Complex shaping of the depth of focus

In this manuscript an exact solution to the inverse problem of axial beam shaping along the focus of a convergent lens is found. This allows to extend, within the framework of the scalar theory of diffraction, the mathematical formalism of complex pupils to include axial phase modulation. Numerical simulations based on Fourier transform as well as convolution operations indicate that amplitude and phase modulation can be performed simultaneously. It is also shown that include or not phase modulation in the beam shaping process can increase its efficiency more than three times. In addition, an analytical expression for the Gouy phase that depends on the introduced phase modulation was also derived. It is expected that obtained results benefit many photonic applications involving the control and manipulation of light along the focal region