Singular Optics in Biomedicine

murmerlastUrban and Civil

Nov 16, 2013 (3 years and 4 months ago)

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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