Figure 1 shows the imaging chain with both the physical optical system and the image processing steps that allow the recovery of an incoherent background image in the presence of a coherent laser source. The optical system consists of a phase mask, , in the pupil plane of the imaging system and an adjacent imaging lens, followed by a focal plane imaging sensor in the plane. A vortex phase element has a complex transmittance Display Formula
(1)where is a nonzero integer known as the topological charge. An axicon phase element has a complex transmittance of Display Formula
(2)where is a constant scaling length. In this report, we ignore any wavelength dependence of and . The coordinates are the circular coordinates in the plane. The irradiance in the sensor plane may be expressed , where the contribution from the laser source and the background scene are given by Display Formula
(3)and Display Formula
(4)where and are the constants, and are the geometrically imaged fields of the laser source and background scene, respectively, and is the complex PSF of the optical system given by the Fourier transform of the pupil function Display Formula
(5)and the aperture function is assumed to be circular with a radius ; i.e., . Analytical closed form expressions of the PSF of the vortex and axicon phase masks can be found in Refs. 1112–13. For simplicity, we will make all calculations at the center wavelength, , and drop the use of and .