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The Talbot effect is a
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
effect first observed in 1836 by
Henry Fox Talbot William Henry Fox Talbot FRS FRSE FRAS (; 11 February 180017 September 1877) was an English scientist, inventor, and photography pioneer who invented the salted paper and calotype processes, precursors to photographic processes of the later ...
. When a
plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...
is incident upon a periodic
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
, the image of the grating is repeated at regular distances away from the grating plane. The regular distance is called the Talbot length, and the repeated images are called self images or Talbot images. Furthermore, at half the Talbot length, a self-image also occurs, but phase-shifted by half a period (the physical meaning of this is that it is laterally shifted by half the width of the grating period). At smaller regular fractions of the Talbot length, sub-images can also be observed. At one quarter of the Talbot length, the self-image is halved in size, and appears with half the period of the grating (thus twice as many images are seen). At one eighth of the Talbot length, the period and size of the images is halved again, and so forth creating a
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
pattern of sub images with ever-decreasing size, often referred to as a Talbot carpet. Talbot cavities are used for coherent beam combination of laser sets.


Calculation of the Talbot length

Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
showed that the Talbot effect was a natural consequence of
Fresnel diffraction In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern ...
and that the Talbot length can be found by the following formula: :z_\text = \frac, where a is the period of the diffraction grating and \lambda is the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
of the light incident on the grating. However, if wavelength \lambda is comparable to grating period a, this expression may lead to errors in z_\text up to 100%. In this case the exact expression derived by Lord Rayleigh should be used: :z_\text = \frac.


Fresnel number of the finite size Talbot grating

The number of Fresnel zones N_\text that form first Talbot self-image of the grating with period p and transverse size N \cdot a is given by exact formula N_\text = N^2. This result is obtained via exact evaluation of Fresnel-Kirchhoff integral in the near field at distance z_\text = \frac.


The atomic Talbot effect

Due to the
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
wave nature of
particles In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
, diffraction effects have also been observed with
atoms Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, an ...
—effects which are similar to those in the case of light. Chapman ''et al.'' carried out an experiment in which a collimated beam of
sodium Sodium is a chemical element with the symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable iso ...
atoms was passed through two diffraction gratings (the second used as a mask) to observe the Talbot effect and measure the Talbot length. The beam had a mean velocity of corresponding to a
de Broglie wavelength Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
of \lambda_\text = . Their experiment was performed with 200 and gratings which yielded Talbot lengths of 4.7 and respectively. This showed that for an atomic beam of constant velocity, by using \lambda_\text, the atomic Talbot length can be found in the same manner.


Nonlinear Talbot effect

The nonlinear Talbot effect results from self-imaging of the generated periodic intensity pattern at the output surface of the periodically poled LiTaO3 crystal. Both integer and fractional nonlinear Talbot effects were investigated. In cubic nonlinear Schrödinger's equation i\frac + \frac \frac + , \psi, ^2 \psi = 0, nonlinear Talbot effect of
rogue waves Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
is observed numerically. The nonlinear Talbot effect was also realized in linear, nonlinear and highly nonlinear surface gravity water waves. In the experiment, the group observed that higher frequency periodic patterns at the fractional Talbot distance disappear. Further increase in the wave steepness lead to deviations from the established nonlinear theory, unlike in the periodic revival that occurs in the linear and nonlinear regime regime, in highly nonlinear regimes the wave crests exhibit self acceleration, followed by self deceleration at half the Talbot distance, thus completing a smooth transition of the periodic pulse train by half a period.


Applications of the optical Talbot effect

The optical Talbot effect can be used in imaging applications to overcome the diffraction limit (e.g. in structured illumination
fluorescence microscopy A fluorescence microscope is an optical microscope that uses fluorescence instead of, or in addition to, scattering, reflection, and attenuation or absorption, to study the properties of organic or inorganic substances. "Fluorescence microscop ...
). Moreover, its capacity to generate very fine patterns is also a powerful tool in Talbot
lithography Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German a ...
. In experimental fluid dynamics, the Talbot effect has been implemented in Talbot
interferometry Interferometry is a technique which uses the ''interference'' of superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, fiber opt ...
to measure displacements and temperature, and deployed with
laser-induced fluorescence Laser-induced fluorescence (LIF) or laser-stimulated fluorescence (LSF) is a spectroscopic method in which an atom or molecule is excited to a higher energy level by the absorption of laser light followed by spontaneous emission of light. It was f ...
to reconstruct free surfaces in 3D, and measure velocity.


See also

*
Angle-sensitive pixel An angle-sensitive pixel (ASP) is a CMOS sensor with a sensitivity to incoming light that is sinusoidal in incident angle. Principles of operation ASPs are typically composed of two gratings (a diffraction grating and an analyzer grating) abo ...


References

{{Reflist


External links


Talbot's 1836 paper via Google BooksRayleigh's 1881 paper via Google BooksUndergraduate thesis by Rob Wild (PDF)Talbot effect observed over space-time for the first time
Diffraction