Kee, Chul-Sik

Kim, Jae-Eun

Lee, Sun-Goo

Oh, Sang Soon

Park, Hae Yong

Online Research @ Cardiff

This is a n  Op e n  Acces s  doc u m e n t  dow nloa d e d  fro m  ORCA, Ca r diff U nive r si ty 'sins ti t u tion al r e posi to ry: h t t p s://o rc a .c a r diff.ac.uk/id/e p rin t/10 7 6 7 4/This  is t h e  a u t ho r’s ve r sion  of a  wo rk  t h a t  w as  s u b mi t t e d  to  / a c c e p t e d  forp u blica tion.Cit a tion  for  final p u blish e d  ve r sion:Lee,  S u n-Goo, Oh,  S a n g  Soon  ORCID: h t t p s://o rcid.o r g/00 0 0-0 0 0 3-3 0 9 3-7 0 1 6,Kim, Jae-E u n,  Pa rk,  H a e  Yong a n d  Kee, Ch ul-Sik 2 0 0 5.  Line-d efec t-ind uc e db e n din g  a n d  s pli t ting  of s elf-collim a t e d  b e a m s  in t wo-di m e n sion al p ho tonicc rys t als . Applied  P hysics  Le t t e r s  8 7  (18) , 1 8 1 1 0 6.  1 0.1 06 3/1.21 1 2 1 8 6  file P u blish e r s  p a g e:  h t t p://dx.doi.o rg/10.10 6 3/1.211 2 1 8 6< h t t p://dx.doi.o rg/10.10 6 3/1.21 1 2 1 8 6 >Ple a s e  no t e:  Ch a n g e s  m a d e  a s  a  r e s ul t  of p u blishing  p roc e s s e s  s uc h  a s  copy-e di ting,for m a t ting  a n d  p a g e  n u m b e r s  m ay no t  b e  r eflec t e d  in t his  ve r sion.  For  t h ed efini tive  ve r sion  of t his  p u blica tion,  ple a s e  r ef e r  to  t h e  p u blish e d  sou rc e.  Youa r e  a dvise d  to  cons ul t  t h e  p u blish e r’s ve r sion  if you  wish  to  ci t e  t his  p a p er.This ve r sion  is b ein g  m a d e  av ailable  in  a cco r d a n c e  wit h  p u blish e r  policie s.S e e  h t t p://o rc a .cf.ac.uk/policies.h t ml for  u s a g e  policies.  Copyrigh t  a n d  m o r al  r i gh t sfor  p u blica tions  m a d e  available  in ORCA a r e  r e t ain e d  by t h e  copyrig h thold e r s .Line-defect-induced bending and splitting of self-collimated beamsin two-dimensional photonic crystalsSun-Goo Lee, Sang Soon Oh, Jae-Eun Kim, and Hae Yong ParkaDepartment of Physics, Korea Advanced Institute of Science and Technology,Daejon 305-701, KoreaChul-Sik KeebAdvanced Photonics Research Institute, GIST, Gwangju 500-712, KoreaReceived 20 June 2005; accepted 14 September 2005; published online 26 October 2005We show that line defects can give rise to the bending and splitting of self-collimated beams intwo-dimensional photonic crystals from the equifrequency contour calculations and thefinite-difference time-domain simulations. The power ratio between two split self-collimated beamscan be controlled systematically by varying the radii of rods or holes in the line defect. We alsoshow that the bending and controllable splitting of self-collimated beams can be useful in steeringthe flow of light in photonic crystal integrated light circuits. © 2005 American Institute of Physics.DOI: 10.1063/1.2112186Photonic crystals PCs are dielectric materials whoserefractive index is periodically modulated in space. With aproper design, PCs can exhibit photonic band gaps PBGs,1the frequency ranges in which light propagation is com-pletely prohibited in any direction. There have been a num-ber of attempts made to control the flow of light by utilizingthe effect of PBG.2–6In recent years, there has been a grow-ing interest in anomalous dispersion properties of PCs.7–9One of the most interesting phenomena originating fromcomplex spatial dispersion is the self-collimated propagationof light beam in PCs. This interesting phenomenon, an inci-dent light propagating with almost no diffraction along adefinite direction,9could provide a new way to control theflow of light in PCs.Kosaka et al. experimentally demonstrated the self-collimated propagation of light in PCs and Witzens et al.theoretically studied the propagation properties of self-collimated beams in detail.9,10In recent studies, it has beenreported that the bending and splitting of self-collimatedbeams are possible.11–14It has been also reported that thefrequency range in which the self-collimation phenomenonoccurs is sufficient for this effect to be a basis for photonicintegrated circuits PICs.14 To implement PICs based on theself-collimation phenomena, however, a simple and efficientway to bend and split self-collimated beams in planar