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Use the forward-slope convention established in part a, with $x^{\prime }=Ax+B\theta$ and ${\theta }^{\prime }=Cx+D\theta$, where $AD-BC=1$. Assume $B\ne 0$. Solve for the endpoint slopes:Fermat's principle makes the optical path stationary against variations at internal interfaces. Consequently only endpoint terms survive when its endpoints change. For optical path $l(x,x^{\prime })$, these areIntegrate the first equation with respect to $x$, then match the second and impose $l(0,0)=L_0$. The resulting ABCD optical path generating function isThus the geometrical kernel defined in the source would have the formThe printed minus sign is inconsistent with forward propagation under these conventions. A concrete counterexample is free space: $A=D=1$, $B=L>0$, $C=0$. Its true geometrical path iswhereas the printed expression predicts a path shorter than $L$ for nonzero transverse separation. For example, $L=1\mathrm{m}$, $x=0$, $x^{\prime }=1\mathrm{mm}$ gives $1\mathrm{m}+0.5\mu \mathrm{m}$, rather than the printed $1\mathrm{m}-0.5\mu \mathrm{m}$. Reversing the slope convention would also reverse the free-space $B$ entry and must be done consistently; it cannot justify the minus sign with $P(L)$ from part a.

The printed form can be recovered by explicitly using the opposite slope coordinate $\tilde{\theta }=-dx/dz$. Then $\tilde{B}=-B$, free space has $\tilde{B}=-L$, and the same physical path is $l=L_0-(A{x}^2+D{x}^{\prime 2}-2x{x}^{\prime })/(2\tilde{B})$. Thus the sign can be a convention choice, but the printed minus sign must not be combined with the positive-$B$ free-space ray transfer matrix used here.

For physical field propagation, the paraxial diffraction kernel retains this quadratic optical-path phase while using the leading spreading amplitude. In one transverse dimension the normalized linear canonical transform iswith a consistent square-root branch and $\lambda =2\pi /k$. The source's prefactor and $1/l$ kernel are a schematic one-dimensional version of the Fresnel-Kirchhoff diffraction integral, not this normalized general-system propagator. The quadratic denominator is normally replaced by its leading value when making the Fresnel approximation. When $B=0$, propagation is an imaging limit rather than an ordinary integral with division by $B$.

For a circular structure, cylindrical symmetry reduces the scalar Fresnel-Kirchhoff diffraction integral to concentric annuli. With plane-wave illumination, radius $a$ and distance $z$, the relevant Fresnel number is $N_F=a^2/(\lambda z)$. Successive Fresnel half-period zones have radii $r_j^2\simeq j\lambda z$: their areas are approximately equal while their phases alternate by $\pi$, causing neighbouring-zone cancellation.

For Fresnel diffraction by a circular aperture, the on-axis field follows directly from the radial integral,Odd zone counts reinforce on axis; even counts cancel in this ideal scalar model. An annular opening gives the difference of the corresponding two edge exponentials. A zone plate alternately blocks or phase-shifts zones so that otherwise cancelling contributions reinforce at a focus, with $r_j^2\simeq j\lambda f$ for its principal focal distance.

Babinet's principle gives $U_{\mathrm{disk}}=U_{\mathrm{free}}-U_{\mathrm{ap}}=U_{\mathrm{free}}{e}^{i\pi N_F}$ for Fresnel diffraction by an opaque disk. Thus its ideal on-axis intensity equals the unobstructed intensity, the Arago spot, even within the geometric shadow. This concerns a narrow axial feature, not the whole shadow. Finite source size, imperfect circularity, bandwidth and alignment reduce its visibility. When $N_F\ll 1$, the circular-aperture far-field pattern becomes the Airy disk intensity pattern $[2J_1(u)/u{]}^2$, with $u=ka\sin \theta$ and first minimum near $\theta =1.22\lambda /(2a)$.