Improvement of minimum paint film thickness for THz
paint meters by multiple-regression analysis
Takashi Yasuda,1,3 Tetsuo Iwata,2 Tsutomu Araki,1 and Takeshi Yasui1,*
1

Graduate School of Engineering Science, Osaka University, 1–3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan
2Department of Mechanical Engineering, University of Tokushima, 2–1 Minamijousanjima-cho,
Tokushima 770-8506, Japan
3Present address: Central Research Laboratory, Hamamatsu Photonics K.K., 5000 Hirakuchi, Hamakita-ku, Hamamatsu,
Shizuoka 434-8601, Japan*Corresponding author: t-yasui@me.es.osaka-u.ac.jp
Received 18 April 2007; revised 22 July 2007; accepted 23 July 2007;
posted 27 July 2007 (Doc. ID 82193); published 18 October 2007

We propose a numerical parameter fitting method to determine the time delay between two temporally
overlapped echo pulses in terahertz (THz) tomography measurements. The method is based on multipleregression analysis with the least-squares method and is applied to decrease the minimum paint film
thickness for THz paint meters. Applying multiple-regression analysis to paint thickness measurements
is five times more sensitive with regard to the minimum thickness than numerical Fourier deconvolution.
We apply the proposed method to determine the optical thickness, geometrical thickness, and grouprefractive index of dry paint film and wet paint film. The proposed method is useful for decreasing theminimum thickness for a THz paint meter and other THz tomography measurements. © 2007 OpticalSociety of America
OCIS codes: 110.6960, 120.4290, 310.1620.

1. Introduction

In the last few decades, terahertz (THz) electromagnetic pulses have received interest as a new probe
for nondestructive testing. In addition to excellent
transmittance in nonmetal materials, THz pulses
provide attractive features for nondestructive testing, such as free-space propagation, extremely low
invasion, beam coherence, ultrashort pulse duration, broad spectral width, and the availability ofimaging and兾or spectroscopy. Of the various THz
imaging techniques, THz tomography [1,2] can visualize internal structures of an object as a crosssectional image. THz tomography is realized by atime-of-flight measurement of THz echo pulses
when a THz pulse is incident upon a sample in areflection configuration. The internal structure of
the sample can then be visualized from the distribution of the group refractive index. Since THz tomography acts as a noncontact and nonionizing
0003-6935/07/307518-09$15.00/0© 2007 Optical Society of America
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probe, in contrast to ultrasound and x ray methods,
it has been used to visualize the cross section ofinternal structures in various applications, such as
for floppy disks [2], pharmaceutical tablet coatings[3], skin burn [1], and skin cancer [4].
Another interesting application of THz tomography is nondestructive testing of paint film. The
painting of industrial products, such as cars, ships,
and aircraft, is important for rust prevention, waterproofing, color effects, light weight, and so on.
Hence, quality control of the paint film using apaint meter (test equipment for measuring painting
thickness) is required to maintain these functions.
Furthermore, in certain industries, such as the car
industry, in-process monitoring methods for paintfilms are required to control the painting spray inreal time based on monitoring results, to achieve
strict quality control of the paint film. In the field ofnondestructive testing of paint film, many methodologies have been established. Contact-type thickness meters with handheld scanners, such as
ultrasonic testing [5] and eddy-current testing [5],are powerful tools for simple and portable applica-

