## Abstract

Based on the theory of scalar diffraction, the diffraction of gratings in the deep Fresnel diffraction region is developed, and the general formula of the diffraction intensity of the one-dimensional grating is presented by using the Hankel function. Through numerical calculations, some interesting diffraction phenomena are found. In the deep Fresnel diffraction region, the dominant effects, with increasing propagation distance from the grating, are, in order, the geometrical effect, the quasi-geometrical effect, and the interference and diffraction effects. Furthermore, the diffraction intensities vary periodically in the diffraction effect region with increasing propagation distance. Quasi-Talbot imaging of the grating exists in the interference and diffraction regions, and the intensity distributions most similar to the structure of the grating are not at the exact Talbot distances. These phenomena in the deep Fresnel diffraction region are distinct from those in the Fresnel diffraction region. The formation origin of quasi-Talbot imaging of the grating is also discussed, and the numerical calculations powerfully verify the theoretical results.

© 2007 Optical Society of America

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### Equations (23)

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(1)
$$t\left({\mathbf{r}}_{0}\right)=a\left({\mathbf{r}}_{0}\right)\mathrm{exp}\left[inkh\left({\mathbf{r}}_{0}\right)\right],$$
(2)
$$u\left(\mathbf{r}\right)=\frac{1}{4\pi}\int {\int}_{S}\{{u}_{0}\left({\mathbf{r}}_{0}\right)\frac{\partial G(\mathbf{r},{\mathbf{r}}_{0})}{\partial n}-G(\mathbf{r},{\mathbf{r}}_{0})\frac{\partial {u}_{0}\left({\mathbf{r}}_{0}\right)}{\partial n}\}\mathrm{d}S,$$
(3)
$$u\left(\mathbf{r}\right)=A\int t\left({\mathbf{r}}_{0}\right)K(\mathbf{r},{\mathbf{r}}_{0})\mathrm{d}{\mathbf{r}}_{0},$$
(4)
$$K(\mathbf{r},{\mathbf{r}}_{0})=\frac{1}{4\pi}\frac{\partial G(\mathbf{r},{\mathbf{r}}_{0})}{\partial n}$$
(5)
$$G(\mathbf{r},{\mathbf{r}}_{0})=i\pi {H}_{0}^{\left(1\right)}\left(kr\right),$$
(6)
$$K(x,{x}_{0})=-\frac{i}{4}k(z-h){H}_{1}^{\left(1\right)}\left(kr\right)\u2215r,$$
(7)
$$I(x,z)=\frac{{k}^{2}{A}^{2}}{16}{\mid \int [t\left({x}_{0}\right)(z-h){H}_{1}^{\left(1\right)}\left(kr\right)\u2215r]\mathrm{d}{x}_{0}\mid}^{2}.$$
(8)
$$t\left({x}_{0}\right)=\sum _{n=0}^{\mathrm{N}}\mathrm{rect}\left(\frac{{x}_{0}-nd}{d\u2215M}\right),$$
(9)
$$t\left({x}_{0}\right)=\sum _{m=-\infty}^{\infty}{C}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(\frac{i2\pi m{x}_{0}}{d}\right),$$
(10)
$$K(\mathbf{r};{\mathbf{r}}_{0})=\frac{1}{iz\lambda}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikz\right)\mathrm{exp}\left(i\frac{k}{2z}{\mid \mathbf{r}-{\mathbf{r}}_{0}\mid}^{2}\right).$$
(11)
$$u(x,z)=\frac{A}{iz\lambda}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikz\right){\int}_{-L\u22152}^{L\u22152}\sum _{m}{C}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\frac{2\pi m{x}_{0}}{d}\right)\mathrm{exp}\left[i\frac{\pi {(x+{x}_{0})}^{2}}{\lambda z}\right]\mathrm{d}{x}_{0}.$$
(12)
$$u(x,z)=\frac{A}{i\sqrt{z\lambda}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikz\right)\sum _{m}{C}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\frac{2\pi mx}{d}\right)\mathrm{exp}(-i\frac{\pi \lambda {m}^{2}z}{{d}^{2}}).$$
(13)
$$I(x,z)=\frac{{A}^{2}}{z\lambda}{\mid \mathrm{exp}\left(ikz\right)\sum _{m}{C}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\frac{2\pi mx}{d}\right)\mathrm{exp}(-i\frac{\pi \lambda {m}^{2}z}{{d}^{2}})\mid}^{2}.$$
(14)
$$K(\mathbf{r},{\mathbf{r}}_{0})=-\frac{i}{4}\sqrt{\frac{2k}{\pi}}\frac{z}{{r}^{3\u22152}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(ikr-i3\pi \u22154).$$
(15)
$$I(x,z)=\frac{k{A}^{2}}{8\pi}{\mid {\int}_{-L\u22152}^{L\u22152}\sum _{m}{C}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\frac{2\pi m{x}_{0}}{d}\right)\frac{z\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikr\right)}{{r}^{3\u22152}}\mathrm{d}{x}_{0}\mid}^{2}.$$
(16)
$$I(x,z)=\frac{k{A}^{2}}{8\pi}{\mid \sum _{m}{C}_{m}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\frac{2\pi mx}{d}\right){\int}_{x-L\u22152}^{x+L\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-i\frac{2\pi m\xi}{d})\frac{z\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left[ik{({\xi}^{2}+{z}^{2})}^{1\u22152}\right]}{{({\xi}^{2}+{z}^{2})}^{3\u22154}}\mathrm{d}\xi \mid}^{2}.$$
(17)
$${F}_{m}={\int}_{x-L\u22152}^{x+L\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-i\frac{2\pi m\xi}{d})\frac{z\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left[ik{({\xi}^{2}+{z}^{2})}^{1\u22152}\right]}{{({\xi}^{2}+{z}^{2})}^{3\u22154}}\mathrm{d}\xi $$
(18)
$${z}_{T}^{\prime}=\frac{\lambda}{\sqrt{1-\left({m}^{2}{\lambda}^{2}\right)\u2215{d}^{2}}-\sqrt{1-\left({n}^{2}{\lambda}^{2}\right)\u2215{d}^{2}}},$$
(19)
$${P}_{m}={\int}_{x-L\u22152}^{x+L\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(-i\frac{2\pi m\xi}{d})\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikz\right)\mathrm{exp}\left[ik{({\xi}^{2}+{z}^{2})}^{1\u22152}\right]\mathrm{d}\xi ,$$
(20)
$$\frac{1}{z}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(ikz\right)\mathrm{exp}(-i\frac{\pi \lambda {m}^{2}z}{{d}^{2}})$$
(21)
$$z+{\xi}^{2}\u22152z-{\xi}^{4}\u22158{z}^{3}+{\xi}^{6}\u221516{z}^{5}-{\xi}^{8}\u2215128{z}^{7},$$
(22)
$$z+{\xi}^{2}\u22152z-{\xi}^{4}\u22158{z}^{3}+{\xi}^{6}\u221516{z}^{5},$$
(23)
$$z+{\xi}^{2}\u22152z-{\xi}^{4}\u22158{z}^{3},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}z+{\xi}^{2}\u22152z.$$