General methods for designing single-mode planar photonic crystal waveguides in hexagonal lattice structures

Transcription

1 Generl methods for designing single-mode plnr photonic crystl wveguides in hexgonl lttice structures N. Wu, M. Jvnmrd, B. Momeni, M. Soltni nd A. Adibi School of Electricl nd Computer Engineering, Georgi Institute of Technology, Atlnt, GA USA Y. Xu nd R. K. Lee Deprtment of Applied Physics, Cliforni Institute of Technology, Psden, Cliforni 91125, USA Abstrct: We systemticlly investigte nd compre generl methods of designing single mode photonic crystl wveguides in two-dimensionl hexgonl lttice of ir holes in dielectric mteril. We pply the rther generl methods to dielectric-core hexgonl lttice photonic crystls since they hve not been widely explored before. We show tht it is possible to obtin single mode guiding in limited portion of the photonic bndgp of hexgonl lttice structures. We lso compre the potentils of different photonic crystl lttices for designing single-mode wveguides nd conclude tht tringulr lttice structures re the best choice Opticl Society of Americ OCIS codes: ( ) Wveguides; ( ) Guided wves References nd links 1. E. Yblonovitch, Inhibited spontneous emission in solid stte physics nd electronics, Phys. Rev. Lett. 58, (1987) 2. S. John, Strong locliztion of photons in certin disordered dielectric superlttices, Phys. Rev. Lett. 58, (1987). 3. A. Mekis, J. C. Chen, I. Kurlnd, S. Fn, P. R. Villeneuve, nd J. D. Jonnopoulos, High trnsmission through shrp bends in photonic crystl wveguides, Phys. Rev. Lett. 77, (1996). 4. S. Lin, E. Chow, V. Hietl, P. R. Villeneuve, nd J. D. Jonnopoulos, Experimentl demonstrtion of guiding nd bending electromgnetic wves in photonic crystl, Science 282, (1998). 5. T. Bb, N. Fuky, nd J. Yonekur, Observtion of light propgtion in photonic crystl opticl wveguides with bends, Electron. Lett. 35, (1999). 6. M. Tokushim, H. Kosk, A. Tomit, nd H. Ymd, Lightwve propgtion through 120 degrees shrply bent single-line-defect photonic crystl wveguide, Appl. Phys. Lett. 76, (2000). 7. A. Adibi, Y. Xu, R. K. Lee, A. Yriv, nd A. Scherer, Properties of the slb modes in photonic crystl opticl wveguides, J. Lightwve Technol. 18, (2000). 8. A. Adibi, R. K. Lee, Y. Xu, A. Yriv, nd A. Scherer, Design of photonic crystl opticl wveguides with singlemode propgtion in the photonic bndgp, Electron. Lett. 36, (2000). 9. A. Adibi, Y. Xu, R. K. Lee, A. Yriv, nd A. Scherer, Guiding mechnism in dielectric-core crystl opticl wveguides, Phys. Rev. B 64, (1-4) (2001). 10. A. Adibi, Y. Xu, R. K. Lee, M. Loncr, A. Yriv, nd A. Scherer, Role of distributed Brgg reflection in photonic-crystl opticl wveguides, Phys. Rev. B 64, (1-4) (2001). 11. D. Cssgne, C. Jounin, nd D. Bertho, Hexgonl photonic-bnd-gp structures, Phys. Rev. B 53, (1996). 12. D. Cssgne, C. Jounin, nd D. Bertho, Photonic bnd gps in two-dimensionl grphite structure, Physics. Rev. B 52, R2217-R2220 (1995). 13. K. S. Yee, Numericl solution of initil boundry vlue problems involving Mxwell s equtions in isotropic medi, IEEE Trns. Antenns Propg. AP-14, (1966). 14. C. T. Chn, Q. L. Yu, nd K. M. Ho, Order-N spectrl method for electromgnetic-wves, Phys. Rev. B 51, (1995). (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1371

2 15. J. P. Brenger, A perfectly mtched lyer for the bsorption of electromgnetic wves, J. Comput. Phys. 114, (1994). 16. A. Tflov nd S. C. Hgness, Computionl electrodynmics: The finite-difference time-domin method, (Artech House, 2000). 1. Introduction Recently, photonic crystls [1-2] hve inspired much interest due to their potentil for controlling the propgtion of light. In prticulr, mny studies hve been devoted to the possibilities of using line defects in photonic crystls s opticl wveguides. Mny interesting properties such s efficient guiding of light through shrp bends hs been recently proposed [3] nd demonstrted t microwve [4] nd opticl [5-6] frequencies. In these studies, the PBG wveguides were generlly constructed by removing one row of either ir columns or dielectric rods, which usully results in multimode guiding. Single mode wveguides, however, re more desirble for prcticl pplictions. We recently studied [7] more generl type of PBG wveguide, which consists of dielectric slb plced between two PBG mirrors. Such PBG wveguide cn be mde single mode