Light scattering by oblate particles near planar interfaces: on the validity of the T-matrix approach

Amos Egel; Dominik Theobald; Yidenekachew Donie; Uli Lemmer; Guillaume Gomard

doi:10.1364/OE.24.025154

1. Introduction

Many applications in nano optics rely on the interaction of light with wavelength scale particles near planar surfaces, for example plasmonic structures supported by a substrate [1], or scattering layers for light management in optoelectronic thin film devices, such as organic light emitting diodes [2, 3]. The T-matrix method [4–6], one of the most powerful tools for the modelling of light scattering by compact particles, has been adapted to include the scattering interaction between the particles and planar interfaces [7–11]. It relies on an expansion of the particle’s scattered field in terms of outgoing spherical waves relative to the particle center.

However, this outgoing spherical wave expansion (SWE) is in general only valid outside the smallest sphere that includes the whole particle, whereas inside that sphere it may diverge. If an oblate particle is located near a planar surface, the surface intersects the particle’s circumscribing sphere, see Fig. 1. Then, the SWE may not converge everywhere on the planar surface [12], which puts the general validity of the T-matrix approach for such systems into question. This apparent restriction might be one reason why, despite its numerous benefits [5], the T-matrix formalism is not employed more frequently in application-oriented studies of light scattering by photonic structures near interfaces.

Fig. 1 Oblate scattering particle near a planar surface

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The objective of the present paper is to demonstrate that the scattered field can be accurately computed using the T-matrix formalism - even if the circumscribing sphere and the planar surface intersect - thereby paving the way for a wider use of that method in the modelling of photonic structures supported by a substrate. The main steps in the line of argument are the following:

First, we briefly summarize the general procedure for the computation of the scattered field of a particle near a planar surface (section 2). In order to evaluate the reflection from the surface, the SWE of the scattered field is temporarily transformed into a plane wave expansion (PWE) [7]. The whole approach thus depends on the validity of the PWE (and not of the SWE) at the planar surface.
In section 3, we assess the validity of the scattered field’s PWE. We conclude that it is valid even in the near field region of the particle where the outgoing SWE breaks down.
A discussion of the convergence mechanism reveals that the truncation of the scattered field’s PWE and the truncation of its SWE should not be chosen independently. For a fixed truncation multipole order of the SWE, the method can only provide accurate results if the PWE is truncated within a certain range of the in-plane wavenumber. Surprisingly, a larger PWE truncation wavenumber leads to a decrease in accuracy.
The analysis is finally illustrated with two case studies: For the scattering of a plane wave by five spheres (treated as a single composite particle) near a perfectly conducting substrate, the accuracy of the T-matrix formalism is examined by comparing the scattered field coefficients to the exact solution of that system (section 4). Finally, we give numerical results for an oblate spheroid near a planar surface and compare the computed near fields to results obtained with the finite element method (FEM) in section 5.

This paper does not cover the computation of the T-matrix for a given scattering particle. Instead, we use third-party code and algorithms to compute the T-matrix for the application examples discussed in sections 4 and 5, whereas in sections 2 and 3 we just assume that it is precisely known. Under that assumption, the scope of the analysis provided in this paper is not restricted to a certain class of particle shapes. It holds for particles with and without axial symmetry. In this spirit, we use the term “oblate particle” in the broadest possible sense for all objects that extend further into the lateral directions than into the direction orthogonal to the planar surface.

2. Particles near interfaces: the general procedure

Scattering by particles near planar interfaces has been well studied in the framework of the T-matrix formalism [7–11]. For the sake of clarity, we briefly summarize the general procedure, considering the following scattering configuration: A planar interface I at z = z_I < 0 separates two regions of constant refractive index, n₀ in the upper region and n₁ in the lower region. For simplicity, we assume that these materials are linear, nonmagnetic and isotropic. A scattering particle S is located inside the upper region and its center coordinate coincides with the coordinate origin. We denote the volume occupied by the scatterer with V_S. The particle is illuminated by a monochromatic initial field (e.g., a plane wave, a Gaussian beam or the field of a point dipole source) with angular frequency ω. In the upper half space, we can write the total electric field as

E (r) = E_{0} (r) + E_{S} (r) + E_{S}^{R} (r),

where the initial field E₀ (r) solves Maxwell’s equations for the half-space problem without the particle S, and the scattered field E_S (r) as well as its reflection

E_{S}^{R} (r)

from the planar interface fulfill the Silver-Müller radiation boundary condition at infinity.

