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14) is if the third-order time-dependent response is known then the cascaded signal can be completely simulated (Fig. 19). Generally speaking the third-order Raman response for simple liquids is well known and can be (3) fit to analytical functions which can then be used in place of Rcas . To properly simulate the various possible tensor elements it is necessary to have the complete set of third-order polarization tensor elements. Third-order spectroscopy is represented by a fourth-rank polarization tensor of which only four elements are necessary to describe the third-order polarization in the sample [78].

The proposed response (5) measured is R60◦ 60◦ 11−60◦ −60◦ where the polarizations are 60◦ counterclockwise for the first pump pair, 0◦ for the second pump pair, and 60◦ clockwise for the probe/analyzer. We have referred to this polarization setting as the “Dutch Cross” configuration in honor of its origin. This configuration provides essentially complete elimination of the cascades by taking advantage of the sign (3) (3) change between the R1111 and R1122 tensor elements. The different polarization angles effectively lead to cancellation of the third-order contribution.

In the case shown, diffraction would only involve approximately three fringes in the fifth-order pattern, yielding poor diffraction, whereas the simple-period third-order polarization components would have many fringes. The result is that the inherent contrast of the phase matching geometry would be significantly reduced experiments [21,74] but the fifth-order phase-matching geometry is much more complicated and requires a custom diffractive optical element (DOE) [11]. To generate the desired beam pattern for proper phase matching and high contrast against lower order cascades (Fig.

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