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Poincaré sphere representation for the 

anisotropy of phase singularities and its 

applications to optical vortex metrology for fluid 

mechanical analysis 

Wei Wang, 1* Mark R. Dennis, 2 Reika Ishijima, 1 Tomoaki Yokozeki, 1 Akihiro Matsuda, 1 

Steen G. Hanson, 3 and Mitsuo Takeda 1 

1 

Laboratory for Information Photonics and Wave Signal Processing, Department of Information and Communication 

Engineering, The University of Electro-Communications, 1-5-1, Chofugaoka, Chofu, Tokyo 182-8585, Japan 

2 

School of Mathematics, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom 

3 

Optics and Plasma Research Dept., Risoe National Laboratory, Technical University of Denmark, 

Dk-4000 Roskilde, Denmark 

* 

Corresponding author: weiwang@ice.uec.ac.jp 

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 

OCIS codes: (100.5070) phase retrieval; (120.4290) nondestructive testing; (150.4620) optical 

flow; (260.5430) polarization; (280.7250) velocimetry; (280.2490) flow diagnostics. 

References and links 

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). 

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). 

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). 

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). 

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). 

9. M. R. Dennis, “Local structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, 

S202-S208 (2004). 

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). 

11. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. 

Commun. 213, 201-221 (2002). 

12. I. Freund, “Stokes-vector reconstruction,” Opt. Lett. 15, 1425-1427 (1990). 

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). 

#82058 - $15.00 USD Received 12 Apr 2007; revised 7 Jun 2007; accepted 27 Jun 2007; published 17 Aug 2007 

(C) 2007 OSA 20 August 2007 / Vol. 15, No. 17 / OPTICS EXPRESS 11008

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). 

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). 

17. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A, 456, 

2059-2079 (2000). 

18. Y. Y. Schechner and J. Shamir, “Parameterization and orbital angular momentum of anisotropic 

dislocations,” J. Opt. Soc. Am. A 13, 967-973 (1996). 

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). 

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). 

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). 

1. Introduction 

Velocity is one of the most important parameters in fluid mechanics due to its cardinal 

influence on transport phenomena. Velocity measurements by using speckle and specklerelated 

techniques have been explored extensively, and various techniques have been 

developed [1, 2]. Though they differ in specific technical details, these speckle-related 

methods have at least two common features, namely, to seed the flow with particles, and to 

detect the velocity vectors through cross-correlation of the intensity distributions for the 

recorded images. The seeding particles moving with a fluid flow provide kinematic 

information at the recording plane, and the local fluid velocity is derived from the ratio 

between the measured spacing for the same tracers and the time between exposures. Although 

many attempts have been made to extend its applications during the past decades, the most 

serious problem for particle image velocimetry lies in the lack of an autonomous method for 

high precision tracking of the seeding particles from the recorded speckle-like patterns. Since 

the resulting velocity vectors are derived through a cross-correlation over small interrogation 

areas of the flow, the spatial resolution for particle image velocimetry has been restricted due 

to the fact that the resulting velocity distribution is a spatially averaged representation for the 

actual velocity field. 

Meanwhile, the concept of singularities in optical fields has been known for a long time, 

and their basic properties have received more and more attention since the seminal work of 

Nye and Berry in the early 1970s [3,4]. Due to the well-known fact that the phase singularities 

are well-defined geometrical points where the corresponding amplitudes always equal zero, 

and also due to the fact that the least photon noise is present at the locations of zero optical 

field, these topological defects of the wavefronts are optimal encoders for position marking. 

Noting that many randomly distributed phase singularities exist in an analytic signalof a 

speckle-like pattern, we have proposed a method referred to as Optical Vortex Metrology 

(OVM), which makes use of the pseudophase singularities in an analytic signal representation 

of the speckle-like pattern for optical metrology [5-7]. In addition to an improved 

performance for micro-displacement measurement with nano-scale resolution as demonstrated 

in a previous paper [6], the technique based on the pseudophase can be applied to a wider 

class of general speckle-like random markings due to the fact that the pseudophase can be 

obtained without recourse to interferometry, and thereby opens up new possibilities in a wide 

range of applications beyond those known for laser speckle metrology [8]. 

