## Abstract

We present numerical experiments of light scattering by a circular dielectric cylinder embedded in a stratified background, using the Green’s tensor technique. The stratified background consists of two or three dielectric layers, the latter forming an anti–reflection system. We show movies of the scattered field as a function of different parameters: polarization, angle of incidence, and relative position of the cylinder with respect to the background interfaces.

©2001 Optical Society of America

## 1 Introduction

The optical properties of cylindrical particles have generated a lot of interest, as they provide a simple model for specific physical systems. The scattering on a cylinder situated in an infinite homogeneous background was for example already treated by Lord Rayleigh in 1881 [1].

However, most real situations are more complex, the particles are located on a substrate or embedded in an even more complicated stratified background. To investigate such geometries many analytical and numerical approaches were developed. They include, for example, image methods [2–4], applications of the extinction theorem [5, 6], expansions of cylindrical waves [7, 8] or integral–equation methods [9–13]. Most of these methods only consider a background medium consisting of two half spaces [2–11] or restrict the choice of the material or geometry parameters [2–5, 7].

Recently, we presented a general technique for light propagation and scattering in two–dimensional (2D) structures formed by a stratified background with embedded scatterers [14]. This approach is based on the Green’s tensor associated with the stratified background and provides a rigorous solution of the vectorial wave equation with the boundary conditions given at the different material interfaces.

In this paper we apply our approach to study the light scattered by a dielectric circular cylinder in the presence of a stratified dielectric background. We investigate the different phenomena that can occur, depending on the relative cylinder position with respect to the background interfaces, as well as on the illumination direction.

In Sec. 2 we briefly recall our computational approach and in Sec. 3 we show calculations of the total electric field amplitude $A=\sqrt{\mathbf{E}\xb7{\mathbf{E}}^{*}}$ as a function of different geometrical and illumination parameters. Our results are summarized in Sec. 4.

## 2 Model

Throughout the paper we consider a circular cylinder with radius
*r*=150 nm and permittivity
*ε*
_{Cyl}=2 embedded in a two– or three–layer
background. This system is illuminated from the top with a plane wave under
different angles of incidence Θ. The illumination wavelength is
λ=633 nm and two different polarizations are considered: s polarization,
with the electric field **E**
^{0} in *y* direction,
and p polarization with the electric field **E**
^{0} lying in the
*xz* plane.

Since the scattering geometry under study is translation invariant in
*y* direction we can restrict the investigation to a cross section in
the *xz* plane [14]. When such a system formed by a stratified background
(*L* layers, permittivity
_{εl}
, *l*=1, …, *L*) with embedded
scatterers is illuminated with an incident field
**E**
^{0}(**r**) propagating in the stratified background
[we consider harmonic fields with time dependence
exp(-*iωt*)], the total field
**E**(**r**) is given by the electric field integral equation:

where **G**(**r**, **r′**) is the
Green’s tensor associated with the stratified background,
${k}_{0}^{2}$=*ω*
^{2}
*ε*
_{0}
*µ*
_{0}
the vacuum wave number and Δ*ε*(**r**)
the dielectric contrast:

Whereas the Green’s tensor for an infinite homogeneous background can be
expressed analytically [15], this is not possible when the background is stratified. In
this case **G**(**r**, **r′**) must be
computed numerically with Sommerfeld type integrals [14].

Since the integration in Eq. (1) runs only over the scatterer section A, the discretization is limited to this volume, the stratified background being accounted for by the Green’s tensor. Another advantage of this method is that the boundary conditions at the different material interfaces, as well as at the edges of the computation window are perfectly and automatically fulfilled [14, 15].

## 3 Results

#### 3.1 Homogeneous space

Before studying complex effects that can arise when the cylinder is placed in a
stratified background, it is educational to review the simpler case of the
scattering by a cylinder in an infinite homogeneous space
*ε*
_{H}=1. Figure 1 shows the total electric field amplitude in such
a system for both s and p polarizations, when the cylinder is illuminated with
an incident electric field **E**
^{0} propagating in
-*z* direction:

where **A**
^{0} is the amplitude vector with
|**A**
^{0}|=1,
**k**=(*k*_{x}
,
*k*_{z}
) is the wave vector of the incident wave and
*k*_{z}
=-|**k**|.
(For all the calculations we normalize the electric field amplitude of the
incident field
|**A**
^{0}|=(**E**
^{0}·**E**
^{0}
^{*})^{1/2}=1.)

