Second-order nonlinear polarization

Second harmonic generation (SHG) is one of the most intensively studied nonlinear optical effects that have ever been combined with near-held scanning optical microscopy (Shen et al. 2000 Zayats and Sandoghdar 2000 Zayats and Sandoghdar 2001 Takahashi and Zayats 2002). SHG, which is an even-order nonlinear process, is forbidden in centrosymmetric media under the dipole approximation (Shen 1984). Non-centrosymmetric molecules and lattices are allowed to exhibit SHG light. The second-order nonlinear polarization for SHG (T shg) is given in a scalar form by... 

Second-Order Nonlinear Polarization of Matter and Second-Order NLO Effects... 

For second-order nonlinear polarization, the problem becomes more complex. As can be seen in Figure 13 the anharmonic polarization shows the largest deviation from the linear polarization with large distortion values. Therefore, if the material is not polarizable (i.e., if the electrons can only be perturbed a small distance from their equilibrium positions), then the anharmonicity will not be manifested. For large second-order nonlinearities we need a material that offers both a large linear... 

The first two terms are electric quadrupole in character while the last term is magnetic dipolar. Under excitation by a single plane wave, the first term vanishes. In a homogeneous medium the second term vanishes by Gauss Law. The third term describes the induced polarization which is along the propagation direction. It can only radiate at the discontinuity of the surface. The full expression for the second-order nonlinear polarization in an isotropic medium is then written as the sum of the surface and bulk polarizations [78] ... 

Eq. 12), Eq. (25) is valid in the Fourier domain and relates the components of the susceptibility tensor elements and the complex components of the electric field and polarization. All quantities are frequency dependent. The general expression of the second-order nonlinear polarization is given by [10] ... 

Other important factors that should be accounted for are the local field corrections and the anisotropy of the molecules. In general, Uquid crystalUne molecules are complex and possess permanent dipole moments. Furthermore, quadrupole moments could also contribute significantly in second-order nonlinear polarizations, and thus the treatment presented in the preceding sections, which ignores these points, has to be appropriately modified. 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

Fig. 1. Representative device configurations exploiting electrooptic second-order nonlinear optical materials are shown. Schematic representations are given for (a) a Mach-Zehnder interferometer, (b) a birefringent modulator, and (c) a directional coupler. In (b) the optical input to the birefringent modulator is polarized at 45 degrees and excites both transverse electric (TE) and transverse magnetic (TM) modes. The appHed voltage modulates the output polarization. Intensity modulation is achieved using polarizing components at the output.

Turning to the investigation of the anisotropy of second-order nonlinearity, one must be cautious of evaluation of the magnitude of X zxx Performed as just described by simply rotating the plane of polarization of the laser beam. We tried this operation and saw a drastic decrease in I2<0. However, this may merely reflect the smaller magnitude of lQ in this SHG process. In this case... 

The large molecular hyperpolarizability of the merocyanine chromophore (4,5) and the highly polar environment of the quasicrystals has prompted studies of the second order nonlinear optical properties of these materials (6). 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

In the electric dipole approximation, the polarization can in general (for any second-order nonlinear optical process) be written as9... 

Nonlinear optics deals with physical systems described by Maxwell equations with an nonlinear polarization vector. One of the best known nonlinear optical processes is the second-harmonic generation (SHG) of light. In this section we consider a well-known set of equations describing generation of the second harmonic of light in a medium with second-order nonlinear susceptibility %(2 The classical approach of this section is extended to a quantum case in Section IV. 

Figure 14. (a) Plots of the electric field of the applied light wave (solid) and the induced polarization wave (dotted), as a function of time, for a second-order nonlinear material (b) cartoon depicting the polarization of the material as a function of time (c) plots of induced polarization vs. applied field for both linear and second-order nonlinear materials. 

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) ... 

In this paper, an overview of the origin of second-order nonlinear optical processes in molecular and thin film materials is presented. The tutorial begins with a discussion of the basic physical description of second-order nonlinear optical processes. Simple models are used to describe molecular responses and propagation characteristics of polarization and field components. A brief discussion of quantum mechanical approaches is followed by a discussion of the 2-level model and some structure property relationships are illustrated. The relationships between microscopic and macroscopic nonlinearities in crystals, polymers, and molecular assemblies are discussed. Finally, several of the more common experimental methods for determining nonlinear optical coefficients are reviewed. 

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