Largest contiguous Fresnel aperture ($w_{\ast }\simeq 1.20938$)

The largest radius of the normalized Cornu spiral occurs on its first outer turn at $w_{\ast }\simeq 1.20938$, where $\mathrm{Re}[F(w{)}^{\ast }{F}^{\prime }(w)]=0$. Oddness makes the chord from $F(-w_{\ast })$ to $F(w_{\ast })$ attain twice that radius, while the triangle inequality bounds every other chord by the same length. The interval is therefore the strongest freely selectable connected aperture at that observation point, with normalized point power about1.80142. Signed physical limits follow from $x=w\sqrt{\lambda z/2}$. Existing openings can constrain this optimization.

A Fresnel diffraction aperture can increase the detected intensity at one selected observation point by removing contributions that would otherwise interfere destructively there. The increased local signal is a redistribution of transmitted radiant power, not an increase in total energy supplied by the incident beam.

For a passive absorbing screen under fixed illumination, closing part of an aperture cannot increase the total transmitted radiant power. Nevertheless the remaining complex amplitudes can have a larger sum at a particular point. Comparing the single-point response with a measurement integrated over the whole transmitted field distinguishes the largest contiguous Fresnel aperture optimization objective from an efficiency claim.

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.