Abstract

Off-axis scattering of laser beams propagating in the atmosphere has been imaged by two separated cameras. We give a theoretical analysis and report experiments that show how these images can be used to reconstruct the position and orientation of the beam relative to the cameras. The information from a single image of the beam only determines the beam within a plane. However, the intersection of these planes of ambiguity using images from two cameras can determine the beam uniquely. When the two planes are nearly parallel, an independent method based on the relative radiance at each camera can be used to determine the beam direction.

© 2013 Optical Society of America

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References

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  1. S. G. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 220157 (1983).
    [CrossRef]
  2. N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47, 086002 (2008).
    [CrossRef]
  3. J. K. Michulec and R. Schleijpen, “Influence of aerosols on off-axis laser detection capabilities,” Proc. SPIE 7463, 1–12 (2009).
  4. O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
    [CrossRef]
  5. F. Hanson, C. Bendall, C. Deckard, and H. Haidar, “Off-axis detection and characterization of laser beams in the maritime atmosphere,” Appl. Opt. 50, 3050–3056 (2011).
    [CrossRef]
  6. T. Moons, L. V. Gool, and M. Vergauwen, “3D reconstruction from multiple images part 1: principles,” Found. Trends Comput. Graph. Vis. 4, 287–404 (2008).
    [CrossRef]
  7. A. M. J. van Eijk, J. T. Kusmierczyk-Michulec, and J. J. Piazzola, “The Advanced Navy Aerosol Model (ANAM): validation of small-particle modes,” Proc. SPIE 8161, 816108 (2011).
    [CrossRef]
  8. J.-Y. Bouguet, “Camera calibration toolbox for Matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#updates .
  9. R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1995).

2011

F. Hanson, C. Bendall, C. Deckard, and H. Haidar, “Off-axis detection and characterization of laser beams in the maritime atmosphere,” Appl. Opt. 50, 3050–3056 (2011).
[CrossRef]

A. M. J. van Eijk, J. T. Kusmierczyk-Michulec, and J. J. Piazzola, “The Advanced Navy Aerosol Model (ANAM): validation of small-particle modes,” Proc. SPIE 8161, 816108 (2011).
[CrossRef]

2009

J. K. Michulec and R. Schleijpen, “Influence of aerosols on off-axis laser detection capabilities,” Proc. SPIE 7463, 1–12 (2009).

2008

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47, 086002 (2008).
[CrossRef]

T. Moons, L. V. Gool, and M. Vergauwen, “3D reconstruction from multiple images part 1: principles,” Found. Trends Comput. Graph. Vis. 4, 287–404 (2008).
[CrossRef]

2002

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

1983

S. G. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 220157 (1983).
[CrossRef]

Bendall, C.

Deckard, C.

Dubovik, O.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Eck, T. F.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1995).

Gathman, S. G.

S. G. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 220157 (1983).
[CrossRef]

Gool, L. V.

T. Moons, L. V. Gool, and M. Vergauwen, “3D reconstruction from multiple images part 1: principles,” Found. Trends Comput. Graph. Vis. 4, 287–404 (2008).
[CrossRef]

Haidar, H.

Hanson, F.

Holben, B.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1995).

Kaufman, Y. J.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

King, M. D.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Kusmierczyk-Michulec, J. T.

A. M. J. van Eijk, J. T. Kusmierczyk-Michulec, and J. J. Piazzola, “The Advanced Navy Aerosol Model (ANAM): validation of small-particle modes,” Proc. SPIE 8161, 816108 (2011).
[CrossRef]

Michulec, J. K.

J. K. Michulec and R. Schleijpen, “Influence of aerosols on off-axis laser detection capabilities,” Proc. SPIE 7463, 1–12 (2009).

Moons, T.

T. Moons, L. V. Gool, and M. Vergauwen, “3D reconstruction from multiple images part 1: principles,” Found. Trends Comput. Graph. Vis. 4, 287–404 (2008).
[CrossRef]

Piazzola, J. J.

A. M. J. van Eijk, J. T. Kusmierczyk-Michulec, and J. J. Piazzola, “The Advanced Navy Aerosol Model (ANAM): validation of small-particle modes,” Proc. SPIE 8161, 816108 (2011).
[CrossRef]

Reid, F.

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47, 086002 (2008).
[CrossRef]

Roy, N.

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47, 086002 (2008).
[CrossRef]

Schleijpen, R.

J. K. Michulec and R. Schleijpen, “Influence of aerosols on off-axis laser detection capabilities,” Proc. SPIE 7463, 1–12 (2009).

