= Solution
<Fresnel diffraction> is interference during propagation when curvature of the optical path across an <aperture> matters. It provides a continuous account of fringes near geometric edges and their evolution into the far-field pattern. A screen that is many <aperture> widths away is not automatically in the <Fraunhofer diffraction> limit; <wavelength> also enters the comparison.
For plane-wave illumination of a transmission $h(\xi,\eta)$ at $z=0$, the <Huygens-Fresnel principle> in the scalar <paraxial approximation> gives
$$
U(X,Y,z)\simeq\frac{U_{\rm in}e^{ikz}}{i\lambda z}
\iint h(\xi,\eta)
\exp\!\left[\frac{i\pi}{\lambda z}
((X-\xi)^2+(Y-\eta)^2)\right]d\xi\,d\eta.
$$
It follows by expanding the propagation distance as $z+[(X-\xi)^2+(Y-\eta)^2]/(2z)$ while treating the amplitude prefactor as slowly varying. Small propagation angles justify dropping higher path terms. Finite coherent illumination, vector edge fields and very small <apertures> can require a more complete theory. The <Fresnel number> $N_F=a^2/(\lambda z)$, with a clearly declared <aperture> scale $a$, measures whether the <aperture>'s quadratic phase is significant. When it is negligible, the remaining linear phase produces the <aperture> <Fourier transform>.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Scalar Diffraction Optics, Fresnel Approximation and Fresnel and Fraunhofer Approximations, equations381–392]{ref}
Define the optical <Fresnel integrals>
$$
C(u)=\int_0^u\cos(\pi s^2/2)\,ds,\qquad
S(u)=\int_0^u\sin(\pi s^2/2)\,ds,\qquad
F(u)=C(u)+iS(u).
$$
Their limiting values are $C(\infty)=S(\infty)=1/2$, with odd continuation to negative arguments. The <Cornu spiral> plots $(C(u),S(u))$: the field contributed by a finite interval is proportional to the chord between its endpoints. Intensities require squared chord lengths; summing zone intensities would lose the cancellation mechanism.
For an opaque straight edge transmitting $\xi>0$ and all $\eta$, let $v=X\sqrt{2/(\lambda z)}$, so $X>0$ is the geometrically illuminated side. Integrating in the unrestricted transverse direction and using the half-line in the other gives
$$
\frac{U}{U_{\rm in}e^{ikz}}
=\frac{1-i}{2}\left[\frac{1+i}{2}+F(v)\right],
$$
hence
$$
\boxed{\frac{I(v)}{I_{\rm in}}=
\frac12\left[\left(\frac12+C(v)\right)^2+
\left(\frac12+S(v)\right)^2\right].}
$$
Far into the shadow, $I\to0$; far into the illuminated region, $I\to I_{\rm in}$. At the geometric shadow boundary, \b[$I(0)=I_{\rm in}/4$], not half the incident intensity: the field is halved there. On the illuminated side, alternating constructive and destructive contributions give bright and dark fringes whose contrast declines away from the edge. Their positions scale with $\sqrt{\lambda z}$ and are not equally spaced. A finite detector averages fine fringes and tends to recover the geometric-optics step. In a controlled experiment, translate a straight edge through a broad collimated beam, scan a small photodetector across the screen and normalize against an unobstructed reference; vary $z$ to test the square-root scaling. Source size, spectral bandwidth, edge roughness and detector <aperture> wash out contrast and must be distinguished from a failure of the propagation law.
For a rectangular <aperture> of full widths $a,b$, define
$$
u_\pm=\sqrt{\frac2{\lambda z}}\left(\pm\frac a2-X\right),\qquad
v_\pm=\sqrt{\frac2{\lambda z}}\left(\pm\frac b2-Y\right).
$$
The two transverse integrals separate:
$$
\boxed{\frac{U}{U_{\rm in}e^{ikz}}
=\frac1{2i}[F(u_+)-F(u_-)][F(v_+)-F(v_-)],}
$$
and
$$
\boxed{\frac I{I_{\rm in}}=
\frac14|F(u_+)-F(u_-)|^2|F(v_+)-F(v_-)|^2.}
$$
A long slit leaves only one varying Fresnel factor. For a square centred on axis, $a=b$ and $w=(a/2)\sqrt{2/(\lambda z)}$ give
$$
\frac{I(0,0)}{I_{\rm in}}
=4[C(w)^2+S(w)^2]^2.
$$
Opposite edge contributions overlap; changing <aperture> size or propagation distance changes their relative phase. Consequently a geometrically illuminated centre can brighten or darken, rather than remaining a fixed top-hat plateau. The original rectangular-aperture map shows fringes parallel to both edges and their crossing structure. Outside the geometric rectangle the <wave> also extends into the shadow.
At large Fresnel number an interior region approaches geometric illumination while edge fringes remain on a scale $\sqrt{\lambda z}$. At intermediate values the opposite-edge fringes interact across the whole <aperture> image. In the far field, the formula tends to the product of sinc-squared factors,
$$
I(X,Y)\propto
\operatorname{sinc}^2\!\left(\frac{\pi aX}{\lambda z}\right)
\operatorname{sinc}^2\!\left(\frac{\pi bY}{\lambda z}\right).
$$
The far-field lobe scale $\lambda z/a$ grows linearly with $z$, unlike the local near-edge scale. The transition is controlled by the neglected phase, not an abrupt physical boundary.
To test the main effect, record normalized profiles at several distances, calculate the <Fresnel integral> with the measured <aperture> dimensions and compare fringe positions before fitting a free intensity scale. Use the same transverse coordinate origin and check that the incident beam is sufficiently uniform. <Babinet's principle> relates complementary <aperture> and obstruction fields through $U_a+U_o=U_{\rm free}$; \b[their near-field intensities are not generally equal], because the unobstructed field remains present and interferes with the diffracted field. This is why a far-field obstruction argument cannot simply be carried over as equality of Fresnel intensities.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2016/paper-2-fresnel.png]
{title=Original Fresnel-integral calculations for a straight edge, slits, rectangular <aperture> and <Cornu spiral>}
{description=The plots use the stated scalar paraxial model and chosen dimensionless <aperture> parameters. The straight-edge boundary value is one quarter. The <aperture> panels show quadratic-phase propagation and the Cornu panel identifies field amplitudes as chords rather than summed intensities.}
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