Abstract

A dual-wedge scanner has potential applications in laser imaging radar. To realize fast scanning imaging without a blind region, the rotation rates of the wedges have to be controlled to perform beam scanning along appropriate track paths. The first-order paraxial approximation method is employed to investigate the 2D scan patterns and path density for different angular frequency ratios of the wedges rotating steadily in the same and opposite directions. The frame rate of no-blind-region scanning imaging is estimated in terms of the imaging coverage requirement. The internal relations between the rotation rates, the instantaneous field of view (IFOV), and the imaging velocity are revealed. The results show that the spiral scanning trace, resulting from co-rotating wedges, is dense in the center and sparse at the edge of the scanning field. The reverse results can be obtained for the rosette scanning trace, resulting from counter-rotating wedges. The denser the scanning trace is, the longer the scan period is. The faster the wedges rotate and the wider the IFOV is, the higher the frame rate is. When the ratio of the width of IFOV to the angular radius of the scanning field is 0.15, the frame rate of no-blind-region spiral scanning imaging can be up to 18 fps for wedge rotation rate of 12000r/min, and that for rosette scanning imaging can be up to 20 fps.

© 2014 Optical Society of America

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References

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    [CrossRef]
  3. Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).
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    [CrossRef]
  5. G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999).
    [CrossRef]
  6. J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. M. Sanchez and D. Gutow, “Control laws for a three-element Risley prism optical beam pointer,” Proc. SPIE 6304, 630403 (2006).
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2013 (2)

2012 (1)

X. Jin, Y. Yan, J. Sun, Y. Zhou, and L. Liu, “Angle-Doppler resolved reflective tomography laser imaging radar,” Acta Opt. Sinica 32, 0828001 (2012).
[CrossRef]

2011 (1)

Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).

2008 (1)

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

2007 (1)

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

2006 (2)

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

J. Khoury, C. L. Woods, J. Lorenzo, J. Kierstead, D. Pyburn, and S. Sengupta, “Resolution limits in imaging ladar systems,” Appl. Opt. 45, 697–704 (2006).
[CrossRef]

1999 (1)

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

Blask, S.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Bu, H.

Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).

Cai, Y.

Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).

Cannata, R.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Carson, R.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Clifton, W.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Davidsohn, D.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Degnan, J.

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

Fan, D.

Field, C.

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

Gutow, D.

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

Hei, M.

Higgins, T.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Jin, X.

X. Jin, Y. Yan, J. Sun, Y. Zhou, and L. Liu, “Angle-Doppler resolved reflective tomography laser imaging radar,” Acta Opt. Sinica 32, 0828001 (2012).
[CrossRef]

Jodor, G.

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

Khoury, J.

Kierstead, J.

Lawrence, D.

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

Leventhal, E.

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

Liu, G.

Liu, L.

X. Jin, Y. Yan, J. Sun, Y. Zhou, and L. Liu, “Angle-Doppler resolved reflective tomography laser imaging radar,” Acta Opt. Sinica 32, 0828001 (2012).
[CrossRef]

Lorenzo, J.

Lu, Y.

Machan, R.

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

Marshall, G. F.

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

Pfannenstiel, J.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Pyburn, D.

Reynolds, V.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Sanchez, M.

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

Sengupta, S.

Shu, R.

Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).

Sun, J.

X. Jin, Y. Yan, J. Sun, Y. Zhou, and L. Liu, “Angle-Doppler resolved reflective tomography laser imaging radar,” Acta Opt. Sinica 32, 0828001 (2012).
[CrossRef]

Tong, X.

Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).

Vaidyanathan, M.

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

Woods, C. L.

Yan, Y.

X. Jin, Y. Yan, J. Sun, Y. Zhou, and L. Liu, “Angle-Doppler resolved reflective tomography laser imaging radar,” Acta Opt. Sinica 32, 0828001 (2012).
[CrossRef]

Zhou, Y.

Acta Opt. Sinica (1)

X. Jin, Y. Yan, J. Sun, Y. Zhou, and L. Liu, “Angle-Doppler resolved reflective tomography laser imaging radar,” Acta Opt. Sinica 32, 0828001 (2012).
[CrossRef]

Appl. Opt. (3)

J. Astronaut. (1)

Y. Cai, X. Tong, H. Bu, and R. Shu, “Study on image deformation of spaceborne three-dimensional LIDAR imaging in no-equilibrium state,” J. Astronaut. 32, 407–413 (2011).