PCs isrequired.In this letter, we show that self-collimated beams canbe easily bent and split by introducing line defects in PCstructures and moreover, the power ratio between two splitself-collimated beams can be controlled systematically byvarying the radii of rods or holes in the line defect. We alsoshow that self-collimated beams can be effectively steered byarranging line defects appropriately.When light is incident from an optically dense mediumhigh refractive index, nh onto a less dense medium lowrefractive index, nl, the incident wave is totally reflectedback into the denser medium at the interface provided thatthe incident angle is larger than the critical angle given byc=arcsinnl /nh. Our idea is based on the fact that self-collimated beams can also be totally reflected at the interfaceof a PC and air, where a PC and air corresponds to an opti-cally dense and a less dense medium, respectively.11Whenlight undergoes total internal reflection, the field amplitudeof light decays exponentially into the less dense medium.Thus, it is expected that self-collimated beam would be to-tally reflected at the interface of a PC and an air layer offinite thickness, the line defect created by removing a fewrods in a row. Such total reflection can cause self-collimatedbeam to bend inside a PC. What is more, self-collimatedbeam can also be partially reflected by a line defect formedby reducing the radii of rods instead of eliminating themtotally, due to the tunnelling of the evanescent wave. Hencewe expect that an incoming self-collimated beam can be splitinto the transmitted and reflected beams by the line defect.In order to verify our conjecture, we first consider a 2Dsquare lattice PC composed of dielectric rods in air and thecase of a 2D square lattice PC composed of air holes in adielectric background follows next. The radius and dielectricconstant of rods are taken as 0.35a and 12.0, where a is thelattice constant. We restrict our attention to the case of theE-polarized mode whose electric-field is parallel to the rodaxes. To find the frequency range in which the self-collimation phenomenon occurs and the propagation direc-tion of light beam, we calculated the equifrequency contoursEFCs as a function of k. It is known that, in an inhomoge-neous medium, the propagation direction of light is identicalto the direction of group velocity of light given by vg=kk, which means that the group velocity is perpen-dicular to the EFC.15We employed the plane wave expansionmethod to plot the EFCs of the 2D PC.16We found from theEFC calculations that the self-collimation phenomenon oc-curs when lights of frequencies around f =0.194c /a, where cis the speed of light in free space, propagate along the 11direction of square lattice M direction. We also found thatself-collimated beam can be totally reflected at the PC-airinterface created by truncating a PC along the 10 directionof square lattice X direction because of the conservationof momentum component parallel to the interface. Thus self-collimated beams can be bent at the 10 PC-air interface.aElectronic mail: hypark@kaist.ac.krbElectronic mail: cskee@gist.ac.krAPPLIED PHYSICS LETTERS 87, 181106 20050003-6951/2005/8718/181106/3/$22.50 © 2005 American Institute of Physics87, 181106-1Downloaded 27 Oct 2006 to 143.248.16.71. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspFrom the finite-difference time-domain FDTD simulationswith the perfectly matched layer absorbing boundaryconditions,17,18we observed that the 10 PC-air interfaceyields the 90° bending of self-collimated beam at the fre-quencies in the vicinity of f =0.194c /a. We also observedthat, when self-collimated beam undergoes total internal re-flection at a 10 PC-air interface, the field amplitude decaysvery rapidly into air, becoming negligible at a distancewithin one lattice constant. Thus an air layer created by re-moving a few rods in a row, a line defect, is expected to giverise to the total internal reflection. Figure 1 shows that a linedefect created by removing 15 rods in the X direction al-most totally reflects self-collimated beam propagating alongthe M direction. However, we observed that when the airlayer is doubled, i.e., two