Abstract

Laser radar cross sections have been evaluated for a number of ideal targets such as cones, spheres, paraboloids, and cylinders by use of different reflection characteristics. The time-independent cross section is the ratio of the cross section of one of these forms to that of a plate with the same maximum radius. The time-dependent laser radar cross section involves the impulse response from the object shape multiplied by the beam’s transverse profile and the surface bidirectional reflection distribution function. It can be clearly seen that knowledge of the combined effect of object shape and reflection characteristics is important for determining the shape and the magnitude of the laser radar return. The results of this study are of interest for many laser radar applications such as ranging, three-dimensional imaging–modeling, tracking, antisensor lasers, and target recognition.

© 2000 Optical Society of America

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References

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  1. A. V. Jelalian, Laser Radar Systems (Artech House, Boston, Mass., 1992).
  2. O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
    [CrossRef]
  3. O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
    [CrossRef]
  4. B. Yons, M. Timmins, “Upgrades to the DELTASNRCTM (Defense Laser Target Signatures Code) for the evaluation of advanced LADAR technologies,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 164–175 (1998).
    [CrossRef]
  5. M. Wellfare, L. Love, K. McCarley, L. Prestwood, “Ladar image synthesis with comprehensive sensor model,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 208–219 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. J. C. Stover, Optical Scattering—Measurements and Analysis (McGraw-Hill, New York, 1990).
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. O. Steinvall, “Theory for laser systems performance modelling,” (Defence Research Establishment, Linköping, Sweden, 1997).
  14. R. E. Walker, J. W. McLean, “Lidar equations for turbid media with pulse stretching,” Appl. Opt. 38, 2384–2397 (1999).
    [CrossRef]
  15. D. Letalick, I. Renhorn, O. Steinvall, “Measured signal amplitude distributions for a coherent FM-cw CO2 laser radar,” Appl. Opt. 25, 3927–3938 (1986).
    [CrossRef] [PubMed]

1999 (1)

1998 (1)

1994 (1)

1991 (1)

1986 (1)

1982 (1)

1981 (1)

1979 (1)

Ahlberg, S.

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

Bolander, G.

O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
[CrossRef]

Capron, B. A.

Carlsson, C.

O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
[CrossRef]

Ginneken, B.

Harney, R. C.

Jelalian, A. V.

A. V. Jelalian, Laser Radar Systems (Artech House, Boston, Mass., 1992).

Jungert, E.

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

Koenderink, J. J.

Kologo, N.

Leader, J. C.

Letalick, D.

D. Letalick, I. Renhorn, O. Steinvall, “Measured signal amplitude distributions for a coherent FM-cw CO2 laser radar,” Appl. Opt. 25, 3927–3938 (1986).
[CrossRef] [PubMed]

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
[CrossRef]

Love, L.

M. Wellfare, L. Love, K. McCarley, L. Prestwood, “Ladar image synthesis with comprehensive sensor model,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 208–219 (1996).
[CrossRef]

McCarley, K.

M. Wellfare, L. Love, K. McCarley, L. Prestwood, “Ladar image synthesis with comprehensive sensor model,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 208–219 (1996).
[CrossRef]

McLean, J. W.

Nerry, F.

Olsson, H.

O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
[CrossRef]

Prestwood, L.

M. Wellfare, L. Love, K. McCarley, L. Prestwood, “Ladar image synthesis with comprehensive sensor model,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 208–219 (1996).
[CrossRef]

Renhorn, I.

Sandberg, M.

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

Shapiro, J. H.

Söderman, U.

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

Stavridi, M.

Steinvall, O.

D. Letalick, I. Renhorn, O. Steinvall, “Measured signal amplitude distributions for a coherent FM-cw CO2 laser radar,” Appl. Opt. 25, 3927–3938 (1986).
[CrossRef] [PubMed]

O. Steinvall, “Theory for laser systems performance modelling,” (Defence Research Establishment, Linköping, Sweden, 1997).

O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
[CrossRef]

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

Stoll, M. P.

Stover, J. C.

J. C. Stover, Optical Scattering—Measurements and Analysis (McGraw-Hill, New York, 1990).

Timmins, M.

