## Abstract

In this study, we propose a two-dimensional (2D) dielectric structure tailored by a systematic design approach on the exit side of a metallic aperture to enhance the off-axis electromagnetic (EM) wave. We adopted a phase field method based topology optimization scheme and designed an arbitrary 2D dielectric structure in order to steer outward beaming through an aperture to a specific direction. Beyond previous one-dimensional structure, we proposed an arbitrary 2D dielectric structure through the introduced design process defining not only *x*- but also *y*-directional dielectric structural boundaries simultaneously and experimentally confirmed enhanced EM wave transmission to a desired direction.

© 2014 Optical Society of America

## 1. Introduction

Since the extraordinary transmission of light was measured experimentally through subwavelength hole arrays fabricated in thin metal films [1], intensive researches have been triggered for theoretical and experimental studies on optical field diffraction through a subwavelength aperture. The work by Lezec et al. [2] showed that an enhanced and directional beaming from subwavelength apertures is allowed via the excitation of surface plasmons on corrugated metallic surfaces. Although EM radiation cannot induce the surface plasmon modes on a smooth metallic surface, the presence of periodic grating structure on the flat surface of a perfect conductor provides the surface bound states which mimic the surface plasmon polaritons of a real metal [3]. Optical components have been also used for the shaping of anomalous reflected beams [4] and it is expansible to metamaterial design for the purpose of reducing the radar cross section for stealth applications [5].

The directional beaming effect through a subwavelength metallic aperture combined with dielectric gratings was also demonstrated [6] and such dielectric gratings were designed to couple the surface plasmon modes induced by the metallic aperture with surrounding medium along a specific off-axis direction [7]. Similar phenomena in the microwave spectrum have been studied for the subwavelength aperture surrounded by using metallic [8] or dielectric [9] grating on the exit side. The emitted beam can be confined to an off-axis specific region when the parameters of the grating structure are appropriately selected.

Regarding the redirection of the output beam, the properties related with its direction and intensity are deeply dependent on geometric factors. The geometrical factors that influence the optical properties of the dielectric corrugations are numerous such as grating period, the aspect ratios and shape of the exit side of a single aperture, profile of the corrugations and so forth. Recent results show that gradient-based 2D design methods [10] were applied to optical device design that hides a cylinder by finding the optimal permittivity distribution [11] or to an EM wave cloaking device composed of a single dielectric medium [12]. To our knowledge any applications of 2D design methodology to structure design for an off-axis beaming have not been reported in scientific literature so far.

In this study, we introduce a systematic approach for designing a 2D structure composed of
conventional isotropic dielectric media on the exit side of a single subwavelength aperture to
enhance the off-axis EM wave. Figure 1 explains the
concept of this study schematically. Beyond previous parametric studies [6–9] which can handle a couple of
parameters at a time for one-dimensional regular or asymmetric grating design [Fig. 1(a)], we propose an arbitrary 2D dielectric structure as
schematically described in Fig. 1(b) through a design
process defining not only *x*- but also *y*-directional dielectric
structural boundaries including air region simultaneously. Finally, we experimentally confirmed
that the systematic design result makes the exit beam steer to a desired direction.

## 2. Numerical analysis modeling

We compute the scattered field in a simulation region composed of a subwavelength aperture in a metal surface having a perfect electrically conducting (PEC) condition and dielectric gratings on the exit surface. Figure 2 shows FE analysis computation area composed of a PEC metal slit surrounded by dielectric grating design area. The electromagnetic finite element (FE) simulations were carried out for normal incident transverse electric (TE)-polarized Gaussian beam in the X-band (10GHz) frequency. The design area is composed of a rectangular area with 180mm width and the thickness of 4.8mm designated as a dotted rectangular. The measuring region is defined as 30mm square area whose center is located 40mm below and 25mm left from the bottom center of the aperture.

The scattered field is governed by time-harmonic Maxwell’s equation and it is formulated as a Helmholz equation for the TE-polarized wave assuming a nonmagnetic metal condition (${\mu}_{r}\approx 1.0$).

*E*represents the scattered field and

_{z}*ε*is the relative permittivity. The analysis of the TE field propagation was carried out by the FE analysis using the RF Module of COMSOL

_{r}^{TM}, with a monochromatic plane wave as excitation [13].

## 3. Design process

The design objective is defined as the integration of the norm scattered field in the measuring region set on an off-axis area in the simulation region. We evaluate the energy flux at the measuring area based on the Poynting vector value *P*; therefore, the optimization problem is formulated as follows:

*ϕ*) and

*V*represent the equality constraint on the volume and the required volume fraction, respectively. The analysis is performed for 2D cases and H

_{req}**-**field parts need to be changed to E

**-**field because only

*E*is measured. Therefore, the state variable

_{z}*u*expressed in Eq. (2) becomes

*E*.

_{z}During the design process, the design variable *ϕ* is updated and it determines the element density in the design area. The design area is set on the dielectric area as designated in Fig. 2. We adopted a topology optimization approach based on the phase field method [14] to find single material distribution in the entire design area. The boundary shape of the dielectric part is described by phase field variables which work as design parameters during the design process. The variable is defined as *ϕ*(*x, y*) in 2D cases and it is restricted in the interval of [0, 1]. The variable is applied to determine the isotropic dielectric constant as *ε _{r} = ε_{r-air} + ϕ*(

*ε*) where

_{r-diel}- ε_{r-air}*ε*has the relative permittivity of 2.2 of paraffin [15] composing the dielectric gratings.