by choosing n pproprite thickness for the dielectric slb. However, the djustment of the slb thickness generlly perturbs the lttice symmetry of the photonic crystls nd renders the design of wveguide bends difficult. The centers of the ir holes in single-mode wveguide for prcticl pplictions (e.g., ll-opticl circuits) must be on the sme lttice, which is not the cse for these single-mode wveguides. To ddress this problem, we recently proposed nother method for designing single mode photonic crystl wveguides which involves incresing the rdii of the ir holes next to the guiding region [8]. This method is bsed on the observtion tht the propgtion of the guided modes of PBG wveguide is minly controlled by the two rows of ir holes next to the guiding region [7,9,10]. Three types of two-dimensionl photonic crystl lttices tht re of prcticl interests re squre lttice, tringulr lttice, nd hexgonl lttice. Although the first two cses hve been widely investigted, there hve been only few reports regrding hexgonl lttice structures [11-12]. These existing reports re focused mostly on the hexgonl lttice of dielectric rods in ir. For plnr wveguiding pplictions, it is more pproprite to use the two-dimensionl photonic crystls formed by etching ir cylinders in plnr dielectric medium, since guiding in the third dimension (perpendiculr to the plne of periodicity) cn be chieved by totl internl reflection. In the serch for n optimum PBG wveguide design, it is necessry to include the cse of dielectric-core hexgonl lttice PBG wveguides in our considertions, which will be investigted in detils in this pper. In prticulr, we demonstrte in this pper tht it is possible to design hexgonl PBG wveguide with only one guided mode in the photonic bndgp. We study nd compre two different methods for chieving this gol, with both prcticl nd fbriction relted issues tken into considertions. Finlly, we compre the potentils of hexgonl lttice photonic crystls for designing single-mode wveguides with those of tringulr nd squre lttice structures. 2. Simultions The min structure we use in the nlysis of this pper is the photonic crystl wveguide formed by introducing line defect in perfect two-dimensionl hexgonl-lttice photonic crystl. The unit cell of the 2D photonic crystl nd one period of the PBG wveguide investigted in this pper re shown in Figs. 1 nd 2 respectively. To nlyze the PBG structures s well s the wveguides, we use two-dimensionl finite difference time domin (FDTD) technique [13], nd n order-n spectrl method [14]. To obtin the photonic crystl bnd structures, we use 2D unit cell of the photonic crystl with Bloch boundry conditions t ll boundries, which is shown in Fig. 1. (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1372

3 0.5 Normlized Frequency, ω/(2πc) 0.4 TM 0.3 TE TM 0.2 TE 2r Normlized Rdius, r/ Fig. 1. Mps of photonic bndgp for TE (electric field perpendiculr to the computtion plne) nd TM polriztions in two-dimensionl hexgonl-lttice structure of ir cylinders in GAs (ε=12.96) s functions of the rtio r/. As shown in Fig. 2, we use Bloch boundry conditions on the left nd right sides to reduce the infinite PBG wveguides into single unit cell, nd use perfectly mtched lyers (PML) [15] on the top nd bottom to simulte the open spce surrounding the guiding structures. In the clcultions of this pper, the speed of light in vcuum (c) is normlized to 1, nd ll sptil dimensions re in the units of clcultion cells. The dielectric mteril is ssumed to be GAs (reltive permittivity ε=12.96) throughout the pper. 3. Design of single mode PBG wveguides PBG wveguides with single-mode propgtion in the photonic bndgp cn be used to route opticl signls nd re essentil for ny photonic crystl circuit with dense integrtion of functionlities such s demultiplexing, guiding, nd filtering. Figure 1 shows the gp mp (i.e., the frequency extent of the PBG for different vlues of the rdius of the ir column) for the first two bndgps of the 2D hexgonl-lttice photonic crystl tht we investigte in this pper. The TE (or TM) polriztion in Fig. 1 represents the electromgnetic modes with electric (or mgnetic) field norml to the computtionl domin. To void the overlp of the ir columns (which complicte the fbriction of the photonic crystl), we limit the rdii of the ir holes (r) to r < 0.5, where is the lttice constnt s shown in Fig. 1. In