Scattering by complex inhomogeneous objects: a first-order reciprocity method

É. Dieudonné; N. Malléjac; S. Enoch

doi:10.1364/OE.22.016558

1. Introduction

This paper addresses scattering by imperfections of optical components. The aim was to quantify the impact of defects, such as inhomogeneities of the materials or roughness, on the behavior of optical components. A rigorous treatment of the problem leads to numerical methods requiring demanding computations [1–3]. Thus, many approximate theories have been developed for simple geometries (rough interfaces, rough thin films...) and the interested reader is referred to [4] for a critical survey.

The scattering losses have been shown to be one of the main limiting factors of the performances of optical components [5]. Scattering losses have been thoroughly studied in the optical thin film domain from both theoretical and experimental points of view [6–8] and represent an important limiting factor of the performances of a device [5].

More complex geometries have also been investigated, such as gratings [9–11], including those for specific applications such as demultiplexers [12]. But, taking into account unavoidable inhomogeneities and roughness in an arbitrary optical component may lead to very large numerical problems. However, with the emergence of nanophotonics devices, this has become a topical subject of practical interest and can be taken into account on its own. As mentioned above, first-order approximate methods have proven their efficiency in the context of optical thin-film components and have recently been extended to magneto-dielectric materials [13] as well as to the case of diffraction by a magneto-dielectric inclusion [14]. We here propose to develop a numerical method based on first-order approximation and reciprocity to compute the first-order scattered field by inhomogeneities of complex optical components.

Within a first-order approximation, inhomogeneities can be seen as equivalent sources proportional to the ideal local field in the structure [13]. Thus the problem of computing the scattered field consists in calculating, first the ideal field without inhomogeneities, then the radiated field by the equivalent sources. For this second step, we propose to use the reciprocity theorem. Finally, the computational burden of evaluating the scattered field relies on local field calculations in the ideal structure illuminated by plane waves. Thanks to this two-step method, finite-element techniques such as any numerical method whose output is the local field can be used to efficiently evaluate scattering in any structure.

Section 2 presents RECY, a method based on reciprocity and first-order approximation. Section 3 describes an application of the method for a simple case: the inhomogeneous slab. Section 4 demonstrates a more complex case: the array. An application of the method to CMOS image sensors is finally presented in Section 5 as an industrial example.

2. Description of the method RECY

Consider an object made up of N materials with an arbitrary geometry according to Fig. 1. The permittivity ε and permeability μ may slightly fluctuate around a mean value for each material. Here, we investigate the scattered field due to these fluctuations.

Fig. 1 Schematic representation of an arbitrary object. The permittivity and permeability are given by Eqs. 2a and 2b, respectively.

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The materials are assumed to be linear and isotropic. The electromagnetic field that may exist in the presence of such structure is noted (E, H). The harmonic regime is considered with a temporal dependence exp(−ιωt), where ω = 2πc/λ (c and λ is respectively the speed of light and the wavelength in vacuum). The fields satisfy the Maxwell equations for each medium i (where i = 1 to N):

\nabla \times E_{i} (x, y, z) = ι ω {\tilde{μ}}_{i} H_{i} (x, y, z),

\nabla \times H_{i} (x, y, z) = - ι ω {\tilde{ε}}_{i} E_{i} (x, y, z) .

Fluctuating media have permittivities and permeabilities given by:

{\tilde{ε}}_{i} = ε_{i} [1 + p_{i} (ρ)],

{\tilde{μ}}_{i} = μ_{i} [1 + q_{i} (ρ)],

where,

ε_i = ε₀ε_i,r where ε₀ = (36π × 10⁹) ⁻¹ F/m is the vacuum permittivity and ε_r the relative permittivity of the medium;
μ_i = μ₀μ_i,r where μ₀ = 4π × 10⁻⁷ H/m is the vacuum permeability and ε_r the relative permeability of the medium;
p_i(ρ) is a function describing the permittivity fluctuation in medium i (p_i(ρ) = 0 if ρ ∉ V_i); and
q_i(ρ) is a function describing the permeability fluctuation in medium i (q_i(ρ) = 0 if ρ ∉ V_i).

