Abstract

This article presents several fundamental formulas for ray tracing in optical systems used in 3D optical scanners. A procedure for numerical modeling of one-mirror and two-mirror optical systems is presented, and the calculation of positioning and accuracy of the laser beam spot in a detection plane is carried out. Finally, a point position and accuracy depending on a transit time is evaluated.

© 2014 Optical Society of America

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References

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  1. G. F. Marshall and G. E. Stutz, Handbook of Optical and Laser Scanning, Vol. 147 (CRC, 2011).
  2. G. Vosselman and H.-G. Maas, Airborne and Terrestrial Laser Scanning (Whittles, 2010).
  3. “Control System—Laser Scanning—Surveying Works,” 2013, http://www.controlsystem.cz/en .
  4. “Surphaser 3D Scanners,” 2013, http://www.surphaser.com .
  5. “Leica Geosystems—HDS,” 2013, http://hds.leica-geosystems.com/en/index.htm .
  6. “TOPCON Global Gateway,” 2013, http://global.topcon.com .
  7. “RIEGL Laser Measurement Systems,” 2013, http://www.riegl.com .
  8. “FARO Laser Scanner Focus3D,” 2013, http://www.faro.com/focus/uk .
  9. “MDL,” 2013, http://www.mdl.co.uk/en/14735.aspx .
  10. “Galvos—Scanning Mirrors—Optical Scanners,” 2013, http://www.cambridgetechnology.com .
  11. “Laser Sensors—IR Temperature Sensors—High Precision Displacement, and Position Measurement—Micro-Epsilon Measurement,” 2013, http://www.micro-epsilon.com/index.html .
  12. R. J. Pegis and M. Rao, “Analysis and design of plane-mirror systems,” Appl. Opt. 2, 1271–1274 (1963).
    [CrossRef]
  13. J. Cohen-Sabban, Y. Cohen-Sabban, and A. Roussel, “Distortion-free 2-D space and surface scanners using light deflectors,” Appl. Opt. 22, 3935–3942 (1983).
    [CrossRef]
  14. Y. J. Li and J. Katz, “Laser beam scanning by rotary mirrors. I. Modeling mirror-scanning devices,” Appl. Opt. 34, 6403–6416 (1995).
    [CrossRef]
  15. Y. J. Li, “Laser beam scanning by rotary mirrors. II. Conic-section scan patterns,” Appl. Opt. 34, 6417–6430 (1995).
    [CrossRef]
  16. Y. J. Li, “Single-mirror beam steering system: analysis and synthesis of high-order conic-section scan patterns,” Appl. Opt. 47, 386–398 (2008).
    [CrossRef]
  17. Y. J. Li, “Beam deflection and scanning by two-mirror and two-axis systems of different architectures: a unified approach,” Appl. Opt. 47, 5976–5985 (2008).
    [CrossRef]
  18. Y. Friedman and N. Schweitzer, “Classification of stable configurations of plane mirrors,” Appl. Opt. 37, 7229–7234 (1998).
    [CrossRef]
  19. R. Shinozaki, O. Sasaki, and T. Suzuki, “Fast scanning method for one-dimensional surface profile measurement by detecting angular deflection of a laser beam,” Appl. Opt. 43, 4157–4163 (2004).
    [CrossRef]
  20. P. Pokorny, “Theoretical foundations of one-mirror and two-mirror optical scanners,” bachelor’s thesis (Czech Technical University in Prague, Faculty of Civil Engineering, Prague, 2012).
  21. A. Miks, Applied Optics (Czech Technical University, 2009), p. 230.
  22. M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
  23. A. Miks, Physics 2: Electromagnetic Field (Czech Technical University, 2005), p. 162.
  24. A. Yariv, Quantum Electronics (Wiley, 1967).
  25. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).
  26. J. A. Stratton, Electromagnetic Theory (Adams, 2008).
  27. A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University, 2010), p. 572.
  28. F. Trager, Springer Handbook of Lasers and Optics (Springer, 2007).
  29. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).
  30. E. Madelung, Die mathematischen Hilfsmittel des Physikers (Springer-Verlag, 1964).
  31. K. Rektorys, Survey of Applicable Mathematics (SNTL, 1969).
  32. K.-R. Koch, Parameter Estimation and Hypothesis Testing in Linear Models (Springer, 1999).
  33. L. Mervart and Z. Lukes, Adjustment Calculus (Czech Technical University, 2007).
  34. S. V. Gupta, Measurement Uncertainties: Physical Parameters and Calibration of Instruments (Springer-Verlag, 2012).