tions. However, these meters have limitations withregard to the quality of the material of the paintfilms and substrates, multilayered paint, wet paint,
mapping of thickness distribution, and兾or detectionof painting defects [6]. Furthermore, contact-type
thickness meters cannot be applied to in-processmonitoring because the in-process monitoring requires remote measurement of paint thickness. The
photothermal method [7] is suitable for the remotemeasurement at high spatial resolution. However,since this method is based on the temporal behaviorof heat injected by light, it is difficult to apply to wet
paint films that might be affected by heat. This
limits its use in in-process paint monitoring.
We have previously proposed a THz paint meterbased on THz tomography, as a noncontact tool for
testing paint films [6]. This method is applicable toinvestigating various kinds of paint films and substrates, multilayered paint, mapping of thickness distribution, and兾or detecting painting defects. Since
the THz paint meter uses THz pulses with low average power (typically, nW to ␮W), it can measure wet
paint films without causing effects due to heat induced by the THz pulse. Monitoring of the dryness in
wet paint films has been demonstrated through thespectroscopic difference in the THz region between
wet and dry paint films. However, since THz tomography is typically based on point-by-point measurements, it is necessary to perform two-dimensional
mechanical scanning of the time delay and sample
position to construct a two-dimensional tomographic
image of the sample. The resultant high time consumption of this method has limited its application tostationary objects. Recently, we proposed real-time
two-dimensional THz tomography to shorten themeasurement time drastically, and demonstrated thetechnique in a THz paint meter for real-time monitoring of the paint film of a moving object and thedrying progress of a wet paint film [8]. The availability of real-time monitoring makes THz paint meters
attractive for in-process paint monitoring.
An unsolved problem for in-process paint monitoring is that the minimum thickness of the THz paintmeter is insufficient for measuring thin paint films
used for car bodies (typically less than a few tens of
␮m). In the THz paint meter, when successive THz
echo pulses are well separated in time (i.e., for a thickpaint film), the thickness of the paint film can be
determined with an accuracy of a few ␮m. However,when the successive THz echo pulses are close to eachother and overlap (i.e., for a thin paint film), the timedelay of the THz echo pulses is affected by the temporal overlap. Therefore, a numerical Fourier deconvolution process (i.e., division of the Fourier spectraof the incident and reflected waveforms with a lowpass filter to remove noise) is often applied to extract
an impulse response signal from the THz echo pulse
train [2,6]. This process produces a sharp spike at atime delay corresponding to the position of each reflecting interface. The duration of the spike is determined by the cut-off frequency of the low-pass filter,
which is set to cover the frequency bandwidth of THz

radiation. In this case, the minimum thickness, determined by the spike duration, is constrained to behalf the coherent length of the THz pulse in a sample.
Although it is possible to generate a narrower spikeby using a low-pass filter with a higher cut-off frequency, this increases the noise components at
higher frequencies, and a weak THz echo pulse isburied under the noise. This results in a decrease ofquantitative accuracy in thickness measurements ofthin paint films, and therefore numerical Fourier deconvolution is not effective in improving the minimum thickness below half the coherent length of theTHz pulse. A direct method to decrease the minimum
paint thickness is to shorten the THz pulse duration.
However, even though the pulse, duration of a femtosecond laser for THz generation can be varied in therange 50 –200 fs, the THz electric field is not changed
significantly. Broadband THz radiation generated byan even shorter femtosecond laser is no longer a
monocycle pulse, but instead is a multicycle pulse [9],which is not suitable for THz tomography. Anothermethod to decrease the minimum thickness is to use
a numerical parameter fitting algorithm to separate
superposed THz echo pulses, in place of numerical
Fourier deconvolution. In multi-component spectrum
analysis in which the spectra of each component overlap spectrally, a parameter fitting algorithm based on
multiple-regression analysis is often used to separate
the spectra [10,11]. Multi-component spectrum analysis is analogous to the case of successive THz echopulses close to each other and overlapping temporally(when applying a THz paint meter to a thin paintfilm).
In this paper, we present a method to decrease
the minimum thickness of a THz paint meter using
a parameter fitting method based on multipleregression analysis. We apply the parameter fitting
method to the monitoring of the optical thickness,geometrical thickness, and group refractive index ofdry paint film and wet paint film.
2. Principle

We consider a single-layer paint film on a metal substrate, as shown in Fig. 1. When numerical Fourier
deconvolution is used for a THz paint meter, the minimum paint thickness 共dmin兲 is determined to be as
follows [6]:dmin ⫽ 共c⌬T兲兾共2ng兲,(1)where c is the velocity of light in vacuum, ⌬T is the
temporal width of the THz pulse, and ng is the grouprefractive index of the paint film. The dmin value isequal to half the coherent length of the THz pulse in
the paint film. When the geometrical thickness of thepaint film (d) is sufficiently larger than dmin, the two
THz echo pulses returning from the front and rearsurfaces of the paint film are well separated in time.
The d value is determined by the time separation
between the two echo signals 共⌬t兲 and the ng value as
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7519