In the T-matrix formalism, the incoming field E_in at the particle is expanded in regular spherical vector wave functions, the $M_{n}^{(1)} (r)$ (see appendix 6). It includes the initial field, and the reflection from its own scattered field at the interface I. We denote the coefficients of the incoming field by a_n:

\begin{array}{l} E_{in} (r) & = E_{0} (r) + E_{S}^{R} (r) \\ = \sum_{n} a_{n} M_{n}^{(1)} (r) . \end{array}

In this expansion, the summation runs over the following indices: the degree l and order m of the multipole, as well as the polarization p of the spherical wave. For the sake of a more concise notation, these indices are subsumed into a single index n = (l, m, p).

Further, the scattered field is represented by an outgoing SWE:

E_{S} (r) = \sum_{n} b_{n} M_{n}^{(3)} (r) .

Then, the T-matrix of the particle is defined as the linear operator that maps the incoming field coefficients a_n to the scattered field coefficients b_n,

b_{n} = \sum_{n^{'}} T_{n n^{'}} a_{n^{'}} .

It incorporates the complete scattering behavior of the particle S. Efficient and accurate methods for the computation of the T-matrix have been proposed for a broad class of particles [13], and computer codes are freely available (for a collection, see [14]). In the present paper, we assume that the T-matrix is precisely known. In order to solve for the scattered field coefficients, one needs to first evaluate the incoming field coefficients,

a_{n} = a_{0, n} + a_{S, n}^{R},

where a_0,n and

a_{S, n}^{R}

are the coefficients of a regular SWE of the initial field and the reflected scattered field, respectively. Whereas E₀ is known a priori,

E_{S}^{R}

and therefore the coefficients

a_{S, n}^{R}

depend on the scattered field coefficients. In order to evaluate

a_{S, n}^{R}

, one proceeds as follows. First, the scattered field is expanded in terms of downwards propagating plane vector wave functions (see appendix 6), by replacing

M_{n}^{(3)}

in Eq. (1) with the right hand side of Eq. (11):

E_{S} (r) = \sum_{n^{'}} \sum_{j} \frac{b_{n^{'}}}{2 π} \int_{ℝ^{2}} \frac{d^{2} k_{‖}}{k_{z} k} B_{n^{'} j} (- k_{z} / k) E_{j}^{-} (κ, α; r) e^{i m^{'} α}

The above integral runs over all possible directions of downwards propagating and evanescent plane waves, each of which is characterized by its polarization j as well as the wavevector, expressed in cylindrical coordinates as (κ, α, k_z). Further, k = n₀ω/c denotes the wavenumber in the upper half space and k_‖ = (k_x, k_y) stands for the in-plane components of the wavevector. For each partial plane wave in this expansion, the reflection from the interface is evaluated by means of the Fresnel reflection coefficient ρ_j (κ) to yield

E_{S}^{R} (r) = \sum_{n^{'}} \sum_{j} \frac{b_{n^{'}}}{2 π} \int_{ℝ^{2}} d^{2} k_{‖} \frac{ρ_{j} (κ) e^{- 2 i k_{z} z_{I}}}{k_{z} k} B_{n^{'} j} (- k_{z} / k) E_{j}^{+} (κ, α; r) e^{i m^{'} α} .

The reflected partial plane waves are then transformed back into a regular SWE using Eq. (12). After integrating over α (using d²k_‖ = dακdκ), one obtains

E_{S}^{R} (r) = 4 \sum_{n} \sum_{n^{'}} \sum_{j} δ_{m m^{'}} b_{n^{'}} \int_{0}^{\infty} d κ \frac{κ ρ_{j} (κ) e^{- 2 i k_{z} z_{I}}}{k_{z} k} B_{n^{'} j} (- k_{z} / k) B_{n j}^{†} (k_{z} / k) M_{n}^{(1)} (r)

which can be rewritten as

E_{S}^{R} (r) = \sum_{n} a_{S, n}^{R} M_{n}^{(1)} (r)

for

a_{S, n}^{R} = \sum_{n^{'}} W_{n n^{'}}^{R} b_{n^{'}}

with

W_{n n^{'}}^{R} = 4 \sum_{j} δ_{m m^{'}} \int_{0}^{\infty} d κ \frac{κ ρ_{j} (κ) e^{- 2 i k_{z} z_{I}}}{k_{z} k} B_{n^{'} j} (- k_{z} / k) B_{n j}^{†} (k_{z} / k) .

