Abstract

Rainbow refractometry can measure the refractive index and the size of a droplet simultaneously. The refractive index measurement is extracted from the absolute rainbow scattering angle. Accordingly, the angular calibration is vital for accurate measurements. A new optical design of the one-dimensional rainbow technique is proposed by using a one-dimensional spatial filter in the Fourier domain. The relationship between the scattering angle and the CCD pixel of a recorded rainbow image can be accurately determined by a simple calibration. Moreover, only the light perpendicularly incident on the lens in the angle (φ) direction is selected, which exactly matches the classical inversion algorithm used in rainbow refractometry. Both standard and global one-dimensional rainbow techniques are implemented with the proposed optical design, and are successfully applied to measure the refractive index and the size of a line of n-heptane droplets.

© 2015 Optical Society of America

1. Introduction

A rainbow is a spectacular light phenomenon, resulting from the complicated interactions of light with transparent droplets [1–5 ]. The primary rainbow is created by the light experiencing one internal reflection (p = 2 in van de Hulst’s notation [1] and Debye series). The primary rainbow is supported by a cone in the backward direction, and accordingly the recording of a rainbow with a plane camera reveals a curved structure over the angle (φ). The interference between rays experiencing one internal reflection generates Airy structures. The Airy structure is characterized by a series of peaks of decreasing intensity and an increasing angular frequency. Moreover, the interference between the light experiencing one internal reflection and the light externally reflected produces high-frequency fringes that are superimposed on the Airy structures and called ripple structures. The rainbow angular location is very sensitive to the refractive index, while its intensity and shape are sensitive to the droplet size and light polarization. Thus, rainbows have been employed to measure the refractive index and size of droplets simultaneously through analyzing the light intensity around the rainbow angle. Therefore, temperature, concentration, species, and other parameters can be extracted from a prior relationship between these parameters and refractive index. These techniques, including the standard rainbow, global rainbow and one-dimensional rainbow, are called rainbow refractometry.

Several optical configurations to implement rainbow refractometry have been proposed. The standard rainbow technique (SRT) assumes only one particle or identical particles at the same location. First presented by Roth et al. [6], this configuration records both Airy fringes and ripple structures. To be able to process the signal, SRT usually records the rainbow from only one droplet or droplets of the same refractive index, size and position, limiting its application to a line of monodispersed droplets created by a devoted droplet generator. Besides, the standard rainbow signal is very sensitive to particle shape (non-sphericity) [7–9 ]. Moreover, the ripple structures and other high order light phenomena (such as overlapping of other high order rainbows, or MDR etc.) if they occur, should be taken into account for an accurate inversion of standard rainbow signal (see Saengkaew et al. [10]). Later, Van Beeck et al. [11] extended the SRT to the global rainbow technique (GRT) for measuring polydispersed droplets. The GRT records the superposition of rainbows from numerous droplets by increasing the pinhole size and the exposure time (up to hundreds of milliseconds). In this configuration, the light scattered from the non-spherical droplets is assumed to interfere destructively to form a uniform background, and the ripple structures superimposed on the Airy fringes disappear due to the overlapping of rainbows scattered by droplets with different sizes. Thus, the GRT captures the smooth, low frequency rainbow structures, and shows an apparent advantage in eliminating the ripple structures and in reducing its sensitivity to the nonsphericity of the sampled droplets. Both the above standard and global rainbow techniques are point measurements, and usually only use a few rows of data near the center of the rainbow image captured by array CCD for data processing, accordingly some researchers use a linear CCD camera to record a rainbow signal. Recently, Wu et al. [12] developed a one-dimensional rainbow technique that extended the measured volume from a point volume to a segment. In Wu’s configuration, a laser sheet illuminates the sprayed droplet field, a horizontal slit is added at the front surface of the first lens to create a synthetic image where each line corresponds to the rainbow issuing from a point in the control volume, and a narrow vertical slit positioned at the image plane of the measured volume defines the probed line segment. We call Wu’s one-dimensional rainbow configuration as ORT-1 (type-I) in this paper. The ORT-1 filters the rainbow signal in the spatial domain straightforwardly (i.e. directly on the front of the first lens), and the scattered rainbow light with different heights (as shown in Figs. 1(a) and 1(b)) passing through the horizontal slit has a different incident angle (φy), and thereafter is separated by a Fourier image system and recorded by different rows in the CCD. The advantages of this setup are: the captured rainbow signal has a large range in incident angles (φy) in the y direction and the probe line segment is not limited by the lens diameter. However, this configuration also suffers from several limitations:

  1. The rainbow is curved over the angle (φ), thus there is a phase shift in the theta (θ) direction for different phi (φ) angles. Consequently, the recorded rainbow (90° − Δφ < φ < 90° + Δφ) is also curved.
  2. The calibration for the relation between scattering angle (θ) and CCD pixel depends on the angle (φ).
  3. The inversion algorithm is usually based on the rainbow at φ = 90°, the center sampling point. But for φ ≠ 90°, and applying the standard inversion algorithm will cause inevitable error. The errors can be so high that they are unacceptable for accurate measurement, such as evaporation rate measurements.

 figure: Fig. 1

Fig. 1 Optical setup of ORT-2 and its comparison to ORT-1: side (a) and top (b) view of ORT-1; c-d. optical setup of ORT-2 using Fourier domain filtering, with side (c) and top (d) view of ORT-2.

