Influence of surface termination morphologies on the imaging properties of acomposite two-dimensional photonic crystal lensZhifang Feng, Shuai Feng, Zhi-Yuan Li, Kun Ren, Bing-Ying Cheng, and Dao-Zhong Zhang Citation: Journal of Applied Physics 100, 053702 (2006); doi: 10.1063/1.2336497 View online: http://dx.doi.org/10.1063/1.2336497 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/100/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in 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 Negative refraction and plano-concave lens focusing in one-dimensional photonic crystals Appl. Phys. Lett. 89, 084104 (2006); 10.1063/1.2338644 Strongly frequency dependent focusing efficiency of a concave lens based on two-dimensional photonic crystals Appl. Phys. Lett. 88, 011102 (2006); 10.1063/1.2159105 Improved superlensing in two-dimensional photonic crystals with a basis Appl. Phys. Lett. 86, 061105 (2005); 10.1063/1.1863413

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Influence of surface termination morphologies on the imaging propertiesof a composite two-dimensional photonic crystal lens

Zhifang Fenga�

Optical Physics Laboratory, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, Chinaand Beijing University of Chemical Technology, Beijing 100029, China

Shuai Feng, Zhi-Yuan Li,b� Kun Ren, Bing-Ying Cheng, and Dao-Zhong ZhangOptical Physics Laboratory, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

�Received 31 October 2005; accepted 19 July 2006; published online 6 September 2006�

By the numerical simulation based on the finite-difference time-domain method, we investigate theadjustability of image distance for the same object distance in two-dimensional photonic crystals�PCs�. When we add a fraction of a metallic component to the center of each dielectric bar, the PCslab lens can form a non-near-field image and the image distance changes for different surfaceterminations formed by introducing cylinders at the surface layers whose geometric and physicalparameters are different from those of the PC bulk. Furthermore, the image distance can be furthertuned by combining the two kinds of cylinders at the surface layers with different ratios of slablength. These simulation results clearly show that the imaging properties can be controlledeffectively by changing the surface termination of PC slab lenses. © 2006 American Institute ofPhysics. �DOI: 10.1063/1.2336497�

I. INTRODUCTION

Materials with simultaneously negative dielectric per-mittivity and negative magnetic permeability are called left-hand materials �LHMs�. Veselago analyzed some interestingpeculiar properties of such materials over 30 years ago,1 butonly recently did people demonstrate these materialsexperimentally.2,3 These materials have many unusual elec-tromagnetic �EM� properties, such as negative refraction andsuperlensing effect, inversed Snell’s law,4,5 reversed Dopplershift,6 and reversed Cherenkov radiation.7 These characteris-tics open the door for approaches to a variety of applications.Recently experimental and theoretical works have also indi-cated that negative refraction phenomena in photonic crystals�PCs� are possible.8–22 In particular, Luo et al. have shownthat all-angle negative refraction �AANR� could be achievedat the lowest band of two-dimensional �2D� PCs.13 Whennegative refraction occurs in the lowest valence band, severaladvantages arise, including the single-beam propagation andhigh transmission efficiency. Microsuperlens has been de-signed to realize focusing of EM waves using these charac-teristics. The physical principle that allows negative refrac-tion in PCs lies in the special dispersion characteristics ofwave propagation in a periodic modulation medium, whichcan be well described by analyzing the equal-frequency sur-face �EFS� of the photonic band structures.23–29

There are some reports about the influence of the surfacetermination on the point imaging.19–22 In order to create asurface state,30–33 people usually cut the outer part of theinterface cylinders, leaving one monolayer of hemicylinders.In this paper, we present our design of PC slab lens struc-tures along a different route. The building blocks of the PCsare metal-core dielectric-shell square rods. Even when the

surface layer rods of the slab lens remain the same as in thebulk medium, the slab lens can still focus a point source onone side of the lens into a real point image on the other side.When the size of the bars at the surface layer decreases, theimage distance also decreases. Even if the surface layer iscomposed of several parts of different rods, the imaging be-havior can still be clearly observed. Furthermore, the imagedistance can be continually tuned when the ratios of thesedifferent parts is adjusted.