order to mximize the guiding bndwidth, we choose the prmeters of the hexgonl photonic crystls tht led to the lrgest bndgp, which is the second TM bndgp with r=0.38. As cn be seen from Fig. 1, the second bndgp covers the normlized frequency (ω) rnge of 0.30<ω/(2πc)< Conventionl dielectric-core PBG wveguides in squre nd tringulr lttices, formed by removing one row of ir holes generlly result in more thn one guided mode in the PBG. The multi-mode nture of the conventionl PBG wveguides is due to the lrge thickness of the middle dielectric slb, direct result of removing one row of ir holes [7-8]. The sme sitution holds for conventionl wveguides in hexgonl lttice photonic crystls. Therefore, the simplest wy to form single mode PBG wveguides, which is lso the first pproch we pursue, is to reduce the thickness of the middle slb. Figure 2 shows how this ide cn be used to design single-mode PBG wveguide in hexgonl-lttice photonic crystl. Figure 2() shows one unit cell of PBG wveguide mde by plcing dielectric slb of thickness d between two PBG mirrors, ech consisting of two-dimensionl hexgonl lttice of ir columns in GAs. The period of the hexgonl lttice nd the rdius of ech ir column (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1373

4 re nd r=0.38, respectively. In the simultions of this section, we choose =29 clcultion cells. Figure 2(e) shows the dispersion digrms for the two dominnt TM modes (mgnetic field norml to the computtion plne) of the PBG wveguides with different vlues of slb thickness d. Typicl mgnetic field ptterns of the four modes in Fig. 2(e) re shown in Figs. 2(), (b) (c), nd (d). The field ptterns of Figs. 2() nd (c) hve odd symmetry with respect to the center of the PBG wveguide, while those of Figs. 2(b) n (d) hve even symmetry. The slb thickness of the conventionl PBG wveguide (mde by removing one row of ir holes) is d=0.66. For this thickness, two modes (one with even symmetry nd one with odd symmetry nd represented in Fig. 2(e) s dshed lines) re guided in the PBG. By reducing the slb thickness d both modes re shifted to higher frequencies nd for d>0.52, the odd mode is pushed up nd seprted from the even mode, s shown in Fig. 2(e). Therefore, we cn obtin single-mode guiding in the PBG by reducing the slb thickness to d=0.52. b c d e d c) /(2π Normlized Frequency, ω b c d Slb width=0.66 Slb width=0.52 PBG Single mode Normlized Phse, k z Fig. 2. ()-(d) One period of the wveguide structure nd mgnetic field pttern in PBG wveguide mde by putting dielectric slb of thickness d between two hexgonl lttice PBG mirrors. The rdii of ir holes in ll cses re r=0.38, with being the distnce between the centers of nerest neighbor ir holes. The thickness of the middle slb in () nd (b) is d=0.66, nd tht in (c) nd (d) is d=0.52. The dispersion digrms for the two dominnt guided TM modes of these PBG wveguides clculted by the FDTD technique re shown in (e). The dielectric mteril is GAs with reltive permittivity ε= The distnce between the centers of the nerest neighbor holes nd the rdius of ech hole in the photonic crystl re =29 clcultion cells nd r=0.38, respectively. The frequency rnges of bndgp (PBG) nd single-mode guiding for d=0.52 re shown in the figure. (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1374

5 As discussed previously, mjor disdvntge of reducing the slb thickness of the PBG wveguides is tht it breks the discrete trnsltionl symmetry of the photonic crystl nd complictes the design of wveguide bends, especilly when cscding multiple bends. To void this difficulty, we investigte the second pproch, where the ir hole centers remin on the sme photonic crystl lttice nd the reduction of the slb thickness is chieved vi incresing the rdii of the ir holes djcent to the middle slb [8] s shown in Fig. 3. The PBG wveguide in Fig. 3() is formed by removing the first nd third rows djcent to the slb nd enlrging the rdius of the second row to r=0.73 on both sides (bove nd below) of the middle slb. The field ptterns nd the dispersion digrms of the TM modes re shown in Fig. 3. Figure 3(d) demonstrtes single mode guiding in this PBG wveguide. b c d ir ir 0.35 Corrugted Wveguide 2w ir c) /(2π Normlized Frequency, ω b Hexgonl Lttice PBG w veguide c Normlized Phse, k z PBG Fig. 3. () One unit cell of the wveguide structure nd the mgnetic field ptterns for different