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

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 ( ) or cos ( ). 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. 1. Introduction Various aberrations in the focusing system are known to result in a wider, distorted focal spot with disturbed axial symmetry [1]. Such an effect is normally considered to be a negative factor. However, it has been shown [2, 3] that some types of wave aberrations enable the central focal spot size to be decreased, providing tight focusing. Note that while only primary (axisymmetric) aberrations were dealt with in [2], aberrations associated with vortex phase components on the basis of Zernike polynomials were also discussed in [3]. In Zernike polynomials, the radius dependence is polynomial and the angle dependence is trigonometric (periodic). Optical elements characterized by periodic angular changes were considered in [4, 5]. In [4], such an element was shown to form the zero central intensity, whereas [5] also looked into diffraction-free properties of the generated light beams. Based on the decomposition of a cosine angular dependent phase function in terms of angular harmonics, the transmission function was shown [5] to produce a diffraction pattern composed of light spots arranged on a circumference. The coaxial interference of two vortex beams with identical topological charges and opposite signs was shown to produce a similar result [6, 7]. At the same time, the odd-order aberrations, such as distortion and coma, have been known to appear in distribution patterns with odd symmetry [1, 8, 9]. In particular, the presence of coma ( ) results in distributions with the third-order symmetry [10], similar to the 2D Airy beams [11]. It was also shown that the product of three 1D Airy functions, rotated by the angle of 120° relative to each other