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The inversion of the captured rainbow signal is another critical procedure in rainbow refractory, and several inversion algorithms based on different rainbow theories have been developed for a large number of applications. A series of works by Van Beeck et al. [11, 13–17 ] have proposed an inversion algorithm based on Airy theory, using some particular points of the recorded scattering diagram (extrema, inflection,etc.) and assuming the shape of the size distribution (Gaussian, log-normal, etc.). This approach has been applied to measure the droplet mean temperature [11, 13, 14], size distribution [14], liquid-liquid suspensions [15], and even the refractive index gradient inside a droplet [16] with point global rainbow technique. Vetrano et al. [17] extended Airy theory to the rainbow of nonuniform spheres. Based on Nussenzveig’s complex angular momentum theory for rainbow [18, 19], Saengkaew et al. [10, 20] developed an inversion algorithm to retrieve the optimal refractive index and size distribution by searching for the best fit between recorded and simulated rainbow pattern. This algorithm has no prior assumption on the size distribution, and is fast even in real time since the computation of rainbow using Nussenzveig’s theory is much less time consuming than using Lorenz-Mie theory, and improve the measurement accuracy using optimization. This work has been applied to investigate droplets in a variety of scenarios, such as sprayed droplets [21], the temperature of burning droplets in a flame [22], CO2 capture of MEA droplets [23] and so on. Song et al. [24] proposed an optimization scheme based on the Debye series of p = 2 by searching the minimum objective function (i.e. the square of the difference between the measured and simulated rainbows), and measured the liquid filled capillary [25] and the liquid jet exited from the capillary [24]. GRT was also applied to measure the droplet field which was a mixture of sprays of two different liquids using a genetic algorithm for data processing by Meunier-Guttin-Cluzel et al. [26] and using rainbow separation by Wu et al. [27]. Rainbow refractory has been extended to access nonspherical droplets [7, 28, 29]. All the above algorithms process the rainbow signal by fitting the measured light intensity around rainbow angle with the theoretical rainbow models, and thus the precise calibration of the angular-pixel relationship is key to ensure an accurate measurement of refractive index.

This work proposes an alternative optical design of one-dimensional rainbow technique using Fourier domain filtering (we call this configuration as ORT-2), and aims to facilitate the angular calibration for the 2D rainbow image. First, the details of the optical setup for ORT-2 will be introduced. Then it is followed by the angular calibration for ORT-2. Finally, measurements of a line of droplets with the calibrated ORT-2, which can be operated as a standard or a global rainbow refractometry will be presented.

2. One-dimensional rainbow with Fourier domain filtering

2.1. Optical design

Figure 1 compares the optical designs of ORT-1 and ORT-2. The side and top views of the schematics of optical setup of ORT-2 using Fourier domain filtering are shown in Figs. 1(c) and 1(d), respectively. Note that several coordinates are used to describe the schematics of the two one-dimensional rainbow techniques, as shown in Fig. 1. The laboratory coordinate systems (xyz, εηz, uvz) with the optical axis of the rainbow imaging system as the z axis, are used to describe the rainbow imaging system. The other one is a spherical coordinate system (r,θ, φ) attached to each individual droplet to describe the rainbow light scattered by the droplet, and the θ = 0° direction is the propagation direction of laser beam.

In ORT-2, a laser sheet (λ =532 nm), with a height of about 13 mm in the experiments, illuminates a perfectly vertical line of droplets to produce rainbows. Two spherical lenses (diameter = 75 mm, f = 150 mm), two slits and a CCD form the one-dimensional rainbow imaging system. The first lens is used to collect the rainbow light scattered by the droplets. The distance z from the droplet to the front principal plane of the first lens is 278 mm. The first slit is horizontal and has a width of about 1 mm, and is placed at the focus plane of the first lens. According to Fourier optics, the horizontal slit and the first lens form a one-dimensional spatial filter. According to the Fresnel diffraction, the complex optical field at the horizontal slit plane g 1 (ε,η) is:

g1(ε,η)=iexp[iπε2+η2λf(1zf)]G(ελf,ηλf),with|η|<a,
where G(ελf,ηλf) is the Fourier transform of the rainbow light field g 0 (x,y) at the measure ment plane, a is the half of the horizontal slit width, with a value of 0.5 mm for a standard setup. The incident rainbow light at angle (θxy) converges toward (ε = θx f, η = φy f) at the focal plane under the paraxial approximation, where the (θxy) are the angles of the incident rainbow light respect to the optical axis of the rainbow imaging system. After the pupil mask of the horizontal slit, the rainbow light that passes through the horizontal slit shoud have the incident angle:
|φy|<af.

