Scattering by Large Bubbles: Correction for Debye Coefficient at a Small Number of Sub-Waves

I found that when the Debye theory calculates the far-field scattered light intensity of bubbles, the forward scattered light intensity is quite different from the result calculated by the Mie theory due to the convergence problem, so the expression of the Debye coefficient has been revised. I derived the Debye reflectance and transmittance according to the physical meaning of Debye theory and compared them with Fresnel’s formula. I modified the Debye coefficient expressions for bubbles based on the differences between the Debye reflectance and transmittance from the Fresnel formula. Finally, compared with the far-field scattered light intensity calculated by the original Debye theory, the far-field scattered light intensity calculated based on the modified Debye coefficient can obtain more accurate forward scattered light intensity with fewer sub-waves.


Introduction
Light scattering of particles is very valuable for the study of metasurfaces [1] and optical tweezers [2]. It has been used widely in various subject areas, such as environmental science, biomedicine, and chemical industry, and applied in environmental monitoring, cell detection, and other technologies [3]. The Lorenz-Mie theory (LMT) proposed a century ago by Mie [4] provides a rigorous way to describe the scattering of a linearly polarized plane wave by a homogeneous sphere. On this basis, Gouesbet [5], Lock [6], Gouesbet, and Gréhan [7] Z. T. Liu DOI: 10.4236/opj.2022.124004 54 Optics and Photonics Journal proposed the generalized MLT (GLMT) for the scattering of shaped beams by a spherical particle. However, both LMT and GLMT, which are rigorous electromagnetic theories, give the total effect of scattering, and cannot give the contribution and the physical explanation of various scattering processes. It turns out that writing each term of the Mie infinite series as another infinite series, known as the Debye series, clarifies the physical origins of many effects that occur in electromagnetic scattering [8]- [13].
In ray theory, when a geometrical light ray is an incident upon a dielectric sphere it is partially reflected by the sphere surface, partially transmitted through the sphere, and partially transmitted after making an arbitrary number of internal reflections. Analogously, each term of the Debye-series decomposition of an individual TE or TM partial-wave scattering amplitude may be interpreted as the diffraction of the corresponding spherical multipole wave or its reflection by the sphere surface (p = 0) or as transmission through the sphere (p = 1) or transmission after making 1 p − internal reflections ( 2 p ≥ ). But in the bubble, Debye theory cannot get the correct forward scattered light intensity when the max p is relatively small like geometric optics. This is because the Debye coefficient converges very slowly when n > mx. According to the localization principle, we deduce that the Debye reflectance and transmittance are opposite to the Fresnel formula when the total reflection occurs in geometric optics. Accordingly, we modify the expression of the Debye coefficient to obtain a relatively accurate forward scattered light intensity when max p is relatively small.
The main structure of this paper is as follows. First, the localization Principle is briefly introduced, which explains the relationship between the order n of the spherical Bessel function and the angle of incidence i θ in geometric optics.

The Localization Principle
Debye's progressive equation proposed in 1908 actually implied the principle of localization principle [14], the localization principle means that in the Mie theory, the light beam containing the n order Bessel function can represent the light incident at the center point

Debye Series, Mie Theory and Fresnel Formula
Consider a spherical particle (region 1) with radius a and refractive index m embedded in a vacuum (region 2). The incident wave is a plane wave with wave- with ω being the angular frequency. The geometry of the system is given in Figure 1. When a plane wave is an incident on the particle, the expressions of the incident, internal, and scattered fields can be expanded using vector spherical wave functions (VSWFs) as [15].  where subscripts inc, int, and sca represent the incident, internal, and scattered fields, respectively.  is [17]: where, the scattering angle function is obtained by where the coefficients A will be considered later), the reflected wave is written in The reflection and transmission of light waves when propagating from the inside of the sphere to outside the sphere can be given in the same way as Equation (6): Among them, 121 Hence, for TE waves, Equations (6) and (7) can be obtained as: Similarly, the TM waves can be obtained by Equations (6) Solving Equations (8) and (9) comparing their solutions, we can find that they have the same form.
Here we will define some parameters: The incidence angle of the Fresnel formula in Figure 2 is related to the particle size a and the order n. That is, it is calculated by Equation (13). When n = mx, the incident angle is exactly the critical angle of total reflection. So from n = mx to n = x, the imaginary parts of Debye reflectance and transmittance are  exactly the opposite of those calculated by the Fresnel formula. We found this phenomenon, but couldn't explain it very well. And it will affect the accuracy of Debye's scattered light intensity.
From the literature [18] we can know the relationship between the Debye scattering coefficient and the Mie scattering coefficient: Here, ( ) max 0, p represents the summation of terms from 0 to max p , and , we need to use the parameters of the TM wave.
The results in Figure 3 show that when n > 750, the convergence of the Debye coefficients becomes worse and worse. When n = 1000, max p needs to be great- The average Debye scattered light intensity is The following is mainly to compare the difference in far-field scattered light intensity of Debye theory, geometric optics approximation, and Mie theory. And for bubbles, the scattered light intensity of Debye theory is corrected.
The purpose of this paper is to make corrections for the Debye scattered light intensity of bubbles in the case of fewer sub-waves ( max p ). The geometrical optics approximation is mentioned here mainly because it is based on the Fresnel formula. The basis for the correction of the Debye scattered light intensity in this paper is that the Debye reflectance and transmittance are inconsistent with the Fresnel formula. The geometrical optics approximation will not give exactly the same results as Mie theory which turns out to be accurate since it does not take into account surface waves, etc. [19], and it is the most time-consuming algorithm here.
In the legend of Figure 4, GOA-Yu refers to the far-field scattered light intensity calculated by Yu et al. [19]. based on geometric optics approximation; Mie refers to the far-field scattered light intensity calculated using Mie theory (Equation (5)); Debey refers to the far-field scattered light intensity calculated using Debey theory (Equations (14), (15), and (16)  paper 19. When the refractive index of the particle m < 1, the difference between the Debye theory and the forward scattered light intensity of the two is more obvious, but this phenomenon does not appear in the particle with m > 1. This is easy to understand. As mentioned earlier, when n > mx, the Debye coefficient convergence will get worse and worse. It will not converge when max 10 p = .
Considering that the Debye reflectance and transmittance are conjugated to the Fresnel results when n > mx, we modified Equation (14) to obtain Equation (17).
The calculation results for bubbles show that when max p is relatively small, our improved Debye scattering light intensity algorithm is close to the Mie algorithm. This is a significant improvement in the accuracy of the original Debye algorithm.

Conclusion
In this work, we describe the problem that the imaginary part of Debye reflectance and transmittance is opposite to the Fresnel equation when the relative refractive index is less than 1 and total reflection occurs in geometric optics. Secondly, we explained the conditions for the slow convergence of the Debye partial wave coefficients. And we give the partial wave convergence graph. Finally, we improved the calculation of Debye scattered light intensity, so that it can obtain a more accurate scattered light intensity even when max p is relatively small.
It is worth noting that this improvement is only based on part of the physical meaning of Debye's theory, but the effect is obvious. If you want to get the accurate scattered light intensity through Debye theory, you need to calculate it through the original formula, and max p needs to be large enough, which is undoubtedly very time-consuming and very easy to overflow.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.