Inserting Eq. (5) into Eq. (2) gives a set of self-consistent linear equations for the scattered field coefficients

b_{n} = \sum_{n^{'}} T_{n n^{'}} (a_{0, n} + a_{0, n^{'}}^{R} + \sum_{n^{″}} W_{n^{'} n^{″}}^{R} b_{n^{″}}),

the solution of which yields the scattered field according to Eq. (1).

3. Plane wave expansion of E_S (r) in the near field

A key feature of the formalism described in section 2 is the temporary expansion of the scattered field in downgoing plane waves, which allows employing Fresnel’s reflection formula in order to evaluate the response from the planar interface. In this section, we turn our attention to the domain of applicability of this expansion. In the case of spherical waves, it is a well known fact that the expansion in Eq. (1) is in general only valid outside the smallest circumscribing sphere with radius r_max = max_{r∈V_S} r around the scatterer, whereas for r < r_max it does in general not converge [15]. However, although the PWE in Eq. (3) is constructed using the outgoing SWE, it turns out that the domains of validity for these expansions are not the same. To see this, it is instructive to think of the scattered field as the collective signal radiated by a current density that is induced inside the scatterer S. For the moment, we are only interested in the direct scattered field E_S (r) and not in the reflected field $E_{S}^{R} (r)$ , such that we can temporarily remove the interface I and think of the scatterer being embedded in an infinite homogeneous medium with refractive index n₀. In addition, we assume that at some plane defined by z = z₀ (with z₀ < −r_max, see Fig. 2) the order of summation and integration in Eq. (3) can be interchanged:

E_{S} (r) = \sum_{j} \int_{ℝ^{2}} \frac{d^{2} k_{‖}}{k_{z} k} \sum_{n^{'}} \frac{b_{n^{'}}}{2 π} B_{n^{'} j} (- k_{z} / k) E_{j}^{-} (κ, α; r) e^{i m^{'} α} for z = z_{0} .

Note that the only difference between Eq. (8) and Eq. (3) is that the summation over n′ has been written inside the integral. The above expression defines the so-called angular spectrum representation [16] of E_S (r),

E_{S} (r) = \int_{ℝ^{2}} d^{2} k_{‖} T (k_{‖}) e^{i k^{-} \cdot r}

with

T (k_{‖}) = \sum_{j} \sum_{n^{'}} \frac{b_{n^{'}}}{2 π} \frac{B_{n^{'} j} (- k_{z} / k)}{k_{z} k} e^{i m^{'} α} {\hat{e}}_{j} .

In general, the angular spectrum representation (i.e., the PWE) is valid for all z < z_min where z_min = min_{r∈V_S} z bounds the source region (i.e., the scattering particle S) from below. In particular, it is also valid in a part of the region inside the circumscribing sphere (see Fig. 2) where the outgoing SWE in general breaks down [17].

Fig. 2 The expansion of E_S (r) in outgoing spherical waves is valid for r > r_max. The expansion in downgoing plane waves is valid for z < z_min. In the dashed white region, the expansion in downgoing plane waves is thus valid, even if the expansion in outgoing spherical waves does not converge. The plane z = z₀ is inside the domain of validity for both expansions.

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At first sight, this seems to be in conflict with the fact that the angular spectrum representation T (k_‖) in Eq. (10) was derived starting from an outgoing SWE of E_S (r). How is it possible that a finite expression (9) for E_S (r) emerges from a diverging expression (1)? The reason for this apparent contradiction is related to the change in the order of integration and summation from Eq. (3) to Eq. (8). Whereas by assumption, this operation is justified for z = z₀, it is in general not allowed in the near field region, i.e. for z → z_min. There, the right hand side of Eq. (10) does not converge uniformly, but only point-wise to T (k‖).

In practice, the SWE is always truncated at a maximal multipole order l_trunc, and the PWE is truncated at a maximal in-plane wavenumber κ_trunc. Obviously, for a finite sum and a finite integral, the order of integration and summation plays no role. However, in order to achieve an accurate representation of the scattered near field in the region z < z_min, we need to select the truncation parameters l_trunc and κ_trunc in the spirit of $\int d κ \sum_{n}$ (as in Eq. (8)), rather than $\sum_{n} \int d κ$ (as in Eq. (3)). That means, we need to make sure that the truncation multipole order l_trunc is chosen large enough such that for each |k_‖| ≤ κ_trunc the angular power spectrum has converged to its true value. In contrast, if we select the truncation parameters the other way round, i.e., such that κ_trunc is large enough to make sure that for each l ≤ l_trunc the integral over κ has converged, the resulting field will exactly resemble the original SWE, and in particular show the same divergent behavior for l_trunc → ∞. As a consequence, we are facing the seemingly paradoxical situation that for a fixed truncation multipole order l_trunc, the calculated near field may be more accurate if a small cut-off wavenumber κ_trunc is chosen rather than a large one. This point will be further illustrated by the following case study, see section 4.1.

