1. Introduction

A rainbow is a spectacular phenomenon of light scattering on a transparent particle. In nature, colorful rainbow arcs are caused both by reflection and refraction of sunlight on raindrops. With the use of well collimated, extremely bright, and coherent laser radiation, a monochromatic rainbow may be formed. Figure 1(a) shows a numerical representation of such an experiment in the form of angular scattering intensity for a cylindrical glass particle.

 

Fig. 1. (a) Far-field intensity versus scattering angle over the vicinity at the primary rainbow for a ${\mathrm{SiO}}_2$ fiber with diameter of 125 μm. (b) Same as (a) but as a function of fiber diameter ($\lambda =0.6328\mathrm{\mu m}$, TM polarization, $m(\lambda )=1.45702+i1\mathrm{E}-7$.

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A series of bright and dark fringes, spread over a wide range of scattering angles, forms the primary rainbow, the principal rainbow phenomenon. The origin of the main, bright bow around the angle of 154° and its associated secondary peaks was explained in the 19th century by Sir George Biddell Airy on the basis of numerous observations and experiences by Young, Descartes, Newton, and others [1–3]. The rainbow, in terms of Airy’s description, is a kind of real caustic, i.e., an envelope formed by rays of light bent by second-order refraction upon entering a particle. As seen in Fig. 1(a), superimposed on the Airy’s rainbow are higher frequency ripples. In this way, other scattered components, not taken into account by Airy, manifest themselves, including the part of light reflected by the particle and multiply scattered in its body. What is more, for transparent particles, resonant scattering occurs [4–6], leading to abrupt fringe relocation or even its destruction.

The complex nature of the rainbow causes major difficulties in the mathematical analysis of the scattered pattern, which aims to solve the inverse problem for a noninvasive characterization of a single and transparent particle, such as a fiber, liquid jet, water droplet, etc. To illustrate this question, Fig. 1(b) shows a two-dimensional contour graph of the scattered intensity in the vicinity of the primary rainbow as a function of a glass fiber diameter. Because of strong nonlinearities, a causal inference [7] is complicated, i.e., it is difficult to formulate an unambiguous connection linking a feature of the rainbow (measurement data) and a physical property of the particle under study (diameter). To reproduce all the qualitative and quantitative features of the phenomenon considered, a complex model of scattering needs to be considered, e.g., the exact solution of Maxwell’s equations in the form of a partial-wave series [8]; however, inversion of such a model is problematic. First of all, the inverse problem within the exact scattering theory does not possess a unique solution due to collinearity of parameters in a complex structure [9,10]. Moreover, complex models of scattering, even if identifiable in theory, are often ill-conditioned in computational practice, leading to calculation errors that hinder or even prevent interpretation of the results [11–14].

The vast number of mathematical methods for data inversion from the vicinity of the rainbow focus on spectrum analysis. An illustrative interpretation of the rainbow spectrum, based on geometrical optics, has been reported by van Beeck and Riethmuller [15]. The usual method of spectrum processing is low-pass filtration in order to extract the Airy rainbow pattern [16] so that the rainbow peaks move monotonically with characteristics of the particle (diameter, refractive-index, etc.). Another strategy is to analyze the spectral components that correspond to the ripples superimposed on the Airy pattern, with the help of cross-spectral density function [17]. An important factor, however, is that the spectral components are easily distinguishable only when the particle size is significantly greater than the length of the illuminating wave. In practice, this limiting condition makes these methods feasible for characterization of particles with a diameter of several hundred micrometers.

Other ways to form the Airy pattern with no high-frequency ripples, apart from low-pass filtration, involve influencing either spatial or temporal properties of the incident laser beam. Through scattering of ultrashort laser pulses, the scattered components of different order become time-resolved on the detector [18,19]. A spatial discrimination of these components may be achieved with the use of a highly focused laser beam, which scans across the particle [20]. Both ways are of a rather limited usage due to high experimental requirements.

The method of inverse analysis proposed by the authors, aimed at noninvasive diameter characterization of a transparent fiber, is to influence the spectral properties of the incident radiation to obtain a rainbow pattern free of high-frequency ripples. It will be shown that the rainbow pattern formed in the far zone for the case of low-coherent light may be, under certain conditions, explained in terms of Airy’s theory of rainbow. In consequence, the approximate model of scattering becomes applicable, which may be easily transformed into an inverse relation to specify the fiber diameter. The computational analysis will be carried out with respect to a silica (${\mathrm{SiO}}_2$) fiber with a diameter of 10–200 μm and for the case of incident light with the peak wavelength 0.6328 μm. For such small fibers and for the case of laser incident light, the ripple structure can completely hide the Airy pattern so that conventional low-pass filtering fails.

