## Abstract

We present exact, nonsingular solutions of the scalar-wave equation for beams that are nondiffracting. This means that the intensity pattern in a transverse plane is unaltered by propagating in free space. These beams can have extremely narrow intensity profiles with effective widths as small as several wavelengths and yet possess an infinite depth of field. We further show (by using numerical simulations based on scalar diffraction theory) that physically realizable finite-aperture approximations to the exact solutions can also possess an extremely large depth of field.

© 1987 Optical Society of America

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### Equations (5)

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(1)
$$\left({\nabla}^{2}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}\right)\hspace{0.17em}E(\mathbf{r},t)=0.$$
(2)
$$E(x,y,z\ge 0,t)=\text{exp}[i(\beta z-\omega t)]{\int}_{0}^{2\pi}A(\varphi )\hspace{0.17em}\text{exp}[i\alpha (x\hspace{0.17em}\text{cos}\hspace{0.17em}\varphi +y\hspace{0.17em}\text{sin}\hspace{0.17em}\varphi )]\text{d}\varphi ,$$
(3)
$$\begin{array}{l}I(x,y,z\ge 0)=\xbd\mid E(\mathbf{r},t){\mid}^{2}\\ =I(x,y,z=0),\end{array}$$
(4)
$$\begin{array}{l}E(\mathbf{r},t)=\text{exp}[i(\beta z-\omega t)]{\int}_{0}^{2\pi}\text{exp}[i\alpha (x\hspace{0.17em}\text{cos}\hspace{0.17em}\varphi +y\hspace{0.17em}\text{sin}\hspace{0.17em}\varphi )]\frac{\text{d}\varphi}{2\pi}\\ =\text{exp}[i(\beta z-\omega t)]{J}_{0}(\alpha \rho ).\end{array}$$
(5)
$$\begin{array}{l}{z}_{\text{max}}=r/\text{tan}\hspace{0.17em}\theta \\ =r{[{(2\pi /\alpha \mathrm{\lambda})}^{2}-1]}^{1/2},\end{array}$$