Instrumentation & Measurement Magazine 24-4 - 6

mid-wavelength infrared (MWIR, 3-8 μm) or long-wavelength
infrared (LWIR, 8-15 μm) range drastically limits the maximal
resolution achievable using high magnification objectives.
Initial Heating
Digital lock-in demodulation exhibits the best results when
the measurement is performed under quasi-steady state conditions.
This means, the sample temperature oscillates with
time around a steady mean temperature. Usually, a thermal
relaxation time that depends mainly on the heat transfer resistance
between the sample surface and the surrounding will
take place at the beginning of the experiment (Fig.1b). This
initial period will induce an amplitude and phase shift that affects
the accuracy of the results. In principle, the quasi-steady
state should be reached before demodulating the thermal signal.
Nonetheless, such an approach is not very practical since
the initial period can last several minutes, depending on the
modulation parameters. Different methods have been proposed
in the literature to solve this problem. In particular, the
'gedankenexperiment' introduced by Breitenstein [15], where
the in-phase and in-quadrature images are corrected by subtracting
the demodulation of the non-steady period, possesses
many advantages. The compensated correction takes the form:
Corr0,   
j
x y

2j sin
2
j1
n
T xy, 21
n


x y

2jcos
2
j1
n
T xy, 21
n
Nn   (7)
Corr90,   
j


Nn   (8)
where ΔT (x, y) is the total temperature drift image, and:
  
  
S xy S xy
00
corr
,
S xy S x y, Corr 90 ,
90
corr
,
  xy
90
 , Corr 0 ,
  x y
A xy S xy S xy
N

, ,,corr
2 00

  tan1 S xy
,
90
S xy corr
,
xy,
0



 


corr

 
22
corr
Similar to the method proposed by Gupta and Breitenstein,
this correction can be performed online and avoids
storing all of the images in the computer memory. Nonetheless,
it demonstrates lower performances than the
" gedankenexperiment. "
Non-harmonic Heating
The lock-in demodulation is based on the expectation of
a harmonic stimulation. To avoid abrupt variations of the
stimulation scheme, the initial phase of the sine or cosine
stimulation signal is set to π / 4 . If a non-harmonic stimulation
is chosen (like square-wave), the transient thermal
signal will contain additional harmonics that will be filtered
out if the camera frame rate is much higher that the
stimulation frequency. In our different experimental setups
where the stimulation frequency usually ranges from 0.1
to 0.8 Hz and where the camera frame rate is above 80 Hz,
we did not notice any significant differences between harmonic
and square-wave stimulation. Nonetheless, the shape
of the stimulation signal plays a major role when trying to
mathematically model the experimental setup to extract
quantitative values.
Emissivity Compensation
(9)
(10)
to allow the correct calculation of the amplitude and phase:

The emissivity is a unitless quantity that describes the ability
of the surface of a sample to emit thermal radiation. The
emissivity is wavelength dependent and angle dependent. It
is usually given for a specific wavelength band (for example
LWIR 8-15 μm), and the angle dependencies are neglected. In
LIT, the emissivity will affect identically the in-phase and inquadrature
signal. As a result, the phase remains unaffected by
emissivity variations, whereas it directly influences the amplitude.
When measuring NPs in low concentration, the surface
emissivity will be dominated by the medium in which the NPs
are embedded. For example, the emissivity of SPIONs diluted
in water at different concentrations will remain the surface
emissivity of water at ambient temperature (0.96). A way to
consider emissivity variations is to introduce a reference with
a known emissivity εref
(11)
(12)
where Ta
We proposed an even simpler compensation method where
the initial non-steady-state temperature signal is divided into
several segments that are subtracted from the transient temperature.
Since the digital lock-in demodulation principle is a
linear operation, this subtraction can be achieved after demodulation.
The amplitude image takes the form after correction:
A xy S x y S0 xy
 
,
6
90 fi
  
2
,
TT
, 


n


 
2
(13)
perature Ti of the sample. The radiated power Pref
reference sample is calculated with:
2
P TT

d
ref ref
4 
44
i  a
and measure the absolute initial temin
Watt of the
(13)
the ambient temperature and σ the StephanBoltzmann
constant. The emissivity ε of an unknown sample
can be calculated using:


ref
TT
TT
ia
and assigned for each region of interest in the infrared camera
software to allow computation of the absolute temperature.
Non-homogeneous Stimulation
When performing LIT experiments, the excitation scheme
should be as homogeneous as possible to ensure accuracy of
IEEE Instrumentation & Measurement Magazine
June 2021
44
44
meas  a

(14)
Instrumentation & Measurement Magazine 24-4

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