Spatial superpositions of Gaussian beams

We explore an interferometric beam shaping technique that considers the coaxial superposition of two Gaussian beams. This technique is traditionally implemented in a Mach-Zehnder interferometer; however, to avoid phase shift drift due to vibrations and thermal effects we employ amplitude and phase modulation with a spatial light modulator (SLM) to achieve the beam shaping. We consider two Gaussian beams of equal but opposite curvature that possess the same phase and width incident on a focusing lens. At the plane of the lens we obtain a multi-ringed beam with a central intensity maximum which develops into a multi-ringed beam with a central null at the focal plane of the lens. The interesting feature of this beam is that it possesses two focal spots on either side of the focal plane of the lens. We investigate obstructing the beam at the focal plane of the lens and by carefully selecting the free parameters we obtain an unobstructed second focus while the equivalent Gaussian beam is sufficiently obstructed.