where Ein共t兲 is the incident THz pulse, R is the reflectivity at the paint surface, A is the absorption in thepaint film, and ␣ is a parameter unifying the effectsof surface reflection, absorption, and thickness of thepaint film; ⌬t1 and ⌬t2 are the time delays of the firstand second echoes in the paint film from Ein共t兲, respectively. The difference between them 共⌬t2 ⫺ ⌬t1兲
corresponds to the optical thickness 共ngd兲 of the paintfilm, from which the geometrical thickness (d) can be
determined if ng is known. The overall signal of theTHz echo pulses 关Eall共t兲兴 is expressed as:
Eall共t兲 ⫽ E1st共t兲 ⫹ E2nd共t兲 ⬇ R ⫻ Ein共t ⫹ ⌬t1兲
⫹ ␣ ⫻ Ein共t ⫹ ⌬t2兲. (5)R, ␣, ⌬t1, and ⌬t2 can be calculated by parameter
fitting for measured Eall共t兲 and Ein共t兲 based on Eq. (5).
In this way, the paint thickness can be determinedfrom the calculated ⌬t1 and ⌬t2, even though the two
THz echo pulses overlap significantly. To this end, we
perform multiple-regression analysis with the leastsquares method to minimize the residual sum of thesquares between a measured Eall共t兲 and a simulated
Eall共t兲. We then use the calculated ⌬t 共⫽ ⌬t2 ⫺ ⌬t1兲 todetermine the paint thickness based on Eq. (2).
3. Experimental Setup

Fig. 1. Principle of the THz paint meter. (a) Single-layer paintfilm and (b) temporal waveforms of THz echo pulse.

follows:d ⫽ 共c⌬t兲兾共2ng兲.

A typical setup for THz tomography based on pointby-point measurements was prepared, as shown in
Fig. 2. The laser light from a mode-locked (ML)Ti:sapphire laser oscillator (ML Ti:S laser; pulse
duration ⫽ 100 fs, central wavelength ⫽ 800 nm, repetition rate ⫽ 80 MHz, average power ⫽ 1 W) is split
into a pump light and a probe light with a beam
splitter (BS). The pump light is focused by an objective lens (L) onto a dipole-shaped, low-temperaturegrown GaAs photoconductive antenna for THz
generation (THz-EM). To improve the signal-to-noise(2)

However, when d ⬍ dmin, the two THz echo pulses
are not well separated and are superposed in time. In
this case, it is difficult to determine d from ⌬t correctly because the degree of the temporal overlapchanges the value of ⌬t.
We next consider the first and second THz echopulses [E1st共t兲 and E2nd共t兲] returning from the front
and rear surfaces of the paint film. In the following
discussions, we neglect the effect of dispersion in therefractive index and absorption of the paint because
the effect of dispersion is very small in a thin film.
We also neglect the effect of multiple reflections of theTHz pulse in the paint film for simplicity of the parameter fitting calculation. E1st共t兲 and E2nd共t兲 are expressed as follows:
E1st共t兲 ⫽ R ⫻ Ein共t ⫹ ⌬t1兲,(3)

E2nd共t兲 ⫽ 共1 ⫺ R兲2 ⫻ exp共⫺2d ⫻ A兲 ⫻ Ein共t ⫹ ⌬t2兲
⬇ ␣ ⫻ Ein共t ⫹ ⌬t2兲,
(4)
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APPLIED OPTICS 兾 Vol. 46, No. 30 兾 20 October 2007

Fig. 2. Experimental setup. ML Ti:S laser: mode-locked
Ti:sapphire laser; BS: beam splitter; L: objective lens; THz-EM:
photoconductive antenna for THz generation; Si-L: hemisphericalsilicon lens; OAP1, OAP2, OAP3: off-axis parabolic mirrors; Si-BS:
hemispherical silicon beam splitter; THz-DT: photoconductive antenna for THz detection.

nesses (d ⫽ 60, 50, 40, 30, 20, and 10 ␮m), all lessthan dmin.
A temporal waveform of the THz echo pulses measured in the 60 ␮m thick paint film is shown in Fig.
4(a). A comparison between Figs. 3 and 4(a) indicatesthat it is difficult to directly distinguish E1st共t兲 andE2nd共t兲 in the 60 ␮m thick paint film. In the multipleregression analysis with the least-squares method,
the temporal waveform in Fig. 4(a) was used as an

Fig. 3. Temporal waveform of an incident THz pulse 关Ein共t兲兴 obtained when setting an aluminum substrate at the sample position.

