# Near-to-far field approximation of scattering by spheres: bridging the gap between the Lorenz-Mie theory and the Fresnel diffraction

Lilian Chabrol, Fabrice R.A. Onofri

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Lilian Chabrol, Fabrice R.A. Onofri. Near-to-far field approximation of scattering by spheres: bridging the gap between the Lorenz-Mie theory and the Fresnel diffraction. Optics Express, 2025, 33 (18), pp.38926-38940. 10.1364/OE.570717. hal-05344506

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# Near-to-far field approximation of scattering by spheres: bridging the gap between the Lorenz-Mie theory and the Fresnel diffraction

LILIAN CHABROL<fnref n="1" />,*, ID AND FABRICE ONOFRI<fnref n="1" />,<fnref n="2" /> ID

**Abstract:** First, we construct a faster approximation of the Lorenz-Mie theory (LMT) that is valid in both the near and far fields and for all scattering directions. Inspired by the Fresnel diffraction and the localization principle, it uses new approximations of spherical Hankel functions. It is much more accurate than the usual far-field approximation of the LMT, with a similar computational time. It can be applied directly to coated and multilayered spheres, and is most likely generalizable to shaped electromagnetic and acoustic beams. Secondly, we restrict the previous approximation to the near-forward scattering and develop a new formulation of diffraction for spheres, which we call the Lorenz-Mie-Fresnel (LMF) diffraction. Similar in aspect to the Fresnel diffraction, it is more accurate and discrete, though not necessarily faster. Finally, we derive the Fresnel diffraction by a disk from the LMT series using a discrete-to-continuous approximation, thereby establishing the connection between these two century-old theories.

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

In order to characterize the properties of particle systems [1–4] or to control the motion of particles [5–10] from their interaction with electromagnetic waves, it is often necessary to strike a balance between accuracy and numerical efficiency. For spherical homogeneous particles illuminated by a plane wave, the Lorenz-Mie theory (LMT) [11–15] and the Debye decomposition [16,17] provide exact electromagnetic solutions. Unfortunately, for particles with large size parameters, calculations can become excessively time-consuming to perform real-time analyses [2,3,18], feed database or train machine learning methods used in solving inverse problems [18–24]. This computational issue is further exacerbated in the case of inhomogeneous particles [25,26], particles whose shapes deviate slightly (e.g. [27,28]) or completely (e.g. [1,4,29,30]) from a sphere, and when the illumination field gradients can no longer be neglected [31–33].

This explains the renewed interest in developing numerically efficient alternatives, based on the complex angular momentum (CAM) method [34–36], geometrical and/or physical optics approximations [37–44]. In most of these approaches, the sub-process called “diffraction” is usually described with Fraunhofer, Fresnel or Rayleigh-Sommerfeld integrals of planar surfaces - an opaque disk in the case of spherical particles [45].

In the far-field, the link between the Fraunhofer diffraction and the LMT series has been discussed theoretically by e.g. by van de Hulst [46] and by Nussenzveig [47,48]. In the case of Fresnel numbers close to one, the proximity of the Fresnel diffraction by a disk to the LMT solutions has been studied numerically [49,50]. However, to the best of our knowledge, no explicit derivation from the LMT has ever been documented. This question, which lies at the very heart of the present work, has proved more fruitful than expected, as we shall show in the remainder of this article.

<footnote n="1">¹Aix-Marseille University, CNRS, UMR 7343, IUSTI, 13453 Marseille, France</footnote>

<footnote n="2">²fabrice.onofri@univ-amu.fr</footnote>

<footnote>*lilian.chabrol@univ-amu.fr</footnote>
After providing a few essential reminders about the Fresnel diffraction integral in Section 2, we derive a radial approximation of the LMT that is applicable in both the near and far fields in Section 3. We provide numerical results on the accuracy and computational efficiency of the approximation for a wide range of parameters, as well as perspectives for other scatterers and scattering problems. In Section 4, the former approximation is further restricted to the near-forward direction. This allows us to derive a new discrete formulation of light diffraction by spheres which we call the Lorenz-Mie-Fresnel diffraction. Finally, using a discrete-to-continuous approximation, we derive the Fresnel diffraction integral explicitly from the Lorenz-Mie series, thus revealing the underlying connection between these two fundamental theories.

## 2. Fresnel diffraction by a disk

Although the diffraction by a sphere of radius $a$ is related to that of an opaque disk of the same radius, for simplicity we will discuss the case of a circular hole which we will refer to as a “hollow disk”, see Fig. 1. According to Babinet’s principle, the results for the hollow and opaque cases are simply opposite in sign [51].