The purpose of this paper is to propose yet another technique for velocity measurement, 

which tracks the pseudophase singularities based on their anisotropic core structures. Rather 

than directly correlating the raw intensity distributions of specklegram-like random patterns, 

which suffers from the reduced resolution for velocity measurements due to spatial averaging 



̃ 

, 

̃ ̃ 

, 

̃ 

through 

and the influence of the rotation of the seeding particles, we first generate the 2-D isotropic 

complex signal representation for the particle-seeded flow pattern by Laguerre-Gauss filtering. 

Next, we detect the core anisotropy of the pseudophase singularities, which serves as the 

fingerprint for their identification. Then, we conduct the fluid mechanics measurements by 

tracking the registered phase singularities. The key idea of the newly proposed technique is 

the unique identification of the matching phase singularities through the geometry of their 

anisotropy. In our previous investigation, we arbitrarily adopted the ellipticity, zero crossing 

angle and vorticity as the singularity fingerprints [7]. Mathematically, however, these ad hoc 

parameters are not the most natural measures, nor are they optimally chosen for the best 

performance. Noting that the fingerprint of a pseudophase singularity depends only on the 

gradient field of the complex signal, we propose the use of an alternative set of field 

parameters, which are mathematically more natural and elegant with an analogy to the Stokes 

parameters in polarimetry [9-14]. After parametrizing the anisotropy ellipse of each 

pseudophase singularity onto a Poincaré sphere, we can uniquely identify that singularity even 

in a complicated environment. We will experimentally demonstrate that even a random 

displacement of an object in a disturbed flow can be determined by tracing the registered 

pseudophase singularities based on their fingerprints represented by the Stokes-like field 

parameters and the Poincaré sphere. 

2. Principle 

2.1 Complex signal representation of a speckle pattern by isotropic Laguerre-Gauss filtering 

Before explaining the proposed technique for labeling the particles used in velocity 

measurements, we first briefly review the pseudophase retrieval from the complex-valued 

signal representation of a speckle-like pattern along the lines of our previous paper [7]. 

Let g( xy , ) be the original intensity distribution of the speckle pattern, and let its Fourier 

spectrum be G( fx, f 

y 

) . We can relate g( xy , ) to its isotropic complex signal g( xy , ) 

a Laguerre-Gauss (L-G) filter. Thus, our definition of a 2-D isotropic complex signal 

representation for the speckle-like pattern is 

+∞ +∞ 

g( xy , ) = H ( f, f) ⋅ G( f, f)exp j2 π ( fx+ 

fy) 

dfdf 

−∞ 

∫∫ 

−∞ 

⎡ 

⎤ 

LG x y x y x ⎣ 

⎦ y x y 

(1) 

⎡ 

⎤ 

⎣ ⎦ , (2) 

where HLG( fx, f 

y) 

is a Laguerre-Gauss filter in the frequency domain defined as follows: 

H ( f , f ) = ( f + jf )exp 

2 2 2 

− ( f + f ) ω 

2 

= ρexp( −ρ 2 

ω )exp( jβ 

) 

LG x y x y x y 

2 2 

where ρ = fx 

+ fy 

, β = arctan( fy 

fx 

) are the polar coordinates in the spatial frequency 

domain. The spiral phase function, also used in the Riesz transform (vortex transform) [15], 

has the unique property that any section through the origin is a signum function with a π 

phase jump, and so can be understood as a radial Hilbert transform. The amplitude of the L-G 

filter has a typical doughnut-like structure [7,16], serving as a band-pass filter by suppressing 

the high spatial frequency components, which are known to be responsible for creating 

unstable phase singularities. Therefore, the density of pseudophase singularities can be 

controlled by choosing a proper bandwidth of the Laguerre-Gauss function, ω , to adjust the 

average size for the speckle-like pattern. After straightforward algebra, we find 

g( xy , ) = gxy ( , ) exp[ jθ 

( xy , )] = gxy ( , ) ∗hLG 

( xy , ) 

(3) 

where ∗ denotes the convolution operation, and hLG 

( xyis , ) again a Laguerre-Gauss function 

in the spatial signal domain 

−1 2 4 2 2 2 2 

hLG ( xy , ) = { HLG ( fx , fy 

)} = ( jπω)( x+ jy)exp[ − πω( F 

x + y)] 

(4) 

2 4 2 2 2 

= ( jπω)[ rexp( −πr ω)exp( jα)] . 



g̃ 

̃ ̃ 

, 

that 

̃ 

̃ 

−1 

2 2 

where is the inverse Fourier transform, and r = x + y , α = arctan( y x) 

are F the 

spatial polar coordinates defined as usual. Because the complex signal gxy ( , ) was generated 

from the Laguerre-Gauss filter, we call it a complex Laguerre-Gauss signal, or an L-G signal 

for short, to stress its difference from the conventional analytic signal derived from Hilbert 

transform. The phase θ ( xy , ) of the L-G signal of a speckle-like pattern is referred to as the 

pseudophase to distinguish it from the true phase of the optical field responsible for the 

speckle pattern. Although it is not the true phase of the complex optical field, the pseudophase 

does provide useful information about the object, as will be shown in Section 3. 

