TheTalbot effect is adiffraction effect first observed in 1836 byHenry Fox Talbot.^[1] When aplane wave is incident upon a periodicdiffraction grating, the image of the grating is repeated at regular distances away from the grating plane. The regular distance is called theTalbot 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 afractal pattern of sub images with ever-decreasing size, often referred to as aTalbot carpet.^[2]Talbot cavities are used forcoherent beam combination of laser sets.

Lord Rayleigh showed that the Talbot effect was a natural consequence ofFresnel diffraction and that the Talbot length can be found by the following formula (page 204):^[3]

${\displaystyle z_{\text{T}}=\frac{\lambda }{1-\sqrt{1-\frac{{\lambda }^2}{a^2}}},}$

where${\displaystyle a}$ is the period of the diffraction grating and${\displaystyle \lambda }$ is thewavelength of the light incident on the grating. For${\displaystyle \lambda \ll a}$, the Talbot length is approximately given by:

${\displaystyle z_{\text{T}}\approx \frac{2a^2}{\lambda }.}$

The number of Fresnel zones${\displaystyle N_{\text{F}}}$ that form first Talbot self-image of the grating with period${\displaystyle p}$ and transverse size${\displaystyle N\cdot a}$ is given by exact formula${\displaystyle N_{\text{F}}=(N-1{)}^2}$.^[4] This result is obtained via exact evaluation of Fresnel-Kirchhoff integral in the near field at distance$z_{\text{T}}=\frac{2a^2}{\lambda }$.^[5]

Due to thequantum mechanical wave nature ofparticles, diffraction effects have also been observed withatoms—effects which are similar to those in the case of light. Chapmanet al. carried out an experiment in which a collimated beam ofsodium atoms was passed through two diffraction gratings (the second used as a mask) to observe the Talbot effect and measure the Talbot length.^[6] The beam had a mean velocity of1000 m/s corresponding to ade Broglie wavelength of${\displaystyle {\lambda }_{\text{dB}}}$ =0.017nm. Their experiment was performed with 200 and300 nm gratings which yielded Talbot lengths of 4.7 and10.6 mm respectively. This showed that for an atomic beam of constant velocity, by using${\displaystyle {\lambda }_{\text{dB}}}$, the atomic Talbot length can be found in the same manner.

The nonlinear Talbot effect results from self-imaging of the generated periodic intensity pattern at the output surface of theperiodically poledLiTaO₃ crystal. Both integer and fractional nonlinear Talbot effects were investigated.^[7]

In cubic nonlinear Schrödinger's equation${\displaystyle i\frac{\mathrm{\partial }\psi }{\mathrm{\partial }z}+\frac{1}{2}\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{x}^2}+|\psi {|}^2\psi =0}$, nonlinear Talbot effect ofrogue waves is observed numerically.^[8]

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, 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.^[9]

The optical Talbot effect can be used in imaging applications to overcome the diffraction limit (e.g. in structured illuminationfluorescence microscopy).^[10]

Moreover, its capacity to generate very fine patterns is also a powerful tool in Talbotlithography.^[11]

TheTalbot cavity is used for the phase-locking of the laser sets.^[12]

In experimental fluid dynamics, the Talbot effect has been implemented in Talbotinterferometry to measure displacements^[13][14] and temperature,^[15][16] and deployed withlaser-induced fluorescence to reconstruct free surfaces in 3D,^[17] and measure velocity.^[18]

• Angle-sensitive pixel

• Talbot's 1836 paper via Google Books
• Rayleigh's 1881 paper via Google Books
• Undergraduate thesis by Rob Wild (PDF)
• Talbot effect observed over space-time for the first time

• Diffraction

• Articles with short description
• Short description matches Wikidata