# Light tunneling and anomalous reflection from the diffraction of complex annuli: modeling the forward light scattering by spherical particles

Lilian Chabrol, Fabrice R.A. Onofri

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Lilian Chabrol, Fabrice R.A. Onofri. Light tunneling and anomalous reflection from the diffraction of complex annuli: modeling the forward light scattering by spherical particles. Optics Letters, 2025, 50 (6), pp.2061. 10.1364/OL.557822. hal-05344484

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Lilian Chabrol and Fabrice R. A. Onofri, "Light tunneling and anomalous reflection from the diffraction of complex annuli: modeling the forward light scattering by spherical particles," Opt. Lett. 50, 2061-2064 (2025)

# Light Tunneling and Anomalous Reflection from the Diffraction of Complex Annuli: Modeling the Forward Light Scattering by Spherical Particles

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

Compiled March 3, 2025

Anomalous reflection and tunneling contributions of the high-frequency forward light scattering by non-absorbing spheres are modeled by the scalar diffraction of two complex annuli - one for TM and TE polarizations. Their complex radii, which depend on the refractive indices and particle size, are derived from the complex angular momentum (CAM) method, thus suggesting possible extensions to non-electromagnetic scattering. Coupled with the diffraction of a disk, and a geometrical optics approximation (GOA) for the refractive contributions, we propose an accurate hybrid model that requires one to two orders of magnitude less computational time than the Lorenz-Mie theory (LMT) in the near-field, a crucial improvement for e.g. inverse methods.

http://dx.doi.org/10.1364/ao.XX.XXXXXX

Accurate assessment of light scattered by nano- to micro-particles is a cornerstone of numerous research and industrial fields ranging from atmospheric and plasma physics, to dispersed flows and biology. Rigorous electromagnetic models exist to predict its properties, e.g. [1–3]. However, they remain constrained by the shape and composition of the particles, or by the prohibitive computational resources that are required for large scatterers. This explains the renewed interest in the development of fast and accurate asymptotic light scattering models [4–11]. Spherical particles play a significant role in this endeavor as they are defined by only two parameters: the particle size parameter and a ratio of refractive indices. They are also ubiquitous in particulate media (clouds, sprays, bubbling flows) and are assumed in most optical particle characterization instruments [12–14].

In the LMT, the electromagnetic field scattered by a spherical particle of radius $a$, when illuminated by a plane wave with wave vector $k_1$, is expressed as an infinite sum over electric and magnetic vector spherical harmonics. The latter come with external scattering coefficients $a_n$ and $b_n$ for $n \ge 1$ [1] - corresponding to transverse magnetic (TM) and transverse electric (TE) modes respectively. According to the localization principle, in the high-frequency limit, i.e. with the size parameter $a = k_1a \gg 1$, the

expansion index $n$ corresponds to the incident rays with impact parameter $h_n = (n + 1/2)/k_1$ [15], see Fig. 1. The coefficients $a_n$ and $b_n$ vanish rapidly when $h_n \ge a$, so that, for $\alpha \in [0.02, 20000]$, the LMT series can be truncated at [16]:

$$n_{\max} = \alpha + \frac{\delta a}{a} \alpha + 2 \quad \text{with} \quad \delta a \equiv 4.05 \frac{\alpha^{\frac{1}{3}}}{\alpha} a. \quad (1)$$

The radii $a \pm \delta a$ define an edge-domain, spanning across the particle surface, where edge and surface effects occur [17]. The Debye series provides a more detailed decomposition of the external coefficients with respect to the scattering orders $p \in \mathbb{N}$ [18]:

$$\left\{ \begin{array}{l} a_n \\ b_n \end{array} \right\} = \frac{1}{2} \left[ 1 - R_{n,\chi}^{(22)} - \sum_{p=1}^{+\infty} T_{n,\chi}^{(21)} (R_{n,\chi}^{(11)})^{p-1} T_{n,\chi}^{(12)} \right] \quad (2)$$