In this work, |φy| < 0.2°. The rainbow light which is almost perpendicularly incident on the first lens in the height direction (angle |φ − 90°| <|φy|max) passes through the horizontal slit, while the other rainbow light (angle |φ − 90°| <|φy|max) is blocked by the horizontal slit (see the schematic in Fig. 1(c)). A vertical slit, with a width b of 1 mm, is placed at the image plane of the line of droplets. The optical field g 2 (u, v) at the vertical slit is:

g2(u,v)=1Mg0(uM,vM),
where M represents the transverse magnification of the droplet image at the vertical slit plane. Typically, M = f/(fz), is about 1.17 in this setup. The width of the slit is set to be as narrow as possible to define a narrow probe volume. The width Δx of the measurement line volume is b/M, with the value of 0.85 mm in this experimental setup. The CCD is located at the image plane of the front surface of the first lens in the rainbow imaging system. The filtered rainbows from droplets at different heights in the measurement volume are recorded by different rows of the CCD. Thus, the scattered light from a probed line in a spray around the rainbow angle is recorded as the one-dimensional rainbow signal. Note that in the ORT-2 configuration, the second lens conjugates the first lens entrance surface on the CCD. Accordingly, to avoid the spatial shift of the angle (θ) direction of the sampled droplets, this requires the sampling volume to be so narrow that is only a few times of the droplet size. In this work, the angular shift of the rainbow poisiton in the (θ) direction due to the droplet position inside the measurement volume could be up to:
ΔθΔx2z0.086°.

The liquid used in the experiments is n-heptane (C7H16), and has a refractive index of 1.390 (at λ = 532 nm). Consequently, the uncertainty of the retrieved refractive index of the n-heptane droplet caused by the rainbow imaging system is about 0.0007.

2.2. Angular calibration

The accurate angular-pixel calibration is essential to retrieve precisely the droplet information in rainbow refractometry, since the rainbow signal is very sensitive to the angle (θ). In the ORT-2, the calibration procedures are similar to those of standard and global point rainbow techniques developed in previous studies [27, 30, 31]. A mirror mounted on the center of a precision rotation platform, is positioned exactly at the probed line segment. The parallel laser sheet is reflected by the mirror, perpendicularly incidents on the first lens and propagates to the CCD as the recorded rainbow light does. The image positions on the CCD sensor of the laser sheet with different incident angles in the angle (θ) direction are obtained by rotating the mirror. Accordingly, the scattering angle for each pixel at different rows on the recorded rainbow image can be determined.

Figure 2(a) shows a panorama of five typical images of the reflected laser sheet at different angles recorded by the CCD during the angular-pixel calibration. The angular step between two adjacent laser sheet images is 2°. The laser sheet remains vertical in the center region of the CCD, while it is distorted in the boundary region of the CCD due to the aberrations introduced by the imaging lens. The intensity weighted center of each laser sheet image at each row is computed, as shown by the colored lines in Fig. 2(a). Then the 2D dispersed center points are used to retrieve the 2D angle-pixel relation in the full image field using linear interpolation. The calibrated angle-pixel relationship is shown in Fig. 2(b), and the calibration results present good linearity between the scattering angle (θ) and the CCD column, while the angle varies slightly along the row direction at the boundaries. The calibrated angle-pixel will be used for the rainbow signal processing in the next section.

 figure: Fig. 2

Fig. 2 Typical rainbow calibration signal. (a) A panorama of images of the reflected laser sheet at five different angles in angular-pixel calibration; (b) The calibrated 2D angle-pixel relation.

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3. Results and discussion

3.1. Processing strategies

To retrieve droplet information from the rainbow signal, different strategies have been developed, which are based on minimization between a recorded and a simulated signal. These strategies can be summarized as follows.

For a GRT signal, an optimization algorithm has been proposed by Saengkaew et al. [10, 20]. The equation, E(θ) = T(n,d,θ)N(d) +σ, is solved using the non-negative least squares algorithm, where the intensity vector E is the samplings of the recorded rainbow curve by CCD, the scattering matrix coefficient T is computed using the Nussenzveig theory [18–20 ] (taking or not the ripple structure into account), and N(d) is the size distribution. σ is a uniform background noise, and its value should be no more than the minimum of the rainbow signal, and ranges from 0 to about 5 for the 8 bits rainbow image in this work. The solutions (refractive index and size distribution) are optimized by iteratively searching for the minimum absolute difference between the sampled rainbow curve and the fitted rainbow signal based on golden section search and parabolic interpolation. The optimized results are achieved until the absolute difference is less than the termination tolerance, and usually it converges after fewer than 10 iterations.