4. First case study: five dielectric spheres

We study the scattering of light at a chain of five identical dielectric spherical particles. This system is chosen because it allows for an exact solution, and at the same time it is a representative example of an oblate particle if the spheres are regarded as a single composite object. For that reason, it is an ideal test case for the formalism described in section 2.

The spheres have a refractive index n_S = 2.4 and radius R, and are located at r_i = (2 (i − 3) R, 0, 0), i = 1, . . . , 5. The system is illuminated by a plane wave, polarized along the y-direction and propagating into the negative z-direction with a wavenumber k = 1/R (i.e., each sphere has a dimensionless size parameter of kR = 1). First, we will examine the scattered near-field in the absence of a planar interface in order to demonstrate the validity of the plane wave representation in Eq. (8), and then consider the case for which spheres are located near a perfectly conducting substrate.

4.1. Five spheres in free space

The scattering problem can be exactly solved treating the spheres as individual scattering particles. Then, the scattered field is expressed by means of a SWE relative to each of the five sphere centers. The coefficients of that expansion can be constructed following, e.g., [18]. Because the circumscribing sphere is identical to the particle when individual spheres are considered, that representation of the fields converges everywhere outside the particles. This solution will serve us as an exact reference in the following analysis, see Fig. 3(A). Alternatively, one can treat the five spheres as a single, composite (i.e., non-connected) particle. Its T-matrix can then be constructed using the superposition T-matrix formalism [19].

Fig. 3 Scattering of a downward propagating plane wave at five dielectric spheres in free space. A: The norm of the exact scattered near-field. B: Error of the field constructed with the superposition T-matrix for the composite particle, truncated at l_trunc = 20. C: Error of the field constructed with the superposition T-matrix, truncated at l_trunc = 20, then transformed into the angular spectrum representation, Eq. (8), truncated at κ_trunc = 2 k. D: The same as C, but with κ_trunc = 50 k. The dashed circles and lines indicate the boundary of the domains of validity for the SWE and the PWE, respectively. The color scale reaches from 0 (dark blue) to 0.8 (dark red).

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It is important to note the difference between the two approaches that we are comparing: In the individual-particles picture, the scattered field is given by a sum of five expansions in spherical waves relative to the respective sphere centers. In the composite-particle picture, the scattered fields are regrouped into a single spherical wave expansion relative to the sphere in the middle of the chain. The difference between these approaches is an instructive illustration of the philosophy that underlies the T-matrix method in general: the whole particles’ scattered field is represented in terms of spherical waves emerging from the particle center - although the actual source of radiation (the induced current distribution) is distributed over the whole particle volume.

Figure 3(B) shows the error of the scattered near field, which has been calculated according to Eq. (1) and Eq. (2) with a truncation of the SWE at a polar multipole order of l_trunc = 20. It can be verified that the SWE of the scattered field in the composite-particle approach accurately resembles the exact near-field, except in the domain near to the coordinate origin, where it diverges. It is worth noting that the domain of convergence of Eq. (1) in fact overlaps the circumscribing sphere of the composite particle. It reaches down to the radius of the smallest sphere containing all singularities of the analytic continuation of the scattered field into the scattering particle (compare [15]), which are, in this example, located at the centers of the five spheres.

Figures. 3(C) and (D) show the error of the scattered field expressed in terms of a PWE. The coefficients of the PWE were calculated from the scattered field coefficients b_n of the composite particle using Eq. (8). For Fig. 3(C), the integral over the in-plane wavenumber was truncated at κ_trunc = 2 k, whereas for Fig. 3(D) κ_trunc = 50 k was chosen. As expected, above the particles, the field diverges for both cases, because a downgoing PWE is in general only valid for z < z_min. Below the spheres, the achieved accuracy differs, which is a consequence of the different truncation in-plane wavenumber of the PWE, respectively. In Fig. 3(C), the error of the scattered near-field is acceptable for all z < z_min (and partly even above). In contrast, Fig. 3(D) reveals that for large κ_trunc, the field inherits the divergent behavior near the coordinate from the SWE of the composite particle, compare Fig. 3(B).

