We perform four-wave mixing (FWM) experiments using Laguerre-Gauss (LG) beams and give a description of the signal generation.

Usual FWM setup use non-collinear beams that are characterized by their wave-vectors **k**_{2} and **k**_{1}. The emitted signal is discriminated by its propagation direction given by the phase matching condition **k**_{s}=2**k**_{2}-**k**_{1}. This situation forbids a strong focusing which would need a large numerical aperture

The configuration we propose uses LG beams. They show a phase profile, which is associated to an orbital angular momentum (OAM) that is characterized through the quantized azimuthal number l. The FWM signal is discriminated using a new phase matching condition that writes l_{s}=2l_{2}-l_{1}. In particular, l_{2}= 1 and l_{1}=2 beams show a donut spot profile with a zero-intensity at the center, where the l_{s}=0 signal (which has a Gaussian beam profile) is emitted: it can thus be easily spatially filtered. It is then possible to use collinear beams that can thus be tighter focused. This method paves the way to nonlinear spectroscopy experiments performed with an improved spatial resolution that allows for a measurement of sample inhomogeneities.

We have performed* time-resolved experiments on CdTe quantum wells and compared the dynamics measured in a usual configurations with non-collinear Gaussians beams to that of the configuration involving collinear LG beams. The identical decay that is obtained shows that the OAM is transferred to and then conserved by the electronic excitations, allowing to measure their dynamics.

A general description of the interaction between LG modes or Hermite-Gauss (HG) modes can be also given. It involves the calculation of the spatial profile products as nonlinear processes are determined by the products of the incident electric fields. We exploit in particular the richness of the operatorial description of generalized Hemite-Laguerre-Gauss beams.

* D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, Phys. Rev. B 92, 115312 (2015)

*Figure caption: Beam profiles. Middle: l*_{b}=2 and l_{a}=1 incident beams.

Left, l_{out}=2l_{b} -l_{a}=3 and right, l_{out}=2l_{b} -l_{a}=0 FWM signals

Optical properties of nanostructures , Nonlinear nano-optics , Enhanced spectroscopy and sensing