Abstract

A new technique for displacement measurement is proposed that makes use of phase singularities in the complex signal generated by a Laguerre-Gauss filter operation applied to a speckle pattern. The core structures of phase singularities are used as unique fingerprints attached to the object surface, and the displacement is determined by tracing the movement of registered phase singularities with their correspondence being identified by the fingerprints. Experimental results for translational and rotational displacement measurements are presented that demonstrate large dynamic range and high spatial resolution of the proposed optical vortex metrology.

© 2006 Optical Society of America

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  1. A. E. Ennos, "Speckle interferometry," in Laser Speckle and Related Phenomena, J. C. Dainty ed. (Springer -Verlag, Berlin, 1976).
  2. R. K. Erf, Speckle Metrology, (Academic Press, New York, 1978).
  3. R. S. Sirohi, Speckle Metrology, (Marcel Dekker Inc., New York, 1993).
  4. N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements, (Springer-Verlag, Berlin, 1998).
  5. S. J. Kirkpatrick and D. D. Duncan, "Optical assessment of tissue mechanics," in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).
  6. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
    [CrossRef]
  7. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain," Appl. Opt. 44, 4909-4915 (2005).
    [CrossRef] [PubMed]
  8. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006).
    [CrossRef] [PubMed]
  9. J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. Roy. Soc. Lond. A 336165-190 (1974).
    [CrossRef]
  10. J. W. Goodman, Statistical Optics, (Wiley-Interscience, New York, 2000).
  11. D. Gabor, "Theory of communications," J. Inst. Electr. Eng. 93, 429-457 (1946).
  12. M. Riesz, "Sur les fonctions conjuguees," Math. Zeitschrift 27, 218-244 (1927).
    [CrossRef]
  13. K. G. Larkin, D. J. Bone and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001).
    [CrossRef]
  14. T. Yokozeki, "Displacement measurement using phase singularities in the Riesz transform of a speckle pattern," Master Thesis, The University of Electro-Communications (January 30, 2006).
  15. C. -S. Guo, Y. -J. Han, J. -B. Xu, and J. Ding, "Radial Hilbert transform with Laguerre-Gaussian spatial filters," Opt. Lett. 31, 1394-1396 (2006).
    [CrossRef] [PubMed]
  16. M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A,  456, 2059-2079 (2000).
    [CrossRef]
  17. W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
    [CrossRef] [PubMed]
  18. M. R. Dennis, "Local structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6,S202-S208 (2004).
    [CrossRef]
  19. Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "The fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6, S217-S228 (2004).
    [CrossRef]
  20. M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  21. A. Asundi and H. North, "White-light speckle method- Current trends," Opt. Laser Eng. 29, 159-169 (1998).
    [CrossRef]
  22. See, for example, X. Dai, O. Sasaki, J. E. Greivenkamp, and T. Suzuki, "Measurement of small rotation angles by a parallel interference pattern," Appl. Opt. 34, 6380-6388 (1995).
    [CrossRef] [PubMed]
  23. B. Rose, H. Imam, S. G. Hanson, H. Y. Yura, and R. S. Hansen, "Laser speckle angular displacement sensor: theoretical and experimental study" Appl. Opt. 37, 2119-2129 (1998).
    [CrossRef]

2006

2005

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
[CrossRef]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain," Appl. Opt. 44, 4909-4915 (2005).
[CrossRef] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

2004

M. R. Dennis, "Local structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6,S202-S208 (2004).
[CrossRef]

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "The fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

2002

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
[CrossRef]

2001

2000

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A,  456, 2059-2079 (2000).
[CrossRef]

1998

1995

1974

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. Roy. Soc. Lond. A 336165-190 (1974).
[CrossRef]

1946

D. Gabor, "Theory of communications," J. Inst. Electr. Eng. 93, 429-457 (1946).

1927

M. Riesz, "Sur les fonctions conjuguees," Math. Zeitschrift 27, 218-244 (1927).
[CrossRef]

Asundi, A.

A. Asundi and H. North, "White-light speckle method- Current trends," Opt. Laser Eng. 29, 159-169 (1998).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A,  456, 2059-2079 (2000).
[CrossRef]

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. Roy. Soc. Lond. A 336165-190 (1974).
[CrossRef]

Bone, D. J.