At this step, the field is the total field (E, H) within the object. The scattered field is by definition the field created by the permittivity and permeability fluctuations. Thus, the total field is the sum of the ideal field (E⁰, H⁰) corresponding to the object without fluctuation (p_i(ρ) = 0 and q_i(ρ) = 0) and the scattered field (E^d, H^d):

(E, H) = (E^{0}, H^{0}) + (E^{d}, H^{d}) .

2.1. First-order approximation

Replacing Eqs. (2) and (3) in Eq. (1), assuming that p_i(ρ) and q_i(ρ) are much smaller than unity, and keeping only first-order terms ( $q_{i} H_{i}^{d}$ and $p_{i} E_{i}^{d}$ are second-order terms and thus are ignored), scattered field equations for fluctuating media are given [13] by:

\nabla \times E_{i}^{d} = ι ω μ_{i} H_{i}^{d} + M_{i},

\nabla \times H_{i}^{d} = - ι ω ε_{i} E_{i}^{d} + J_{i},

with

M_{i} = ι ω μ_{i} q_{i} H_{i}^{0},

J_{i} = - ι ω ε_{i} p_{i} E_{i}^{0} .

In Eq. (4), the source terms which results from the first-order approximation, are known and proportional to the ideal field (i.e. the field in the object without fluctuation). Thus, the inhomogeneous medium can be replaced by a homogeneous medium containing sources. These sources are the key element for the RECY method based on the reciprocity theorem.

2.2. Reciprocity theorem

The principle of reciprocity is given by a combination of electromagnetic fields existing in two configurations. Attributed to Lorentz [15] in 1896, many demonstrations of this theorem have been given [16]. It has been applied mainly with dipoles and the constraints on sources for an easy application have been clarified [17–26]. It is generally presented in an integral form for magnetic and electric sources:

∭_{V} (E^{b} \cdot J^{a} - H^{b} \cdot M^{a}) d ρ = ∭_{V} (E^{a} \cdot J^{b} - H^{a} \cdot M^{b}) d ρ .

In Fig. 2, the source in configuration a and the field in configuration b is in the medium 1. The source in configuration b and the field in configuration a is in the medium 2. As shown in the next section, to apply the principle to the calculation of the scattered field, special configurations a and b are considered.

Fig. 2 Illustration of the reciprocity principle for the electric sources only.

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2.3. Scattered field using first-order approximation and the reciprocity theorem: RECY

In configuration a, the source is an electric (or magnetic) dipole at ρ_d. It is oriented along a unit norm vector d and is given by:

J^{a} = - Il d δ (ρ - ρ_{d}),

M^{a} = Il d δ (ρ - ρ_{d}),

where Il is the amplitude of the dipole.

We have considered two cases according to the position of the dipole. In example of Section 5, we calculate the field in ideal structure created by a dipole ( $E_{i}^{dip}$ or $H_{i}^{dip}$ ), we can then obtain the scattered field at the position of this dipole. In Sections 3 and 4, the dipole is located at infinity and illuminate the ideal structure with the plane wave created by it. We thus obtain the scattered field at infinity.

In configuration b, the inhomogeneous medium i is replaced by a homogeneous medium containing sources M^b and J^b (located in V_i) obtained by the first-order approximation of Maxwell equations (see Section 2.1).

J^{b} = - ι ω ε_{i} p_{i} (ρ) E_{i}^{0} ℋ (ρ - V_{i})

M^{b} = ι ω μ_{i} q_{i} (ρ) H_{i}^{0} ℋ (ρ - V_{i})

where the Heaviside function ℋ (ρ − V_i) = 1 if ρ ∈ V_i and ℋ (ρ − V_i) = 0 if ρ ∉ V_i.