<figure><img src="image_3.png" /><figcaption>Fig. 1. Diffraction by a 2D circular aperture.</figcaption></figure>

Consider then the diffraction of a scalar plane wave $E(z) = E_0e^{ik_1z}$ by a hollow disk of radius $a$ where $k_1 = m_1 \frac{2\pi}{\lambda_0} = \frac{2\pi}{\lambda_1}$ is the incident wavenumber in the surrounding medium with real refractive index $m_1$, see Fig. 1. Let $P$ be a point on the disk and $M$ be the observer. Using the first Rayleigh-Sommerfeld diffraction integral, the scalar field at $M$ for $z>0$ is [52,53]:

$$E_{\text{Diff}}^{\text{R-S}}(M) = \frac{1}{2\pi} \int_{P \in \text{Disk}} E_0 \left( \frac{k_1}{i} + \frac{1}{PM} \right) \frac{e^{ik_1 PM}}{PM} \cos \theta \, d^2S(P). \quad (1)$$

Assuming thereafter that the distance $PM$ is large enough so that $k_1PM \gg 1$, one recovers the Huygens-Fresnel interpretation of diffraction in which spherical secondary waves - terms in the integral proportional to $\frac{e^{ik_1PM}}{PM}$ - are emitted at each point on the disk, with an obliquity factor $\cos \theta$ [53].

The Fresnel diffraction integral is often obtained from the Rayleigh-Sommerfeld (or Fresnel-Kirchhoff) diffraction integral in the limit where $HM \ll OM$ and $\rho \equiv OP \ll OM$ so that $PM$ is approximated by a first-order expansion around $OH$ [50,52]. Here we prefer to apply the
expansion with respect to OM [53,54] so that, if we use the angle $\kappa = (\widehat{PO}, \widehat{OM})$, we have:

$$ 
\begin{aligned}
PM = |\widehat{PO} + \widehat{OM}| &= \sqrt{\rho^2 + r^2 + 2r\rho \cos \kappa} \\
&= r + \frac{\rho^2 + 2r\rho \cos \kappa}{2r} - \frac{r}{8} \left( \frac{\rho^2 + 2r\rho \cos \kappa}{r^2} \right)^2 + \dots \\
&\approx r + \frac{\rho^2 + 2r\rho \cos \kappa}{2r}.
\end{aligned} 
\quad (2) $$

This approximation of the distance PM is appropriate in the Rayleigh-Sommerfeld integral (Eq. (1)) if the second-order term in the limited expansion of the square root in Eq. (2) does not significantly contribute to the phase of secondary waves. This can be verified if:

$$ \frac{\rho^4}{8r^3\lambda_1} \ll 1, \quad \frac{\rho^2\cos^2\kappa}{2r\lambda_1} \ll 1. \quad (3) $$

If one defines the Fresnel number $F = \frac{\lambda_1^2}{\lambda_1 r}$, the two previous conditions become:

$$ F^2 \frac{\lambda_1}{8r} \ll 1, \quad \cos^2\left(\frac{\pi}{2} - \theta\right) = \sin^2\theta \ll \frac{2}{F}. \quad (4) $$

For a given particle size and incident wavelength, the first condition in Eq. (4) establishes a lower limit on the observer's distance from the scatterer. The second condition is a constraint on the polar angle $\theta$, which restricts the observer's distance from the optical axis.

Now, assuming that these two conditions are met, and using the cylindrical symmetry of the problem, the Fresnel diffraction integral of a hollow disk can be written as follows:

$$ E_{\text{Diff}}^{\text{Fre}}(M) = E_0 \frac{k_1 e^{ik_1 r}}{i r} \cos \theta \int_{\rho=0}^{a} \rho d\rho e^{ik_1 \frac{\rho^2}{2r}} J_0(k_1 \rho \sin \theta). \quad (5) $$

Keeping in mind the two conditions in Eq. (4) and the expression in Eq. (5) of the Fresnel diffraction by a hollow disk, we now turn to the LMT.

