Focusing properties of a rectangular-rod photonic-crystal slabShuai Feng, Zhi-Yuan Li, Zhi-Fang Feng, Kun Ren, Bing-Ying Cheng, and Dao-Zhong Zhang Citation: Journal of Applied Physics 98, 063102 (2005); doi: 10.1063/1.2058190 View online: http://dx.doi.org/10.1063/1.2058190 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/98/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhancement of second harmonic generation in NaNO2-infiltrated opal photonic crystal using structural lightfocusing Appl. Phys. Lett. 105, 051902 (2014); 10.1063/1.4892363 Imaging properties of a metallic photonic crystal J. Appl. Phys. 101, 113105 (2007); 10.1063/1.2737771 Engineering the imaging properties of a metallic photonic-crystal slab lens Appl. Phys. Lett. 88, 031104 (2006); 10.1063/1.2166470 Highly efficient wide-angle transmission into uniform rod-type photonic crystals Appl. Phys. Lett. 87, 111107 (2005); 10.1063/1.2048823 Improved superlensing in two-dimensional photonic crystals with a basis Appl. Phys. Lett. 86, 061105 (2005); 10.1063/1.1863413

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Focusing properties of a rectangular-rod photonic-crystal slabShuai Feng, Zhi-Yuan Li,a� Zhi-Fang Feng, Kun Ren, Bing-Ying Cheng, andDao-Zhong ZhangInstitute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China

�Received 22 April 2005; accepted 9 August 2005; published online 23 September 2005�

The focusing properties of a photonic-crystal �PC� slab consisting of a square lattice of rectangulardielectric rods in the air background are studied theoretically. We employ the finite-differencetime-domain method to investigate the field patterns of a point source placed in the vicinity of thePC slab and find that an image can form in the opposite side of the slab in a frequency windowlocated slightly below the fundamental band gap. We change the orientation of the rectangular rodsand find that when the rods are arranged asymmetrically with respect to the surface normal of theslab, the image spot can show a vertical shift relative to the point source. The influence of the PCslab thickness on the quality of the image is also analyzed. From these simulation results and theequifrequency-surface contour analysis, we find that the dominant physical mechanism that shapesthe focusing behavior of these rectangular-rod PC slabs in the ground photonic band is theself-collimation effect instead of the negative refraction effect. © 2005 American Institute ofPhysics. �DOI: 10.1063/1.2058190�

I. INTRODUCTION

Negative refractive-index materials �NIMs� have at-tracted great renewed interest due to their experimental real-izations several years ago.1–6 Negative refraction is the foun-dation for a variety of phenomena and potential applications.A NIM slab can amplify the evanescent wave componentsinvolved in the radiation from a point source and accomplishexcellent retrieval of the fine spatial information that is nec-essary for the formation of a high-resolution image. Pendrypredicted that a NIM slab could make a superlens with aresolution far beyond the diffraction limit that is inherent inthe conventional far-field lens.1

It has been shown that negative refraction can also occurin photonic crystals �PCs�, which are inhomogeneous mediamade from periodic arrays of dielectric or metallic buildingblocks with a lattice constant comparable with thewavelength.7–14 In particular, Luo et al. showed that all-anglenegative refraction �AANR� could be achieved at the lowestband of two-dimensional �2D� square-lattice PCs and at awavelength four times larger than the lattice constant, and asa result of the amplification of evanescent waves and thisall-angle negative refraction, a subwavelength image couldform in the near-field region of the PC slab.7,8 Li and Linshowed that other physical mechanisms such as the self-collimation effect and near-field scattering effect might playa dominant role in the formation of the near-field subwave-length image, although the negative refraction effect alsotook place in the structure.9 From another point of view, Kuoand Ye attributed the near-field imaging characteristic to theanisotropic wave propagation behavior due to the partialband gap in the square-lattice PC structure in the concernedfrequency window.10

The PC-based NIMs that most authors have investigatedare built from 2D square or triangular lattices of dielectric or

metallic cylinders. Very recently, Moussa et al. have shownboth theoretically and experimentally that a triangular latticeof rectangular dielectric rods can behave as a NIM in a fre-quency window located within the second photonic band.11

In this paper, we will show that negative refraction and su-perlensing effect can also be achieved in a square lattice ofrectangular dielectric rods embedded in the air backgroundin a frequency window located at the lowest photonic band.

