Ambiguity Function Representation of a Aberration Invariant System

The ambiguity function is a polar display of a 1-D optical transfer function (OTF) of an incoherent system as a function of misfocus.

Ambiguity functions for the standard systems.

Ambiguity functions for the aberration-invariant systems.

Ambiguity functions for the windowed aberration-invariant systems.

With the standard rectangular aperture system, notice that the width (vertical dimension) of this ambiguity function is very narrow. Since this ambiguity function is a polar display of the OTF, the standard system should be very sensitive to mis-focus.

However, notice that with the aberration-invariant cubic-phase system, the width (vertical dimension) of this ambiguity function is quite broad compared to that of the standard system. Hence, the cubic-phase system is expected to be much less sensitive to mis-focus aberration than the standard system.

The ambiguity function of a windowed aberration-invariant cubic-phase system is much smoother than that of the unwindowed cubic-phase system. The smoothness of this ambiguity function results in less change of the OTF as a function of mis-focus. Or, the sidelobe level of the combined optical/digital system can be smaller with a windowed cubic phase mask than with an unwindowed mask.