## Abstract

A new two-stage axiconic reflector called a reflaxicon is discussed. This device consists essentially of a primary conical mirror and a larger secondary conical mirror located coaxially with respect to the primary mirror. The secondary mirror is truncated at such a location that the inner diameter of the secondary mirror (the diameter of the hole) exceeds the base diameter of the primary mirror. The function of this device is to convert a solid light beam (such as a Gaussian intensity distribution laser beam) into a hollow one in an essentially lossless manner. The solid (or Gaussian) light beam is incident upon the outside of the primary mirror and is reflected therefrom to the inside of the secondary mirror, from which it is in turn reflected in the direction desired, such as to a target or receiver. Proper selection of the half-angle of the secondary cone, in comparison with the half-angle of the primary cone, will result in a converging, parallel, or diverging beam. Some modifications to the basic reflaxicon and potential applications in laser bore sighting and collimation are also discussed. In addition, a holographic technique using reflaxicons is considered.

© 1973 Optical Society of America

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### Equations (26)

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(1)
$${x}_{1}={r}_{1}\hspace{0.17em}\text{cot}{\theta}_{1}.$$
(2)
$${x}_{2}=(h-{r}_{1})\hspace{0.17em}\text{cot}2{\theta}_{1}.$$
(3)
$$z=y\hspace{0.17em}\text{tan}\varphi .$$
(4)
$$z=y\hspace{0.17em}\text{tan}2({\theta}_{1}-{\theta}_{2}).$$
(6)
$$y=(h-{r}_{3})\hspace{0.17em}\text{cot}2({\theta}_{1}-{\theta}_{2}).$$
(7)
$$D={x}_{1}+{x}_{2}+y.$$
(8)
$$D={r}_{1}\hspace{0.17em}\text{cot}{\theta}_{1}+(h-{r}_{1})\hspace{0.17em}\text{cot}2{\theta}_{1}+(h-{r}_{3})\hspace{0.17em}\text{cot}2({\theta}_{1}-{\theta}_{2}).$$
(9)
$${x}_{3}=d/(\text{cot}\sigma -\text{cot}w).$$
(10)
$${x}_{3}=d/(\text{tan}\hspace{0.17em}2{\theta}_{1}-\text{tan}{\theta}_{2}).$$
(11)
$$p={x}_{3}\hspace{0.17em}\text{tan}{\theta}_{2},$$
(13)
$$p=d/(\text{tan}\hspace{0.17em}2{\theta}_{1}-\text{tan}{\theta}_{2})\hspace{0.17em}\text{cot}{\theta}_{2}.$$
(15)
$$c+d={r}_{2}-{r}_{1},$$
(16)
$$c=a\hspace{0.17em}\text{tan}2{\theta}_{1}.$$
(17)
$$p=({r}_{2}-{r}_{1}-c)/(\text{tan}\hspace{0.17em}2{\theta}_{1}-\text{tan}{\theta}_{2})\hspace{0.17em}\text{cot}{\theta}_{2},$$
(18)
$$p=({r}_{2}-{r}_{1}-a\hspace{0.17em}\text{tan}2{\theta}_{1})/(\text{tan}\hspace{0.17em}2{\theta}_{1}-\text{tan}{\theta}_{2})\hspace{0.17em}\text{cot}{\theta}_{2}.$$
(19)
$$h={r}_{2}+\{[{r}_{2}-{r}_{1}-(k-{r}_{1}\hspace{0.17em}\text{cot}{\theta}_{1})\hspace{0.17em}\text{tan}2{\theta}_{1}]/(\text{tan}\hspace{0.17em}2{\theta}_{1}-\text{tan}{\theta}_{2})\hspace{0.17em}\text{cot}{\theta}_{2}\}.$$
(20)
$$D={r}_{1}\hspace{0.17em}\text{cot}{\theta}_{1}+(h-{r}_{1})\hspace{0.17em}\text{cot}2{\theta}_{1}+(h-{r}_{3})\hspace{0.17em}\text{cot}2({\theta}_{1}-{\theta}_{2}),$$
(21)
$${D}_{0}={r}_{1}\hspace{0.17em}\text{cot}{\theta}_{1}+({h}_{0}-{r}_{1})\hspace{0.17em}\text{cot}2{\theta}_{1}+{h}_{0}\hspace{0.17em}\text{cot}2({\theta}_{1}-{\theta}_{2}),$$
(22)
$${h}_{0}={r}_{2}+\{[{r}_{2}-{r}_{1}-(k-{r}_{1}\hspace{0.17em}\text{cot}{\theta}_{1})\hspace{0.17em}\text{tan}2{\theta}_{1}]/(\text{tan}\hspace{0.17em}2{\theta}_{1}-\text{tan}{\theta}_{2})\hspace{0.17em}\text{cot}{\theta}_{2}\}.$$
(23)
$${D}_{0f}={h}_{f}\hspace{0.17em}\text{cot}2{\theta}_{1}+{h}_{f}\hspace{0.17em}\text{cot}2({\theta}_{1}-{\theta}_{2}),$$
(24)
$${h}_{f}={r}_{2}+[({r}_{2}-k\hspace{0.17em}\text{tan}2{\theta}_{1})/(\text{tan}2{\theta}_{1}-\text{tan}{\theta}_{2})\hspace{0.17em}\text{cot}{\theta}_{2}].$$
(25)
$${I}_{a}(r)={I}_{b}(r)\{({{r}_{a}}^{2}-{{r}_{b}}^{2})\hspace{0.17em}\text{sin}{\theta}_{1}/[{({r}_{1}+{r}_{a})}^{2}-{({r}_{1}+{r}_{b})}^{2}]\hspace{0.17em}\text{sin}{\theta}_{2}\},$$
(26)
$${I}_{a}(r)={I}_{b}({r}_{0})[{r}_{a}+{r}_{b})\hspace{0.17em}\text{sin}{\theta}_{1}/(2{r}_{1}+{r}_{a}+{r}_{b})\hspace{0.17em}\text{sin}{\theta}_{2}].$$