Existing-opening constraint in Fresnel aperture optimization

If a screen already has an interval hole, additional removal cannot reduce that hole. A connected final aperture must contain it, imposing lower/upper endpoint constraints on its Cornu spiral chord. The unrestricted longest chord need not be feasible. For an existing interval $0\le w\le 1.76777$, constrained chord maximization gives approximately $w_{-}=-1.21578$, $w_{+}=2.34054$, with point-power ratio1.60257. The two aperture endpoints, rather than just a nominal Fresnel-zone width, determine the received field.