Third-order polarization

B1.3.2.4 TIME EVOLUTION OF THE THIRD ORDER POLARIZATION BY WAVE MIXING ENERGY LEVEL (WMEL) DIAGRAMS. THE RAMAN SPECTROSCOPIES CLASSIFIED... 

The general task is to trace the evolution of the third order polarization of the material created by each of the above 12 Raman field operators. For brevity, we choose to select only the subset of eight that is based on two colours only—a situation that is connnon to almost all of the Raman spectroscopies. Tliree-coloiir Raman studies are rather rare, but are most interesting, as demonstrated at both third and fifth order by the work in Wright s laboratory [21, 22, 23 and 24]- That work anticipates variations that include infrared resonances and the birth of doubly resonant vibrational spectroscopy (DOVE) and its two-dimensional Fourier transfomi representations analogous to 2D NMR [25]. 

Wliatever the deteetion teehnique, the window stage of the 4WM event must eonvert these evolved vibrational wavepaekets into the third order polarization field that oseillates at an ensemble distribution of optieal frequeneies. One must be alert to the possibility that the window event after doorway ehaimel B may involve resonanees from eleetronie state manifold e to some higher manifold, say r. Thus ehaimel B followed by an e (ket) or a (bra) event might be enlianeed by an e-to-r resonanee. However, it is nonnal to eonfine the... 

A typical noisy light based CRS experiment involves the splitting of a noisy beam (short autocorrelation time, broadband) into identical twin beams, B and B, tlnough the use of a Michelson interferometer. One ami of the interferometer is computer controlled to introduce a relative delay, x, between B and B. The twin beams exit the interferometer and are joined by a narrowband field, M, to produce the CRS-type third order polarization in the sample ([Pg.1209]

The focal helds set up a polarization in the material. In the case of CARS, we are interested in the polarization resulting from the combined action of the pump (of frequency co ) and Stokes (of frequency coj beams, which induce motions in the electron clouds that oscillate at frequency 2co - co, the anti-Stokes frequency. The ability of the material to oscillate at the anti-Stokes frequency when the pump and Stokes helds are present is given by the third-order nonlinear susceptibility The strength of the polarization is furthermore determined by the amplitude of the pump (E ) and Stokes (E ) driving helds. In the tensorial notation, where and I denote the polarization components of the nonlinear susceptibility, the third-order polarization in the polarization direction i is given as... 

In the next sections, we shall see that by manipulating the spatial as well as temporal phase of the third-order polarization at the anti-Stokes frequency the properties of the far-field CARS signal can be favorably influenced. We will first discuss phase manipulations in terms of temporal interference in Section III and then zoom into spatial phase manipulations in Section IV. 

The interference process in this collinear approach is, however, different from the interference realized by mixing the local oscillator and the CARS field on a beam splitter. Interference takes place in the sample, which, in the presence of multiple frequencies, mediates the transfer of energy between the beams that participate in the nonlinear process. The local oscillator mixes with the anti-Stokes polarization in the focal volume, and is thus coherently coupled with the pump and Stokes beams in the sample through the third-order polarization of the material. In other words, the material s polarization, and its ability to radiate, is directly controlled in this collinear interferometric scheme. Under these conditions, energy from the local oscillator may flow to the pump and Stokes fields, and vice versa. For instance, when the local oscillator field is rout of phase with the pump/Stokes-induced anti-Stokes polarization in the focal interaction volume, complete depletion of the local oscillator may occur. The energy of the local oscillator field is not redistributed in terms... 

In this case the analytic result for the low frequency asymptotics of the third order polarization operator (see Fig. 3.6) [35] is used... 

How this additional field will alter the magnitude of the various tensor elements and the form of the rotational anisotropy has been examined by Koos et al. [122]. X0) has the properties of a fourth-rank tensor such that the third order polarization... 

The third-order polarization energy can written as the sum of three distinct contributions,... 

Calculations of the third-order exchange contributions have not been performed thus far. The results reported in Ref. (81) for the total third-order exchange effect for the helium dimer suggest that they quench a large part of the third-order polarization contribution. 

The measured CARS signal Scoh is proportional to the time integral over the absolute value squared of the total third-order polarization, P = Piso + Paniso + Pnr> because of the slow intensity response of the detector ... 

The third-order polarization is obtained by convolution of the response functions with the electric fields of the three laser pulses ... 

In a homodyne detection scheme, such as in the stimulated photon echo experiments described in the next paragraph, the detector measures the t-integrated intensity of the square of the third-order polarization... 

Figure 3 (a) Time sequence of the (stimulated) three-pulse photon echo experiment. The times ti, t2, and t3 represent the time coordinates used in the response functions [Equations (7)—(12)] while r, T, and t measure the delay times with respect to the peak positions of the light pulses. For 5-shaped light pulses, both sets of times would be equivalent, (b) The so-called box configuration, (101) which allows the spatial separation of the third-order polarization generated in the —ki + k2 + kj and the +ki — k2 + k3 phase matched directions. 

In a self-heterodyne experiment, however, there is no independent control over the phase of the local oscillator field, so that the complete information on the complex third-order polarization of Equation (32) cannot be obtained. It is necessary to analyze in more detail the measurement process in order to determine the accessible information. In the actual experiment the spectrometer performs the Fourier transform of the generated third-order field of Equation (31) with respect to time coordinate t2, generating the field components of El3,(ti oy, ) given by ... 

Similarly to the phase matching condition, also the induced third-order polarization can be written in a general form for all the nonlinear coherent Raman processes discussed above. The induced polarization which radiates a signal field of angular frequency... 

In the presence of strong absorption at co and 3(o, the intensity of the third harmonic can be written in terms of the third order polarization as (19) ... 

The dominant macroscopic third-order polarization element in DRl is of the form... 

We must now once more return to the perturbation expansion of the molecular polarization and consider the third-order polarization in Eq. (36) from which we will identify a formula for the second-order hyperpolarizability in analogy with... 

We have used that [ "] = E and the fact that Wp M2, and 0)3 are dummy summation indices that run over both positive and negative frequencies. None of Eqs. (59)-(62) is symmetric in the tensor indices y, and 8. As pointed out in connection with Eq. (7), we normally choose our hyperpolarizability tensors to possess intrinsic symmetry, and it is clear that we can accomplish this without changing the polarization of the molecule by taking the average of the six terms generated by permuting pairs of the dummy indices (j3, Wj), (% M2), and (8, 0)3) we denote this operation with the symbol l/6J2 i,2,3> where tlie factor of one sixth is required to maintain the same value of the polarization. The third-order polarization in Eq. (36) can then be written as... 

The generated conjugated beam is proportional to tlie third-order polarization oscillating at frequency a> (Eq. 91). Since kf = -kf, and kp = -k, the backward-geometry is always phase-matched. 

Of course, the frequencies and wave vectors fulfil the phase-matching conditions. The third-order susceptibility Xijw is a fourth-rank tensor having a priori 81 elements. In an isotropic material, there remain 21 non-vanishing elements, among which only three are independent [69]. The simplest case consists in a unique incident plane wave, linearly polarized. Indeed, the third-order polarization vector is then parallel to the electric field and reduces to the sum of two propagating terms, one oscillating at the wave circular frequency co, and another at the circular frequency 3(o. The amplitudes of these two contributions write, respectively. 

The polarization induced in a molecule by n successive interactions with a field E t) is termed the nth order polarization. Each interaction involves the field coupling to a transition dipole p. The 2D IR involves a calculation of the third order polarization, rr pP (t)p, which requires a quantum dynamics derivation of the third order term, in the expansion of the density operator as a... 

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