B. Yons, M. Timmins, “Upgrades to the DELTASNRCTM (Defense Laser Target Signatures Code) for the evaluation of advanced LADAR technologies,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 164–175 (1998).
[CrossRef]

Walker, R. E.

Wellfare, M.

M. Wellfare, L. Love, K. McCarley, L. Prestwood, “Ladar image synthesis with comprehensive sensor model,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 208–219 (1996).
[CrossRef]

Yons, B.

B. Yons, M. Timmins, “Upgrades to the DELTASNRCTM (Defense Laser Target Signatures Code) for the evaluation of advanced LADAR technologies,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 164–175 (1998).
[CrossRef]

Yura, H. T.

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

Other (7)

J. C. Stover, Optical Scattering—Measurements and Analysis (McGraw-Hill, New York, 1990).

O. Steinvall, “Theory for laser systems performance modelling,” (Defence Research Establishment, Linköping, Sweden, 1997).

A. V. Jelalian, Laser Radar Systems (Artech House, Boston, Mass., 1992).

O. Steinvall, H. Olsson, G. Bolander, C. Carlsson, D. Letalick, “Gated viewing for target detection and recognition,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 432–448 (1999).
[CrossRef]

O. Steinvall, U. Söderman, S. Ahlberg, M. Sandberg, D. Letalick, E. Jungert, “Airborne laser radar: systems and methods for reconnaissance and terrain modelling,” in Laser Radar Technology and Applications IV, G. W. Kamerman, C. Werner, eds., Proc. SPIE3707, 12–26 (1999).
[CrossRef]

B. Yons, M. Timmins, “Upgrades to the DELTASNRCTM (Defense Laser Target Signatures Code) for the evaluation of advanced LADAR technologies,” in Laser Radar Technology and Applications III, G. W. Kamerman, ed., Proc. SPIE3380, 164–175 (1998).
[CrossRef]

M. Wellfare, L. Love, K. McCarley, L. Prestwood, “Ladar image synthesis with comprehensive sensor model,” in Laser Radar Technology and Applications, G. W. Kamerman, ed., Proc. SPIE2748, 208–219 (1996).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Geometry for calculation of laser radar cross sections.

Fig. 2
Fig. 2

Geometry for definition of the BRDF.

Fig. 3
Fig. 3

Illustration of the shapes of BRDFdiff and BRDFspek for several values of parameters m and s.

Fig. 4
Fig. 4

(a) Surface profile of surface 1 together with (b) a histogram of the surface slopes. The rms of the surface height is 2.13 µm, and that of the surface slope is 0.2958.

Fig. 5
Fig. 5

(a) Surface profile of surface 2 together with (b) a histogram of the surface slopes. The rms of the surface height is 1.58 µm, and that of the surface slope is 0.2349.

Fig. 6
Fig. 6

Monostatic BRDF measured and fitted from Eq. (12) for surface 1.

Fig. 7
Fig. 7

Monostatic BRDF measured and fitted from Eq. (12) for surface 2. The rms surface height and slope are smaller than for surface 1, as is reflected in the slightly more pronounced peak, which is due to multiple glints.

Fig. 8
Fig. 8

(a) Ratio between the laser radar cross section for the cone and that of the flat plate (σ/σplate) with the same r max and BRDF of the type B cos(θ) m . (b) Ratio between the laser radar cross section for a cone and that of a flat plate (σ/σplate) with the same r max and BRDF of the type [A/cos6(θ)]exp[-tan2(θ)/s 2].

Fig. 9
Fig. 9

(a) Ratio between the laser radar cross section for a sphere and that for a flat plate (σ/σplate) with the same r max and BRDF of the type B cos(θ) m . (b) Specular-type target and (σ/σplate) with the same r max and BRDF of the type [A/cos6(θ)]exp[-tan2(θ)/s 2].

Fig. 10
Fig. 10

(a) Ratio of beam size to target size for a laser radar cross section of a paraboloid. Diffuse type of target. (b) Same ratio for a specular type of target.

Fig. 11
Fig. 11

(a) Relative cross sections of a cylinder of the diffuse type and (b) specular targets.