II. NEGATIVE REFRACTION BY 2D COATEDMETALLODIELECTRIC PC

In this paper, we consider a 2D triangular lattice ofmetal-core dielectric-shell square bars immersed in the airbackground. As have been shown in many literatures, bothmetallic photonic crystals13 and dielectric photoniccrystals14,21–26 each can exhibit negative refraction in somecertain frequency windows for some appropriate physicaland structural parameters. It is expected that combining thetwo elements together into a single crystal structure will al-low for a larger freedom to control the negative refractionproperties. For instance, the all-angle negative refraction fre-quency window can be expanded and the image distance canbe tuned in a larger space range beyond the near-field regionof the flat lens.17,27 The edge lengths of the metal cores anddielectric bars are 0.44a and 0.8a, respectively, where a isthe lattice constant. The dielectric constant of the dielectriccoating is 11.4. When the frequency of the EM wave is low�much lower than the collision frequency of the electrons inthe metal�, the metal can be described by a frequency-independent conductivity. From Maxwell’s equations, theequivalent relative permittivity for the metal is ����=1+ i� /�. We assume that the metal has a constant �=5.8�107 S/m, corresponding to the copper conductivity. Mean-

a�Electronic mail: fengzff@aphy.iphy.ac.cnb�Electronic mail: lizy@aphy.iphy.ac.cn

JOURNAL OF APPLIED PHYSICS 100, 053702 �2006�

0021-8979/2006/100�5�/053702/6/$23.00 © 2006 American Institute of Physics100, 053702-1

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while, we only consider the TM polarization, where the elec-tric field is kept parallel to the extension axis of the metallicrods.

The photonic band structures and the EFS contours areimportant to the investigation of negative refraction and fo-cusing effect. We employ the finite-difference time-domain�FDTD� method with periodic boundary conditions to calcu-late the band structure as well as the EFS contours of thesecond band of the PCs. The results are shown in Fig. 1. Dueto the geometrical anisotropy of the square bars, the EFScontours are not perfect circles, but ellipses. It is clear fromthe figure that the EFS contours move inwards when thefrequency increases from 0.42 to 0.46�2�c /a�, and thismeans that the group velocity and the phase velocity areopposite to each other in this frequency range. This EFSfeature indicates that the propagation of the EM wave in thePC structures follows the left-handed behavior. The directionof the refracted wave inside the PCs can be estimated fromthe EFS. The conservation of the surface parallel wave vec-tor would result in the negative refraction effect in thesecases.

An important application of negative refraction materialis the superlens, which can focus a point source on one sideof the lens into a real point image on the other side even fora flat slab of material. In order to see whether the PC mate-rial can make a superlens, we consider a slab of the sample

36a wide and seven layers thick. The surface normal of theslab is along the �K direction of the crystal. A continuous-wave point source is placed at a distance 5a from the leftsurface of the slab and resonant at a frequency of0.425�2�c /a�, which is chosen to lie within the negativerefraction frequency window �see Fig. 1�. The FDTD methodis used to calculate the propagation of the TM-polarizedwave through the PC slab.

We investigate the field distribution of the point sourceagainst a PC slab where all the bars �those at the two inter-faces and within the bulk region� have the same inner andouter edge lengths. A typical field distribution pattern is plot-

FIG. 2. The intensity distribution of point source and its image at frequency�=0.425�2�c /a� for S wave. �a� The edge lengths of the metal cores anddielectric bars at the PC-air interface are 0.44a and 0.8a, respectively. Theobject distance is 5a. �b� The edge lengths of the metal cores and dielectricbars at the PC-air interface are 0.38a and 0.7a, respectively. The otherparameters are the same.

