860 OPTICS LETTERS / Vol. 33, No. 8 / April 15, 2008

Approximate optical cloaking in an axisymmetricsilicon photonic crystal structure

D. Xiao* and H. T. JohnsonDepartment of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign,

Urbana, Illinois 61801, USA*Corresponding author: dxiao@uiuc.edu

Received December 13, 2007; revised March 4, 2008; accepted March 5, 2008;posted March 12, 2008 (Doc. ID 90846); published April 14, 2008

Axisymmetric photonic crystal structures may be designed to possess interesting optical properties, particu-larly when the photonic band structure of the material is highly anisotropic. We use finite element calcula-tions to demonstrate an approximate electromagnetic cloaking effect imparted by a structure consisting ofconcentric silicon photonic crystal layers. The results show that it is possible to bend light around an objectby simply using anisotropy. The calculations show that the cloaking mechanism is fundamentally differentfrom Pendry’s approach. This design may work as a practical solution for optical cloaking. © 2008 OpticalSociety of America

OCIS codes: 230.3205, 230.5298, 350.5500.

Over the past two years, several ideas have been pro-posed to realize an electromagnetic cloaking device[1–8]. With a general recipe developed by Leonhardt[1], it is now theoretically possible to engineer a con-tainer that provides perfect invisibility according tothe work of Pendry et al. [2]. Two-dimensional (2D)cloaks at microwave frequencies have been experi-mentally realized recently [3], and intensive researchis being conducted to extend its range into the opticalregion. For example, optical cloaking has been de-signed with metamaterials based on the same idea[4].

However, analysis has shown that the cloaking de-sign based on this idea is very sensitive to small per-turbations [5]. Here we demonstrate the device sen-sitivity with some sample calculations. First, weassume that the cloaking device will have an effec-tive refractive index that exactly follows the formal-ism proposed in [1–3,6]. Then we add a small effec-tive conductivity of 500 S/m into such a perfectlydesigned system, with perfectly matched layers(PMLs) surrounding a domain of 12 �m�9 �m. Thefield distribution upon incidence of a plane wave of0.7 �m wavelength, from the left, is shown inFig. 1(a) and is similar to what has been presented in[3,6]. Also, we show the normal component ofthe time-averaged total power flow Sn�x ,y�= n · �E� �x ,y�H� �x ,y�� incident at the right boundary(left-hand line), compared with the power distribu-tion when the device is not present (right-hand line).Since we are most interested in light circumventingthe device, we replace the plane wave with a delta-function-shaped Gaussian beam (H� =e−4.5y2

eikxz,placed outside the left PML layer to avoid spuriousreflections owing to the source itself), as shown inFig. 1(b). We find that the transmitted power at theright boundary in this case is only �8% of the powerone would measure in the absence of such a cloakingdevice.

Next, we demonstrate via a numerical experimentthat the ratio of a metal object’s physical dimensionto the light wavelength controls the effective conduc-

0146-9592/08/080860-3/$15.00 ©

tivity of a system. Here we construct a domain of9 �m by 7 �m, with a gold particle in the center po-sition and with PMLs around the perimeter. Figure 2shows a plane-wave propagation of 0.6 �m wave-length with a gold particle radius of (a) 6 and (b)60 nm. The optical properties of gold at 0.6 �m (�r=−9.86 and �=3.89�104 S/m) are interpolated from[9]. When the particle radius is only 1% of the lightwavelength, we note that the difference between theinput power and the transmitted power is less than0.01%. In Fig. 2(b), with a particle radius of 10% of

Fig. 1. (Color online) Field and power distribution of a 2Dcloaking device with an effective index that follows theideal formalism, but with an effective conductivity of500 S/m under (a) a plane wave and (b) a Gaussian beamH� =e−4.5y2

eikxz illumination of 0.7 �m. At right, the left-hand line represents the distribution of the transmittedpower Sn through the cloaking device and the right-handline represents the power distribution when the device is

not present.

2008 Optical Society of America

April 15, 2008 / Vol. 33, No. 8 / OPTICS LETTERS 861

the light wavelength, we observe some distortion ofthe plane wave, such that the transmitted power tothe right is �99.2% of the input power and �0.023%is lost owing to material absorption. Also, increasingthe wavelength to 200 nm with a fixed particle radiusof 60 nm leads to a significant increase in transmis-sion. We note from these simple calculations that theeffective conductivity of the system is controlled bythe metal particle size relative to the light wave-length. In [3], the copper split-ring resonators’ fea-ture size (thickness) is only 17 �m, or 0.05% of theoperating microwave wavelength, so the structureappears to be uniform with minimal effective conduc-tivity. However, technical limits will hinder the use ofthe index-profile-fitting method in the visible range.Other approaches are also adopted to achieve an ap-proximate cloaking in the optical region [7,8]. How-ever, the material lossiness of metallic particles inthe visible range may still be an issue to limit thesedesigns, especially when one tries to make the struc-ture very large compared to the wavelength.

In this Letter, we report on a simple axisymmetricphotonic crystal (PC) structure that may be used toachieve an approximate cloaking effect. The struc-ture is composed of thin concentric axisymmetric lay-ers of silicon and air, surrounding a container com-posed of air, as shown in Fig. 3. Since the geometry ofthe axisymmetric cloaking structure does not dependon the tangential coordinate �, the optical properties

Fig. 2. (Color online) Field distribution for a system con-taining one gold particle whose radius is (a) 6 and (b) 60 nmunder plane-wave illumination of 0.6 �m wavelength.

