Focal shift and axial dispersion of binary pure-phase filters in focusing systems

The focal shift effect and axial dispersion property of binary pure-phase filters (BPFs) in focusing systems are described in the regime of the scalar Debye theory. By expanding the formula of the electric field in focal region into a summation of a polynomial series, the axial behavior and the general rule of the focal shift effect of BPFs are analytically studied. The small focal shift formula of BPFs is derived based on the second-order approximation, and its scope of validity is also discussed. Based on this small focal shift formula, the property of axial dispersion for a given BPF is also analytically discussed. Furthermore, numerical results of 2-zone and 3-zone BPFs with various normalized radii and phase shift values are also given for verifying these analytical results. At last, as an example, the application of BPFs with negative axial dispersion in compensation of chromatic aberration of a single lens in an ultrashort laser focusing system is presented. The numerical results show that the chromatic aberration induced by the single lens is compensated well nearly for the whole spectral bandwidth of the ultrashort laser. Therefore, this focal shift effect induced by BPFs should be of great interest for its potential applications in compensation of chromatic aberration and compact tunable focal modulation in some special cases.