Some features of the field distributions in Fig. 1 are common to both polarizations: in backward
direction stationary waves are created due to the interference of the incident
and the reflected waves. In the forward direction we recognize the diffraction
pattern of the cylinder. However, the near field distribution at close vicinity
of the cylinder strongly depends on the polarization of the incident field. For
s polarization a maximum of the intensity can be observed inside the cylinder
whereas for p polarization two peaks occur on the left and right sides of the
scatterer. This behavior can be explained with the boundary conditions resulting
from Maxwell’s equations. When the electric field **E** is
parallel to a material interface it must be continuous [16]. When it is normal to the interface, it is the
displacement field **D**=*ε*
**E**
which must be continuous. Hence, for a p-polarized excitation the total electric
field amplitude is discontinuous by a factor*
ε*
_{Cyl}/*ε*
_{H}
wherever the incident field is normal to a material interface (i.e. on the left
and right sides of the cylinder in Fig. 1(b)). Such depolarization effects do not occur at
the top and bottom sides of the cylinder for p polarization [Fig. 1(b)] or on the entire circumference for s
polarization [Fig. 1(a)], since in those locations the incident field
is parallel to the material interfaces [17].

As a consequence of the depolarization effects for a p-polarized excitation, the
forward scattered light is focussed by the cylinder and reaches a maximum at
*z*≈-300 nm. For s polarization, on the other
hand, the field maximum is situated inside the cylinder.

The total scattering cross section of a dielectric circular cylinder is greater for s polarization than for p polarization, the difference increasing with a larger dielectric contast [18]. As a matter of fact, we observe in Fig. 1(a) a more pronounced interference pattern than in Fig. 1(b). This polarization dependence of the total scattering cross section is also observable for all further configurations under study.

#### 3.2 Scattering in the presence of a surface

Now we divide the background into two half spaces
*ε*
_{1}=1
(*z*≥0) and
*ε*
_{2}=2 (*z*<0),
in such a way that the lower layer has the same permittivity as the cylindrical
scatterer,
*ε*
_{2}=*ε*
_{Cyl}.

The movies in Fig. 2 show the response of the system under normal
incidence (Θ=0) for s- and p-polarized illumination. Each movie frame
corresponds to a given distance *h* between the cylinder center
and the material interface.

Since the incident wave is now reflected back at the plane interface, the illumination electric field becomes in the top layer

and in the bottom layer

where *k*_{iz}
, *i*=1, 2, is the
*z* component of the wave vector in layer *i*,
*R* is the Fresnel reflection coefficient and
*T* the corresponding transmission coefficient for normal
incidence [19]:

Note that *R* and *T* are real for absorptionless
media. The negative reflection coefficient obtained in our system leads to a
phase shift of 180° for the reflected field.

The illumination amplitude in the upper half space reads now

and a stationary wave with period
Δ=*π*/*k*
_{1z}=λ/2
is created in the upper half space (Fig. 2). Because of *R*<0 the
amplitude is minimal directly at the interface.

The cylinder is not anymore illuminated by a homogeneous plane wave but moves in the field of the stationary wave. Hence, in the top layer the total field distribution is a superposition of the stationary wave and a diffraction pattern similar to that in Fig. 1. This pattern determines also the field distribution in the bottom layer depending on the transmittance of the interface.

As in the case of the homogeneous background the near field of the cylinder
depends on the polarization of the excitation. Additionally, the location and
strength of the scattered field maxima are modulated by the cylinder altitude
*h*.

The stationary wave itself is further amplified in the region between the cylinder and the interface. This is caused both by the focusing provided by the cylinder and by multiple reflections between the surface and the cylinder. Note also that the interaction between the excitation and the field scattered by the cylinder creates an interference pattern in x direction.

For *h*=-15 nm, when half of the cylinder is included in the
surface, most of the back-scattering disappears in a narrow sector of the upper
halfspace. Finally, for *h*=-165 nm the cylinder is entirely
immersed in the lower layer with identical permittivity, and the total electric
field is simply given by the excitation in the stratified background. Since
without the cylinder s and p polarization are indistinguishable for normal
incidence, the field distributions are identical in both frames.

Let us now rotate the angle of incidence Θ with respect to the surface
normal. Figure 3 shows again the scattering as a function of the
cylinder altitude *h*, but for Θ=30°.

In this case, the illuminating electric field in the top layer becomes

where the reflection coefficients correspond to the polarization of the given
component: *R*_{x}
=-*R*
^{p},
*R*_{y}
=*R*
^{s}, and
*R*_{z}
=*R*
^{p}. In this
case of non–normal illumination, the Fresnel coefficients
*R*
^{s/p} depend on the polarization of the incident
wave [19]:

The illumination amplitude in the upper half space is then

$$+{A}_{z}^{{0}^{2}}\left[1+{R}^{{p}^{2}}+{2R}^{p}\mathrm{cos}\left({2k}_{1z}z\right)\right]{\}}^{\frac{1}{2}},\phantom{\rule{.2em}{0ex}}z\ge 0.$$

Again, a stationary wave excites the cylinder, but the period is now increased to
Δ=*π*/*k*_{z}
=λ/(2
cosΘ)=λ/√3.

The diffraction patterns and position of the near field maxima rotate accordingly to the change of the incidence angle (compare Figs. 2 and 3). For p polarization, one of the two field maxima previously visible on the lateral sides of the cylinder (Fig. 1) disappears and only a maximum is now visible on the right, in the direction of the illumination momentum (Fig. 3).