(S兾N) ratio using lock-in detection, we chopped theTHz pulse by electric bias modulation of the THz-EM(amplitude ⫽ 20 Vp-p, offset ⫽ 10 V, freq. ⫽ 5 kHz [12].
The radiated THz beam is focused on a sample via
a hemispherical silicon lens (Si-L), two off-axis parabolic mirrors (OAP1 and OAP2), and a silicon
beam splitter (Si-BS). The THz echo pulse returned
from the sample is recollimated and focused onto
another dipole-shaped, photoconductive antenna
for THz detection (THz-DT) with mirrors OAP2 andOAP3 and another Si-L. The effective focal lengths
of OAP1, OAP2, and OAP3 are 76.2 mm. The probe
light is incident on the THz-DT via a motor-drivenmechanical delay line. When the THz pulse and theprobe pulse are incident on the THz-DT simultaneously, the electric field of the THz pulse causes atransient photocurrent in the THz-DT. The temporal
waveform of the THz electric field is obtained with alock-in amplifier (time constant ⫽ 100 ms) after passing through a current preamplifier. To determine theprofile of Ein共t兲, we set an aluminum mirror at the
sample position and measured the temporal waveform of the THz electric field. We obtained a pulsed
THz electric field with 1.2 ps pulse duration and an
S兾N ratio over 800, as shown in Fig. 3.
4. Results
A.

Dry Paint Film

As a sample we used an oil-based, acrylic paint film
(black color) on an aluminum substrate. The grouprefractive index 共ng兲 of the dry paint sample was determined beforehand using thick paint film samples
from the relationship between the geometrical thickness (d) measured with a contact-type thickness
meter (eddy-current type, precision ⫽ ⫾3% of the actual thickness) and the optical thickness 共ngd兲 measured with a THz paint meter according to theprocedure given in a previous paper [6]. The ng valueof the sample was determined to be 1.66, and dmin in
the present system was estimated to be 108 ␮mbased on Eq. (1). We selected six single-layer dry
paint film samples with different geometrical thick-Fig. 4. Temporal waveform of the THz echo pulse of a single-layer
dry paint film with 60 ␮m thickness. (a) Signal measured with
THz paint meter, (b) signal simulated by multiple-regression analysis, and (c) residual between the measured signal and simulated
signal.
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Table 1. Result of Multiple Regression Analysis for Dry Paint Films

Paint Thickness
(␮m)

k0

k1
(ps)

k2

k3
(ps)

Loop
Number⌬t (ps)[⫽ k3 ⫺ k1]

d
(␮m)

60
50
40
30
20
10

044
0.48
0.48
0.50
0.50
NA

8.091
2.144
8.501
0.850
2.745
NA

0.56
0.52
0.52
0.50
0.50
NA

8.774
2.682
8.911
1.190
3.008
NA

5
2
4
3
10
NA

0.683
0.538
0.410
0.340
0.263
NA

61
48
37
31
23
NA

objective function X. We defined a model function Y tosimulate the objective function X as follows:Y ⫽ k0 ⫻ Ein共t ⫺ k1兲 ⫹ k2 ⫻ Ein共t ⫺ k3兲,(6)

where t is an independent variable and ki 共i ⫽ 0,
1, 2, 3兲 is a regression parameter. The following constraint conditions are applied for the regression parameters ki:
k0, k2 ⬎ 0
k 1 ⬍ k3
k0 ⫹ k2 ⫽ 1.

(7)

Before starting the multiple-regression analysis,
we set the initial values k0, k1, k2, and k3 to simulate
X with as good an approximation as possible so thatthe analysis converges to a characteristic global
minimum and not to incorrect local minima. The
multiple-regression analysis was performed using
computing software (IGOR PRO, WaveMetrics, Inc.) and
was automatically repeated until the residual sum of
the squares (␹2) between X and Y reached the global
minimum. Table 1 shows the result of the multipleregression analysis for the 60 ␮m thick paint film.
Figures 4(b) and 4(c) show the simulated Y resulting
from the multiple-regression analysis and the residua (␹) between X and Y, respectively. We can confirm from these results that the simulated Y well
reproduces the measured X. The results for the othersamples are also summarized in Table 1. We obtained
good results for paint films with thicknesses of morethan 20 ␮m.
However, for the 10 ␮m thickness paint film, theanalysis did not converge even to a local minimum,
let alone the global minimum, even though k0, k1, k2,
and k3 were varied over the whole available range.
Hence, the present parameter fitting method is notapplicable to paint films with thicknesses of less than
10 ␮m. For the present experimental setup, this limitation is due to the THz pulse width 共⫽ 1.2 ps兲 andthe probe pulse width 共⫽ 100 fs兲, because a paintthickness of 10 ␮m is equivalent to a time separationof 111 fs.
For the parameter fitting algorithm, we consider
that the minimum thickness is limited by two effects
that were neglected in the analysis. One effect is7522

