Level set based topology optimization for optical cloaksGaruda Fujii, Hayato Watanabe, Takayuki Yamada, Tsuyoshi Ueta, and Mamoru Mizuno Citation: Applied Physics Letters 102, 251106 (2013); doi: 10.1063/1.4812471 View online: http://dx.doi.org/10.1063/1.4812471 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/102/25?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Design of a two-dimensional metamaterial cloak with minimum scattering using a quadratic transformationfunction J. Appl. Phys. 116, 124501 (2014); 10.1063/1.4893480 Superscattering of light optimized by a genetic algorithm Appl. Phys. Lett. 105, 011109 (2014); 10.1063/1.4887475 Localized transformation optics devices Appl. Phys. Lett. 103, 214104 (2013); 10.1063/1.4833279 Spherical cloaking using multilayer shells of ordinary dielectrics AIP Advances 3, 112111 (2013); 10.1063/1.4831942 Topology optimized low-contrast all-dielectric optical cloak Appl. Phys. Lett. 98, 021112 (2011); 10.1063/1.3540687

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Level set based topology optimization for optical cloaks

Garuda Fujii,1,a) Hayato Watanabe,2 Takayuki Yamada,3 Tsuyoshi Ueta,4

and Mamoru Mizuno1

1Faculty of Systems Science and Technology, Akita Prefectural University, Yurihonjo 015-0055, Japan2Department of Mechanical Science and Engineering, Nagoya University, Nagoya 464-8603, Japan3Department of Mechanical Engineering and Science, Kyoto University, Kyoto 615-8540, Japan4Physics Laboratory, The Jikei University School of Medicine, Tokyo 182-8570, Japan

(Received 17 December 2012; accepted 9 June 2013; published online 25 June 2013)

This letter presents a level set-based topology optimization method that provides discrete cloaking

configurations with superior performance. In some cases, the amount of light scattered around the

two-dimensional cloaking structures is less than one-tenth that obtained in previous studies. Optimal

configurations that express different geometrical characteristics can be obtained by adjusting a

regularization parameter. The obtained configurations are free from grayscales, areas of intermediate

density between that of dielectric materials or air, and the use of a level set method provides clear

structural boundaries. The level set functions are given on grid points and the dielectric boundaries

are interpreted as lines on the iso-surface of the level set functions. The finite element method is used

for light scattering analyses, computations of the adjoint variable field, and when updating the level

set functions. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4812471]

Photonic crystals1,2 have been applied in a range of opti-

cal devices such as waveguides,3 lasers,4–9 cavities,10–12 and

other devices. Optical cloaks13,14 that can control the propa-

gation of light around an object, so that little or no scattering

occurs, have attracted the attention of many researchers.

Both experimental15 and theoretical studies16 for optical

cloaks have been presented during the past several years.

Various optical devices with unique structures, such as

dielectric carpet cloaks17–20 and metamaterials, composed of

split ring resonators21,22 or fishnet structures23 have been

demonstrated. The performances of the above optical devices

strongly depend on their nanosized structures, hence the im-

portance of appropriate structural designs for the realization

of effective optical cloaks.

Shape24 and topology25 optimization methods are

powerful numerical methods that can be applied when

designing optical devices. Andkjær et al. applied gradient-

based26 and density-based topology optimizations27 in the

design of all-dielectric cloaks. In their advanced work,26

rotationally symmetric configurations were developed, but

these included grayscales, areas of intermediate density

between that of air and the dielectric material used to con-

struct the cloak. Ideally, such grayscales should be removed

because their presence makes practical fabrication difficult.

Andkjær et al. later succeeded in removing grayscales27

from optimal configurations by using a Heaviside projection

filter,28,29 and obtained clear configurations for polarization-

independent cloaks.

However, the use of a filtering scheme in topology opti-

mizations is not an ideal solution to the problem of gray-

scales because the filtered configurations, being altered from

the originally obtained optimal configurations that included

grayscales, are no longer optimal. The presence of grayscales

profoundly affects device properties, hence the difficulty

of realizing optical device designs that provide desired

performances when using topology optimization methods

that must deal with the presence of grayscales. Experience

has shown that the objective functional values determining

the amount of light scattering for optical cloaks increase con-

siderably when grayscales are filtered out to provide clear

optimal configurations. These increases imply that the initial

intrusion of grayscales significantly and deleteriously

affected the performance of the obtained optical cloak

designs. Thus, an optimization method that can obtain clear

optimal configurations directly would facilitate the develop-

ment of more advanced optical cloaks.

