## Abstract

A vectorial equation that describes the Huygens principle was reported, and an expression for the secondary-spherical-wave energy density was found. With a vectorial formulation, we performed an exact calculation for the relative axial intensity of the wave diffracted by an illuminated circular aperture. The off-axis intensity in Fresnel’s and Fraunhofer’s approximations was calculated. The zone plate was also studied by vectorial formulation. We showed that with increasing number of rings, the intensity maxima magnify as ${(2n+2)}^{2}$, their full widths decrease, their positions remain unchanged, and secondary maxima appear, in a behavior similar to that for diffraction gratings.

© 2006 Optical Society of America

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### Equations (27)

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(1)
$$\mathrm{d}U({\mathbf{r}}^{\prime},\mathbf{r})=\frac{-z}{2\pi}\frac{U({x}^{\prime},{y}^{\prime},0)}{\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\frac{\partial}{\partial \mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\left[\frac{\mathrm{exp}(\mp ik\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid )}{\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\right]\mathrm{d}{s}^{\prime},$$
(2)
$$\mathrm{d}{U}_{x}({\mathbf{r}}^{\prime},\mathbf{r})({x}^{\prime}-x)+\mathrm{d}{U}_{y}({\mathbf{r}}^{\prime},\mathbf{r})({y}^{\prime}-y)-\mathrm{d}{U}_{z}({\mathbf{r}}^{\prime},\mathbf{r})z=0.$$
(3)
$$\mathbf{d}\mathbf{U}({\mathbf{r}}^{\prime},\mathbf{r})=[\mathbf{d}{\mathbf{s}}^{\prime}\times \mathbf{U}({x}^{\prime},{y}^{\prime},0)]\times \frac{{\mathbf{r}}^{\prime}-\mathbf{r}}{\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\frac{\partial}{\partial \mid \mathbf{r}-\mathbf{r}\mid}\left[\frac{\mathrm{exp}(\mp ik\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid )}{2\pi \mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\right],$$
(4)
$$\mathbf{U}({x}^{\prime},{y}^{\prime},0)={U}_{x}({x}^{\prime},{y}^{\prime},0){\mathbf{e}}_{x}+{U}_{y}({x}^{\prime},{y}^{\prime},0){\mathbf{e}}_{y}.$$
(5)
$$\mathbf{E}(z,t)={\mathbf{E}}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\{\pm i[wt-kz+\alpha ]\}.$$
(6)
$$\mathbf{U}({x}^{\prime},{y}^{\prime},0)=\mathbf{E}(0,t),$$
(7)
$$\mathbf{d}\mathbf{E}({\mathbf{r}}^{\prime},\mathbf{r},t)=[\mathbf{d}{\mathbf{s}}^{\prime}\times {\mathbf{E}}_{p}\left(t\right)]\times \frac{{\mathbf{r}}^{\prime}-\mathbf{r}}{\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\frac{\partial}{\partial \mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\left[\frac{\mathrm{exp}(\mp ik\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid )}{2\pi \mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}\right],$$
(8)
$${\mathbf{E}}_{p}\left(t\right)={\mathbf{E}}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[\pm i(wt+\alpha )].$$
(9)
$$\mathbf{d}\mathbf{E}({\mathbf{r}}^{\prime},\mathbf{r},t)=\Gamma \left({\mathbf{r}}^{\prime}\right)\mathbf{\u03f5}({\mathbf{r}}^{\prime},\mathbf{r},t)\mathrm{d}{s}^{\prime},$$
(10)
$${\mathbf{\u03f5}}_{e}({\mathbf{r}}^{\prime},\mathbf{r},t)=[\frac{\mathbf{k}}{2\pi k\Gamma \left({\mathbf{r}}^{\prime}\right)}\times {\mathbf{E}}_{p}\left(t\right)]\times \frac{{\mathbf{r}}^{\prime}-\mathbf{r}}{\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}(\mp ik-\frac{1}{\mid {\mathbf{r}}^{\prime}-\mathbf{r}\mid}),$$
(11)
