Talbot effect

Calculation of the Talbot length

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 (page 204):

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

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

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

Fresnel number of the finite size Talbot grating

The number of Fresnel zones N_\text{F} 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{F} = (N-1)^2.{{cite journal This result is obtained via exact evaluation of Fresnel-Kirchhoff integral in the near field at distance z_\text{T} = \frac{2 a^2}{\lambda}.{{cite journal

Atomic Talbot effect

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.{{cite journal The beam had a mean velocity of corresponding to a de Broglie wavelength of \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 \lambda_\text{dB}, 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.{{cite journal

In cubic nonlinear Schrödinger's equation 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.{{cite journal

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.{{cite journal

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).{{cite arXiv

Moreover, its capacity to generate very fine patterns is also a powerful tool in Talbot lithography.{{cite journal

The Talbot cavity is used for the phase-locking of the laser sets.{{cite journal |title=The effect of roughness of optical elements on the transverse structure of alight field in a nonlinear Talbot cavity

In experimental fluid dynamics, the Talbot effect has been implemented in Talbot interferometry to measure displacements {{cite journal and temperature,{{cite journal and deployed with laser-induced fluorescence to reconstruct free surfaces in 3D,{{cite journal

References

References

1. (1836). "LXXVI. Facts relating to optical science. No. IV". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.
2. (1881). "XXV. On copying diffraction-gratings, and on some phenomena connected therewith". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.