2.2 Poincaré sphere representation for phase singularity anisotropy 

Just as a random speckle intensity pattern imprints marks on a coherently illuminated object 

surface, randomly distributed phase singularities in the pseudophase map of the speckle 

pattern imprint unique marks on the object surface. Based on the information about the 

locations and the core structures of the phase singularities, we have proposed the use of 

Optical Vortex Metrology (OVM) for optical metrology [5-7]. To uniquely identify the 

corresponding pseudophase singularities before and after displacement, we proposed to use 

the ellipticity of the contour ellipse, zero crossing angle, and vorticity as the singularity 

fingerprints in our previous investigation. Though we demonstrated the performance of OVM 

with large dynamic range and high spatial resolution, these three parameters are not the most 

mathematically natural measures. Noting that the local structure of a pseudophase singularity 

̃ 

depends only on the gradient field of the L-G signal ∇g( xy , ) , we propose yet another 

fingerprint for the purpose of unique identification. 

Usually, the change of the pseudophase around the phase singularities are non-uniform as 

a function of the azimuthal angle, and the amplitude contours, which is a typical core structure 

around the pseudophase singularity, displays a strongly anisotropic ellipse [17-18]. In the 

immediate vicinity of a pseudophase singularity, the real and imaginary parts of the L-G 

signal representation of a 2-D speckle-like pattern can be expressed as 

Re[ gxy ( , )] = ax+ by+ c, Im[ gxy ( , )] = ax+ by+ 

c 

(5) 

r r r i i i 

where the coefficients: ak, bk, ck 

( k = r, i) 

can be obtained by the least-square fitting method 

from the detected complex values at the pixel grids surrounding the phase singularity[19]. 

̃ 

Meanwhile, the complex vector field ∇g 

, which is the associated gradient of the L-G signal 

, can be expressed as 

∇ g = ( a ) ˆ ( ) ˆ 

r 

+ jai x+ br + jbi 

y 

(6) 

= ( axˆ+ byˆ) + jax ( ˆ+ 

byˆ 

), 

r r i i 

where, ˆx and ŷ are unit vectors. Recently, a set of Stokes-like parameters has been proposed 

̃ 

for the complex vector field ∇g 

shares similar geometric features associated with the 

vector polarization field [9]. Therefore, the parameters, describing the anisotropy ellipse, may 

be defined as: 

2 2 2 2 

S0 = ar + br + ai + bi 

, (7.1) 

2 2 2 2 

S1 = ar + ai −br − bi 

, (7.2) 

S2 = 2( ab 

r r 

+ ab 

i i) 

, (7.3) 

S3 = 2( ab 

r i 

− ab 

i r) 

, (7.4) 

̃ 

These parameters describing the anisotropic ellipse related to ∇g 

, are mathematically 

analogous, but physically unrelated, to the Stokes parameters in polarization. For instance, the 

sign of S 3 gives the sign (topological charge) of the vortex (i.e. the sense of phase increase 



around the singularity), analogous to the handedness of polarization. Meanwhile, it follows 

from Eq. (7) that only three of them are independent since they are related by the identity: 

S = S + S + S . (8) 

2 2 2 2 

0 1 2 3 

Following the same procedure as used for polarization [20], we can represent the 

ellipsoidal parameters as in the following equations: 

⎛ ⎞ ⎛ ⎞ 

⎜ ⎟ ⎜ ⎟ 

⎜ ⎟ ⎜ ⎟ 

⎝ ⎠ ⎝ ⎠ 

S S 1 0cos(2 )cos(2 ) (9.1) 

S S 2 0cos(2 )sin(2 ) (9.2) 

S S sin(2 ) 3 0 (9.3) 