Dai, X.

Dennis, M. R.

M. R. Dennis, "Local structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6,S202-S208 (2004).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
[CrossRef]

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A,  456, 2059-2079 (2000).
[CrossRef]

Ding, J.

Egorov, Y. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "The fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

Fadeyeva, T. A.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "The fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

Gabor, D.

D. Gabor, "Theory of communications," J. Inst. Electr. Eng. 93, 429-457 (1946).

Greivenkamp, J. E.

Guo, C. -S.

Han, Y. -J.

Hansen, R. S.

Hanson, S. G.

Imam, H.

Ishii, N.

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain," Appl. Opt. 44, 4909-4915 (2005).
[CrossRef] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
[CrossRef]

Ishijima, R.

Larkin, K. G.

Miyamoto, Y.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006).
[CrossRef] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain," Appl. Opt. 44, 4909-4915 (2005).
[CrossRef] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
[CrossRef]

W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

North, H.

A. Asundi and H. North, "White-light speckle method- Current trends," Opt. Laser Eng. 29, 159-169 (1998).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. Roy. Soc. Lond. A 336165-190 (1974).
[CrossRef]

Oldfield, M. A.

Riesz, M.

M. Riesz, "Sur les fonctions conjuguees," Math. Zeitschrift 27, 218-244 (1927).
[CrossRef]

Rose, B.

Sasaki, O.

Suzuki, T.

Takeda, M.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006).
[CrossRef] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
[CrossRef]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain," Appl. Opt. 44, 4909-4915 (2005).
[CrossRef] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

Volyar, A. V.

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "The fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

Wada, A.

Wang, W.

W. Wang, T. Yokozeki, R. Ishijima, A. Wada, S. G. Hanson, Y. Miyamoto, and M. Takeda, "Optical vortex metrology for nanometric speckle displacement measurement," Opt. Express 14, 120-127 (2006).
[CrossRef] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement observation based on phase-only correlation in the signal domain," Appl. Opt. 44, 4909-4915 (2005).
[CrossRef] [PubMed]

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
[CrossRef]

W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

Xu, J. -B.

Yokozeki, T.

Yura, H. Y.

Appl. Opt.

J. Inst. Electr. Eng.

D. Gabor, "Theory of communications," J. Inst. Electr. Eng. 93, 429-457 (1946).

J. Opt. A: Pure Appl. Opt.

M. R. Dennis, "Local structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6,S202-S208 (2004).
[CrossRef]

Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, "The fine structure of singular beams in crystals: colours and polarization," J. Opt. A: Pure Appl. Opt. 6, S217-S228 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Math. Zeitschrift

M. Riesz, "Sur les fonctions conjuguees," Math. Zeitschrift 27, 218-244 (1927).
[CrossRef]

Opt. Commun.

W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement," Opt. Commun. 248, 59-68 (2005).
[CrossRef]

M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
[CrossRef]

Opt. Express

Opt. Laser Eng.

A. Asundi and H. North, "White-light speckle method- Current trends," Opt. Laser Eng. 29, 159-169 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

W. Wang, S. G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A

M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A,  456, 2059-2079 (2000).
[CrossRef]

Proc. Roy. Soc. Lond. A

J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. Roy. Soc. Lond. A 336165-190 (1974).
[CrossRef]

Other

J. W. Goodman, Statistical Optics, (Wiley-Interscience, New York, 2000).

A. E. Ennos, "Speckle interferometry," in Laser Speckle and Related Phenomena, J. C. Dainty ed. (Springer -Verlag, Berlin, 1976).

R. K. Erf, Speckle Metrology, (Academic Press, New York, 1978).

R. S. Sirohi, Speckle Metrology, (Marcel Dekker Inc., New York, 1993).

N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements, (Springer-Verlag, Berlin, 1998).

S. J. Kirkpatrick and D. D. Duncan, "Optical assessment of tissue mechanics," in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed., (SPIE Press, Bellingham, 2002).