Proc. SPIE (4)

M. Vaidyanathan, S. Blask, T. Higgins, W. Clifton, D. Davidsohn, R. Carson, V. Reynolds, J. Pfannenstiel, and R. Cannata, “Jigsaw phase III: a miniaturized airborne 3-D imaging laser radar with photo-counting sensitivity for foliage penetration,” Proc. SPIE 6550, 65500N (2007).
[CrossRef]

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

J. Degnan, R. Machan, E. Leventhal, D. Lawrence, G. Jodor, and C. Field, “Inflight performance of a second generation, photon counting, 3D imaging lidar,” Proc. SPIE 6950, 695007 (2008).
[CrossRef]

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

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Figures (11)

Fig. 1.
Fig. 1.

Schematic diagram of the rotational double-prism beam-steering system. Notation and coordinate systems for Risley prism system. The unit vector s^1i for the incident ray is collinear and negative with the Z axis, which is also the optical axis of the system. The zero orientation of both prisms is described with the opening angle pointing up and the positive rotational angle will follow the right hand rule around the Z axis.

Fig. 2.
Fig. 2.

Steering mechanism study with the first-order paraxial approximation method. (a) Pointing position prediction. The total ray deviation of the prisms can be obtained with the vector sum of δ^1 and δ^2. (b) Orientations’ inverse solution. Two triangle diagrams can be graphed for a pointing position. Therefore, there are two groups of inverse oriental solutions.

Fig. 3.
Fig. 3.

Scanning patterns of the co-rotating wedges. The angular frequency ratio k=T1/T2=f2/f1 and the symbol Ns represents the number of spiral lines.

Fig. 4.
Fig. 4.

Number of spiral lines and scanning period for the single frame scanning imaging of co-rotating wedges.

Fig. 5.
Fig. 5.

Coverage diagrams of the scanning fields of co-rotating wedges for different angular frequency ratio k. The gray squares represent the IFOVs.

Fig. 6.
Fig. 6.

Relation between imaging frame and angular frequency of wedges θ/θm for spiral scanning imaging without blind region.

Fig. 7.
Fig. 7.

Scanning patterns of the counter-rotating wedges. The angular frequency ratio and the symbol represents the scanning period.

Fig. 8.
Fig. 8.

Density distribution of rosette scanning lines. (a) Density description of scanning. (b) Radial distribution of the maximum tangential space lm of scanning lines.

Fig. 9.
Fig. 9.

Imaging analysis of rosette scanning without blind region.

Fig. 10.
Fig. 10.

Imaging region without a blind region for rosette scanning. The gray squares represent the IFOVs and the circle areas represent the imaging regions without blind region.

Fig. 11.
Fig. 11.

Relation between imaging frame and angular frequency of wedges, θ/θm for rosette scanning imaging without blind region.

Equations (29)

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{δ1=α1(n11)δ2=α2(n21).
{δX=δ1cosθ1+δ2cosθ2δY=δ1sinθ1+δ2sinθ2.
{θ1=2πf1t+θ10θ2=±2πf2t+θ20,
{X=Pδ[cos(2πf1t+θ10)+cos(±2πf2t+θ20)]Y=Pδ[sin(2πf1t+θ10)+sin(±2πf2t+θ20)].
Ts=nT1=mT2,
R=2Pδ|cos[π(f1f2)t+θ10θ202]|
Θ=π(f1±f2)t+θ10+θ202.
TR={1/|f1f2|for the same rotational direction1/(f1+f2)for inverse rotational direction.
Ts={|nm|TRfor the same rotational direction(n+m)TRfor inverse rotational direction.
ΔΘ=π(f1+f2)Ts=πf1(1+k)Ts=πn(1+k).
ΔΘ=ΔΘ+|nm|π={2nπ,whenf1>f22mπ,whenf1<f2.
Ns=Δθ2π={n;whenf1>f2m,whenf1<f2.
TF=12TR=12|nm|Ts=12|1k|T1,
Ns=π(f1+f2)TF2π=1+k4|1k|.
T=2ππ(f1+f2)=2(f1+f2).
r=2Pδsin[π|f1f2|T]=2Pδsin[2|1k|π(1+k)].
θrP=2δsin[2|1k|π1+k].
τθ=θθmsin[2|1k|π1+k],
{k2πarcsin(θ/θm)2π+arcsin(θ/θm),forf1>f2k2π+arcsin(θ/θm)2πarcsin(θ/θm),forf1<f2.
{fF4arcsin(θ/θm)2π+arcsin(θ/θm)f1,forf1>f2fF4arcsin(θ/θm)2π+arcsin(θ/θm)f2,forf1<f2.
No=TsTR=n+m=n(1+k).
w=2R1k1+karccos(R2Pδ).
s=|2NπRNow|.
Nl=floor(wNo2πR),Nl=ceil(wNo2πR),
l=|2NlπRNow|,l=|2NlπRNow|,lm=max(l,l),
wm=2.237(1k)Pδ(1+k).
s=2πRNow=2πRn(1+k)w.
k2.237Pδw2.237Pδ+w
n2πR(w+w)(1+k).

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