Abstract

The interaction between a rotating object and laser beams has been studied in the field of laser Doppler velocimetry, where two incident laser beams are focused on one small spot of the rotating surface and interferometry is used. In the case of a single incident laser beam illuminating a relatively large area of the rotating surface, both the Doppler broadening and the reflected-power level are dictated by points distributed over a wide curved area at varying angles of incidence. An analytical model of spectra in backscatter from cones and cylinders rotating around their axes is presented. This analytical solution may contribute to laser Doppler velocimetry as well as ladar applications.

© 2000 Optical Society of America

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References

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  1. L. Lading, “Principles of laser anemometry,” in Optical Diagnostics for Flow Processes, L. Lading, P. Buchhave, G. Wigley, eds. (Plenum, New York, 1994), pp. 84–126.
  2. J. D. Briers, “Laser Doppler and time-varying speckle: a reconciliation,” J. Opt. Soc. Am. A 13, 345–350 (1996).
    [CrossRef]
  3. S. G. Hanson, L. Lading, “Generics of systems for measuring linear and angular velocities of solid objects,” in Optical Velocimetry, M. Pluta, K. Jobczynski, eds., Proc. SPIE2729, 81–90 (1996).
    [CrossRef]
  4. E. G. Arik, “Recent developments in fiber optic and laser sensors for flow, surface vibration, rotation, and velocity measurements,” in Fiber Optic and Laser Sensors IX, R. P. DePaula, E. Odd, eds., Proc. SPIE1584, 202–211 (1991).
    [CrossRef]
  5. H. T. Yura, B. Rose, S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
    [CrossRef]
  6. H. T. Yura, S. G. Hanson, L. Landing, “Laser Doppler velocimetry:  an analytical solution to the optical system including the effects of partial coherence of the target,” J. Opt. Soc. Am. A 12, 2040–2047 (1995).
    [CrossRef]
  7. R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
    [CrossRef]
  8. C. G. Bachman, Laser Radar Systems and Techniques (Artech House, Dedham, Mass., 1979).
  9. A. V. Jelalian, Laser Radar Systems (Artech House, Boston, Mass., 1992).
  10. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978), pp. 24–28.
  11. J. C. Stover, Optical Scattering:  Measurement and Analysis (McGraw-Hill, New York, 1990), pp. 223–227.
  12. R. C. McMillan, R. B. Davidson, R. L. Robertson, W. Farmer, “Light-weight, low-volume CO2 ladar technology,” in Applied Laser Radar Technology II, G. W. Kamerman, ed., Proc. SPIE2472, 132–141 (1995).
    [CrossRef]
  13. M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
    [CrossRef]
  14. A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

1998 (2)

H. T. Yura, B. Rose, S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
[CrossRef]

R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
[CrossRef]

1996 (1)

1995 (1)

Arik, E. G.

E. G. Arik, “Recent developments in fiber optic and laser sensors for flow, surface vibration, rotation, and velocity measurements,” in Fiber Optic and Laser Sensors IX, R. P. DePaula, E. Odd, eds., Proc. SPIE1584, 202–211 (1991).
[CrossRef]

Bachman, C. G.

C. G. Bachman, Laser Radar Systems and Techniques (Artech House, Dedham, Mass., 1979).

Briers, J. D.

Czyzak, S.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Davidson, R. B.

R. C. McMillan, R. B. Davidson, R. L. Robertson, W. Farmer, “Light-weight, low-volume CO2 ladar technology,” in Applied Laser Radar Technology II, G. W. Kamerman, ed., Proc. SPIE2472, 132–141 (1995).
[CrossRef]

Farmer, W.

R. C. McMillan, R. B. Davidson, R. L. Robertson, W. Farmer, “Light-weight, low-volume CO2 ladar technology,” in Applied Laser Radar Technology II, G. W. Kamerman, ed., Proc. SPIE2472, 132–141 (1995).
[CrossRef]

Ghoshroy, S.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Hanson, S. G.

Hasson, V.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Jelalian, A. V.

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

Kawasaki, N.

R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
[CrossRef]

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978), pp. 24–28.

Kovacs, M.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Lading, L.

S. G. Hanson, L. Lading, “Generics of systems for measuring linear and angular velocities of solid objects,” in Optical Velocimetry, M. Pluta, K. Jobczynski, eds., Proc. SPIE2729, 81–90 (1996).
[CrossRef]

L. Lading, “Principles of laser anemometry,” in Optical Diagnostics for Flow Processes, L. Lading, P. Buchhave, G. Wigley, eds. (Plenum, New York, 1994), pp. 84–126.

Landing, L.

McMillan, R. C.

