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A scalar aperture approximation that sums the secondary waves reaching an observation point. For a point source with incident spherical amplitude $A{e}^{iks}/s$, one common forward-normal convention isHere $s$ and $r$ are source-to-aperture and aperture-to-observer distances and $K$ is an obliquity factor. The Fresnel diffraction approximation retains quadratic path phase; Fraunhofer diffraction additionally neglects the quadratic phase across the aperture. Kirchhoff boundary assumptions do not give an exact electromagnetic solution at a sharp edge.

The obliquity factor weights the directional contribution of an aperture element in a scalar diffraction integral. In a fixed observation direction in the Fraunhofer diffraction regime, its variation across a sufficiently small aperture can be neglected along with the variation of the geometric amplitude denominator. Its angular dependence can still matter when comparing widely separated observation directions.

For time dependence $e^{-i\omega t}$, a scalar Fresnel-Kirchhoff aperture formula for a point-source amplitude $A$ isHere $s=|S-Q|$, $r=|P-Q|$, and $K$ is the obliquity factor, approximately the mean of the incident and outgoing direction cosines measured with the forward aperture normal. The factors describe incident spherical spreading, outgoing spreading and accumulated phase. This is the scalar Kirchhoff aperture approximation; it does not supply an exact electromagnetic edge boundary solution.

In the Fresnel regime, expand path lengths to quadratic order in aperture coordinates while replacing slowly varying amplitude and obliquity factors by their axial values. In the Fraunhofer regime the quadratic aperture phase is negligible, leaving the aperture Fourier transform. For characteristic aperture size $D$, the latter requires $D^2/(\lambda R)\ll 1$ in the relevant finite-source geometry; a lens can supply the Fourier-plane configuration.

For the one-dimensional Cornu construction, put $S,O,P$ on the unobstructed straight ray, with the aperture plane perpendicular at $O$. An element is displaced transversely by $x$, with $|x|\ll a,b$, where $a=SO$ and $b=OP$. A long straight edge or slit makes the other transverse direction separable. Then

For knife-edge diffraction at the fence, the edge is $h=0.5\mathrm{m}$ above the direct ray, with $a=5\mathrm{m}$, $b=45\mathrm{m}$, $\lambda \simeq 0.100\mathrm{m}$, and $R=4.5\mathrm{m}$. The unblocked aperture extends upward from $h$, so its Cornu chord runs from $u_h=h\sqrt{2/(\lambda R)}=1.054$ to infinity. Using $C(\infty )=S(\infty )=1/2$ and normalizing by the full unobstructed chord givesThe approximate chord coordinates are $C=0.775$, $S=0.492$. This is the standard single-knife-edge Fresnel normalization.^[ref]

For geometric spreading in a moving-receiver diffraction measurement, set $b=d$ while retaining $a=5$ and $h=0.5$:Normalized transmission decreases toward the fence. Absolute received intensity initially increases, because the free-space spherical-wave intensity rises as source–receiver distance decreases. In this model it reaches a maximum near $d\simeq 2.3\mathrm{m}$, then falls. The formal paraxial small-$d$ behavior is $I\propto d/(5+d{)}^3$; extrapolation to $d\lesssim h$ is outside the small-angle approximation. The sketch distinguishes the absolute signal from its ratio to the unobstructed signal at each position.

Take the aperture in $z=0$ and the observation point $P=(0,0,b)$, with $b>0$. Under scalar diffraction theory, the Huygens-Fresnel principle sums complex amplitudes from area elements, rather than summing their intensities. Apart from a common normalization,Here $h$ is the complex aperture transmission, $b$ is the axial distance, and $\theta$ is the angle of the secondary ray to the aperture normal. The obliquity factor $K$ describes the direction dependence of the secondary contribution; one Kirchhoff convention is $K=(1+\cos \theta )/2$ for normal illumination. Other scalar boundary approximations have slightly different angular weights, all tending to unity near the axis.

Put $r^2=x^2+y^2$. The paraxial approximation gives $R\simeq b+r^2/(2b)$ in the phase and $1/R\simeq 1/b$ in the amplitude. Removing common $e^{ikb}/b$ yieldsAssumptions are uniform normally incident plane-wave illumination, coherent monochromatic propagation, aperture dimensions large enough for the scalar description, $r/b\ll 1$, and phase accuracy $k{r}^4/(8b^3)\ll 1$ over the relevant aperture. The last condition justifies dropping the quartic distance term; geometric smallness alone does not guarantee phase accuracy.

For a circular aperture, $h=1$ for $r\le a$ and zero elsewhere. Rotational symmetry and $s=r^2$ give $dxdy=rdrd\phi$, $rdr=ds/2$, andThe constant $\pi$ can be absorbed into the proportionality factor, producing the required one-dimensional integral.

At a focus a distance $f$ away, half-period-zone boundaries obeyWith slowly varying obliquity and illumination, each zone has almost the same area and its integrated phasor has the opposite sign to its neighbour. A zone plate blocks alternate zones, so the remaining phasors add instead of nearly cancelling. Choosing an open central zone, the first three open regions have outer boundaries $n=1,3,5$:These are outer radii, not zone widths; the second open region is the annulus between $r_2$ and $r_3$.

For the specified outer radius, $R^2/(\lambda f)=49$: there are approximately 49 half-period zones, of which 25 odd zones are open. Normalize to the field ${\psi }_0$ of the unobstructed plane wave. The normalized Fresnel aperture integral isFor any odd open annulus this gives $2$, so the whole plate gives $\psi /{\psi }_0\simeq 50$ andThe axial intensity gain of an amplitude zone plate grows as the square of the number of open zones. Treating the number of zones continuously gives $49^2\simeq 2.40\times 10^3$, the same accuracy as an estimate. This large local intensity gain is consistent with conservation of energy because light is concentrated into a small focal spot. It is not a power efficiency; the ideal first-order power efficiency of a binary amplitude zone plate is only $1/{\pi }^2$.^[ref]

“Without the zone plate” has here meant removing the finite plate and its outer stop, giving the unobstructed incident wave. If instead a clear circular aperture of radius $R$ is retained, its axial field is ${\psi }_{\mathrm{clear}}/{\psi }_0=1-e^{i49\pi }=2$, so the ratio to that finite-aperture reference is $2500/4=625$. The two reference experiments differ; neither ratio should be confused with total transmitted power.

For the normally incident plane wave, the incident complex amplitude and phase are uniform across the aperture. Treat the field using scalar diffraction, with the aperture represented by its prescribed amplitude transmittance $h(x,y)$. Let $R$ be the distance from the aperture centre to the observation point and let ${\ell }_x,{\ell }_y$ be the transverse direction cosines of that point.

Across an aperture of characteristic size $a$, the Fraunhofer diffraction assumptions are that $a/R\ll 1$, so $1/r\simeq 1/R$ and the obliquity factor is nearly constant, and that the quadratic contribution to the propagation phase is negligible:The remaining path-length dependence is linear:A coherent monochromatic wave and an aperture large enough relative to wavelength for the scalar aperture model complete the usual physical assumptions. The complex amplitude becomeswhere $C$ includes the common propagation phase, distance factor and nearly constant obliquity factor. For the central part of a planar screen in the paraxial approximation, ${\ell }_x\simeq X/R$ and ${\ell }_y\simeq Y/R$. Small observation angles are useful for that screen-coordinate formula; a general far-field Fourier transform can instead use the exact direction cosines. A lens can also produce this pattern in its focal plane without requiring a physically very distant screen.