FIG. 1. The calculated photonic band structure �a� and equal-frequencysurface �b� of a triangular lattice of metal-core dielectric-shell square bars inair for the S wave. The edge lengths of the metal cores and dielectric barsare 0.44a and 0.8a, respectively. The dielectric constant of the coating ma-terial is 11.4. The inset shows the geometry of the microstructure.

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ted in Fig. 2�a�. One can clearly observe a high quality imagein the opposite side of the slab and the focusing in the middleof the slab. The distance of the image spot from the rightsurface of the slab is about 11a, clearly located beyond thenear-field region of the PC slab. The transverse size �full sizeat half maximum� of the image spot is about 0.5�. Then wechange the surface termination by replacing all the bars at thetwo interface layers by different metal-core dielectric-shellsquare bars whose inner and outer edge lengths are 0.38a and0.70a, respectively. The calculated field pattern of the pointsource against such a PC slab is displayed in Fig. 2�b�. Ob-viously the slab can also form a high quality image. From the

figure, we can see that the distance of image decreases toabout 7.6a. The change of image distance is very large whenthe termination of the slab lens is modified.

To have a more clarified idea about the surface termina-tion effect on the imaging feature of the PC slab lens, weturn to look at the refraction behavior of the slab upon inci-dent EM wave beams. As schematically depicted in Fig. 3�a�,a Gaussian beam propagates horizontally along the x-axisdirection and impinges upon the PC slab. The slab is inclinedfrom the horizontal direction by 40°, which means that the

FIG. 3. Simulations of negative refraction of an EM wave Gaussian beamthrough the PC slab. The shape of the sample and a snapshot of the refrac-tion process are shown on top of the figure. The intensities of electric fieldfor incidence and refraction are shown. The frequency of the incident waveis 0.425�2�c /a�. The crystals and parameters in �a� and �b� correspond tothose in Figs. 2�a� and 2�b�, respectively. The incident angle is 40°.

FIG. 4. The angles of refraction ��� vs angles of incidence ��0� at thefrequency 0.425�2�c /a�. The parameters in �a� and �b� correspond to thestructures in Figs. 3�a� and 3�b�.

FIG. 5. The structures of the PC-air interface are displayed. a1 and a2

represent different ranges. The edge lengths of the metal cores and dielectricbars in the a1 region are 0.44a and 0.8a, while the edge lengths of the metalcores and dielectric bars in the a2 region are 0.38a and 0.7a, respectively.The dotted line represents the image place.

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incident angle of the wave beam is 40°. The surface normalof the PC slab is along the �K direction of the triangularlattice. The half-width of the incident wave beam at the waistis 1.5a. When the incident wave goes through the PC slab,refraction will take place at the two interfaces and the trans-mission wave beam will have a parallel shift with respect tothe incident beam. From this parallel shift, the effective re-fraction index of the slab material can be deduced. TheFDTD method is used to retrieve the results. In order to

avoid the edge diffraction effects, the width of the slabsamples is much larger than the thickness. The calculatedfield patterns across the slab samples corresponding to thestructure studied in Figs. 2�a� and 2�b� are displayed in Figs.3�a� and 3�b�, respectively. It can be clearly seen that theenergy flux of the refraction wave travels following the nega-tive refraction law. The summation results of the refractionangles ��� versus incident angles ��0� at 0.425�2�c /a� cor-responding to the structure studied in Figs. 3�a� and 3�b� are

FIG. 6. The field distributions for the samples with dif-ferent ratios of a1 /2a2 are shown. The ratios of a1 /2a2

are 4/32, 8 /28, 12/24, 14/22, 16/20, and 18/18 inpanels �a�, �b�, �c�, �d�, �e�, and �f�, respectively. Thepositions of the image are denoted by the arrows.

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summarized in Figs. 4�a� and 4�b�. From this figure, we cansee that the relation of � vs �0 is not linear, indicating theexistence of spatial dispersion of the negative refraction be-havior. These simulation results clearly show that the effec-tive refractive index of the PC slab lens may vary by chang-ing the surface termination morphology.