This paper is organized as follows. In Sec. II we analyzethe photonic band structures and equifrequency-surface�EFS� configuration of the rectangular-rod PC structure andtheir connection with the negative refraction and imagingproperties of a PC slab. We will first discuss the focusingproperties of PC slabs where the rectangular rods are orien-tated so that the structures are symmetrical with respect tothe surface normal of the slabs. In Sec. III we change theorientation direction of the rectangular rods with respect tothe crystalline axis and the surface normal of the slab and seewhat happens to the negative refraction and focusing behav-iors of the PC slab in the asymmetric configuration. Finally,we summarize this paper in Sec. IV.

II. FOCUSING PROPERTIES OF PC SLABS ATSYMMETRICAL ORIENTATION CONFIGURATIONS

In this paper, we consider a 2D PC slab consisting of asquare lattice of rectangular dielectric rods immersed in theair background with a lattice constant a. The geometry of thePC slab is depicted in Fig. 1. The length and width of therectangular rods are l1 and l2, respectively. For thisdielectric-rod structure, we only consider the TM polariza-tion, where the electric field Ez is kept parallel to the exten-sion axis of the rectangular rods �assumed to be the z axis�.We make the dielectric constants of the rectangular rods tobe �=14, l1=0.75a, and l2=0.375a. These parameters areused in all the simulations. Under this condition, the PCanalyzed here has the same dielectric constant and an almosta�Electronic mail: lizy@aphy.iphy.ac.cn

JOURNAL OF APPLIED PHYSICS 98, 063102 �2005�

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identical filling fraction of rods as the circular-rod NIMstructure investigated in Refs. 7 and 9. Also, similarly, weexamine negative refraction and imaging against a PC slabwhose surface normal is parallel to the �M direction of thesquare-lattice crystal.

The well-established routine towards the exploration ofnegative refraction and focusing effect starts from the inves-tigation of the photonic band structure and EFS contours.7–24

In this paper, we employ the conventional plane-wave expan-sion method to calculate the band structure and the EFS con-tours of the rectangular-rod PC.25,26 441 plane waves areadopted in all the calculations, and a good numerical conver-gence has been reached. In order to obtain a high enoughresolution of the fine structure of the curve, more than10 000 Bloch-wave vectors within the first Brillouin zone�BZ� of the square lattice are used in constructing the EFScontour.

We start from a PC slab where the long side of the rect-angular rods is along the �M direction, and the geometry isshown in Fig. 1�a�. The calculated band structures for theseveral lowest photonic bands along the high-symmetriclines in the first BZ are shown in Fig. 2�a�, while the EFScontours of the fundamental �lowest� band for this PC areshown in Fig. 2�b�. Notice that the long side of the rods iscollinear with the �11� ��M� direction, thus the band alongthe �01� and �10� directions is symmetric, while it is asym-

metric along the �11� and �1̄1� directions. This asymmetrycan be clearly seen from the second and higher bands be-tween the �M and �M� lines, but is negligibly small for thefirst band, perhaps due to the long wavelength nature of thefirst band. The EFS contour of the first band is exactlymirror-symmetric with respect to the �M direction. FromFig. 2�a� we can see that a complete band gap opens betweenthe first and second bands and lies at frequencies 0.221–0.265 �in unit of 2�c /a�. The light line for this PC slab isshown by the dotted line. The dashed line represents the

frequency of 0.196, which is one of the frequencies that weconsider in this paper. The other considered two frequenciesare 0.186 and 0.206, respectively. These three frequenciesare all located in the fundamental band but are close to theband gap edge. Therefore some strong scattering effects areexpected to exist. From Fig. 2�a� we also see that the fre-quency 0.206 is above the light cone. According to thetheory of Luo et al.7 this frequency is outside the AANRregion.

