1
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:[email protected])
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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
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
0.00 1.00 2.00 3.00 4.00 5.00
Frequency THz
Ampl
itude
a.u
.
Incident THz Spectrum
Transmitted THz
Spectrum
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2.00E-04
3.00E-04
4.00E-04
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Frequency THz
Ampl
itude
a.u
.
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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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0.00E+00 1.00E+12 2.00E+12 3.00E+12 4.00E+12 5.00E+12
Frequency Hz
Ind
ex o
f Ref
ract
ion
water
propanol-1
0
50
100
150
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250
300
350
0.00E+00 1.00E+12 2.00E+12 3.00E+12 4.00E+12 5.00E+12
Frequency Hz
abso
rpti
on
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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1.00E-04
2.00E-04
3.00E-04
4.00E-04
0.00 1.00 2.00 3.00 4.00 5.00
Frequency THz
Ampl
itude
a.u.
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
0.00 1.00 2.00 3.00 4.00 5.00
Frequency THz
Ampl
itude
a.u.
Transmitted spectraIncident spectra
1 000 000 photons
a1 a2
a2/a1*1 000 000 photons
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Results - Full Spectrum
-0.00002
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0 1E+12 2E+12 3E+12 4E+12 5E+12
Frequency Hz
am
pli
tud
e
thin film matrix model
experiment
-2000
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18000
0 1E+12 2E+12 3E+12 4E+12 5E+12
Frequency Hz
Nu
mb
er o
f p
ho
ton
s
Monte Carlo Model
Experiment
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Results - Up to 1 THz
-0.00002
0
0.00002
0.00004
0.00006
0.00008
0.0001
0.00012
0.00014
0 2E+11 4E+11 6E+11 8E+11 1E+12
Frequency Hz
amp
litu
de
thin film model
experiment
-2000
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18000
0 2E+11 4E+11 6E+11 8E+11 1E+12
Frequency Hz
Nu
mb
er o
f p
ho
ton
s
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
-2.00E-05
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
1.00E-04
-0.00002 0.00003 0.00008 0.00013
model
Exp
erim
ent
Identity
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%
-0.00005
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0.0002
0.00025
0 5E+11 1E+12 1.5E+12 2E+12 2.5E+12 3E+12
Frequency Hz
Am
plit
ud
e
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.