where $\chi$ stands for TM or TE. The order $p$ accounts for partial waves (rays for a GOA) that interact $p + 1$ times with the particle surface. $R_{n,\chi}^{(ii)}$ are the reflection coefficients in the medium $i$ and $T_{n,\chi}^{(ij)}$ are the transmission coefficients from the medium $i$ to $j$. The coefficients $(1 - R_{n,\chi}^{(22)})/2$ account for different scattering processes which we denote collectively by $p = 0_a$ (subscript $a$ meaning all). They include pure diffraction ($p = 0_d$, subscript $d$) with $a_n = b_n = 1/2$ and with the sum caped to $n = [\alpha - 1/2]$ [18]. The other contributions correspond to surface waves, tunneling and reflection phenomena [17, 19, 20]. In a recent study [21], the present authors have shown, with a rigorous but time consuming approach relying on Debye series calculations, that the orders $p = 0_a$ and $p = 1$ clearly dominate in the near-forward region and high-frequency limit. In the Fraunhofer regime (i.e. for small Fresnel numbers, $F = a^2/(\lambda z) \ll 1$), the divergence of the energy of order $p = 1$ makes it negligible and the term $p = 0_d$ becomes dominant.

In this letter we present a model, numerically more efficient than the LMT, that captures most of the details of the forward scattering patterns of large spherical particles with a relative refractive index greater than one. Valid in the near- and the far-fields, it accounts for diffraction, grazing reflection and tunneling effects ($p = 0_a$), all described in the framework of scalar

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

<footnote>*lilian.chabrol@univ-amu.fr</footnote>

<footnote n="2">²fabrice.onofri@univ-amu.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Geometry of the scattering problem.</figcaption></figure>

diffraction theory, as well as pure refraction ($p = 1$) described by a GOA.

Consider a spherical particle with radius $a$ and relative refractive index $m = m_2/m_1 > 1$, where $m_2$ and $m_1$ are the real refractive indices of the homogeneous particle and the surrounding medium, respectively (see Fig. 1). An incident plane wave with wavelength in vacuum $\lambda_0$ illuminates the particle centered in a spherical ($\text{Or}\theta\varphi$) and a Cartesian ($\text{Oxyz}$) coordinate system. The wave vector in the medium 1 is directed along the positive z-axis with norm $k_1 = m_1(2\pi/\lambda_0)$. For finite distance observations, a circular screen of radius $L$ is defined in the plane at $z = z_s$.

To model the contribution $p = 0_a$, we begin with the earlier approximations of the Debye solution derived by Nussenzveig [22] (TM mode) and Khare [23] (TE and TM modes) for the high-frequency forward scattering of scalar and electromagnetic plane waves by non-absorbing spherical scatterers, respectively. There, they apply the CAM method to the first leading term of the Debye series. This method allows for slow convergent series to be recast as integrations in the complex plane and residue calculations, corresponding to Regge-Debye poles, in different scattering regions [24]. In accordance with the notation of Fig. 1, their original expression for the contributions of $p = 0_a$ to the forward scattered electric field for an x-polarized incident plane wave with component $E(z \ll 0) = E_0e^{ik_1z}$ becomes:

$$E_{0a\chi}^{\text{Nus}}(\alpha, \theta, z_s, m) = -E_0 \frac{k_1 a^2 e^{ik_1 OM}}{i \alpha OM} \frac{z_s}{OM} \left[ \frac{J_1(\alpha\theta)}{\alpha\theta} + A_{\chi}(\alpha, m) J_0(\alpha\theta) - B_{\chi}(\alpha, m)\theta J_1(\alpha\theta) \right], \quad (3)$$

where $J_0$ and $J_1$ are Bessel functions of the first kind of order zero and one respectively; $OM$ is the distance between the particle center $O$ and the observer position $M$. The Eq. (3) was derived under the following assumptions: large size parameter with $a^{1/3} \gg 1$; refracting particles with $|m-1|^{1/2}a^{1/3} \gg 1$ and $m^2\gamma/|m^2-1|^{1/2} \ll 1$; maximum scattering angle $\theta_{\text{max}} \simeq \sin(\theta_{\text{max}}) \ll \gamma = (2/a)^{1/3}$; and far-field regime. These limits correspond to the non-weak forward scattering of large sphere.

