Abstract

Scan blind zone and control singularity are two adverse issues for the beam scanning performance in double-prism Risley systems. In this paper, a theoretical model which introduces a third prism is developed. The critical condition for a fully eliminated scan blind zone is determined through a geometric derivation, providing several useful formulae for three-Risley-prism system design. Moreover, inverse solutions for a three-prism system are established, based on the damped least-squares iterative refinement by a forward ray tracing method. It is shown that the efficiency of this iterative calculation of the inverse solutions can be greatly enhanced by a numerical differentiation method. In order to overcome the control singularity problem, the motion law of any one prism in a three-prism system needs to be conditioned, resulting in continuous and steady motion profiles for the other two prisms.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  7. X. Tao and H. Cho, “Variable view imaging system and its application in vision based microassembly,” Proc. SPIE 6719, 67190L (2007).
    [Crossref]
  8. J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  15. P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
    [Crossref]
  16. M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
    [Crossref]
  17. A. Li, Q. Zuo, W. Sun, and W. Yi, “Beam distortion of rotation double prisms with an arbitrary incident angle,” Appl. Opt. 55(19), 5164–5171 (2016).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2016 (2)

2015 (1)

2014 (1)

A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
[Crossref]

2013 (1)

2011 (1)

2010 (1)

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

2008 (2)

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

X. Tao, H. Cho, and F. Janabi-Sharifi, “Active optical system for variable view imaging of micro objects with emphasis on kinematic analysis,” Appl. Opt. 47(22), 4121–4132 (2008).
[Crossref] [PubMed]

2007 (4)

W. C. Warger and C. A. DiMarzio, “Dual-wedge scanning confocal reflectance microscope,” Opt. Lett. 32(15), 2140–2142 (2007).
[Crossref] [PubMed]

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

X. Tao and H. Cho, “Variable view imaging system and its application in vision based microassembly,” Proc. SPIE 6719, 67190L (2007).
[Crossref]

2006 (2)

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

2005 (2)

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

2003 (1)

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

1999 (1)

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

1985 (1)

1960 (1)

Amirault, C. T.

Bian, Y.

Bos, P. J.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

Cho, H.

DiMarzio, C. A.

Ding, Y.

A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
[Crossref]

Doughty, N.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Duncan, B. D.

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

Fan, D.

Gao, X.

Garcia, H.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

Gutow, D.

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Hafez, M.

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

Harford, S.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Hei, M.

Hoffman, C.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Janabi-Sharifi, F.

Lavigne, V.

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

Li, A.

Li, Y.

Liu, D.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Liu, G.

Liu, H.

Liu, L.

A. Li, X. Gao, W. Sun, W. Yi, Y. Bian, H. Liu, and L. Liu, “Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution,” Appl. Opt. 54(33), 9981–9989 (2015).
[Crossref] [PubMed]

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Lu, Y.

Luan, Z.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Marshall, G. F.

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

Ostaszewski, M.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Pierce, R.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Régnier, S.

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

Richard, B.

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

Rosell, F. A.

Sanchez, M.

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

Sánchez, M.

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

Sergan, V.

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

Souvestre, F.

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

Sun, J.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Sun, W.

Tao, X.

Wang, L.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Warger, W. C.

Xu, N.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Yan, A.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Yi, W.

Yun, M.

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Zhang, M.

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Zhong, X.

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Zhou, Y.

Y. Zhou, Y. Lu, M. Hei, G. Liu, and D. Fan, “Motion control of the wedge prisms in Risley-prism-based beam steering system for precise target tracking,” Appl. Opt. 52(12), 2849–2857 (2013).
[Crossref] [PubMed]

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

Zhu, L.

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

Zuo, Q.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

B. D. Duncan, P. J. Bos, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasure applications,” Opt. Eng. 42(4), 1038–1047 (2003).
[Crossref]

P. J. Bos, H. Garcia, and V. Sergan, “Wide-angle achromatic prism beam steering for infrared countermeasures and imaging applications: solving the singularity problem in the two-prism design,” Opt. Eng. 46(11), 113001 (2007).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttg.) (1)

J. Sun, L. Liu, M. Yun, L. Wang, and M. Zhang, “The effect of the rotating double-prism wide-angle laser beam scanner on the beam shape,” Optik (Stuttg.) 116(12), 553–556 (2005).
[Crossref]

Proc. SPIE (8)

L. Liu, L. Wang, J. Sun, Y. Zhou, X. Zhong, Z. Luan, D. Liu, A. Yan, and N. Xu, “An integrated test-bed for PAT testing and verification of inter-satellite lasercom terminals,” Proc. SPIE 6709, 670904 (2007).
[Crossref]

X. Tao and H. Cho, “Variable view imaging system and its application in vision based microassembly,” Proc. SPIE 6719, 67190L (2007).
[Crossref]

