Talbot effect

The Talbot effect is visible in near-field diffraction by a ( diffraction ) grating: The brightness distribution in certain Talbot distances behind a lattice corresponds exactly to the structure of the grid itself, the effect was in 1836 by William Henry Fox Talbot with a fine stream sunlight discovered.

Description

In the basic case (see illustration), a lattice of flat monochromatic wave (for example, an expanded laser beam) irradiated:

• If, an observation screen at a distance LTalbot (or an integral multiple thereof) behind the grid, the result is an image of the grid.
• In the distance LTalbot / 2 = d2 / λ (or an odd multiple thereof) shifted by d / 2 image is obtained ( in the figure, respectively indicated by black dots for the intensity maxima ).
• Between two self-images you can find more images of the grating with increased period (if the grid has sufficiently narrow columns). If in very narrow columns, the position of the observation screen changes continuously and plotted along the side of the image produced results in a Talbot carpet showing the structure of the self-images.
• Changes the direction of the incident light, then the picture changes as opposed to if it were a shadow of the grid.

For the Talbot distance LTalbot applies when illuminated with the wavelength λ:

The effect can be used for wavelengths of completely different magnitudes, because a change of λ by a factor of x can be compensated by a change in the grating period by only ( with unchanged Talbot distance ).

Of the simple expression is an approximation when the structure size ( the grating period ) is much larger than the wavelength. The correct term is also true for small structures is:

Demarcation

If the Talbot distance exceeded significantly under - or dominate other effects:

• Shorter distances: Objects from daily life (eg, ridges ), we often observe from dominant to short distances where the shadows.
• Longer distances: The result of Fraunhofer diffraction in which the light shines in each case as a single beam in certain directions. In the everyday world, corresponding to the viewing of a CD.

Applications

The Talbot effect will be applied in the research:

• Wavefront measurement: because the image moves when changing the direction of the incident light, so the wave fronts and propagation directions of light rays can be measured, comparable with a Hartmann- Shack sensor. To ensure that small refractive indices can be determined precisely. In the visible range can be for example by hot air ( candle) produced minimal refractions prove.
• X-ray: X-rays usually show a shadow ( " absorption " ), eg bone. However, X-rays are deflected by different refractive indices. By measuring the deflection of images can be generated, for example, Pose tissue detail.
• Matter waves: According to the de Broglie wavelength can matter a wavelength to be assigned. In beams of atoms or even massive organic molecules of the Talbot effect could be demonstrated:
• Other applications and literature, see for example

Variants

• Two-dimensional lattice: The effect occurs along both grid dimensions. If, for example, the grid of a checkerboard pattern arises this again in the Talbot distance.
• Kugelwelllen: If, for any laser beam, then differences in distance before adding interfering light beams and after the grid by growing the phase differences. Is the grid exactly between the light source and viewing screen, so the doubled phase differences must be compensated by greater distances, the Talbot distance is doubled.

• A surface light source: incandescent or focal point in X-ray tubes have a certain size. This size leads to superposition of stripe patterns, it is no longer recognizable stripe pattern ( more precisely: the size of the light source reduces the (spatial ) coherence of the light source too ). One placed near the light source, an additional grid ( with roll ' coherence grid ' ), so that overlap the frames of the individual grid columns constructive, of the Talbot effect occurs again.
• Polychromatic light source: with multi-colored light of the Talbot distance exactly true only for a certain wavelength. The wider the lattice columns, the better deviations are tolerated by the design distance. Talbot himself discovered the effect on the basis of a narrow beam of sunlight.

• Phase grating: In contrast to (amplitude ) gratings (lands opaque), the entire grid to be transparent, the webs delay the light it by half a wavelength. Here, the lattice spacing is greatly reduced to. Furthermore, the effect occurs only in the odd-numbered multiple of this distance. Especially for the 0 - times the distance in the phase grating is itself the brightness evenly distributed. Phase gratings are often used in X-rays.

Derivation

Narrow columns

Is considered the case of the monochromatic planar light wave ( wavelength). The grating period was, the distance to the screen was.

Preliminary 1st order Taylor expansion:

Comparing a direct light beam (which is perpendicular to the grating, k = 0) to the length a, the lattice column of the grid occurs (length, k = 1), above or below. Constructive interference occurs when the difference in length, for example, is:

Result:

• This constructive interference occurs at. ( The geometric derivation in the drawing shows this situation. )
• The extension of the path is the square of the grid column = 0, 1, 2, ...: . This interferes constructively in the light of all grid columns same point.

For other distances L, the situation is slightly different. Example: The image on the screen is offset, the beam passes through one of the two directly adjacent column has the length, the rays of the next- distant column lengths.

The path length difference between the column and is then

The differences between the path lengths are thus again multiples of and there is constructive interference.

Width columns (gap fraction, for example 50%)

( Still missing. / Goes with the Fourier series? )