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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
Investigation of a self-calibration method for global rainbow
refractometry
Kailin Cao1, Xuecheng Wu1,*, Haoyu Jiang1, Yingchun Wu1, 2, Can
Li1, Kefa Cen1, and Gerard Grehan2
1: State Key Lab of Clean Energy Utilization, Zhejiang
University, Hangzhou 310027, China 2: UMR 6614/CORIA, CNRS, BP12
76801 Saint Etienne du Rouvray, France
* Correspondent author: [email protected]
Keywords: Global rainbow refractometry, Self-calibration,
Scattering angle, Dual-wavelength laser
ABSTRACT
Accurate measurement and control of key parameters of droplets
in the complex atomization or spray flow field play an instructive
role in the mechanism study of multiphase flows and optimization of
specific industrial process. Among numerous advanced optical
measurement techniques for characterizing droplets, global rainbow
refractometry is able to measure the size distribution and
refractive index of droplets simultaneously. Temperature,
concentration, solution components and any other relevant parameter
can be determined according to the relationship between refractive
index and target parameter. Since the information of refractive
index and droplet size contained in the rainbow intensity
distribution is highly sensitive to the scattering angle, accuracy
of calibration for the relationship between CCD pixel column and
scattering angle greatly affects the measurement accuracy. A novel
self-calibrated global rainbow refractometry based on
dual-wavelength laser is proposed in this paper. The angular
calibration and the refractive index measurement under two
wavelengths can be realised simultaneously without additional
measure devices. The actual refractive indices of droplets under
two wavelengths and calibration equation can be quickly and
simultaneously obtained by searching the minimum of root mean
square (RMS) errors between the hypothesizing refractive indices
and the ones from inversion. The results of simulation and
experiment show the feasibility and high precision of self
calibration method for global refractometry. This self-calibrated
global rainbow system was also successfully applied to measure the
refractive index of water–ethanol solutions with volume
concentration from 10% to 60%. What’s more, this design can be
extended to calibration for one-dimensional rainbow technique.
1. Introduction Liquid atomization and spray has a broad range
of applications in the energy, chemical and many other industrial
fields. Accurate measurement and control of key parameters of
droplets in the complex atomization or spray flow field play an
instructive role in the mechanism study of multiphase flows and
optimization of specific industrial process. Among numerous
advanced optical measurement techniques for characterizing
droplets, global rainbow refractometry (Van Beeck and Giannoulis,
1999) has been shown to be a powerful tool for its advantages in
simultaneously measuring the size distribution and refractive index
for both individual droplets,
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
and spray of polydispersed droplets. Temperature (Hom and
Chigier, 2002), concentration (Wu and Wu, 2012), solution
components (Zhao and Qiu, 2006) and any other characterization
parameter can be determined according to the relationship between
refractive index and target parameters. By analyzing the
distribution of scattered light around the primary rainbow angle,
rainbow technique extracts the absolute angular location and the
shape of the rainbow pattern, which can be used to inverse the
refractive index and the droplet size respectively. Standard
rainbow technique (Roth and Anders, 1991) was firstly presented to
measure a spherical single droplet or individual monodispersed
droplets in a line. Later, the standard rainbow technique was
progressed to the global rainbow technique (Van Beeck and
Giannoulis, 1999) for measuring spray droplets. Global rainbow
technique requires larger size of the pinhole and longer exposure
time of the CCD camera to get superposition of rainbows from
numerous droplets that are illuminated by laser. This superposition
of rainbows avoids the effects of the non-sphericity of the
droplets and eliminates the high-frequency component of the rainbow
signals to obtain a steady and smooth rainbow structure. Recently,
a one-dimensional rainbow technique (ORT-1) (Wu and Jiang, 2014)
was proposed that extended a point volume measurement to a
one-dimensional segment measurement by using slit apertures. Then,
an alternative optical design of one-dimensional rainbow technique
(ORT-2) (Wu and Promvongsa, 2015) was developed. ORT-2 aims to
facilitate the angular calibration in the full image field by using
Fourier domain filtering. Since the information of refractive index
and droplet size contained in the rainbow intensity distribution is
highly sensitive to the scattering angle, precise calibration of
the relationship between CCD pixel column (pix) and scattering
angle (θ) is the key part to guarantee the measurement accuracy.
