Non-collinear interaction of photons with orbital angular momentum

Abstract

We study the nonlinear interaction between two non-collinear light beams that carry orbital angular momentum (OAM). More specifically, two incident beams interact at an angle in a medium with a second order nonlinearity and thus generate a third, non-collinear beam at the second harmonic frequency that experiences a reduced conversion efficiency in comparison to that expected based on conventional phase-matching theory. This reduction scales with the input beam OAM and, differently from previous spiral bandwidth calculations, is due to a geometric effect whereby the input OAM is projected along the non-collinear interaction direction. The effect is relevant even at small interaction angles and is further complicated at large angles by a non-conservation of the total OAM in the nonlinear interaction. Experiments are performed under different conditions and are in excellent agreement with the theory. Our results have implications beyond the specific case studied here of second-harmonic generation, in particular for parametric down-conversion of photons or in general for phase-matched non-collinear interactions between beams with different OAM.

Figure 1. Theoretical plot of normalised second harmonic power as a function of the vortex beams winding number ℓ.

The curves are plotted according to the non-collinear interaction of a Gaussian and vortex beam using the phase-matching solutions given by Eq. 6 (black line) and Eq. 7 (red line). The angle of incidence was chosen to be θ = 5° and all other parameters are the same as in experiments.

Figure 2. Experimental layout for case (i), described in detail in the methods section.

Input pulses are split by a beamsplitter (S), the reflected component, with Gaussian profile (L = 0), passes a tuneable delay stage while the second traverses a fixed optical path in which the pulse's phase and amplitude are corrected (by a series of half-wave plates (HWP) and a spatial light modulator (SLM)) to impart OAM onto the beam (L). A spatial filter (an aperture (SF) in the fourier plane of a 1:1 telescope (T)) removes higher order OAM modes and the beams (L = 0 and L) are loosely focussed (F) and overlapped non-collinearly onto a BBO crystal. The non-collinear output is measured with a photodetector.

Figure 3

Spatial intensity profiles of the (a) spiral (ℓ = 20) and (b) Gaussian beams at the overlap position (within the BBO crystal) measured with a beam profiler. The spiral and Gaussian beams have diameters of ~0.5 mm and ~1.2 mm respectively. The size of the spiral beam is kept constant as the OAM number ℓ is varied. (c) a photograph of the second harmonic (SH) generated beams projected onto a surface shortly after the BBO crystal position, from left to right the spiral SH, the non-collinear SH and Gaussian SH.

Figure 4. Non-collinear second harmonic power as a function of the winding number ℓ of the OAM beam.

(a) A Gaussian and vortex beam are overlapped in a non-collinear geometry onto a BBO crystal (Case (i)). The angle of incidence between the two non-collinear beams was 6°. (b) Two vortex beams of opposing winding numbers ℓ and −ℓ respectively, are overlapped onto a BBO crystal (Case (ii)). The non-collinear second harmonic beam is detected. The data are fit using Eq. (7) in (a) and Eq. (9) in (b) using only the experimental conditions. The dashed blue line shows the dependence assuming collinear geometry according to Eq. (6). Note that the collinearly detected beams do not vary with winding number (only shown in (a)).