T. Yokozeki, "Displacement measurement using phase singularities in the Riesz transform of a speckle pattern," Master Thesis, The University of Electro-Communications (January 30, 2006).

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Figures (5)

Fig. 1.
Fig. 1.

Core structure around a phase singularity with zero crossings of real and imaginary parts of complex signal representation for a speckle pattern. (a) Amplitude contours and zero crossing lines; (b) Pseudophase structure.

Fig. 2.
Fig. 2.

(a). Displacement of phase singularities, ◦: before displacement; ■: after displacement; Histograms of coordinate changes of phase singularities, (b) x -direction;(c) y -direction.

Fig. 3.
Fig. 3.

Results of phase singularities after fine identification (a) Displacement of phase singularities; Histograms of coordinate changes of phase singularities, (b) x -direction;(c) y -direction.

Fig. 4.
Fig. 4.

Schematic diagram for in-plane rotation measurement by optical vortex metrology

Fig. 5.
Fig. 5.

Results of optical vortex metrology applied for rotational displacement. Left column: rough searching over whole probe area; Right column: local searching for fine identification. (a), (b) Displacement of phase singularities; (c), (d) Histogram of rotation angle of phase singularities (Note the difference in the scales.)

Equations (21)

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I ˜ ( x , y ) = + + LG ( f x , f y ) · ( f x , f y ) exp [ j 2 π ( f x x + f y y ) ] d f x d f y ,
LG ( f x , f y ) = ( f x + j f y ) exp [ ( f x 2 + f y 2 ) / ω 2 ] = ρ exp ( ρ 2 / ω 2 ) exp ( ) .
I ˜ ( x , y ) = I ˜ ( x , y ) exp [ ( x , y ) ] = I ( x , y ) * ( x , y ) ,
( x , y ) = 1 { LG ( f x , f y ) } = ( j π 2 ω 4 ) ( x + jy ) exp [ π 2 ω 2 ( x 2 + y 2 ) ]
= ( j π 2 ω 4 ) [ r exp ( π 2 r 2 ω 2 ) exp ( ) ] ,
Re [ I ˜ ( x , y ) ] = a r x + b r y + c r , Im [ I ˜ ( x , y ) ] = a i x + b i y + c i ,
e = 1 ( a r 2 + a i 2 + b r 2 + b i 2 ) ( a r 2 + a i 2 b r 2 b i 2 ) 2 + 4 ( a r b r + a i b i ) 2 ( a r 2 + a i 2 + b r 2 + b i 2 ) + ( a r 2 + a i 2 b r 2 b i 2 ) 2 + 4 ( a r b r + a i b i ) 2 ,
θ RI = { arctan [ ( a r b i a i b r ) / ( a r a i + b r b i ) ] θ RI < π / 2 π arctan [ ( a r b i a i b r ) / ( a r a i + b r b i ) ] θ RI > π / 2 ,
Ω = a r b i a r b r ,
q = sgn ( Ω · e z ) = sgn ( a r b i a i b r ) .
q = q′ ,
Δ e = e e′ < ε 1 ,
ΔΩ = ( Ω Ω ) / ( Ω + Ω′ ) < ε 2 ,
Δ θ RI = θ RI θ′ RI < ε 3 ,
E = ( e e′ ) 2 + ( Ω Ω′ Ω + Ω′ ) 2 + [ 2 π ( θ RI θ′ RI ) ] 2 .
A i x + B i y + C i = 0
A i = 2 ( x i x′ i ) , B i = 2 ( y i y′ i ) , and C i = ( x′ i 2 + y′ i 2 x i 2 y i 2 ) ,
x c = ( i B i 2 ) ( i A i C i ) ( i A i B i ) ( i B i C i ) ( i A i 2 ) ( i B i 2 ) ( i A i B i ) 2 ,
y c = ( i A i 2 ) ( i B i C i ) ( i A i B i ) ( i A i C i ) ( i A i 2 ) ( i B i 2 ) ( i A i B i ) 2 .
x = x c + ( x i x c ) cos ϕ + ( y i y c ) sin ϕ ,
y = y c ( x i x c ) sin ϕ + ( y i y c ) cos ϕ .

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