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, $t(x,y)$, in the pupil plane of the imaging system and an adjacent imaging lens, followed by a focal plane imaging sensor in the $(x\u2032,y\u2032)$ plane. A vortex phase element has a complex transmittance Display Formula
$tv(x,y)=exp(im\theta ),$(1)
where $m$ is a nonzero integer known as the topological charge. An axicon phase element has a complex transmittance of Display Formula$ta(x,y)=exp(ir/a),$(2)
where $a$ is a constant scaling length. In this report, we ignore any wavelength dependence of $m$ and $a$. The coordinates $(r,\theta )$ are the circular coordinates in the $(x,y)$ plane. The irradiance in the sensor plane may be expressed $I(x\u2032,y\u2032,\lambda )=IL(x\u2032,y\u2032,\lambda L)+Ib(x\u2032,y\u2032,\lambda b)$, where the contribution from the laser source $IL(x\u2032,y\u2032,\lambda L)$ and the background scene $Ib(x\u2032,y\u2032,\lambda b)$ are given by Display Formula$IL(x\u2032,y\u2032,\lambda L)=\alpha |Ug(x\u2032,y\u2032,\lambda L)*h(x\u2032,y\u2032,\lambda L)|2,$(3)
and Display Formula$Ib(x\u2032,y\u2032,\lambda b)=\beta [bg(x\u2032,y\u2032,\lambda b)*|h(x\u2032,y\u2032,\lambda b)|2],$(4)
where $\alpha $ and $\beta $ are the constants, $Ug(x\u2032,y\u2032,\lambda L)$ and $bg(x\u2032,y\u2032,\lambda b)$ are the geometrically imaged fields of the laser source and background scene, respectively, and $h(x\u2032,y\u2032,\lambda )$ is the complex PSF of the optical system given by the Fourier transform of the pupil function Display Formula$h(x\u2032,y\u2032,\lambda )=FT{A(x,y)t(x,y)}=\u222b\u2212\u221e\u221e\u222b\u2212\u221e\u221eA(x,y)t(x,y)\xd7exp[\u2212i2\pi \lambda f(x\u2032x+y\u2032y)]dx\u2009dy,$(5)
and the aperture function is assumed to be circular with a radius $R$; i.e., $A(x,y)=circ(r/R)$. Analytical closed form expressions of the PSF of the vortex and axicon phase masks can be found in Refs. ^{11}^{12}–^{13}. For simplicity, we will make all calculations at the center wavelength, $\lambda $, and drop the use of $\lambda L$ and $\lambda b$.