The EFS contours at several relevant frequencies aredemonstrated in Fig. 2�b�. In most part of the 0.186 contour,which is centered around the �M direction, the curve is alittle flat, although it is concave with respect to the M point,while in the other small parts near the BZ edges �line XMand line X�M�, the surface normal is directed towards the Mpoint. In most part of the 0.196 contour, the curve is quite flatand has the surface normal pointing to the �M direction. Inthe small parts near the BZ edges, the surface normal is alsodirected towards the M point. The whole 0.206 contour isconvex with respect to the M point, although the curve is stillsomewhat flat in the middle of this contour. The group ve-locities centered around the �M direction are different fromeach other for these three frequencies. They corresponding topositive refraction direction, self-collimation, and negativerefraction direction for frequencies of 0.186, 0.196, and0.206, respectively.

FIG. 1. The schematic configuration of rectangular rods in a PC slab withthe surface normal along the �M direction. �a� The long side of the rods isalong the �M direction. �b� The long side of the rods is along the �Xdirection.

FIG. 2. TM mode band structures �a� and EFS contours �b� of therectangular-rod PC. The PC is composed of a square lattice of rectangulardielectric rods in the air background. The long and short sides of the rect-angular rods are of lengths l1=0.75a and l2=0.375a, respectively. The di-electric constant of the rods is 14. The long side of the rods is along the �Mdirection. The light cone shifted to M is shown by the dotted line. Thedashed line represents the frequency of 0.196.

063102-2 Feng et al. J. Appl. Phys. 98, 063102 �2005�

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To visualize the propagation of electromagnetic �EM�waves through this kind of rectangular-rod PC slabs, we em-ploy the finite-difference time-domain method with a bound-ary treatment of perfectly matched layers.27–29 A point sourceof a continuous wave is placed at the left side of the PC slaband fixed at x=−a, y=0. Here we assume that the center ofthe rectangular rods in the first layer is located at x=0, andthe central row of the PC slab is centered at y=0. The PCslab is 8 layers thick and 60 layers wide. The calculated Ez

field patterns for this PC slab at frequencies of 0.186, 0.196,0.206 are displayed in Figs. 3�a�–3�c�, respectively. Thedashed lines show the two surfaces of the slab. From Fig.3�a� we see that an image spot is formed in the right side ofthe PC slab, but the center of this image is very close to theslab surface, so the image spot is more like half a circle inthe region outside the slab surface. When the frequency be-comes 0.196, a high-quality image spot is shown in Fig. 3�b�.Note that this frequency is located within the AANR region,and the image spot is a little further away from the surface ofthe PC slab than at the frequency of 0.186. When we in-crease the frequency continuously to 0.206, the image dis-tance gets further away from the slab surface, but the image

is a little elongated in the lateral direction �the y axis� whilecontracted in the longitudinal direction �the x axis�. Compar-ing Figs. 3�a�–3�c�, we find that with the increasing of thefrequency, the image spot continuously moves away from thesurface of the slab. The distances from the surface of the slabto the center of the image spot are about 0.2a, 1.0a, and 4.0a,respectively, at frequencies 0.186, 0.196, and 0.206. At thesame time, the quality of the image degrades. We find thatthe image spot has sizes �full size at half maximum� in thelateral direction about 2.0a, 2.4a, and 3.1a, corresponding to0.32�, 0.47�, and 0.64�, respectively, for the above threefrequencies.

Then we change the thickness of the above PC slab to 24layers, while keeping other parameters unchanged. No appar-ent image spot is formed in Fig. 4�a�. Two elongated imagespots form in the opposite sides of the PC slabs in Figs. 4�b�and 4�c�; the lateral sizes of these images are about 3.8a and4.0a, corresponding to 0.74� and 0.82�, respectively. Thesetwo image spots are also located in the near-field region ofthe opposite side of the slab, and the image distance does nothave a simple dependence on the slab thickness expected fora NIM lens that works dominantly under the well-knownwave-beam refraction law. With the increasing of the slabthickness, we can clearly see a bright channel connecting thepoint source and the image spot inside the PC slab. No ap-parent focusing of the EM wave is found inside the slab. InFig. 4�a� we see that the bright channel inside the PC slabbecomes wider with the propagation of the EM wave, andthis clearly indicates that the usual positive refraction effecthas played an active role at this frequency, as has been found

FIG. 3. Ez field patterns of a point source placed at x=−a, y=0, and itsimage across a rectangular-rod PC slab at frequencies of �a� 0.186, �b�0.196, and �c� 0.206 �2�c /a�, respectively. The geometry of the PC slab isdepicted in Fig. 1�a�. The PC slab is 8 layers thick and 60 layers wide,whose surfaces are represented by two dashed lines. The dark and brightregions correspond to negative and positive Ez, respectively.