III. IMAGING CONTROL BY MEANS OF DIFFERENTCOMPOSITE BUILDING BLOCKS

Now we proceed to consider a more complicated PClens structure that is composed of the above two different

kinds of metal-dielectric bars. The geometry of the resultingsample is displayed in Fig. 5. The whole lens structure is stillsymmetrical about the image point. The bars at the PC-airinterfaces in the a1 and a2 regions have different geometricparameters. The coated bars in the a1 and a2 regions haveinner and outer edge lengths of 0.44a and 0.8a, and 0.38aand 0.7a, respectively. On the other hand, the bars within theinner bulk region have the same parameters as in Figs. 2�a�and 3�a�. Namely, the inner and outer edge lengths are 0.44aand 0.8a. We have considered many different ratios ofa1 /2a2 and see what happens to the imaging properties of the

FIG. 7. Panels �a�–�f� are for the in-tensity distributions along the Y direc-tion in the focal plane correspondingto Figs. 6�a�–6�f�.

053702-5 Feng et al. J. Appl. Phys. 100, 053702 �2006�

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lens. The typical results of the field patterns are plotted inFig. 6. The ratios of a1 /2a2 are 4 /32, 8 /28, 12/24, 14/22,16/20, and 18/18 in �a�, �b�, �c�, �d�, �e�, and �f�, respec-tively. From these patterns, we can find that the image dis-tance can be adjusted by adopting different interface mor-phologies of the lens. They are 5a, 4.7a, 5.7a, 7a, 8.3a, and9.7a, respectively. The corresponding intensity distributionsalong the Y direction are plotted in Figs. 7�a�–7�f�. The datareveal that the transverse half-widths of the central peak are0.7�, 0.65�, 0.5�, 0.6�, 0.55�, and 0.55�, respectively. Thisbehavior can be qualitatively attributed to the change of theeffective refractive index under different fractions of thecomposite metallodielectric bars. As we have seen in Sec. II,the lens with the a1 region at the interface has a differenteffective refractive index from the lens with the a2 region atthe interface. It is thus understandable that the effective re-fractive index under different composite fractions of thesetwo regions is different from each other. Variation of theeffective refractive index directly influences the focusing ofthe slab, in particular, the image distance. The numericalsimulation results as shown in Figs. 6�a�–6�f� indicate thatthere is no simple dependence of the overall effective refrac-tive index on the a1 /2a2 ratio. The same complex situationholds for the image distance. This might be qualitatively un-derstood by the complex spatial dispersion behavior of theeffective refraction index of the two individual buildingblocks, as shown in Fig. 4. As can be envisioned from Fig. 5,different spatial-frequency �or angular� wave componentsemitting from the point source will impinge upon differentregions �either a1 or a2� of the slab lens and witness differenteffective refractive indices. Therefore, the overall refractionbehavior is quite complex. The existence of multiple scatter-ing effect between the two composite regions will furthercomplicate the wave propagation properties. Despite thiscomplexity, our studies have clearly shown that modificationof the surface morphology of a negative refraction lens is aneffective way to engineer the imaging properties of the lens.The adoption of composite building blocks at the surfacelayer of the lens represents a freedom towards engineering ofthe surface morphology in addition to the usual way of cut-ting outermost layers.

IV. SUMMARY

In conclusion, by the exact numerical simulations basedon the finite-difference time-domain method, we study thefocus of superlens that is built from a triangular lattice ofmetal-core dielectric-shell square bars. A high quality imagecan form against the PC lens with appropriate geometricsizes of the core-shell bars. The image distance can be ad-justed by changing the surface layer geometry of the lens.Adoption of different building bars at the surface layers andcontrol of their volume ratio can offer an additional freedomto engineer the imaging properties of the PC lens.

ACKNOWLEDGMENTS

This work was supported by the National Key Basic Re-search Special Foundation of China at Grant Nos.2004CB719804 and 2001CB610402, and the National Natu-ral Science Foundation of China at Grant Nos. 10404036 and10525419. The support from Supercomputing Centre, CNIC,CAS is also acknowledged.

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