_{r-diel}The field variable is updated by the reaction diffusion equation combined with the double well potential functions to converge its value to the boundary value 0 or 1 [16].

**R**

*(*

^{N}*N*= 2 for 2D cases) is a bounded domain with boundary ∂Ω and

*ϕ*(

*t*) represents a phase order parameter that is the design variable of our optimization problem.

**n**is the outer unit vector normal to ∂Ω and

*T*is the time required for the convergence of the optimization problem. The domain Ω is separated into solid material and void phases by a diffuse interfacial layer. The diffusion coefficient

*β*determines the thickness of the interfacial layer between the two phases and generally it is represented as a small constant related to the interfacial energy.

The double well potential *F* is assumed to have two global minima at 0 and 1, and is defined taking the equality constraint into account:

*λ*is the Lagrange multiplier and

*γ*is a penalty parameter.

*μ*is a parameter for normalizing the design sensitivity $\partial \Gamma /\partial \varphi $.

*ϕ*is assumed to evolve during a virtual time

*t*. The design sensitivity $\partial \Gamma /\partial \varphi $ at the time

*t*is regarded as a constant value. Due to the usage of double well potential functions the sensitivity at outside of the diffuse interface becomes zero so that the approach provides the interface tracking property. The adjoint method [17] is used to compute the sensitivities necessary for the update scheme.

_{1}## 4. Numerical and experimental results

Optimization has been started from a regular paraffin grating with 9.15mm period and 4.8mm
thickness to *y*-direction. The width and the thickness of the aperture in the
PEC are 6mm and 12mm, respectively. The convergence history of the objective function and shape
change of the design area at each of iterations are displayed in Fig. 3.The shape on the exit side is converging to an arbitrary 2D structure from a regular
grating as displayed with the enhancement of the design objective value. The minus sign was
originated from minimizing the negative value of the design cost function, i.e., maximizing the
field intensity in the measuring region. The transmission boost ratio to the desired direction
is expected about three times in comparison with the result by the initial regular grating as
can be confirmed in the plot.

Figures 4 shows electric field distributions at 10
GHz by numerical simulation with the optimized 2D structure as well as the regular grating
structure and the structure with the proposed grating shape shows the transmission boost to the
desired direction. The transmission intensity plot of Fig.
4(b) estimates that a full-width at half-maximum divergence from the derived structure
is about ± 14°. We also confirmed that the proposed dielectric shape gives an
enhanced transmission to a desired direction based on intensity plots displayed in the bottom of
Figs. 4(a) and 4(b) measured along the designated dotted line parallel to *x*-axis.

Figure 5(a) represents the 2D dielectric shape
derived by the design process. Contrary to ordinary shapes we observed that the exit region on
the outlet of the aperture is filled with both dielectric and empty space along the
*y*-axis direction. To investigate the effect of the dielectric portion, we
carried out parametric studies for *t _{1}* and

*t*as designated in the inset of Fig. 5(a). Geometric parameters

_{2}*t*and

_{1}*t*are defined on the air portion to evaluate the size dependence of the 2D dielectric structure on the exit side of the metallic aperture. Figure 5(b) shows that the integrated field intensity in the measuring region is dependent on

_{2}*t*and its value showing the most enhanced performance matches well with the value of the proposed structure. It is important that the derived 2D structure through the proposed approach clearly offers the co-existence of dielectric and air parts exited on the outlet of the metallic aperture and their appropriate sizes at the same time without depending on ordinary parameter study methodologies.

_{2}To verify the design result for the off-axis directional beaming, we carried out the experimental
measurement in microwave frequency for the structure composed of subwavelength metallic aperture
with the dielectric layer having the proposed shape. We made two test samples as illustrated in
Fig. 6.First sample [Fig. 6(a)] has regular paraffin
grating while second sample [Fig. 6(b)] has the optimal
grating structure described in Fig. 5(a). We normally
launched 10 GHz microwave on the *x*-*y* plane waveguide and
mapped the TE field distribution in the plane. We used a phase-sensitive, nearfield microwave
scanning system inside a planar waveguide and captured the TE field mapping image on the x-y
plane [18]. The radiation source was located 22cm away
from the metal slit and the experimental setup consisted of a scanning spectrometer to measure
the electric field strength and a network analyzer as schematically explained in Fig. 7.

Figures 8(a) and 8(b) show electric field distributions at 10 GHz from experimental results for the
regular grating and the proposed grating structure, respectively. We confirm good agreement with
the simulation results in Fig. 4. From both results, our
proposed dielectric shape gives an enhanced transmission boost to a desired direction and its
intensity plots [bottom of Figs. 8(a) and 8(b)] along the line parallel to *x*-axis at
the measuring location also confirms the off-axis beaming phenomenon. Those plots give similar
tendency as numerical simulation based intensity plots displayed in Fig. 4.

## 5. Conclusion

This study proposes a 2D structure through a simulation based systematic design approach for a dielectric structure attached on the exit side of a metallic aperture to guide an off-axis directional beaming at a microwave frequency. We verified the proposed structure through experiments and proved that the dielectric structure transmits light to a desired direction. The suggested design process for controlling the transmission direction has the potential expansible to structural design of metamaterial at any frequency range such as IR or visible region.

## Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the South Korean government (MEST) (NRF-2011-0017512).

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