For standard rainbow signal processing, an alternative strategy has been developed without using matrix to permit a continuous research on the diameter (see Saengkaew et al. [10]). Nevertheless, in this paper, the same code, developed for GRT, is used to process SRT and GRT signals. The only difference is that, in order to fit the ripple structures, the scattering matrix coefficient T take into account both the refraction light with one interval reflection (p = 2) and the external reflection light (p = 0) on the droplet surface. Besides, the droplet size distribution can be set to be very narrow, since the droplets are of almost the same size.

3.2. One-dimensional standard rainbow technique (OSRT)

Standard rainbow refers to the scattering light around the rainbow angle by droplets assumed to be identical (in refractive index, particle size, position). In the standard rainbow, the primary rainbow due to one internal refraction (p = 2) as well as the ripple structures resulting from the interference between the light externally reflected and the light experimenting one internal refraction, are clearly visible. In such experiments, to produce a stable standard rainbow, the droplets are produced by a monodispersed droplet generator. In this paper the monodispersed droplets generator from FMP technology is used with a working frequency of 18.35 kHz. The n-heptane droplets move upward to form a stable vertical line of droplets after being ejected from the nozzle, and the droplets are located at the same position inside the measurement volume. Thus, the vertical slit is not necessary for the measurement in this experiment, as the line of droplet are set to be perfectly vertical. Accordingly, the transverse displacement of the droplets at a given height is very small and smaller than its diameter, i.e. 100 μm. Thus, according to Eq. (4), the uncertainty of the refractive index measurement caused by the rainbow imaging system during the experiment is less than 0.0001.

Figure 3 shows a typical standard one-dimensional rainbow image. Both the primary rainbow (i.e. the first and second blows) and ripple structure are clearly recorded. The Airy structures keep almost unchanged. The ripple structures slightly tilt along the height (y) direction, and this is due to the phase shift of the ripple structures which is mainly caused by the tiny change of the droplet diameter at nanometer scale as a result of evaporation [32]. The height of the rainbow image is about 400 pixels, corresponding to a measurement along a line of 8 mm length. The rainbow intensity decreases with the distance away from the center in the vertical direction. It is caused by the laser sheet illumination which is an elliptical Gaussian beam with a Gaussian profile of intensity. Each rainbow intensity-angular (θ) curve is averaged from a ROI (region of interest) with 50 pixels in height, and a series of rainbow curves are obtained with a step of 25 pixels in the vertical direction. A wavelet denoise algorithm is applied to the rainbow curve to filter the embedded high frequency noise, yielding a smoothed rainbow curve, as shown in Fig. 4(a).

 figure: Fig. 3

Fig. 3 Standard one-dimensional rainbow signal.

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 figure: Fig. 4

Fig. 4 Results of measurements of the droplet line with one-dimensional standard rainbow technique. (a) Comparison of the measured and inversed rainbow curve, refractive index n = 1.3940, diameter d = 99.7μm; (b) Statistics on the measured refractive indices; (c) Statistics on the measured diameters.

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Figure 4(a) demonstrates the processing of the experimental rainbow signal captured by the one-dimensional standard rainbow setup. The measured optimized refractive index and size are 1.3940 and 99.7 μm, respectively. The inverse rainbow fits the experimental rainbow very well, as shown in Fig. 4(a). Comparing the fitted rainbow to the experimental rainbow, the first and second peaks of the Airy structures in the primary rainbow are located exactly at the same angular position. The ripple structures of both rainbows match each other in both angular location and frequency. Note there might be small discrepancies in the absolute light intensity of some ripple structures between the experimental and inversed rainbows, which might slightly increase with the angular value. For example, the experimental rainbow signal is a little lower than the theoretical inverse rainbow in the second peak of the Airy structure, as shown in Fig. 4(a). Besides various noises and the sampled droplets not being exactly identical, it can be partially attributed to one factor that the rainbow signal is slightly distorted which leads to a decrease in its intensity, as shown in the calibrated laser sheet images in Fig. 2(a). However, the relative light intensity (or the light intensity fluctuation) in these ripple structures is almost the same. The agreements of the experimental and inverse rainbows in both Airy peaks and ripple structures indicate that the refractive index and droplet size have been accurately retrieved. A series of rainbow images of n-heptane droplets at atmospheric conditions (20 °C and 1 bar) were processed for statistical analysis, and the results are shown in Figs. 4(b) and 4(c). The measured refractive index is from 1.3941 to 1.3945, with an accuracy of up to the forth digit. The mean value of the refractive index is 1.3943, with the standard derivation of 0.00005. The measured refractive index is the same as that provided by standard database (1.394). The retrieved droplet diameter ranges from 98.5 μm to 101.5 μm. The averaged diameter is 99.8 μm, and the standard deviation is within 1.5 μm. The droplet sizes also show a good agreement with the calibrated value (100 μm) provided by the monodispersed nozzle manufacture.