## 3. Near-to-far field approximation of the LMT for all scattering angles

### 3.1. Theory

Let an x-polarized incident plane wave of wavelength in vacuum $\lambda_0$ propagate in a homogeneous medium of real refractive index $m_1$ (see Fig. 2). After interaction with a Mie scatterer of radius a and refractive index $m_2$ (possibly complex for an absorbing particle), the scattered external electric field observed in M is:

$$ \mathbf{E}^{\text{LMT}}(M) = \sum_{n=1}^{\infty} i^n E_0 \frac{2n+1}{n(n+1)} \times \left( ia_n \mathbf{N}_{e1n}^{(3)}(M) - b_n \mathbf{M}_{01n}^{(3)}(M) \right). \quad (6) $$

with the time convention $e^{-iwt}$ and where $a_n$ and $b_n$ are the external scattering coefficients [13]. In the spherical coordinate system $(\theta, \varphi)$ the electric and magnetic vector harmonics $\mathbf{N}_{e1n}^{(3)}$ and
$M_{01n}^{(3)}$ are:

$$ \mathbf{N}_{e1n}^{(3)} = \cos \varphi n(n+1)\sin\theta\pi_n(\cos\theta)\frac{h_n^{(1)}(k_1r)}{k_1r}\mathbf{e}_r + \cos\varphi\tau_n(\cos\theta)\frac{[k_1r h_n^{(1)}(k_1r)]'}{k_1r}\mathbf{e}_\theta \\ - \sin\varphi\pi_n(\cos\theta)\frac{[k_1r h_n^{(1)}(k_1r)]'}{k_1r}\mathbf{e}_\varphi, \qquad (7) $$

and:

$$ \mathbf{M}_{01n}^{(3)} = \cos \varphi \pi_n (\cos \theta) h_n^{(1)}(k_1 r) \mathbf{e}_{\theta} - \sin \varphi \tau_n (\cos \theta) h_n^{(1)}(k_1 r) \mathbf{e}_{\varphi}, \qquad (8) $$

where:

$$ \begin{aligned} \tau_n(\cos \theta) &= \frac{1}{\sin \theta} P_n^1(\cos \theta), \\ \pi_n(\cos \theta) &= \frac{d}{d\theta} P_n^1(\cos \theta) \end{aligned} \qquad (9) $$

with $P_n^m$ the associated Legendre functions of degree $n$ and order $m$ (Eq. 4.25 of [13]), while $h_n^{(1)}$ are the spherical Hankel functions of the first kind. Note that in the near-field, the polar angle $\theta$ is not necessarily the scattering angle. The sum in Eq. (6) will be evaluated up to the expansion parameter [55]:

$$ n_{\max} = [\alpha + 4.05\alpha^{\frac{1}{3}} + 2] \qquad (10) $$

where $\alpha = k_1 a$ is the particle size parameter in its surrounding medium.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Electromagnetic plane-wave with wavenumber $k_1$ interacting with a homogeneous sphere of radius $a$.</figcaption></figure>

For large values of $k_1 r$, i.e. at large distances from the scatterer compared to the wavelength, it is common to replace the spherical Hankel functions with their asymptotic - or far-field (FF) - expressions [10,13,14,46,55]:

$$ h_n^{(1)}(k_1 r) \stackrel{k_1 r \gg n^2}{\approx} (-i)^n \frac{e^{ik_1 r}}{ik_1 r} \equiv h_{\text{FF},n}^{(1)}(k_1 r) \qquad (11) $$
so that the far-field approximation of the LMT reads as:

$$ 
\begin{aligned}
\mathbf{E}_{\text{FF}}^{\text{LMT}}(r, \theta, \varphi) &\equiv E_0 \frac{e^{ik_1 r}}{k_1 r} \sum_{n=1}^{n_{\text{max}}} \left(n + \frac{1}{2}\right) \\
&\quad \left\{ i \cos \varphi \left[ (a_n + b_n) \frac{\pi_n + \tau_n}{n(n+1)} + (b_n - a_n) \frac{\pi_n - \tau_n}{n(n+1)} \right] \mathbf{e}_{\theta} \right. \\
&\quad \left. - i \sin \varphi \left[ (a_n + b_n) \frac{\pi_n + \tau_n}{n(n+1)} - (b_n - a_n) \frac{\pi_n - \tau_n}{n(n+1)} \right] \mathbf{e}_{\varphi} \right. \\
&\quad \left. + \frac{2}{k_1 r} \cos \varphi \sin \theta [a_n \pi_n] \mathbf{e}_r \right\}
\end{aligned}
\quad (12) $$

Note that, although we wrote the radial component in Eq. (12) for future considerations, it can be neglected here. The far-field approximation is classically used to simplify the full LMT series equations and accelerate the calculation of the far-field scattering patterns. It is also central to van de Hulst's derivation of the Fraunhofer diffraction by a disk from the LMT, see chapter 12 of [46].

