Diffraction processes affecting waves are amenable to quantitative description and analysis. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. Numerical approximations may be used, including the Fresnel and Fraunhofer approximationss.

General diffraction

Because diffraction is the result of addition of all waves (of given wavelength) along all unobstructed paths, the usual procedure is to consider the contribution of an infinitesimally small neighborhood around a certain path (this contribution is usually called a wavelet) and then integrate over all paths (= add all wavelets) from the source to the detector (or given point on a screen).

Thus in order to determine the pattern produced by diffraction, the phase and the amplitude of each of the wavelets is calculated. That is, at each point in space we must determine the distance to each of the simple sources on the incoming wavefront. If the distance to each of the simple sources differs by an integer number of wavelengths, all the wavelets will be in phase, resulting in constructive interference. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. Usually, it is sufficient to determine these minima and maxima to explain the observed diffraction effects.

The simplest descriptions of diffraction are those in which the situation can be reduced to a two-dimensional problem. For water waves, this is already the case, as water waves propagate only on the surface of the water. For light, we can often neglect one dimension if the diffracting object extends in that direction over a distance far greater than the wavelength. In the case of light shining through small circular holes we will have to take into account the full three-dimensional nature of the problem.

Several qualitative observations can be made of diffraction in general:

• The angular spacing of the features in the diffraction pattern is inversely proportional to the dimensions of the object causing the diffraction. In other words: the smaller the diffracting object, the wider the resulting diffraction pattern, and vice versa. (More precisely, this is true of the sines of the angles.)

• The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object.

• When the diffracting object has a periodic structure, for example in a diffraction grating, the features generally become sharper. The fourth figure, for example, shows a comparison of a double-slit pattern with a pattern formed by five slits, both sets of slits having the same spacing between the center of one slit and the next.

Approximations

The problem of calculating what a diffracted wave looks like, is the problem of determining the phase of each of the simple sources on the incoming wave front. It is mathematically easier to consider the case of far-field or Fraunhofer diffraction, where the point of observation is far from that of the diffracting obstruction, and as a result, involves less complex mathematics than the more general case of near-field or Fresnel diffraction away from the object. At some point on the screen the path length to one side of the object is given by the Pythagorean theorem

Depending on the size of the diffraction object, the distance to the object and the wavelength of the wave, the Fresnel approximation, the Fraunhofer approximation or neither approximation may be valid. As the distance between the measured point of diffraction and the obstruction point increases, the diffraction patterns or results predicted converge towards those of Fraunhofer diffraction, which is more often observed in nature due to the extremely small wavelength of visible light.

Multiple narrow slits

A simple quantitative description

Maxima in the intensity occur if this path length difference is an integer number of wavelengths.

where

• is an integer that labels the order of each maximum,

• is the wavelength,

• is the distance between the slits, and

• is the angle at which constructive interference occurs.

For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper, as can be seen in the image.

Mathematical description

is the phase of the wave at the slits at time t

. In the Fraunhofer limit

The sum has the form of a geometric sum

denotes the complex conjugate.

Single slit

As an example, an exact equation can now be derived for the intensity of the diffraction pattern as a function of angle in the case of single-slit diffraction.

A mathematical representation of Huygens' principle can be used to start an equation.

Consider a monochromatic complex plane of wavelength λ incident on a slit of width a.

If the slit lies in the x-y

Let be a point inside the slit over which it is being integrated. If .

The distance r

Assuming Fraunhofer diffraction. In other words, the distance to the target is much larger than the diffraction width on the target. By the binomial expansion rule, ignoring terms quadratic and higher, the quantity on the right can be estimated to be:

It can be seen that 1/r in front of the equation is non-oscillatory, i.e. its contribution to the magnitude of the intensity is small compared to our exponential factors. Therefore, we will lose little accuracy by approximating it as 1/z.

To make things cleaner, a placeholder C is used to denote constants in the equation. It is important to keep in mind that C can contain imaginary numbers, thus the wave function will be complex. However, at the end, the ψ will be bracketed, which will eliminate any imaginary components.

participates in this exponential and it is being integrated).

)

It can be noted through Euler's formula and its derivatives that

and from the geometry that

1. .

Therefore, we have

where the (unnormalized) sinc function.

of the diffracted waves at an angle θ

Multiple slits

Let us again start with the mathematical representation of Huygens' principle

General case for far field

The following section uses a more mathematical approach where treating the problem as a summation over the spherical waves mentioned earlier is derived from the relevant wave equation; see for instance Born and Wolf for details. that is given by the solution of the frequency domain wave equation for a point source (the Helmholtz equation is the 3-dimensional delta function. The delta function has only radial dependence, so the Laplace operator (a.k.a. scalar Laplacian) in the spherical coordinate system

(See del in cylindrical and spherical coordinates.) By direct substitution, the solution to this equation can be readily shown to be the scalar Green's function, which in the spherical coordinate system

, then we may represent the scalar Green's function

Therefore, if an electric field is incident on the aperture, the field produced by this aperture distribution is given by the surface integral

as can be seen in the adjacent figure.

the Fraunhofer region field of the planar aperture assumes the form of a Fourier transform

In the far-field / Fraunhofer region, this becomes the spatial Fourier transform of the aperture distribution. 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 is equivalent to doing a Fourier transform on the gaps in the barrier.

See also

• Diffraction grating

• Envelope (waves)

• Fourier analysis

• N-slit interferometer

• Radio telescopes

References