The computer experiments reported in Fig. 3 also illustrate the polarization dependence of the
boundary conditions at the interface between the two layers: for s polarization
the electric field must be continuous, whereas p polarization enforces a
discontinuity proportional to the ratio
*ε*
_{1}/*ε*
_{2}.

To further investigate the effect of the angle of incidence Θ, we show
in Fig. 4 movies for a fixed cylinder altitude
*h*=615 nm and a varying illumination angle Θ.

As expected, the period of the stationary wave increases for larger angles
Θ and the diffraction pattern becomes more and more twisted (Fig. 4). Since the reflection coefficients
*R*
^{s/p} of the interface approach 1 in the limit
Θ=90°, the field transmitted into the lower half space
vanishes for larger angles and the amplitude maxima of the stationary wave
converge towards their upper value for total reflection *A*=2
(see Eq. (12) with *k*_{z}
→0
and *R*
^{s/p}→1). This produces a strong
scattering both in forward and in backward direction when the maximum of the
stationary wave coincides with the cylinder altitude
(Θ=75°, Fig. 4).

At first sight, it may be surprising that in the case of p polarization the stationary wave in the top layer vanishes for Θ≈40°-60°, and only the field scattered by the cylinder can be observed [Fig. 4(b)]. Depending on the angle, two different mechanisms contribute to this behavior. First, for Θ=45° the x and z components of the p-polarized illumination electric field are identical, ${A}_{x}^{0}$ =${A}_{z}^{0}$ , and Eq. (12) becomes

Hence, the *z* dependence of
**E**·**E*** is cancelled out.

To illustrate the second mechanism we report in Fig. 5 the amplitude Arefl reflected by a plane interface
between two half spaces *ε*
_{1}=1 and
*ε*
_{2}=2, as a function of the angle of
incidence Θ. At Θ≈55°, the Brewster
angle of the material system under study, *A*
_{refl}=0.
At this angle, the reflectivity vanishes for p-polarized plane waves and all the
energy of the incident field is transmitted into the lower half space. Note
however that zero reflectivity does *not* imply that the
amplitude of the transmitted light *T* in the bottom layer is 1 [19]. A straightforward calculation gives $T=\sqrt{{\epsilon}_{1}\u2044{\epsilon}_{2}}$, which yields in our case *T*≈0.75,
in perfect agreement with our calculations. Both above–mentioned phenomena do
not exist for s polarization.

In Fig. 5 we can further observe that on the entire angle range s polarization leads to a larger reflectivity. This behavior is similar to the polarization dependence of the scattering on a cylinder (see Sec. 3.1).

#### 3.3 Anti–reflection layer

In the following, we add a third layer to create a structure with
*ε*
_{1}=1,
*ε*
_{2}=1.334, and
*ε*
_{3}=1.78. This structure represents an
air/water system, separated by an intermediated slab acting as anti–reflection
layer [19]. For normal incidence on the material interfaces, the
reflectivity is indeed strictly zero only if the permittivities are connected by

and the thickness *d* of the slab is

Taking *n*=2 in Eq. (15), we use for our calculations
*d*=411 nm, so that the cylinder fits within the antireflection
layer.

The movies in Fig. 6 clearly illustrate that, independent of the
cylinder location, there is no stationary wave in the top layer due to
reflection of the illuminating field by the slab. Striking in these movies is
the fact that the antireflection layer remains very efficient, even in the
presence of the cylinder. Again, it is the s–polarized illumination which
produces a weak interference pattern. To explain this behavior, in Fig. 5 we also report the reflected amplitude
*A*
_{refl} for the three–layer system under study.
Similar to the case of the single interface, s polarization leads to a stronger
scattering for all angles Θ and, consequently, to an enhanced
backreflection of the field components scattered by the cylinder and impinging
at non–normal incidence on the interface. Note also that
*A*
_{refl}=0 for Θ=0° since we
designed the slab to be reflectionless. Further, in Fig. 5 we observe that for p polarization the Brewster
angle is slightly shifted to Θ≈53°, and the
corresponding reflected amplitude does not vanish completely for that
three–layers system.

## 4 Summary

When a cylindrical particle is located in a stratified background, the resulting electric field distribution is determined by the interaction between the cylinder and the different material interfaces. In this paper, we used the Green’s tensor technique to illustrate the scattering phenomena occurring in such structures. The Green’s tensor technique is very well adapted to this kind of geometry because the optical processes in the layered material system are accurately described by the Green’s tensor. Further, only the cylinder needs to be discretized, irrespective of the number of layers forming the stratified background. Our calculations illustrate the complexity of the phenomena that can occur in such a rather simple system, depending on the relative position of the cylinder to the different material interfaces.

## Acknowledgments

It is a pleasure to thank B. Michel (IBM) for his support of the project. We gratefully acknowledge funding from the Swiss National Science Foundation.

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