This behavior could be expected with regard to the discussion presented in the last paragraph of section 3. We take a closer look at the plane wave spectrum for different truncation multipole orders in the SWE. Figure 4 shows the norm of the integrand of Eq. (9) for r = (0, 0, −1.5R), i.e., T (κê_x) exp (1.5ik_z R), as a function of the in-plane wavenumber κ. Each line corresponds to a different truncation multipole order l_trunc up to which the sum over n was executed in the right hand side of Eq. (10). The solid black line depicts the exact spectrum, constructed from the reference solution. With growing l_trunc, the curves converge point-wise to the exact spectrum. For each curve, the spectral axis can be roughly divided in three domains: In the interval [0, κ₁], the curve is in good agreement with the exact spectrum and deviates less than some specified tolerance (e.g., less than 5 %). This interval of convergency is followed by an interval [κ₁, κ₂] in which the curve deviates strongly from the exact spectrum and takes huge values. Finally, for κ > κ₂, the curve drops to very small values (e.g., below 10⁻⁵), as the suppression by the factor exp (−ik_z z) becomes dominant, because k_z = (k²− κ²)^1/2 is an imaginary quantity of growing modulus. The position of the interval [κ₁, κ₂] thereby depends on the truncation multipole order and on the position where the field is evaluated. For growing l_trunc, the interval is shifted to higher κ, and at the same time the maximum of the curves in the interval of divergence grows to larger and larger values. This is why for field locations inside the circumscribing sphere of the scatterer, the curves do not converge uniformly to the exact spectrum, but only point-wise with respect to κ.

Fig. 4 Plane wave expansion of the scattered field for five spheres in free space. The y-axis shows the weight of the respective partial plane waves to the scattered electric field evaluated at 1.5R below the particle centers, |T (κêx) exp(1.5ik_z R)|. The solid curve corresponds to the exact PWE constructed from the reference solution. The dashed, dotted and dash-dotted lines correspond to the PWE in the composite-particle approach for different truncation multipole orders l_trunc in the right hand side of Eq. (8). The vertical grey lines indicate the location of κ₁ and κ₂ for the respective truncation multipole orders. We have also marked the points up to which the PWE was considered in the evaluation of the scattered fields shown in Figs. 3(C) and (D), respectively, as black dots.

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For the convergency in real space, these spectral regions have the following meaning: From the waves in the first interval, [0, κ₁], the scattered near- and far-field can be constructed. The computed field approximates the exact field well if l_trunc was chosen large enough to allow for a sufficiently high κ₁, such that all relevant evanescent waves are included. In contrast, the interval [κ₁, κ₂] includes those evanescent waves that build up the divergent field inside the circumscribing sphere of the scatterer. If the PWE is truncated at κ ≥ κ₂, it resembles the SWE up to the multipole order l_trunc. In particular, the so constructed field may diverge inside the circumscribing sphere of the scatterer, just as the SWE. The truncation parameters l_trunc and κ_trunc that were used for the construction of the fields displayed in Fig.s 3(C) and (D) are marked in Fig. 4. For the case of Fig. 3(C), κ_trunc corresponds to κ₁ whereas for Fig. 3(D), κ_trunc comes near κ₂. This explains, why Fig. 3(D) shows a divergent near-field below the particle, whereas the field in Fig. 3(C) is a good approximation to the exact field for all z < z_min. Note that if the test plane (in our case z = −1.5R) is shifted towards z = 0, the suppression factor exp (−ik_z z) becomes weaker, and κ₂ grows to larger values. For that reason, Fig. 3(D) exhibits a large error for z ≳ −1.5R. There, κ₂ is larger than the chosen κ_trunc = 50 k of the PWE which thus has not converged to the field given by the SWE.

4.2. Five spheres near a perfectly conducting substrate

We now introduce the case of five spheres located near a planar interface which separates the upper half space with n₀ = 1 from a perfectly conducting substrate, with a distance of Δz between the interface and the spheres, see Fig. 5. Accordingly, the initial field also includes the reflection of the incident plane wave with a reflection coefficient of ρ_TE (κ) = −1. Again, there are different possible approaches to model that system. In the following, we will compare three solutions.

Fig. 5 Five dielectric spheres near a perfectly conducting substrate.

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First, we model the spheres as individual particles and account for the particles-interface scattering by switching to the equivalent image particle problem (compare [8]) with ten individual spheres in free space, which can be solved with high precision [19]. For a perfectly conducting substrate, the image method is exact such that we can use it as a reference solution.