R. C. McMillan, R. B. Davidson, R. L. Robertson, W. Farmer, “Light-weight, low-volume CO2 ladar technology,” in Applied Laser Radar Technology II, G. W. Kamerman, ed., Proc. SPIE2472, 132–141 (1995).
[CrossRef]

Mita, G.

R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
[CrossRef]

Omori, R.

R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
[CrossRef]

Pohle, R.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Robertson, R. L.

R. C. McMillan, R. B. Davidson, R. L. Robertson, W. Farmer, “Light-weight, low-volume CO2 ladar technology,” in Applied Laser Radar Technology II, G. W. Kamerman, ed., Proc. SPIE2472, 132–141 (1995).
[CrossRef]

Rose, B.

Ruffatto, D.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

Stover, J. C.

J. C. Stover, Optical Scattering:  Measurement and Analysis (McGraw-Hill, New York, 1990), pp. 223–227.

Suzuki, A.

R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
[CrossRef]

Wendt, R.

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

Yura, H. T.

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys., Part 1 (1)

R. Omori, G. Mita, N. Kawasaki, A. Suzuki, “Two dimensional measurement of rotational speed of a diffuse object based on laser Doppler anemometry,” Jpn. J. Appl. Phys., Part 1 37, L450–L452 (1998).
[CrossRef]

Other (10)

C. G. Bachman, Laser Radar Systems and Techniques (Artech House, Dedham, Mass., 1979).

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

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978), pp. 24–28.

J. C. Stover, Optical Scattering:  Measurement and Analysis (McGraw-Hill, New York, 1990), pp. 223–227.

R. C. McMillan, R. B. Davidson, R. L. Robertson, W. Farmer, “Light-weight, low-volume CO2 ladar technology,” in Applied Laser Radar Technology II, G. W. Kamerman, ed., Proc. SPIE2472, 132–141 (1995).
[CrossRef]

M. Kovacs, S. Ghoshroy, V. Hasson, R. Pohle, D. Ruffatto, S. Czyzak, R. Wendt, “Phase 1 high performance CO2 ladar surveillance sensor description and test results,” in Gas and Chemical Lasers, R. C. Sze, ed., Proc. SPIE2702, 84–94 (1996).
[CrossRef]

A. E. Siegman, Lasers (University Science, Sausalito, Calif., 1986).

S. G. Hanson, L. Lading, “Generics of systems for measuring linear and angular velocities of solid objects,” in Optical Velocimetry, M. Pluta, K. Jobczynski, eds., Proc. SPIE2729, 81–90 (1996).
[CrossRef]

E. G. Arik, “Recent developments in fiber optic and laser sensors for flow, surface vibration, rotation, and velocity measurements,” in Fiber Optic and Laser Sensors IX, R. P. DePaula, E. Odd, eds., Proc. SPIE1584, 202–211 (1991).
[CrossRef]

L. Lading, “Principles of laser anemometry,” in Optical Diagnostics for Flow Processes, L. Lading, P. Buchhave, G. Wigley, eds. (Plenum, New York, 1994), pp. 84–126.

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Figures (12)

Fig. 1
Fig. 1

Block diagram of main components in the considered laser transmitter and receiver system.

Fig. 2
Fig. 2

Cone and cylinder in coordinate axis used for the model. The laser source is considered to be below the xy plane, and the collimated laser beam is parallel to the z axis.

Fig. 3
Fig. 3

(a) Meridians used in the definition of the lateral surfaces of cones and cylinders, (b) surface of revolution obtained when α is not zero, (c) same surface as in (b) but after a clockwise rotation of γ in the yz plane.

Fig. 4
Fig. 4

Angles of surface normals in the yz plane.

Fig. 5
Fig. 5

Position of the edge of the illuminated region as a function of aspect angle.

Fig. 6
Fig. 6

Relationship between the angle of the illuminated-region edge (δ) and the aspect angle (γ) for three different half-cone angles (α). The diagonal (dashed line) is the case of the cylinder. The value of α for each of the three curves is equal to the value of γ when δ is zero, i.e., 5°, 10°, and 20°, respectively, from the diagonal.

Fig. 7
Fig. 7

Vector relations for a point rotating around an arbitrary axis in the yz plane.

Fig. 8
Fig. 8

Normalized-power spectra for cylinders and cones with diffuse Lambertian reflectance. In each plot the curves correspond to the aspect angles of 10°, 20°, 30°, 40°, 50°, and 60° from bottom to top. (a) Cylinder with a radius of 0.1 m, (b) cylinder with a radius of 0.2 m, (c) cone with a half-cone angle of 5.7°, (d) cone with a half-cone angle of 11.3°.