However, using Eq. (11) also has drastic consequences that we can better explain by invoking the localization principle. This principle states that, for large spheres, the partial waves of order $n \gg 1$ in the LMT series correspond to the geometrical optics contributions of rays with impact parameters:

$$ \rho_n = \frac{n + \frac{1}{2}}{\alpha} a. \quad (13) $$

Thus, the far-field approximation of the outgoing spherical Hankel function $h_n^{(1)}$ in Eq. (11) effectively eliminates the dependence on the radial parameters $\rho_n$, see Fig. 2. Consequently, all partial-waves in Eq. (12) are outgoing spherical waves generated at the origin, reducing the scatterer to a point.

For sufficiently large values of $k_1 r$, we can relax the radial asymptotic expressions in Eq. (11) by considering a less restrictive approximation of the functions $h_n^{(1)}$ (equation 10.49.6 of [56]):

$$ 
\begin{aligned}
h_n^{(1)}(k_1 r) &= (-i)^n \frac{e^{ik_1 r}}{ik_1 r} \sum_{m=0}^{n} \frac{i^m}{m!(2k_1 r)^m} \frac{(n+m)!}{(n-m)!} \\
&\approx (-i)^n \frac{e^{ik_1 r}}{ik_1 r} \left[ 1 + \frac{i}{2k_1 r} [n(n+1)] \right] \\
&\equiv h_{\text{dev},n}^{(1)}(k_1 r).
\end{aligned}
\quad (14) $$

Although the approximations of $h_n^{(1)}$ by $h_{\text{dev},n}^{(1)}$ are already more accurate approximations than $h_{\text{FF},n}^{(1)}$, we can improve them further. Inspired by the quadratic term in the Fresnel's diffraction integral (Eq. (5)) and the localization principle (Eq. (13)), we introduce the "near-to-far field" (N2F) approximations of spherical Hankel functions:

$$ 
\begin{aligned}
h_n^{(1)}(k_1 r) &\approx (-i)^n \frac{e^{ik_1 r}}{ik_1 r} e^{i \frac{(n+\frac{1}{2})^2}{2k_1 r}} \\
&= (-i)^n \frac{e^{ik_1 r}}{ik_1 r} e^{i \frac{k_1 \rho_n^2}{2r}} \equiv h_{\text{N2F},n}^{(1)}(k_1 r)
\end{aligned}
\quad (15) $$

and consider:

$$ \left[ k_1 r h_n^{(1)}(k_1 r) \right]' \approx (-i)^n e^{ik_1 r} e^{i \frac{k_1 \rho_n^2}{2r}}. \quad (16) $$
The derivation of $h_{\text{N2F},n}^{(1)}$ is substantiated by identifying the first-order expansion of the exponential and using the approximation $n(n+1) \approx (n + \frac{1}{2})^2$ for large values of the expansion parameter $n$. In addition to its physical significance, it is also more numerically efficient, as will be seen in the next section. Note that, depending on the time convention, the use of spherical Hankel functions of the second kind $h_n^{(2)}$ is possible. In this case, one should consider the conjugate of Eq. (15) to be the correct approximation. Finally, using the approximations in Eq. (15) and in Eq. (16), we derive a new near-to-far field approximation of the Lorenz-Mie series:

$$ \mathbf{E}_{\text{N2F}}^{\text{LMT}}(r, \theta, \varphi) \equiv E_0 \frac{e^{ik_1 r}}{r} \sum_{n=1}^{n_{\text{max}}} \rho_n e^{\frac{ik_1 \rho_n^2}{2r}} \times \\ \left\{ i \cos \varphi \left[ (a_n + b_n) \frac{\pi_n + \tau_n}{n(n+1)} + (b_n - a_n) \frac{\pi_n - \tau_n}{n(n+1)} \right] \mathbf{e}_{\theta} \right. \\ - i \sin \varphi \left[ (a_n + b_n) \frac{\pi_n + \tau_n}{n(n+1)} - (b_n - a_n) \frac{\pi_n - \tau_n}{n(n+1)} \right] \mathbf{e}_{\varphi} \\ \left. + \frac{2}{k_1 r} \cos \varphi \sin \theta [a_n \pi_n] \mathbf{e}_r \right\}. \quad (17) $$

By construction, Eq. (17) accounts for the spatial extension of the spherical scatterer in a manner analogous to Fresnel's diffraction by a disk in Eq. (5).

## 3.2. Numerical comparisons

All of the following calculations were performed in the Matlab environment [57] with 32 significant decimal digits of precision. The complete LMT code, used as a reference, is a transcription by the authors of the Fortran code and subroutines available from Barber and Hill [58].

