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Abstract
Accurate quantification of scattering angle versus detector pixel strongly determines the measurement accuracy of rainbow refractometry. This is an emerging measurement technique operating at backscatter angles and characterizing droplets or complex droplets in terms of size and refractive index. A novel method for calibration of the rainbow scattering angle using a monodisperse droplet stream is introduced and the achievable accuracy is estimated. The assumption of a linear pixel-to-angle relation is derived, and a calibration procedure is proposed based on global fit of calibration data to the theoretically known rainbow signal. The accuracy of this method was examined by simulations and experiments, where the uncertainties of a priori parameters of droplets were also considered and validated using shadowgraphy as a ground truth. The results confirm the feasibility of this method with a maximum absolute error of 0.032°and 3.9E-5°/pixel respectively for the intercept and slope of the linear relationship. These values translate into maximum uncertainties in diameter and refractive index of approx. 0.67% and 2.8 × 10−4.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Numerous techniques of optical particle/drop sizing are available and these offer the known advantages of being non-intrusive and capable of achieving high data rates, which are especially valuable features in the characterisation of spray processes. Most of these techniques involve solving an inverse problem, deducing particle characteristics (e.g. size and refractive index) from detected scattered light and comparing this with the known Lorenz-Mie scattering from spherical particles. However this approach requires an accurate measure of where the detector is positioned relative to the scattering center, i.e. an angle-to-pixel transformation is necessary for any given optical system.
Several scattering angle calibration methods are adopted for techniques using forward scattering. For instance, for X-ray and small angle scattering, pixel-to-angle conversion is usually performed by fitting theoretical diffraction expressions to several experimental diffraction peaks obtained from selected standard samples [1] or polystyrene spheres [2]. The spatial positions in the Fourier plane represent the scattering angles in the object plane, and several discrete scattering angles can be determined by placing a diffraction grating (a calibration standard) in the object plane. Since only the peak and valley values are used for fitting, the calibration is inadequate when high precision measurements are required. To overcome this problem in the angle calibration of an aerosol measurement system, several nanoscale standard polystyrene latex microspheres have been used as a standard [3], adding additional data points and covering a wider size range. To exclude intense, non-scattered light from near-forward scattered light, a micron-sized pinhole and global fit to its diffraction pattern are used to establish a mapping between each pixel in the sensor and its corresponding angular coordinate [4,5]. With some other techniques, such a high-precision absolute angle calibration is not necessary. As an example, with interferometric particle imaging [6], the scattering angles can be adequately estimated simply using a ruler to measure the distance between the transmitting and receiving elements of the optical system.
The present study examines rainbow refractometry, which operates using light scattered from a droplet in the backward direction around the primary rainbow angle (${\theta _{{\rm {rg}}}} = 2{\cos ^{ - 1}}\left [ {{3^{{{ - 3} \mathord {\left / {\vphantom {{ - 3} 2}} \right.} 2}}}{n^{ - 2}}{{\left ( {4 - {n^2}} \right )}^{{3 \mathord {\left / {\vphantom {3 2}} \right.} 2}}}} \right ]$, where $n$ is the refractive index of droplet relative to the surrounding). A detector (usually a camera) records the light scattered by droplets in the form of rainbow signals/images (as shown in Fig. 1(a) and (c)) using a Fourier lens in front of the measurement volume. Standard rainbow signal from a single droplet or monodisperse droplets, is composed of a low frequency structure (Airy rainbow from the self-interference of $p$=2 rays shows a series of wave peaks with decreasing intensity) and a high frequency structure (ripple fringes formed by the interference of $p$=2 with $p$=0 rays). Here the nomenclature of [7] is adopted, where $p$ denotes the number of internal chords travelled by the ray in the droplet. On the other hand, a global rainbow signal, arising from the superposition of standard rainbow signals from numerous droplets, usually exhibits only a single smooth primary peak with an apparent background intensity in the baseline.
Fig. 1. Example rainbow signals: (a) standard rainbow image and (b) its derived intensity signal as a function of scattering angle transformed from detector pixel; (c) global rainbow signal and (d) its derived intensity signal.