Here, ϕ is the azimuth angle between the major semi-axis of the ellipse and the x -axis, and 

χ is the ellipticity angle, which is the arctangent of the ratio between the length of the two 

axes of the ellipse as shown in Fig. 1(a). Equations (9) indicate a simple geometrical 

representation for all the parameters Si 

( i = 1,2,3) as components in a three-dimensional 

sphere surface occupied in space by the unit vector: 

s1 S1 

−1 

S = s2 = S0 S2 

. (10) 

s S 

3 3 

Mathematically, the Poincaré sphere representation for polarization, phase singularity 

anisotropy, and elsewhere, such as optical angular momentum [21], can be interpreted the 

Hopf fibration (Hopf map) as follows [22]. A normalized, complex two-dimensional vector 

can be represented by a point on the 3-sphere (hypersphere) in 4-D. The set of points on this 

hypersphere corresponding to the complex vectors equivalent up to phase factors is a circle; 

the Hopf mapping identifies these points, and the image space is the 2-sphere in 3-D space. 

The Cartesian coordinates of the Hopf map are simply the Stokes parameters [11, 23]. 

ŝ 3 

s 3 

 

S 

χ 

ϕ 

s 1 

2χ 

2ϕ 

s 2 

ŝ 2 

ŝ 1 

(a) 

(b) 

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. 

Figure 1(b) shows the Poincaré sphere representation for the anisotropic core structure of 

the pseudophase singularity. Here, s 

1 

, s 

2 

, s3 

may be regarded as the Cartesian coordinates of a 

 

point S on a surface of the sphere with unit radius, such that 2ϕ and 2χ are the spherical 

angular coordinates of this point. Thus, to every possible state of the anisotropy ellipse for the 



core structure of a phase singularity, there corresponds one point on the sphere surface, and 

vice versa. It is this uniqueness of the core structure serving as the fingerprint of a 

pseudophase singularity that enables the correct identification and tracking of the pseudophase 

singularity through a complicated movement. 

To find the correctly matching pseudophase singularities for the object after displacement, 

it is necessary to introduce a similarity measure that gives a criterion for correct identification. 

As shown in Fig. 2, the distance along a geodesic line can be chosen as the figure of merit for 

the best matching of the pseudophase singularities. The geodesic line is the shortest path 

between two points on the sphere surface, also known as an orthodrome, which is a segment 

of a great circle. After a straight forward algebra, the merit function based on the distance 

along the geodesic line is given by 

Δ S = arccos( S⋅S′ 

) ≤ε 

, (11) 

where primed parameters are related to the pseudophase singularities after displacement. After 

setting an appropriate threshold value ε , and calculating the merit function for each pair of 

pseudophase singularities, we can identify the correct counterparts based on the minimum 

 

value of ΔS . Thus, the in-plane displacement of a particle can be estimated from the 

coordinate change ( Δx, Δ y) 

of the corresponding pseudophase singularity attached to the 

particle, and the linear velocity of the measured flow at a given position can be estimated from 

the rate of change for the coordinate of each singularity between two sequential exposures. 

Meanwhile, when observed with focus on the seeded particle surface, the change of azimuth 

angle ϕ can be directly related to the rotation or spin of each pseudophase singularity 

associated with the particle. Therefore, we can also estimate the angular velocity and the 

applied torque of the particle spin through the change of the azimuth angle Δ ϕ . 

ŝ 3 

 

Δs 

 

s 

 

s′ 

ŝ 2 

ŝ 1 

Fig. 2. Orthodrome on Poincaré sphere as the merit function of the best matching for 

pseudophase singularities. 

3. Experiments 

Experiments have been conducted to demonstrate the validity of the proposed principle. In 

experiments, tiny pieces of tea leaves were used as the micro-particles floating on the water 

surface. The tea leaves moving with the flow are imaged by a Nikon Macro lens ( f = 55mm, 

F2.8) onto the image sensor plane of a high-speed camera, FASTCAM-NET500/1000/Max 

