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Two methods for modelling the propagation of terahertz

radiation in a layered structure.

GILLIAN C. WALKER1*, ELIZABETH BERRY1, STEPHEN W. SMYE2, NICK N. ZINOV’EV3, ANTHONY J. FITZGERALD1, ROBERT. E. MILES3, MARTYN CHAMBERLAIN3 AND MICHAEL A. SMITH1 1Academic Unit of Medical Physics, University of Leeds, UK2Department of Medical Physics and Engineering, Leeds Teaching Hospitals NHS Trust, UK3Institute of Microwaves and Photonics, University of Leeds, UK(*Author for correspondence, email:gcw@medphysics.leeds.ac.uk)

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Objectives

Create a modelling tool to simulate the passage of THz radiation through biological tissue.– Biological tissue is highly inhomogeneous.– The interactions that occur with THz and

biological tissue are complex.

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Outline

Modelling Biological Tissue In Vitro Phantom Thin Film Matrix Model Monte Carlo Model Results Discussion Future Work

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Modelling the interaction of THz radiation with biological tissue. THz radiation is being investigated as an

imaging tool for skin. It has been shown that TPI can resolve the

stratum corneum, epidermis and dermis. (Cole et al. Laser Plasma Generation and Diagnostics, SPIE Proc 2001; 4286.)

A three-layer system of parallel sided slabs, each with frequency dependent physical properties could be used to simulate human skin.

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The modelling problem

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Frequency THz

Ampl

itude

a.u

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Incident THz Spectrum

Transmitted THz

Spectrum

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In vitro phantom

Water/Propanol-1 solution

Spacer - 180 m

TPX - 2 mm

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Physical Properties of Water/Propanol -1 The absorption coefficient and index of

refraction of water and propanol-1 were calculated using the Cole-Cole model. (Kindt et al. Journal of Physical Chemistry 1996;100 :10373-9)

These were averaged using volume weighting to give the physical properties of the specific water/propanol-1 solution.

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Physical Properties of Water/Propanol-1

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ex o

f Ref

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ion

water

propanol-1

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abso

rpti

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co

effi

cien

t cm

-1

water

Propanol-1

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Thin Film Matrix Model

A method for calculating the change in electric field as it passes through the layered medium.

Implementation of the boundary conditions for the electric and magnetic components of the incident radiation at each boundary result in a matrix formulation of the problem.

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Thin Film Matrix Model

i

iTPXPWTPX

t

t

H

EMMM

H

E/

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Thin Film Matrix Model

TPXTPXTPX

TPXTPX

TPX

TPX

hnhnin

hnn

inh

M

2cos

2sin

2sin

2cos

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Thin Film Matrix Model

4

2cos/

22/

11 inhMM wpPWPW

4

2sin

4

/12 inh

in

iM wp

wp

PW

4

2sin

4/

21 inhiniM wpwpPW

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Monte Carlo Model

Creating a Photon Distribution.– A THz pulse is recorded in units

proportional to electric field. – A photon distribution is created by

randomly sampling the spectrum and a Poisson distribution to account for the coherent nature of the radiation.

– One million photons were included in an incident ensemble.

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The Monte Carlo Model

The position of each photon is tracked in the sample.

The probability of a photon crossing a boundary within the sample is determined by the Fresnel coefficients.

In the water/propanol-1 solution the Beer-Lambert law is sampled as the probability distribution for an absorption event.

The number of photons transmitted and reflected is counted to give the output spectra.

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Presentation of Results

The results of the Monte Carlo simulation were in expressed as a photon distribution while the experimental results were in arbitrary units proportional to electric field.

The experimental results were converted into photon distributions. The number of photons included in each respective ensemble were calculated as a fraction of one million, the fraction given by the experimental area to the incident spectrum area.

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Graphical comparison

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itude

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Transmitted spectraIncident spectra

1 000 000 photons

a1 a2

a2/a1*1 000 000 photons

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Results - Full Spectrum

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thin film matrix model

experiment

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Monte Carlo Model

Experiment

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Results - Up to 1 THz

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experiment

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Monte Carlo Model

Experiment

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Results - Full Spectrum

Monte Carlo Model

y = 0.5863x + 8.0586

R2 = 0.7836

02000400060008000

10000120001400016000

-2000 3000 8000 13000 18000

model

exp

erim

ent

Identity

Best fit line

Thin Film Matrix Model

y = 0.5911x - 6E-08

R2 = 0.9023

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model

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erim

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Best Fit Line

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Discussion

The model results show good agreement with the experimental results reproducing all major features.

Up to 1 THz, where the physical parameters have been verified there is close agreement with model and simulated amplitude.

Generally the Thin Film Matrix Model more closely reproduces the experimental results.

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Future Work

Investigations into how the absorption coefficient and index of refraction affect the simulated results are being carried out.

Implementation of a fitting routine to extract physical parameters from the tissue.

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Future Work

Variation of absorption coefficient from 50-250%

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50%-250% of the absorption coefficient

Experimental result

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Future Work

The Monte Carlo model is to be used to investigate scattering of THz radiation.

The models are to be used to simulate results from human skin for both transmission and reflection data, in vitro and in vivo.

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Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council (GR/N39678)

We are grateful for the contributions of the members of the EU Teravision project (IST-1999-10154), especially W. Th. Wenckebach, T.U. Delft.