Key theoretical studies include discussion on the nature of the rainbow for the case of low-coherent light and inversion analysis for characterization of a glass fiber with the use of a mathematical formula based on the Airy integral, corrected by comparison with the solution according to the complex angular momentum method.

2. Nature of the Rainbow for the Case of Low-Coherent Incident Light

A. Scattering Model

Consider the geometry shown in Fig. 2. A well-collimated incident beam of low-coherent light propagates in the negative $x$ direction. Normal incidence is assumed where the wave vector ${\mathbf{k}}^{\text{inc}}=-k{\mathbf{e}}_x=-(2\pi /\lambda ){\mathbf{e}}_x$, with $\lambda$ denoting the wavelength, is normal to the axis of the fiber. A Gaussian distribution is assumed for the emission line, which is normal for sources exhibiting moderate spectral broadening, such as visible-spectrum light-emitting diodes (LEDs):

 

Fig. 2. Scattering geometry.

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The fiber with diameter $d$ is assumed to be infinitively long, axisymmetric, and linear. The material is dispersive, i.e., the real part of the complex index of refraction is a function of wavelength $\lambda$ according to the Sellmeier formula suitable for glasses and polymers:

In order to determine the scattered field in the far-zone distance in the $x–y$ plane (i.e., $2\pi r/\lambda \gg 1$), the emission spectrum according to Eq. (1) is divided into $N$ monochromatic, equally spaced components. The complex amplitudes of all waves from the spectrum are assumed to vary in a random fashion and independently from one another. Scattering of each from $N$ waves by the fiber is considered as a separate event. As the fiber is assumed to be linear, superposition applies, and the total solution, i.e., the far-field intensity, is the incoherent (i.e., phase-independent) sum of the solutions for the plane waves comprising the incident field.

The elastic scattering of light at wavelength $\lambda$ and angle $\theta$ at large distance from the fiber is considered using the standard separation-of-variables solution of Maxwell’s equations [8]. As for the case where normally incident light polarization is maintained, the problem of scattering may be restricted to the electric component polarized parallel to the fiber axis, $E_{\parallel }^{\text{sca}}$. Following the solution by van de Hulst [8], the relation between the scattered and the incident intensity is

This paper also encompasses an analysis devoted to a multilayered fiber (optical fiber), as well as homogeneous fiber in terms of Debye series expansion of the scattered wave discussed elsewhere [24,25], to analyze the validity of geometric optics approach to low-coherence light scattering. The expansion coefficient for a multilayered fiber, written in a recursive form, has been proposed by Kai and D’Alessio [26] and also by Kerker and Matijeviĉ [27]. For the sake of Debye analysis, the $b_n$ coefficient may be found in Li et al. [25].

B. Numerical Analysis

An initial stage of numerical analysis has been devoted to studying the effects of low-coherent incident radiation on the scattered field in the vicinity of the primary rainbow. The calculations assumed dispersion coefficients $A_i$, ${\ell }_i$ in Eq. (3) suitable for silica (${\mathrm{SiO}}_2$) glass [22]: $A_i=(0.6961663,0.4079426,0.8974794)$, ${\ell }_i=(0.0684043,0.1162414,9.896161)$, together with extinction coefficient equal to $1\mathrm{E}-07$. The emission line of the light source according to Eq. (1) with the peak-wavelength of 0.6328 μm was divided into $N=1345$ components spaced evenly at 0.0001 μm in the spectral range from 0.5656 to 0.7 μm.