2008

2004

1998

1995

1983

1963

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Cohen-Sabban, J.

Cohen-Sabban, Y.

Friedman, Y.

Gupta, S. V.

S. V. Gupta, Measurement Uncertainties: Physical Parameters and Calibration of Instruments (Springer-Verlag, 2012).

Katz, J.

Koch, K.-R.

K.-R. Koch, Parameter Estimation and Hypothesis Testing in Linear Models (Springer, 1999).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

Li, Y. J.

Lipson, A.

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University, 2010), p. 572.

Lipson, H.

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University, 2010), p. 572.

Lipson, S. G.

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University, 2010), p. 572.

Lukes, Z.

L. Mervart and Z. Lukes, Adjustment Calculus (Czech Technical University, 2007).

Maas, H.-G.

G. Vosselman and H.-G. Maas, Airborne and Terrestrial Laser Scanning (Whittles, 2010).

Madelung, E.

E. Madelung, Die mathematischen Hilfsmittel des Physikers (Springer-Verlag, 1964).

Marshall, G. F.

G. F. Marshall and G. E. Stutz, Handbook of Optical and Laser Scanning, Vol. 147 (CRC, 2011).

Mervart, L.

L. Mervart and Z. Lukes, Adjustment Calculus (Czech Technical University, 2007).

Miks, A.

A. Miks, Physics 2: Electromagnetic Field (Czech Technical University, 2005), p. 162.

A. Miks, Applied Optics (Czech Technical University, 2009), p. 230.

Pegis, R. J.

Pokorny, P.

P. Pokorny, “Theoretical foundations of one-mirror and two-mirror optical scanners,” bachelor’s thesis (Czech Technical University in Prague, Faculty of Civil Engineering, Prague, 2012).

Rao, M.

Rektorys, K.

K. Rektorys, Survey of Applicable Mathematics (SNTL, 1969).

Roussel, A.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Sasaki, O.

Schweitzer, N.

Shinozaki, R.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (Adams, 2008).

Stutz, G. E.

G. F. Marshall and G. E. Stutz, Handbook of Optical and Laser Scanning, Vol. 147 (CRC, 2011).

Suzuki, T.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Trager, F.

F. Trager, Springer Handbook of Lasers and Optics (Springer, 2007).

Vosselman, G.

G. Vosselman and H.-G. Maas, Airborne and Terrestrial Laser Scanning (Whittles, 2010).

Wolf, E.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, 1967).

Appl. Opt.

Other

P. Pokorny, “Theoretical foundations of one-mirror and two-mirror optical scanners,” bachelor’s thesis (Czech Technical University in Prague, Faculty of Civil Engineering, Prague, 2012).

A. Miks, Applied Optics (Czech Technical University, 2009), p. 230.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

A. Miks, Physics 2: Electromagnetic Field (Czech Technical University, 2005), p. 162.

A. Yariv, Quantum Electronics (Wiley, 1967).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

J. A. Stratton, Electromagnetic Theory (Adams, 2008).

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University, 2010), p. 572.

F. Trager, Springer Handbook of Lasers and Optics (Springer, 2007).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover, 2000).

E. Madelung, Die mathematischen Hilfsmittel des Physikers (Springer-Verlag, 1964).

K. Rektorys, Survey of Applicable Mathematics (SNTL, 1969).

K.-R. Koch, Parameter Estimation and Hypothesis Testing in Linear Models (Springer, 1999).

L. Mervart and Z. Lukes, Adjustment Calculus (Czech Technical University, 2007).

S. V. Gupta, Measurement Uncertainties: Physical Parameters and Calibration of Instruments (Springer-Verlag, 2012).

G. F. Marshall and G. E. Stutz, Handbook of Optical and Laser Scanning, Vol. 147 (CRC, 2011).

G. Vosselman and H.-G. Maas, Airborne and Terrestrial Laser Scanning (Whittles, 2010).

“Control System—Laser Scanning—Surveying Works,” 2013, http://www.controlsystem.cz/en .