Neglecting contributions from Regge-Debye poles, as justified in [23], we approximate the coefficients as:

$$A_{\chi} = \frac{1}{\alpha} \left[ \frac{M_0}{\gamma} - \frac{i}{Me_{\chi}} + \frac{8}{15} M_1 \gamma - i \frac{M_0(4m^2e_{\chi}^2 - 2 - e_{\chi}^2)}{6M^3} \gamma^2 \right] \quad (4)$$

and:

$$B_{\chi} = \frac{1}{\alpha\gamma} \left[ \frac{M_1}{\gamma} - \frac{iM_0}{Me_{\chi}} + \left[ \frac{3}{10} M_2 - \frac{1}{2M^2e_{\chi}^2} \right] \gamma \right], \quad (5)$$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Module and argument of the complex radii for both polarization ($a = 0.1\text{mm}$, $m = 1.05 - 2$).</figcaption></figure>

with $e_{\text{TM}} = 1$ and $e_{\text{TE}} = 1/m^2$, $M = (m^2 - 1)^{1/2}$. The constants $M_0, M_1$ and $M_2$ can be found in [25].

Nussenzveig observed that in the limit where $\sin(\theta) \simeq \theta$, the term $J_1$ in Eq. (3) corresponds to the Fraunhofer approximation (Fra) of the scalar diffraction of a hollow disk of radius $a$ centered in the plane $z = 0$. This fixes the coefficient in front of the square brackets (missing in [22] and [23]), the minus sign being justified by Babinet's principle.

Surprisingly, however, Nussenzveig did not notice or report that the expression between brackets in Eq. (3) resembles a second order Taylor expansion [26] of $J_1$ for $\epsilon\theta \ll 1$: $J_1((\alpha + \epsilon)\theta)/\theta \simeq (J_1(\alpha\theta)/\theta)(1 - \epsilon/\alpha + \epsilon^2/\alpha^2) + J_0(\alpha\theta)(\epsilon - \epsilon^2/2\alpha) - \theta J_1(\alpha\theta)e^2/2$. After some derivation, this remark, though simple, has allowed us to rewrite Eq. (3) in terms of the Fraunhofer diffraction of a hollow disk (H. Disk$_a$) with radius $a$ interfering with that of an annulus ($\text{Ann}_{\tilde{a}_{\chi}^{\pm}}$) with complex radii $\tilde{a}_{\chi}^{-}$ and $\tilde{a}_{\chi}^{+}$, with $|\tilde{a}_{\chi}^{-}| < |\tilde{a}_{\chi}^{+}|$:

$$E_{0a\chi}^{\text{Nus}} \simeq - \left( E^{\text{Fra}} \{ \text{H. Disk}_a \} + E^{\text{Fra}} \{ \text{Ann}_{\tilde{a}_{\chi}^{\pm}} \} \right), \quad (6)$$

where:

$$\tilde{a}_{\chi}^{\pm}(\alpha, m) = a + \frac{1}{2k_1} \left( \frac{2\alpha B_{\chi}}{\alpha A_{\chi} - B_{\chi}} \pm (\alpha A_{\chi} - B_{\chi}) \right), \quad (7)$$

with, for the two-dimensional hollow disk:

$$\begin{align} E^{\text{Fra}} \{ \text{H. Disk}_a \} (\alpha, \theta, z_s) & \nonumber \\ & = E_0 \frac{k_1 a^2 e^{ik_1 OM}}{i \alpha OM} \frac{z_s}{OM} \frac{J_1(\alpha \sin(\theta))}{\alpha \sin(\theta)}, \tag{8} \end{align}$$

and the annuli:

$$\begin{align} E^{\text{Fra}} \{ \text{Ann}_{\tilde{a}_{\chi}^{\pm}} \} (\alpha, \theta, z_s, m) & \nonumber \\ & \equiv E_0 \frac{k_1 e^{ik_1 OM}}{i \alpha OM} \frac{z_s}{OM} \int_{\rho=\tilde{a}_{\chi}^{-}}^{\rho=\tilde{a}_{\chi}^{+}} \rho d\rho J_0 (k_1 \rho \sin(\theta)). \tag{9} \end{align}$$