In this work, we use a topology optimization method

based on a level set method30 for the design of optical cloaks

in air, made of a dielectric material. The use of level set

expressions for the dielectric structures guarantees that the

boundaries of the optimal configurations are clear, and ena-

bles us to design such configurations directly. Additionally,

our method enables obtaining a variety of related optimal

configurations, by adjusting a regularization parameter, s,

that represents the ratio of the objective functional to a ficti-

tious interface energy term.30 We provide numerical exam-

ples of optimal cloaking configurations with different

geometries, obtained using different values of s.

We optimize dielectric configurations for optical cloaks

operating in the transverse magnetic (TM) mode, and assume

harmonic oscillation of the electric field. Figure 1(a) shows a

schematic image of the optical cloaks. Optical cloaks are

designed and transformed in a design domain Xdesign with ra-

dius Rdesign. Dielectric structures of optical cloaks are repre-

sented by domains Xdielectric and the boundaries of Xdielectric

are Cdielectric. Optical cloaks minimize the amount of scattered

light in an outer domain Xout with side length 2Rout. The finite

element method (FEM), a perfectly matched layer (PML)

absorbing boundary condition,31 and an optimized absorbing

function32 are used to simulate the scattering of light in an

open region. PMLs are represented by domains XPML. A

perfect electric conductor (PEC) boundary condition isa)Electronic mail: garuda_fujii@akita-pu.ac.jp

0003-6951/2013/102(25)/251106/5/$30.00 VC 2013 AIP Publishing LLC102, 251106-1

APPLIED PHYSICS LETTERS 102, 251106 (2013)

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implemented on CPEC, the boundary of a circular PEC with

radius RPEC, to model a strongly scattering object placed at

the center of the Xdesign domain. The total electric field Ez can

be expressed as the sum of respective scattering and incident

fields, Es and Ei

Ez ¼ Es þ Ei :

The incident field Ei is assumed to be plane waves propagat-

ing in the positive x direction. The wavelength of the inci-

dent plane waves in free space is fixed at Rdesign=2:5. We

solve a simple Helmholtz equation as follows:

r2Es þx2

c2�ðxÞEs ¼ �

x2

c2½�ðxÞ � �background�Ei ;

where c is the velocity of light in a vacuum and �background is

the relative permittivity of the background, which is assumed

to be 1. The position-dependent relative permittivity �ðxÞ is

defined in both Xdesign and Xout as follows:

�ðxÞ ¼ �background þ vð�dielectric � �backgroundÞ x 2 Xdesign

�background x 2 Xout;

�

where �dielectric is the relative permittivity of the dielectric

material and v is a characteristic function defined in

Xdesign as

vð/ðxÞÞ ¼ 1 if x 2 Xdielectric;0 if x 2 Xdesign =Xdielectric;

�

where /ðxÞ are level set functions defined as piecewise

constant values to the dielectric material boundaries, such

that

�1 � /ðxÞ < 0 for 8x 2 Xdesign =Xdielectric;/ðxÞ ¼ 0 for 8x 2 Cdielectric;0 < /ðxÞ � 1 for 8x 2 Xdielectric =Cdielectric:

8<:

The values of the level set functions are given with respect

to grid points as shown in Fig. 1(b). The functions are line-

arly interpolated at each location and have a value of 0 on

the dielectric boundaries. Finite elements are created based

on the boundaries. The interval between neighboring grid

points, Lgrid, is equal to Rdesign=100. The numbers of nodes

and elements in the FEM models are approximately 350 000

and 700 000, respectively.

In order to realize optical cloaks, Ez needs to be equal to

Ei; that is, Es ¼ 0. The objective functional for minimizing

light scattering around the dielectric structures is, therefore,

defined as

F ¼ 1

F0

ðXout

EsE�s dX; (1)

where E�s is the complex conjugate of Es and F0 is the inte-

grated intensity of the scattering field in Xout when no cloak

exists.

Topology optimizations need regularization to obtain

optimal configurations because such optimizations are ill-

posed problems. Based on the formulation of the level

set-based topology optimization method,30 the objective func-

tional (1) is regularized here by adding a fictitious interface

energy term derived from the phase field model as follows:

Fregularized ¼1

F0

ðXout

EsE�s dXþ

ðXdesign

1

2sjr/j2dX;

where s is a positive regularization parameter that represents

the ratio of the objective functional to the fictitious interface

energy term. The above regularization, with its fictitious

energy term, also acts a perimeter control, a type of geomet-

rical constraint. The geometrical constraint becomes weaker,

and optimal configurations can become more complex. As sis set to larger values, the fictitious interface energy is

increased, the perimeter becomes shorter, the geometrical

constraint becomes stronger, and optimal configurations

become more simple. Therefore, by adjusting the value of s,

optimal configurations that express different geometries can

be obtained. The length of the perimeter of the dielectric

structure, Lp, thus inversely indicates the degree to which the

geometrical constraint is active. The operation of this perim-

eter control enables us to obtain several different optical

cloak configurations, each of which is optimal. Thus, the

obtained optimal solutions are not unique, but give the local

minimum of the functional in the configuration space for

each value of s. Parameter s is never set to 0 because this

would negate the regularization, so the topology optimiza-

tion would revert to being an ill-posed problem.

