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 (φ_y), 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 (φ_y) in the y direction and the probe line segment is not limited by the lens diameter. However, this configuration also suffers from several limitations:

 

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:

In this work, |φ_y| < 0.2°. The rainbow light which is almost perpendicularly incident on the first lens in the height direction (angle |φ − 90°| <|φ_y|_max) passes through the horizontal slit, while the other rainbow light (angle |φ − 90°| <|φ_y|_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:

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.

 

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).

 

Fig. 3 Standard one-dimensional rainbow signal.

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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.

 

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:

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.