This technique has been progressively developed to simultaneously measure diameter, refractive index (temperature) and transient evaporation rate of droplets [8–16]. Recently, the technique has been demonstrated in a planar version [17] and extended to characterize more complex droplets, such as oscillating [18] and colloidal droplets [19]. Also for these advancements a reliable transformation between the scattering angles and the sensor pixels (as shown in Fig. 1(b) and (d)) is a key element in achieving accurate measurements. The conventional approach for determining the transformation uses a rotating mirror, usually used in a point or line rainbow refractometry system, involves manually placing and rotating a mirror in the measurement volume using a co-axially mounted rotary stage. An illuminated laser beam or sheet is then detected at certain scattering angles on the corresponding pixel positions of the camera, one after the other. This is tedious and too complex to perform in scenarios that require extensive calibration, such as scattering angle calibration over an entire plane. Alternatively, a calibration method using a cylindrical scatterer detects the rainbow signal of a liquid jet or a transparent cylinder with known diameter and refractive index [20]. By matching the theoretical rainbow pattern with the experimentally obtained image, the scattering angles and their corresponding pixels at the first few peaks of the Airy pattern can be extracted. This results in a linear fit between angle and pixel, yielding the desired calibration relation. However, the liquid column or the transparent cylinder is usually neither perfectly straight nor of constant diameter, due to surface waves and gravitational effects during their generation. Recently a dual-wavelength calibration method [21] was proposed. A droplet illuminated by a dual-wavelength laser beam has two different refractive indices, resulting in an angular shift of the rainbow signal, while the angular frequency is constant. It enables calibration not only with monodisperse droplets, but also with polydisperse droplets. One drawback of this method is that it is difficult to guarantee the accuracy of the calibration by relying only on the two calibration points at the two wavelengths.
Scattering angle calibration has been extensively studied, but most approaches rely only on a few discrete points or require special expertise and experience. The present goal is to further develop a previous method based on a calibration using a droplet stream traversed through the illuminating laser sheet of a planar rainbow refractometry [17]. This technique is very straightforward and allows simultaneous measurement of droplet size, refractive index and position within the illuminated plane. In the present study, the accuracy of the calibration is more rigorously examined than in previous work, by taking into account also uncertainties in the diameter and refractive index of the calibrating droplets. A ground truth is provided by shadowgraphy, which provides highly accurate estimates of the calibrating droplet sizes.
The description begins with an introduction to the calibration approach in Section 2, including a derivation of the linear pixel-to-angle relationship. Rainbow signals with various typical calibration coefficients are then simulated, and the proposed method is evaluated with or without the variation of a priori knowledge of droplet diameter and refractive index. Finally, the method is validated experimentally using a monodisperse droplet stream, measured also using shadowgraphy.
2. Calibration approach
The necessary angle calibration is pictured in Fig. 2, where the scattering angle $\theta = {180^{\circ}} -\theta _{\rm {i}}+\theta _{0}$ is to be related to the pixel position $x$ on the detector, all in the plane of the figure. It will now be shown that this relation is linear, taking the form of $\theta = a$ + $b \times x$. The calibration intercept $a$ and slope $b$ denote respectively the starting scattering angle and angular sampling frequency of the camera.
Fig. 2. Ray path depicting the relation between scattering angle $\theta = 180^{\circ} -\theta _{\rm {i}}+\theta _0$ and detector pixel position $x_3$
For a given offset $x_0$ of the particle from the optical axis and a given incident angle $\theta _{\rm {i}}$ of the illuminating laser sheet to the optical axis of the detector, the ABCD optical matrix relates the scattering angle to the pixel position $x_3$ in the form
In deriving this equation the paraxial approximation has been assumed, since the half scattering angle of the primary rainbow is typically less than 10°. Moreover, lens aberrations have been neglected, which in the present study was justified because an achromatic compound lens was used. The calibration therefore consists of determining the intercept $a$ and slope $b$ of the linear relationship.
The calibration procedure then proceeds according to the flowchart pictured in Fig. 3. The theoretical rainbow signal ${I_{\rm {T}}}\left ( {{\theta _j}} \right )$ over a wide range of scattering angles ${\theta _j}$ is first predicted by specifying the droplet diameter $D$ and refractive index $n$ in a Lorenz-Mie code [22]. The approximate and initial values of $a_0$ and $b_0$ are computed by matching the primary and secondary peaks of the theoretical and experimental signals. The iteration domains for the two calibration coefficients are then set to [$a_0$-1°, $a_0$+1°], [0.5$b_0$, 1.5$b_0$], generating $N$=5000 initial calibration equations $(\theta = {a_i}$ + ${b_i} \cdot x,{\rm {\ }}i \le N)$ by the multistart algorithm [23]. This algorithm can find multiple local minima to numerous local minimum problems arising from different starting points, and runs for global minimization. For the $k$th local problem, experimental scattering angles ${\theta _{{\rm {e,\ }}j}} = {a_k} + {b_k} \cdot {x_j}$ at various ${x_j}$ are calculated, and the nearest theoretical scattering angles ${\theta _j}$ are sought by minimizing $\left | {{\theta _{{\rm {e,\ }}j}} - {\theta _j}} \right |$. The intensity difference of the experimental and theoretical rainbow signals are thus summed as ${SID_k} = \sum _{j = 1}^m {{{\left [ {{I_{{\rm {exp}}}}\left ( {{\theta _{{\rm {e}},{\rm {\ }}}}_j} \right ) - {I_{\rm {T}}}\left ( {{\theta _j}} \right )} \right ]}^2}{\rm {\ }}}$ over the target series of pixel positions covering at least the entire primary peak of the rainbow. The minimization search for the summed intensity difference $SID$ of the $N$ local problems yields the desired coefficients $a$ and $b$ in the calibration procedure.
