# Talbot effect

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Near-field diffraction effect

The optical Talbot effect for monochromatic light, shown as a "Talbot carpet". At the bottom of the figure the light can be seen diffracting through a grating, and this pattern is reproduced at the top of the picture (one Talbot length away from the grating). At regular fractions of the Talbot length the sub-images form.

The **Talbot effect** is a [diffraction](/source/Diffraction) effect first observed in 1836 by [Henry Fox Talbot](/source/Henry_Fox_Talbot).[1] When a [plane wave](/source/Plane_wave) is incident upon a periodic [diffraction grating](/source/Diffraction_grating), 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](/source/Fractal) pattern of sub images with ever-decreasing size, often referred to as a **Talbot carpet**.[2] [Talbot cavities](/source/Talbot_cavity) are used for [coherent beam combination](/source/Coherent_addition) of laser sets.

## Calculation of the Talbot length

[Lord Rayleigh](/source/Lord_Rayleigh) showed that the Talbot effect was a natural consequence of [Fresnel diffraction](/source/Fresnel_diffraction) and that the Talbot length can be found by the following formula (page 204):[3]

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

where a {\displaystyle a} is the period of the diffraction grating and λ {\displaystyle \lambda } is the [wavelength](/source/Wavelength) of the light incident on the grating. For λ ≪ a {\displaystyle \lambda \ll a} , the Talbot length is approximately given by:

- z T ≈ 2 a 2 λ . {\displaystyle z_{\text{T}}\approx {\frac {2a^{2}}{\lambda }}.}

### Fresnel number of the finite size Talbot grating

The number of Fresnel zones N F {\displaystyle N_{\text{F}}} that form first Talbot self-image of the grating with period p {\displaystyle p} and transverse size N ⋅ a {\displaystyle N\cdot a} is given by exact formula N F = ( N − 1 ) 2 {\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 T = 2 a 2 λ {\textstyle z_{\text{T}}={\frac {2a^{2}}{\lambda }}} .[5]

## Atomic Talbot effect

Due to the [quantum mechanical](/source/Quantum_mechanical) wave nature of [particles](/source/Elementary_particle), diffraction effects have also been observed with [atoms](/source/Atoms)—effects which are similar to those in the case of light. Chapman *et al.* carried out an experiment in which a [collimated beam](/source/Collimated_beam) of [sodium](/source/Sodium) 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 of 1000 m/s corresponding to a [de Broglie wavelength](/source/De_Broglie_wavelength) of λ dB {\displaystyle \lambda _{\text{dB}}} = 0.017 [nm](/source/Nanometre). 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 λ dB {\displaystyle \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](/source/Periodic_poling) [LiTaO3](/source/Lithium_tantalate) crystal. Both integer and fractional nonlinear Talbot effects were investigated.[7]

In cubic nonlinear Schrödinger's equation i ∂ ψ ∂ z + 1 2 ∂ 2 ψ ∂ x 2 + | ψ | 2 ψ = 0 {\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](/source/Rogue_waves) is observed numerically.[8]

The nonlinear Talbot effect was also realized in linear, nonlinear and highly nonlinear [surface gravity water waves](/source/Gravity_wave). 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]

## 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](/source/Fluorescence_microscope)).[10]

Moreover, its capacity to generate very fine patterns is also a powerful tool in Talbot [lithography](/source/Photolithography).[11]

The [Talbot cavity](/source/Talbot_cavity) is used for the phase-locking of the laser sets.[12]

In experimental [fluid dynamics](/source/Fluid_dynamics), the Talbot effect has been implemented in Talbot [interferometry](/source/Interferometry) to measure displacements [13][14] and temperature,[15][16] and deployed with [laser-induced fluorescence](/source/Laser-induced_fluorescence) to reconstruct free surfaces in 3D,[17] and measure velocity.[18]

## See also

- [Angle-sensitive pixel](/source/Angle-sensitive_pixel)

## References

1. **[^](#cite_ref-1)** Talbot, H.F. (1836). ["LXXVI. Facts relating to optical science. No. IV"](https://zenodo.org/record/1431005). *The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*. **9** (56): 401–407. [doi](/source/Doi_(identifier)):[10.1080/14786443608649032](https://doi.org/10.1080%2F14786443608649032).

