Scanning lens design for minimum mean-square phase error

Microwave lenses of the constraining or binormal type having superior wide-angle scanning properties can be achieved by suitable choices of lens contours and of the manner in which the index of refraction varies from point to point across the aperture plane. The phase error (i.e., departure from a plane phase front at the exit surface of the lens) can be expressed in terms of three unknown functions: the equations of the two lens surfaces and of the index of refraction. The square of this error is integrated over the desired angle of scan and also over the aperture area of the lens. The three unknown functions which minimize the phase-error integral are then determined from the Euler-Lagrange equations of the calculus of variations. Graphs of phase errors for several typical lens designs are included.