Slutsker, I.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Smirnov, A.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Tanré, D.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

van Eijk, A. M. J.

A. M. J. van Eijk, J. T. Kusmierczyk-Michulec, and J. J. Piazzola, “The Advanced Navy Aerosol Model (ANAM): validation of small-particle modes,” Proc. SPIE 8161, 816108 (2011).
[CrossRef]

Vergauwen, M.

T. Moons, L. V. Gool, and M. Vergauwen, “3D reconstruction from multiple images part 1: principles,” Found. Trends Comput. Graph. Vis. 4, 287–404 (2008).
[CrossRef]

Appl. Opt.

Found. Trends Comput. Graph. Vis.

T. Moons, L. V. Gool, and M. Vergauwen, “3D reconstruction from multiple images part 1: principles,” Found. Trends Comput. Graph. Vis. 4, 287–404 (2008).
[CrossRef]

J. Atmos. Sci.

O. Dubovik, B. Holben, T. F. Eck, A. Smirnov, Y. J. Kaufman, M. D. King, D. Tanré, and I. Slutsker, “Variability of absorption and optical properties of key aerosol types observed in worldwide locations,” J. Atmos. Sci. 59, 590–608 (2002).
[CrossRef]

Opt. Eng.

S. G. Gathman, “Optical properties of the marine aerosol as predicted by the Navy aerosol model,” Opt. Eng. 22, 220157 (1983).
[CrossRef]

N. Roy and F. Reid, “Off-axis laser detection model in coastal areas,” Opt. Eng. 47, 086002 (2008).
[CrossRef]

Proc. SPIE

J. K. Michulec and R. Schleijpen, “Influence of aerosols on off-axis laser detection capabilities,” Proc. SPIE 7463, 1–12 (2009).

A. M. J. van Eijk, J. T. Kusmierczyk-Michulec, and J. J. Piazzola, “The Advanced Navy Aerosol Model (ANAM): validation of small-particle modes,” Proc. SPIE 8161, 816108 (2011).
[CrossRef]

Other

J.-Y. Bouguet, “Camera calibration toolbox for Matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#updates .

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1995).

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Figures (8)

Fig. 1.
Fig. 1.

Off-axis imaging of a laser beam (B) projecting from a source (S) with two cameras (C1) and (C2).

Fig. 2.
Fig. 2.

Scattering geometry when the laser beam (B), source (S), and cameras (C1) and (C2) all lie in the same plane. The scattering angle ψ+θ is the same at points β1 and β2 when θ1=θ2+ϕC.

Fig. 3.
Fig. 3.

Errors in the calculated beam elevation angle for the laboratory experiment.

Fig. 4.
Fig. 4.

Errors in the calculated beam azimuth angle for the laboratory experiment.

Fig. 5.
Fig. 5.

Errors in the calculated beam elevation angle for the outdoor experiment.

Fig. 6.
Fig. 6.

Errors in the calculated beam azimuth angle for the outdoor experiment.

Fig. 7.
Fig. 7.

Radiance (dashed) versus sensor angle (θ) and scaled radiance (solid) versus scattering angle (ζ=θ+ψ) in arbitrary units for camera 1 (gray) and camera 2 (black). The laser beam azimuth angle ϕB=1.9°.

Fig. 8.
Fig. 8.

Error functions based on scaled radiance ratios for eight different beam azimuth angles at zero elevation angle. True beam angles are shown as dotted vertical lines.

Equations (10)

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u=n1×n2/|n1×n2|,
S=C1+s1[(C2C1)·n2]/[s1·n2],
L(θ)=P0T(z,r)β(ψ+θ)Rsin(ψ),
T(z,r)=exp[α(z+r)],
L1(ζψ1)L2(ζψ2)=R2sin(ψ2)R1sin(ψ1)exp[α(z1z2+r1r2)].
χ2(ϕB)=ζaζbdζ[LS1(ζψ1)LS2(ζψ2)R1sin(ψ1)R2sin(ψ2)1]2/(ζbζa).
w=Ry(θy±ϕC/2)Rx(θx)Rz(δθz)·v,
Rx(θ)=[1000cos(θ)sin(θ)0sin(θ)cos(θ)]Ry(θ)=[cos(θ)0sin(θ)010sin(θ)0cos(θ)]Rz(θ)=[cos(θ)sin(θ)0sin(θ)cos(θ)0001].
tan(θx)=txsin(δθz)tycos(δθz),
tan(θy)=2[txcos(δθz)tysin(δθz)][2+tx2+ty2+(ty2tx2)cos(2δθz)+2txtysin(2δθz)]1/2,

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