Each local minimum problem, classified as a constrained nonlinear minimization, is solved iteratively by the SQP algorithm [24]. The iteration terminates when the step size is less than the tolerance of $10^{-6}$. The tolerance is set to a small value to ensure a high accuracy of the inversed calibration coefficients. The expected theoretical signal is computed using only the $p=2$ contribution to the scattered signal, which avoids the large number of local minima arising from the ripple structure. This is computed using the Debye series decomposition of the Lorenz-Mie theory which describes scattering of a plane wave from a spherical particle [7]. Another approach is to introduce an estimation of forward angular frequency [25]. The theoretical scattering angle resolution $({\theta _{j + 1}} - {\theta _j})$ used in this procedure is 0.001°, which is accurate enough to capture all features in the experimentally detected signal. The ranges of the calibration coefficients and the value of $N$ are determined as a compromise between accuracy and computational time.
3. Validation through simulation
To examine the performance of this calibration method, rainbow signals of droplets of diameter $D$=80 $\mathrm{\mu}$m, 100 $\mathrm{\mu}$m, 120 $\mathrm{\mu}$m and refractive index $n$=1.3350 have been calculated. For each rainbow signal, nine pairs of typical values for $a$ and $b$ are then specified, as presented in Table 1.
Table 1. List of simulation parameters for calibration coefficients.
The sampling of the simulated rainbow signal using the specified calibration coefficients in Table 1 is depicted for $D$=80 $\mathrm{\mu}$m and $n$=1.3350 in Fig. 4(a). The sampling range of the scattering angle is equal to $b$*$num_{\rm d}$, where $num_{\rm d}$ is the number of pixels of the detector and remains constant with the value 2000.
Fig. 4. (a) Detecting position and range of the detector varies with different combinations of calibration coefficients $a$ and $b$ for a typical rainbow signal ($D$=80 $\mathrm{\mu}$m, $n$=1.3350), (b) matching of simulated and predicted calibration signals.
The simulated preset diameter $D$ and refractive index $n$ of the droplet is input into the calibration procedure. The best matching of simulated and predicted signals, as depicted in Fig. 4(b), yields the final measured calibration coefficients. The errors of calibration coefficients retrieved from the above simulated calibration rainbow signals of droplets of different diameters $D$=80/100/120 $\mathrm{\mu}$m are graphically summarized in Fig. 5. For all choices of calibration coefficients, the maximum absolute errors of $a$ and $b$ are respectively less than 0.032°and 2.7E-5°/pixel. Since the two coefficients have opposite positive and negative errors, the calibration method yields a maximum scattering angle error of 0.032°, resulting in a refractive index uncertainty of $2.3\times 10^{-4}$. Hence, these simulation results indicate a satisfactory accuracy using this scattering angle calibration procedure.
Fig. 5. Errors of calibration coefficients retrieved from simulated rainbow signals from droplets of different diameters (a) $D$=80 $\mathrm{\mu}$m, (b) $D$=100 $\mathrm{\mu}$m, (c) $D$=120 $\mathrm{\mu}$m.
In the case of $D$=80 $\mathrm{\mu}$m and 100 $\mathrm{\mu}$m, the errors of the two retrieved coefficients and their stability are comparable, with mean values of 0.021°, 1.4E-5°/pixel and 0.020°, 1.6E-5°/pixel respectively. However, in the case of $D$=120 $\mathrm{\mu}$m, the two values of 0.018°, 1.3E-5°/pixel are slightly lower and more stable over different values of slope. This is due to the richer and finer (ripple) structure of the rainbow signal of larger sized droplets, enabling the fitting process more accurate. In addition, since the scattering intensity in the rainbow region is proportional to droplet diameter $D^{7/3}$ [26], the signals arising from the $D$=120 $\mathrm{\mu}$m droplets have a higher signal-to-noise ratio, which also increases accuracy.