(PHOTRON) with a pixel size of 7.4μm× 7.4μm . Based on the nominal magnification of the 

imaging optics and the pixel separation of the high-speed camera, we estimate that the 

displacement by an amount of unit pixel in the image plane corresponds to an object 



displacement of 42.4μm . Under the illumination by a halogen lamp, sequences of the water 

flow seeded by tea leaves were recorded at a frame rate of 125 frames per second. Though we 

have referred to the observed random pattern as a speckle-like pattern, strictly it is not a 

generic speckle pattern of interference nature, but rather represents the image of the floating 

particles with random shapes. This means that our experiment also serves as a demonstration 

that the proposed technique can be applied to a wider class of random patterns including nongeneric 

speckle patterns observed on an object illuminated by incoherent light [24]. From 

every recorded image, we generated an isotropic complex L-G signal by Laguerre-Gauss 

filtering, and retrieved the pseudophase information. In the experiment, we carefully adjusted 

the density of the pseudophase singularities by choosing a proper bandwidth of the Laguerre- 

Gauss filter ω = 10 pixels in Eq. (2), so that the pseudophase singularities on the water surface 

are clearly separated, and each tea leaf corresponds to one pseudophase singularity. After 

identifying the corresponding pseudophase singularities through their fingerprints as described 

above, and determining their coordinates for every two consecutive pictures, we conducted 

the velocity measurement and the flow analysis. 

(a) 

(b) 

(c) 

(d) 

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. 

Two examples of the recorded images for the floating tea leaves at different instants of 

time are shown in Figs. 3(a) and 3(b). The image in Fig. 3(b) was recorded after 8 ms from the 

recording of Fig. 3(a). The corresponding amplitude distributions of the generated L-G signals 

of the recorded images are shown in Figs. 3(c) and 3(d), where the locations of the positively 

and negatively signed pseudophase singularities are indicated by points in red and green, 

respectively. In these figures, we can observe that every tea leaf has experienced a different 



amount of movement during the recording period, and the pseudophase singularities have 

maintained their spatial structures even after being translated with respect to each other by an 

amount corresponding to its complicated movement. As expected, for most of the 

pseudophase singularities we were able to find their correct counterparts through their unique 

fingerprints represented on the Poincaré sphere. The coordinate differences between the 

corresponding pseudophase singularities gave a good estimate for the movements of the 

floating leaves, and can be used for kinematic analysis of fluids, provided that we use only 

successfully matched phase singularities. 

However, we also found that some pseudophase singularities failed to find their matching 

counterparts. This phenomenon may be attributed to the following: because of the distortion 

of the appearance of the tea leaves during flow, the recorded images begin to change their 

shapes; in other words, speckle decorrelation increases when the time interval between two 

recorded images is large. This gives rise to the creation or annihilation of singularities in the 

pseudophase map. Meanwhile, the local shape change in the recorded images also introduces 

the distortions of the core structures of the pseudophase singularities, which occurs 

particularly near annihilations. These newly created or annihilated pseudophase singularities 

have no counterparts to match in the pseudophase map. Furthermore, the distortion of the core 

structure for the pseudophase singularities increases the difficulty in finding the correct 

counterparts for calculation of their coordinate difference. 

Fig. 4. Distribution of anisotropic pseudophase singularities on Poincaré sphere 

Based on the generated complex L-G signal representation of recorded tea images, we can 

obtain the Stokes-like parameters around each pseudophase singularities and displayed them 

on the surface of Poincaré sphere as shown in Fig. 4. As anticipated, each pseudophase 

singularity has its unique anisotropic core structures with different ellipticity and azimuth 

angles from each other, and these ellipticity and azimuth angles have almost uniform 

distribution on Poincaré sphere with slight concentration at the two poles. It is this uniqueness 

of the anisotropy that enables the correct identification and the tracking of the complicated 

movement of pseudophase singularities through its fingerprints expressed by the Stokes-like 

parameter. Figure 5 shows an example for the in-plane movements of phase singularities, 

where the locations of phase singularities before and after displacement are indicated by open 

circles and filled squares, respectively. In this identification process to find the correct 

counterparts for pseudophase singularities, the threshold value in Eq. (11) was selected 

as ε = 0.1 , and the search was performed in the neighborhood area within 30× 30 square 

pixels; this is because the recording speed of the high-speed camera is much faster than the 



fluid velocity so that the phase singularities do not move more than 30 pixels between 

successive frames. After identifying the corresponding pseudophase singularities for each pair 

of consecutive images making use of their Poincaré sphere representation of the core 

structures as fingerprints, we traced the complicated movement of the pseudophase 

singularities. Figure 6 shows an example of the trajectories for the selected four tea leaves for 

99 frames of recorded images, where the arrows indicate their movement directions at 

different instants of time. As expected when blowing the water surface, the central flow had a 

large velocity and pushed the tea leaves aside, so that these floating particles at the left and 

right part of the water have counterclockwise and clockwise movements, respectively, being 

pushed back by the wall of the circular container. We also found that some phase singularities 

changed the directions of their movement abruptly due to collision between leaves, as shown 

trajectory C in Fig. 6. 