“Surphaser 3D Scanners,” 2013, http://www.surphaser.com .

“Leica Geosystems—HDS,” 2013, http://hds.leica-geosystems.com/en/index.htm .

“TOPCON Global Gateway,” 2013, http://global.topcon.com .

“RIEGL Laser Measurement Systems,” 2013, http://www.riegl.com .

“FARO Laser Scanner Focus3D,” 2013, http://www.faro.com/focus/uk .

“MDL,” 2013, http://www.mdl.co.uk/en/14735.aspx .

“Galvos—Scanning Mirrors—Optical Scanners,” 2013, http://www.cambridgetechnology.com .

“Laser Sensors—IR Temperature Sensors—High Precision Displacement, and Position Measurement—Micro-Epsilon Measurement,” 2013, http://www.micro-epsilon.com/index.html .

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Figures (14)

Fig. 1.
Fig. 1.

Refraction and reflection on the plane.

Fig. 2.
Fig. 2.

Reflection on the planar mirror.

Fig. 3.
Fig. 3.

One-mirror scanner.

Fig. 4.
Fig. 4.

System of two mirrors.

Fig. 5.
Fig. 5.

System of two mirrors in the plane (x, y).

Fig. 6.
Fig. 6.

Reflection on second mirror.

Fig. 7.
Fig. 7.

Point P in the detection plane for φ1={45°,43°,,43°,45°} and φ2={40°,38°,,38°,40°} with parameters from example 1.

Fig. 8.
Fig. 8.

Coordinate standard deviation in the detection plane—example 1.

Fig. 9.
Fig. 9.

Simulation of the laser beam spot in the detection plane for the expected values of the angles φ1=20° and φ2=10° and 10,000 repetitions.

Fig. 10.
Fig. 10.

Progression of sample coordinate standard deviation with an increase in the number of repetitions for angles φ1=20° and φ2=10°.

Fig. 11.
Fig. 11.

Error ellipsoids (scaled 5001) for different transit time t, φ1=0° and φ2={15°,10°,,10°,15°} with parameters from example 1.

Fig. 12.
Fig. 12.

Scheme of two-mirror scanner and point P in the detection plane for φ1={30°,29°,,29°,30°}, φ2={18°,17°,,17°,18°} with parameters from example 2.

Fig. 13.
Fig. 13.

Coordinate standard deviation in the detection plane—example 2.

Fig. 14.
Fig. 14.

Error ellipsoids (scaled 10001) for transit time t={0.33·108s,0.67·108s}, φ1={30°,20°,,20°,30°} and φ2={15°,5°,5°,15°} with parameters from example 2.

Equations (64)