The coefficients $A_\chi$ and $B_\chi$ - and thus the complex radii $a_\chi^{\pm}$ - represent an average of the contributions of the non-diffractive part of the order $p = 0_a$, i.e. tunneling and anomalous reflection (surface wave contributions being neglected in Eq. (4) and Eq. (5)). Fig. 2 shows the evolution of the module and the phase of the complex radii for $a = 0.1\text{mm}$ and $m = 1.05 - 2$. More generally, it can be observed that for particle radii ranging from $a = 0.1\text{mm}$ to $1\text{mm}$ (not shown), the modules exhibit a discrepancy with respect to $a$ of approximately two percent of $\delta a$ for $|\tilde{a}_{\chi}^{-}|$, and ten percent for $|\tilde{a}_{\chi}^{+}|$. Their phase is less than one degree, with a tendency to increase with decreasing particle radius, indicating stronger edge effects for smaller particles.

Nussenzveig's Eq. (3), and thus the Eq. (6), are solely applicable within the Fraunhofer regime. Nevertheless, in light
of our reformulation Eq. (6), it is compelling to consider a less restrictive scalar diffraction formalism to attempt a near-field approximation of the scattered field:

$$E_{0a,\chi}^{\text{R-S}} = - \left( E^{\text{R-S}}\{\text{H. Disk}_a\} + E^{\text{R-S}}\{\text{Ann}_{\hat{a}_{\chi}^{\pm}}\} \right), \quad (10)$$

using the Rayleigh-Sommerfeld (R-S) integral, where, with $E_0$ the incident electric field on a diffracting surface $S$ in the plane $z=0$:

$$E^{\text{R-S}}(M) = \frac{1}{2\pi} E_0 \iint_{P \in S} dS_P \left( \frac{k_1}{i} + \frac{1}{PM} \right) \frac{z_S}{PM} \frac{e^{ik_1 PM}}{PM}. \quad (11)$$

Note that the Rayleigh-Sommerfeld diffraction of a complex annulus means that one integrates between two complex radii within a cylindrical coordinate system. The analog of Eq. (10) for Fresnel diffraction ($E_{0a,\chi}^{\text{Fre}}$) can be obtained from a Taylor expansion of PM evaluated around OM (see e.g. chapter 7 of [27]).

Rayleigh-Sommerfeld's formulation has the advantage of providing an intuitive secondary wave interpretation of the contributions from a complex annulus. Let $P$ be one of its points defined as $\overrightarrow{OP} = (|\rho|e^{i\sigma}\cos(\kappa), |\rho|e^{i\sigma}\sin(\kappa), 0)$ in a complex coordinate system with $\sigma \in [-\pi, \pi]$. To $P$ we associate a real point $P_0$ in the particle plane $z=0$ with $\overrightarrow{OP_0} = (|\rho|\cos(\kappa), |\rho|\sin(\kappa), 0)$, (see Fig. 3). The contribution of the complex surface element $dS_P$ near $P$ to the scattered field observed at a point $M$ is:

$$\delta E_P(M) = \frac{E_0}{2\pi} \left( \frac{k_1}{i} + \frac{1}{PM} \right) \frac{z_S}{PM} \frac{e^{ik_1 PM}}{PM} dS_P. \quad (12)$$

