Abstract

Abstract

A new technique for fluid mechanics measurement is proposed that makes use of the elliptic anisotropy of phase singularities in the complex signal representation of a speckle-like pattern. Based on the formal analogy between the polarization field of a vector wave and the gradient field of the complex signal, the Poincaré sphere representation has been used to characterize the phase singularities that serve as unique fingerprints attached to the seeding particles moving with the flow. Experimental results for flow velocity and acceleration measurement are presented that demonstrate the validity of the proposed optical vortex metrology for fluid mechanics measurement.

© 2007 Optical Society of America

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References

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  1. N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements (Springer-Verlag, 1998), Chap. 8.
  2. M. Raffel, C. Willert, and J. Kompenhans, Particle Image Velocimetry, (Springer-Verlag, 2002), Chap. 5.
  3. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
    [CrossRef]
  4. M. S. Soskin and M. V. Vasnetsov, "Singular Optics," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2001).
  5. 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]
  6. 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]
  7. W. Wang, T. Yokozeki, R. Ishijima, S. G. Hanson, and M. Takeda, "Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern," Opt. Express 14, 10195-10206 (2006).
    [CrossRef] [PubMed]
  8. 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]
  9. M. R. Dennis, "Local structure of wave dislocation lines: twist and twirl," J. Opt. A: Pure Appl. Opt. 6, S202-S208 (2004).
    [CrossRef]
  10. 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]
  11. M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002).
    [CrossRef]
  12. I. Freund, "Stokes-vector reconstruction," Opt. Lett. 15, 1425-1427 (1990).
    [CrossRef] [PubMed]
  13. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27, 545-547 (2002).
    [CrossRef]
  14. A. I. Konukhov and L. A. Melnikov, "Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser," J. Opt. B: Quantum Semiclass. Opt. 3, S139-S144 (2001).
  15. 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]
  16. 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]
  17. M. V. Berry and M. R. Dennis, "Phase singularities in isotropic random waves," Proc. R. Soc. Lond. A,  456, 2059-2079 (2000).
    [CrossRef]
  18. Y. Y. Schechner and J. Shamir, "Parameterization and orbital angular momentum of anisotropic dislocations," J. Opt. Soc. Am. A 13, 967-973 (1996).
    [CrossRef]
  19. 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]
  20. M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1999), Chap. 1.
  21. M. J. Padgett and J. Courtial, "Poincaré sphere equivalent for light beams containing orbital angular momentum," Opt. Lett. 24, 430-432 (1999).
    [CrossRef]
  22. N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, 1951), Chap. 20.
  23. I. Madsen and J. Tornehave, From Calculus to Cohomology (Cambridge University Press, 1997), Chap. 14.
  24. A. Asundi and H. North, "White-light speckle method- Current trends," Opt. Laser Eng. 29, 159-169 (1998).
    [CrossRef]

2006 (3)

2005 (3)

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]

2004 (2)

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 (2)

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

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, "Stokes singularity relations," Opt. Lett. 27, 545-547 (2002).
[CrossRef]

2001 (2)

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]

A. I. Konukhov and L. A. Melnikov, "Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser," J. Opt. B: Quantum Semiclass. Opt. 3, S139-S144 (2001).

2000 (1)

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

1999 (1)

1998 (1)

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

1996 (1)

1990 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Angelsky, O. V.

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, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Bone, D. J.

Courtial, J.

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]

Freund, I.

Guo, C. -S.

Han, Y. -J.

Hanson, S. G.

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.

Konukhov, A. I.

A. I. Konukhov and L. A. Melnikov, "Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser," J. Opt. B: Quantum Semiclass. Opt. 3, S139-S144 (2001).

Larkin, K. G.

Melnikov, L. A.

A. I. Konukhov and L. A. Melnikov, "Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser," J. Opt. B: Quantum Semiclass. Opt. 3, S139-S144 (2001).

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, "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]

Mokhun, A. I.

Mokhun, I. I.

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, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Oldfield, M. A.

Padgett, M. J.

Schechner, Y. Y.

Shamir, J.

Soskin, M. S.

Takeda, M.

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.

Xu, J. -B.

Yokozeki, T.

