Arbitrarily high focal depth with a quasioptimum real and positive transmittance apodizer

Focal depth is assessed by the average value of the square modulus of the slope associated with the complex amplitude along the optical axis. Then, the calculus of variations is used for identifying the optimum apodizer, characterized by a Strehl ratio vs defocus with high focal depth, for a specified light throughput. We show that a certain Lorentzian profile is a quasioptimum solution for the above requirements. This apodizer has real and positive transmittance, and it can be modified to achieve arbitrarily high focal depth. A closed formula relates focal depth to light throughput.