Babinet's principle
The Babinet's principle ( also Babinet's theorem) is a set of optics and states that the diffraction patterns of two mutually complementary aperture (eg, slit and a wire of the same thickness) outside the range in which the geometrical- optical imaging falls (the function without diffraction effects), are the same. For example, the diffraction pattern of a single slit hardly differs from that of a wire and a circular disk not from the a hole in the size of the circle.
The Babinet's principle applies to both Fresnel and Fraunhofer diffraction.
The name goes back to the French physicist Jacques Babinet (1794-1872), who established the theorem in 1837.
Explanation and Application
In the simple picture of geometrical optics, light rays travel from a straight line. In fact, it may occur due to diffraction for the deflection of light, for example, when the light passes through a shutter. At the edges of the aperture, the light waves are diffracted and interference between the diffracted light waves leads to diffraction phenomena, ie, the image of the aperture on a screen differs from that from what you would expect for a purely geometric- optical beam path. The image on a screen, which would (without diffraction effects) arise in a purely geometric- optical beam path shall be referred to hereinafter as " geometric- optical imaging ".
The established principle of Babinet 1837 now states that mutually complementary aperture - ie aperture, at which openings and opaque areas are reversed - outside the range that the geometric- optical imaging would occupy, produce the same diffraction phenomena.
Complementary panels are, for example, a gap, and a wire thickness as well as the gap width, or a circular aperture and a circular plate of the same diameter. The Babinet's principle therefore makes it possible to trace the diffraction by an opaque obstacle on the same at an opening outline ( see corona - where the diffraction is returned to the water droplets of clouds on the diffraction at a circular aperture ).
Derivation
If a beam of light onto a screen, it generates is a bright area. If there is no obstacle between the light source and the screen, the beam propagates in a straight line - so unbowed - from. The bright area on the screen therefore corresponds to the geometric- optical imaging (pictured right; above). The distribution of brightness and darkness on the screen corresponds to a distribution of the amplitude of the light wave. The amplitude will be referred to here. In the bright area, the amplitude is large, outside the bright area of the screen is dark, there is thus the amplitude of zero. Now two complementary screens shall be introduced successively into the beam (both panels together to completely cover the beam ), as shown in the figure on the right, center and bottom, presented without consideration of diffraction effects.
However, both the pinhole and the opaque obstacle course produce a diffraction pattern. For these diffraction patterns of pinhole and obstacle and falling onto the screen amplitude can be decomposed into a respective geometric and diffraction share:
Since a diaphragm each transmits the light, which the other cuts, both diffraction patterns must add up to the geometric- optical image of the light source with no bezel. The sum of the total amplitudes behind the pinhole and the obstacle must therefore be equal to:
The sum of the geometric proportions of the two amplitude distributions must be equal to the geometric proportion of the image without visor to be - so it must also be the same, since there is only the geometric proportion without cover:
If we now Eq. (1) and (2) in Eq. (3), we obtain:
With Eq. ( 4) yields:
And thus:
The shares of the amplitudes which account for diffraction, and are therefore the same for pinhole and obstacle, but have opposite sign - the amplitude distribution of the pinhole is therefore outside the imaging geometric- optical opposite to that of the complementary obstacle: Where the pinhole a negative amplitude generated, the obstacle results in a magnitude equal positive amplitude and vice versa. To superimposing two amplitudes that are equal but have opposite signs, the amplitude of the total wave zero, it comes to extinction. So we superimposed the amplitude distributions of both complementary aperture, one obtains ( outside the geometric- optical imaging ) extinction, and thus darkness, as one would expect for the case in which there is no aperture. The superposition of the amplitude after the two complementary diaphragm thus results in the amplitude distribution of the array without apertures.
For the perception of the diffraction patterns, the amplitude is now but not decisive, but the intensity. The intensity of light is proportional to the square of the amplitude - the intensity of the diffraction maxima so it is indifferent whether the amplitude is positive or negative, it just comes down to their absolute value of size. Creates the pinhole at one point a large positive amplitude, the complementary aperture ensures at the same location for the same magnitude large negative amplitude - and both have the same intensity result. For this reason, the diffraction pattern of a pinhole and obstacle outside the figure geometrical optics are the same.
Fraunhofer diffraction
The Babinet's theorem applies to both Fresnel and Fraunhofer diffraction. In the Fraunhofer diffraction, the light source is located in an infinitely large distance, ie, the light source must be sufficiently small and the distance between it and the observation screen sufficiently large ( or light source and screen must be " moved " by lenses in infinite distance ). In the case of an approximately point -shaped light source and the field of geometric- optical imaging is very small and plays little role in the diffraction pattern. To see in the Fraunhofer diffraction, the diffraction patterns of the complementary aperture in total ( almost) the same. ( Therefore, it is justified to replace in the formation of the corona significant diffraction of water droplets through a diffraction at circular aperture. )