First, we compare the three approximations $h_{\text{FF},n}^{(1)}$, $h_{\text{dev},n}^{(1)}$ and $h_{\text{N2F},n}^{(1)}$ with the exact spherical Hankel functions $h_n^{(1)}$ for $n = 10, n = 100$ and $n = 1000$. The corresponding relative errors with respect to the parameter $k_1 r$ are shown in Fig. 3. As $n$ increases, the N2F approximations become more and more accurate compared to the others, while remaining good approximations for small $n$. Note that for $k_1 r \approx n = 10^3$, the relative error of all approximations is large. This is because, in this case, orders greater than one in the expansion of the Hankel function Eq. (14) cannot be neglected. The average time gains of each approximation of the spherical Hankel functions relative to the $h_n^{(1)}$ calculations are presented in the Table 1. The averages are calculated over the range of $k_1 r$ shown in Fig. 3. We notice that though the approximations $h_{\text{N2F},n}^{(1)}$ and $h_{\text{dev},n}^{(1)}$ are much more precise than $h_{\text{FF},n}^{(1)}$, they are not significantly slower.

<table><caption>**Table 1.** Average time gain for the calculations of the approximations with respect to $h_n^{(1)}$ for the parameter range of the Fig. 3.</caption><thead><tr><th></th><th>$h_{\text{FF},n}^{(1)}$</th><th>$h_{\text{dev},n}^{(1)}$</th><th>$h_{\text{N2F},n}^{(1)}$</th></tr></thead><tbody><tr><td>$n = 10$</td><td>17.1</td><td>9.8</td><td>11.0</td></tr><tr><td>$n = 100$</td><td>39.3</td><td>24.0</td><td>26.1</td></tr><tr><td>$n = 1000$</td><td>45.0</td><td>29.2</td><td>31.0</td></tr></tbody></table>

Now, we compare the performance of the N2F and FF approximations numerically with that of the full LMT. We begin with Fig. 4 which shows the scattering patterns calculated with the full LMT and with our N2F approximation, in the near-field, for a 5-micrometer radius water droplet in air and a 10-micrometer radius air bubble in water. We do not present the FF approximation
<figure><img src="image_5.png" /><figcaption>Fig. 3. Relative error of three approximations (Eq. (11), Eq. (14) and Eq. (15)) of spherical Hankel functions.</figcaption></figure>

results because the observation distances are equal to only five radii, making them irrelevant. Clearly, the N2F results match quite perfectly those of the full LMT.

To conduct a more exhaustive and quantitative study, we now compare the full LMT with the N2F and FF approximations over a wide range of particle sizes and observer distances. We calculate the Mean Relative Error (MRE) with respect to the modulus of the field $E_{\theta}^{\text{LMT}}$ in the $e_{\theta}$ direction:

$$ \text{MRE}_{\theta} = \text{MEAN} \left| \frac{|E_{\text{Approx},\theta}^{\text{LMT}}| - |E_{\theta}^{\text{LMT}}|}{|E_{\theta}^{\text{LMT}}|} \right|, \qquad (18) $$

for the TM polarization (with $\varphi = 0$) and where “Approx” can be either “N2F” of “FF”. The MRE represents the average relative error of the approximations compared to the full LMT in the $\theta$ direction. MRE$_\theta$ calculations are performed for $\theta \in [0, \pi]$ with arcminute intervals. Note that the results for the directions $e_r$, and for $e_\varphi$ (TE polarization), are nearly identical and are therefore not presented. The results for the N2F approximation are shown in Fig. 5(a) for particle radii between 1 and 500 µm (with particle size parameters ranging from $\alpha \approx 10$ to $\alpha \approx 4965$ respectively) and with ratio of observer distance to particle radius ranging from $r/a = 2$ to $10^5$. Similar results for the FF approximation are shown in Fig. 5(b). Note that the values of MRE$_\theta$ of Fig. 4(a) and (b) are 1.2% and 0.8% respectively. As can be expected, both approximations improve with greater observer distance. As the particle grows larger for fixed $r/a$, the accuracy of both approximations decreases. This can be easily explained by considering the approximations of the spherical Hankel functions with $n \approx \alpha$ for which $\rho_n \approx a$, see Fig. 2. In these cases, the amplitude of the first order in the limited expansion of Eq. (14) is approximately $\pi \frac{a^2}{\lambda_1 r} \equiv \pi F$
where F is the Fresnel number. For a fixed $\frac{r}{a}$ and an increasing a, the limited expansion becomes less and less accurate. Note that the authors chose to use the ratio $\frac{r}{a}$ instead of the Fresnel number on the y-axis of Fig. 5 because the MRE is approximately constant with respect to the radius a for sufficiently large values of the distance ratio $\frac{r}{a}$.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Normalized scattered electric field components from the full LMT (Eq. (6)) and its N2F approximation (Eq. (17)). Wavelength in vacuum $\lambda_0 = 632.8$ nm. (a) water drop in air with $m = 1.33$; $a = 5$ µm; $r = 5a = 25$ µm; $F \equiv a^2/(\lambda_1 r) \approx 1.58$. (b) air bubble in water with $m = 1/1.33$; $a = 10$ µm; $r = 5a = 50$ µm; $F \equiv a^2/(\lambda_1 r) \approx 4.20$. For drawing considerations, an offset of $\times 10$ has been introduced for the TE component.</figcaption></figure>

