Ruled surfaces generated by Risley prism pointers: I. Pointing accuracy evaluation and enhancement based on a structural analysis of the scan field inside the pointer

Yajun Li

doi:10.1364/JOSAA.435621

Journal of the Optical Society of America A
Vol. 38,
Issue 12,
pp. 1884-1892
(2021)
•https://doi.org/10.1364/JOSAA.435621

Ruled surfaces generated by Risley prism pointers: I. Pointing accuracy evaluation and enhancement based on a structural analysis of the scan field inside the pointer

Yajun Li

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Yajun Li, "Ruled surfaces generated by Risley prism pointers: I. Pointing accuracy evaluation and enhancement based on a structural analysis of the scan field inside the pointer," J. Opt. Soc. Am. A 38, 1884-1892 (2021)

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Abstract

Based on the elementary concept of ruled surfaces, a new method relating to a structural analysis of the scan field distributed over the interior of the Risley prisms is proposed that opens the way to a re-examination of the pointing accuracy that predicted the existing noniterative solution of the Risley prism pointers for precision tracking of targets [Li, J. Opt. Soc. Am. A 28, 667 (2011) [CrossRef] and Li, Appl. Opt. 50, 4302 (2011) [CrossRef] ]. After taking the thickness and the air gap of the two prisms into consideration, the refined version of the closed-form analytic solution is obtained, which predicts the correct results for detecting and tracking targets traversing across both the near and far zones of the beam-steering field in front of the pointer if the linear size of the target, the diameter of the laser beam, or their combination is greater than a small fraction of the central thickness of the prism. For very tiny targets, a method for pointing accuracy enhancement is proposed based on tracking a virtual target.

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Corrections

3 January 2022: A typographical correction was made to Eq. (1a).

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

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	Cone Section #3	Interface 22^(*)
Ruled surface directrix	$r_{# 30} = (0, 0, Δ z_{3})$ ^(**)	$r_{220} = (0, 0, D)$
Director of rulings	${\hat{s}}_{22}^{(i)} = (k_{22}^{(i)}, l_{22}^{(i)}, m_{22}^{(i)})$ ^(***)	${\hat{s}}_{22}$
(*) This is a double ruled surface. For simplicity, a point directrix at the center of interface 22 is assumed, from which the radiation rays are the rulings in direction specified by ${\hat{s}}_{22}$ , that is, a unit vector lying in 22, and hence ${\hat{n}}_{22} \cdot {\hat{s}}_{22} = 0$ , where ${\hat{n}}_{22} = (- \sin α_{2} \cos θ_{2}, - \sin α_{2} \sin θ_{2}, \cos α_{2})$ is the surface normal.
(**) $Δ z_{3} = d_{a i r} (\sqrt{(n_{1}^{2} - a_{1}^{2}) / (1 - a_{1}^{2})} - 1)$ where $a_{1} = (\cos α_{1} - \sqrt{n_{1}^{2} - \sin^{2} α_{1}}) \sin α_{1}$ and $a_{2} = - \sqrt{n_{2}^{2} - a_{1}^{2}}$ .
(***) $k_{22}^{(i)} = (a_{1} / n_{2}) \cos θ_{1}$ , $l_{22}^{(i)} = (a_{1} / n_{2}) \sin θ_{1}$ and $m_{22}^{(i)} = - \sqrt{1 - {(a_{1} / n_{2})}^{2}}$ .

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Journal of the Optical Society of America A