The second approach corresponds to the method described in section 2. We model the five spheres as a single composite particle through the superposition T-matrix. The particle/substrate interaction is accounted for through a PWE of the scattered field. For the numerical evaluation of the Sommerfeld integral in Eq. (6), we use the trapezoidal rule with a sampling of 10⁻⁴k and a truncation scale κ_trunc. In the vicinity of the branch point at κ = k, we deflect the integral contour slightly into the negative imaginary for a better accuracy.

Finally, in the third approach, we again employ the image method but for the composite particle, i.e., solving a two-particle system. For small particle-interface distances, when the circumscribed spheres of the composite particle and of its image intersect, this method can in general not be expected to converge to the correct field.

We measure the accuracy of a computed solution by comparing the scattered field coefficients b_n to those of the exact reference solution, b_exact,_n. For that purpose, we use the spherical vector wave addition theorem [20] to construct an outgoing SWE relative to the coordinate origin for the total scattered field of all five spheres. The relative error is defined as

relative error = \frac{‖ b - b_{exact} ‖}{‖ b_{exact} ‖},

where ‖·‖ denotes the Euclidean norm. Figure 6(A) shows the error for the composite-particle approach as a function of the truncation parameters l_trunc and κ_trunc. The distance between the planar interface and the particle was set to Δz = R. The best accuracy was achieved for a truncation multipole order of l_trunc = 25 and a truncation in-plane wavenumber around κ_trunc = 2.8 k. For larger κ_trunc, the accuracy drops drastically, because for those in-plane wavenumbers, the point-wise convergency of the PWE has not yet taken place for l_trunc ≤ 30, compare section 4.1. Interestingly, the best accuracy is not observed for the highest considered SWE truncation order, l_trunc = 30, but for l_trunc = 25. We suppose that this is because the linear system in Eq. (7) can contain large coefficients and might be ill-conditioned for large l_trunc. For the case of light scattering by two overlapping spheres, chaotic behavior with respect to the truncation order of the SWE has been reported [21]. Possibly, the deeper reasons for the here observed instability is related to that issue.

Fig. 6 A) Achieved accuracy as a function of the truncation in-plane wavenumber and truncation multipole order. The distance between the particles and the planar interface was fixed to Δz = R. B) Minimal error (i.e., for optimal κ_trunc), as a function of the distance between the particle centers and the planar interface. The lines refer to different truncation multipole orders (see annotation). The dashed grey line refers to the accuracy that results from the image method for the composite particle for l_trunc = 15.

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Further, Fig. 6(B) shows the best achieved error (i.e. the minimum with respect to κ_trunc) as a function of the particle-substrate distance Δz. If the five spheres touch the substrate, the achievable accuracy for the composite-particle approach is around 10⁻². For an increasing distance between the particles and the planar interface the error drops fast and a satisfying accuracy is observed even if the circumscribing sphere of the composite particle significantly intersects the substrate, i.e., for Δz < 4R. For comparison, we also show the results for the composite-particle approach using the image method, i.e., for solving a two-composite particle problem in free space. Because the circumscribing sphere of the composite particle and its image intersect, the accuracy of that approach is poor for Δz ≲ 3R.

5. Second case study: oblate spheroid

The second numerical example is an oblate spheroid with a semi-major axis A = 250 nm in the x- and y-direction and a semi-minor axis C = 50 nm in the z-direction, located in vacuum above a perfectly conducting substrate with a distance Δz. We consider the cases of a dielectric particle (with a refractive index of 2.4) and of a perfectly conducting particle, see Fig. 7. As before, the system is illuminated from above by a plane wave with unit amplitude, polarized in the y-direction and with a vacuum wavelength 2π/k = 550 nm.

Fig. 7 Oblate spheroid near a perfectly conducting substrate.

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The T-matrix for the spheroid is computed by means of the NFM-DS code which is included in [22] and can also be downloaded from [14]. We compute the direct and reflected scattered field $E_{S} + E_{S}^{R}$ according to the procedure described in section 2. The SWE truncation multipole order was fixed to l_trunc = 20 and for the truncation of the PWE (i.e., of the Sommerfeld integrals) we considered κ_trunc = 3 k and 20 k. In an additional simulation, we used the image method (compare [23]) rather than a PWE in order to account for the particle/substrate interaction.

As a reference solution, we used the FEM-based commercial software COMSOL [24] in order to compute the scattered field. We used the built-in far field projection tool (based on the Stratton-Chu formula) in order to evaluate the far field power flux. The mesh size was selected to provide a good convergence of the far field results.