3.3. one-dimensional global rainbow technique (OGRT)

The optical setup for global one-dimensional rainbow is the same as that for the standard one-dimensional rainbow, but the droplets are surrounded by a propane gas flame. The propane gas is supplied to an annular slit burner, and is ignited to form a diffusion flame. The droplets are in the center by the diffusion flame and are heated up through convection and radiation. The droplet temperature increases due to the heating, which causes a decrease of the droplet refractive index. Meanwhile, it speeds up droplet evaporation, leading to a loss of mass and thus a decrease of diameter. To minimize the effect of the luminous flame, a band-pass filter (532 nm) was placed before the CCD. The exposure time of the CCD is about 67.6 ms, and about 1240 droplets are sampled during the exposure time for each rainbow signal in the CCD image. A detailed description of the experimental setup can be found in [32]. Figure 5(a) shows a typical global rainbow image. The ripple structure disappears, and only smooth Airy peaks are observed in the rainbow signal, and in this image the first and second peaks are recorded. The ripple structure is smoothed by a combination of the effects of the tiny changes of refractive index and diameter caused by heating on the few 50 rows of integration. Both the first and second Airy peaks are tilted, and there is an obvious angular shift in rainbow signal along the height (y) direction. This is caused by the decrease of the droplet refractive index as a result of the temperature increase due to the heating along its upward movement.

 figure: Fig. 5

Fig. 5 Demonstration of one-dimensional global rainbow technique. (a) Global rainbow image of a line of droplet in a flame; (b) Rainbow signal at different heights and an example (subfigure) of rainbow signal processing; (c) Measured refractive index along the height.

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Each rainbow signal is averaged from a rectangular window with fifty rows (corresponding to about 1 mm), as shown in Fig. 5(a), and fifteen rainbow signals are derived from each one-dimensional rainbow image with a step of 25 rows along the height. Figure 5(b) shows the normalized rainbow signals at the height from 1 mm to 7 mm. The rainbow signals present an obvious and stable shift in the angular direction. This angular shift is due to the changes of the refractive index. The global rainbow is processed to obtain the refractive index and size distribution using an inversion algorithm based on Nussenzveig’s theory, which has been validated in [20, 27]. A typical comparison of the experimental rainbow curve with the rainbow of the inversed droplets is shown in the subfigure in Fig. 5(b). The inverse rainbow fits the experimental one accurately. One hundred rainbow images have been processed. The averages of measured refractive indices of the droplets at different heights are between 1.387 to 1.392, and are stable at each position, with the standard deviation about 0.001, as shown in Fig. 5(c). The droplet refractive index decreases along the height, with a slope of 0.0008/mm. Accordingly, the droplet temperature increases about 1.5 °C per mm along its trajectory in the measurement volume [33]. The decrease of the refractive index is due to the temperature increase, caused by the heating effect of the flame. The mean droplet size is stable along the height in the rainbow image. With an accuracy of about 1 μm on the size measurement, the extracted droplet diameter corresponds to a single class of size with a value of 100 ± 1 μm.

4. Conclusion

A new optical configuration based on signal filtering in Fourier domain is proposed to implement the one-dimensional rainbow technique (ORT-2). In ORT-2, the horizontal slit is placed at the focal plane of the first lens, and the second lens conjugates the first lens entrance surface on the CCD plane. The system uncertainty of the refractive index measured by the ORT-2 is formulated. This optical configuration has two advantages:

  1. Only the light impinging perpendicularly to the lens in the angle (φ) direction passes through the horizontal slit, while other light with oblique incident in the angle (φ) direction is removed. This fits the classical rainbow signal inversion algorithm developed for point measurement rainbow refractometry.
  2. The angular-pixel calibration of the full rainbow image field can be easily performed. The calibration procedures for one-dimensional rainbow technique with parallel laser sheet illumination are the same as those in the rainbow techniques for point measurement.

Both standard and global one-dimensional rainbow techniques are demonstrated with the proposed optical configuration, and they are applied to measure the evolution (in size and refractive index) of a line of monodispersed droplets. The measured refractive index and size (for OSRT) or size distribution (for OGRT) agree well with the exact values. This rainbow configuration is an effective tool to simultaneously measure the refractive index and size of droplet along segment, and can be applied to quantitatively evaluate the evolution of droplets, such as droplet evaporation.

Acknowledgments

The authors gratefully acknowledge financial supports from LABEX EMC3-3D, the National Nature Science Foundation of China (grant 51176162), the Major Program of the National Natural Science Foundation of China (grant 51390491), National Basic Research Program of China ( 2015CB251501), the Program of Introducing Talents of Discipline to University ( B08026).