1. **[^](#cite_ref-2)** Case, William B.; Tomandl, Mathias; Deachapunya, Sarayut; Arndt, Markus (2009). ["Realization of optical carpets in the Talbot and Talbot–Lau configurations"](https://doi.org/10.1364%2FOE.17.020966). *Opt. Express*. **17** (23): 20966–20974. [Bibcode](/source/Bibcode_(identifier)):[2009OExpr..1720966C](https://ui.adsabs.harvard.edu/abs/2009OExpr..1720966C). [doi](/source/Doi_(identifier)):[10.1364/OE.17.020966](https://doi.org/10.1364%2FOE.17.020966). [PMID](/source/PMID_(identifier)) [19997335](https://pubmed.ncbi.nlm.nih.gov/19997335).

1. **[^](#cite_ref-3)** Rayleigh, Lord (1881). ["XXV. On copying diffraction-gratings, and on some phenomena connected therewith"](https://zenodo.org/record/1629943). *The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*. **11** (67): 196–205. [doi](/source/Doi_(identifier)):[10.1080/14786448108626995](https://doi.org/10.1080%2F14786448108626995).

1. **[^](#cite_ref-Okulov,_A_Yu_1993_4-0)** Okulov, A. Yu. (1993). "Scaling of diode-array-pumped solid-state lasers via self-imaging". *Opt. Commun*. **99** (5–6): 350–354. [Bibcode](/source/Bibcode_(identifier)):[1993OptCo..99..350O](https://ui.adsabs.harvard.edu/abs/1993OptCo..99..350O). [doi](/source/Doi_(identifier)):[10.1016/0030-4018(93)90342-3](https://doi.org/10.1016%2F0030-4018%2893%2990342-3).

1. **[^](#cite_ref-Okulov,_A_Yu_1990_5-0)** Okulov, A. Yu. (1990). "Two-dimensional periodic structures in nonlinear resonator". *JOSA B*. **7** (6): 1045–1050. [Bibcode](/source/Bibcode_(identifier)):[1990JOSAB...7.1045O](https://ui.adsabs.harvard.edu/abs/1990JOSAB...7.1045O). [doi](/source/Doi_(identifier)):[10.1364/JOSAB.7.001045](https://doi.org/10.1364%2FJOSAB.7.001045).

1. **[^](#cite_ref-6)** Chapman, Michael S.; Ekstrom, Christopher R.; Hammond, Troy D.; Schmiedmayer, Jörg; Tannian, Bridget E.; Wehinger, Stefan; Pritchard, David E. (1995). "Near-field imaging of atom diffraction gratings: The atomic Talbot effect". *Physical Review A*. **51** (1): R14–R17. [Bibcode](/source/Bibcode_(identifier)):[1995PhRvA..51...14C](https://ui.adsabs.harvard.edu/abs/1995PhRvA..51...14C). [doi](/source/Doi_(identifier)):[10.1103/PhysRevA.51.R14](https://doi.org/10.1103%2FPhysRevA.51.R14). [PMID](/source/PMID_(identifier)) [9911659](https://pubmed.ncbi.nlm.nih.gov/9911659).

1. **[^](#cite_ref-7)** Zhang, Yong; Wen, Jianming; Zhu, S. N.; Xiao, Min (2010). "Nonlinear Talbot Effect". *Physical Review Letters*. **104** (18) 183901. [Bibcode](/source/Bibcode_(identifier)):[2010PhRvL.104r3901Z](https://ui.adsabs.harvard.edu/abs/2010PhRvL.104r3901Z). [doi](/source/Doi_(identifier)):[10.1103/PhysRevLett.104.183901](https://doi.org/10.1103%2FPhysRevLett.104.183901). [PMID](/source/PMID_(identifier)) [20482176](https://pubmed.ncbi.nlm.nih.gov/20482176).

