Babinet's principle

The Babinet'sche principle (also Babinet'sches theorem ) is a set of the optics , and states that the diffraction pattern of two mutually complementary aperture (for example, slit diaphragm and a wire of the same thickness) outside the region in which the geometric-optical imaging falls ( i.e. the image without diffraction effects) are the same. For example, the diffraction pattern of a single slit hardly differs from that of a wire and that of a circular disk does not differ from that of a hole in the size of the circle.

The Babinet principle applies to both Fresnel and Fraunhofer diffraction .

The name goes back to the French physicist Jacques Babinet (1794–1872), who set up the theorem in 1837.

Explanation and application

In the simple picture of geometrical optics , rays of light spread in a straight line. In fact, however, diffraction can cause the light to be deflected, for example if the light passes through a diaphragm . The light waves are diffracted at the edges of the diaphragm and interference between the diffracted light waves leads to diffraction phenomena, ie the image of the diaphragm on a screen deviates from what one would expect for a purely geometrical-optical beam path. The image on a screen that would result from a purely geometrical-optical beam path (i.e. without diffraction effects) will be referred to below as “geometrical-optical imaging”.

The principle established by Babinet in 1837 now states that mutually complementary diaphragms - i.e. diaphragms in which openings and opaque areas are interchanged - produce the same diffraction phenomena outside the area that the geometric-optical image would occupy.

Complementary diaphragms are, for example, a gap and a wire that is just as thick as the gap is wide, or a circular diaphragm and a circular plate of the same diameter. Babinet's principle therefore allows the diffraction at an opaque obstacle to be traced back to that at an opening of the same outline (see corona - there the diffraction at the water droplets of the clouds is traced back to the diffraction at a circular aperture).

Application in electrodynamics

Analogous to optics, Babinet's principle can also be used for electrodynamics, so that useful relations result. For example, electromagnetic effects can be calculated through apertures in conductive planes. Knowledge of the electrical fields tangential to the surface is sufficient to calculate the transmitted fields in the aperture. For example, it can be used to determine the electromagnetic field of a hole in an infinitely extensive, conductive surface in which the current distribution cannot be calculated. According to Babinet's principle, the hole becomes a conductive surface. The surface currents then only have to be calculated over the hole and the inverse transformation finally provides the result for a hole in an infinitely large, conductive surface.

Derivation

Geometric-optical beam propagation (i.e. without diffraction) through complementary diaphragms (schematic)

If a ray of light falls on a screen, it creates a bright area there. If there is no obstacle between the light source and the screen, the beam spreads in a straight line - i.e. unbending. The light area on the screen therefore corresponds to the geometrical-optical image (figure on the right; above). The distribution of lightness and darkness on the screen corresponds to a distribution of the amplitude of the light wave. The amplitude should be referred to here with . In the light area the amplitude is large, outside the light area the screen is dark, there the amplitude is zero. Now two complementary diaphragms are to be introduced into the beam one after the other (both diaphragms together should completely cover the beam), as shown in the figure on the right, center and below, without taking diffraction effects into account. ${\ displaystyle E_ {0} (x)}$

Both the pinhole and the opaque obstacle naturally produce a diffraction pattern. For these diffraction patterns from pinhole and obstacle, the amplitude falling on the screen can be broken down into a geometric and a diffraction component:

(1)

{\ displaystyle E _ {\ text {loch}} (x) = E _ {\ text {lochgeo}} (x) + E _ {\ text {lochbeug}} (x)}

(2)

{\ displaystyle E _ {\ text {hind}} (x) = E _ {\ text {hindgeo}} (x) + E _ {\ text {hindbeug}} (x)}

{\ displaystyle E (x)}

= Amplitude distribution on the screen; = Coordinate along the screen; hole = pinhole; hind = obstacle complementary to the pinhole; geo = geometric part; diff = diffraction fraction

{\ displaystyle x \,}

Since one aperture allows the light to pass through which the other cuts out, both diffraction images must together produce the geometrical-optical image of the light source without aperture. The sum of the total amplitudes behind the pinhole and the obstacle must therefore be the same : ${\ displaystyle E _ {\ text {hole}} (x) + E _ {\ text {hind}} (x)}$ ${\ displaystyle E_ {0} (x)}$