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30. M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006). [CrossRef]  

31. Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

32. J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

33. A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013). [CrossRef]  

References

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  1. H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Corporation, 1957).
  2. P. L. Marston and G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander’s dark band,” Appl. Opt. 33, 4702–4713 (1994).
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  5. Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of rainbow scattering by a chiral sphere,” Opt. Express 21, 21879–21888 (2013).
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  8. H. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transfer 126, 105–112 (2013).
    [Crossref]
  9. H. Yu, F. Xu, and C. Tropea, “Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement,” Opt. Express 21, 25761–25771 (2013).
    [Crossref] [PubMed]
  10. S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
    [Crossref]
  11. J. Van Beeck, D. Giannoulis, L. Zimmer, and M. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
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  12. X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
    [Crossref] [PubMed]
  13. J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
    [Crossref]
  14. J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Char. 18, 196–204 (2001).
    [Crossref]
  15. M. R. Vetrano, J. Petrus Antonius Johannes van Beeck, and M. L. Riethmuller, “Global rainbow thermometry: improvements in the data inversion algorithm and validation technique in liquid-liquid suspension,” Appl. Opt. 43, 3600–3607 (2004).
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  16. M. R. Vetrano, J. P. A. Johannes van Beeck, and M. L. Riethmuller, “Assessment of refractive index gradients by standard rainbow thermometry,” Appl. Opt. 44, 7275–7281 (2005).
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  17. M. R. Vetrano, J. P. Antonius Johannes van Beeck, and M. L. Riethmuller, “Generalization of the rainbow Airy theory to nonuniform spheres,” Opt. Lett. 30, 658–660 (2005).
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  18. H. Nussenzveig, “High-frequency scattering by a transparent sphere. II. theory of the tainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
    [Crossref]
  19. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [Crossref]
  20. S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
    [Crossref]
  21. P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
    [Crossref]
  22. C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
    [Crossref]
  23. M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
    [Crossref]
  24. F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
    [Crossref]
  25. F. Song, C. Xu, and S. Wang, “Twin primary rainbows scattering by a liquid-filled capillary,” Opt. Commun. 332, 96–102 (2014).
    [Crossref]
  26. S. Meunier-Guttin-Cluzel, S. Saengkaew, and G. Gréhan, “Réfractométrie d’Arc-en-ciel Global: Analyse de mélanges de gouttes par algorithme génétique,” in Congrès Francophone de Techniques Laser, (2010).
  27. X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
    [Crossref]
  28. S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
    [Crossref]
  29. J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
    [Crossref]
  30. M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
    [Crossref]
  31. Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).
  32. J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.
  33. A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
    [Crossref]

2015 (1)

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

2014 (2)

2013 (7)

Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of rainbow scattering by a chiral sphere,” Opt. Express 21, 21879–21888 (2013).
[Crossref] [PubMed]

H. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transfer 126, 105–112 (2013).
[Crossref]

H. Yu, F. Xu, and C. Tropea, “Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement,” Opt. Express 21, 25761–25771 (2013).
[Crossref] [PubMed]

F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
[Crossref]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
[Crossref]

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

2012 (1)

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

2011 (1)

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

2010 (1)

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

2009 (1)

S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[Crossref]

2008 (1)

2006 (3)

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
[Crossref]

2005 (2)

2004 (1)

2002 (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[Crossref]

2001 (1)

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Char. 18, 196–204 (2001).
[Crossref]

1999 (1)

1996 (1)

1994 (1)

1991 (1)

1979 (1)

1969 (1)

H. Nussenzveig, “High-frequency scattering by a transparent sphere. II. theory of the tainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

Adam, J. A.

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[Crossref]

Adler, C. L.

Anders, K.

Antonius Johannes van Beeck, J. P.

Atthasit, A.

J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
[Crossref]

Bai, L.

Biscos, Y.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
[Crossref]

Blaisot, J.

S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[Crossref]

Blanco, A.

A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
[Crossref]

Bouilloux, L.

P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
[Crossref]

Boulet, P.

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

Buchlin, J. M.

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

Cen, K.

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

Charinpanikul, T.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

Charinpanitkul, T.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

Chen, L.

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

Estel, L.

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

Fleet, R. W.

Fournier-Salaün, M.-C.

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

Frohn, A.

Fungtammasan, B.

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

Gao, X.

Gauthier, S.

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

Gayol, A.

A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
[Crossref]

Giannoulis, D.

Godard, G.

S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[Crossref]

Goméz, D.

A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
[Crossref]

Gong, L.

Gouesbet, G.

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

Gréhan, G.

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[Crossref]

P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
[Crossref]

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

S. Meunier-Guttin-Cluzel, S. Saengkaew, and G. Gréhan, “Réfractométrie d’Arc-en-ciel Global: Analyse de mélanges de gouttes par algorithme génétique,” in Congrès Francophone de Techniques Laser, (2010).

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

Han, Y.

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

Hong, Q.

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

Hulst, H. C.

H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Corporation, 1957).

Jiang, H.

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

Johannes van Beeck, J. P. A.

Kaduchak, G.

Laurent, C.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

Lavergne, G.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
[Crossref]

Lemaitre, P.

P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
[Crossref]

Letty, C.

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

Li, Z.-J.

Lock, J. A.

Marston, P. L.

Mees, L.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

Meunier-Guttin-Cluzel, S.

