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Singular Optics in Biomedicine. Sean J.Kirkpatrick , Ph.D. Department of Biomedical Engineering Michigan Technological University 1400 Townsend Dr. Houghton, MI 49931 USA [email protected] Optical Vortex Metrology.

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Sean J.Kirkpatrick , Ph.D. Department of Biomedical Engineering Michigan Technological University

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Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Singular Optics in Biomedicine

Sean J.Kirkpatrick, Ph.D.

Department of Biomedical Engineering

Michigan Technological University

1400 Townsend Dr.

Houghton, MI 49931 USA

[email protected]


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Speckle field In terms of a random walk

IM

In the case where:

A

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

Re

Complex addition of elementary phasor contributions


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Vortices and Topological Charge, nt

Singularities come in pairs

zero contour of real part

I

-

1

+

2

3

R

4

zero contour of

imaginary part

2

+

1

-

4

3

“negative” charge

I

-

1

+

2

3

R

2

4

-

1

+

4

3

“positive” charge


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Topological charge,

where is the local phase, and the line integral is taken over path on a closed loop around the vortex. 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

We can describe the phase singularities in in terms of topological charge:


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Locating phase singularities

Simple physical scenario:

n1

n2 ≠ n1

By defining the gradient of the phase in terms of a wave vector

Topological charge can then be re-written as


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Locating phase singularities

(and therefore the topological charge) can be estimated by a convolution operation of a phase image with a series of Nabla windows:

Since phase is a continuous function (and has continuous first derivatives), by Stokes theorem

Efficient computation through a series of convolution operations:

and is the convolution operator


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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.


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Applications

A speckle pattern….

What does look like?


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Applications

What about ?

The locations of the vortices are obvious and indicated on the next slide


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Applications

What about


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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.


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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.


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

  • Question:

  • What isthe 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

  • In an attempt to address this, we performed numerical simulations


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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?


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

Speckle (solid lines) and Vortex Field (dotted lines) Decorrelation Behavior

  • Vortex fields always decorrelatein an exponential fashion

  • Not unexpected as vortex locations are in either one pixle or another, not both

  • That defines exponential behavior


Sean j kirkpatrick ph d department of biomedical engineering michigan technological university

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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