We obtain Eq. (10), the electric scattered field created by a slightly permittivity fluctuation of the medium i at ρ_d by replacing Eqs. (6) and (8) in the reciprocity theorem (5) (in this case M = 0 in both configurations). Respectively, we obtain Eq. (11), the magnetic scattered field created by a slightly permeability fluctuation of the medium i at ρ_d by replacing Eqs. (7) and (9) in Eq. (5) (with J = 0).

E^{d} (ρ_{d}) \cdot d = ι ω ε_{i} ∭_{V_{i}} p_{i} (ρ) E_{i}^{dip} (ρ) \cdot E_{i}^{0} (ρ) d ρ

H^{d} (ρ_{d}) \cdot d = ι ω μ_{i} ∭_{V_{i}} q_{i} (ρ) H_{i}^{dip} (ρ) \cdot H_{i}^{0} (ρ) d ρ

The advantage of (10) and (11) is their simplicity. The fields in the integrals may be obtained by any analytical or numerical method before integration. Most numerical methods (finite-element, integral equation,...) coupled with RECY may be efficiently used to estimate the scattered field due to inhomogeneities without meshing the inhomogeneities. Moreover, once the calculation of the ideal field has been done, the first-order scattered field of any fluctuation can be evaluated by a simple integration.

3. Simple application: case of the slab

Consider a slab infinite in (x, y) of slightly fluctuating permittivity and permeability in air as shown in Fig. 3.

Fig. 3 Schematic representation of the studied media. k is the wavevector and it is defined by the two angles θ (normal angle) and ϕ (polar angle) as well as its norm k. Layers 1 and 3 are infinite medium air (ε_r = 1, μ_r = 1). Layer 2 of slightly fluctuating permittivity and permeability (ε̃, μ̃) has a thickness e.

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In this case Maxwell’s equations can be solved analytically to obtain the field in the ideal structure. To do so, we need to work in the Fourier space, and consequently we must demonstrate the principle of reciprocity in this space.

3.1. Reciprocity theorem in the Fourier space

Equations (12a) and (12b) define the 2D Fourier transformations used in this paper for an arbitrary vector X. The Fourier transformation is defined for temperate distributions. This is the case of our sources: Dirac and Heaviside generalized functions (see Eqs. (6), (7), (8) and (9)).

X (r, z) = \int_{σ} \hat{X} (σ, z) exp (ι σ \cdot r) d σ,

\hat{X} (σ, z) = \frac{1}{4 π^{2}} \int_{r} X (r, z) exp (- ι σ \cdot r) d r .

The choice of the 2D transform is due to obvious discontinuities along the z-direction (a 3D Fourier transform would have complicated the analytic part of calculation). So r and σ are two-dimensions vector.

σ is the projection of the wave-vector k in the plan (xOy), it has two components: σ_x = k₁ sinθ cosφ and σ_y = k₁ sinθ sinφ. The two angles θ and φ are defined in Fig. 3 and the amplitude k₁ is the amplitude of the wave-vector in the air k₁ = 2π/λ.

The wave-vector is not same in each medium i of refractive index n_i, it is given by: k_i = σ_xx + σ_yy + α_iz, where $α_{i} = {[k_{i}^{2} - (σ_{x}^{2} + σ_{y}^{2})]}^{1 / 2}$ and k_i = 2πn_i/λ.

α_i can be imaginary or real, it has no consequences on the establishment of the reciprocity theorem in Fourier space. Moreover, only medium with refractive index n_i ≥1 are studied and the scattered field are calculated at infinity for angle θ = [0, 90]°, so the α_i is always real.