Fig. 12
Fig. 12

(a) Cross section of a plate relative to those of various objects versus the target-to-beam radius x max. Diffuse target, m = 1. (b) Cross section of a plate relative to those of various objects versus the target-to-beam radius x max. Specular target with s = 0.1 and s = 0.01.

Fig. 13
Fig. 13

Impulse response for a diffuse cone with a 45-deg half-angle: (a) r max = 1 m, (b) r beam = 1 m. X max = r max/r beam.

Fig. 14
Fig. 14

(a) Impulse response for a fixed beam radius r beam = 1 m. (b) Pulse shortening for several spectral reflections. Higher values of m mean more specular reflection; r beam = 1 m.

Fig. 15
Fig. 15

Response from a sphere with a fixed radius of 1 m. (a) Note the pulse shortening for X max = 0.1 for a diffuse sphere, meaning that only a limited part of the sphere is illuminated. For a specular sphere (b) the pulse is short for all beam sizes.

Fig. 16
Fig. 16

Impulse response for a paraboloid. z = kr 2, k = 1. (a) Diffuse reflection; m = 1. (b) Specular case shown on a log–log scale for clarity.

Fig. 17
Fig. 17

Impulse response for a long cylinder. (a) Diffuse reflection; m = 1. (b) Specular case.

Fig. 18
Fig. 18

Impulse response for a long cylinder. (a) Diffuse reflection; m = 1. (b) Specular case, which gives a much shorter pulse, as expected. The cylinder radius is constant, 1 m. Note the small difference in the pulse responses for beam radii of 1 and 10 m, especially for specular reflection.

Fig. 19
Fig. 19

Comparison of the waveforms (in arbitrary amplitude units) of the three shapes, with r beam/R max = 0.1, 0.333, 1. The large footprint (r beam/R max = 1) results in a large difference in shape between the waveforms and indicates the target recognition potential from waveform analysis.

Equations (26)

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Pr=PTGT4πR2σ4πR2πD24 ηatmηsys,
Pr=ηsys4PTπθT2R2 ρreflATa×ARΩTaR2 Tatm2
Pr=ηsysPTρreflARΩTaR2 Tatm2.
Ir=target Ix, y, z, tρbx, y, zdSR2,
Ix, y, z, t=I0gx, ysz=I0gx, yst-2Rx, y/c,
Prt=ηsystIrArR2 Tatm2=ηsystI0×target gx, yρbx, y, zst-2zx, y/cdxdyR2ArR2 Tatm2.
σ=4π target gx, yρbx, y, zst-2zx, y/cdxdy.
σ=4πρbAtarget cosθ,
dIr=I0R2 ρbθexp-r22rbeam2cosθdS.
Ir=I0R20rmax,T ρbrexp-r22rbeam22πr1+dz/dr21/2dr,
BRDFdifferential radiancedifferential irradiancedPs/dΩsPi cos θsPs/ΩsPi cos θs.
BRDF=BRDFspec+BRDFdiff=Acos6 θexp-tan2θ/s2+B cosm θ,
Ir=I0R2sinαρbπ/2-α2π 0rma,T r exp-r22rbeam2dr, =I0R2sinαρbπ/2-α2πrbeam2Qxmax.
Qxmax=1-exp-xmax2/2.
σplate=4πρb02πrbeam2Qq=4πρb0πw02Qxmax.
Ir=I0R2 2πrbeam2Q1, Q1=0xmax x exp-x2/21-x/xmax20.5ρbxdx,
Q1=0xmax x exp-x2/21+2kx/xmax2-0.5ρbxdx,
Ir=I0R20Lexp-y2/2rbeam2dy 0xmax 1 exp-x2/21-x2/x2 max0.5ρbxdx,
IcylIplate=σcylσplate=0xmax1-x2/xmax2exp-x2/2ρbxdxerfxmax/2.
σ=4π target gx, yρbx, y, zst-2zx, y/cdxdy=stδt=ht.
st=- hτpt-τdτ.
St=ptarget pscintSst,
PtargetA=aσ2exp-A22σ2+QI02QAσ,
PtargetS=12σ2exp-S2σ2+QI02SQσ.
Pscint=1S2πσln I2exp-ln(S/Ssav+1/2σln I222σln I2.
Pt=t2τ2exp-t/τ.

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