Engineering the imaging properties of a metallic photonic-crystal slab lensShuai Feng, Zhi-Yuan Li, Zhi-Fang Feng, Bing-Ying Cheng, and Dao-Zhong Zhang Citation: Applied Physics Letters 88, 031104 (2006); doi: 10.1063/1.2166470 View online: http://dx.doi.org/10.1063/1.2166470 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/88/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Direct characterization of focusing light by negative refraction in a photonic crystal flat lens Appl. Phys. Lett. 93, 191114 (2008); 10.1063/1.3027069 Experimental demonstration of a wavelength demultiplexer based on negative-refractive photonic-crystalcomponents Appl. Phys. Lett. 91, 091117 (2007); 10.1063/1.2779927 Influence of unit cell rotated on the focusing in a two-dimensional photonic-crystal-based flat lens J. Appl. Phys. 101, 123112 (2007); 10.1063/1.2745374 Imaging properties of a metallic photonic crystal J. Appl. Phys. 101, 113105 (2007); 10.1063/1.2737771 Focusing properties of a rectangular-rod photonic-crystal slab J. Appl. Phys. 98, 063102 (2005); 10.1063/1.2058190

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Engineering the imaging properties of a metallic photonic-crystal slab lensShuai Feng, Zhi-Yuan Li,a� Zhi-Fang Feng, Bing-Ying Cheng, and Dao-Zhong ZhangInstitute of Physics, Chinese Academy of Sciences, P. O. Box 603, Beijing 100080, China

�Received 11 July 2005; accepted 23 November 2005; published online 18 January 2006�

Focusing properties of a two-dimensional metallic photonic crystal �PC� slab lens are studiedthrough the finite-difference time-domain technique. The PC consists of a square lattice of metalliccylinders immersed in a dielectric background. A good-quality image can form in the opposite sideof the slab in both the lowest and second higher transverse magnetic-polarized photonic bands,consistent with the analysis of equifrequency-surface contours. The all-angle negative refraction canbe realized and controlled by simply adjusting the background material permittivity,. © 2006American Institute of Physics. �DOI: 10.1063/1.2166470�

Left-handed materials and negative refraction of electro-magnetic �EM� waves have attracted a great deal of interestin the past several years.1–8 Negative refraction is the foun-dation for a variety of novel phenomena and potential appli-cations, for instance, fabrication of a superlens with a reso-lution far beyond the diffraction limit.1 Negative refractioncan occur in metamaterials6,7 and photonic crystals �PCs�.9–16

In particular, Luo et al.9,10 showed that all-angle negativerefraction �AANR� could be achieved at the lowest band oftwo-dimensional �2D� square-lattice PCs, leading to a sub-wavelength image formed in the near-field region of the PCslab. Zhang12,13 studied a 2D square lattice of coated cylin-ders in air by introducing a metallic component in the centerof every dielectric cylinder and found that AANR could oc-cur in the lowest band for both the transverse magnetic �TM�and transverse electric polarization modes. In this letter, wewill show that negative refraction and superlensing effect canbe achieved in a square lattice of metallic cylinders im-mersed in a dielectric background in both the lowest andsecond higher TM-polarized photonic bands.

The PC slab is placed in the air environment. The me-tallic cylinders are assumed to be perfectly conducting, sothe electric field within the metal material and tangential tothe metal surface is equal to zero. It turns out that metallicPCs with different dielectric backgrounds have the sameband structures and equifrequency-surface �EFS� contourswhen they are measured with respect to the normalized fre-quency n�. Here, n is the refractive index of the backgroundmaterial, and � is the angular frequency of the EM wave. Inall of the simulations throughout this letter, the radius of themetallic cylinders remains r=0.15a.

