## Abstract

We present an analytical method for systematic optical design of a double-pass axicon that shows almost no astigmatism in oblique illumination compared to a conventional linear axicon. The anastigmatic axicon is a singlet lens with nearly concentric spherical surfaces applied in double pass, making it possible to form a long narrow focal line of uniform width. The front and the back surfaces have reflective coatings in the central and annular zones, respectively, to provide the double pass. Our design method finds the radii of curvatures and axial thickness of the lens for a given angle between the exiting rays and the optical axis. It also finds the optimal position of the reflecting zones for minimal vignetting. This method is based on ray tracing of the real rays at the marginal heights of the aperture and therefore is superior to any paraxial method. We illustrate the efficiency of the method by designing a test axicon with optical parameters used for a prototype axicon, which was manufactured and experimentally tested. We compare the optical characteristics of our test axicon with those of the experimental prototype.

© 2007 Optical Society of America

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### Equations (22)

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(1)
$${\theta}_{1}={\alpha}_{1}-{\alpha}_{2}=\mathrm{arcsin}\left(h/{r}_{1}\right)-\mathrm{arcsin}\left(h/n{r}_{1}\right)\text{,}$$
(2)
$${z}_{1}={r}_{1}-{\left({{r}_{1}}^{2}-{h}^{2}\right)}^{1/2}\text{,}$$
(3)
$${y}_{2}={\mathrm{cos}}^{2}\text{\hspace{0.17em}}{\theta}_{1}(h+\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{1}\left\{d-{r}_{2}+{z}_{1}+{\left[{{r}_{2}}^{2}-{h}^{2}-2h\left(d-{r}_{2}+{z}_{1}\right)\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{1}-\left(d+{z}_{1}\right)\left(d-2{r}_{2}+{z}_{1}\right){\mathrm{tan}}^{2}\text{\hspace{0.17em}}{\theta}_{1}\right]}^{1/2}\right\})\text{,}$$
(4)
$${z}_{2}={r}_{2}-{\left({{r}_{2}}^{2}-{{y}_{2}}^{2}\right)}^{1/2}\text{,}$$
(5)
$${\theta}_{2}=2{\alpha}_{3}+{\theta}_{1}=2\text{\hspace{0.17em}}\mathrm{arcsin}\left({y}_{2}/{r}_{2}\right)-{\theta}_{1}\text{,}$$
(6)
$${y}_{3}={\mathrm{cos}}^{2}\text{\hspace{0.17em}}{\theta}_{2}\left({y}_{2}-\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{2}\left\{d+{r}_{1}-{z}_{2}+{\left[{{r}_{1}}^{2}-{{y}_{2}}^{2}+2{y}_{2}\left(d+{r}_{1}-{z}_{2}\right)\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{2}+\left(d-{z}_{2}\right)\left(d+2{r}_{1}-{z}_{2}\right){\mathrm{tan}}^{2}\text{\hspace{0.17em}}{\theta}_{2}\right]}^{1/2}\right\}\right)\text{,}$$
(7)
$${z}_{3}={r}_{1}-{\left({{r}_{1}}^{2}-{{y}_{3}}^{2}\right)}^{1/2}\text{,}$$
(8)
$${\theta}_{3}={\theta}_{2}-2{\alpha}_{4}=2\text{\hspace{0.17em}}\mathrm{arcsin}\left({y}_{3}/{r}_{1}\right)-{\theta}_{2}\text{,}$$
(9)
$${y}_{4}={\mathrm{cos}}^{2}\text{\hspace{0.17em}}{\theta}_{3}({y}_{3}+\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{3}\left\{d-{r}_{2}+{z}_{3}+{\left[{{r}_{2}}^{2}-{{y}_{3}}^{2}-2{y}_{3}\left(d-{r}_{2}+{z}_{3}\right)\mathrm{tan}\text{\hspace{0.17em}}{\theta}_{3}-\left(d+{z}_{3}\right)\left(d-2{r}_{2}+{z}_{3}\right){\mathrm{tan}}^{2}\text{\hspace{0.17em}}{\theta}_{3}\right]}^{1/2}\right\})\text{,}$$
(10)
$${z}_{4}={r}_{2}-{\left({{r}_{2}}^{2}-{{y}_{4}}^{2}\right)}^{1/2}\text{,}$$
(11)
$${\alpha}_{5}=\mathrm{arcsin}\left({y}_{4}/{r}_{2}\right)-{\theta}_{3}\text{,}$$
(12)
$$\alpha =\mathrm{arcsin}\left(n\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}{\alpha}_{5}\right)-\mathrm{arcsin}\left({y}_{4}/{r}_{2}\right)\text{,}$$
(13)
$$\rho ={y}_{4}-{z}_{4}\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{,}$$
(14)
$$z={y}_{4}\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\alpha -{z}_{4}\mathrm{.}$$
(15)
$${r}_{2}=\frac{2n\epsilon {r}_{1}\left(n+1\right)}{n\left(4n{\epsilon}^{2}-n+2\right)-1}\text{,}$$
(16)
$$d=\frac{n{r}_{1}\left(1-\epsilon \right)}{1-n+2n\epsilon}\text{,}$$
(17)
$${r}_{1}=\frac{{h}_{\mathrm{max}}\left(n-1\right)\left[n\left(4n{\epsilon}^{2}-n+2\right)-1\right]}{2.4\alpha {n}^{2}\left(n+1\right)}\text{,}$$
(18)
$$\epsilon =0.6-\frac{1.06n}{1.874+{n}^{2}}+\left(0.4+\frac{1.9}{{n}^{2}}\right)\sqrt{\alpha}.$$
(19)
$$w=\frac{2.4048\lambda}{2\pi}\text{\hspace{0.17em}}\frac{z\left(h\right)}{\rho \left(h\right)}\text{,}$$
(20)
$$I\left(z\right)=\frac{4{\pi}^{2}}{\lambda}\text{\hspace{0.17em}}I\left(\rho \right)\frac{{\rho}^{2}\left(z\right)}{z\left|1-z\phi \u2033\left(\rho \right)\right|}\text{,}$$
(21)
$$I\left(z\right)=\frac{4{\pi}^{2}}{\lambda}\text{\hspace{0.17em}}h/\left(\frac{\mathrm{d}z}{\mathrm{d}h}\right)/.$$
(22)
$${\epsilon}_{\mathrm{max}}=-1/9+n/\left({n}^{2}+2.73\right)+\left(7.476-6.9n+2{n}^{2}\right)\sqrt{\alpha}\text{,}$$