The derivation of a reciprocity theorem in the Fourier space (σ, z) may now be established: Assuming that (E^a, H^a) and (E^b, H^b) correspond to the field generated by the monochromatic sources (J^a, M^a) and (J^b, M^b) respectively, it satisfies Maxwell’s equations for generalized functions. In the case of the isotropic homogeneous slab infinite in (x, y), the permittivity ε and permeability μ are only functions of z in the whole space. In the Fourier space, the electric and magnetic field have the same dependence in z. Consequently, Maxwell equations can be written for both configurations:

k \times {\hat{E}}^{a} = {\hat{M}}^{a} + ι ω μ {\hat{H}}^{a},

k \times {\hat{H}}^{a} = - {\hat{J}}^{a} - ι ω ε {\hat{E}}^{a},

k \times {\hat{E}}^{b} = {\hat{M}}^{b} + ι ω μ {\hat{H}}^{b},

k \times {\hat{H}}^{b} = - {\hat{J}}^{b} - ι ω ε {\hat{E}}^{b},

The following operations were performed:

{\hat{H}}^{b} \cdot (13 a) + {\hat{H}}^{a} \cdot (14 a) + {\hat{E}}^{b} \cdot (13 b) + {\hat{E}}^{a} \cdot (14 b),

by using the property (16):

(C \times A) \cdot B + (C \times B) \cdot A = 0,

to obtain:

{\hat{E}}^{b} \cdot {\hat{J}}^{a} + {\hat{H}}^{b} \cdot {\hat{M}}^{a} = {\hat{E}}^{a} \cdot {\hat{J}}^{b} + {\hat{H}}^{a} \cdot {\hat{M}}^{b},

and finally integrating over z, we obtain :

\int_{z} ({\hat{E}}^{b} \cdot {\hat{J}}^{a} + {\hat{H}}^{b} \cdot {\hat{M}}^{a}) d z = \int_{z} ({\hat{E}}^{a} \cdot {\hat{J}}^{b} + {\hat{H}}^{a} \cdot {\hat{M}}^{b}) d z .

Equation (18) is the integral form of reciprocity theorem in Fourier space.

3.2. RECY for slab in Fourier space

As the slab in Fig. 3 is infinite in (x, y), plane waves are a natural basis to illuminate the slab. So, the configuration a is chosen with a dipole located at infinity. In Fourier space, the dipole generates a plane wave with an amplitude given by the Green’s function.

{\hat{E}}_{0}^{a} (σ, z) = \frac{1}{4 π^{2}} Il e^{- ι σ \cdot r_{\infty}} \frac{ω μ_{0}}{2 α_{1}} e^{- ι α_{1} z_{\infty}} \bar{Γ} d e^{ι α_{1} z}

where Γ̄ is given by :

\bar{Γ} = (\begin{matrix} 1 - \frac{σ_{x}^{2}}{k_{1}^{2}} & - \frac{σ_{x} σ_{y}}{k_{1}^{2}} & - \frac{σ_{x} α_{1}}{k_{1}^{2}} \\ - \frac{σ_{y} σ_{x}}{k_{1}^{2}} & 1 - \frac{σ_{y}^{2}}{k_{1}^{2}} & - \frac{σ_{y} α_{1}}{k_{1}^{2}} \\ - \frac{α_{1} σ_{x}}{k_{1}^{2}} & - \frac{α_{1} σ_{y}}{k_{1}^{2}} & 1 - \frac{α_{1}^{2}}{k_{1}^{2}} \end{matrix}) .

Using linearity of Maxwell equations, we can illuminate the slab by a unit norm plane wave oriented along Γ̄d and then multiply the resulting field by the amplitude of the plane wave generated by the dipole

\frac{1}{4 π^{2}} Il exp (- ι σ \cdot r_{\infty}) \frac{ω μ_{0}}{2 α_{1}} exp (- ι α_{1} z_{\infty}) .

The field in the slab for this illumination is noted ${\hat{E}}_{2}^{a}$ and is given by the homogeneous solution of the Maxwell equations.

In configuration b, the inhomogeneous slab is replaced by a homogeneous slab containing sources M^b and J^b obtained by the first-order approximation of Maxwell equations.