We employ the finite-difference time-domain �FDTD�method17,18 to calculate the band structures and EFS contoursof the above PC at the TM polarization mode. The calculatedband structures are shown in Fig. 1�a�. We can see that themetallic cylinders lift the first TM-polarized photonic bandand cause an omnidirectional stop band at the lowest fre-quencies. The cutoff normalized frequency of this metallicPC is 0.463 �in units of 2�c /a�. A complete band gap opensbetween the first and second bands, and lies at normalizedfrequencies 0.719–0.796. Figure 1�b� shows several EFScontours in the fundamental band. The 0.62 contour appearsto be convex around the M point. Consequently, the group

velocities of the excited Bloch wave modes at normalizedfrequency 0.62 correspond to an apparent negative refractiondirection for a PC slab whose surface normal is parallel tothe �M direction.

We then use the FDTD technique to visualize the propa-gation of the EM waves through this kind of metallic PCslab. We first consider a PC slab consisting of metallic cyl-inders immersed in the air background. The rectangular me-tallic PC slab is 61 layers wide and 8 layers thick, and thesurface normal is along the �M direction. The edges of therectangle are collinear with the coordinate axes, and the cen-tral metallic cylinder in the first layer is located at x=0 andy=0. A point source of continuous-wave resonant at�=0.62 is placed at the left side of the PC slab and the fieldpattern is displayed in Fig. 2�a�. The radiated EM wavetransmits through the PC slab and travels forward, like analmost parallel light beam. The width of this light beam firstnarrows, and then remains unchanged at a beam size of about3a, corresponding to 1.86�. This means that negative refrac-tion still works in this situation. The reason why a pointsource cannot form an image is that the excitation frequencyis located much above the light line of the air environment,as can be seen from Fig. 1�a�, thus strong diffraction effectsexist and strongly degrade the image quality.

To reduce or avoid the diffraction effect, one method weuse is to raise the permittivity of the background material.Now, we set the refractive index of background material tobe n=2.4. The distances from the four slab edges to thecenter of the adjacent metallic cylinders are all 0.2a, whileall the other parameters of the PC slab are the same as thosedepicted above. For the same normalized frequencyn�=0.62, the corresponding frequency becomes 0.62/2.4�0.2583. This frequency is still above �but already close to�the light line of the air environment. The calculated fieldpattern is shown in Fig. 2�b�. A visible image spot is nowformed in the right side of the PC slab. This image is a littleelongated in the lateral direction and contracted in the longi-tudinal direction. The width of the image spot is about 2a,corresponding to 0.5166�. To the best of our knowledge, thisis the first observation of a visible image forming at a fre-quency above the light line at the fundamental band of a PC.

We further increase the refractive index of the back-ground material to 3.74. For the same normalized frequencyn�=0.62, the corresponding excitation frequency now be-comes about 0.1567. All the other parameters are the same asin Fig. 2�b�. The calculated field pattern is displayed in Fig.a�Electronic mail: lizy@aphy.iphy.ac.cn

APPLIED PHYSICS LETTERS 88, 031104 �2006�

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2�c�, where we can see that a high-quality image spot isformed. The width of the image spot is about 1.9a, corre-sponding to 0.2977�, far below the diffraction limit. Com-paring Figs. 2�a�–2�c�, we can see that in order to get agood-quality image, the frequency must located below thelight line of the air environment so that AANR. can workwell. On the other hand, even if the frequency is above thelight line, negative refraction can also work to some extentso that we still can find a slightly degraded image spot or analmost parallel light beam stemming from a point sourceagainst the PC slab.

Now, we proceed to see the focusing properties of themetallic PC slabs in the second band. From Fig. 1�a�, we can

see that the second band spans from 0.796 to 1.026, and apartial band gap appears in the �X direction. Several EFScontours of the second band are shown in Fig. 1�c�. We seethat in the most parts of the 0.88 contour, which are centeredaround the �X line, the curve is quite flat and has the surfacenormal pointing to the X� direction. As the group velocity vgis antiparallel to the x component of the wave vector k, theeffective refractive index becomes negative, and negative re-fraction happens too. In the other parts centered around the

FIG. 1. The TM-polarized photonic band structures �a� and the EFS con-tours in the lowest �b� and the second �c� bands of the metallic PC withrespect to the normalized frequency n�. The dashed line in the �X regionrepresents the light line of the air environment when the EM wave travelsalong the �10� direction of the PC slab. The dashed line in the M� regionand shifted to the M point represents the light line when the EM wavetravels along the �11� direction of the PC slab.