APPLIED OPTICS 兾 Vol. 46, No. 30 兾 20 October 2007

multiple reflections of the THz echo pulse in the paintfilm. When the Fresnel reflection at the paint surface
is 0.25 from the group refractive indices of air 共⫽ 1兲
and the paint 共⫽ 1.66兲, the electric field of the THz
echo pulse returning from the rear surface of thepaint film is 56% of the THz incident pulse for no
multiple reflections, 14% for one multiple reflection,and 3.5% for two multiple reflections. Although theeffect of multiple reflections is neglected to simplify
the parameter fitting calculation in this paper, forprecise determination of the paint thickness the effect of multiple reflections must be considered. The
second effect is the dispersion of the absorption andrefractive index in the paint film. The dispersion effect distorts the temporal shape of the THz echopulse. It is difficult to accurately apply the present
fitting algorithm assuming that there is no temporal
distortion of the THz echo pulse.
Figure 5 shows the relationship between the actual
paint thickness (d) determined by a contact-type thickness meter (eddy-current type, precision ⫽ ⫾3% of
the actual thickness) and that (dsim) simulated bymultiple-regression analysis of each paint sample.
The relationship between d and dsim can be confirmed
as being d ⫽ dsim (fitting error: ␹2 ⫽ 24, R ⫽ 0.986),

Fig. 5. Relationship between actual paint thickness measured
with a contact-type thickness meter and simulated painting thickness by multiple regression analysis for five dry paint film samples
with different thicknesses. Solid line represents d ⫽ dsim.

and hence we can conclude that the minimum thickness of the present THz paint meter is correctly attained down to 20 ␮m using multiple-regression
analysis that improves the temporal resolution numerically. Defining the uncertainty as the standard
deviation of the difference between d and dsim, an
uncertainty of 2.4 ␮m was obtained. The uncertainty
was due to the limitation of the present fitting
method, the precision of the contact-type thickness
meter, and the inconsistency of the measurement
spot between the THz paint meter and the contacttype thickness meter.
We next investigated the effect of the S兾N ratio in
the measured THz echo signal (X). The noise in theTHz paint meter is due to two sources, environmental
artifacts and background noise. When the THz paintmeter is used in a factory environment, the main
noise source is environmental artifacts, such as air
turbulence or mechanical vibration. Conversely, in astable condition with less environmental artifacts,
background noise, such as the noise of the detector or
amplifier, is dominant. The effect of the two types ofnoise on the THz signal is that different environmental artifacts have a multiplying effect, while background noise has an additive effect. Our objective for
the THz paint meter is for use in a factory environment with various artifacts. Therefore, we successively decreased the S兾N ratio to 40 for the THz echo
signal (X) measured in the 20 ␮m paint film by multiplying the THz signal by white noise. We then determined the minimum ␹2 for X at each S兾N ratio
while varying k0, k1, k2, and k3 over the available
range to achieve accurate parameter fitting. Figure 6
shows the relationship between the S兾N ratio andthe simulated paint thickness. The simulated paintthickness converges well to the actual paint thickness
共20 ␮m兲 when the S兾N ratio ⬎ 100. However, when
the S兾N ratio ⬍ 100, the simulated values deviatesignificantly from the actual paint thickness. Therefore, an S兾N ratio of more than 100 is required todetermine the paint thickness correctly with the proposed parameter fitting method. This is because noisedue to environmental artifacts is dependent on theelectric field of the THz echo signal by a multiplicative factor, and is enhanced largely around the THz
echo pulse. When adding white noise to the THz signal (i.e., simulating a stable condition with less environmental artifacts), the proposed fitting algorithm
could be used at a lower S兾N ratio. The THz paintmeter can easily achieve an S兾N ratio over 100 because the THz absorption of a paint film is low in
general. Hence, the proposed method is useful for
decreasing the minimum paint thickness in a THz
paint meter.
We plot the k0 and k2 parameters with respect toeach paint sample in Fig. 7. The k0 parameter should
be fixed at a certain level for all the samples if
the surface of each paint film is always placed at thesame position with high precision. For example,
the k0 parameter is calculated to be 0.25 from theFresnel reflection formula for the group refractive
indices of air 共⫽ 1兲 and the paint 共⫽ 1.66兲. The actualk0 parameter deviates greatly from this value and is
not constant. The deviation and fluctuation are due
to variations of the focal condition of the THz beamfor each sample, caused by positioning imprecisionand兾or tilting of the sample. On the other hand, thek2 parameter is influenced by THz absorption of thepaint film [see Eq. (4)]. If the THz absorption is dominant over variations of the focal condition of the THz
beam, the k2 parameter will be dependent on thepaint thickness. However, the actual k2 parameter is
independent of the paint thickness, which implies
that THz absorption in the paint film is negligible.
We consider that the k0 and k2 parameters act asbuffers to cancel the fluctuation of the focal conditionof the THz beam. Since the paint thickness can be
determined by the k1 and k3 parameters, the numberof fitting parameters can be reduced for faster convergence of the multiple regression analysis. This
will expand the range of measurable film thickness.