According to Eq. (18), the scattered field Ê^b(σ, z) must be calculated at the same localization of the source in configuration a so it must be calculated at z_∞. Thus the scattered field become ${\hat{E}}_{1}^{b} (σ, z_{\infty}) = {\hat{A}}_{1}^{b} (σ) exp (- ι α_{1} z_{\infty})$

Applying the reciprocity theorem, we then obtain the amplitude ${\hat{A}}_{1}^{b} (σ)$ :

{\hat{A}}_{1}^{b} (σ) \cdot d = ι ω ε \frac{ω μ_{0}}{2 α_{1}} \hat{p} (σ - σ_{0}) \int_{z = 0}^{e} {\hat{E}}_{2}^{a} (σ, z) \cdot {\hat{E}}_{2}^{0} (σ_{0}, z) d z

3.3. Validation

A numerical code calculating the scattered fields based on the method described in Section 2 has been developed. The validation was performed by comparing results obtained with this code and those given by the exact calculation of COMSOL Multiphysics^® [27].

Consider a single layer illuminated by a plane wave of wavelength λ = 7.5 cm, in air, as shown in Fig. 4. The incident electric field is S polarized, so E_x = E_y = 0 and E_z ≠ 0. The length and the thickness of this layer are respectively L = 10λ for COMSOL calculation and L = ∞ for RECY and e = 2.7 × 10⁻²λ for both. The relative homogeneous permittivity of the layer is ε_r = 5 and the permeability is μ_r = 2.

Fig. 4 Schematic representation of the considered medium in S polarization. Here, e is the thickness, λ is the wavelength, L is the length of the layer and θ is the observation angle between 0 and 90.

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Two cases are studied:

Case I (cf. Fig. 5(a)), a relative permittivity fluctuating with p(x): ε̃_r = ε_r(1 + p(x)), μ̃_r = μ_r,
Case II (cf. Fig. 5(b)), a relative permeability fluctuating with p(x): μ̃_r = μ_r(1 + p(x)), ε̃_r = ε_r,

where p(x) is given by Fig. 6 in both case.

Fig. 5 Comparison between the first-order method and the exact calculation.

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Fig. 6 Representation of the function p(x) in [−0.375, 0.375], p(x) = 0 if x ∉ [−0.375, 0.375].

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Between 0° and 70°, the first-order method is valid to 16% in case I and to 6% in case II. Obviously, the relative variation of the optical index n = (ε_rμ_r)^1/2 is more important in the first case than in the second. Thus, the obtained result shows mainly that the first-order method fails when the relative variations of the optical index are too important. When the variation is low enough, (typically in the second case), the first-order method gives accurate results. The remaining difference in this case can be explained by the finite length of the layer in the COMSOL^® case (exact calculation) while the first-order method assumes an infinite layer.

4. A more complex case: array

An array of square section cylinders has been considered (cf. Fig. 7). It is infinite in y-direction (2D array). It is constituted by a 35-square of permittivity ε_c = 15, permeability μ_c = 1, and length e = 0.03 m. The cylinders are distributed periodically along the x-direction of period p_r = 0.09 m.

Fig. 7 Configuration for the application example: a 35-square cylinder, width e = 0.03 m, period p_r = 0.09 m, permittivity ε_c = 15, permeability μ_c = 1, S polarization, normal incidence i₀ = 0, wavelength λ = 1 m.

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This structure could be seen as a generic example of periodic structures, such as photonic crystals, metamaterials, etc. The impact of a defect localized in this array has been investigated.

This array is illuminated by a normal incident plane wave (i₀ = 0) at wavelength λ = 1 m in S polarization, so E_x = E_z = 0 and E_y ≠ 0. In this case, the reciprocity principle can be written in the following form:

E_{y}^{d} (i_{0}, θ) = ι ω ε_{c} E_{dip}^{2 D} \tilde{p} \int_{x_{c}}^{x_{c} + e} \int_{z_{c}}^{z_{c} + e} E_{c y}^{a} (x, z, θ) E_{c y}^{0} (x, z, i_{0}) d x d z

where:

$E_{c y}^{0}$ is the homogeneous field in the structured material in the incidence direction i₀
$E_{c y}^{a}$ is the homogeneous field in the structured material in the incidence direction θ
p̃ is the percentage of permittivity fluctuations in the square cylinder,
$E_{dip}^{2 D} = \frac{ω μ}{4 π} \sqrt{λ}$ is the amplitude of the plane wave in 2D.