Fig. 9
Fig. 9

Normalized-power spectra for cylinders and cones with diffuse exponential reflectance. In each plot the curves correspond to the aspect angles of 20°, 30°, 40°, 50°, and 60° from bottom to top. (a) Cylinder with a radius of 0.1 m, (b) cylinder with a radius of 0.2 m, (c) cone with a half-cone angle of 5.7°, (d) cone with a half-cone angle of 11.3°.

Fig. 10
Fig. 10

Doppler-broadened spectrum of the backscatter from a rotating cone with half-cone angle 12.7°, aspect angle 30°, and angular velocity 0.54 rad/s. The solid curve corresponds to experimental data, and the dashed curve is the model outcome.

Fig. 11
Fig. 11

Normalized-interval-power spectra for cylinders and cones with diffuse Lambertian reflectance. (a), (b) Cylinder with a radius of 0.1 m at aspect angles of 10° and 60°, respectively; (c), (d) cone with a half-cone angle of 5.7° at aspect angles of 10° and 60°, respectively.

Fig. 12
Fig. 12

Normalized-interval-power spectra for cylinders and cones with diffuse exponential reflectance. (a), (b) Cylinder with a radius of 0.1 m at aspect angles of 10° and 60°, respectively; (c), (d) cone with a half-cone angle of 5.7° at aspect angles of 10° and 60°, respectively.

Equations (77)

