Expansion coefficients, nonlinear optics

It is important to note that the coefficients fp, gp, and hs are always nonvanishing, for both achiral and chiral isotropic films. On the other hand, fs, gs, and hp can only be nonvanishing if the isotropic film is chiral (nonracemic) because they completely depend on the chiral susceptibility components. Note that gs is always equal to zero within the electric dipole approximation. The sign of the chiral expansion coefficients changes between enantiomers, while that of the achiral expansion coefficients stays the same. Experimental determination of all expansion coefficients fully characterizes the nonlinearity and nonlinear optical activity of the sample. Once all expansion coefficients are... 

This effect, which is in a loose sense the nonlinear analog of linear optical rotation, is based on using linearly polarized fundamental light and measuring the direction of the major axis of the ellipse that describes the state of polarization of the second-harmonic light. For a simple description of the effect, we assume that the expansion coefficients are real, as would be the case for nonresonant excitation within the electric dipole approximation.22 In this case, the second-harmonic light will also be linearly polarized in a direction characterized by the angle... 

From the knowledge of the order parameter from the measurement of the variation of the optical absorption spectrum due to poling, one can estimate the x parameter (Eq. (43)) intervening in the above developments. For poled polymers one can use also another alternative description of linear and nonlinear optical susceptibilities by expanding the orientation distribution function in the series of Legendre polynomials, where the expansion coefficients are order parameters ... 

The macroscopic optical responses of a medium are given by its linear and nonlinear susceptibilities, which are the expansion coefficients of the material polarization, P, in terms of the Maxwell fields, 1 3]. For a dielectric or ferroelectric medium under the influence of an applied electric field, the defining equation reads... 

We have seen how the molecular properties in nonlinear optics are defined by the expansion of the molecular polarization in orders of the external electric field, see Eq. (5) beyond the linear polarization this definition introduces the so-called nonlinear hyperpolarizabilities as coupling coefficients between the two quantities. The same equation also expresses an expansion in terms of the number of photons involved in simultaneous quantum-mechanical processes a, j3, y, and so on involve emission or absorption of two, three, four, etc. photons. The cross section for multiphoton absorption or emission, which takes place in nonlinear optical processes, is in typical cases relatively small and a high density of photons is required for these to occur. 

The finite field method is the simplest method for obtaining nonlinear optical properties of molecules. This method was first used by Cohen and Roothaan to calculate atomic polarizabilities at the Hartree-Foclc level. The basic idea is to truncate the expansion of the energy (Eq. [6]) and solve for the desired coefficients by numerical differentiation. For example, if the expression is truncated after the quadratic term, the result is E(P) = E[0) — — iot yF,Fy. 

Note that the equations describing nonlinear optical effects can have different forms depending on the system of units used (esu or SI), the inclusion of the 1/n factors of the power expansion in the coefficients or using them in Ifont of the coefficients, and inclusion (or not) of the degeneracy factors that may appear in the equations, depending on the number of identical fields in a given nonlinear interaction. See, for example, [11]. 

Cross-linked polymeric liquid crystals offer a wide variety of unique and in-tere.sting properties. Because of the interaction between the mesogens and the network backbone in liquid crystal elastomers, mechanical deformations can align the director, and these materials are piezoelectric. Industrial applications of liquid crystalline thermosets are driven by additional properties such as toughness, a tunable coefficient of thermal expansion, ferroelectricity, and nonlinear optical properties. Reviews on this topic are given by Barclay and Ober [4] and by Warner and Terentjev [5]. 

Explicit formulas for the nonlinear susceptibilities xljL Xuu derived by working out the coefficients c (t), c (r),..., respectively, in the time-ordered expansion (1.96). Straightforward evaluation of the integrals in the time-ordered expansions rapidly becomes unwieldy, and an efficient diagrammatic technique is developed in Section 11.1 for writing down the contributions to cj r) that are pertinent to any multiphoton process of interest. In Sections 11.2 and 11.3, we apply this technique to obtaining the nonlinear susceptibilities for two important nonlinear optical processes, SHG and CARS. Experimental considerations that are unique to such coherent optical phenomena are also discussed. 

Our theoretical understanding of third-order optical nonlinearity at the microscopic level is really in its infancy. Currently no theoretical method exists which can be reliably used to predict, with reasonable computational time, molecular and polymeric structures with enhanced optical nonlinearities. The two important approaches used are the derivative method and the sum-over-states method (7,24). The derivative method is based on the power expansion of the dipole moment or energy given by Equations 3 and 4. The third-order nonlinear coefficient Y is, therefore, simply given by the fourth derivative of the energy or the third derivative of the induced dipole moment with respect to the applied field. These... 

The types of expression in Eqs. (2a) and (2b) are generally valid when the optical fields are weak compared to the electric field that binds the electrons in the material and when the coefficients of the various terms in Eqs. (2a) and (2b) are constant over the range of frequencies contained in the individual incident and generated fields. In addition, the wavelength of the radiation must be long compared to the dimension of the scattering centers (the atoms and molecules of the nonlinear medium), so that the charge distributions can be accounted for with a multipole expansion. 

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