A numerical code calculating the scattered field based on the method described in Section 2 has been developed. It was the validated by comparing results obtained by this code with those given by the exact calculation of COMSOL Multiphysics [27]. This method was applied for the geometry presented in Fig. 7.

Figure 8 shows the scattered field for the case where the permittivity of the defect square cylinder was 10% higher than for the other square cylinders.

Fig. 8 Comparison between reciprocity and exact calculation - Localized property defect with a 10% higher permittivity on one cylinder.

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According to results presented in Fig. 8 the method of reciprocity predicts the scattered field with an excellent agreement (the average difference between both calculation was 1.59%). However, the main advantage of this reciprocity method is that it only requires a field calculation in the ideal structure. This could be reduced to one single period for periodic structures, leading to a greatly decreased computation burden, especially for 3D periodic structures. The scattered field can thus be deduced for any spatial defect distributions.

For other permittivities and other percentages of permittivity fluctuation, this method appears to be quite robust (cf. Table 1).

Table 1. The average difference between RECY and exact calculation for differents permittivities ε_c and differents perecentage of defect.

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The error between exact calculation and our method increases quite fast with the property fluctuation. However, this error still remains sufficiently low for an engineer use even for fluctuation as large as a factor two. Such result is unexpected for a first-order approximation. One may ask why multiple scattering does not impact more significantly the results. It is important to emphasize that what is called Born approximation is usually applied in case where the field acting on the fluctuation is approximated by the incident field. In the first-order approximation we use [8, 13] the field acting on the fluctuation is the total field in the ideal structure (which can only be calculated numerically in most of the case). This field takes into account the interactions of the different inclusions as if these were all identical, which is already a significant part of multiple interaction. Only the interactions of the scattered field with the defect himself are then missed in this approximation: this term may grow significantly if there were many defects on which this scattered field could interact in the geometry.

5. Reciprocity application - CMOS image sensors

To illustrate the versatility of our method, the principle of reciprocity was applied to a CMOS [28] structure, shown in Fig. 9. On this kind of devices, each CMOS pixel has to be independent from the others, and it is of interest to evaluate how a defect on one of them would induce signals on the others. This known effect, called cross-talking, has important consequences on the performances of the devices.

Fig. 9 Two-dimentional CMOS pixel model with layer materials.

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Each pixel has a width of L = 1.75 μm and is constituted of the layer stack describe in Table 2.

Table 2. Height and permittivity of the components of the CMOS structure.

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The permittivity of the color filters is given by:

ε_{r}^{filter} = 2.25 - 10 ι | \frac{λ_{f} - λ}{λ} |

where λ is the studied wavelength, and λ_f is the corresponding wavelength of the filter:

λ_f = 450 nm for a blue color filter,
λ_f = 555 nm for a green color filter,
λ_f = 650 nm for a red color filter.

We studied the structure of the green wavelength λ = 555 nm, and choose to take as a defective microlens the one above the green filter, for which the permittivity is given as:

ε_{r}^{deflens} = ε_{r}^{lens} (1 + p (x)),

with

ε_{r}^{lens} = 2.56

and

p (ρ) = 0.1 sin (\frac{π x}{L})

.

It was of interest to compute the scattered field at the bottom of each pixel (averaged on the width). Thus a scattered near-field computation has to be conducted. Through our method bosed on the reciprocity theorem, it was possible to obtain the scattered field at point s using the calculation of the field in the ideal structure illuminated by a dipole oriented along a unit norm vector d at s. The scattered field E^d(ρ_s) is given by (25).

E^{d} (ρ_{s}) \cdot d = ι ω ε_{r}^{lens} ∭ p (ρ) E^{dip} (ρ) \cdot E^{0} (ρ) d ρ

where

p(ρ) is a function describing the defect,
E^dip(ρ) is a field create by the dipole in the ideal structure,
E⁰(ρ) is a field in the ideal structure illuminated by the incident wave.