In practice, the $D$ and $n$ values of the calibrating droplet obtained by measurement or simulation are not exactly the correct values. This fact is taken into account by postulating a certain deviation between the input and the preset values in the simulation. The relative and absolute variations of $D$ and $n$ are assumed to be 1% and $1\times 10^{-4}$ respectively, and the reasons for this choice are given in the following description.
Since only a single parameter change of the droplet is not common, the simultaneous variation of $D$ and $n$ are now considered. The following error analysis is only carried out for the case of $D_0$=120 $\mathrm{\mu}$m, $n_0$=1.3350. There are thus four combinations of the droplet parameters, namely 118.8 $\mathrm{\mu}$m & 1.3349, 118.8 $\mathrm{\mu}$m & 1.3351, 121.2 $\mathrm{\mu}$m & 1.3349, 121.2 $\mathrm{\mu}$m & 1.3351. Figure 6 shows the errors of calibration coefficients from simulated rainbow signals of droplets with simultaneous variations in $D$ and $n$. The maximum errors of $a$ and $b$ are 0.030°, 3.9E-5°/pixel. The effect of these variations on the retrieved $a$ and $b$ values is so small that they can be neglected, independent of their initial values.
Fig. 6. Errors of calibration coefficients retrieved from simulated rainbow signals ($D$=120 $\mathrm{\mu}$m, $n$=1.3350) with different calibration slopes (a) $b$=0.004°/pixel, (b) $b$=0.005°/pixel, (c) $b$=0.006°/pixel in the cases where droplet size and refractive index vary simultaneously.
Figure 7 compares the normalized scattering intensity of $p=2$ of a droplet ($D$=120 $\mathrm{\mu}$m, $n$=1.3350) and droplets with simultaneous variations in $D$ and $n$. From the primary peak in the magnified view (inset), only scattering angle shifts of -0.004°, 0.020°, -0.020°and 0.004°can be observed on these four theoretical signals, respectively. The potential uncertainty in the two parameters of the known droplet appears to be quite limited, providing an explanation for the results in Fig. 6.
Fig. 7. Panoramic and magnified view (inset) of normalized scattering intensity of $p$=2 of a droplet ($D$=120 $\mathrm{\mu}$m, $n$=1.3350) and of droplets with simultaneous variations in size and refractive index.
4. Experimental verification
The experiments in this study involve two steps; the first is the generation of a monodisperse droplet stream, another is the validation of scattering angle calibration for a given point rainbow measurement system.
A droplet stream has been used as the calibration standard instead of glass beads or other known scatterers for the following reasons. First off, the proposed calibration method requires a spherical scatterer which can be calculated theoretically using the Lorenz-Mie theory. Due to surface tension, droplets smaller than 200 $\mathrm{\mu}$m (this threshold value may vary slightly under different conditions and will also depend on the surface tension) in size remain highly spherical. Moreover, the above simulation results indicate that a droplet of a larger diameter helps to improve the calibration accuracy. Therefore, a droplet diameter of 180–190 $\mathrm{\mu}$m has been found to be an excellent compromise. Using the droplet stream, the droplet shape can be monitored using backlight shadowgraphy and adjusted before the calibration if necessary. Such an adjustment is not possible with glass beads, which are inevitably not exactly spherical. Not only that, but the translation of glass beads through the illuminating plane of light is mechanically difficult if a planar measurement is required.
4.1 Experimental setup
The above calibration method was applied using a monodisperse droplet stream with known size and refractive index. The use of a droplet stream is common in the experimental verification of the standard calibration model. The principle of the droplet generation is based on the Rayleigh-Plateau instability of a laminar liquid jet under a periodic disturbance imposed by a piezoelectric ceramic driven by the signal generator [27]. As illustrated in Fig. 8(a), a droplet generation system comprises a monodisperse droplet generator (MTG-01-G1, FMP Inc) with an orifice diameter of 75 $\mathrm{\mu}$m, a high-pressure syringe pump (NE-1010, accuracy $\leq$ 0.5%), and a function signal generator (TOE 7404). The droplet diameter $D$ can be predicted as a function of liquid flow rate $Q$ given by the syringe pump and the disturbance excitation frequency $f$ controlled by the signal generator, expressed as
The shadowgraphy system was used to quantify the uncertainty of diameter predicted by Eq. (3) and its layout is shown in Fig. 8(a). A CMOS camera (3.75 $\mathrm{\mu}$m/pixel, $659\times 493$ pixels) captures the shadow images of droplet stream illuminated by a diffuse backlight.
