Fresnel diffraction
However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below.
Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as
The multiple Fresnel diffraction at closely spaced periodical ridges (ridged mirror) causes the specular reflection; this effect can be used for atomic mirrors.
[2] Some of the earliest work on what would become known as Fresnel diffraction was carried out by Francesco Maria Grimaldi in Italy in the 17th century.
He uses the Principle of Huygens to investigate, in classical terms, what transpires.
The wave front that proceeds from the slit and on to a detection screen some distance away very closely approximates a wave front originating across the area of the gap without regard to any minute interactions with the actual physical edge.
The result is that if the gap is very narrow only diffraction patterns with bright centers can occur.
If the gap is made progressively wider, then diffraction patterns with dark centers will alternate with diffraction patterns with bright centers.
As the gap becomes larger, the differentials between dark and light bands decrease until a diffraction effect can no longer be detected.
MacLaurin does not mention the possibility that the center of the series of diffraction rings produced when light is shone through a small hole may be black, but he does point to the inverse situation wherein the shadow produced by a small circular object can paradoxically have a bright center.
(p. 219) In his Optics,[4] Francis Weston Sears offers a mathematical approximation suggested by Fresnel that predicts the main features of diffraction patterns and uses only simple mathematics.
By considering the perpendicular distance from the hole in a barrier screen to a nearby detection screen along with the wavelength of the incident light, it is possible to compute a number of regions called half-period elements or Fresnel zones.
If the diameter of the circular hole in the screen is sufficient to expose two Fresnel zones, then the amplitude at the center is almost zero.
That means that a Fresnel diffraction pattern can have a dark center.
where The analytical solution of this integral quickly becomes impractically complex for all but the simplest diffraction geometries.
The main problem for solving the integral is the expression of r. First, we can simplify the algebra by introducing the substitution
[5] Let us substitute this expression in the argument of the exponential within the integral; the key to the Fresnel approximation is to assume that the third term is very small and can be ignored, and henceforth any higher orders.
In order to make this possible, it has to contribute to the variation of the exponential for an almost null term.
If this condition holds true for all values of x, x', y and y', then we can ignore the third term in the Taylor expression.
Thus, as a practical matter, the required inequality will always hold true as long as
The condition for validity is fairly weak, and it allows all length parameters to take comparable values, provided the aperture is small compared to the path length.
This is valid in particular if we are interested in the behaviour of the field only in a small area close to the origin, where the values of x and y are much smaller than z.
This is the Fresnel diffraction integral; it means that, if the Fresnel approximation is valid, the propagating field is a spherical wave, originating at the aperture and moving along z.
The integral modulates the amplitude and phase of the spherical wave.
Analytical solution of this expression is still only possible in rare cases.
Unlike Fraunhofer diffraction, Fresnel diffraction accounts for the curvature of the wavefront, in order to correctly calculate the relative phase of interfering waves.
The integral can be expressed in other ways in order to calculate it using some mathematical properties.
then we can express the integral in terms of the two-dimensional Fourier transform.
That is, first multiply the field to be propagated by a complex exponential, calculate its two-dimensional Fourier transform, replace
From the point of view of the linear canonical transformation, Fresnel diffraction can be seen as a shear in the time–frequency domain, corresponding to how the Fourier transform is a rotation in the time–frequency domain.