By azimuthal symmetry let $\overrightarrow{OM} = (d, 0, z_S)$. Assuming that $\sigma \ll 1$, based on Fig. 2, and that $z_S$ is sufficiently large so that $(|\rho|^2 + d|\rho|\sigma) \ll P_0 M^2$, we get $PM \simeq P_0 M + i\sigma (|\rho|^2 - d|\rho|\cos(\kappa)) / P_0 M$. Now, consider a first-order approximation of PM with respect to $\sigma$ in the exponential of Eq. (12), and a zeroth-order approximation otherwise, it follows that:

$$\delta E_P(M) \approx \delta E_{P_0}(M) \exp[-k_1 \sigma |\rho| \sin(\beta)], \quad (13)$$

where $\delta E_{P_0}(M)$ is the conventional secondary wave generated at the real point $P_0$, observed in $M$, and with $\sin(\beta) \equiv (|\rho| - d\cos(\kappa))/P_0 M = \pm P_0 K / P_0 M$. The angle $\beta$, shaded red in Fig. 3, is the deviation angle in the direction $e_\rho$ in order for an incoming ray at $P_0$ to reach $M$ ($\beta$ positive if the ray converges toward the optical axis, negative otherwise). Consider now $z_S \gg \rho + d$ so that $\sin(\beta) \simeq \beta$. We then find that the secondary wave generated at the complex point $P$ in the complex annulus is proportional, in our limits, to the one generated by the real point $P_0$: its amplitude is either attenuated or amplified depending on the phase associated to $P$ and the characteristic distance $|\rho\beta|$ the ray interacts with the sphere (Fig. 3, in green). This distance is proportional to the angle of deviation undergone by the ray toward (or away) from the center of the particle.

As a first illustration of the benefits of generalizations of Eq. (6), Fig. 4 compares the (a) normalized intensity and (b) phase difference of the scattered near-field by a water droplet ($a = 100 \mu\text{m}$, $m = 1.33$) in air when illuminated by a plane wave with $\lambda_0 = \lambda_1 = 632.8 \text{nm}$. The observation plane is at $z_S = 3.16 \text{mm}$ (or $F = 5$) with maximum colatitude angle $\theta_{\text{max}} = \gamma/2 \simeq 3.62^\circ$, relaxing Nussenzveig's conditions. Due to the close proximity of the two solutions, only the TM polarization is shown. $E_{0a,\text{TM}}^{\text{Deb}}$ and $E_{0d}^{\text{Deb}}$ are obtained using LMT and

<figure><img src="image_4.png" /><figcaption>Fig. 3. (a) Observer's view of the particle. (b) Side views of the scattering. The angle $\beta$ is represented in red.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Comparison of scattered fields (a) normalized intensity and (b) phase differences. $a = 0.1 \text{mm}$; $m = 1.33$; $\lambda_1 = 632.8 \text{nm}$; $F = 5$; $z_S = 3.16 \text{mm}$; $\theta_{\text{max}} = 3.62^\circ$.</figcaption></figure>

Debye's exact solution, evaluated from Barber and Hill's Fortran code [28] implemented in Matlab by the present authors [21, 29]. Fresnel diffraction is easily recast as a radial integral and $E_{0a}^{\text{Fre}}$ is evaluated using Matlab's integral function. It is clear from Fig. 4 that our model significantly improves the description of the oscillatory behavior of the order $p = 0a$, while reducing the computational time by a factor of $G = 28$ compared to the exact solutions.

Now, for the contribution $p = 1$, we start with the GOA introduced by van de Hulst for far-field conditions [15]. To extend the latter to the near-field case, it is necessary to modify the expressions of the curvature radii of the wave front and the phases of the complex rays. After some derivations that will be detailed in a future work, the scalar field for a $p \ge 1$ ray with impact parameter $h = a \cos(\tau)$ observed in $M$ can be written with the following vectorial complex ray model equation (similarly to [8]):