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nIsinαI=nTsinαT,
N=gradF(r)|gradF(r)|.
AT=nInTAI1nTN(nT2nI2sin2αInIcosαI).
AR=AI+2NcosαI,AR=AI2N(N·AI).
N(φ)=Ncosφ+C(C·N)(1cosφ)+(C×N)sinφ,
N(dφ)=N+(C×N)dφ.
A2=A12N(A1·N)=MA1.
M=(12Nx22NxNy2NxNz2NxNy12Ny22NyNz2NxNz2NyNz12Nz2),A1=(A1xA1yA1z),A2=(A2xA2yA2z),N=(NxNyNz).
Ak+1=|A12(A1·N1)2(A1·N2)2(A1·N3)2(A1·Nk)N112(N1·N2)2(N1·N3)2(N1·Nk)N2012(N2·N3)2(N2·Nk)N3001Nk0001|,
A3=|A12(A1·N1)2(A1·N2)N112(N1·N2)N201|=A12N1(A1·N1)2N2(A1·N2)+4N2(N1·N2)(A1·N1).
N(φ1)=N(0)cosφ1+C1(C1·N(0))(1cosφ1)+(C1×N(0))sinφ1,
A2=A12N(φ1)(A1·N(φ1)),
Ni(φi)=Ni(0)cosφi+Ci(Ci·Ni(0))(1cosφi)+[Ci×Ni(0)]sinφi,Ai+1(φi)=Ai(φi)2Ni(φi)(Ai(φi)·Ni(φi)),
A3=A2cosφ2+C2(C2·A2)(1cosφ2)+(C2×A2)sinφ2
A3T=Ry(φ2)A2T,
Ry(φ2)=(cosφ20sinφ2010sinφ20cosφ2)
ξ((rbq)·q)=0,
rP=pA3.
rP=bA3·qA3.
rP=dA3,
dm=ct=2d+dS+dD,
rP=12(ctdSdD)A3.
rP=r2+p3A3(φ2),
r2=ajaj·N2(φ2)A2(φ1)·N2(φ2)A2(φ1),
p3=b+r2·qA3(φ2)·q,
A3(φ2)=A2(φ1)2N2(φ2)(A2(φ1)·N2(φ2)),
A2(φ1)=A1(0)2N1(φ1)(A1(0)·N1(φ1)),
dm=ct=2(d+a)+dS+dD.
rP=r2+dA3(φ2),
d=12(ctdSdD)a.
a=|r2aj|=aj·N2(φ2)A2(φ1)·N2(φ2).
rP=aj+aj·N2(φ2)A2(φ1)·N2(φ2)[A3(φ2)A2(φ1)]+12(ctdSdD)A3(φ2).
σxy=σx2+σy22,sxy=sx2+sy22,
si=jδij2n,
Σf=JΣxJT.
σfi2=j=1n(fi/xj)2σxj2.
A2=jsinφ1+kcosφ1,A3=icosφ1sinφ2jsinφ1+kcosφ1cosφ2,rP=ibtanφ2jbtanφ1cosφ2+kb.
xP=btanφ2,yP=btanφ1cosφ2,zP=b.
yP=tanφ1sinφ2xP.
xPbφ2+b3φ23,yPbφ1b2φ1φ22b3φ13,
δxPxP=13φ22,δyPyP=12φ22.
σxy=σx2+σy22,
σx2=(xPφ1)2σφ12+(xPφ2)2σφ22=(bcos2φ2)2σφ22,
σy2=(yPφ1)2σφ12+(yPφ2)2σφ22=(bcos2φ1cosφ2)2σφ12+(btanφ1sinφ2cos2φ2)2σφ22.
xP=12(ctdSdD)cosφ1sinφ2,yP=12(ctdSdD)sinφ1,zP=12(ctdSdD)cosφ1cosφ2.
σct=cσt,
σx2=[12(dS+dDct)sinφ1sinφ2]2σφ12+[12(ctdSdD)cosφ1cosφ2]2σφ22+[12cosφ1sinφ2]2σct2,
σy2=[12(dS+dDct)cosφ1]2σφ12+[12sinφ1]2σct2,
σz2=[12(dS+dDct)sinφ1cosφ2]2σφ12+[12(dS+dDct)cosφ1sinφ2]2σφ22+[12cosφ1cosφ2]2σct2.
A1(0)=i,C1=i,C2=k,q=i,α1=α2=45°,N1(0)=(i+j)/2,N2(0)=(i+j)/2,
N1(φ1)=12(i+jcosφ1+ksinφ1),
A2(φ1)=(jcosφ1+ksinφ1),
N2(φ2)=i2(cosφ2sinφ2)+j2(cosφ2+sinφ2),
A3(φ2)=icosφ1(12sin2φ2)+jcosφ1sin2φ2ksinφ1.
r2=aktanφ1,
rP=ib+jbtan2φ2ktanφ1(a+bcos2φ2).
xP=b,yP=btan2φ2,zP=tanφ1(a+bcos2φ2).
zP=(tanφ1sin2φ2)yPatanφ1.
yP2bφ2+83bφ23,zPφ1(a+b)2bφ1φ22,
δyPyP=43φ22,δzPzP21+a/bφ222φ22,
σyz=σy2+σz22,
σy2=(yPφ1)2σφ12+(yPφ2)2σφ22=(2bcos22φ2)2σφ22,
σz2=(zPφ1)2σφ12+(zPφ2)2σφ22=(b+acos2φ2cos2φ1cos2φ2)2σφ12+(2btanφ1tan2φ2cos2φ2)2σφ22.
xP=[12(ctdSdD)acosφ1]cosφ1cos2φ2,yP=[12(ctdSdD)acosφ1]cosφ1sin2φ2,zP=12[(dS+dDct)sinφ1].

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