Inverse solutions for tilting orthogonal double prisms

Anhu Li; Ye Ding; Yongming Bian; Liren Liu

doi:10.1364/AO.53.003712

Applied Optics
Vol. 53,
Issue 17,
pp. 3712-3722
(2014)
•https://doi.org/10.1364/AO.53.003712

Inverse solutions for tilting orthogonal double prisms

Anhu Li, Ye Ding, Yongming Bian, and Liren Liu

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Anhu Li, Ye Ding, Yongming Bian, and Liren Liu, "Inverse solutions for tilting orthogonal double prisms," Appl. Opt. 53, 3712-3722 (2014)

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Abstract

An analytical reverse solution and actual examples are given to show how to direct a laser beam from a pair of orthogonal prisms to given targets in free space. Considering the influences of double-prism structural parameters, a lookup table method to seek the numerical reverse solution of each prism’s tilting angle is also proposed for steering the double-prism orientation to track a target position located in the near field. Some case studies, as well as a specified elliptical target trajectory scanned by the cam-based driving double prisms, exhibit the significant application values of the theoretical derivation. The analytic reverse and numerical solutions can be generalized to investigate the synthesis of scanning patterns and the controlling strategy of double-prism tilting motion, the potentials of which can be explored to perform the orientation and position tracking functions in applications of precision engineering fields.

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Equations (8)

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$D_{2}$ (mm)	$θ_{t 1} = θ_{t 2} (°)$	$x_{t p} (mm)$	$y_{t p} (mm)$	$x_{t p}^{'} (mm)$	$y_{t p}^{'} (mm)$	$d_{s} (mm)$
3,000,000	0	$- 278068.661$	$- 275114.057$	$- 278055.805$	$- 275111.334$	13.141
	5	$- 285622.880$	$- 278661.021$	$- 285611.100$	$- 278656.819$	12.507
	10	$- 297578.521$	$- 285900.119$	$- 297567.685$	$- 285894.316$	12.292
300	0	$- 40.662$	$- 30.234$	$- 27.806$	$- 27.511$	13.141
	5	$- 40.342$	$- 32.068$	$- 28.561$	$- 27.866$	12.507
	10	$- 40.593$	$- 34.392$	$- 29.757$	$- 28.589$	12.292

	Prism $Π_{1}$ ( $i = 1$ )	Prism $Π_{2}$ ( $i = 2$ )
$C_{i 0}$	0.007750828068567	0.011846416396477
$C_{i 1}$	$- 0.003104467584725$	$- 0.004974452731015$
$C_{i 2}$	0.095018622748549	0.097672796471536
$C_{i 3}$	$- 0.046632319391502$	$- 0.059342099150951$
$C_{i 4}$	0.010291469536128	0.019886852037600
$C_{i 5}$	$- 0.001257346067799$	$- 0.004430221888414$
$C_{i 6}$	0.000066773274916	0.000570360616791
$C_{i 7}$		$- 0.000026726099159$
$C_{i 8}$		$- 0.000001856841762$
$C_{i 9}$		0.000000198737488

$D_{2}$ (mm)	$θ_{t 1} = θ_{t 2} (°)$	$x_{t p} (mm)$	$y_{t p} (mm)$	$x_{t p}^{'} (mm)$	$y_{t p}^{'} (mm)$	$d_{s} (mm)$
3,000,000	0	$- 278068.661$	$- 275114.057$	$- 278055.805$	$- 275111.334$	13.141
	5	$- 285622.880$	$- 278661.021$	$- 285611.100$	$- 278656.819$	12.507
	10	$- 297578.521$	$- 285900.119$	$- 297567.685$	$- 285894.316$	12.292
300	0	$- 40.662$	$- 30.234$	$- 27.806$	$- 27.511$	13.141
	5	$- 40.342$	$- 32.068$	$- 28.561$	$- 27.866$	12.507
	10	$- 40.593$	$- 34.392$	$- 29.757$	$- 28.589$	12.292

	Prism $Π_{1}$ ( $i = 1$ )	Prism $Π_{2}$ ( $i = 2$ )
$C_{i 0}$	0.007750828068567	0.011846416396477
$C_{i 1}$	$- 0.003104467584725$	$- 0.004974452731015$
$C_{i 2}$	0.095018622748549	0.097672796471536
$C_{i 3}$	$- 0.046632319391502$	$- 0.059342099150951$
$C_{i 4}$	0.010291469536128	0.019886852037600
$C_{i 5}$	$- 0.001257346067799$	$- 0.004430221888414$
$C_{i 6}$	0.000066773274916	0.000570360616791
$C_{i 7}$		$- 0.000026726099159$
$C_{i 8}$		$- 0.000001856841762$
$C_{i 9}$		0.000000198737488

Abstract

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Equations (8)

Applied Optics