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

S. Meunier-Guttin-Cluzel, S. Saengkaew, and G. Gréhan, “Réfractométrie d’Arc-en-ciel Global: Analyse de mélanges de gouttes par algorithme génétique,” in Congrès Francophone de Techniques Laser, (2010).

Navaza, J. M.

A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
[Crossref]

Nussenzveig, H.

H. Nussenzveig, “High-frequency scattering by a transparent sphere. II. theory of the tainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

Nussenzveig, H. M.

Ouboukhlik, M.

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

Petrus Antonius Johannes van Beeck, J.

Porcheron, E.

P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
[Crossref]

Promvongsa, J.

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

Qu, T.

Renou, B.

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

Reveillon, J.

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

Riethmuller, M.

J. Van Beeck, D. Giannoulis, L. Zimmer, and M. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[Crossref]

J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
[Crossref]

Riethmuller, M. L.

Roth, N.

Saengkaew, S.

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[Crossref]

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

S. Meunier-Guttin-Cluzel, S. Saengkaew, and G. Gréhan, “Réfractométrie d’Arc-en-ciel Global: Analyse de mélanges de gouttes par algorithme génétique,” in Congrès Francophone de Techniques Laser, (2010).

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

Shang, Q.-C.

Song, F.

F. Song, C. Xu, and S. Wang, “Twin primary rainbows scattering by a liquid-filled capillary,” Opt. Commun. 332, 96–102 (2014).
[Crossref]

F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
[Crossref]

Song, J.

Tanthapanichakoon, W.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

Tropea, C.

H. Yu, F. Xu, and C. Tropea, “Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement,” Opt. Express 21, 25761–25771 (2013).
[Crossref] [PubMed]

H. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transfer 126, 105–112 (2013).
[Crossref]

Vallikul, P.

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

van Beeck, J.

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

J. Van Beeck, D. Giannoulis, L. Zimmer, and M. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[Crossref]

J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
[Crossref]

van Beeck, J. P. A. J.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Char. 18, 196–204 (2001).
[Crossref]

J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[Crossref] [PubMed]

Van De Hulst, H.

H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Corporation, 1957).

Vanisri, H.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

Vetrano, M.

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

Vetrano, M. R.

Wang, J.

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

Wang, S.

F. Song, C. Xu, and S. Wang, “Twin primary rainbows scattering by a liquid-filled capillary,” Opt. Commun. 332, 96–102 (2014).
[Crossref]

F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
[Crossref]

Wu, X.

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

Wu, Y.

X. Wu, H. Jiang, Y. Wu, J. Song, G. Gréhan, S. Saengkaew, L. Chen, X. Gao, and K. Cen, “One-dimensional rainbow thermometry system by using slit apertures,” Opt. Lett. 39, 638–641 (2014).
[Crossref] [PubMed]

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

Wu, Z.-S.

Xu, C.

F. Song, C. Xu, and S. Wang, “Twin primary rainbows scattering by a liquid-filled capillary,” Opt. Commun. 332, 96–102 (2014).
[Crossref]

F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
[Crossref]

Xu, F.

H. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transfer 126, 105–112 (2013).
[Crossref]

H. Yu, F. Xu, and C. Tropea, “Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement,” Opt. Express 21, 25761–25771 (2013).
[Crossref] [PubMed]

Yan, Y.

F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
[Crossref]

Yu, H.

H. Yu, F. Xu, and C. Tropea, “Optical caustics associated with the primary rainbow of oblate droplets: simulation and application in non-sphericity measurement,” Opt. Express 21, 25761–25771 (2013).
[Crossref] [PubMed]

H. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transfer 126, 105–112 (2013).
[Crossref]

Zimmer, L.

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Char. 18, 196–204 (2001).
[Crossref]

J. Van Beeck, D. Giannoulis, L. Zimmer, and M. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[Crossref]

Acta Physica Sinica (1)

Y. Wu, X. Wu, S. Saengkaew, H. Jiang, Q. Hong, G. Gréhan, and K. Cen, “Concentration and size measurements of sprays with global rainbow technique,” Acta Physica Sinica 62, 90703 (2013).

Appl. Opt. (6)

Can. J. Chem. Eng. (1)

M. Ouboukhlik, S. Saengkaew, M.-C. Fournier-Salaün, L. Estel, and G. Gréhan, “Local measurement of mass transfer in a reactive spray for CO2 capture,” Can. J. Chem. Eng. 93, 419–426 (2015).
[Crossref]

Combust. Flame (1)

C. Letty, B. Renou, J. Reveillon, S. Saengkaew, and G. Gréhan, “Experimental study of droplet temperature in a two-phase heptane/air V-flame,” Combust. Flame 160, 1803–1811 (2013).
[Crossref]

Exp. Fluids (4)

S. Saengkaew, G. Godard, J. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[Crossref]

J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique: sensitivity to non-sphericity of droplets,” Exp. Fluids 51, 149–159 (2011).
[Crossref]