Figures 3(a)–3(c) show the angular distribution of the scattered light, calculated for a set of fiber diameters within the range of 10,12–200 μm. Each plot refers to a different $fwhm$ of the spectrum, i.e., 0.1, 1, and 15 nm. The results from Fig. 3(a) permit some general statements about the nature of the rainbow for the source exhibiting small spectral broadening. The main rainbow peak and its supernumeraries are strongly affected by the ripple structure, especially at lower $d$. However, if the source with a broader spectrum is used, the ripples undergo attenuation. As spatial frequency of the ripples is proportional to $d$ (or, equivalently, to the size parameter $x$), we may observe that the actual amount of attenuation varies depending on it, i.e., the attenuation is less effective at lower $d$. To interpret these observations it is important to note that phase-independent superposition of $N$ plane waves in the far zone corresponds to the operation of spatial averaging, which is a particular kind of low-pass filtering. In fact, no physical separation of the scattered components forming the rainbow occurs. Such a low-pass filter yields a smoother form of the rainbow excited for the incident wave of length ${\lambda }_0$, provided that the emission spectrum is symmetrical with respect to ${\lambda }_0$.

 

Fig. 3. Far-field scattered intensity $I_{\parallel }^{\text{sca}}$ versus scattering angle $\theta$ in the vicinity of the primary rainbow for a ${\mathrm{SiO}}_2$ fiber with diameter $d$ of 10, 12–200 μm. The case of incident light with the spectral line half-width $fwhm$ and the peak wavelength of 0.6328 μm is considered: (a) $fwhm=0.1\mathrm{nm}$, (b) 1 nm, and (c) 15 nm.

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An inherent feature of light scattered by a transparent particle with a high degree of rotational symmetry, such as a fiber, sphere, etc., is a highly nonlinear effect called morphology-dependent resonances (MDRs), occurring for certain combinations of $m$ and $x$. Physically, a particle exhibits resonance when a standing wave travels along its concave surface [4–6]. The salient features of MDRs for a glass fiber are illustrated in Fig. 4(a), where the scattered intensity over the main rainbow peak as a function of very small diameter changes ($\mathrm{\Delta }d=0.0001\mathrm{\mu m}$) and for the incident wave of length 0.6328 μm is shown. It is worth mentioning that MDRs are typically not observed in experiments with multiphase/polydisperse systems [28–33], as each particle contributes to the scattered field differently so that the resonance peaks are smoothed. Similar calculations carried out for the case of incident low-coherent beam do not reveal the existence of MDRs [see Fig. 4(b)]. This, however, does not mean that no resonant scattering occurs, but its effect is imperceptible in the far zone.

 

Fig. 4. Scattered intensity over the first rainbow peak as a function of very small diameter changes ($\mathrm{\Delta }d=0.0001\mathrm{\mu m}$) of a ${\mathrm{SiO}}_2$ fiber for the case of: (a) incident monochromatic light ($\lambda =0.6328\mathrm{\mu m}$) and (b) incident low-coherent light ($\lambda =0.6328\mathrm{\mu m}$, $fwhm=20\mathrm{nm}$).

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Figures 5(a)–5(d) show how selected fringes of the rainbow shift in the scattering angle with $d$ for the case of incident light with the spectral linewidth $fwhm$ of 5–40 nanometers. Although the oscillations tend to fade with $fwhm$, the bright fringes $({\theta }_{1+},{\theta }_{2+})$ are more affected than the dark ones $({\theta }_{1-},{\theta }_{2-})$. The residual fluctuations arise from incident radiation near the edge of the fiber, which, suffering one internal reflection, contributes dominantly to the rings of the glory [34–36].

 

Fig. 5. Angular position of the first and second dark $({\theta }_{1-},{\theta }_{2-})$ and bright $({\theta }_{1+},{\theta }_{2+})$ rainbow fringes versus ${\mathrm{SiO}}_2$ fiber diameter for the case of incident light with the spectral line half-width $fwhm$ and the peak wavelength of 0.6328 μm.

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Considering the above observations, it is worth asking whether the smooth pattern of the main rainbow peak together with its associated supernumeraries, as observed for the case of low-coherent light, is equivalent to the Airy rainbow, within its physical interpretation as well as mathematical representation. Let us recall that the Airy’s theory accounts for geometrical light rays suffering one internal reflection, i.e., of order $p=2$, see the ray trajectory sketch in Fig. 6. The figure includes the scattered intensity due to a vector sum of these rays, computed with the help of Debye series expansion for the incident monochromatic wave of length 0.6328 μm, against the intensity for the case of low-coherent light with the peak wavelength of 0.6328 μm. The angular positions of dark fringes are in agreement with accuracy of 0.001° (simulation step), whereas bright fringes differ in position by a maximum of 0.024°, confirming earlier observations that bright fringes are more affected by residual fluctuations. Therefore, it is legitimate to claim that the effects of low-coherent light scattering may be considered in terms of simplified physical notions that address the problem of a plane monochromatic wave scattering. This equivalence is valid only if the emission spectrum of the source is symmetrical with respect to the peak wavelength.