1. **[^](#cite_ref-8)** Zhang, Yiqi; Belić, Milivoj R.; Zheng, Huaibin; Chen, Haixia; Li, Changbiao; Song, Jianping; Zhang, Yanpeng (2014). "Nonlinear Talbot effect of rogue waves". *Physical Review E*. **89** (3) 032902. [arXiv](/source/ArXiv_(identifier)):[1402.3017](https://arxiv.org/abs/1402.3017). [Bibcode](/source/Bibcode_(identifier)):[2014PhRvE..89c2902Z](https://ui.adsabs.harvard.edu/abs/2014PhRvE..89c2902Z). [doi](/source/Doi_(identifier)):[10.1103/PhysRevE.89.032902](https://doi.org/10.1103%2FPhysRevE.89.032902). [PMID](/source/PMID_(identifier)) [24730908](https://pubmed.ncbi.nlm.nih.gov/24730908). [S2CID](/source/S2CID_(identifier)) [41885399](https://api.semanticscholar.org/CorpusID:41885399).

1. **[^](#cite_ref-9)** Rozenman, Georgi Gary; Schleich, Wolfgang P.; Shemer, Lev S. N.; Arie, Ady (2022). "Periodic Wave Trains in Nonlinear Media: Talbot Revivals, Akhmediev Breathers, and Asymmetry Breaking". *Physical Review Letters*. **128** 214101. [Bibcode](/source/Bibcode_(identifier)):[2022PhRvL.128u4101R](https://ui.adsabs.harvard.edu/abs/2022PhRvL.128u4101R). [doi](/source/Doi_(identifier)):[10.1103/PhysRevLett.128.214101](https://doi.org/10.1103%2FPhysRevLett.128.214101). [PMID](/source/PMID_(identifier)) [35687471](https://pubmed.ncbi.nlm.nih.gov/35687471). [S2CID](/source/S2CID_(identifier)) [249140572](https://api.semanticscholar.org/CorpusID:249140572).

1. **[^](#cite_ref-10)** Chowdhury, S.; Chen, J.; Izatt, J.A. (2018). "Structured illumination fluorescence microscopy using Talbot self-imaging effect for high-throughput visualization". [arXiv](/source/ArXiv_(identifier)):[1801.03540](https://arxiv.org/abs/1801.03540) [[physics.optics](https://arxiv.org/archive/physics.optics)].

1. **[^](#cite_ref-11)** Isoyan, A.; Jiang, F.; Cheng, Y. C.; Cerrina, F.; Wachulak, P.; Urbanski, L.; Rocca, J.; Menoni, C.; Marconi, M. (2009). "Talbot lithography: Self-imaging of complex structures". *Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena*. **27** (6): 2931–2937. [Bibcode](/source/Bibcode_(identifier)):[2009JVSTB..27.2931I](https://ui.adsabs.harvard.edu/abs/2009JVSTB..27.2931I). [doi](/source/Doi_(identifier)):[10.1116/1.3258144](https://doi.org/10.1116%2F1.3258144).

1. **[^](#cite_ref-Okulov,_A_Yu_1991_12-0)** Okulov, A Yu (1991). "The effect of roughness of optical elements on the transverse structure of alight field in a nonlinear Talbot cavity". *J. Mod. Opt*. **53** (11): 1887–1890. [Bibcode](/source/Bibcode_(identifier)):[1991JMOp...38.1887O](https://ui.adsabs.harvard.edu/abs/1991JMOp...38.1887O). [doi](/source/Doi_(identifier)):[10.1080/09500349114551991](https://doi.org/10.1080%2F09500349114551991).

1. **[^](#cite_ref-13)** Spagnolo, G Schirripa; Ambrosini, D.; Paoletti, D. (2002). "Displacement measurement using the Talbot effect with a Ronchi grating". *Journal of Optics A: Pure and Applied Optics*. **4** (6): S376–S380. [Bibcode](/source/Bibcode_(identifier)):[2002JOptA...4S.376S](https://ui.adsabs.harvard.edu/abs/2002JOptA...4S.376S). [doi](/source/Doi_(identifier)):[10.1088/1464-4258/4/6/383](https://doi.org/10.1088%2F1464-4258%2F4%2F6%2F383).