(3)

{\ displaystyle E _ {\ text {hole}} (x) + E _ {\ text {hind}} (x) = E_ {0} (x)}

The sum of the geometric components of both amplitude distributions must be the same as the geometric component of the image without a diaphragm - it must therefore also be the same , since there is only the geometric component without a diaphragm: ${\ displaystyle E _ {\ text {lochgeo}} (x) + E _ {\ text {hindgeo}} (x)}$ ${\ displaystyle E_ {0} (x)}$

(4)

{\ displaystyle E _ {\ text {lochgeo}} (x) + E _ {\ text {hindgeo}} (x) = E_ {0} (x)}

If one now sets Eq. (1) and (2) in Eq. (3) one gets:

(5)

{\ displaystyle E _ {\ text {loch}} (x) + E _ {\ text {hind}} (x) = E _ {\ text {lochgeo}} (x) + E _ {\ text {lochbeug}} (x) + E _ {\ text {hindgeo}} (x) + E _ {\ text {hindbeug}} (x) = E_ {0} (x)}

With Eq. (4) results in:

(6)

{\ displaystyle E_ {0} (x) + E _ {\ text {lochbeug}} (x) + E _ {\ text {hindbeug}} (x) = E_ {0} (x)}

and thus:

(7)

{\ displaystyle E _ {\ text {lochbeug}} = - E _ {\ text {hindbeug}}}

The proportions of the amplitudes that are attributable to the diffraction are therefore the same for the pinhole and the obstacle, but have opposite signs - the amplitude distribution of the pinhole is therefore opposite to that of the complementary obstacle outside the geometrical-optical image: where the pinhole has a negative amplitude generated, the obstacle results in a positive amplitude of the same magnitude and vice versa. If two amplitudes that are equal but have opposite signs overlap, the amplitude of the overall wave is zero, and cancellation occurs. If one superimposes the amplitude distributions of both complementary diaphragms, one obtains (outside of the geometrical-optical image) extinction, and thus darkness, as one would expect in the case where there are no diaphragms at all. The superposition of the amplitudes behind the two complementary diaphragms thus results in the amplitude distribution of the arrangement without diaphragms.

For the perception of the diffraction images, however, it is not the amplitude that is decisive, but the intensity . The intensity of the light is proportional to the square of the amplitude - for the intensity of the diffraction maxima it does not matter whether the amplitude is positive or negative, it only depends on its magnitude. If the pinhole generates a large positive amplitude at one point, the complementary diaphragm at the same point ensures a negative amplitude of the same magnitude - and both result in the same intensity. For this reason, the diffraction patterns of the pinhole and the obstacle are the same outside of the geometrical-optical image.

Fraunhofer diffraction

Diffraction patterns of a pin and a gap between two razor blades, the width of which was generated with the help of the pin. It was illuminated with a red laser pointer. Because of the inaccuracy with which the width of the gap and needle could (not) be brought into agreement, the positions of the maxima differ from one another.

Babinet's theorem applies to both Fresnel and Fraunhofer diffraction . With Fraunhofer diffraction, the light source is at an infinite distance, ie the light source must be sufficiently small and the distance between it and the observation screen large enough (or the light source and screen must be "moved" to an infinite distance by lenses). In the case of an approximately point-shaped light source, the area of the geometrical-optical image is also very small and hardly plays a role in the diffraction image. With Fraunhofer diffraction, the diffraction images of the complementary diaphragms look (almost) the same overall. (It is therefore justified to replace the diffraction on water droplets, which is decisive in the formation of the corona, with diffraction on circular apertures.)

literature

Bergmann / Schaefer: Textbook of Experimental Physics, Volume 3: Optics. 10th edition, 2004, Walter de Gruyter, Berlin, p. 368

Web links

Individual evidence

↑ Leugner, Dietmar .: Calculation of the electromagnetic coupling through apertures with analytical methods and the moment method . As Ms. gedr. VDI-Verl, Düsseldorf 2004, ISBN 3-18-336021-7 .

[1] Leugner, Dietmar .: Calculation of the electromagnetic coupling through apertures with analytical methods and the moment method . As Ms. gedr. VDI-Verl, Düsseldorf 2004, ISBN 3-18-336021-7 .