M. Vetrano, S. Gauthier, J. van Beeck, P. Boulet, and J. M. Buchlin, “Characterization of a non-isothermal water spray by global rainbow thermometry,” Exp. Fluids 40, 15–22 (2006).
[Crossref]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[Crossref]

J. Math. Phys. (1)

H. Nussenzveig, “High-frequency scattering by a transparent sphere. II. theory of the tainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

H. Yu, F. Xu, and C. Tropea, “Spheroidal droplet measurements based on generalized rainbow patterns,” J. Quant. Spectrosc. Radiat. Transfer 126, 105–112 (2013).
[Crossref]

Meas. Sci. Technol. (2)

P. Lemaitre, E. Porcheron, G. Gréhan, and L. Bouilloux, “Development of a global rainbow refractometry technique to measure the temperature of spray droplets in a large containment vessel,” Meas. Sci. Technol. 17, 1299–1306 (2006).
[Crossref]

X. Wu, Y. Wu, S. Saengkaew, S. Meunier-Guttin-Cluzel, G. Gréhan, L. Chen, and K. Cen, “Concentration and composition measurement of sprays with a global rainbow technique,” Meas. Sci. Technol. 23, 125302 (2012).
[Crossref]

Opt. Commun. (3)

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakoon, L. Mees, G. Gouesbet, and G. Gréhan, “Rainbow refractrometry: on the validity domain of Airy’s and Nussenzveig’s theories,” Opt. Commun. 259, 7–13 (2006).
[Crossref]

F. Song, C. Xu, S. Wang, and Y. Yan, “An optimization scheme for the measurement of liquid jet parameters with rainbow refractometry based on Debye theory,” Opt. Commun. 305, 204–211 (2013).
[Crossref]

F. Song, C. Xu, and S. Wang, “Twin primary rainbows scattering by a liquid-filled capillary,” Opt. Commun. 332, 96–102 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Part. Part. Syst. Char. (1)

J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Char. 18, 196–204 (2001).
[Crossref]

Phys. Chem. Liq. (1)

A. Blanco, A. Gayol, D. Goméz, and J. M. Navaza, “Temperature dependence of thermophysical properties of ethanol + n-hexane + n-heptane,” Phys. Chem. Liq. 51, 381–403 (2013).
[Crossref]

Phys. Rep. (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Rep. 356, 229–365 (2002).
[Crossref]

Other (4)

H. C. Hulst and H. Van De Hulst, Light Scattering by Small Particles (Courier Corporation, 1957).

J. van Beeck, M. Riethmuller, G. Lavergne, Y. Biscos, and A. Atthasit, “Processing droplet temperature measurement data obtained with rainbow thermometry,” in Optical Diagnostics for Fluids, Solids, and Combustion (SPIE, 2001), 251–264.
[Crossref]

J. Promvongsa, Y. Wu, G. Gréhan, S. Saengkaew, B. Fungtammasan, and P. Vallikul, “One-Dimensional Rainbow Technique to Characterize the Evaporation at Ambient Temperature and Evaporation in a Flame of Monodis-persed Droplets,” in 7th European Combustion Meeting (ECM2015), pp. 1–5.

S. Meunier-Guttin-Cluzel, S. Saengkaew, and G. Gréhan, “Réfractométrie d’Arc-en-ciel Global: Analyse de mélanges de gouttes par algorithme génétique,” in Congrès Francophone de Techniques Laser, (2010).

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Figures (5)

Fig. 1
Fig. 1 Optical setup of ORT-2 and its comparison to ORT-1: side (a) and top (b) view of ORT-1; c-d. optical setup of ORT-2 using Fourier domain filtering, with side (c) and top (d) view of ORT-2.
Fig. 2
Fig. 2 Typical rainbow calibration signal. (a) A panorama of images of the reflected laser sheet at five different angles in angular-pixel calibration; (b) The calibrated 2D angle-pixel relation.
Fig. 3
Fig. 3 Standard one-dimensional rainbow signal.
Fig. 4
Fig. 4 Results of measurements of the droplet line with one-dimensional standard rainbow technique. (a) Comparison of the measured and inversed rainbow curve, refractive index n = 1.3940, diameter d = 99.7μm; (b) Statistics on the measured refractive indices; (c) Statistics on the measured diameters.
Fig. 5
Fig. 5 Demonstration of one-dimensional global rainbow technique. (a) Global rainbow image of a line of droplet in a flame; (b) Rainbow signal at different heights and an example (subfigure) of rainbow signal processing; (c) Measured refractive index along the height.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

g 1 ( ε , η ) = i exp [ i π ε 2 + η 2 λ f ( 1 z f ) ] G ( ε λ f , η λ f ) , w i t h | η | < a ,
| φ y | < a f .
g 2 ( u , v ) = 1 M g 0 ( u M , v M ) ,
Δ θ Δ x 2 z 0.086 ° .

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