Talbot effect

Lord Rayleigh showed that the Talbot effect was a natural consequence of Fresnel diffraction and that the Talbot length can be found by the following formula:[3]

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

where ${\displaystyle a}$ is the period of the diffraction grating and ${\displaystyle \lambda }$ is the wavelength of the light incident on the grating. However, if wavelength ${\displaystyle \lambda }$ is comparable to grating period ${\displaystyle a}$, this expression may lead to errors in ${\displaystyle z_{\text{T}}}$ up to 100%.[4] In this case the exact expression derived by Lord Rayleigh should be used:

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

Due to the quantum mechanical wave nature of particles, diffraction effects have also been observed with atoms—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 atoms was passed through two diffraction gratings (the second used as a mask) to observe the Talbot effect and measure the Talbot length.[7] The beam had a mean velocity of 1000 m/s corresponding to a de Broglie wavelength of ${\displaystyle \lambda _{\text{dB}}}$ = 0.017 nm. Their experiment was performed with 200 and 300 nm gratings which yielded Talbot lengths of 4.7 and 10.6 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 the periodically poled LiTaO₃ crystal. Both integer and fractional nonlinear Talbot effects were investigated.[8]

In cubic nonlinear Schrödinger's equation ${\displaystyle i{\frac {\partial \psi }{\partial z}}+{\frac {1}{2}}{\frac {\partial ^{2}\psi }{\partial x^{2}}}+|\psi |^{2}\psi =0}$, nonlinear Talbot effect of rogue waves is observed numerically.[9]

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.[10]

The optical Talbot effect can be used in imaging applications to overcome the diffraction limit (e.g. in structured illumination fluorescence microscopy).[11]

Moreover, its capacity to generate very fine patterns is also a powerful tool in Talbot lithography.[12]

The Talbot cavity is used for the phase-locking of the laser sets.[13]

In experimental fluid dynamics, the Talbot effect has been implemented in Talbot interferometry to measure displacements [14][15] and temperature,[16][17] and deployed with laser-induced fluorescence to reconstruct free surfaces in 3D,[18] and measure velocity.[19]

• 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