FIG. 4. The same as Fig. 3, except that the thickness of the PC slab is 24layers.

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from the above EFS contour analysis shown in Fig. 2�b�. Thebright channel in Fig. 4�b� keeps a good rectangular shape. Itseems that in the situation the wave transmitting into the slabis first collimated, then travels across the slab via this con-fined wave front, much like a wave transmitting through aPC waveguide. This feature is also consistent with our EFScontour analysis. From Fig. 4�c� we see that the centralbright channel still keeps a rectangular shape, but the wavefront of the EM wave in the wider angle shows a narrowingbehavior when it transverses the slab, a clear signature of astrong negative refraction effect for those large angle wavecomponents. The result is also consistent with the EFS con-tour analysis.

The shapes of the above three bright channels, whichmean how the EM wave transmits through the PC slab, aredifferent from each other, representing positive refraction,self-collimation, and negative refraction in Figs. 4�a�–4�c�,respectively. The simulation also clearly shows that a high-quality image is formed only in a small frequency rangewhose EFS contour along the �M direction is quite flat. Apoint source with the frequency within this range travelsthrough the PC slab mainly through a bright channel, mean-ing through the self-collimation effect. The reason that thethickness of the slab can influence the quality of the image isprobably the complex near-field scattering effect in these fre-quencies near the band gap.

III. FOCUSING PROPERTIES OF PC SLABS ATASYMMETRICAL ORIENTATION CONFIGURATIONS

In the above cases, the long side of the rectangular rodsis parallel to the surface normal of the PC slab. Now we keepl1 and l2 unchanged, while rotating the rectangular rods inthe square lattice and letting the long side of the rectangularrods point in the �X direction. The schematic configurationof this kind of PC slab with the surface normal along the �Mis depicted in Fig. 1�b�. The way how the rectangular rodsare oriented and arrayed in the PC slab can be seen clearly.Figure 5�a� shows the calculated band structures of such aPC, from which we see that a complete band gap lies atfrequencies 0.222–0.236 and the first band is asymmetricbetween the �X and �X� crystalline directions. This is con-sistent with the unit-cell symmetry analysis. The EFS con-tours of the lowest band are shown in Fig. 5�b�. We see thatthe 0.1 and 0.15 contours are also very close to a perfectcircle, but the 0.186 contour is completely distorted from acircle. When the 0.186 contour moves from the �X to the�X� direction, the distance �k� from point � at first keeps asa constant, then increases and finally goes beyond the rangeof the first BZ before reaching the �X� direction. At the verysmall part of this contour, which is located near the BZ edgeX�M, the curve is convex and has a surface normal pointingto a negative refraction direction. The 0.196 contour is alsonot symmetrical about the �M direction as the 0.186 contour,but is symmetrical about a particular direction somewherebetween the �X and �M direction �about 10° from the �Mdirection�. In most part of the 0.196 contour, which is cen-tered around that particular direction, the curve is almost flatand the surface normal points at this direction but not the�M direction. In other small parts near the BZ edge X�M, the

surface normal of the contour points at a negative refractiondirection. The 0.206 contour is similar to the 0.196 contour,except that the flat part of this curve is shorter than that ofthe 0.196 curve. The surface normal of the flat part in thesetwo contours points at the same direction.

Then we calculate the propagation of EM waves radiat-ing from a point source through this kind of PC slab with thesurface normal along the �M direction. The point source isalso located at x=−a and y=0, whereas the center of rectan-gular rods in the first layer is also located at x=0 and thecentral row of rods is also centered at y=0. The calculated Ez

field pattern is shown in Fig. 6�b� when the frequency of thecontinuous wave is 0.196. We can see that an image spot isformed in the opposite side of the slab. Interestingly, theimage spot is not located at the same horizontal position asthe point source �y=0�, but is downshifted some distancefrom y=0. In other words, the image and object are not col-linear with the surface normal of the PC slab. When thefrequency of the point source becomes 0.186 and 0.206, theEz field patterns are shown in Figs. 6�a� and 6�c�, respec-tively. A similar interesting phenomenon of downshifted im-age spot from the point source is also obvious. On the otherhand, the image spot is elongated in the lateral directionwhen the frequency is 0.186. The result of imaging proper-ties can be well explained from the EFS contours shown inFig. 5�b�. The above three EFS contours are not symmetricalabout the �M direction. In the region centered around the�M direction, the surface normal of the EFS contours pointsto a particular direction somewhere between the �X and �M

FIG. 5. The same as Fig. 2, except that the long side of the rods is along the�X direction.