More concretely, for the N2F approximation, an observer placed at a distance $r = 2$ cm from a particle with radius $a \le 0.5$ mm guarantees $MRE_{\theta} \le 10^{-3}$. For $r = 10$ cm one even obtains $MRE_{\theta} \le 10^{-5}$. On the other hand, for the FF approximation, an observer-particle distance of 50 cm does not yield an average MRE of 1% for particles with radii greater than 10 micrometers.

Now, we turn to the time gain of the N2F and FF approximations with respect to the full LMT:

$$G \equiv T^{\text{LMT}}/T^{\text{LMT}}_{\text{Approx}} \qquad (19)$$

with $T^{\text{LMT}}$ the computer time needed to evaluate both the TM polarization along $e_{\theta}$ and $e_{r}$ (with $\varphi = 0$) and the TE polarization along $e_{\varphi}$ (with $\varphi = \pi/2$). Results for N2F and FF approximations are shown in Fig. 6(a) and Fig. 6(b) respectively. We find that both time gains G tend to increase with the particle radius, while they tend to decrease with the observer distance.

For large particles, $a \ge 100$ µm, and a rather close proximity of the observer, $r \le 10$ cm, the time gain is globally greater than G = 10 for the N2F approximation. Though the FF approximation is faster in this case, the corresponding average error exceeds 30% so that one cannot hope to use this approximation in practice. For small radii ($a \le 5$ µm), the time gains are similar, except for $\frac{r}{a} = 10$ for which the FF approximation is clearly invalid ($MRE_{\theta} \ge 10\%$). For
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Mean Relative Error of the (a) N2F and (b) FF approximations with respect to the LMT calculations over all scattering angles for the *e*θ component of the scattered electric field. Water drop in air *m* = 1.33; wavelength *λ*₁ = 632.8 nm.</figcaption></figure>

intermediate radii (10 ≤ *a* ≤ 50 µm), the FF approximation is approximately two times faster
than the N2F approximation, but much less accurate.

Overall, with the exception of optical configurations for which the observer is at infinity
(behind a Fourier lens for instance), the N2F approximation is superior to the FF one in terms of
accuracy and to the full LMT in terms of computational time. Note that, although not shown,
the results presented here do not vary significantly for other particle refractive indices, with or
without absorption.

**3.3. Perspectives of the N2F approximation**

It is straightforward to extend the use of the N2F approximation to coated [25] and multilayered
[26,59] spherical particles because the approximation does not affect the external scattering
coefficients *a*<sub>n</sub> and *b*<sub>n</sub> which account for the particle’s material properties. Generalizing it to
homogeneous, coated and multilayered infinite cylinders should also be rather trivial as the
external scattered field is expressed in terms of vector cylindrical harmonics, which are defined
using spherical Hankel functions, see e.g. part 8.4 of [13].

For shaped illumination beams, the generalized Lorenz-Mie theory expresses the scattered
field in terms of Ricatti-Bessel functions which can be expressed in terms of spherical Hankel
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Time gain of the (a) N2F and (b) FF approximations with respect to the LMT calculations over all scattering angles. Both TM and TE polarization are calculated. Water drop in air $m = 1.33$; wavelength $λ_1 = 632.8$nm.</figcaption></figure>

functions (see e.g. part III.5 of [60] and Eq.25 of [61]). In the case of spheroidal particles, Hankel functions appear in the definition of the vector spheroidal wavefunctions [62,63]. Although we expect the N2F approximation to be valid in these two instances, we suspect the time savings will be less important due to the increased calculation complexity.

The N2F approximation is not useful for calculating optical cross-sections of Mie scatterers because they can be evaluated directly from the external scattering coefficients $a_n$ and $b_n$. However, it should speed up the calculation of the radiation pressure cross-section of electromagnetic waves [6, 8, 10] and, consequently, of optical forces.

