## Abstract

A new method of generating nondiffracting beams is presented. It consists of focusing a Gaussian beam in the vicinity of an opaque disk. A beam is generated whose central peak is surrounded by a wide number of bright rings $(\sim 250)$. After collimation, the beam propagates without changing the rings’ radii, similar to a diffraction-free beam. The central peak can conserve its dimension over more than $5\text{\hspace{0.17em}}\mathrm{m}$. The diameter of the central peak is adjusted by choosing the focal length of the collimating lens. Experimental results are well predicted by our theoretical developments that simulate exactly the paraxial diffraction.

© 2007 Optical Society of America

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### Equations (19)

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(1)
$$E(\xi ,\eta ,z)={E}_{0}\left(z\right)\mathrm{exp}(-\frac{{\xi}^{2}+{\eta}^{2}}{\omega {\left(z\right)}^{2}})\mathrm{exp}(-\frac{i\pi}{\lambda}\frac{{\xi}^{2}+{\eta}^{2}}{R\left(z\right)}),$$
(2)
$$\omega \left(z\right)=\sqrt{{\omega}_{0}^{2}[1+{\left(\frac{z}{{z}_{0}}\right)}^{2}]},$$
(3)
$$R\left(z\right)=-z[1+{\left(\frac{{z}_{0}}{z}\right)}^{2}],$$
(4)
$$E(x,y,{z}_{c}-{z}_{q})=\frac{\mathrm{exp}(i2\pi {z}_{c}\u2215\lambda )}{i\lambda {z}_{c}}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}E(\xi ,\eta ,-{z}_{q})[1-T(\xi ,\eta )]\mathrm{exp}\left\{\frac{i\pi}{\lambda {z}_{c}}[{(\xi -x)}^{2}+{(\eta -y)}^{2}]\right\}\mathrm{d}\xi \mathrm{d}\eta ,$$
(5)
$$T(\xi ,\eta )=\{\begin{array}{ll}1& \text{if}\phantom{\rule{0.3em}{0ex}}\sqrt{{\xi}^{2}+{\eta}^{2}}<D\u22152,\\ 0& \text{otherwise}.\end{array}\phantom{\}}.$$
(6)
$$I(x,y,{z}_{c}-{z}_{q})=E(x,y,{z}_{c}-{z}_{q}).E{(x,y,{z}_{c}-{z}_{q})}^{*},$$
(7)
$${A}_{1}={K}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left[\beta {r}^{2}(iM-N)\right],$$
(8)
$${A}_{2}=\frac{\pi {D}^{2}}{2}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}\left(i\beta {r}^{2}\right){T}_{0}\left(r\right).$$
(9)
$${T}_{0}\left(r\right)=\mathrm{exp}(-iu\u22154){\left(\frac{2\pi}{u}\right)}^{1\u22152}\sum _{s=0}^{\infty}{K}_{s}{(-1)}^{s}\frac{{J}_{2s+1}\left(\beta Dr\right)}{\beta Dr}$$
(10)
$${K}_{s}={(-i)}^{s}(2s+1){J}_{s+1\u22152}(u\u22154).$$
(11)
$$\beta =\pi \u2215\lambda {z}_{c},$$
(12)
$$u=\beta {D}^{2}\u22152(-1+\lambda {z}_{c}{b}_{1})-i{a}_{1}({D}^{2}\u22152),$$
(13)
$$K={\left[\frac{\pi {\omega}_{q}^{2}}{1+i\beta {\omega}_{q}^{2}(\frac{{z}_{c}}{{R}_{q}}-1)}\right]}^{1\u22152},$$
(14)
$$N=\frac{\beta {\omega}_{q}^{2}}{1+{\pi}^{2}{\omega}_{q}^{4}\u2215{\lambda}^{2}{(\frac{1}{{R}_{q}}-\frac{1}{{z}_{c}})}^{2}},$$
(15)
$$M=1+N\frac{\pi {\omega}_{q}^{2}}{\lambda}(\frac{1}{{R}_{q}}-\frac{1}{{z}_{c}}),$$
(16)
$${a}_{1}=\frac{1}{{\omega}_{q}^{2}},$$
(17)
$${b}_{1}=\frac{1}{\lambda {R}_{q}}.$$
(18)
$$t(x,y)=\mathrm{exp}(-\frac{i\pi ({x}^{2}+{y}^{2})}{\lambda {f}_{{\mathrm{L}}_{2}}}),$$
(19)
$$E({x}^{\prime},{y}^{\prime},{z}^{\prime})=\frac{\mathrm{exp}(i2\pi {z}^{\prime}\u2215\lambda )}{i\lambda {z}^{\prime}}\int {\int}_{{L}_{2}}E(\xi ,\eta ,{z}_{c}-{z}_{q})\mathrm{exp}[-\frac{i\pi}{\lambda {f}_{{\mathrm{L}}_{2}}}({\xi}^{2}+{\eta}^{2})]\mathrm{exp}\left\{\frac{i\pi}{\lambda {z}^{\prime}}[{(\xi -{x}^{\prime})}^{2}+{(\eta -{y}^{\prime})}^{2}]\right\}\mathrm{d}\xi \mathrm{d}\eta .$$