Nonlinear optical susceptibility described with a spherical formalism applied to coherent anti-Stokes Raman scattering

We describe coherent Raman scattering in a complete spherical formalism allowing a better understanding of the coherent Raman process with respect to its symmetry properties, which is especially helpful in polarized coherent Raman microscopy. We describe how to build the coherent Raman tensor from spontaneous Raman tensor for crystalline and disordered media. We introduce a distribution function for molecular bonds and show how this distribution function results in a new macroscopic symmetry which can be very different from the symmetry of vibrational modes. Finally, we explicitly show polarization configurations for coherent anti-Stokes Raman scattering to probe specific vibration symmetries in crystalline samples and lipid layers.

Figure 1

General process for the passage from microscopic tensor to macroscopic tensor with the spherical formalism.

Figure 2

Spherical tensors generated with three different circular polarization schemes according to Eq. (12). $m_F=0$ is read out when all the fields have the same circular polarization and the order $m_F=2$ when the Stokes or the anti-Stokes beam have the opposite polarization handiness with respect to the pump beam. $m_F=4$ is read when the Stokes and the anti-Stokes polarizations have the opposite handiness with respect to the pump.

Figure 3

Symmetry-resolved CARS and spontaneous Raman spectrum of HT8 zeolite crystal. (a), (b) $T_{2g}$ and $E_g$ resonance in the $O-H$ bending mode region. (c) $T_{2g}$ and $A_{1g}$ in the $O-H$ stretching mode region. The order 0 is the black line, order 2 the dashed green line, and order 4 the blue (gray) line.

Figure 4

Symmetry-resolved CARS of multilamellar vesicles in the carbon-carbon bending region. (a) Order 0 (black upper line), 2 (green middle line), and 4 (blue lower line) of MLVs with $0\%$ of cholesterol. (b) Order 0 (black upper line), 2 (green middle line), and 4 (blue lower line) of MLVs with $5\%$ of cholesterol.