Acquiring global rainbow images on a set of sprays of known
temperature is one of the calibration methods (Lemaitre and
Porcheron, 2006). This method is susceptible to environmental
perturbations, such as temperature change. The most conventional
calibration process is completed by using a mirror mounted on the
rotation platform (Vetrano and Gauthier, 2006), which we call as
the mirror calibration method. However, errors caused by manual
operation and additional precise instrument of mirror calibration
method is unavoidable. When it comes to harsh industrial
environments such as spray in a container, calibration for the
mirror inaccessible occasions is a challenge. To solve the problem,
we propose a self-calibration method based on dual-wavelength
lasers for global rainbow refractometry. This method enables us to
calibrate the scattering angle along with the refractive index
measurement under two wavelengths illuminations.
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
2. Numerical and Experimental Methods As the material’s natural
property, the refractive index changes with the wavelength of the
light when other conditions keep the same. With the increase of the
wavelength of the light, refractive index of common liquid is
decreased, such as water and alcohol. For spray droplets
illuminated by two different wavelengths of laser beams that are
modulated into a coincident one, two rainbow patterns of different
wavelengths can be obtained on method of appropriate signal
separation. Obviously, these two rainbows correspond to the same
particle size distribution and the same temperature. If the
equation for the refractive index as a function of wavelength and
temperature is known in advance, according to coupling relationship
of refractive index between this pair of rainbows, both the
calibration equation and the droplets temperature (T) might be
yielded simultaneously. In this study, two vertically placed lasers
of different wavelengths (e.g. λ1= 532 nm, λ2= 633 nm) are
modulated to get one coincident laser beam by using a beam
splitter, as shown in Fig. 1. The coincident laser beam
horizontally illuminates the spray with unknown temperature.
Fig. 1. Optical setup for the self-calibrated global rainbow
refractometry.
Two spherical lenses, one aperture and a color CCD form the
imaging system for self-calibrated global rainbow refractometry.
Instead of using a black-white CCD sensor to record monochromatic
rainbow signals in classical global rainbow system, here a color
CCD sensor is applied to record dual-wavelength scattered light
signals. The mixed signals can be separated in red, green, and blue
channels, so two global rainbow curves of two wavelengths can be
extracted
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
respectively from the Green and Red channel, as the simulated
global rainbow curves shown in Fig. 2(a). The global rainbow curves
are simulated based on the Lorenz–Mie theory. The pixels
corresponding to the geometric rainbow angles (pix1, pix2) can be
determined by the angles where the normalized scattering light
intensities reach a value of RI (RI = Irg /IMAX) before the primary
rainbow maximum. Irg is the scattered light intensity corresponding
to the geometric rainbow angle. IMAX is the scattered light
intensity that reaches the maximum. The geometric rainbow angle,
θrg only depends on the refractive index (n) that varies with the
wavelength of light, as given by Eq. (1) (Hulst and Van De Hulst,
1957).
2 21 1
2
4 44sin 2sin3 3rgn nn
θ π − −− −
= − + (1)
As the equation for the refractive index of spray as a function
of wavelength and temperature should be known in advance, here we
choose deionized water whose physical characteristics have been
fully studied, as given by Eq. (1) (Quan and Fry, 1995).
6 2 1 2 6 3( , ) 1.31405 2.02 10 (15.868 0.00423 ) 4382 1.1455
10n T T Tλ λ λ λ− − − −= − × + − − + × (2)
When the spray temperature (T) has been hypothesized, the
refractive indices of calibration medium under two wavelengths (n1,
n2) can be determined by Eq. (2), and geometric rainbow angles can
be calculated by Eq. (1). In Fig. 2(a), pix1 and pix2 correspond to
θrg1 and θrg2 respectively.
Fig. 2. Simulated global rainbow curves under two
wavelengths
Since the calibrated angle-pixel relationship presents good
linearity between the scattering angle θ and the CCD pixel, the
calibration equation is usually obtained by linear fitting (θ = a +
b × pix)
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
and has been shown to ensure the accuracy. By both hypothesizing
the value of RI and spray temperature T, two characteristic
calibration point (pix1, θrg1) and (pix2, θrg2) can be used to
obtain a and b, the characterizing parameters of linear calibration
equation. Then we get the dual-wavelength distributions of
intensity on scattering angle, as shown in Fig. 2(b). These
distributions are used for the rainbow signal processing to inverse
the refractive indices (m1, m2) by using an inversion algorithm
based on the modified Nussenzveig’s theory (Saengkaew and
Charinpanikul, 2010). By using the non-negative least-squares
algorithm, the refractive indices (m1, m2) and size distribution
are inversed respectively by solving the equations, as given by Eq.