Fig. 5. Displacement of pseudophase singularities; singularities before and after displacement 

are indicated by open circles and filled squares, respectively. 

B 

C 

D 

A 

Fig. 6. Trajectories of different tea leaves on the water surface. 

From the coordinate changes ( Δx, Δ y) 

for each registered pseudophase singularity, we 

obtained the 2-D displacement distances along x - and y -directions for each time interval 

between exposures Δ t . Therefore, the horizontal and vertical components of velocity can be 

expressed as: vx 

=Δx Δ t , and v y =Δy Δ t , where the time interval is Δ t = 8ms in this 

example. Figure 7(a) shows an example of the velocity history for tea leaf A in Fig. 6, where 

2 2 1 2 

the dash dot line, dashed line, and the solid line indicate v x , v y , and v = ( vx 

+ vy) 

, 



espectively. As anticipated, the tea leaf velocity has a fluctuating structure due to the 

advection with the water. At the beginning of the measurement, the vertical component of the 

velocity v y played a dominant role with a relatively larger value than the horizontal part of 

velocity. As time goes by, the two velocity components decrease, and the tea leaf finally cease 

their movement due to fluid damping. After calculating the finite difference for the horizontal 

and vertical velocity components, we obtained the variation of the tea leaf acceleration as 

shown in Fig. 7(b). Under the reasonable assumption that the mass of the particle is constant 

during the recording process, Fig. 7(b) provides a local force diagrams for the tea leaf A at the 

different instants of time. From the solid curve for the acceleration sum, we can also observe 

the local acceleration peaks stemming from the collision force that the tea leaf received from 

other tea leaves during its random movements. 

v = v + v 

2 2 

x y 

v x 

v y 

(a) 

2 2 

x y 

a = a + a 

a x 

a y 

(b) 

Fig. 7. Measurement results of the velocity (a) and acceleration (b) for the tea leaf A in Fig. 6. 

Meanwhile, we also obtained the spinning angular displacement Δ ϕ around the center of 

the phase singularity from the change of azimuth angle for each registered pseudophase 

singularity, and estimated the spin angular velocity based on: Ω=Δϕ 

Δ t as shown in Fig. 

8(a). After the calculation of the finite difference for the spin angular velocity, we obtained 

the history of the spin angular acceleration in Fig. 8(b), that is proportional to the applied 



torque on the tea leaf A because of the constant moment of inertia. Therefore, Figs 7 and 8 

serve as an experimental demonstration of the validity of the proposed Optical Vortex 

Metrology for fluid mechanics measurement based on the anisotropic core structure of 

pseudophase singularities. 

(a) 

(b) 

Fig. 8. Measurement results of the spin angular velocity (a) and spin angular acceleration (b) 

for the tea leaf A in Fig. 6. 

4. Conclusions 

In summary, we have proposed a technique for fluid kinematic analysis based on the 

improved Optical Vortex Metrology, which makes use of the elliptic anisotropy of the 

pseudophase singularities in the complex L-G signal representation of a speckle-like pattern. 

In analogy with the polarization of the vector wave field, a set of Stokes-like parameters has 

been applied for the description of the gradient field of the complex L-G signal around a 

pseudophase singularity, and the Poincaré sphere representation has been used as unique 

fingerprints attached to the seeding particles moving with the flow. Experiments have been 

performed that demonstrate the validity of the proposed technique applied for fluid dynamical 

measurements. Furthermore, the registration of the pseudophase singularities based on their 

anisotropic core structures facilitate quantitative characterization of the local structures of the 



individual particles, and thus introduces new opportunities to explore other potential 

applications of the concept of phase singularities in experimental mechanics. 

Acknowledgment 

Part of this work was supported by Grant-in-Aid of JSPS B(2) No.18360034, Grant-in-Aid of 

JSPS Fellow 15.52421, and by The 21st Century Center of Excellence(COE) Program on 

“Innovation of Coherent Optical Science” granted to The University of Electro- 

Communications. MRD is a Royal Society University Research Fellow.

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