Plasmonics DOI 10.1007/s11468-016-0438-4 Inverse Design of Dielectric Resonator Cloaking Based on Topology Optimization Yongbo Deng1 · Zhenyu Liu2 · Yongmin Liu3,4 · Yihui Wu1 Received: 12 August 2016 / Accepted: 9 November 2016 © Springer Science+Business Media New York 2016 Abstract In many applications, a cloaked resonator is respectively. The derived results demonstrate that the res- highly desired, which can harvest and maximize the energy onator cloaking can be categorized into three types, which within the resonator without being detected. This paper are the Fabry-Perot´ resonance cloaking, Mie resonance presents the resonator cloaking achieved by topology cloaking and hybrid resonance cloaking. optimization-based inverse design methodology. The res- onator cloaking is inversely designed by solving the topol- Keywords Resonator cloaking · Inverse design · Topology ogy optimization problem with minimizing the ratio of the optimization · Fabry-perot´ resonance cloaking · Mie scattering field energy outside the cloak and the cloaked res- resonance cloaking · Hybrid resonance cloaking onating field energy. By inversely designing the resonator cloaking with relative permittivity 2 for both the resonator and cloak, the topology optimization-based inverse design Introduction methodology is demonstrated, where the incident angle sensitivity is considered to derive incident angle insensi- Many different applications are in pressing need of effec- tive design. Then, the proposed methodology is applied tively cloaking resonators (or sensors and detectors), which for the cases with resonator and cloak materials chosen can efficiently detect signals but has negligible disturbance from dielectrics with low, moderate and high permittivity, on the surrounding environment. For example, in physics and engineering experiments, this means that a probe, e.g. the tip of a near-field scanning optical microscope or a microwave antenna, may have a minimal scattering effect on the quantity it is designed to measure [1, 2]. Yongbo Deng With the development of transformation optics, the old [email protected] dream of a device which render an object invisible to the human eye is already within reach [3, 4]. By trans- 1 State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP), Chinese formation optics, the cloak/anticloak interaction has been Academy of Sciences, Changchun, 130033, China investigated to realize the sensor cloaking [5]. However, the derived cloak/anticloak has extreme electromagnetic prop- 2 Changchun Institute of Optics, Fine Mechanics and Physics erties, permittivity and permeability. And they normally (CIOMP), Chinese Academy of Sciences, Changchun, 130033, China are implemented by exotic metamaterials [6]. The tai- lored microstructure of such metamaterials has to be much 3 Department of Mechanical and Industrial Engineering, smaller than the wavelength, and this makes it very chal- Northeastern University, Boston, MA, 02115, USA lenging to realize the desired magnetic properties at optical 4 Department of Electrical and Computer Engineering, frequencies. Would it be possible to design a cloaked res- Northeastern University, Boston, MA, 02115, USA onator using conventional simple isotropic dielectric readily Plasmonics available in nature instead of using metamaterials with extreme electromagnetic properties? To address this question, we adopt an inverse design approach based on topology optimization to find the geo- metrical configuration of the conventional nonmagnetic isotropic dielectric cloak for a resonator. Besides, the metasurfaces-based electromagnetic illusion or virtual shap- ing has also been demonstrated to be an alternative approach [7–9]. Topology optimization is a full-parameter method used to inversely determine the geometrical configuration, which represents distribution of materials [10]. It can be used to implement the structural design for the cases where Fig. 1 Sketch for the computational domain. The resonator r is the scale is large enough to ensure the reasonability for located at the center of the computational domain; the ring-shaped using physical parameters of materials fitting in with sta- design domain d for the cloak surrounds the resonator; and the tistical hypothesis or continuum hypothesis. In contrast to outside surrounding s is set to be vacuum designing devices by tuning a handful of structural param- eters in size and shape optimization, topology optimization to be vacuum. For transverse electric (TE) polarization, the method utilizes the full-parameter space to design structures waves are described by the governing equation as follows solely based on the user’s desired performance specifi- cation. Therefore, topology optimization is more