Figures 2 and 3, respectively, show optimal configura-

tions and normalized total electric fields for a variety of

cases using different values of the regularization parameter

s. The objective functional values F for corresponding values

of s are shown in Fig. 3 subparts. As s is made smaller, the

normalized perimeter value Lp=Rdesign, representing the

degree of geometrical constraint, increases as shown in

Fig. 2, and the value of F decreases as shown in Fig. 4. In

particular, when s � 1:2� 10�5, the value of F is less than

one-tenth of that presented in a previous study [Fig. 3(d) in

Ref. 27: Cloak designs and associated objective values U for

Ez polarization]. The performances of the above cloaking

configurations are considerably improved in comparison

with those of the prior configuration. The sizes of the struc-

tural differences between optimal configurations illustrated

in Fig. 2 are obviously smaller than the wavelength of light,

but such small structures play an important role in preventing

light scattering around the cloak structures.

FIG. 1. (a) Cloak scheme. The domains sizes are RPEC ¼ Rdesign=3 and

Rout ¼ 2Rdesign. (b) Level set functions, dielectric boundary, and finite ele-

ments in a grid. Lgrid is the interval between neighboring grid points.

251106-2 Fujii et al. Appl. Phys. Lett. 102, 251106 (2013)

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FIG. 2. Optimal configurations and nor-

malized perimeter values, Lp=Rdesign, of

each optimal configuration for different

values of s. Blue, cyan, and the inner

white domains represent dielectric

material, background, and PEC,

respectively.

FIG. 3. Normalized total electric field

ReðEzÞ=jEij examples and correspond-

ing objective functional values. The rel-

ative permittivity values are �dielectric

¼ 2:0 and �background ¼ 1:0. The circles

represent the design domains.

251106-3 Fujii et al. Appl. Phys. Lett. 102, 251106 (2013)

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We consider that ease or difficulty of fabricating an opti-

mal structure mainly depends on the size of the smallest

structural feature, and roughly estimate the accuracy required

to fabricate the obtained optimal configurations. Comparing

the optimal configuration when s ¼ 2:0� 10�5 with that

when s ¼ 3:0� 10�5 in Fig. 2, we find two small circular

structures, indicated with arrows in Fig. 2(g), which are not

present in Fig. 2(h). We regard the presence and lack of these

two circular structures as the most important difference

between these two optimal configurations, and evaluate the

size of these features. The difference between the normalized

perimeters, Lp=Rdesign, for these two configurations is 0.575.

Assuming that the wavelength of incident light is 400 nm,

the difference between the perimeters becomes DLp ¼ 575

nm. The radii of the two circular structures Rsd are then esti-

mated as Rsd ¼ 45:7 nm, which is equivalent to the maxi-

mum fabrication tolerance required to realize the obtained

optimal configuration. However, such small structures are

not present in the optimal configuration obtained when

s ¼ 8:0� 10�5, shown in Fig. 2(l). By exercising perimeter

control, we can obtain desirable configurations that not only

provide superior performance and are optimal but also have

appropriate geometries that can be fabricated using practical

and expedient techniques.

We now investigate the frequency dependencies of

cloaking performance for the obtained optimal configura-

tions, as shown in Fig. 5. Good cloaking performance is only

obtained at the frequency xRdesign=2pc ¼ 2:5 assumed in the

optimization presented here, although designs for wideband

cloaks are sought. However, obtaining designs that provide

effective cloaking performance across a range of frequen-

cies, especially in higher frequency ranges, is currently quite

difficult. In higher frequency ranges, the size of scattering

objects becomes relatively large compared with the wave-

length of incident light, and this relative enlargement makes

it difficult to minimize light scattering. Obtaining cloaking

structure designs for higher and wideband frequencies is

essential, and a future goal of our research.

In conclusion, we developed a level set-based topology

optimization method for optical cloaks and demonstrated

optimal configurations with advanced performance in the

TM mode. The use of level set functions provided clear opti-

mal configurations, whose fabrication is practical, without

the problem of grayscales. Optical cloak performance was

considerably improved compared with previous results, and

close to that of a perfect optical cloak in some cases. A vari-

ety of optimal configurations for optical cloaks were

obtained by adjusting a regularization parameter, enabling

the design of optimal configurations that are appropriate for

the maximum tolerance of an existing practical fabrication

technique.

This work was supported by JSPS Grant-in-Aid for

Young Scientists (Start-up) (Grant No. 24860050).

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