 

Fig. 6. Normalized scattering intensity around the rainbow angle for a ${\mathrm{SiO}}_2$ fiber of 125 μm diameter according to (—) Lorenz-Mie theory calculations for low-coherent light with the spectral half-width of 20 nm, (···) Debye series calculations for monochromatic incident beam due to contribution of $p=2$ order rays (see inset).

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As the rainbow is formed by rays that undergo a single internal reflection, it is sensitive to any change in the particle morphology. Table 1 contains information on the angular positions of the rainbow peaks, related to three types of silica (${\mathrm{SiO}}_2$) fibers different in structure, but with the same outer diameter (125 μm), i.e., (i) homogeneous fiber, (ii) standard, single-mode optical fiber (SMF) with the core of diameter 8.2 μm, and (iii) standard, graded-index multi-mode optical fiber (MMF) with 50 μm core diameter. A core-dopant is assumed to offset the dispersion curve according to Eq. (3) by a constant $+0.014$. The refractive index profile for a graded-index fiber is parabolic and approximated by 250 layers for the sake of numerical analysis. The incident low-coherent light is assumed with ${\lambda }_0=0.6328\mathrm{\mu m}$ and $fwhm=15\mathrm{nm}$. According to Table 1, even trivial inhomogeneity, as for the case of SMF, causes fringe displacement that cannot be justified by the limited resolution of the scattering angle readings. This displacement is caused by a second (twin) rainbow scattered by the core [37–39].

Table 1. Angular Position of the First and Second Dark $({\mathsf{\theta }}_{\mathsf{1}-},{\mathsf{\theta }}_{\mathsf{2}-})$ and Bright $({\mathsf{\theta }}_{\mathsf{1}+},{\mathsf{\theta }}_{\mathsf{2}+})$ Rainbow Fringes for Different Fibers and the Case of Low-Coherent Incident Beam (${\mathsf{\lambda }}_{\mathsf{0}}=\mathsf{0.6328}\mathsf{\mu m}$, $\mathsf{f}\mathsf{w}\mathsf{h}\mathsf{m}=\mathsf{15}\mathsf{nm}$)

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3. Inverse Analysis for Characterization of a Glass Fiber

Two-step inversion procedure for a glass fiber characterization is discussed in this section. First, a straightforward mathematical formula is introduced, based on the Airy integral, and corrected in a further step by comparison with the solution according to the complex angular momentum (CAM) method. Such a correction provides better approximation of the rainbow formed by low-coherent light scattering. Afterward, the causal (direct) formula is transformed into an inverse function, which converts measurement data from the scattered pattern into unambiguous information on the fiber diameter.

A. Rainbow in Airy’s Theory with Correction According to CAM

In the Airy theory, the wavefront in the vicinity of the primary, monochromatic rainbow is approximated by a cubic function [1]. The amplitude in the far zone is found by solving the scalar wave equation using the theory of Fraunhofer diffraction. Adopting the notation proposed by Lock [40], an expression for the rainbow intensity may be written as follows:

 

Fig. 7. Normalized scattering intensity around the rainbow angle for a ${\mathrm{SiO}}_2$ fiber with diameter of 125 μm according to: (—) Lorenz-Mie theory calculations for low-coherent light with spectral half-width of 20 nm, (- - -) Airy theory with correction, for monochromatic incident beam, and (···) classical Airy theory, monochromatic incident beam.

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B. Inverse Model

The principal strategy for solving the inverse problem is to convert the measurement data, i.e., the angular difference between the first two successive minima, into unambiguous information about the fiber diameter, with the use of an inverse function to Eq. (14). There are two issues that determined the choice of measurement data. First, as shown by Wang and Hulst [45], the angular spacing is affected mainly by $d$ and, to a much lesser extent, by $n$. Second, the dark fringes are less affected by residual fluctuations than bright ones, as shown in Section 2.

