Physics 114: Lecture 17
More Fitting to Arbitrary
Functions
John F. Federici
NJIT Physics Department
Star Wars quotes and jokes….
The problem…
A current problem with applying paint to vehicles such as automobiles, planes, buildings,
etc. is the need to monitor the thickness and composition of paint layers. For example, to
paint a car, one typically requires at least four layers
• Primer – this acts as an adhesive layer to the metal (or composite) body so that
subsequent layers stick
• Color layer – This gives the ‘color’ to the body of the car.
• Top Coat – This usually is a protective layer to protect the color layer underneath.
For example, it usually has TiO2 (essentially sunscreen) to prevent UV radiation from
dulling the color.
metal
Color
TopCoat
Primer
Primer
The problem … Cont.

Of course, there are also other types of coatings which automobile manufacturers
use… Rust inhibitors, Corrosion protection coatings (particularly on the undercarriage
of the car).

And of course the Military has ‘other’ types of coatings which is uses to enhance the
‘stealthy’ properties of its aircraft and helicopters and ground vehicles.

The key point is that in order to get the proper performance out of the coatings its is
important to control the THICKNESS of the various layers as they are deposited.
Essentially, variations in thickness can lead to variations in performance of the
coatings.

As a simple example, if the corrosion protection layers are too thin, then the car
corrodes to quickly. If the layers are too thick, you just used too much paint….
Multiply the wasted cost in the paint times the number of parts you have to spray to
make ~million cars per year and it turns into a non-negligible loss.
Monitoring thicknesses of Paint layers
How can you measure the thickness of paint layers NONINVASIVELY?
• Electromagnetic probing a good choice!
• Can not use ANY color of light since these paints are
generally opaque in the visible range.
• Let’s use THE FORCE LUKE!
• Sorry, I don’t know how to use the FORCE, but I can use
TERAHERTZ Radiation.
Terahertz Time-Domain Data – Recall the basic experimental layout
Sample
Terahertz
Source
Terahertz
Detector
t
Reference
Terahertz
Source
Terahertz
Detector
Think of THz pulses of radiation as similar to a Radar pulse
HOWEVER, THz radiation will NOT propagate through metal substrate
which is painted. Experiment MUST be done in reflection.
Transmission Vs. Reflection
Transmission
Terahertz
Source
Tera
h
Sou ertz
rce
ertz
Terah or
Detect
Sample
Reflection
Sample
Terahertz
Detector
Methods and Materials
Incident
THz Pulse
MultiLayer
Coating
Substrate
The basic method is to illuminate the multilayer coating with a short pulse
(several picoseconds) of THz radiation. Whenever there is a refractive index
change from one layer to another layer, a portion of the THz pulse is reflected
from the boundary.
The total reflectance includes contributions from each layer of the
coating: FRESNEL reflections from each boundary.
8
Thick and Thin Paint
For THICK Paint layers, Reflection from BACK and FRONT Interface
are WELL SEPARATED in time.
Reflection
Reflection
Sample
Sample
t
t
But for thin layers, the reflected pulses overlap in time and you can not
separate them easily. Pulses INTERFERE with each other
Frequency Domain Analysis
For a single paint layer on a
substrate material.
L
n
Metal
n  nr  ini
Complex Refractive Index
r( )  Es ( ) / Er ( )
Fourier Transformed Reflected
THz electric field from Sample
Fourier Transformed Reflected
THz electric field from Reference
e  i n 2 L / c  n  1  ei n 2 L / c  n  1
r ( , n )   i n 2 L / c
e
 n  1  e i n 2 L/c  n  1
Frequency of THz
wave
10
Results – Single Paint layers
FFT
The (left) time domain and (right) frequency dependent reflectivity plots for the
rain erosion coating layer on the aluminum substrate before any degradation
occurred. A fit to the measured reflectivity using the reflectivity equation on
previous slide gives a best fit of n  2.00  i 5.233  102
for an assumed thickness of 10.25 mil.
11
Results – Multilayer stack
Acquire a THz image of sample coupon by scanning sample pixel by pixel.
real refractive index
imaginary refractive index
In analysis, it is assumed that thickness of sample is FIXED.
• Variation in refractive index over coupon due to changing THICKNESS of paint
Optical Path
L  Lonreal
Length
12
Class Exercize
Your OVERALL GOAL (it will be in final project) will be to create a 2D map of the thickness
of painted sample. You will extract the optical path length from experimental data. From the
2D map of thickness you will then determine the AVERAGE thickness of the pain.
As a first step, let’s develop a code which extracts a best fit refractive index from the
experimental data.
(a) Download the REFERENCE file and SAMPLE file from WEEK 13 (in-class project)
(b) Convert data from time domain to frequency domain…. Lecture 09.
(c) The data which you will fit will be the magnitude of the reflectivity.
e  i n 2 L / c  n  1  ei n 2 L / c  n  1
r ( , n )   i n 2 L / c
e
 n  1  e i n 2 L/c  n  1
Assume a thickness of 12.75 mils, fitting parameters are the REAL refractive
index and the IMAGINARY refractive index
r
i
n  n  in