1. **[^](#cite_ref-14)** Shakher, Chandra; Agarwal, Shilpi (2018). "Low-frequency in-plane vibration monitoring/measurement using circular grating Talbot interferometer". *Optical Engineering*. **57** (5) 054112. [Bibcode](/source/Bibcode_(identifier)):[2018OptEn..57e4112A](https://ui.adsabs.harvard.edu/abs/2018OptEn..57e4112A). [doi](/source/Doi_(identifier)):[10.1117/1.OE.57.5.054112](https://doi.org/10.1117%2F1.OE.57.5.054112). [S2CID](/source/S2CID_(identifier)) [125924183](https://api.semanticscholar.org/CorpusID:125924183).

1. **[^](#cite_ref-15)** Shakher, Chandra; Daniel, A. J. Pramila (1994). "Talbot interferometer with circular gratings for the measurement of temperature in axisymmetric gaseous flames". *Applied Optics*. **33** (25): 6068–6072. [Bibcode](/source/Bibcode_(identifier)):[1994ApOpt..33.6068S](https://ui.adsabs.harvard.edu/abs/1994ApOpt..33.6068S). [doi](/source/Doi_(identifier)):[10.1364/AO.33.006068](https://doi.org/10.1364%2FAO.33.006068). [PMID](/source/PMID_(identifier)) [20936022](https://pubmed.ncbi.nlm.nih.gov/20936022).

1. **[^](#cite_ref-16)** Agarwal, Shilpi; Kumar, Manoj; Shakher, Chandra (2015). "Experimental investigation of the effect of magnetic field on temperature and temperature profile of diffusion flame using circular grating Talbot interferometer". *Optics and Lasers in Engineering*. **68**: 214–221. [Bibcode](/source/Bibcode_(identifier)):[2015OptLE..68..214A](https://ui.adsabs.harvard.edu/abs/2015OptLE..68..214A). [doi](/source/Doi_(identifier)):[10.1016/j.optlaseng.2015.01.004](https://doi.org/10.1016%2Fj.optlaseng.2015.01.004).

1. **[^](#cite_ref-17)** Florou, Eirini I.; Fort, Charles; Habukawa, Masayuki; André, Matthieu A.; Bardet, Philippe M. (2023). "Surface reconstruction in three-dimensional space using structured illumination". *Experiments in Fluids*. **64** (4): 70. [Bibcode](/source/Bibcode_(identifier)):[2023ExFl...64...70F](https://ui.adsabs.harvard.edu/abs/2023ExFl...64...70F). [doi](/source/Doi_(identifier)):[10.1007/s00348-023-03608-9](https://doi.org/10.1007%2Fs00348-023-03608-9). [S2CID](/source/S2CID_(identifier)) [257604959](https://api.semanticscholar.org/CorpusID:257604959).

1. **[^](#cite_ref-18)** Fort, Charles; André, Matthieu A.; Pazhand, Hatef; Bardet, Philippe M. (2020). "Talbot-effect structured illumination: pattern generation and application to long-distance μ-MTV". *Experiments in Fluids*. **61** (2): 40. [Bibcode](/source/Bibcode_(identifier)):[2020ExFl...61...40F](https://ui.adsabs.harvard.edu/abs/2020ExFl...61...40F). [doi](/source/Doi_(identifier)):[10.1007/s00348-019-2870-7](https://doi.org/10.1007%2Fs00348-019-2870-7). [S2CID](/source/S2CID_(identifier)) [213543686](https://api.semanticscholar.org/CorpusID:213543686).

## External links

- [Talbot's 1836 paper via Google Books](https://books.google.com/books?id=ewBbAAAAQAAJ&dq=1836%20philosophical%20magazine&pg=PA401)

- [Rayleigh's 1881 paper via Google Books](https://books.google.com/books?id=O5EOAAAAIAAJ&dq=philosophical%20magazine%20rayleigh%201881%20diffraction%20gratings&pg=PA196)

- [Undergraduate thesis by Rob Wild (PDF)](http://www.physics.arizona.edu/~cronin/Research/Lab/wildthesis2.pdf)

- [Talbot effect observed over space-time for the first time](https://aip.scitation.org/doi/10.1063/10.0005054)

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Adapted from the Wikipedia article [Talbot effect](https://en.wikipedia.org/wiki/Talbot_effect) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Talbot_effect?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.