## Abstract

Solutions of the scalar-wave equation that are based on modified Bessel functions are introduced. These functions are radially nonoscillating, unbound, and nondiffractive. Their propagation constant is larger than that of vacuum, meaning that there is an inverse Guoy effect or a phase velocity smaller than c. The beams are physically realized by apodization of the modified Bessel profiles by means of suitable window functions. When Gaussian apodization functions are used, axial decay is slower than in the Gaussian case and the case of ordinary Bessel counterparts. Super-Gaussian and circular apodization are also examined. In the latter case a persistent narrow axial lobe that also has a nondecaying character is encountered.

© 1994 Optical Society of America

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### Equations (13)

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(1)
$$E(x,y,z,t)=(1/2\pi )\text{exp}[i(\omega t-\beta z)]\times {\int}_{0}^{\infty}A(\varphi )\text{exp}[i\alpha (x\hspace{0.17em}\text{cos}\hspace{0.17em}\varphi +y\hspace{0.17em}\text{sin}\hspace{0.17em}\varphi )]\text{d}\varphi ,$$
(2)
$$E(x,y,z,t)=\text{exp}[i(\omega t-\beta z)]{I}_{0}(\gamma r),$$
(3)
$${\beta}^{2}={(\omega /c)}^{2}+{\gamma}^{2}.$$
(4)
$$U(r,z)=i\frac{k}{z}{\int}_{0}^{\infty}U({r}_{0},0)\text{exp}\hspace{0.17em}\left[-i\frac{k}{2z}({r}^{2}+{{r}_{0}}^{2})\right]\times {J}_{0}[(k/z)r{r}_{0}]{r}_{0}\text{d}{r}_{0},$$
(5)
$$U({r}_{0},0)={I}_{0}(\gamma {r}_{0})\text{exp}[-{({r}_{0}/{w}_{0})}^{2}].$$
(6)
$$U(0,z)=\sqrt{{A}_{g}(z)}\hspace{0.17em}\text{exp}[{(\gamma z/k{w}_{0})}^{2}{A}_{g}(z)]\times \text{exp}[-i({\gamma}^{2}z/2k){A}_{g}(z)+i{\psi}_{g}(z)],$$
(7)
$$\begin{array}{c}{A}_{g}(z)={[1+{(z/{z}_{R})}^{2}]}^{-1},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{\psi}_{g}(z)=\text{arctan}(z/{z}_{R}),\\ {z}_{R}=k{{w}_{0}}^{2}/2.\end{array}$$
(8)
$$\mathcal{A}=\frac{{\{\text{max}[U(0,z)]\}}_{0}^{\infty}}{{\{\text{max}[U(r,0)]\}}_{0}^{\infty}}.$$
(9)
$$U(0,z)=\text{exp}[-ik{a}^{2}/(2z)]\hspace{0.17em}\sum _{n=1}^{\infty}{\left(\frac{ika}{\gamma z}\right)}^{n}{I}_{n}(\gamma a),$$
(10)
$$U(0,z)=\text{exp}[i{\gamma}^{2}z/(2k)]-\text{exp}[-ik{a}^{2}/(2z)]\times \sum _{n=0}^{\infty}{\left(-\frac{i\gamma z}{ak}\right)}^{n}{I}_{n}(\gamma a).$$
(11)
$$U(0,z)\simeq \text{exp}[i{\gamma}^{2}z/(2k)]-\text{exp}[-ik{a}^{2}/(2z)]{I}_{0}(\gamma a).$$
(12)
$$U(r,z)\approx {I}_{0}(a\gamma )\text{exp}[-ik{a}^{2}/(2z)]\times \text{exp}[-ik{r}^{2}/(2z)]{J}_{0}(akr/z).$$
(13)
$${G}_{I}(\rho )=\frac{a}{{\gamma}^{2}+{(2\pi \rho )}^{2}}[2\pi \rho {I}_{0}(\gamma a){J}_{1}(2\pi \rho a)-\gamma {I}_{1}(\gamma a){J}_{0}(2\pi \rho a)],$$