rows of line defects 15 rods eachare introduced, the structure yields total internal reflection. Inthe FDTD simulations, we considered 202a202a of thesquare PC composed of dielectric rods with r=0.35a and aGaussian beam with the full width at half maximum 5a wasused. We excited a monochromatic wave of the frequencyf =0.194c /a and launched it into the PC along the M direc-tion. Then we monitored the propagation of input beam byobserving the spatial variation of the electric- and magnetic-field distributions as a function of time.Based on the line defect induced beam bending phenom-enon, we constructed a line defect beam splitter shown inFig. 2a, which is composed of 15 rods aligned in the Xdirection with the radii rd different from those of host rods.One can expect a gradual transition from no reflection tototal reflection to occur as rd is varied from 0.35a to zero. Inorder to study the quantitative relation between the value ofrd and the power ratio between two split beams, we mea-sured the time averaged power before and after the beamsplitting by integrating the Poynting vector across the beamcross section. The reflected and transmitted powers are nor-malized with respect to the input power and the results areshown in Fig. 2b. The result clearly shows that an incomingself-collimated beam can be split into the reflected and trans-mitted beams with an arbitrary power ratio by adjusting thevalue of rd. The sum of the reflected and transmitted power isalways less than the input power by about 3%. This smallamount of propagation loss may have been caused by thescattering in the splitting structure which has been alsoobserved in the previous study.12We also studied the beam splitting phenomenon inducedby a line defect in hole-type PCs. We employed a 2D squarelattice of air holes with the hole radius r=0.25a and thedielectric constant of high index background =12.0. Thecalculations were done for the H-polarized mode which hasthe magnetic-field parallel to the hole axes. We observed thata line defect can cause the beam splitting phenomena for thelight of frequencies near f =0.183c /a propagating in the Mdirection. The time averaged power before and after thebeam splitting is obtained for the light of f =0.183c /a andthe resulting normalized power is shown in Fig. 3. The insetrepresents the top view of the beam splitting structure. Thisresult also shows that the power ratio between the reflectedand transmitted beams can be controlled systematically byvarying the radii of air holes rd in the line defect.The line defect induced beam bending and splitting ef-fects can be useful in routing light beams in 2D PCs. Wedesigned a one-to-four beam splitter composed of four linedefects in rod-type PC as shown in Fig. 4. Radii of rods inthe line defects are chosen to be 0.301a, 0.292a, 0.275a, and0, successively, so that an input beam is divided into fouroutput beams of an equal power. We found that the power ofeach output beam can be easily controlled by varying theradii of rods in the corresponding line defect.As concluding remarks, as the self-collimation phenom-enon occurs due to the dispersion properties of propagationmodes in PCs, not due to the existence of PBGs, low indexFIG. 1. Color online Simulated spatial distribution of the steady-stateelectric-field of the E-polarized mode at f =0.194c /a when light propagatesalong the M direction. Electric-field is parallel to the rod axes. Note thatthe line defect created by removing 15 rods in the X direction acts as atotal internal reflection mirror for self-collimated beams propagating alongthe M direction. The outer dark gray region means the absorbing regionwith the PML boundary condition and arrows indicate the propagationdirection.FIG. 2. Color online a Simulated spatial distribution of the steady-stateelectric-field of the E-polarized mode at f =0.194c /a in the line defect beamsplitter composed of 15 defect rods with the radii rd aligned in the Xdirection. b Reflected and transmitted powers which are normalized withrespect to the input power as a function of rd.181106-2 Lee et al. Appl. Phys. Lett. 87, 181106 2005Downloaded 27 Oct 2006 to 143.248.16.71. