Analyzing the Symmetry Properties of a Distribution in the Focal Plane for a Focusing Element with Periodic Angle Dependence of Phase

Abstract

We analyze the symmetry properties of the focal plane distribution when light is focused with an element characterized by a periodic angular dependent phase, sin (mϕ) or cos (mϕ). The majority of wave aberrations can be described using the said phase function. The focal distribution is analytically shown to be a real function at odd values of m, which provides a simple technique for generating designed wave aberrations by means of binary diffractive optical elements. Such a possibility may prove useful in tight focusing, as the presence of definite wave aberrations allows the focal spot size to be decreased. The analytical computations are illustrated by the numerical simulation, which shows that by varying the radial parameters the focal spot configuration can be varied, whereas the central part symmetry is mainly determined by the parity of m: for even the symmetry order is 2m and for odd is m.

Diffraction patterns withmth order symmetry generated by sectional spiral phase plates

“…If we use the sectional SPP, we should take into account the periodicity of the complex transmission function m 2 . p DOEs having periodic trigonometric angular dependence were investigated in [35,36]. The diffraction patterns obtained in [36] are very similar to the intensity distributions in figure 3.…”

“…p DOEs having periodic trigonometric angular dependence were investigated in [35,36]. The diffraction patterns obtained in [36] are very similar to the intensity distributions in figure 3.…”

Generating asymmetric aberration laser beams with controlled intensity distribution

“…(18) The condition 0 < w 2 < 2σ 2 makes sure that the asymmetric effects are well aligned with an input Gaussian beam. It should be noted that the symmetry of trigonometrically modulated phase is decided by parity of m [54]. As sin(m(θ + π)) = ± sin(mθ), the diametrically opposite points on phase distribution will be in and out of phase for even and odd parity, respectively.…”