TM modes of: (), (b) specilly designed wveguide s discussed in the text, nd (c) the equivlent corrugted wveguide mde by considering only two rows of the photonic crystl djcent to the middle slb with ir bove these two rows. (d) Dispersion digrms of the dominnt guided modes corresponding to hexgonl lttice wveguide nd corrugted wveguide shown in (), (b), nd (c). The dielectric mteril in ll wveguides is GAs (εs=12.96). The cldding region of the corrugted wveguides is ir (ε=1). The period of the hexgonl lttice nd the rdius of ech ir column in the PBG wveguide re =29 clcultion cells nd r=0.38, respectively. (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1375

6 An interesting feture of the PBG wveguide in Fig. 3 is the substntil mount of electromgnetic field outside the middle slb. We believe this is due to the cretion of multiple line defects in the PBG structure tht re coupled to ech other. As described erlier, the PBG wveguide in Fig. 3() is mde by removing three rows of ir holes nd incresing the rdii of ll ir holes in two other rows. The enlrgement of the ir holes in Fig. 3() simultneously cretes three defect chnnels: one between the two middle (lrge) holes, one between the lrge holes nd the djcent (norml) holes bove the middle slb, nd one between the lrge holes nd the djcent holes below the middle slb. These line defects forms three PBG wveguides tht re strongly coupled together due to their close proximity. As result, the overll modes of the structure re extended over these three wveguides. Further incresing the rdii of the ir holes next to the middle slb cn shift the second mode upwrd nd increse the single-mode guiding bndwidth. However, fbriction of the structure with too lrge of hole dimeter my be difficult, due to the smll spcing between djcent holes. From the discussion bove, it is cler tht the min drwbck of the hexgonl lttice photonic crystls for designing single-mode wveguides is the reltive positions of the ir holes, where removing one row of holes cuse n symmetric nd highly corrugted structure. In order to mke less corrugted structure, either hole periodicity below or bove the wveguide slb must be broken (method 1, Fig. 2) or n extensive line defect over severl rows of ir holes (method 2, Fig. 3) must be introduced. In Fig. 3(b), the guided mode of the PBG wveguide is minly confined within the middle slb, which suggests tht for these modes the PBG wveguide cn be pproximted by n equivlent corrugted wveguide tht only retins the two rows of ir holes djcent to the middle slb [10], s shown in Fig. 3(c). The dispersion of the guided mode in this corrugted wveguide, s shown in Fig. 3(d), is similr to tht of the corresponding PBG wveguide mode in portion of the bndgp. The differences between the dispersion digrms of the two wveguide re cused by the multi-line defect nture of the PBG wveguide tht mkes it different from the corrugted wveguide. It is lso interesting to note tht the lower mode of the PBG wveguide (Fig. 3()) does not correspond to the corrugted wveguide mode. This cn be explined by the observtion tht the wveguide mode in Fig. 3() is result of coupling between three defect chnnels in the guiding structure, which mkes it essentilly different from the corrugted wveguide mode. 4. Discussion The results of Section 3 suggest tht hexgonl lttice of ir holes in dielectric mteril cn be used for building integrted photonic circuits, nd it is possible to design singlemode PBG wveguide within the frequency bnd gp. However, there re severl drwbcks in using the hexgonl lttice photonic crystls for integrted optics pplictions. First, single-mode propgtion cn be chieved only within prt of the PBG, which my limit the frequency bndwidth of the integrted optics devices. Second, we demonstrte tht designing single mode wveguide in hexgonl lttice structures relies on the following two pproches: (1) reducing the center slb thickness, which breks the lttice symmetry nd complictes the design of wveguide bends, nd (2) creting n extensive line defect by modifying the rdius of the ir holes djcent to the center slb mteril, which leds to complicted field distribution nd possibly coupling between different guiding chnnels. Hving discussed the benefits nd drwbcks of the hexgonl lttice PBG structures, it is instructive to compre it with other photonic crystl lttices. We begin with the normlized width of the PBG, which is defined s the size of the bndgp divided by the gp center frequency. The mximum normlized gp widths of three different lttices of ir holes in GAs re compred in Tble 1. The normlized gp width depends on the rdius of the ir holes (r), which cnnot be mde too lrge due to fbriction difficulties. In order to ensure enough seprtion between the neighbor ir holes, we only consider the cses with r<0.43, where is the lttice constnt. The mximum vlue of the normlized gp width is then (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1376