Fig. 8. (a) Schematic of the generation of a monodisperse droplet stream and the shadowgraphy system; (b) Typical point rainbow measurement system calibrated by an angular reflector.
A point rainbow measurement system was set up and calibrated to validate the feasibility and accuracy of the proposed method. As depicted in Fig. 8(b), the system was composed of a vertically polarized illumination laser ($\lambda$=532 nm, 200 mW, beam diameter=1.8 mm), two spherical lenses (plano-convex, diameter=50 mm, $f$=75 mm), an aperture and a large target sized camera (4.8 $\mathrm{\mu}$m/pixel, $2048\times 2592$ pixels, 73.0 fps). The camera is mounted in the scattering plane with an azimuthal angle of 0°. More details of this optical setup can be found in [28].
4.2 Uncertainties of a priori known parameters of the droplet stream
The shadowography system was calibrated by illuminating a checkerboard with cell sizes of 250 $\mathrm{\mu}$m (seen as the scale bar), and the calibration image is displayed in Fig. 9(a). The magnification of the system is then calibrated as 4.454. The typical shadow images of the monodisperse droplet stream excited at frequencies of 17–34 kHz are recorded and illustrated in Fig. 9(b). These shadow images are of very high quality with clear and sharp droplet contour edges. According to the procedure from previous work [12], the images are processed to extract the droplet diameters shown in Fig. 9(c). As expected, the diameter decreases with increasing excitation frequency. The deviation between the measured and predicted droplet diameter is also shown in percentage. The theoretical predictions agree very well with shadowgraphy measurements, with a maximum relative error of 0.9%.
The uncertainty of the signal generator is $\pm$2 digits and estimated to be ${\Delta f}/{f}\;=0.07\%$. The liquid flow rate is controlled by a high-precision syringe pump with an uncertainty less than ${\Delta Q}/{Q}\;=0.5\%$. Therefore, the maximum variation in $D$ caused by the uncertainty in $Q$ and $f$ is
Fig. 9. (a) Calibration image of the shadowgraphy system, (b) shadow images of the monodisperse droplet stream excited at frequencies from 17 to 34 kHz, (c) droplet diameters measured by shadowgraphy and their relative errors compared with theoretical predictions.
The uncertainty of refractive index of the droplet, mainly from its temperature, is less than $0.7\times 10^{-4}$ using a T-type thermocouple for the room temperature measurement with an accuracy of $\pm$0.5°C. Therefore, the 1% and $1\times 10^{-4}$ variations in $D$ and $n$ for the above simulation settings are reasonable estimates.
4.3 Comparison of calibration results
The system was first calibrated using the traditional rotating mirror method. A rotary stage co-axially mounted under an angular reflector and rotated around its central axis to adjust the incident angle of the laser beam. The scattering angle is calculated by subtracting the rotational angle measured by the stage from 180°. As displayed in Fig. 10(a), eight discrete calibration points corresponding to different scattering angles, were separately recorded at different pixel positions of the image. An adaptive binarization algorithm was used to identify the horizontal pixel positions of the center of the light spot on the detector. Figure 10(b) shows the linear fitting of the calibration points. The scattering angle is very linear with the pixel position, and its fitted correlation coefficient $R^{2}$ is 0.99869, confirming the aforementioned theoretical linear relation between scattering angle and pixel position. The fitted calibration coefficients are in this case $a_c$=135.35014, $b_c$=0.003101.
Fig. 10. (a) Calibration points at different pixels of the recorded images and (b) their linear fit.
A deionized water jet with a flow rate of $Q$=4.167 ml/min was excited by a sinusoidal periodic signal ($f$=21.4 kHz) at a room temperature of 13°C. The diameter and refractive index of the generated droplet stream are respectively predicted to be 183.6 $\mathrm{\mu}$m, and 1.3357 according to the provided formulation in [29].
To test the angle calibration method in experiment, a series of standard rainbow patterns were recorded, one of which is displayed in Fig. 11(a). The clearly visible primary rainbow, and its 1$^{st}$ and 2$^{nd}$ supernumerary bows are superimposed by numerous high frequency fringes (ripple structure). The 100 rows of pixels in the middle of the pattern are averaged into the measured calibration signal and then processed with the calibration procedure shown in Fig. 3. The calibration coefficients are iteratively adjusted to match the experimental and predicted signals, as illustrated in Fig. 11(b). The two signals are very consistent in angular position and frequency, indicating a good fit.
