Sean J.Kirkpatrick, Ph.D.Department of Biomedical Engineering

Michigan Technological University1400 Townsend Dr.

Houghton, MI 49931 USAsjkirkpa@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

ka

N

Re

IM

A

In the case where: Im( )

0 , is undefinedRe( )

U

U

Such a point is referred to as a phase singularity or vortex

Vortices and Topological Charge, nt

zero contour of real part

zero contour ofimaginary 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

Singularities come in pairs

We can describe the phase singularities in in terms of topological charge: ,U x y

1, ,

2t cn x y dl

Topological charge, tn

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

lc

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

n2 ≠ n1

n1

,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 jx y

Topological charge can then be re-written as 1,

2t cn 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

1t

t

n

n

if , then region does not encircle a singularity

if , then region encircles a singularity

1,

2t cn 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

tn 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 2t c an 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 solutionsCytometryMonomer-to-Polymer conversionetc

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

1g

• 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