Focal shifts in focused beams with an elliptical diffracting screen

The approximate analytical formula of the focal shift has been derived for an elliptical diffracting screen that consists of a circular lens and an elliptical aperture. It is found that the focal shift of the beam focused by this elliptical diffraction screen depends not only on the product of the Fresnel number of the focusing geometry and the standard deviation of a mapped function, but also on the ellipticity (the ratio between the minor and the major axes) of the elliptical aperture. The focal shift increases as the decrease in the ellipticity of elliptical aperture. By using an approximate analytical formula and the diffraction integral formula, some numerical simulation comparisons are done. The presented analyses show that the actual analytical expression for the focal shift of a rotationally asymmetric diffraction screen cannot generally be obtained by using the azimuthal average of the screen amplitude transmittance.