FIG. 2. Ez field patterns of a point source placed in the vicinity of the PCslab. The refractive index of the background materials are �a� n=1, �b�n=2.4, and �c� n=3.74, respectively. All the three rectangular PC slabs are61 layers wide and 8 layers thick. The slab surface normal is along the �Mdirection. The normalized frequency of the EM wave is n�=0.62�2�c /a�.The point source is placed at �a� x=−0.5a ,y=0, �b� x=−a ,y=0, and �c�x=−a ,y=0, respectively. The slab surfaces are represented by two dashedlines. Dark and bright regions correspond to negative and positive Ez,respectively.

031104-2 Feng et al. Appl. Phys. Lett. 88, 031104 �2006�

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�M line, the curve is convex with respect to the � point,which also means a negative refraction direction.

We calculate the propagation of EM waves radiated froma point source through metallic PC slabs whose surface nor-mal is parallel to the �X direction. The PC slabs are all 51layers wide and 8 layers thick. We also consider three differ-ent dielectric materials in which the metallic cylinders areembedded. The refractive indices of these materials aren=1, 2.4, and 3.74, respectively. The PC slabs are all placedin the air environment. The distances from the four edges ofthe rectangular PC slab to the centers of the adjacent metalliccylinders are all 0.2a.

A point source of continuous-wave resonant at the samenormalized frequency n�=0.88 is placed in the vicinity ofthe PC slab. The calculated field patterns across the slabs aredisplayed in Figs. 3�a�–3�c� for the three background dielec-tric materials, respectively. At n=1 �Fig. 3�a��, the radiatedEM wave is resonant at frequency 0.88 and transmits

through the PC slab in the shape of an almost parallel lightbeam. The width of the light beam is about 1.8a, correspond-ing to 1.584�. The image is not formed because the excita-tion frequency is much above the light line of the air envi-ronment, and the diffraction effect is strong. At n=2.4 Fig.3�b��, the excitation frequency of the point source is 0.3667,a little bit above the light line of the air environment. We cansee that a visible image spot is formed in the right side of thePC slab. This image is distorted from a circle, just like theimage shown in Fig. 2�b�. The width of this elongated imagespot is about 1.6a, corresponding to 0.5867�. At n=3.74�Fig. 3�c��, the excitation frequency of the point source isabout 0.2352 and is found to locate within the AANR win-dow. An excellent-quality image spot is formed in the oppo-site side of the PC slab. The size is about 1.5a, correspond-ing to 0.353�, which is also well below the diffraction limit.These results clearly indicate that the square-lattice PC slabcan also behave as a high-quality lens for EM waves withfrequencies located in the second photonic band.

We have also considered a nonperfectly conducting PCslab lens made from silver, and found that the imaging prop-erties remain close to the microwave cases in the wavelengthregime from 1550 nm down to 500 nm. This indicates that,for good metals where the absorption effect is mild, the de-signed metallic PC lens can also work well in the opticalregion.

In summary, we have investigated the focusing proper-ties of EM waves by 2D PC slabs consisting of a squarelattice of metallic cylinders immersed in different dielectricbackgrounds. We find that the AANR region can be realizedand controlled both in the lowest and second photonic bandsby simply adjusting the background material permittivity.This offers a powerful way to engineer the negativerefraction and focusing properties of PC slab lens.

This work was supported by the National Key BasicResearch Special Foundation of China �Nos. 2004CB719804and 2001CB610402�, and the National Natural ScienceFoundation of China �No. 10404036�.

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FIG. 3. The same as Fig. 2, except that the PC slab is 41 layers wide, 8layers thick, and its surface normal is along the �X direction. The incidentwave is at the normalized frequency n�=0.88�2�c /a�.

031104-3 Feng et al. Appl. Phys. Lett. 88, 031104 �2006�

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