Figure 8 shows the scattered far field intensity in the xz-plane as a function of the polar angle. For Δz = 300 nm, the circumscribing sphere of the particle does not intersect the substrate. Then, the results of all four computations match very well. However, for Δz = 0 and 50 nm, the image method does not yield accurate results, because it is based only on the SWE which diverges inside the circumscribing sphere. Likewise, using the T-matrix approach with a rather large Sommerfeld integral truncation order of κ_trunc = 20 k leads to an equally low accuracy. The reason for that can be seen in Fig. 4, where κ = 20 k is well beyond the regime of convergence of the angular spectrum for l_trunc = 20, such that the diverging behavior of the SWE affects the results also after the transformation into a PWE. In contrast, the T-matrix results with κ_trunc = 3 k are in good agreement with the reference FEM results for all cases. This again confirms that a smaller truncation scale of the Sommerfeld integrals can result in a higher accuracy.

Fig. 8 Scattered far field intensity in the xz-plane as a function of the polar angle for a dielectric (n = 2.4, left column) or perfectly conducting (right column) oblate spheroid near a perfectly conducting substrate. The particles are illuminated under normal incidence by a plane wave set at 550 nm. Each row of graphs corresponds to a fixed particle-substrate distance. We compare T-matrix results with a PWE truncation of κ_trunc = 3 k (solid line) and 20 k (dash-dotted line) to FEM results (black dots). In addition, T-matrix results using the image method are shown, too (dashed line). For Δz = 0, the results for the image method and for κ_trunc = 20 k have been scaled down to fit into the axes box.

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6. Discussion and conclusions

The T-matrix approach for the computation of the scattered field of a particle near a planar interface can yield accurate results, even if the circumscribing sphere of the particle intersects the planar interface. We have confirmed this claim by theoretical reasoning and numerical examples.

In a related paper, Doicu et al. have already demonstrated that the non-intersection condition can be relaxed up to a certain degree [12]. They pointed out that the smallest sphere containing all singularities of the scattered field’s analytic continuation can be smaller than the circumscribing sphere of the scatterer - such that the SWE is (partly) valid inside the circumscribing sphere.

In another previous study, Cappellin et al. [17] emphasized that the plane wave expansion of an antenna in free space is valid in the near field, where the spherical wave expansion diverges.

In the present contribution, we specifically analyzed the convergence mechanism for the case of scattering by oblate particles near interfaces, with a discussion of the transformation properties between spherical and plane vector waves. Mathematically, the coefficients of this plane wave expansion, i.e., the angular spectrum, can be constructed from the spherical wave coefficients by interchanging the order of integration and summation. In practice, this means that for a given truncation multipole order in the T-matrix formalism, the truncation in-plane wavenumber of the plane wave expansion (i.e., the truncation of the Sommerfeld integrals) should be chosen inside the regime where the angular spectrum has converged. Furthermore, increasing the truncation multipole order to very large values again leads to a decrease in the accuracy. We attribute this behavior to an increase in the condition number of the linear system in Eq. (7). The overall achievable accuracy of the scattered field is thus limited for small l_trunc by the truncation error and for large l_trunc by numerical instability of the linear system. The numerical instability for large l_trunc has also been pointed out in [12].

We conclude that the range of applicability of the T-matrix method for the electromagnetic scattering by oblate particles near planar interfaces is larger than one might think from looking at the near field convergency of the spherical wave expansion alone. Instead, the convergency of the plane wave expansion is relevant, and its domain of validity extends into the particle’s circumscribing sphere.

The method can be employed for both dielectric and metallic particles or substrates. It is also not restricted to planar interfaces but can be generalized to the scattering by particles near corrugated interfaces [7,22]. Further, the discussion presented in this paper might also be relevant for the problem of electromagnetic scattering by multiple particles with intersecting circumscribing spheres. In that case, it can be advantageous to compute the particle coupling using a (correctly truncated) plane wave expansion of the scattered field instead of the spherical vector wave addition theorem.