Finally, it is believed that any scattering process whose solution involves spherical Hankel functions of the first kind can potentially be approximated using our N2F approximation Eq. (15). This is notably the case with the scattering of acoustic waves by homogeneous spheres or cylinders [64] and spherical or cylindrical shells [65,66].

## 4. From LMT to Fresnel's diffraction

Now that we have provided a radial approximation of spherical Hankel functions directly inspired by Fresnel diffraction, we continue our investigation to recover the Fresnel diffraction by a disk (Eq. (5)) from the LMT. The first problem we face is defining the scattering process called
“diffraction”. To address this issue, we refer again to van de Hulst’s book [46], in which Fraunhofer diffraction by an opaque disk is derived from the LMT by assuming that, for large particles:

$$ \text{Diffraction: } \begin{cases} a_n = b_n = \frac{1}{2} \text{ if } \rho_n \le a \\ a_n = b_n = 0 \text{ otherwise.} \end{cases} \qquad (20) $$

This definition is based on two main arguments. First, within the Debye decomposition (e.g. [17]), the coefficients $a_n$ and $b_n$ are naturally decomposed as the sum of a single material-independent term (equal to $\frac{1}{2}$) and other material-dependent ones. In the high-frequency limit, these material-dependent terms are traditionally interpreted as contributions to refraction and reflection processes. However, one cannot simply set $a_n = b_n = 1/2$ for all n because the LMT series Eq. (6) then diverges [17]. Second, then, the localization principle distinguishes rays that ‘hit’ the sphere from others, similar to the separation of secondary waves that are or are not blocked in the Huygens-Fresnel view of the diffraction process. Thus, the LMT series is capped to account only for rays that interact with the sphere in the Geometrical Optics limit.

In this light, we now consider the following near-forward approximations of the angle-dependent functions $\pi_n$ and $\tau_n$, for $\theta \ll 1$ which, for large n and fixed $n\theta$, reads as [46]:

$$ \begin{aligned} 2 \frac{\pi_n(\cos \theta)}{n(n+1)} &\approx \left[ J_0\left(\left(n+\frac{1}{2}\right)\theta\right) + J_2\left(\left(n+\frac{1}{2}\right)\theta\right) \right] \\ 2 \frac{\tau_n(\cos \theta)}{n(n+1)} &\approx \left[ J_0\left(\left(n+\frac{1}{2}\right)\theta\right) - J_2\left(\left(n+\frac{1}{2}\right)\theta\right) \right] \end{aligned} \qquad (21) $$

where $J_0$ and $J_2$ are Bessel functions of the first kind and order 0 and 2 respectively. The diffraction subpart of the N2F approximation in Eq. (17), evaluated in the near-forward direction, is then:

$$ \mathbf{E}_{\text{Diff}}^{\text{N2F}}(r, \theta \ll 1, \varphi) \approx [\cos \varphi \mathbf{e}_{\theta} - \sin \varphi \mathbf{e}_{\varphi}] i E_0 \frac{e^{ik_1 r}}{r} \sum_{n=1}^{\lfloor \frac{\alpha-1}{2} \rfloor} \rho_n e^{-\frac{ik_1 \rho_n^2}{2r}} J_0 \left[ \left(n + \frac{1}{2}\right) \theta \right], \qquad (22) $$

where we used Eq. (17), Eq. (20), Eq. (21), and assumed that the radial part of the electric field is negligible ($k_1 r \gg 1$ and $\sin \theta \approx 0$). The Eq. (22) can be reformulated in terms of what we decide to call the “Lorenz-Mie-Fresnel (LMF)” scalar diffraction:

$$ E_{\text{Diff}}^{\text{LMF}}(r, \theta) \equiv i E_0 \frac{e^{ik_1 r}}{r} \sum_{n=1}^{\lfloor \frac{\alpha-1}{2} \rfloor} \rho_n e^{-\frac{ik_1 \rho_n^2}{2r}} J_0[k_1 \rho_n \theta]. \qquad (23) $$

Identification with the one-dimensional Fresnel diffraction integral of an opaque disk ((-1)× Eq. (5)) is now obvious when considering the near-forward approximation:

$$ \cos \theta \approx 1 \quad , \quad \sin \theta \approx \theta, \qquad (24) $$

and the discrete-continuum approximation:

$$ \sum_n \leftrightarrow \int_{\rho} k_1 d\rho. \qquad (25) $$

From Eqs. (22) and (23), we simply define the vectorial Lorenz-Mie-Fresnel diffraction as:

$$ \mathbf{E}_{\text{Diff}}^{\text{LMF}}(r, \theta, \varphi) \equiv E_{\text{Diff}}^{\text{LMF}}(r, \theta) \times [\cos \varphi \mathbf{e}_{\theta} - \sin \varphi \mathbf{e}_{\varphi}]. \qquad (26) $$

