Nonlinear polarization source term

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 complex quantity, y6br = e (y(3)r) + i Im (x r), represents the nuclear response of the molecules. The induced polarization is resonantly enhanced when the Raman shift wp — ws matches the frequency Qr of a Raman-active molecular vibration (Fig. 6.1A). Therefore, y(3)r provides the intrinsic vibrational contrast mechanism in CRS-based microscopies. The nonresonant term y6bnr represents the electronic response of both the one-photon and the two-photon electronic transitions [30]. Typically, near-infrared laser pulses are used to prevent the effect of two-photon electronic resonances. With input laser pulse frequencies away from electronic resonances, y(3)nr is independent of frequency and is a real quantity. It is important to realize that the nonresonant contribution to the total nonlinear polarization is simply a source for an unspecific background signal, which provides no chemical contrast in some of the CRS microscopies. While CARS detection can be significantly effected by the nonresonant contribution y6bnr [30], SRS detection is inherently insensitive to it [27, 29]. As will be discussed in detail in Sects. 6.3 and 6.4, this has major consequences for the image contrast mechanism of CARS and SRS microscopy, respectively. 

As examples, the nonlinear polarization which is the source term for the observed signal generated in the overlap of the incident laser beams (with e.g. parallel polarizations)... 

As we shall discuss later in a detailed fashion, the nonlinear polarization associated with the nonlinear susceptibility of a medium acts as a source term for radiation at the second harmonic (SH) frequency 2co. 

Here Pi is the amplitude of the nonlinear polarization with frequency tw,. It is determined by an appropriate combination of optical fields and nonlinear susceptibilities according to the expansion in Eq. (2a). It serves as a source term for the optical field with frequency < ,. The wave-vector of the nonlinear polarization fcf is in general different from the wave-vector of the optical field at the same frequency. This difference plays a very important role in determining the effectiveness of many nonlinear optical interactions. 

As is the case with molecular quantities, Fourier components of E and P are accompanied by frequency-dependent, complex susceptibilities % The macroscopic susceptibilities are used in the physical description of NLO effects, such effects typically being analyzed using wave equations in which the nonlinear polarization produced by a given type of interaction constitutes a source term. Quantities other than the susceptibilities are often used for describing specific NLO interactions. The most useful of these are the electro-optic coefficient r related to co ca,0), the nonlinear refractive index ti2, related to the real component of the degenerate third-order susceptibility Re(x —m,ai)), and the two-photon absorption coefficient jSg, related to the corresponding imaginary component Im(x —[Pg.66]

The derivation of formulae for the frequency-dependent nonlinear susceptibilities of nonlinear optics from the time-dependent response functions can be found in a number of sources, (Bloembergen, Ward and New, Butcher and Cotter, Flytzanis ). Here it is assumed that the susceptibilities can be expressed in terms of frequency-dependent quantities that connect individual (complex) Fourier components of the polarization with simple products of the Fourier components of the field. What then has to be shown is how the quantities measured in various experiments can be reduced to these simpler parameters. 

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