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ΔP=JρΔAAcR2T2,
ΔP(β)=Kρ(β)ΔA cos β,
y=z tan α+c.
x2+y2=z tan α+c,
y  y cos γ-z sin γ,z  z cos γ+y sin γ,
f(x,y,z)=x2+y2(cos2 γ-tan2 α sin2 γ)+z2(sin2 γ-tan2 α cos2 γ)-2cy tan α sin γ-2cz tan α cos γ-2yz(sin γ cos γ)(1+tan2 α)-c2=0.
fx=f(x, y, z)x,fy=f(x, y, z)y,fz=f(x, y, z)z.
n=1(fx2+fy2+fz2)1/2(fx, fy, fz),
cos β=fz(fx2+fy2+fz2)1/2.
fx=f(x, y, z)x=2x,
fy=f(x, y, z)y=2y(cos2 γ-tan2 α sin2 γ)-2c tan α sin γ-2z(sin γ cos γ)(1+tan2 α),
fz=f(x, y, z)z=2z(sin2 γ-tan2 α cos2 γ)-2c tan α cos γ-2y(sin γ cos γ)(1+tan2 α).
fy=±2[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2,
fz=2z tan2 α+2c tan α cos γ±2(sin γ cos γ)(1+tan2 α)[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2tan2 α sin2 γ-cos2 γ.
fx2+fy2+fz2=4(1+tan2 α){(cos γ)(z tan α+c cos γ)±(tan α sin γ)[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2}2(tan2 α sin2 γ-cos2 γ)2.
cos β=-(sin α cos α)(z sin α+c cos α cos γ)±(sin γ cos γ)[x2(sin2 α-cos2 γ)+(z sin α+c cos α cos γ)2]1/2(cos α cos γ)(z sin α+c cos α cos γ)±(sin γ sin α)[x2(sin2 α-cos2 γ)+(z sin α+c cos α cos γ)2]1/2.
cos β=(sin γ)c2-x2c.
cos β=-sin α cos α±sin γ cos γcos α cos γ±sin γ sin α.
(sin α cos α)(z sin α+c cos α cos γ)=(sin γ cos γ)[x2(sin2 α-cos2 γ)+(z sin α+c cos α cos γ)2]1/2.
x=±(z sin α+c cos α cos γ)(sin2 γ-sin2 α)1/2sin γ cos γ.
z=y(cos γ sin γ)(cos2 γ-sin2 α)-c(sin α cos α)(sin2 γ-cos2 γ)sin2 γ cos2 γ-sin2 α cos2 α,
Δf=2ωr cos θλ.
r=u-a,
a2=a·u,
a=y sin γ+z sin γ.
r2=u2-a2,
r2=x2+y2+z2-(y sin γ+z cos γ)2=x2+(y cos γ-z sin γ)2.
v  a,v  u,v=1.
v=a×u|a×u|,
a×u=aijk0sin γcos γxyz,
a×u=a[i(z sin γ-y cos γ)+j x cos γ-k x sin γ],
|a×u|=a[x2+(z sin γ-y cos γ)2]1/2.
vz2=cos2 θ=x2 sin2 γx2+(y cos γ-z sin γ)2.
r cos θ=±x sin γ,
x=±λΔf2ω sin γ.
ΔA=ΔxΔz|cos η|=ΔxΔz(fx2+fy2+fz2)1/2fy,
ΔA=ΔxΔzg(x, z),
g(x, z)=(cos α cos γ)(z sin α+c cos α cos γ)±(sin α sin γ)[x2(sin2 α-cos2 γ)+(z sin α+c cos α cos γ)2]1/2(cos α)(sin2 α-cos2 γ)[x2(sin2 α-cos2 γ)+(z sin α+c cos α cos γ)2]1/2.
y tan γ+z=hcos γ,
y tan γ+z=0.
CO:
0<z<-y tan γ+hcos γ,cos β<0,
CY:
-y tan γ<z<-y tan γ+hcos γ,
cos β<0.
ρ(β)=kL cos β,
ΔP(β)=KLΔxΔzg(x, z)cos2 β,
P(x)=KLΔxzCYg(x, z)cos2 β dz.
cos β=-sin γ,
ρ(β)=kE exp(-τβ).
ΔP(β)=KEΔxΔzg(x, z)(cos β)exp(-τβ),
ui(z)=1σ2πexp[-(z-zi)2/2σ2],
Pi(x)=KLΔxZCYorzCOui(z)g(x, z)cos2 β dz.
Pi(x)=KLΔxZCYorzCOui(z)g(x, z)(cos β)exp(-τβ)dz.
y=z(sin γ cos γ)(1+tan2 α)+c tan α sin γ±[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2cos2 γ-tan2 α sin2 γ.
x2(sin2 α-cos2 γ)+(z sin α+c cos α cos γ)20.
sin α|sin(90-γ)|,
90-αγ90+α.
|x|z sin α+c cos α cos γ(cos2 γ-sin2 α)1/2,
fy=±2[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2,
fz=2z tan2 α+2c tan α cos γ±2(sin γ cos γ)(1+tan2 α)[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2tan2 α sin2γ-cos2 γ.
x2+z2=c2.
cos β=zx2+z2.
fx=f(x, y, z)x=2x,
fy=f(x, y, z)y=-2cos γsin γ(z+c sin γ),
fz=f(x, y, z)z=(z+c sin γ)2(sin2 γ-cos2 γ)-x2 sin2 γ(sin2 γ)(z+c sin γ).
y=(x2-c2)sin2 γ+z2(sin2 γ-cos2 γ)-2cz sin γ cos2 γ2(sin γ cos γ)(c sin γ+z).
cos β=±x2 sin2γ+(z+c sin γ)2-2(sin2 γ)(z+c sin γ)2x2 sin2 γ+(z+c sin γ)2.
fx2+fy2+fz2=N(tan2 α sin2 γ-cos2 γ)2,
N=4x2(tan2 α sin2 γ-cos2 γ)2+4[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2](tan2 α sin2 γ-cos2 γ)2+4 tan2 α(z tan α+z cos γ)2+4(sin2 γ cos2 γ)(1+tan2 α)2×[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]±8(tan α)(z tan α+c cos γ)×(sin γ cos γ)(1+tan2 α)[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]1/2.
4x2 cos4 γ+4x2 tan4 α sin4 γ-8x2 tan2 α sin2 γ cos2 γ.
4[x2(tan2 α sin2 γ-cos2 γ)+(z tan α+c cos γ)2]×(tan4 α sin2 γ+cos2 γ),
4x2 tan6 α sin4 γ-4x2 tan4 α sin2 γ cos2 γ+4(tan4 α sin2 γ)(z tan α+c cos γ)2+4x2 tan2 α sin2 γ cos2 γ-4x2 cos4 γ+4(cos2 γ)×(z tan α+c cos γ)2.
4(z tan α+c cos γ)2(tan4 α sin2 γ+cos2 γ+tan2 α)+4x2(tan4 α sin4 γ)(1+tan2 α)-4x2(tan2 α sin2 γ cos2γ)(1+tan2 α),
tan4 α sin2 γ+cos2 γ+tan2 α=(sin2 γ)(tan4 α-1)+tan2 α+1=[(sin2 γ)(tan2 α-1)+1](tan2 α+1)=(sin2 γ tan2 α+cos2 γ)(tan2 α+1).
4(1+tan2 α){(tan2α sin2 γ)(z tan α+c cos γ)2+(cos2 γ)(z tan α+c cos γ)2+x2(tan2 α sin2 γ)×(sin2 γ tan2 α-cos2 γ)±2(tan α)(z tan α+c cos γ)(sin γ cos γ)×[x2(sin2 γ tan2 α-cos2 γ)+(z tan α+c cos γ)2]1/2},
{(sin γ tan α)[x2(sin2 γ tan2 α-cos2 γ)+(z tan α+c cos γ)2]1/2±(cos γ)(z tan α+c cos γ)}2.

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