Polarization waves, nonlinear

Nonlinear process Nonlinear polarization Wave-vector mismatch Plane-wave phase-matching condition... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

In the theoretical section above, the nonlinear polarization induced by the fundamental wave incident on a planar interface for a system made of two centrosymmetrical materials in contact was described. However, if one considers small spheres of a centrosymmetrical material embedded in another centrosymmetrical material, like bubbles of a liquid in another liquid, the nonlinear polarization at the interface of a single sphere is a spherical sheet instead of the planar one obtained at planar surfaces. When the radius of curvature is much smaller than the wavelength of light, the electric field amplitude of the fundamental electromagnetic wave can be taken as constant over the whole sphere (see Fig. 7). Hence, one can always find for any infinitely small surface element of the surface... 

H.W. Schurmarm, V.S. Serov and Yu.V. Shestopalov, TE-polarized waves guided by a lossless nonlinear three-layer stmctms ,Phys.Rev.E 58, 1040 - 1050 (1998). 

By this nonlinear polarization different incident lightwaves, or lightwaves and phonon waves, are coupled together, resulting in the production of sum and difference frequencies and second or third harmonics. (This is to some extent analogous to rf frequency mixing in a nonlinear diode). 

Recently, there has been much interest in the development and application of multidimensional coherent nonlinear femtosecond techniques for the study of electronic and vibrational dynamics of molecules [1], In such experiments more than two laser pulses have been used [2-4] and the combination of laser pulses in the sample creates a nonlinear polarization, which in turn radiates an electric field. The multiple laser pulses create wave packets of molecular states and establish a definite phase relationship (or coherence) between the different states. The laser pulses can create, manipulate and probe this coherence, which is strongly dependent on the molecular structure, coupling mechanisms and the molecular environment, making the technique a potentially powerful method for studies of large molecules. 

Just as linear polarization leads to linear optical effects, such as refractive index and birefringence, nonlinear polarization leads to other and usually more subtle (nonlinear) effects. It is precisely these effects we hope to understand and exploit. In Figure 14, application of a symmetric field (i.e., the electric field associated with the light wave) to the anharmonic potential leads to an asymmetric polarization response. This polarization wave shows diminished maxima in one direction and accentuated... 

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. 

Nonlinear optics entails the mixing of one (with itself in some cases) or more fields to produce a nonlinear polarization source term, which in turn can radiate a new electromagnetic wave (13,14). This term is usually written as... 

The existence of the nonlinear polarization field does not ensure the generation of significant signal fields. With the exception of phenomena based on an intensity-dependent refractive index, the generation of the nonlinearly produced signal waves at frequency cos can be treated in the slowly varying amplitude approximation with well-known guided wave coupled mode theory (1). As already explicitly assumed in Equation 1, the amplitudes of the waves are allowed to vary slowly with... 

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] ... 

In the study of the KE for a selection of pure liquids [30] the concept of effective polarizabilities was extended to introduce the contribution of the output wave. Radiation at a frequency to induces a macroscopic nonlinear polarization density (pm)NL at the same frequency, the output wave, generating an additional perturbing field. The molecules of the liquid respond with an additional effective polarizability a(—to to), whereby Equation (2.242) becomes... 

In the third-harmonic generation, the third-order susceptibility leads to a nonlinear polarization component which oscillates at the third-harmonic frequency of the incident laser beam. This leads to a light wave at the third-harmonic frequency of the fundamental wave. As optical frequencies are involved and since the output frequency is different from the input frequency only the electronic nonlinearities can participate without any contributions from thermal or orientational effects. Because one needs fast nonlinearities for all-optical signal processing, the main interest is directed towards the fast electronic nonlinearities. Therefore and also due to its simplicity, third-harmonic generation is a very attractive method to characterize newly developed materials. 

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