Fig. 6. Change of simulated paint thickness for the 20 ␮m paintfilm when changing the S兾N ratio.

Fig. 7. Parameters k0 and k2 for five dry paint film samples withdifferent thickness.
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7523

Fig. 8. Temporal evolution of THz echo pulse signal during the
wet-to-dry transformation of a paint film. Temporal waveforms of
the THz echo pulses were measured every 1 min after applying thepaint.

B.

Wet Paint Film

An advantage of a THz paint meter over a conventional thickness meter is the ability to monitor dryness and paint thickness of a wet paint film. There isa discernible difference in the spectroscopic characteristics (index of refraction and absorption) in theTHz range between the wet and dry conditions ofpaint film, and therefore the wet-to-dry transforma-tion causes a temporal change of the THz echo pulse
train [6,8]. We previously reported that two echopulses, from the paint surface and the paintsubstrate boundary, temporally evolve in time delay
and pulse height. We further proposed time delay andecho pulse contrast as drying degree parameters,
which can be directly extracted from the THz echopulse signal. However, the two echo pulses for a thin
wet paint film are not well separated and overlap intime. Such a temporal overlap of two THz echo pulses
may lead to errors in the parameters. Therefore, weapplied the parameter fitting method described in
Subsection 4.A to the correct monitoring of dryness
and paint thickness in a thin wet paint film.
For this examination, we applied another black
acrylic paint film (quick-drying type) on an aluminum
substrate just before starting the measurement. We
then measured the temporal waveform of THz echopulses every 1 min after applying the paint. Figure 8
shows the temporal evolution of the THz echo pulse
signal. The first echo comes from the paint surface
while the second echo is a reflection from the paintsubstrate boundary. The time separation between
the first and second echoes corresponds to the optical
thickness of the paint film. We can confirm that thetwo echo pulses are separated before 2 min but overlap after 3 min. We applied the multiple-regression
analysis based on Eq. (6) for the THz echo pulse
signal of the wet paint film in Fig. 8. The results of themultiple-regression analysis, time delay 共⫽ ⌬t ⫽ k3
⫺ k1兲, and optical thickness 共⫽ c · ⌬t兾2 ⫽ ngd兲 of thewet paint film are summarized in Table 2. If thegroup refractive index 共⫽ ng兲 is known, one can determine the geometrical thickness 共⫽ d兲 from the optical thickness. However, in the case of the wet paintfilm, the group refractive index is unknown because it
varies every moment depending on the drying degree.
Therefore, it is impossible to determine the geometrical thickness from the optical thickness. However, if
we utilize a relationship between geometrical thickness change and optical thickness change accompanying the wet-to-dry transformation, it is possible todetermine the group refractive index of the wet paintfilm at each degree of the drying. Figure 9(a) shows
the temporal changes of the k1 and ⌬t parameterswith respect to the elapsed time, plotting their deviations from the initial values at 0 min. Since theTable 2. Result of Multiple Regression Analysis for a Wet Paint Film

7524

Elapsed Time
(min)

k0

k1
(ps)

k2

k3
(ps)

Loop
Number⌬t (ps)[⫽ k3 ⫺ k1]

ngd
(␮m)