Fig. 3. Axisymmetric silicon PC structure used to achievean approximate cloaking effect, and the computational do-

main used in the finite element calculation.

may be understood approximately in terms of theBloch theorem and thus the photonic band structure(PBS) of the usual one-dimensional (1D) PC with thesame physical parameters, where the x coordinate isreplaced with the radial coordinate and the y coordi-nate is equivalent to the tangential coordinate.

The layered structure is divided into inner andouter parts. The outer part of the axisymmetricstructure bends light via the anisotropy of the struc-ture. To demonstrate this design, we consider a caseof five layers with a radial period (i.e., the distancebetween two layers along the radial direction) of a1=0.06 �m and a cylinder thickness of t1=0.014 �m,for interaction with red light of wavelength 0.7 �m.The refractive index of silicon at this frequency is3.783− i*0.013 [9]. The PBS of the 1D PC with thesephysical dimensions reveals that along the tangen-tial direction the first band has large anisotropy, withlarger anisotropy for the H polarization than for theE polarization. So H-polarized light is used in thecloaking design. For red light of 0.7 �m, the effectiveindex can be calculated from the PBS using neff=ck /� so that along the radial direction nr=2.07 andalong the tangential direction n�=1.13. According toFermat’s principle, this large index anisotropy leadsto bending of the light during propagation throughthis region. The inner part of the anisotropic struc-ture prevents light from penetrating the container,but it does permit light propagation tangentially.This part consists of layers with a radial period ofa2=0.17 �m and a cylinder thickness of t2=0.03 �m.Light in the frequency of interest �a2 /��0.243� is inthe bandgap along the radial direction but in the firstband along the tangential direction. This ensuresthat light propagating toward the center of the con-tainer will be blocked. Eight such layers form the in-ner part of the cloaking structure, and the innermostlayer has a diameter of d1=4.3 �m, as shown in Fig.3. The domain and the device size is the same as inFig. 1. The simulation results, described next, showthat this combination of parameters leads to an ap-proximate cloaking effect.

The magnetic field distribution under the sameGaussian beam illumination is shown in Fig. 4, withthe distribution of the power flow at the right bound-ary. Light sources of wavelengths of (a) 0.7 and (b)0.8 �m impinge on the structure from the left side. Atest object with the refractive index of 3 is placed inthe center. The black arrows in Fig. 4 show the time-averaged power flow S�x ,y� at the specified locations,with their lengths proportional to their magnitudes.The arrows show that the Gaussian beam peaked atthe center position of the horizontal axis propagatesaround the cloaking container, and the field distribu-tions show that the penetration into the cloaking con-tainer is quite small. We note that light of differentfrequencies follows different paths around the struc-ture. For the design considered here, the transmittedpower of 0.8 �m light is much smaller than that of0.7 �m light.

This specific structure shows an approximate

cloaking effect when the light wavelength is between

862 OPTICS LETTERS / Vol. 33, No. 8 / April 15, 2008

0.65 and 0.85 �m. Outside this frequency range, thetransmitted power is either too weak or light willpenetrate the cloak and diminish the cloaking effect.Figure 5 shows the frequency response of the struc-ture with respect to (a) the transmitted power and (b)the scattering width. The dispersion of silicon hasbeen included using data from [9]. In graph (a), thetransmitted power ratio is the ratio of the transmit-ted power at the right boundary with the cloaking de-vice to the transmitted power at the right boundarywithout the device. The top and bottom curves repre-sent our design and the ideal design using the indexfitting method and an the effective conductivity of500 S/m, respectively; this shows clearly that thepresent design may have stronger transmission thanthe index fitting design. In graph (b), we estimate thenormalized scattering width by integrating the scat-tering power at the cloaking surface, as is reported in[6]. We find that over some frequency range the scat-tering width of our design is smaller than the idealdesign with effective conductivity, although in bothcases scattering is not insignificant. Finally, we notethat the effect is strongly dependent on the structureparameters. For example, if the number of the layersin either the inner or the outer part is changed, thestructure should be redesigned to optimize transmis-sion; this is possibly due to the importance of thephase matching between layers as a function of thechange of the device size.

In summary, we propose that a simple axisymmet-ric photonic crystal structure may be used to design adevice that imparts an approximate optical cloakingeffect. The anisotropy of the structure causes inci-dent light to bend around the structure. This effect is

Fig. 4. (Color online) Magnetic field and power distribu-tion for the proposed cloaking structure with H-polarizedincident light of (a) 0.7 and (b) 0.8 �m under the sameGaussian beam incidence. The size of the cloaking deviceand the domain are the same as in Fig. 1. A test object withindex of 3 is placed in the center.

applicable in the visible light range and achieveshigh transmission.

The generous support of the U.S. Department ofEnergy, Division of Materials Sciences, under awardDEFG02-91ER45439, through the Frederick SeitzMaterials Research Laboratory at the University ofIllinois at Urbana-Champaign is gratefully acknowl-edged. D. Xiao thanks Mihai Chiruta for helpful dis-cussions.

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(b)

(a)

Fig. 5. (Color online) (a) Diagram of the transmittedpower ratio as a function of wavelength, which is the ratioof the transmitted power through the proposed siliconcloaking device (top curve) and the ideal cloaking devicewith effective conductivity of 500 S/m (bottom curve) to thetransmitted power without the cloaking device; (b) diagramof the normalized scattering width of the proposed siliconcloaking device (top curve) and the ideal cloaking devicewith effective conductivity of 500 S/m (bottom curve).