Focal characteristics of partially coherent rectangular array beams focused by a bifocal lens system

The propagation characteristics of rectangular array beams through a bifocal lens system have been studied, including irradiance distribution and focal shift. By using Collins formula and diffraction integral, the analytical expressions for irradiance distributions of the array beams have been derived. The results show that the irradiance distributions exhibit diversified variations due to the different focal length in two directions, and the changes of the irradiance distributions for the correlated and uncorrelated superposition array beam are different. There are usually two maxima of the axial irradiance on propagation. The focal shifts demonstrate that the axial maximum irradiance near the larger focal distance is displaced toward the lens system when the Fresnel number is smaller. The focal shift results from the competition of the two irradiance maxima, which can lead to a permutation of the focal points. This work would be useful for the practical design and use of the bifocal lens in focusing array beams.