Sean
J.Kirkpatrick
, Ph.D.
Department of Biomedical Engineering
Michigan Technological University
1400 Townsend Dr.
Houghton, MI 49931 USA
sjkirkpa@mtu.edu
Singular
Optics in Biomedicine
Optical Vortex Metrology
•
Optical vortices
are singular points (phase singularities, or pure dislocations) in
electromagnetic fields (Nye & berry 1974)
•
Characterized by zero intensity, undefined phase, and a rotation of phase along
a closed loop encircling the point (i.e., they are branch

point types of singularities)
•
They are 2

D in nature and propagate in 3

D space in the direction of wave
propagation
•
They exist only in pairs
–
a single vortex can not be annihilated by local
perturbations to the field (i.e., they are stable)
•
Optical
Vortex metrology
has been proposed as an extension of speckle tracking
metrology
•
Assumes that vortex motion can be used as a surrogate for speckle and object
motion
2
Speckle
field
In terms of a random walk
Complex addition of elementary
phasor
contributions
1
Im( )
Phase tan
Re( )
U
U
k
a
N
Re
IM
A
In the case where:
Im( )
0, is undefined
Re( )
U
U
Such a point is referred to as a
phase singularity
or
vortex
V
ortices and Topological Charge,
n
t
zero contour of real part
zero contour of
imaginary part
+

+

●
●
●
●
1
2
3
4
“negative” charge
R
I
●
1
●
2
●
3
●
4
+

+

●
●
●
●
1
2
3
4
“positive” charge
R
I
●
1
●
4
●
3
●
2
S
ingularities come in pairs
We can describe the phase singularities in in terms of topological charge
:
,
U x y
1
,,
2
t
c
n x y dl
Topological charge,
t
n
where is the local phase, and the line integral is taken over path on a closed
loop around the vortex. is the
phase gradient.
,
x y
,
x y
l
c
By defining topological charge in this fashion and identifying
phase singularities based on their topological charge, it is
possible to observe and track the singularities in a dynamic field
By observing and tracking the singularities, it is possible to infer
information from the scattering media
Locating phase singularities
n
2
≠ n
1
n
1
,
u x y
,
U x y
Simple physical scenario:
k
,
U x y
By defining the gradient of the phase in terms of a wave vector
k
,,,,,,,,
k x y t x y t x y t i x y t j
x y
Topological charge can then be re

written as
1
,
2
t
c
n k x y dl
Locating phase singularities
k
(and therefore the topological charge) can be estimated by a
convolution operation of a phase image with a series of
Nabla
windows:
0
1
t
t
n
n
if , then region does not encircle a
singularity
if , then region encircles a singulari
ty
1
,
2
t
c
n x y dl
Efficient computation through a series of convolution operations:
1 2 3 4
1 2 3 4
,,,,
0 1 1 1 1 0 0 0
;;;
0 1 0 0 1 0 1 1
where
t
n x y x y x y x y
and is the convolution operator
Since phase is a continuous function (and has continuous first derivatives), by Stokes theorem
1 1
,
2 2
t
c a
n x y dl x y
x y y x
Locating phase singularities

Example
Speckle pattern (left) with singular points indicated by red circles. Phase of the
speckle pattern (right) with singular points indicated by red and blue circles. Red
circles indicate positive vortices and blue indicates negative vortices.
Applications
–
How can we use vortex behavior?
Quantifying the behavior of dynamic systems:

Brownian motion

Particle sizing

Cellular activity
Colloidal solutions
Cytometry
Monomer

to

Polymer conversion
etc
Applications
What does look like?
(,)
I x y
A speckle pattern….
Applications
What about ?
ˆ
(,,)
x y t
The locations of the vortices are obvious and indicated on the next slide
Applications
What about
ˆ
(,,)
x y t
Applications
Look at just the
spatio

temporal behavior of the vortices: Creation of
vortex paths
These vortex paths trace the
spatio

temporal dynamics of
optical vortices created by light
scattering of slowly moving
particles. Note that the vortex
trails are relatively long and
straight.
Applications
Take a closer look at some of the features of vortex paths
Speed things up a bit: vortex paths from rapidly moving particles
Things to notice:
•
Short lifetime
•
Tangled paths
Notice the differences in the vortex path images from the
previous two slides. Clearly the dynamics of the scattering
medium strongly influences the
spatio

temporal behavior of
the vortex paths.
Question:
What is the relationship between the decorrelation behavior a dynamic speckle
fields and their corresponding vortex fields
•
That is, can the autocorrelation functions of vortex fields be
confidently used as surrogates for the autocorrelation functions of
speckle fields
•
Can we use vortex decorrelation behavior to directly estimate the
motions in a dynamic, scattering medium?
•
•
Photon correlation spectroscopy (DWS)
•
Cellular dynamics
•
Motion and flow
1
g
•
In an attempt to address this, we performed numerical simulations
Numerical Simulations
•
Numerically generate sequences of dynamic speckle patterns with
different
decorrelation
behaviors and speeds
•
Gaussian
•
Exponential
•
Constant sequential correlation coefficient
•
Identify vortex fields and generate autocorrelation function
•
Address question: Is the autocorrelation function of a vortex field
representative of the autocorrelation function of the corresponding
speckle field?
Speckle (solid lines) and Vortex Field (dotted lines)
Decorrelation
Behavior
•
Vortex fields always
decorrelate
in an
exponential fashion
•
Not unexpected as vortex locations
are in either one
pixle
or another, not
both
•
That defines exponential behavior
Results and Discussion
•
The dynamic behavior of the scattering medium influences the behavior of
the vortex field.
•
Rapidly moving particles result in a vortex field that
decorrelates
rapidly and
is characterized by short, convoluted vortex paths. Slow mowing particles
result in the opposite.
•
Vortex fields exhibit exponential
decorrelation
behavior
•
The
decorrelation
behavior of vortex fields is not directly representative of
the behavior of the corresponding speckle fields
•
May still find applications in DWS, cell and tissue dynamics, fluid flows and
other biological dynamics
Thank you
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