Singular Optics in Biomedicine
Sean J.Kirkpatrick, Ph.D.
Department of Biomedical Engineering
Michigan Technological University
1400 Townsend Dr.
Houghton, MI 49931 USA
sjkirkpa@mtu.edu
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  2 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
Speckle field In terms of a random walk
IM
ak
N
A

Re
P hase    tan
1
Im (U )
R e(U )
In the case where:
Im (U ) 
  0 ,  is undefined
R e(U ) 
Such a point is referred to as a phase singularity or vortex
Complex addition of elementary phasor contributions
Vortices and Topological Charge, nt
Singularities come in pairs
zero contour of real part
2
●
3
●
zero contour of
imaginary part
●1
●4
I
+
●
4
-
+
1
●
3
●2
R
●4
R
●
“negative” charge
2
3
●
●
4
●
I
+
●
2
●
1
“positive” charge
●
+
3
●
1
Topological charge, n t
We can describe the phase singularities in U  x , y  in terms of topological charge:
nt 
1
2

c
   x , y   dl
,
where   x , y  is the local phase, and the line integral is taken over path l on a closed
loop c around the vortex.    x , y  is the phase gradient.
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
Simple physical scenario:
n1
u  x, y 
U  x, y 
n 2 ≠ n1
U  x, y 
k
By defining the gradient of the phase in terms of a wave vector k
k  x, y, t      x, y, t  

x
  x, y,t  i 
Topological charge can then be re-written as
nt 
1
2

y
  x, y,t  j
 k  x , y   dl
c
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:
if
n t  0 , th e n re gio n d o e s n o t e n circle a sin gu la rity
if n t   1, th e n re gio n e n circle s a sin gu la rity
nt 
1
2

c
   x , y   dl
Since phase is a continuous function (and has continuous first derivatives), by Stokes theorem
nt 
1
2

c
   x, y   dl 
 


 
2 a   x  y
yx
1

 x  y

Efficient computation through a series of convolution operations:
nt    x , y    1    x , y    2    x , y    3    x , y    4
w h e re
0
1  
0
and

1 
1
; 2  
1
0
is the convolution operator
1
 1
; 3  
0 
 1
0
 0
; 4  
0
 1
0

1
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 I ( x , y ) look like?
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?
•
•
•
•
g 1 

Photon correlation spectroscopy (DWS)
Cellular dynamics
Motion and flow
• 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