The LMF is a demonstration in itself of the link between two century-old theories, the Fresnel diffraction and the LMT. As a numerical illustration of its capabilities, the Fig. 7 compares the
diffraction subpart of the LMT with the conventional Fresnel diffraction by a disk (Eq. (5)) and the LMF diffraction (Eq. (23)). For small polar angles $\theta \le 10^\circ$, the three formulations provide consistent results. However, for larger angles, only the LMF’s predictions remain coherent with those of the LMT. The conventional Fresnel diffraction deviates from the LMT for large angles, partly due to its dependency with respect to $\sin \theta$ rather than $\theta$. Another advantage of the LMF is that it uses a sum instead of an integral, which eliminates non-convergence issues that can arise with large particle size parameters. Unfortunately, numerically estimating the Bessel function $J_0$ with Matlab in Eq. (23) increases the evaluation time compared to the calculations of the $\tau_n$ and $\pi_n$ in Eq. (17). Therefore, as things stand, the current implementation of the LMF does not improve computational efficiency compared to our previous N2F approximation. Please note that we are not providing a statistical analysis similar to that in Fig. 5 because it would depend heavily on the maximum polar angle considered.

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Comparison of the forward electric field of the diffraction calculated from the LMT ($E_{\text{Diff}}^{\text{LMT}}$, Eq. (6) and (20)), the Fresnel diffraction by a disk ($E_{\text{Diff}}^{\text{Fre}}$, Eq. (5)) and the LMF diffraction ($E_{\text{Diff}}^{\text{LMF}}$, Eq. (23)). Parameters: $a = 10 \ \mu\text{m}$, $\lambda_1 = 632.8 \ \text{nm}$, $r = 5a$. Fresnel number $F \approx 3.16$.</figcaption></figure>

Following the derivation of the relationship between the LMT and the Fresnel diffraction, one might be tempted to look for a similar relationship with the Rayleigh-Sommerfeld diffraction by a disk, as it is known to be more accurate for shorter observer-particle distances and for all polar angles $\theta < \pi/2$. However, according to the authors, this seems difficult to achieve for several reasons. First, the radial component of the electric field in the LMT (Eq. (6)) becomes less and less negligible as $r$ decreases. Therefore, a scalar theory of diffraction would not accurately describe this vectorial phenomenon. Second, the Rayleigh-Sommerfeld equation is valid even for large $\theta$ so that the near-forward approximations used in Eq. (21) are not applicable. One would then need to link the polar angle dependency in the Rayleigh-Sommerfeld integral Eq. (1) with the exact $\pi_n$ and $\tau_n$ functions. Third, spherical Hankel functions in Eq. (14), expressed as a series in terms of $r = OM$ (see Fig. 1), would need to be linked to the distances PM in the Rayleigh-Sommerfeld equation. Although PM can be written as a series from the square root expansion in Eq. (2), the authors do not see an obvious link between the square root and spherical Hankel functions series.
**5. Conclusion and perspectives**

Our proposed near-to-far field (N2F) approximation (Eq. (17)) is straightforward to implement and has been shown to be up to an order of magnitude faster than the LMT, as implemented by Barber and Hill [58], for large Mie scatterers. While it is much more accurate than the usual far-field approximation of the LMT (Eq. (12)), its computational time is similar. Therefore, except for spherical particles a few micrometers in size or for an observer at infinity, our N2F approximation is more relevant. It should therefore be used in all other cases, corresponding to many potential applications. The authors believe that the N2F can be directly applied to coated and multilayered spheres and cylinders. It can also be used for shaped beam illuminations and to evaluate optical forces. With the same reasoning, it should also apply to acoustic scattering.

Using the N2F approximation, we successfully derived the Fresnel diffraction integral of a disk from the LMT series and proposed a more accurate discrete alternative for spherical scatterers: the Lorenz-Mie-Fresnel (LMF) diffraction. Discrepancies between Fresnel diffraction and LMF diffraction may provide, for instance, opportunities to improve holographic backpropagation with spherical particles.

**Funding.** Agence Nationale de la Recherche (ANR-23-CE51-0023); Bpifrance (PIA-MIRAGE-DOS0196786/00).

**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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