$$\mid {\mathbf{\u03f5}}_{e}(0,\mathbf{r},t)\mid =\frac{\mid {\mathbf{E}}_{p}\left(t\right)\mid}{2\pi \Gamma \left(0\right)}{[{k}^{2}+\frac{1}{{r}^{2}}]}^{1\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\left(\beta \right),$$
(12)
$$W=\frac{\u03f5{\mid {\mathbf{E}}_{p}\mid}^{2}}{8{\pi}^{2}{r}^{2}{\Gamma}^{2}}({k}^{2}+\frac{1}{{r}^{2}})[1+{\mathrm{cos}}^{2}\left(\varphi \right)],$$
(13)
$$\mathbf{E}(\mathbf{r},t)=\frac{-1}{2\pi}[{\mathbf{e}}_{z}({\mathbf{\rho}}_{o}\bullet {\mathbf{E}}_{p}){I}_{1}+z{\mathbf{E}}_{p}{I}_{2}],$$
(14)
$${I}_{1}=2{\int}_{0}^{\pi}\frac{\mathrm{exp}\{\mp ik{[{\rho}_{o}^{2}+{r}^{2}-2{\rho}_{o}\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)]}^{1\u22152}\}}{{[{\rho}_{o}^{2}+{r}^{2}-2{\rho}_{o}\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)]}^{1\u22152}}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)\mathrm{d}\psi ,$$
(15)
$${I}_{2}=2{\int}_{0}^{\pi}{\int}_{r}^{{[{\rho}_{o}^{2}+{r}^{2}-2{\rho}_{o}\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)]}^{1\u22152}}\frac{\partial}{\partial R}\left[\frac{\mathrm{exp}(\mp ikR)}{R}\right]\{1+\frac{\rho \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)}{{[{R}^{2}-{r}^{2}+{\rho}^{2}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{2}\left(\psi \right)]}^{1\u22152}}\}\mathrm{d}R\mathrm{d}\psi ,$$
(16)
$$I=1+\frac{{z}^{2}}{{z}^{2}+{\rho}_{o}^{2}}-\frac{2z}{{[{z}^{2}+{\rho}_{o}^{2}]}^{1\u22152}}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left\{k[{({z}^{2}+{\rho}_{o}^{2})}^{1\u22152}-z]\right\}.$$
(17)
$$I={I}_{1}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{2}\left(\theta \right)+{a}^{2}{I}_{2},$$
(18)
$${I}_{1}={\mid \frac{1}{\pi}{\int}_{0}^{\pi}\frac{\mathrm{exp}\{\mp ic{[1+{a}^{2}+{b}^{2}-2b\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)]}^{1\u22152}\}}{{[1+{a}^{2}+{b}^{2}-2b\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)]}^{1\u22152}}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)\mathrm{d}\psi \mid}^{2},$$
(19)
$$t={[{l}^{2}+{a}^{2}+{b}^{2}-2bl\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left(\psi \right)]}^{1\u22152},$$
(20)
$${I}_{2}={\mid \frac{1}{\pi}{\int}_{0}^{\pi}{\int}_{0}^{1}\frac{1}{t}\frac{\partial}{\partial t}\left[\frac{\mathrm{exp}(\mp ict)}{t}\right]l\mathrm{d}l\mathrm{d}\psi \mid}^{2},$$
(21)
$$\mathbf{E}(z,t)=z{\mathbf{E}}_{p}\left(t\right)\sum _{l=0}^{2n+1}\frac{{(-1)}^{l}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\{\mp ik{[{\rho}_{l}^{2}+{z}^{2}]}^{1\u22152}\}}{[{\rho}_{l}^{2}+{z}^{2}]}.$$
(22)
$$I=\sum _{\begin{array}{c}l=0\\ j=0\end{array}}^{2n+1}\frac{{(-1)}^{l+j}{z}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left[k({w}_{l}-{w}_{j})\right]}{{w}_{l}{w}_{j}},$$
(23)
$$\text{where}\phantom{\rule{0.3em}{0ex}}{w}_{l}={({\rho}_{l}^{2}+{z}^{2})}^{1\u22152}.$$
(24)
$$k({w}_{l}-{w}_{j})=(2m+1)\pi (l+j),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}m\u220a\mathbb{Z}.$$
(25)
$${\rho}_{l}^{2}={\left(\frac{l\lambda}{2}\right)}^{2}+{\rho}_{0}^{2}+l\lambda {({z}_{0}^{2}+{\rho}_{0}^{2})}^{1\u22152},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}0\u2a7dl\u2a7d2n+1.$$
(26)
$$I(x,y,n)=\sum _{\begin{array}{c}l=0\\ j=0\end{array}}^{2n+1}\frac{{(-1)}^{l+j}{x}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\left[\frac{2\pi}{y}({w}_{l}-{w}_{j})\right]}{{w}_{l}{w}_{j}},$$
(27)
$${w}_{l}={[{\left(\frac{ly}{2}\right)}^{2}+ly+{x}^{2}]}^{1\u22152},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}0\u2a7dl\u2a7d2n+1.$$