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspcontrast PC structures can exhibit the diffractionless propa-gation of light.19Thus, optical devices based on the bendingand splitting of self-collimated beams may be realized byutilizing low refractive index dielectric materials such aspolymers. Recently, it has been demonstrated that nanoim-print lithography is suitable for periodic patterning of variouspolymer thin-films20and passive optical devices for the in-frared and visible wavelength range are realized on 150 mmwafers with feature sizes of about 50 nm.21Hence the peri-odically patterned polymer thin-films fabricated by such atechnique may be useful for the implementation of the opti-cal devices based on self-collimated beams. Besides, pro-posed beam bends and splitters have simple geometric struc-tures and clear operating mechanism. Thus our results mayhave an important role in the realization of PICs based on theself-collimated light propagation.This work was partially supported by the Korea Scienceand Engineering Foundation KOSEF through the QuantumPhotonic Science Research Center.1J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals:Molding the Flow of Light Princeton University Press, Princeton, NJ,1995.2E. Yablonovitch, Phys. Rev. Lett. 58, 2059 1987.3S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 78,3294 1997.4S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos,Science 282, 274 1998.5A. Chutinan, M. Okano, and S. Noda, Appl. Phys. Lett. 80, 1698 2002.6F. Du, Y.-Q. Lu, and S.-T. Wu, Appl. Phys. Lett. 85, 2181 2004.7H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato,and S. Kawakami, Phys. Rev. B 58, R10096 1998.8S. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, Phys. Rev. Lett.90, 107402 2003.9H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato,and S. Kawakami, Appl. Phys. Lett. 74, 1212 1999.10J. Witzens, M. Loncar, and A. Scherer, IEEE J. Sel. Top. QuantumElectron. 8, 1246 2002.11X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 2003.12S. Shi, A. Sharkawy, C. Chen, D. M. Pustai, and D. W. Prather, Opt. Lett.29, 617 2004.13D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman,A. Sharkawy, G. J. Schneider, and J. Murakowaki, Opt. Lett. 29, 502004.14D. M. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. W. Prather, Opt.Express 12, 1823 2004.15P. Yeh, J. Opt. Soc. Am. 69, 742 1979.16K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 31521990.17A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method Artech House, Boston, 1995.18J.-P. Berenger, J. Comput. Phys. 114, 185 1994.19R. Liew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs,D. Schelle, E.-B. Kley, S. Nolte, and A. Tünnermann, Appl. Phys. Lett.85, 5854 2004.20S. Y. Chou, P. R. Krauss, W. Zhang, L. Guo, and L. Zhuang, J. Vac. Sci.Technol. B 15, 2897 1997.21J. Seekamp, S. Zankovych, A. H. Helfer, P. maury, C. M. SotomayorTorres, G. Bottger, C. Ligura, M. Eich, B. Heidari, L. Montelius,and J. Ahopelto, Nanotechnology 13, 581 2002.FIG. 3. The power spectra for the split beams by a line defect in a 2D squarePC composed of air holes with the radius of r=0.25a as a function of rd, theradii of defect holes. The inset represents the top view of the beam splitterstructure. The calculations were done for the H-polarized mode at the fre-quency f =0.183c /a propagating in the M direction with the magnetic-fieldparallel to the hole axes. The dielectric constant of high index backgroundis 12.0.FIG. 4. Color online Simulated spatial distribution of the steady-stateelectric-field of the E-polarized mode at f =0.194c /a in a one-to-four beamsplitter composed of four line defects. Radii of rods in each line defect arechosen to be 0.301a, 0.292a, 0.275a, and 0, successively, so that an inputbeam is divided into four output beams of an equal power. Arrows indicatethe propagation direction of the input and four output beams.181106-3 Lee et al. Appl. Phys. Lett. 87, 181106 2005Downloaded 27 Oct 2006 to 143.248.16.71. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals

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Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals

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