Fig. 11. (a) A recorded standard rainbow pattern of the droplet stream, (b) Matching of experimental and predicted calibration signals, (c) Errors of experimentally retrieved calibration coefficients for different input droplet parameters.
Figure 11(c) summarizes the errors of experimentally retrieved calibration coefficients for different input droplet parameters. Absolute errors are calculated by subtracting the retrieved coefficients from those calibrated by the traditional method. Referring to Fig. 6, there are overall five combinations of the input droplet parameters, 183.6 $\mathrm{\mu}$m & 1.3357, 182 $\mathrm{\mu}$m & 1.3356, 185 $\mathrm{\mu}$m & 1.3356, 182 $\mathrm{\mu}$m & 1.3358, 185 $\mathrm{\mu}$m & 1.3358. The errors of the retrieved $a$ and $b$ values are quite satisfactory, with the maximum values of 0.020°and 2.0E-5°/pixel compared to the results of the traditional method. The scattering angle calibration accuracy by the proposed method is actually close to the angle resolution (0.011°) achievable with a precision rotary stage.
4.4 Measurement error analysis
The droplet diameter $D$ is proportional to the angular frequency, represented by $b^{-1}$. The measurement error of $D$ arising from the retrieval errors of $b$, is then expressed as ($b_0$/$b$ - 1). Based on the retrieved calibration coefficients in Fig. 11(c), the errors in the measured $D$ of the five input droplet parameters are deduced to be 0.28%, 0.42%, 0.26%, 0.67%, 0.25%. It is well known that the geometric rainbow angle is a function of the refractive index $n$, and the measured uncertainty of $n$ is thus related to its deviation. Figure 12 depicts the deviation of scattering angle at different pixel positions. The maximum absolute angle deviations are respectively 0.013, 0.019, 0.012, 0.039, 0.020 for the five cases. These correspond to satisfactory measurement errors of $0.9\times 10^{-4}$, $1.4\times 10^{-4}$, $0.8\times 10^{-4}$, $2.8\times 10^{-4}$, $1.4\times 10^{-4}$ for the $n$. Usually, the rainbow peak is located on the left part of the image for recording more supernumerary arcs, the measurement error will be of course smaller than the above values.
Fig. 12. Deviation of scattering angle at different pixel positions based on the results in Fig. 11(c).
5. Conclusion
A novel method for rainbow scattering angle calibration using a droplet stream is proposed and comprehensively investigated. The hypothesis of the angular calibration relationship expressed in terms of calibration intercept $a$ and slope $b$, is first derived from geometric optics, based on the ABCD optical matrix. A calibration procedure is established based on the best global fit of experimental signals to the theoretically expected signals. Simulations and experiments were carried out and took into account the uncertainties of respectively 1% and $1\times 10^{-4}$ for the droplet diameter $D$ and refractive index $n$, using shadowgraphy measurements as a ground truth comparison. Rainbow signals of droplets of $D$=80/100/120 $\mathrm{\mu}$m, $n$=1.3350 with nine groups of calibration coefficients were simulated and sampled. The simulations indicate a maximum error of 0.032°and 3.9E-5°/pixel respectively in the retrieved $a$ and $b$ calibration coefficients, and the effect of uncertainties in droplet parameters were so small that they can be neglected. Compared to the results of the conventional rotating mirror, the errors of both $a$, $b$ are quite satisfactory and are close to the performance of a precision rotary stage, with maximum values of 0.020°and 2.0E-5°/pixel. Measurement errors of $D$ and $n$ from these calibration deviations are estimated to be within 0.67% and $2.8\times 10^{-4}$.
The calibration procedure is very simple and fast without complex operations, mainly comprising of a droplet stream generation, calibration rainbow signal recording and signal processing. Moreover, this method is flexible to use in confined spaces, as it does not require additional devices inside the flow chamber. For pixel-to-angle calibration with a nonlinear relationship, a more complex fitting formula, based on aberration analysis, is also possible using this approach. This approach of using a sufficiently prominent peak as a matching key point, is also applicable in other scattering regions with dominant peaks, such as critical angle scattering.
Funding
National Natural Science Foundation of China (52206221); Natural Science Foundation of Jiangsu Province (BK20220952); Shuangchuang Program of Jiangsu Province (JSSCBS20210207); National Key Laboratory of Transient Physics (6142604210204).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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