Appendix: Spherical and plane vector waves

The spherical vector wave functions live in the spherical coordinate system (r, θ, ϕ) of the position vector r. We use a similar definition to that given in [22]:

\begin{array}{l} M_{l m 1}^{(ν)} (r) = \frac{1}{\sqrt{2 l (l + 1)}} \nabla \times (r z_{l}^{(ν)} (k r) P_{l}^{| m |} (cos θ) e^{i m ϕ}) \\ M_{l m 2}^{(ν)} (r) = \frac{1}{k} \nabla \times M_{m l 1}^{(ν)} (r) \end{array}

The radial wave function

z_{l}^{(ν)}

stands either for the spherical Bessel function of order l,

z_{l}^{(1)} = j_{l}

, or the spherical Hankel function of first kind,

z_{l}^{(3)} = h_{l}^{(1)}

. Further, k = n₀ω/c is the wave number and

P_{l}^{m}

denote the normalized associated Legendre functions. The indices of

M_{m l p}^{(ν)}

stand for: p the polarization (TE = 1, i.e. E ⊥ r, TM = 2), m = −l . . . l the angular index with respect to ϕ and l = 1, 2, . . . the angular index with respect to θ. A multi index n is introduced to subsume them all, (mlp) → n. The number (ν) indicates if the spherical wave is of regular kind (ν = 1) or represents an outgoing wave (ν = 3).

On the other hand, the plane vector wave functions are defined as

E_{j}^{\pm} (κ, α; r) = exp (i k^{\pm} \cdot r) {\hat{e}}_{j}

with (κ, α, ±k_z) being the cylindrical coordinates of the wave vector k^± for

k_{z} = \sqrt{k^{2} - κ^{2}}

. The plus sign corresponds to waves that propagate / decay in the positive z-direction, whereas the minus sign refers to waves that propagate / decay in the negative z-direction. The index j of E_j indicates the polarization (1 = TE and 2 = TM). The unit vectors ê₁ = ê_α and ê₂ = ê_β belong to the azimuthal and polar angle of k^±, respectively.

The spherical vector wave functions can be expanded in terms of plane vector wave functions and vice versa [20,22]. We use the relations

M_{n}^{(3)} (r) = \frac{1}{2 π} \int_{ℝ^{2}} d^{2} k_{‖} \frac{1}{k_{z} k} \sum_{j = 1}^{2} B_{n j} (\pm k_{z} / k) E_{j}^{\pm} (κ, α; r) e^{i m α} for z ≷ 0

and

E_{j}^{\pm} (κ, α; r) = 4 \sum_{n} e^{- i m α} B_{n j}^{†} (\pm k_{z} / k) M_{n}^{(1)} (r),

where the transformation operator B is given by

B_{n j} (x) = - \frac{1}{i^{l + 1}} \frac{1}{\sqrt{2 l (l + 1)}} (i δ_{j 1} + δ_{j 2}) (δ_{p j} τ_{l}^{| m |} (x) + (1 - δ_{p j}) m π_{l}^{| m |} (x))

with the spherical functions τ and π defined as

\begin{array}{l} π_{l}^{m} (cos θ) = \frac{P_{l}^{m} (cos θ)}{sin θ} \\ τ_{l}^{m} (cos θ) = \partial_{θ} P_{l}^{m} (cos θ) . \end{array}

The matrix

B_{n j}^{†}

has all explicit i in Eq. (13) set to −i.

Funding

Deutsche Forschungsgemeinschaft (DFG), Priority Program 1839; Helmholtz-Gemeinschaft, Postdoctroal Program; Karlsruhe School of Optics & Photonics; Karlsruhe Institue of Technology (KIT) and DFG, Open Access Publishing Fund.

Acknowledgments

The authors would like to thank Thomas Wriedt and Yuri Eremin for organizing the Bremen workshop on light scattering 2016, and its participants for helpful and encouraging discussions. We thank Philipp Brenner and Jonathan Lehr for proof reading of the manuscript.

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Light scattering by oblate particles near planar interfaces: on the validity of the T-matrix approach

Abstract

1. Introduction

2. Particles near interfaces: the general procedure

3. Plane wave expansion of E_S (r) in the near field

4. First case study: five dielectric spheres

4.1. Five spheres in free space

4.2. Five spheres near a perfectly conducting substrate

5. Second case study: oblate spheroid

6. Discussion and conclusions

Appendix: Spherical and plane vector waves

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (22)

Optics Express

Abstract

1. Introduction

2. Particles near interfaces: the general procedure

3. Plane wave expansion of ES (r) in the near field

4. First case study: five dielectric spheres

4.1. Five spheres in free space

4.2. Five spheres near a perfectly conducting substrate

5. Second case study: oblate spheroid

6. Discussion and conclusions

Appendix: Spherical and plane vector waves

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (22)

Optics Express

3. Plane wave expansion of E_S (r) in the near field