (<72). 

In Eqs. 1 and 2. the indices i, j. k. and I refer to the coordinate system of the bulk material and molecule, respectively. Illustrated in Fig. 1 are the linear and nonlinear polarizations with respect to electric field. The Fourier decomposition of this nonlinear polarization comprising components of zero frequency, the fundamental frequency, the second-harmonic frequency, the third-harmonic frequency, etc., is shown in Fig. 2. The effects up to the second order, which are easily observed experimentally, are called the optical rectification. P(0) linear electro-optic effect P((u) second-harmonic generation P(2a>), and third-harmonic generation P(3co). 

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