In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.
The equation was named in honour of Joseph von Fraunhofer although he was not actually involved in the development of the theory.
This article gives the equation in various mathematical forms, and provides detailed calculations of the Fraunhofer diffraction pattern for several different forms of diffracting apertures, specially for normally incident monochromatic plane wave. A qualitative discussion of Fraunhofer diffraction can be found elsewhere.
When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, and light and dark bands are often seen at the edge of the shadow – this effect is known as diffraction. The Kirchhoff diffraction equation provides an expression, derived from the wave equation, which describes the wave diffracted by an aperture; analytical solutions to this equation are not available for most configurations.
The Fraunhofer diffraction equation is an approximation which can be applied when the diffracted wave is observed in the far field, and also when a lens is used to focus the diffracted light; in many instances, a simple analytical solution is available to the Fraunhofer equation – several of these are derived below.
Diffraction geometry, showing aperture (or diffracting object) plane and image plane, with coordinate system.
If the aperture is in x'y' plane, with the origin in the aperture and is illuminated by a monochromatic wave, of wavelength λ, wavenumberk with complex amplitudeA(x ',y '), and the diffracted wave is observed in the x,y,z plane where l,m are the direction cosines of the point x,y with respect to the origin, the complex amplitude U(x,y) of the diffracted wave is given by the Fraunhofer diffraction equation as:
It can be seen from this equation that the form of the diffraction pattern depends only on the direction of viewing, so the diffraction pattern changes in size but not in form with change of viewing distance.
The Fraunhofer diffraction equation can be expressed in a variety of mathematically equivalent forms. For example:
It can be seen that the integral in the above equations is the Fourier transform of the aperture function evaluated at frequencies
where Â is the Fourier transform of A. The Fourier transform formulation can be very useful in solving diffraction problems.
Another form is:
where r and r' represent the observation point and a point in the aperture respectively, k0 and k represent the wave vectors of the disturbance at the aperture and of the diffracted waves respectively, and a0(r' ) represents the magnitude of the disturbance at the aperture.
The time dependent factor is omitted throughout the calculations, as it remains constant, and is averaged out when the intensity is calculated. The intensity at r is proportional to the amplitude times its complex conjugate
These derivations can be found in most standard optics books, in slightly different forms using varying notations. A reference is given for each of the systems modelled here. The Fourier transforms used can be found here.
Computer simulation of Fraunhofer diffraction by a rectangular aperture
When a slit of width W and height H is illuminated normally by a monochromaticplane wave of wavelength λ, the complex amplitude can be found using similar analyses to those in the previous section, applied over two independent dimensions as:
The intensity is given by
where θ and φ are the angles between the x and z axes and the y and z axes, respectively.
In practice, all slits are of finite length and will therefore produce diffraction on both directions. If the length of the slit is much greater than its width, then the spacing of the horizontal diffraction fringes will be much less than the spacing of the vertical fringes. If the illuminating beam does not illuminate the whole length of the slit, the spacing of the horizontal fringes is determined by the dimensions of the laser beam. Close examination of the two-slit pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious vertical fringes.
Intensity of a plane wave diffracted through an aperture with a Gaussian profile
An aperture with a Gaussian profile, for example, a photographic slide whose transmission has a Gaussian variation, so that the amplitude at a point in the aperture located at a distance r' from the origin is given by
This function is plotted on the right, and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings. This can be used in a process called apodization - the aperture is covered by a filter whose transmission varies as a Gaussian function, giving a diffraction pattern with no secondary rings.:
The pattern which occurs when light diffracted from two slits overlaps is of considerable interest in physics, firstly for its importance in establishing the wave theory of light through Young's interference experiment, and secondly because of its role as a thought experiment in double-slit experiment in quantum mechanics.
Assume we have two long slits illuminated by a plane wave of wavelength λ. The slits are in the z = 0 plane, parallel to the y axis, separated by a distance S and are symmetrical about the origin. The width of the slits is small compared with the wavelength.
Solution by integration
The incident light is diffracted by the slits into uniform spherical waves. The waves travelling in a given direction θ from the two slits have differing phases. The phase of the waves from the upper and lower slits relative to the origin is given by (2π/λ)(S/2)sin θ and -(2π/λ)(S/2)sin θ
The complex amplitude of the summed waves is given by:
Solution using Fourier transform
The aperture can be represented by the function:
It can be seen that the form of the intensity pattern is the product of the individual slit diffraction pattern, and the interference pattern which would be obtained with slits of negligible width. This is illustrated in the image at the right which shows single slit diffraction by a laser beam, and also the diffraction/interference pattern given by two identical slits.
The diagram shows the diffraction pattern for a grating with 20 slits, where the width of the slits is 1/5th of the slit separation. The size of the main diffracted peaks is modulated with the diffraction pattern of the individual slits.
The Fourier transform method above can be used to find the form of the diffraction for any periodic structure where the Fourier transform of the structure is known. Goodman uses this method to derive expressions for the diffraction pattern obtained with sinusoidal amplitude and phase modulation gratings. These are of particular interest in holography.
If the aperture is illuminated by a mono-chromatic plane wave incident in a direction (l0,m0, n0), the first version of the Fraunhofer equation above becomes:
The equations used to model each of the systems above are altered only by changes in the constants multiplying x and y, so the diffracted light patterns will have the form, except that they will now be centred around the direction of the incident plane wave.
In all of the above examples of Fraunhofer diffraction, the effect of increasing the wavelength of the illuminating light is to reduce the size of the diffraction structure, and conversely, when the wavelength is reduced, the size of the pattern increases. If the light is not mono-chromatic, i.e. it consists of a range of different wavelengths, each wavelength is diffracted into a pattern of a slightly different size to its neighbours. If the spread of wavelengths is significantly smaller than the mean wavelength, the individual patterns will vary very little in size, and so the basic diffraction will still appear with slightly reduced contrast. As the spread of wavelengths is increased, the number of "fringes" which can be observed is reduced.