Appl. Opt. (1)

J. Opt. A: Pure Appl. Opt. (2)

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. B: Quantum Semiclass. Opt. (1)

A. I. Konukhov and L. A. Melnikov, "Optical vortices in a vector field: the general definition based on the analogy with topological solitons in a 2D ferromagnet, and examples from the polarization transverse patterns in a laser," J. Opt. B: Quantum Semiclass. Opt. 3, S139-S144 (2001).

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

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 (2)

Opt. Laser Eng. (1)

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

Opt. Lett. (4)

Phys. Rev. Lett. (1)

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 (2)

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, "Dislocations in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974).
[CrossRef]

Other (6)

M. S. Soskin and M. V. Vasnetsov, "Singular Optics," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2001).

N. A. Fomin, Speckle Photography for Fluid Mechanics Measurements (Springer-Verlag, 1998), Chap. 8.

M. Raffel, C. Willert, and J. Kompenhans, Particle Image Velocimetry, (Springer-Verlag, 2002), Chap. 5.

M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1999), Chap. 1.

N. Steenrod, The Topology of Fibre Bundles (Princeton University Press, 1951), Chap. 20.

I. Madsen and J. Tornehave, From Calculus to Cohomology (Cambridge University Press, 1997), Chap. 14.

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Figures (8)

Fig. 1.
Fig. 1.

(a). Amplitude contour ellipse for describing the anisotropic core structure of a pseudophase singularity in a L-G signal; (b) Stokes-like parameters for amplitude contour ellipse and the Poincaré’s sphere representation.

Fig. 2.
Fig. 2.

Orthodrome on Poincaré sphere as the merit function of the best matching for pseudophase singularities.

Fig. 3.
Fig. 3.

Recorded images, (a) and (b), of the floating tea leaves on the water surface at different instants of time separated by 8 ms and the corresponding L-G signals, (c) and (d), with positive and negative pseudophase singularities indicated by red and green dots.

Fig. 4.
Fig. 4.

Distribution of anisotropic pseudophase singularities on Poincaré sphere

Fig. 5.
Fig. 5.

Displacement of pseudophase singularities; singularities before and after displacement are indicated by open circles and filled squares, respectively.

Fig. 6.
Fig. 6.

Trajectories of different tea leaves on the water surface.

Fig. 7.
Fig. 7.

Measurement results of the velocity (a) and acceleration (b) for the tea leaf A in Fig. 6.

Fig. 8.
Fig. 8.

Measurement results of the spin angular velocity (a) and spin angular acceleration (b) for the tea leaf A in Fig. 6.

Equations (18)

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g ˜ ( x , y ) = + + H LG ( f x , f y ) · G ( f x , f y ) exp [ j 2 π ( f x x + f y y ) ] d f x d f y ,
H 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 ( j β ) ,
g ˜ ( x , y ) = g ˜ ( x , y ) exp [ j θ ( x , y ) ] = g ( x , y ) * h LG ( x , y ) ,
h LG ( x , y ) = F 1 { H LG ( f x , f y ) } = ( j π 2 ω 4 ) ( x + j y ) exp [ π 2 ω 2 ( x 2 + y 2 ) ]
= ( j π 2 ω 4 ) [ r exp ( π 2 r 2 ω 2 ) exp ( j α ) ] .
Re [ g ˜ ( x , y ) ] = a r x + b r y + c r , Im [ g ˜ ( x , y ) ] = a i x + b i y + c i ,
g ˜ = ( a r + j a i ) x ̂ + ( b r + j b i ) y ̂
= ( a i x ̂ + b r y ̂ ) + j ( a i x ̂ + b i y ̂ ) ,
S 0 = a r 2 + b r 2 + a i 2 + b i 2 ,
S 1 = a r 2 + a i 2 b r 2 b i 2 ,
S 2 = 2 ( a r b r + a i b i ) ,
S 3 = 2 ( a r b i a i b r ) ,
S 0 2 = S 1 2 + S 2 2 + S 3 2 .
S 1 = S 0 cos ( 2 χ ) cos ( 2 φ ) ,
S 2 = S 0 cos ( 2 χ ) sin ( 2 φ ) ,
S 3 = S 0 sin ( 2 χ ) .
S = ( s 1 s 2 s 3 ) = S 0 1 ( S 1 S 2 S 3 ) .
Δ S = arccos ( S · S ' ) ε ,

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