0
1
2
3
4
5
6
7
8

0.44
0.40
0.37
0.36
0.35
0.34
0.34
0.33
0.32

⫺0.133
0.051
0.120
0.153
0.253
0.187
0.187
0.167
0.153

0.56
0.60
0.63
0.64
0.65
0.66
0.66
0.67
0.68

0.823
0.814
0.774
0.743
0.804
0.715
0.708
0.681
0.653

6
6
4
7
11
7
5
5
5

0.956
0.763
0.654
0.589
0.551
0.529
0.521
0.515
0.500

143
115
98
88
83
79
78
77
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APPLIED OPTICS 兾 Vol. 46, No. 30 兾 20 October 2007

an error in the parameter fitting method, indicating
too many loop numbers as compared with otherfitting results. It is clear from Figs. 9(a) and 9(b) thatthe wet-to-dry transformation changes the optical
thickness, geometrical thickness, and average grouprefractive index of the wet paint film.
The average group refractive index is effective formonitoring of the dryness of the wet paint film. In
general, wet paint is composed of a resin (e.g., acrylic
resin), a pigment, and an organic solvent (e.g., paintthinner). The drying process of the wet paint is temporally advanced by the volatilization of the organic
solvent, and the volatilization of the solvent and兾orchemical change of the resin caused the spectroscopic
difference in the THz region between the wet and dry
conditions of a paint film. For example, since the wet
paint has the group refractive index dominated by
the solvent, the group refractive index of the wet
paint film is smaller than that of the dry film andthus it increases depending on the degree of drying. If
the group refractive index of the dry paint film isknown, the difference of the group refractive index
between the wet and dry paint films indicates thedryness of the wet paint. Therefore, the averagegroup refractive index of the wet paint film can beused as a quantitative parameter indicating the drying progress. Since the group refractive index of theblack acrylic paint used is 1.81 for dry film [6], temporal change of the average group refractive index in
Fig. 9(b) indicated that the drying process is not still
completed at 8 min.
5. Conclusions

Fig. 9. Temporal change of each parameter for wet paint filmwith respect to elapsed time. (a) Deviations of ⌬t and k1 from theinitial values at 0 min and (b) average group refractive index.

change of k1 is due to displacement of the paint surface caused by geometrical shrinkage, it is related tochange of the geometrical thickness 关⫽ ⌬dgeo共t兲兴. On
the other hand, the change of ⌬t reflects that of optical thickness of the wet paint film 关⫽ ⌬dopt共t兲兴. The⌬dopt共t兲 is related with the ⌬dgeo共t兲 as follows:⌬dopt共t兲 ⫽ Ng共t兲 · ⌬dgeo共t兲,(8)where Ng共t兲 is an average group refractive index.
With regard to the group refractive index of the wet
paint film, we have to consider its inhomogeneity
along the depth direction because the drying advances from the paint surface. Therefore, we use theaverage group refractive index Ng共t兲 for the wet paintfilm. From Eq. (8) and the results of Fig. 9(a), we
determined the average group refractive index of thewet paint film at each degree of the drying, as shownin Fig. 9(b). We consider that the large fluctuation of
the average group refractive index at 4 min is due to

We proposed a numerical parameter fitting method
based on multiple-regression analysis to separate superposed THz echo pulses in a THz paint meter. The
minimum paint thickness for a THz paint meter was
decreased from 108 to 20 ␮m at an S兾N ratio of over
100 by the use of multiple-regression analysis. A parameter fitting method considering multiple reflections of the THz echo pulse and兾or the dispersive
absorption and refractive index in a paint film willfurther decrease the minimum paint thickness to lessthan 20 ␮m.
We applied the proposed parameter fitting method
to determining the optical thickness and group refractive index of a wet paint film. The results indicatethat optical shrinking occurs, caused by both geometrical shrinking and a change in the group refractive
index in the wet-to-dry transformation of the paintfilm. The proposed method will be useful for decreasing the minimum thickness for THz paint meters andother THz tomography measurements.
This work was supported by the Industrial Technology Research Grant Program 2002 of the New Energy
and Industrial Technology Development Organization (NEDO) of Japan, the Strategic Information andCommunications R&D Promotion Program (SCOPE)
of the Ministry of Internal Affairs and Communications (MIC) of Japan, and Grants-in-Aid for Scientific
Research 18686008 and 18650121 from the Ministry
20 October 2007 兾 Vol. 46, No. 30 兾 APPLIED OPTICS

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of Education, Culture, Sports, Science, and Technology of Japan. We are grateful to Mamoru Hashimotoof Osaka University for fruitful discussions.
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