(3) and Eq. (4).
11 1( ) ( , , ) ( )I T m d D dθ θ= (3)
22 2( ) ( , , ) ( )I T m d D dθ θ= (4)
The intensity vector I is the scattered light intensity
distribution extracted from the CCD recordings. The rainbow
scattering coefficient matrix T is computed by using the modified
Nussenzveig’s theory and the vector D is the size distribution of
the spray droplets. The subscript numbers 1 and 2 correspond to
wavelengths of λ1 and λ2 respectively. For a particular spray, its
actual refractive indices under two wavelengths are definitely
certain values. Any deviation either in preset value of RI or
preset temperature T would lead to an error when comparing the
inverse refractive indices (m1, m2) with the hypothesizing
refractive indices (n1, n2). Thus actual refractive indices of
droplets under two wavelengths and calibration equation can be
quickly and simultaneously obtained by searching the minimum of
root mean square (RMS) errors between the hypothesizing refractive
indices and the ones from inversion. A typical example shows the
RMS error for the case of a water spray at 25℃ under two
wavelengths (λ1= 532 nm, λ2= 633 nm), as shown in Fig. 3. The
actual refractive indices of deionized water under two wavelengths
at each temperature can be calculated by Eq. (2). The actual value
of RI here equals to 0.47, which can be obtained by locating the
geometric rainbow angle in the simulated global rainbow curves.
Tested water sprays at temperatures ranging 20℃–30℃ and value of RI
ranging 0.44–0.49 are inspected. The preset pair of the value of RI
and spray temperature T is used to obtain the hypothetical
refractive indices (n1, n2) and linear calibration equation, which
can be used to inverse the refractive indices (m1, m2) from the
simulated global rainbow curves.
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
Fig. 3. Simulated RMS errors between the preset refractive
indices and the inversion ones
The result in Fig. 3 shows that the RMS error between inverse
refractive indices (m1, m2) and the hypothetical refractive indices
(n1, n2) is significantly decreased when the tested temperature and
the value of RI are close to the target ones. By iteratively
searching for the minimum error, accurate temperature, value of RI
and the calibration equation can be obtained at the same time. The
feasibility of the self-calibration method presented here is
verified by the experimental setup in Fig. 4. Two vertically
emitted semiconductor laser (wavelength: λ1=532 nm) and He-Ne laser
(wavelength: λ2=633 nm) are transformed into one coincident laser
beam by a 560 nm dichroic beam splitter (λ > 560 nm transmit; λ
< 560 nm reflect). The back-scattered light of spray illuminated
by the coincident laser beam is collected by Lens 1 with a diameter
of 10 cm and a focal length of 16 cm. Lens 1 is placed 25 cm away
from the probe volume. An aperture with a diameter of 2mm is
located at the image plane of the Lens 1, in order to select and
define the measured probe volume. Then the collected light passes
through Lens 2 with a diameter of 10 cm and a focal length of 8 cm.
Lens 2 conjugates the image focal plane of Lens 1 on the color CCD
sensor, which records chromatic rainbows with a resolution of 1392
× 1040 pixels (6.45 μm/pixel)).
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
Fig. 4. Setup diagram of the experimental system.
3. Results and Discussion A spray of deionized water is tested.
A typical chromatic global rainbow image of mixed rainbow signals
is shown in Fig. 5(a). Subject to the power of He-Ne laser, the
image presents a yellow-green primary rainbow pattern. The mixed
signals are separated in red, green, and blue (RGB) channels, as
shown in Fig. 5(b) - (d). For droplets illuminated by green and red
lasers, the nearly complete dark image in Fig. 5(d) shows that
signal intensity in blue channel is negligible compared to the
signal intensities in the other two channels.
