## Abstract

Pattern generation in Talbot planes has generally been interpreted in terms of image formation, the repetitive slits are said to make repetitive images of themselves. In this context, Fourier optics developments have correctly predicted the positions of some but not all of the Talbot planes. Now, wave-optics methods are used to obtain general expressions for the positions of all known Talbot planes and the lateral positions of the diffraction fringes within them. These equations predict the key features of the Talbot effect, and they better relate multiple-slit diffraction in the Fresnel and Fraunhofer domains.

© 1992 Optical Society of America

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### Equations (16)

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(1)
$$A={\displaystyle \sum _{h=1}^{H}{A}_{h}\phantom{\rule{0.2em}{0ex}}\text{exp}\left(i{\varphi}_{h}\right)},$$
(2)
$$\Delta ={y}^{2}/\left(2R\right),$$
(3)
$${\Delta}_{\upsilon c}=\left({y}_{\upsilon}^{2}-{y}_{c}^{2}\right)/\left(2R\right)={y}_{a\upsilon}ja/R,$$
(4)
$$\varphi =2\pi m=2\pi \Delta /\lambda ,$$
(5)
$${y}_{a\upsilon}=m\lambda {R}_{2}/ja=ma{R}_{2}/jT=\mathit{\text{max}}/jT,$$
(6)
$$d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =m\lambda .$$
(7)
$${\Delta}_{1}+{\Delta}_{2}=m\lambda .$$
(8)
$$\left(ja\phantom{\rule{0.2em}{0ex}}{y}_{1a\upsilon}\right)/{R}_{1}+\left(j\phantom{\rule{0.2em}{0ex}}a{y}_{2a\upsilon}\right)/{R}_{2}=m\lambda .$$
(9)
$$d\left[\text{sin}\left(i\right)+\text{sin}(\theta )\right]=m\lambda .$$
(10)
$$ja\left({y}_{1a\upsilon -cd}\right)/{R}_{1}+ja\left({y}_{2a\upsilon -cd}\right)/{R}_{2}={m}_{cd}\lambda ,$$
(11)
$$ja\left({y}_{1a\upsilon -de}\right)/{R}_{1}+ja\left({y}_{2a\upsilon -de}\right)/{R}_{2}={m}_{de}\lambda .$$
(12)
$$1/{R}_{1}+1/{R}_{2}=1/\left(nT\right),$$
(13)
$${y}_{2a\upsilon}=mna/j=maj/q.$$
(14)
$${y}_{2a\upsilon}=ma/q.$$
(15)
$${b}^{\prime}=a\frac{j}{q}\left(\frac{{R}_{1}+{R}_{2}}{{R}_{1}}\right).$$
(16)
$$b=\frac{a}{q}\frac{\left({R}_{1}+{R}_{2}\right)}{{R}_{1}}.$$