Diffraction

Single-slit diffraction

An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit. Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit, the interference effects can be calculated. If the incident light is a single wavelength and coherent, these sources all have the same phase. Light in the space downstream of the slit is made up of contributions from each of these point sources. If the relative phases of contributions from each point source vary due to differences in the path lengths, the resulting intensity will vary. Imagining the slit as the y axis with the z across the slit and x pointing downstream, intensity minima and maxima can be seen along z for large values of x.^[1]: 444

For points very close to ${\displaystyle x}$ all the point sources are in phase. This is the undiffracted beam, forming a maximum. Moving away from the ${\displaystyle x}$ axis, the path length from the point sources in the center and those on the edges of the slit differ. When the path difference equals ${\displaystyle \lambda /2}$, the central sources cancel the edge sources in destructive interference. For angles off the ${\displaystyle x}$ axis of ${\displaystyle \theta }$, the path difference is approximately ${\displaystyle {\frac {d\sin(\theta )}{2}}}$ so that the minimum intensity occurs at an angle ${\displaystyle \theta _{\text{min}}}$ given by $${\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}$$ where ${\displaystyle d}$ is the width of the slit and ${\displaystyle \lambda }$ is the wavelength of the light.^[1]: 444

The entire intensity profile can be calculated using the Fraunhofer diffraction equation as

$${\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}$$ where ${\displaystyle I(\theta )}$ is the intensity at a given angle, ${\displaystyle I_{0}}$ is the intensity at the central maximum (${\displaystyle \theta =0}$), which is also a normalization factor of the intensity profile that can be determined by an integration from ${\textstyle \theta =-{\frac {\pi }{2}}}$ to ${\textstyle \theta ={\frac {\pi }{2}}}$ and conservation of energy, and ${\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}$, which is the unnormalized sinc function.^[1]: 443

This analysis applies only to the far field (Fraunhofer diffraction), that is, at a distance much larger than the width of the slit.^[1]: 443

Circular aperture

The far-field diffraction pattern of a plane wave incident on a circular aperture is known as the Airy disk. The Airy disk has the following intensity distribution as a function of angle θ:^[1]: 461 $${\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}$$ where ${\displaystyle a}$ is the radius of the circular aperture, ${\displaystyle k}$ is equal to ${\displaystyle 2\pi /\lambda }$ and ${\displaystyle J_{1}}$ is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams. For aperture diameters close to the wavelength of the light, the Airy disk begins to act like a point source with very large divergence of diffracted beams.^[1]: 461

Babinet's principle

An opaque body and a hole of the same size and shape as the opaque body are called complementary apertures: as diffraction apertures they sum to a completely open space. For example, a screen with a circular hole is complementary to a circular disk the same size as the hole. The optical effect ${\displaystyle E}$ of these apertures add, $${\displaystyle E_{\textrm {opaque}}+E_{\textrm {hole}}=E_{0}}$$ giving the optical effect, ${\displaystyle E_{0}}$ of no obstacle. This is Babinet's principle; it works well in the Fraunhofer diffraction limit. When ${\displaystyle E_{0}\approx 0,}$ ${\displaystyle E_{\textrm {opaque}}=-E_{\textrm {hole}}}$ the intensities in the two patterns are the same everywhere. By imaging a point source through either aperture, the same pattern results, except at the focal point.^{[1]: 500 [11]: §8.3}

Knife edge

The knife-edge effect or knife-edge diffraction is a truncation of a portion of the incident radiation that strikes a sharp well-defined obstacle, such as a mountain range or the wall of a building. Knife-edge diffraction is an outgrowth of the "half-plane problem", originally solved by Arnold Sommerfeld using a plane wave spectrum formulation. A generalization of the half-plane problem is the "wedge problem", solvable as a boundary value problem in cylindrical coordinates. The solution in cylindrical coordinates was then extended to the optical regime by Joseph B. Keller, who introduced the notion of diffraction coefficients through his geometrical theory of diffraction (GTD). In 1974, Prabhakar Pathak and Robert Kouyoumjian extended the (singular) Keller coefficients via the uniform theory of diffraction (UTD).^[33][34]

Gratings

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θ_m which are given by the grating equation^[35][36]$${\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}$$ where ${\displaystyle \theta _{i}}$ is the angle at which the light is incident, ${\displaystyle d}$ is the separation of grating elements, and ${\displaystyle m}$ is an integer which can be positive or negative.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different.

General case for far field

A more mathematical approach involves treating the problem as a summation over spherical waves derived from the relevant wave equation; see for instance Born and Wolf for details.^[11] The wave that emerges from a point source has an amplitude ${\displaystyle \psi }$ at location ${\displaystyle \mathbf {r} }$ that is given by the solution of the frequency domain wave equation for a point source (the Helmholtz equation), $${\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r} ),}$$ where ${\displaystyle \delta (\mathbf {r} )}$ is the 3-dimensional delta function. By direct substitution, the solution to this equation can be shown to be the scalar Green's function, which in the spherical coordinate system (and using the physics time convention ${\displaystyle e^{-i\omega t}}$) is $${\displaystyle \psi (r)={\frac {e^{ikr}}{4\pi r}}.}$$which is a spherical wave emanating from the origin, the mathematical form of Huygen"s wavelets in the Huygens-Fresnel appriach. This solution assumes that the delta function source is located at the origin. If the source is located at an arbitrary source point, denoted by the vector ${\displaystyle \mathbf {r} '}$ and the field point is located at the point ${\displaystyle \mathbf {r} }$, then we may represent the scalar Green's function (for arbitrary source location) as $${\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{4\pi |\mathbf {r} -\mathbf {r} '|}}.}$$

In the far field, where ${\displaystyle r}$ is large the Green's function simplifies to

$${\displaystyle \psi (\mathbf {r} |\mathbf {r} ')={\frac {e^{ikr}}{4\pi r}}e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}}$$The expression for the far (Fraunhofer region) wave then becomes $${\displaystyle \Psi (r)\propto {\frac {e^{ikr}}{4\pi r}}\iint \limits _{\mathrm {aperture} }\!\!E_{\mathrm {inc} }(x',y')e^{-ik(\mathbf {r} '\cdot \mathbf {\hat {r}} )}\,dx'\,dy'.}$$

with an electric field ${\displaystyle E_{\mathrm {inc} }(x,y)}$ incident on the aperture for the case of an electromagnetic wave. In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture. Huygens' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields (see Fourier optics). In the far field, where r is essentially constant, then the equation: $${\displaystyle \Psi =\int _{\mathrm {aperture} }{\frac {i}{r\lambda }}\Psi ^{\prime }e^{-ikr}\,d\mathrm {aperture} }$$is equivalent to doing a Fourier transform on the gaps in the barrier.^[37] Similar forms can be derived for matter and other types of waves.^[15] For instance, with electron diffraction the aperture would be replaced by the electrostatic potential, while for x-ray diffraction it would be the electron charge density.

Dynamical diffraction

In the cases discussed above it is implicitly assumed that the wave encounters some single barrier or obstruction and is then diffraction by it. In reality it may encounter a number of barriers along the direction that it is travelling. The wave diffraction by the first one it encounters can be diffraction by the next, and so forth. The case when only single diffraction occurs is called kinematical diffraction, the more general case is called dynamical diffraction. Dynamical diffraction is quite well developed for x-rays,^[20][21] and also for electrons.^[15][18][19] As discussed extensively in the existing literature and reviews the results with dynamical diffraction can be quite different from those when only single scattering is considered. It can also occur with light, one example being opals and photonic crystals.^[38]