Coherency vector scattered field

Much more important for gas phase spectroscopy than the hyper-Raman effect are the various coherent Raman effects, so we shall develop the theory of coherent Raman scattering in rather more detail. The usual starting point is the bulk polarization of the medium expressed as a function of the electric field vectors of the various light waves present simultaneously in the medium (SI)... 

The polarization characteristics of the incident field can also be described by the coherency and Stokes vectors. Although the ellipsometric parameters completely specify the polarization state of a monochromatic wave, they are difEcult to measure directly (with the exception of the intensity Eq). In contrast, the Stokes parameters are measurable quantities and are of greater usefulness in scattering problems. The coherency vector is defined as... 

As mentioned before, a scattering particle can change the state of polarization of the incident beam after it passes the particle. This phenomenon is called dichroism and is a consequence of the different values of attenuation rates for different polarization components of the incident light. A complete description of the extinction process requires the introduction of the so-called extinction matrix. In order to derive the expression of the extinction matrix we consider the case of the forward-scattering direction, = 6fc, and define the coherency vector of the total field E = Eg + E. by... 

Figure 1 Schematic representation of a time-resolved coherent Raman experiment, (a) The excitation of the vibrational level is accomplished by a two-photon process the laser (L) and Stokes (S) photons are represented by vertical arrows. The wave vectors of the two pump fields determine the wave vector of the coherent excitation, kv. (b) At a later time the coherent probing process involving again two photons takes place the probe pulse and the anti-Stokes scattering are denoted by subscripts P and A, respectively. The scattering signal emitted under phase-matching conditions is a measure of the coherent excitation at the probing time, (c) Four-photon interaction scheme for the generation of coherent anti-Stokes Raman scattering of the vibrational transition.

Momentum also plays a role in ordinary spontaneous Raman spectroscopy. When the pump radiation at 532 nm is passed through a sample, the aE term of Eq. (2) produces scattering and, for the first Stokes case shown in Fig. la, the frequency is i si = where is the Raman-active vibration excited in the sample. It should be noted that there is an exchange between the radiation field and molecule not only of energy but also of momentum, represented by the vector ky. The direction and magnitude of k, are determined by the photon-scattering direction, which is random for this spontaneous event. The result is scattering in all directions so that there is no coherent addition of photon amplitudes, as expressed in the summation /(i si) = C8q The net intensity from this inco-... 

Other, related coherent Raman effects are also represented in Figure 5, such as the case (C) where the signal beam is detected at the Stokes frequency. The Raman-induced Kerr effect (B) may be interpreted as the quadratic influence of an electric field of frequency CO2 on the elastic scattering of radiation at a frequency or vice versa. In this case the phasematching (or wave-vector-matching) condition is fulfilled for any angle between beams 1 and 2, while in cases (A) and (C) it may only be met for certain angles of the beams with respect to each other. 

A = amplitude, d = thickness of crystalline material, D = degeneracy factors, E, Ej, = energy levels. Eg = cohesive electric field intensity, E = laser field amplitude, /j = intensity of the fundamental, = vector sum of the wave vectors, = coherence length, L = interaction length, P = polarization induced by an incident electric field, T = carrier period, = group velocity, a = linear absorption coefficient, = dipolar susceptibility of order ,A = grating period, ft) =fre-quency, tOj = scattered wave frequency, h = Planck s constant/2jt. 

However, there is a major difficulty in carrying out this inversion process. The electric field vector E h,k/) of the scattered x-rays is directly proportional to Sc(hM> at the present there is no method for directly measuring this E-field. For x-rays, we are limited to the intensity I as the observable, which is given by I E / The E is a complex function involving both amplitude and phase, but the phase irrformation is lost unless one uses a coherent source to record the phase information as is done in holography. Since we do not yet have x-ray lasers suitable for this purpose, this is known as the phase problem in crystallography (see McPherson, 1999). 

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