L. Wang, L. Liu, L. Zhu, J. Sun, Y. Zhou, and D. Liu, “The mechanical design of the large-optics double-shearing interferometer for the test of the diffraction-limited wavefront,” Proc. SPIE 7091, 70910S (2008).
[Crossref]

A. Li, X. Gao, and Y. Ding, “Comparison of refractive rotating dual-prism scanner used in near and far field,” Proc. SPIE 9192, 919216 (2014).
[Crossref]

G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
[Crossref]

F. Souvestre, M. Hafez, and S. Régnier, “DMD-based multi-target laser tracking for motion capturing,” Proc. SPIE 7596, 75960B (2010).
[Crossref]

M. Ostaszewski, S. Harford, N. Doughty, C. Hoffman, M. Sanchez, D. Gutow, and R. Pierce, “Risley prism beam pointer,” Proc. SPIE 6304, 630406 (2006).
[Crossref]

M. Sánchez and D. Gutow, “Control laws for a 3-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
[Crossref]

RTO-MP-SET (1)

V. Lavigne and B. Richard, “Step-stare image gathering for high-resolution targeting,” RTO-MP-SET 092, 17 (2005).

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Figures (9)

Fig. 1
Fig. 1

Theoretical model of the three-Risley-prism system.

Fig. 2
Fig. 2

Beam scan trajectories under different angular velocity ratios of three prisms, where ω r1 : ω r2 : ω r3 equals to (a) 1:1:1 , (b) 1:2:1.5 , (c) 1:2:1.5 , (d) 1:1.5:1.5 , (e) 1:1.5:1.5 and (f) 1:1.5:1 , respectively.

Fig. 3
Fig. 3

Schematic diagram illustrating the critical condition for a fully eliminated blind zone.

Fig. 4
Fig. 4

Comparison between beam scan regions (a) in the absence of prism 3 and (b) in the presence of prism 3.

Fig. 5
Fig. 5

The three-prism system separated into single-prism and double-prism subsystems.

Fig. 6
Fig. 6

Prism motion profiles for (a) a given heart-shaped trajectory, including (b) the first set and (c) the second set of inverse solutions.

Fig. 7
Fig. 7

Prism motion profiles for (a) a given pentacle trajectory, including (b) the first set and (c) the second set of inverse solutions.

Fig. 8
Fig. 8

Control singularities caused by (a) a linear scan trajectory passing through the center of scan region, which exist in both (b) the first set and (c) the second set of inverse solutions.

Fig. 9
Fig. 9

The continuous motion profiles of three prisms for (a) the given linear scan trajectory, including (b) the first set and (c) the second set of inverse solutions.

Tables (1)

Tables Icon

Table 1 Operation time for the inverse solutions to given beam scan trajectories

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

N 11 = ( cos θ r1 sinα,sin θ r1 sinα,cosα ) T , N 12 = ( 0,0,1 ) T , N 21 = ( 0,0,1 ) T , N 22 = ( cos θ r2 sinα,sin θ r2 sinα,cosα ) T , N 31 = ( cos θ r3 sinα,sin θ r3 sinα,cosα ) T , N 32 = ( 0,0,1 ) T .
A i =( n i1 n i ) A i-1 +{ 1 ( n i1 n i ) 2 [ 1 ( A i-1 N ) 2 ] ( n i1 n i ) A i-1 N }N= ( x i , y i , z i ) T .
r 1 =dtan δ 2 +( D 1 2d )tan δ 3 ,
r 2 = d r 1 tanα cot δ 5 +tanα ,
r 3 =[ d( r 1 + r 2 )tanα ]tan δ 10 + D 3 tan δ 11 .
R bz = r 1 + r 2 r 3 .
D c ={ ( r 1 + r 2 )[ d( r 1 + r 2 )tanα ]tan δ 2 }cot δ 3 .
λ 2 1( m 1 +1 ) λ 1 m 2 ,
λ 3 m 1 λ 1 + m 2 .
θ r1 =φ180°.
φ={ arccos( x 6 x 6 2 + y 6 2 ), y 6 0 2πarccos( x 6 x 6 2 + y 6 2 ), y 6 <0
[ x p y p ]=[ f x ( θ r2 , θ r3 ) f y ( θ r2 , θ r3 ) ]=F( θ ).
J=[ f x θ r2 f x θ r3 f y θ r2 f y θ r3 ]
L 4 ( θ r2 )= i=s2 s+2 [ j=s2,ji s+2 ( θ r2 θ j θ i θ j )g( θ i ) ].
g'( θ s )= 1 12h [ g( θ s2 )8g( θ s1 )+8g( θ s+1 )g( θ s+2 ) ] h 4 30 g ( 5 ) ( ξ )
θ r1 =( φ180° )±f( σ ).
θ r1 ( k )={ [ φ( k )180° ] f max 90° arccos[ σ( k ) σ max ], 1k<p [ φ( k )180° ]+ f max 90° arccos[ σ( k ) σ max ], p<kq θ r1 ( k1 )+ θ r1 ( k+1 ) 2 , k=p

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