## Abstract

This paper is a continuation of our previous publication on the stationary-phase-method analysis of lens axicons [J. Opt. Soc. Am. A **15**2383 (1998)]. Systems with spherical aberration up to the fifth order are studied. Such lens axicons in their simplest versions can be made either as a setup composed of two separated third-order spherical-aberration lenses of opposite powers or as a doublet consisting of one third-order diverging element and one fifth-order converging element. The axial intensity distribution and the central core width turn out to be improved and become almost constant. The results obtained are compared with the numerical evaluation of the corresponding diffraction integral.

© 1999 Optical Society of America

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### Equations (26)

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(1)
$$\varphi (r)=-{r}^{2}/2s+\beta {r}^{4}-\gamma {r}^{6},\hspace{1em}\hspace{1em}1/s=1/{f}_{1}-1/{f}_{2},$$
(2)
$$I(\rho ,z)={I}_{0}(k/z{)}^{2}{\left|{\int}_{{R}_{1}}^{{R}_{2}}exp[\mathit{ikf}(r)]{J}_{0}(\mathit{kr}\rho /z)r\mathrm{d}r\right|}^{2},$$
(3)
$$f(r)=\varphi (r)+{r}^{2}/2z,$$
(4)
$$I(\rho ,z)\propto 2\pi {\mathit{kr}}_{s}^{2}{J}_{0}^{2}({\mathit{kr}}_{s}\rho /z)/{z}^{2}|{f}^{\u2033}({r}_{s})|,$$
(5)
$${f}^{\prime}({r}_{s})=0.$$
(6)
$${r}_{{s}_{1,2}}={r}_{max}[1\pm \sqrt{1-L(z)/{L}_{max}}{]}^{1/2},$$
(7)
$$L(z)=1/s-1/z,\hspace{1em}\hspace{1em}{L}_{max}=1/s-1/{d}_{max}=2{\beta}^{2}/3\gamma ,$$
(8)
$$I(\rho ,z)\propto \frac{\pi {\mathit{kr}}_{max}^{2}}{2{z}^{2}\{{L}_{max}[{L}_{max}-L(z)]{\}}^{1/2}}\times {J}_{0}^{2}\left(\frac{k\rho {r}_{max}}{z}\{[\sqrt{{L}_{max}}-\sqrt{{L}_{max}-L(z)}]/\sqrt{{L}_{max}}{\}}^{1/2}\right).$$
(9)
$${\rho}_{0}(z)=c\mathrm{\lambda}z/{r}_{max}[1-\sqrt{s({d}_{max}-z)/z({d}_{max}-s)}{]}^{1/2},$$
(10)
$${I}_{\mathrm{norm}}(0,z)=I(0,z)/I(0,{d}_{1})=({d}_{1}/z{)}^{3/2}[({d}_{max}-{d}_{1})/({d}_{max}-z){]}^{1/2},$$
(11)
$${d}_{max}=(d_{2}{}^{4}-d_{1}{}^{4})/(d_{2}{}^{3}-d_{1}{}^{3}).$$
(12)
$${L}_{max}=(d_{1}{}^{2}+d_{2}{}^{2})/{d}_{1}{d}_{2}({d}_{1}+{d}_{2}).$$
(13)
$${r}_{max}={R}_{1,2}/\{[1-\sqrt{1-L({d}_{1,2})/{L}_{max}}]{\}}^{1/2}$$
(14)
$$\beta ={L}_{max}/2r_{m}{}^{2},\hspace{1em}\hspace{1em}\gamma ={L}_{max}/6r_{m}{}^{4}.$$
(15)
$${z}_{1,2}={z}_{c}[1\pm (1-25{\mathit{sd}}_{max}/16z_{c}{}^{2}{)}^{1/2}],$$
(16)
$${z}_{c}=(3s/4+{d}_{max}/2).$$
(17)
$$s=4{d}_{max}/9,\hspace{1em}\hspace{1em}{z}_{\mathrm{inf}}=5{d}_{max}/6.$$
(18)
$$I(0,{d}_{max})/I(0,{d}_{2})=\pi C_{1}{}^{2}{k}^{1/3}{2}^{11/3}d_{2}{}^{2}|{f}^{\prime \prime \prime}({r}_{s}){|}^{-2/3}\times \{{L}_{max}[{L}_{max}-L({d}_{2})]{\}}^{1/2}/{d}_{max}^{2},$$
(19)
$$\varphi (r)=-{r}^{2}/2{f}_{1}-\beta {r}^{4}.$$
(20)
$$\frac{\mathrm{d}\varphi}{\mathrm{d}r}\approx \frac{r-{r}^{\prime}}{d}.$$
(21)
$${r}^{\prime}=r(1-d/{f}_{1})-4\beta {\mathit{dr}}^{3}.$$
(22)
$$r={r}^{\prime}/(1-d/{f}_{1})+{\mathit{Ar}}^{\prime 3}+{\mathit{Br}}^{\prime 5}.$$
(23)
$$A=4\beta d/(1-d/{f}_{1}{)}^{4},\hspace{1em}\hspace{1em}B=48{\beta}^{2}{d}^{2}/(1-d/{f}_{1}{)}^{7}.$$
(24)
$$\frac{\mathrm{d}{\varphi}^{\prime}}{\mathrm{d}{r}^{\prime}}\approx \frac{r-{r}^{\prime}}{d},$$
(25)
$${\varphi}^{\prime}({r}^{\prime})=-{r}^{\prime 2}/2({f}_{1}-d)-{\beta}^{\prime}{r}^{4}-{\gamma}^{\prime}{r}^{6},$$
(26)
$${\beta}^{\prime}=\beta /(1-d/{f}_{1}{)}^{4},\hspace{1em}\hspace{1em}{\gamma}^{\prime}=8{\beta}^{2}d/(1-d/{f}_{1}{)}^{7}.$$