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
Fig. 5. Original global rainbow image (a) and images extracted
from the red (b), green (c), and blue (d) channels
The filtered global rainbow curves extracted from red and green
rainbow images are shown in Fig. 6, where the noise has been
suppressed by fast Fourier transform (FFT) filtering. The value of
RI =0.45 and temperature T=23℃ are used as the initial preset
values. According to the self -calibration method presented above,
we can search the minimum of RMS errors of the preset refractive
indices and the ones from inversion, through iterations of
correcting the preset value of RI equals to 0.47 and temperature.
When iteration is finished, the final preset value of RI equals to
0.47 and temperature T equals to 20℃. The temperature measured by
self-calibration method is consistent with the thermocouple
readings 21.2℃, which is in the error range (±0.75%+1℃) of a k-type
thermocouple. The pixels (pix1 = 206, pix2 = 122) corresponding to
the final preset geometric rainbow angles ( =138.217°, =137.741°)
can be positioned in the filtered global rainbow curves with the
help of RI. And two characteristic calibration point (pix1, θrg1)
and (pix2, θrg2) can be used to obtain the parameters of linear
calibration equation, a=137.05°, b =0.00566°/pixel. The final RMS
error equals to 7.67×10-5 by comparing the preset refractive
indices (n1=1.33503, n2=1.33176) at 20℃ and the ones from inversion
(m1=1.33511, m2=1.33183).
Fig. 6. Dual-wavelength distributions of light intensity
extracted from the green and red rainbow images. The high frequency
noise on the rainbow signals was suppressed by fast
Fourier transform (FFT) filtering.
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
The performance of this calibrated global rainbow system has
been tested and verified further by the water–ethanol solutions
with volume concentration from 10% to 60%. FigureFig. 7 shows
chromatic global rainbow images of water–ethanol solutions. The
location of the bright fringe in rainbow image shifts distinctly
for solutions of volume concentration. Directly inverse the
refractive indices from the extracted rainbow signals in Red and
Green channels, and the measured results are shown in Fig. 8, by
method of self-calibration and mirror calibration. The measurement
results of these two calibration methods, shown by solid and dashed
curves, fit well respectively under two wavelengths.
Fig. 7. Chromatic global rainbow images of water–ethanol
solutions with volume concentration
from 10% to 60%, with the step of 10% FigureFig. 8 shows that
the average discrepancy of refractive index between two methods of
calibration is 0.0005, which is equivalent to discrepancy of
scattering angle of ~0.07°.This difference might be caused by
instrumental error of rotating platform under the mirror (rotating
accuracy: 0.05°) and positioning deviation of geometric rainbow
angles due to residual noise in the filtered rainbow curves
(positioning deviation: ~0.03°). Also, these inversion results show
good agreement with the reported values (Hall and Payne, 1922). For
the method of self-calibration and mirror calibration, the average
deviations of refractive index under each two adjacent wavelength
(532nm and 589nm, 589nm and 633nm) is 0.00198 and 0.0015
respectively, which are both roughly in line with the average
wavelength deviation 0.0021 at 20℃ calculated
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
by Eq. (2). But obviously this deviation of self-calibration
method is closer to the theoretical calculation value than that of
mirror calibration method. This one aspect demonstrates that
self-calibration method achieves a higher precision than that of
mirror calibration method.
Fig. 8. Comparison of the measured refractive index by method of
mirror calibration and self-
calibration with the theoretical data for water–ethanol
solutions at 20℃. Note that when obtaining both the calibration
equation and the refractive index measurement under two
wavelengths, self-calibrated system only works for the liquids with
a prior relationship between refractive index and wavelength,
temperature. A different way of putting it, liquids (such as
deionized water) whose physical parameters have been fully studied,
can be used as calibration medium to complete the angular
calibration for rainbow refractometry without additional precise
measure device for angle, temperature or refractive index. 4.
Conclusion In conclusion, by use of the natural shift of rainbow
pattern under two wavelengths, the relationship between scattering
angles and pixels are calibrated in parallel to the measurement of
the refractive indices under two wavelengths. We call this method a
self-calibrated global rainbow refractometry. The feasibility of
this method has been tested by simulation and experiment, and the
results show that it is an efficient and accurate calibration
method for global rainbow system. This method spares the rainbow
refractometry from extra
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
angle or temperature measuring instrument. It can apply to
mirror inaccessible occasions, and may even be extended to
calibration for one-dimensional rainbow technique.
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18th International Symposium on the Application of Laser and
Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 –
7, 2016
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