7 clculted for different lttices. For ech photonic crystl lttice, the bndgp for both TE nd TM polriztions over the entire rnge of hole rdius (r) hs been clculted, nd the mximum vlue of the gp width nd its corresponding polriztion hs been reported in Tble 1. The rdius of ir holes corresponding to the mximum normlized gp width is lso shown in Tble 1. Tble 1. Comprison of the mximum photonic bndgp size nd the corresponding rdius of the ir holes in different photonic crystl lttices. The rdius of the ir holes nd the distnce between the centers of the nerest neighbor ir holes re represented by r, nd, respectively. Photonic crystl lttice Tringulr Squre Hexgonl Polriztion for mximum bndgp TM TM TM Gp Rtio (size of PBG/center frequency) ~52% ~17% ~10% Rdius of ir holes corresponding to lrgest gp rtio (r/) The results shown in Tble 1 suggest tht tringulr lttice photonic crystls re the best cndidtes for pplictions in integrted photonic circuits, where the mximum PBG size is desired. Furthermore, the lrge PBG in tringulr lttices is obtined t smller hole rdii compred to other types of lttices. This llows for more flexibility in designing PBG wveguides (nd cvities). Recently, there hve been reports [7] tht it is possible to design PBG wveguides in squre nd tringulr lttices with only one mode in the entire PBG. Combining these studies with the results in the previous section, we believe the best choice for designing single-mode dielectric-core plnr wveguides for prcticl pplictions is tringulr lttice photonic crystl. We lso emphsize tht the two methods for designing single-mode PBG wveguides described in this pper cn be pplied to ny photonic crystls structures (tringulr lttice, squre lttice, etc.). Note tht the results of this pper were clculted using 2D-FDTD technique. Thus, the ccurcy of the results is determined by the well-known ccurcy of the technique [16]. More ccurte results cn be obtined by 3D simultions nd tking the finite thickness of the structure into ccount. While 3D simultion results in the modifiction of the PBG nd the single-mode wveguide frequency rnge, the generl conclusions of this pper, i.e., the superiority of tringulr lttice for designing PBG wveguide nd structurl problems of the PBG wveguides in hexgonl lttice structures remins intct. 5. Conclusion We presented here two systemtic methods of designing dielectric-core single-mode PBG wveguides in two-dimensionl hexgonl lttice photonic crystls. We showed tht the properties of the guided modes in these wveguides cn be modified by chnging the geometry of only the ir holes tht re djcent to the middle slb. Single mode guiding cn be obtined in t lest portion of the photonic bndgp by chnging the geometry of the photonic crystl in the guiding region. In n idel structure, extr modes cn be pushed out of the bndgp by either reducing the thickness of the middle slb or by incresing the rdii of the ir holes djcent to the middle slb. The second method is preferble for pplictions tht require wveguide bends s well since it does not modify the loctions of the centers of the ir holes (i.e., it conserves the periodicity of the centers of the ir holes). Both of these methods hve limittions in pplying to hexgonl lttice structures due to the geometry of the ir holes which result in highly corrugted wveguides. The existence of some unwnted modes due to the coupling of multiple line defects in hexgonl lttice structures is n exmple of such limittions. We lso compred different 2D photonic crystl lttices for designing single-mode wveguides. Our results show tht tringulr lttice photonic crystls re the best cndidtes, since they hve the lrgest bndgp nd lrger frequency rnge for single mode guiding. (C) 2003 OSA 16 June 2003 / Vol. 11, No. 12 / OPTICS EXPRESS 1377