063102-4 Feng et al. J. Appl. Phys. 98, 063102 �2005�

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directions �about 10° from the �M direction�. When the sur-face normal of the PC slab is along the �M direction, the EMwaves transmit into this PC slab mainly along this particulardirection.

When the thickness of the PC slab becomes 24 layers,the calculated field patterns are shown in Fig. 7. From Fig.7�a� we can see that the point source resonant at a frequencyof 0.186 cannot form a high-quality image through this PCslab. When the frequency becomes 0.196 and 0.206, an im-age is formed in the right side of this slab, although elon-gated in the lateral direction, as can be found in Figs. 7�b�and 7�c�, respectively. The two images formed in the aboveconditions are not located at the same horizontal position asthe point sources, they downshift some distance from y=0.Comparing with Fig. 6, the downshift distance increaseswhen the PC slab becomes thicker. The image spots are stillvery close to the right surface of the PC slab. These simula-tion results cannot be explained by Snell’s law for an effec-tive homogeneous NIM. So the PC slab we analyze for thepoint imaging in the ground band is not a simple NIM lensthat works dominantly under the well-known wave-beam re-fraction law. In other words, the negative refraction effect,although present, is not the dominant mechanism leading tothe near-field lensing effect of the PC slab. The existence ofa guiding channel within the slab indicates that the self-

collimation effect that has been found in Ref. 9 for a circular-rod PC NIM also plays a key role here. This wave propaga-tion behavior is consistent with the analysis on the EFScontour.

The above discussions show that the focusing propertiesof the square-lattice rectangular-rod PC slabs in the funda-mental photonic band exhibit many similarities with themuch more familiar circular-rod PC structures.7–9 Severalprominent points are worthwhile to mention. First, the phe-nomena take place in the fundamental band of the PC struc-tures, and the EM waves can only be allowed to dominantlypropagate in a certain direction ��M direction here�. Second,a good quality image spot can form but is strongly confinedin the near-field region of the PC slab. Third, the imagequalities and image distances are different for different fre-quencies. Fourth, a guiding channel is formed within the PCslab and connects the near-field object and image. We alsoalter the object distance and find that the image is stillstrongly confined in the near-field region and that the imagedistance has little dependence on the source distance and theslab thickness. When the point source is moved out of thenear-field region of the PC lens, the imaging effect becomessignificantly degraded. These remarkable imaging character-istics indicate that in addition to the negative refraction ef-fect, the existence and action of a self-collimation effect aswell as the complex near-field scattering effect also play animportant role in the formation of a subwavelength near-fieldimage against the rectangular-rod PC slabs, in a way verysimilar to that found in the circular-rod PC slabs.9

FIG. 6. The same as Fig. 3, except that the long side of the rods is along the�X direction.

FIG. 7. The same as Fig. 6, except that the thickness of the PC slab is 24layers.

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IV. SUMMARY

In summary, we have investigated the focusing proper-ties of EM waves by 2D rectangular-rod PC slabs. The cal-culated results show that although the negative refractioneffect might play some roles, the self-collimation effect is thekey role that governs and shapes the formation of a subwave-length near-field image against the rectangular-rod PC slabs.The strong geometric anisotropy inherent in a rectangularrod can offer many interesting features and additional free-doms to control the focusing properties of these PC slabs thatare not possible in a PC structure that is made from isotropiccircular rods. This kind of PC slab can be used to make asuperlens capable of the subwavelength imaging for somespecial purpose.

ACKNOWLEDGMENTS

This work was supported by the National Key Basic Re-search Special Foundation of China under Grants No.2004CB719804 and No. 2001CB610402 and the NationalNatural Science Foundation of China under Grant No.10404036.

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