There has been emerging needs for the quantitative polarization analysis for the Coherent Anti-stokes Raman Spectroscopy and Coherent Anti-stokes Hyper-raman Spectroscopy, as the experimental studies with coherent anti-stokes raman spectroscopy and coherent anti-stokes hyper-raman spectroscopy for the interface and membrane studies being growing. Recently we have demonstrated that orientational analysis of linear and nonlinear spectroscopy from the ordered molecular system, such as molecular interfaces and films, can be carried out with the formulation of the orientational function in simple functional forms. Applications of such formulation for the second order spectroscopy, namely, the Second Harmonic Generation and Sum Frequency Generation Vibrational Spectroscopy, have helped to understand spectral and orientational details of the molecular interfaces and films. In order to employ this formulation for the higher order coherent nonlinear spectroscopy, the detailed expressions of the experimental observables and the macroscopic susceptibility/microscopic polarizability tensors for the third and fourth-order nonlinear spectroscopy for the interface or film is presented with the rotational symmetry. General expressions for the typical third and fourth order spectroscopy, such as the Third Harmonic Generation, the degenerated coherent anti-stokes raman spectroscopy, the Fourth Harmonic Generation and the degenerated coherent anti-stokes hyper-raman spectroscopy, are presented for their future applications. The advantages and limitations of the third and fourth order spectroscopic techniques are also discussed.

Topics
Polarizability, Nonlinear optics, Third-harmonic generation, Second harmonic generation, Fourth-order nonlinear spectroscopy, Sum frequency generation spectroscopy, Raman spectroscopy, Coherent anti Stokes Raman scattering
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