## Abstract

Studying the limitations of sharpness in self-images of the Talbot effect leads us to abandon the use of the paraxial assumption. In this respect, we will clarify the self-image concept and show experimentally and theoretically the influence of “nonparaxial effects” on the self-image. The boundary between paraxial and nonparaxial scalar theory is also clarified in this context. The Rayleigh criterion in aberration diffraction theory is adapted for explicit estimations of this boundary and the maximum sharpness of lines projected by use of the Talbot effect.

© 2004 Optical Society of America

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### Equations (21)

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(1)
$$t(x)=\sum _{p}{c}_{p}exp\left(\mathit{jp}\frac{2\pi}{d}x\right),$$
(2)
$${c}_{p}=\frac{a}{d}sinc\left(\frac{\mathit{pa}}{d}\right).$$
(3)
$$u(x,z)={u}_{0}\sum _{p}{c}_{p}\times exp\left\{j\frac{2\pi}{\mathrm{\lambda}}z{\left[1-{\left(\frac{p\mathrm{\lambda}}{d}\right)}^{2}\right]}^{1/2}\right\}exp\left(\mathit{jp}\frac{2\pi}{d}x\right).$$
(4)
$${\left[1-{\left(\frac{p\mathrm{\lambda}}{d}\right)}^{2}\right]}^{1/2}=1-\frac{1}{2}{\left(\frac{p\mathrm{\lambda}}{d}\right)}^{2}.$$
(5)
$$u={u}_{0}exp\left(j\frac{2\pi}{\mathrm{\lambda}}z\right)\sum _{p}{c}_{p}\times exp\left(-j2\pi \frac{z}{{z}_{T}}{p}^{2}\right)exp\left(\mathit{jp}\frac{2\pi}{d}x\right).$$
(6)
$${u}_{r}=uexp\left(-j\frac{2\pi}{\mathrm{\lambda}}z\right),$$
(7)
$${u}_{r}={u}_{0}\sum _{p}{c}_{p}exp\left(-j2\pi \frac{z}{{z}_{T}}{p}^{2}\right)exp\left(\mathit{jp}\frac{2\pi}{d}x\right),$$
(8)
$${u}_{r}=\sum _{p}{c}_{p}{u}_{p},$$
(9)
$${c}_{p}{u}_{p}={u}_{0}{c}_{p}exp[j{\varphi}_{r}^{(p)}(z)]exp\left(\mathit{jp}\frac{2\pi}{d}x\right),$$
(10)
$${\varphi}_{r}^{(p)}(z)=\frac{2\pi}{\mathrm{\lambda}}z\left\{{\left[1-{\left(\frac{p\mathrm{\lambda}}{d}\right)}^{2}\right]}^{1/2}-1\right\}.$$
(11)
$$(\mathrm{\Delta}\upsilon {)}_{z}=\frac{\int |I(\upsilon ,z){|}^{2}\mathrm{d}\upsilon}{|I(0,z){|}^{2}},$$
(12)
$$(\mathrm{\Delta}\upsilon {)}_{z}=\frac{1}{d}\frac{{\displaystyle \sum _{p}}|{D}_{p}{|}^{2}}{|{D}_{0}{|}^{2}}.$$
(13)
$$C(z)=d(\mathrm{\Delta}\upsilon {)}_{z}=\frac{{\displaystyle \sum _{p}}|{D}_{p}(z){|}^{2}}{|{D}_{0}(z){|}^{2}}.$$
(14)
$$C\left(m\frac{{z}_{T}}{2}\right)=\frac{d}{a},$$
(15)
$${\varphi}_{p}=\frac{2\pi}{\mathrm{\lambda}}z\left\{{\left[1-{\left(\frac{p\mathrm{\lambda}}{d}\right)}^{2}\right]}^{1/2}-1\right\}+p\frac{2\pi}{d}x.$$
(16)
$${\varphi}_{0p}=-2\pi \frac{{\mathit{md}}^{2}}{\mathrm{\lambda}{z}_{T}}{p}^{2}+p\frac{2\pi}{d}x=-\pi {\mathit{mp}}^{2}+p\frac{2\pi}{d}x$$
(17)
$${\varphi}_{\mathrm{ab}(p)}(z)=\frac{2\pi}{\mathrm{\lambda}}z\left\{{\left[1-{\left(\frac{p\mathrm{\lambda}}{d}\right)}^{2}\right]}^{1/2}-1\right\}+\pi {\mathit{mp}}^{2},$$
(18)
$${\varphi}_{{\mathrm{ab}}_{({p}_{max})}}\left(m\frac{{d}^{2}}{\mathrm{\lambda}}\right)=\frac{2\pi}{\mathrm{\lambda}}m\frac{{d}^{2}}{\mathrm{\lambda}}\left\{{\left[1-{\left(\frac{\mathrm{\lambda}}{a}\right)}^{2}\right]}^{1/2}-1\right\}+\pi m{\left(\frac{d}{a}\right)}^{2}.$$
(19)
$${\varphi}_{{\mathrm{ab}}_{({p}_{max})}}\left(m\frac{{z}_{T}}{2}\right)=\frac{\pi}{4}m{\left(\frac{d}{\mathrm{\lambda}}\right)}^{2}{\left(\frac{\mathrm{\lambda}}{a}\right)}^{4}\le \frac{\pi}{2}.$$
(20)
$${a}_{\mathrm{opt}}=(m/2{)}^{1/4}\sqrt{\mathrm{\lambda}d}\approx 0.84\sqrt{\mathrm{\lambda}d}.$$
(21)
$${a}_{\mathrm{opt}}\approx 0.55\sqrt{\mathrm{\lambda}d},$$