Maxwell equations susceptibility

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. 

By analogy with Eq. (278), we (a) suggest the contributions of dipoles and ions into the complex permittivity s of the solution to be additive and (b) consider the ionic component of this permittivity to be proportional to the ionic susceptibility /v(on( )- From Maxwell equations we have (see Appendix) the following relation between the ionic conductivity cr = a + ia" and susceptibility X on ... 

The Maxwell equations become wave equations for ( and ,. In the absence of externally applied currents, conductivity, and externally inserted charges, with scalar electric and magnetic susceptibilities s and // that are constant in each region, we have10... 

Hence, transformations of Maxwell equations, the change of orientational distribution functions to the form, close to the Boitzmann distribution with account of classical equations of motions yield the complex susceptibility /(co) determined by unperturbed collision-free motion of an individual particle in a given static potential well. In our approach, the complex permittivity e(co) is found as a simple rational function of this susceptibility /(co). 

Macroscopic polarization, 52 Macroscopic polarization field, 53, 54 Macroscopic susceptibility, 33, 35, 53-55 Magnetic dipole moment, 14 Maxwell electric field, 53, 54 Maxwell equation, 53 Maxwell field, 27, 33, 34 Mixed electric magnetic hyper-magnetizabili-ty, 28... 

The susceptibility tensor is the proper vari able to describe the Ught-matter interactions, si nee it links directly the electromagnetic field and the polarization effects. Nevertheless, other variables such as the index of refraction or the dielecUic function are very often used in the literature. The Maxwell equations provide the connection between the dielectric tensor, and the linear susceptibility tensor, Xij (in literature sometimes called liquid polarizability and reported as Ilij) eij = 1 + 4Trxij in the CGS unit system. It is also steaightforward the connection with the index of refraction, since 77 = ffe. So when only the fluctuating or time-dependent part is relevant, the three functions are substantially equivalent ixij oc 6e 877... 

Introduction 124 Maxwell s equations 125 Constitutive relations 125 Linear optics linear susceptibility 126... 

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... 

In equations 32 and 33, x is the susceptibility of the medium. Expressed in the Fourier space, the Maxwell field E and the polarization P are related by... 

The interaction of laser radiation with the medium occurs through the third-order, non-linear, electric susceptibility denoted by and gives rise to an induced polarization field, which acts as a source term in Maxwell s wave equation. On solving the wave equation, one arrives at the following expression for the intensity of the CARS signal ... 

Based upon Faraday s work, James Clerk Maxwell published his famous equations in 1873. He more specifically calculated the resistance of a homogeneous suspension of uniform spheres (also coated, two-phase spheres) as a function of the volume concentration of the spheres. This is the basic mathematical model for cell suspensions and tissues still used today. However, it was not Maxwell himself who in 1873 formulated the four equations we know today as Maxwell s equations. Maxwell used the concept of quaternions, and the equations did not have the modern form of compactness he used 20 equations and 20 variables. It was Oliver Heaviside (1850—1925) who first expressed them in the form we know today. It was also Heaviside who coined the terms impedance (1886), conductance (1885), permeability (1885), admittance (1887), and permittance, which later became susceptance. 

On the other hand, the polarization can be derived in classical electrodynamics from Maxwell s equations using the dielectric constant q or the susceptibility X, i.e.,... 

The contributions to the fifth-order nonlinear optical susceptibility of dense medium have been theoretically estimated by using both the local-field-corrected Maxwell-Bloch equations and Bloembergen s approach. In addition to the obvious fifth-order hyperpolarizability contribution, the fifth-order NLO susceptibility contains an extra term, which is proportional to the square of the third-order hyperpolarizability and which originates purely from local-field effects, as a cascaded contribution. Using as model the sodium 3s 3p transition system, it has been shown that the relative contribution of the cascaded term to the fifth-order NLO susceptibility grows with the increase of the atomic density and then saturates. 

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