Let ${\theta }_i$, ${\theta }_j$ be the angular positions of two arbitrary fringes and $z_i$, $z_j>0$—the Airy function arguments in Eq. (14), corresponding to ${\theta }_i$, ${\theta }_j$, respectively. The difference $z_i-z_j$ can be expressed as follows:

 

Fig. 8. Fiber diameter limiting error versus (a) spectral line half-width of the incident light with the peak wavelength 0.6328 μm for three measurement ranges (see inset) and (b) measurement range for a fixed spectral line half-width. Empty symbols refer to the inversion formula related to the Airy theory; the filled ones, to the Airy theory with correction.

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The accuracy of diameter estimation depends on two factors: (i) approximate nature of the inverse model and (ii) noise on the measurement data due to a limited resolution of fringe position readings (to a lesser extent). As shown in Fig. 8(a), $\delta d$ may be as low as 0.5% for the case of inversion formula with correction. Furthermore, $\delta d$ decreases along the spectral line half-width to become comparable in the measurement ranges considered. As for the inverse formula with no correction ($B=0$), the limiting error decreases with the increase in fiber diameter. There are no excessive growths in the error value in Figs. 8(a) and 8(b), which, if present, could indicate the resonant scattering.

It is worth noting that to estimate the fiber diameter according to Eq. (17), no detailed knowledge of the refractive index changes due to material dispersion, i.e., $n(\lambda )$, is required. Only the refractive index corresponding to the peak wavelength, i.e., $n({\lambda }_0)$, is necessary. Nevertheless, if the emission line profile of the incident light is different from Gaussian, this will result in an additional systematic error. As far as a light-emitting diode is considered, a common source of low-coherent light, its spectral characteristics, including energy distribution and peak wavelength, depend on emission band (color) and temperature, as well as the way of driving and, in general, differ slightly from natural Gaussian [46–48]. Table 2 contains information on the angular positions of selected rainbow peaks, calculated for the case of silica (${\mathrm{SiO}}_2$) fiber and for two different spectral profiles of the incident light: (i) Gaussian spectrum according to Eq. (1) and (ii) experimentally determined spectrum of a high-power LED (CREE Inc.). The emission peak of LED has been thermally adjusted ($T_j=294.15\mathrm{K}$) and stabilized at ${\lambda }_0=0.6328\mathrm{\mu m}$. LED’s junction has been directly coupled to a spectrometer input. The spectral bandwidth at half intensity was 14.5 nm. Gaussian profile characteristics, i.e., ${\lambda }_0$ and $fwhm$, have been adjusted to fit the experimental curve. According to Table 2, only minor differences are present. Making a diameter assessment across the measurement range of 120–130 μm, with the use of Eq. (17) and data from Table 2, it turns out that for the case of LED spectrum the limiting error $\delta d$ increases to 0.56% in comparison to the case of Gaussian spectrum, $\delta d=0.46\%$. For both cases, however, the accuracy may be improved through fitting an empirically obtained relation ${\widehat{d}}_{\text{emp}}({\theta }_i,{\theta }_j,\lambda ,n)$.

Table 2. Angular Position of the First and Second Dark $({\mathsf{\theta }}_{\mathsf{1}-},{\mathsf{\theta }}_{\mathsf{2}-})$ and Bright $({\mathsf{\theta }}_{\mathsf{1}+},{\mathsf{\theta }}_{\mathsf{2}+})$ Rainbow Fringes for Different Spectral Distributions of the Incident Light (${\mathsf{\lambda }}_{\mathsf{0}}=\mathsf{0.6328}\mathsf{\mu m}$, $\mathsf{f}\mathsf{w}\mathsf{h}\mathsf{m}=\mathsf{14.5}\mathsf{nm}$)

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4. Conclusion

The intention of the research was to provide qualitative rather than quantitative results. Theoretical considerations and numerical analysis show that it is possible to form a smooth pattern of the main rainbow peak together with its associated supernumeraries, which is equivalent to the Airy rainbow. Incident light of low temporal coherence has been used as a measurement tool. With the help of a straightforward inverse relation derived on the basis of the Airy theory, it has become possible to characterize a silica fiber uniquely. The applicability of this method has been verified computationally with respect to fibers of relatively small diameter (in reference to the wavelength), beginning from 50 μm. Fiber diameter limiting error for the measurement range of 120–130 μm is less than 0.5%. Application of the technique to the noninvasive characterization of transparent objects with the use of highly efficient LEDs will require further theoretical and empirical research.