$$E_{p \ge 1, \chi}^{\text{GOA}}(\tau) = E_0 \times D(\tau)^{\frac{1}{2}} \frac{a}{\sqrt{|R_x(M)R_y(M)|}} \left(1 - r_{\chi}^2(\tau)\right) \left(-r_{\chi}(\tau)^{p-1}\right) e^{i\left[\frac{\pi}{2}(-1+n_{\text{FL}})+k\delta\right]}, \quad (14)$$

where $r_\chi$ are the Fresnel coefficients of reflection at the particle interface; $R_x(M)$ and $R_y(M)$ are the curvature radii of the wave-front in the observation plane [30]; $k\delta$ is the phase shift due to the optical path length $\delta$ and $n_{\text{FL}}$ the number of focal lines; $D$ is the divergence of the rays in the far-field limit introduced in [15], chapter 12. Note that this model is not expected to hold in the vicinity of the forward caustic [31].
<figure><img src="image_6.png" /><figcaption>Fig. 5. Comparison of the intensities of the total electrical fields. Inset: corresponding quarter holograms. (a) $m = 1.33$ and (b) $m = 1.05$. $a = 0.1$mm; $\lambda_1 = 632.8$nm; $F = 5$; $z_S = 3.16$mm; $\theta_{max} = 3.62^{\circ}$.</figcaption></figure>

Finally, our hybrid model for the orders $p = 0_d$ and $p = 1$ reads as:

$$E_{0_d 1, \chi}^{\text{Hyb-Diff}} = E_{0_u \chi}^{\text{Diff}} + E_{1, \chi}^{\text{GOA}}, \quad (15)$$

where "Diff" can be "Fra", "Fre" or "R-S" depending on the scalar diffraction model considered. Its predictions for the TM mode and in the Fresnel case are compared with LMT ($E_{\text{TM}}^{\text{LMT}}$), pure diffraction ($E_{0_d}^{\text{R-S}}$), and pure diffraction combined with refraction ($E_{0_d 1, \text{TM}}^{\text{R-S}}$) in Fig. 5. This is done in the context of holography - where the scattered and incident fields interfere (which requires an accurate description of the phases). We consider a close proximity of the observer plane ($F = 5$) and $\theta_{\text{max}} = \gamma/2$ as a more difficult test case, with $a = 0.1$mm, $m = 1.33$ or $m = 1.05$. We find that the Fresnel integral formulation gives very satisfactory results and its time gain with respect to LMT calculations is $G \approx 25$ for $a = 0.1$mm. Its computational efficiency increases with the particle size with $G \approx 75$ for $a = 0.3$mm and $G \approx 120$ for $a = 0.5$mm (their high-frequency oscillations make them unsuitable for representation). The Rayleigh-Sommerfeld method gives the same accuracy as the Fresnel method, but does not improve the calculation time as much.

In conclusion, we showed that tunneling and anomalous reflection can be accounted for by the scalar diffraction of two complex annuli for TM and TE polarization. Combined with the usual scalar diffraction of a disk, and refraction described by a GOA, this hybrid model provides accurate predictions of the scattered and total fields in both the near- and far-fields. The a priori key assumptions, i.e. large sphere ($\alpha^{1/3} \gg 1$), non-weak scattering ($|m - 1|^{1/2} \alpha^{1/3} \gg 1$, $m^2 \gamma / |m^2 - 1|^{1/2} \ll 1$) and small scattering angle ($\theta_{\text{max}} \ll (2/\alpha)^{1/3}$), can be partially mitigated and will be further investigated. One to two orders of magnitude faster than the LMT in the high-frequency regime, our

model is a powerful alternative for predicting forward scattering patterns and holograms used e.g. in inverse methods and machine-learning based diagnostics. The generalization of our model to bubbles is another perspective that is currently being explored.

**Funding.** This work was supported by French government grants managed by the Agence Nationale de la Recherche and Bpifrance, references ANR-23-CE51-0023 and PIA-MIRAGE-DOS0196786/00.

**Disclosures.** The authors declare no conflicts of interest.

**Data Availability Statement.** 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.

**Supplemental document.** See Supplement 1 for supporting content.

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