Fraunhofer diffraction
The diffraction integral makes it possible to calculate in optics the diffraction of light through an arbitrarily shaped aperture. Specifically, this, starting from an incident wave and the aperture unit a function that describes the transmittance of the aperture, which is calculated at a point of the observation screen incident intensity of the light.
Two limiting cases of the diffraction integral are the approximations for the far field ( Fraunhofer diffraction ) and for the near field ( Fresnel diffraction ). See the relevant sections.
The accompanying sketch shows the experimental arrangement, consisting of a light source, an aperture at which the incident light is diffracted, and an observation screen on which the incident light intensity is investigated on. The shape and the characteristics of the panel thereby determine how the intensity distribution appears in the observation screen.
Does the aperture, for example the shape of a double slit, we obtain as the intensity distribution of known interference pattern. Further applications of the diffraction integral are, for example, the Airy disk and clothoid.
Kirchhoff diffraction integral
Called Kirchhoff diffraction integral, and Fresnel diffraction Kirchhoff's integral, is
In this case, denote
- The amplitude of the source,
- The magnitude of the wave vector,
- The wavelength of the light,
- An infinitesimal surface element of the aperture,
- The fader
- The slope factor, and finally
- The amplitude at the point on the observation screen.
Since the distances and are in most applications sufficiently perpendicular to the aperture, the slope factor in these cases, equal to one set. In this case, and the angle between the respectively marked with lines and a solder to the diaphragm plane at the intersection of the lines.
The intensity at a point is calculated as the absolute square of
Fraunhofer and Fresnel diffraction
For the light paths and the geometric relationships apply ( see map)
Under the assumptions and the roots may be approximated by a Taylor expansion.
This approximation corresponds precisely to the case, that is, for these considerations can be approximated be set to 1 the slope factor. This is the diffraction integral
It can also be set in the denominator because of the approximation. The exponent contains the essential information for the interference phase and can not be simplified in this way. It follows
The approximation for the terms and explicitly designed to 2nd order results,
As well as
Expressed by the coordinates, and produces the
And
Fraunhofer approximation
The Fraunhofer approximation corresponds to a far-field approximation, which means that not only the aperture size as a small, but also the distance of the observation screen are assumed to be large. As diffraction integral results here essentially just the Fourier transform of the aperture function. That is why it is called as part of the Fraunhofer diffraction by the Fourier optics.
According to these assumptions, only terms are taken into account, which are linear and in which is
In this case, the diffraction integral simplifies to
If we define a new wave vector, we obtain for the integral
This is just the Fourier transform of the aperture function.
Fresnel approximation
The Fresnel approximation corresponds to a near-field approximation. Here also quadratic terms are taken into account in the exponent. The diffraction integral has no longer the simple form of a Fourier transform and is in general only be solved numerically.
Taking into account quadratic terms in and results
In this case, the diffraction integral is
Introduction of with and then gives the diffraction integral in near-field approximation
Derivation
From the source with amplitude in the spherical wave whose amplitude decreases inversely with the distance () occurs. Wavevector times distance is the phase displacement of the shaft in place, angular frequency times time, the phase shift at a time. Said shaft is defined by the phase at the location of at the moment:
At the point when the wave hits in the distance on the bezel. It is the intensity of the wave at the point.
According to Huygens' principle, the point is the starting point of an elementary wave of the secondary shaft.
The amplitude of is proportional to the source amplitude and diaphragm function. The aperture function indicates the permeability of the diaphragm. In the simplest case, when the shutter is open, and when the aperture is closed. is the infinitesimal surface element of the aperture at the point.
The secondary wave is generated at the point on the screen at the wave intensity. Is infinitesimal, since only the contribution of all other points is not considered on the diaphragm.
The time dependence can be neglected, since they already disappears later in calculations with intensities by the absolute square. Substituting we get:
From any point on the shield is a secondary wave goes out. The intensity in the point of observation is generated by the superposition of all individual contributions:
This equation already strongly reminiscent of the diffraction integral given above. The proportionality factor results (angle of inclination neglected):
- Optics