The finite element package COMSOL can give us the transmission t across the silicon layer below each pixel. So to validate our method we have to calculate the same transmission. To do so the scattered filed on a set of dipoles placed in the silicon layer has been calculated. The diffracted light was computed by a defect on the central pixel (more precisely on the lens) and provided the amplitude of the diffracted light at the bottom of each pixel as a measure of the cross-talk induced by the defect (cf. Table 3).

Table 3. Scattered transmission of the defective CMOS structure.

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We can notice that the RECY gives a quite good estimation of the cross-talking. Besides, it has the advantage that any kid of defect can now be estimated without performing another finite element calculation: the fields of the ideal structure once obtained can be reused to estimate with Eq. (25) the effect of other function p(ρ), i.e. any other defect in structure.

6. Conclusion

By using a first-order approximation and the reciprocity theorem, we have shown that a scattered-field calculation can be obtained with only one ideal field computation in this structure for each incident angles. The developped method enables us to handle complex structures and evaluate the effects of defects and inhomogeneities. Any numerical code, including commercial packages such as COMSOL, could be used to apply the method, wherefore it may be applied in various areas such as integrated photonics, optical telecommunications, etc. As shown in the chosen examples, near and far diffracted fields can be studied. A parametric investigation of defects can easily be conducted as the method will require only one computation of the local field in the defectless structure. For periodic structures, the method enables to drastically reduce the required numerical resources by restricting the computational domain to a single period.

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Defect	ε = 1.3	ε = 7	ε = 15
+10%	0.36%	0.50%	1.59%
+20%	0.57%	1.60%	2.86%
+30%	0.75%	2.55%	4.72%
+40%	0.93%	3.51%	6.55%
+50%	1.12%	4.42%	8.31%
+60%	1.31%	5.45%	10.10%
+70%	1.49%	6.41%	11.82%
+80%	1.68%	7.37%	13.48%
+90%	1.87%	8.32%	15.10%

Layer	Height	Permittivity
Microlens	0.4 μm	2.56
Silicon dioxide	0.4 μm	2.1316
Color filter	0.9 μm	$ε_{r}^{filter}$ (23)
Silicon nitride	0.2 μm	4
Silicon dioxide	2.5 μm	2.1316
Silicon	0.1 μm	16

Color	\|t\|	\|t\| RECY	Relative difference %
Blue	1.95 × 10⁻²	1.69 × 10⁻²	13.37
Green	14.64 × 10⁻²	13.75 × 10⁻²	6.12
Red	1.95 × 10⁻²	1.69 × 10⁻²	13.28

Defect	ε = 1.3	ε = 7	ε = 15
+10%	0.36%	0.50%	1.59%
+20%	0.57%	1.60%	2.86%
+30%	0.75%	2.55%	4.72%
+40%	0.93%	3.51%	6.55%
+50%	1.12%	4.42%	8.31%
+60%	1.31%	5.45%	10.10%
+70%	1.49%	6.41%	11.82%
+80%	1.68%	7.37%	13.48%
+90%	1.87%	8.32%	15.10%

Layer	Height	Permittivity
Microlens	0.4 μm	2.56
Silicon dioxide	0.4 μm	2.1316
Color filter	0.9 μm	$ε_{r}^{filter}$ (23)
Silicon nitride	0.2 μm	4
Silicon dioxide	2.5 μm	2.1316
Silicon	0.1 μm	16

Scattering by complex inhomogeneous objects: a first-order reciprocity method

Abstract

1. Introduction

2. Description of the method RECY

2.1. First-order approximation

2.2. Reciprocity theorem

2.3. Scattered field using first-order approximation and the reciprocity theorem: RECY

3. Simple application: case of the slab

3.1. Reciprocity theorem in the Fourier space

3.2. RECY for slab in Fourier space

3.3. Validation

4. A more complex case: array

5. Reciprocity application - CMOS image sensors

6. Conclusion

References and links

Cited By

Figures (9)

Tables (3)

Equations (33)

Optics Express