flexible ∇· −1∇ + + 2 + = μr (Ezs Ezi) k0r (Ezs Ezi) 0, in and robust, because of its low dependence on initial struc- ture and implicitly expression of the material distribution −1 −1 μ ∇Ezs · n + jk0 r μr Ezs = 0, on ∂ in structures. Further, topology optimization can inversely r determine the geometrical configuration representing simul- (1) taneously the structural topology, shape and size; and it is a where E is the scattering TE field; E is the incident TE more general computational design methodology. Topology zs zi field; and μ are the relative permittivity and permeabil- optimization has been applied to multiple physical prob- r r ity, respectively; k is the free space wave number; j is the lems, such as elastics, acoustics, electromagnetics, fluidics, 0 imaginary unit; is the computational domain with trace optics, thermal dynamics, and material design problems ∂. This paper considers the inverse design case for uni- [10]. In electromagnetics, it has been applied in the inverse form plane incident waves with the incident TE wave Ezi set design of cloaks, splitter, photonic crystal, plasmonic nanos- − · to be e jk0k x,wherek is the normalized wave vector and x tructures, dielectric metamaterials [18–22], to name the is the spatial coordinate. most prominent. Therefore, topology optimization is one Topology optimization approach is based on the mate- reasonable approach to achieve resonator cloaking. rial interpolation between two different materials. And the material interpolation is implemented with the binary dis- tribution defined in the design domain, where the binary Methodology distribution with values 0 and 1 respectively represent two material phases. This paper considers nonmagnetic mate- An infinitely long cylinder domain is illuminated in the rials with unity relative permeability. Then, the inverse free space with monochromatic propagating wave. Due to design for the resonator cloaking is focused on the geomet- the invariance of the electromagnetic properties along the rical configuration corresponding to the spatial distribution cylinder axis, the problem can be formulated in a plane of materials with two different relative permittivity. The perpendicular to the cylinder axis. A firs- order absorb- binary distribution is set to be the design variable, which ing boundary condition is used as an approximation to the is relaxed to vary in the interval [0, 1] in the gradient Sommerfeld radiation condition in order to truncate the infi- information-based topology optimization. To regularize the nite domain. Thus, the computational domain is preset as relaxed design variable continuously valued in [0, 1] and shown Fig. 1 with one circularly-shaped resonator at the converge to one binary distribution at the end of the topol- center. A time-harmonic electromagnetic wave propagates ogy optimization procedure, the design variable is filtered from the left boundary through the computational domain. using the Helmtoltz equation-based PDE filter [23] In the computational domain, the resonator cloak is located in a ring-shaped domain with the same center as the res- −∇· r2∇γ + γ = γ, in onator, and it is inversely determined using the topology f f d optimization approach. The rest surrounding medium is set ∇γf · n = 0, on ∂d (2) Plasmonics where γ is the design variable defined on the design domain, used to determine the evolution direction of the design vari- and d is the design domain set to be the ring-shaped cloak able. The adjoint analysis-derived gradient for the quotient domain that surrounds the resonator. The filtered design in Eq. 7 is variable is projected by the threshold method [24] δEs Es δq = − δEr (8) tanh (βξ) − tanh β γf − ξ 2 γ = (3) Er Er p tanh (βξ) − tanh (β (1 − ξ)) In Eq. 8, δEs is the first-order variational of Es where r is the filter radius chosen based on numerical exper- iments [23]; γf is the filtered design variable; γp is the δEs =− γ˜f δγ d (9) projected design variable named physical density represent- d ing the geometrical configuration [25]; ξ ∈ [0, 1] and β with γ˜f derived by solving the adjoint equations are the threshold and projection parameters for the thresh- ∗ 2Ezs ˜ −1 ˜ ∗ ˜ ˜ ∗ ˜ old projection, respectively: ξ is set to be 0.5, and β is set Ezs − μ ∇E ·∇Ezs + k0r E Ezs d r zs zs with initial value 1 and it is doubled every 40 iterations [26]. Es0 − −1 ˜ ∗ ˜ = Then, the material interpolation is implemented using the jk0 r μr EzsEzs d 0, projected design variable as ∂ 2 ˜ ˜ ∂r ∂γp = + γ ( − ) (4) r ∇˜γf ·∇γ˜f +˜γf γ˜f + k0 Re (Ezs + Ezi) r rl p ru rl ∂γ ∂γ d p f where and are the relative permittivity of two differ- ∗ ∗ rl ru Re E˜ − Im(E + E ) Im E˜ γ˜ d = 0, ent dielectrics, respectively. zs zs zi zs f In this inverse design