1500 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 8, AUGUST 1999

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1 1500 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 8, AUGUST 1999 Analysis of Finite 2-D Photonic Crystals of Columns and Lightwave Devices Using the Scattering Matrix Method Jun Yonekura, Mitsutaka Ikeda, and Toshihiko Baba, Member, IEEE Abstract The scattering matrix method was applied to the analysis of finite two-dimensional photonic crystals and lightwave devices. Results indicated that 1) the light transmission at the photonic band gap (PBG) is suppressed to less than 030 db in the densely packed and honeycomb crystals, both of which are composed of only four rows of unit cells of semiconductor columns and 2) this PBG effect is weakened to half when the nonuniformity from 10 to 30% is brought to the diameter of columns. Also, the light propagation in defect waveguides with abrupt bends, a branch and a directional coupler was demonstrated by this method. It was found that the coupling loss at the input end of the waveguide is drastically changed by the shape of the input end. The reflection loss at 60 bends was estimated to be less than 1 db, and the excess loss at an abrupt Y-branch was estimated to be db, depending on the frequency of the input wave. The demultiplexing and power dividing functions were expected in a directional coupler with a submicron coupling length, which is considered to be due to antiguide characteristics of the waveguides. Index Terms Lightwave circuit, optical components, optical waveguides, photonic crystal, scattering matrices. I. INTRODUCTION PHOTONIC crystals are artificial periodic structures in which optical atoms are arranged with a period of the order of half the optical wavelength [1]. The most attractive property of photonic crystals is the photonic bandgap (PBG), frequency range that inhibits light emission and propagation. When an irregular atom is introduced into a uniform crystal, a resonant mode appears in the PBG. Similarly, when a series of irregular atoms are introduced, a guided mode appears in the PBG due to the coupling of resonant modes. These properties are expected to provide novel light emitters and waveguides, respectively. Manuscript received December 18, 1998; revised April 30, This work was supported in part by a Grant-in-Aid # , the Priority Area The fabrication of photonic crystals and the control of electromagnetic radiation in photonic crystals, by the Ministry of Education, Science, Sports, and Culture, and Support Center for Advanced Telecommunications Technology Research. J. Yonekura was with the Division of Electrical and Computer Engineering, Baba Laboratory, Yokohama National University, Yokohama Japan. He is now with Canon, Inc., Tokyo Japan. M. Ikeda was with the Division of Electrical and Computer Engineering, Baba Laboratory, Yokohama National University, Yokohama Japan. He is now with Fujitsu Ltd., Kawasaki Japan. T. Baba is with the Division of Electrical and Computer Engineering, Baba Laboratory, Yokohama National University, Yokohama Japan. Publisher Item Identifier S (99) So far, propagation characteristics of a photonic crystal waveguide have been theoretically studied with the demonstration of 90 bends. However, to apply photonic crystals to a functional lightwave circuit, it is necessary to investigate those for various waveguide patterns. So far, the following methods have been used for the analysis of photonic crystals: 1) the plane wave expansion (PWE) method for the photonic band analysis of infinite and uniform crystals [2] [4] and for the resonant mode analysis of crystals with irregular atoms [5], 2) the transfer matrix method [6] and the scattering matrix method [7], [8] for the light transmission analysis of finite crystals, and 3) the finite difference time domain (FDTD) method [9] for the analysis of photonic crystal waveguides. Generally, the PWE and FDTD methods need a large amount of computer resource and a long computation time, and moreover the accuracy of results for round structures, which are often seen in photonic crystals, is still a subject for discussion. The transfer matrix method cannot be used for the calculation of electromagnetic fields but only for the calculation of transmission spectra, so that it is not helpful for the physical understanding of photonic crystal waveguides. Concerned with these points, the scattering matrix method is attractive, although the application of this method is limited to two-dimensional (2-D) crystals. It gives both field distributions and transmission spectra against a static field excitation. According to our comparison, the computation time for a field distribution in a finite 2-D crystal by this method is shorter than one-tenth of that by the FDTD method. In addition, this method almost guarantees precise results, as explained in [7]. In [8] has been reported the application of this method to finite and uniform 2-D crystals with and without defects. However, various structures of crystal with arbitrary polarization, nonuniform crystals and photonic crystal waveguides have not been analyzed by this method yet. In this study, we applied this method to several different crystals and waveguides with two orthogonal polarizations, and found some unique properties. In addition, we modified several conventional lightwave devices to those composed of photonic crystal waveguides and demonstrated their functions. In Section II, the scattering matrix method described in [7] and [8] is reviewed. Then, in Section III, field distributions and transmission spectra in finite crystals composed of uniform columns are presented. In Section IV, how the nonuniformity in columns, which is brought in actual fabrications, affects transmission characteristics is discussed. In Section V, res /99$ IEEE

2 YONEKURA et al.: ANALYSIS OF FINITE 2-D PHOTONIC CRYSTALS 1501 located outside the columns, is generally expressed as (1) Fig. 1. General calculation model of 2-D photonic crystal. onator and waveguide characteristics in a uniform crystal with defects as irregular atoms are presented. Finally, in Section VI, lightwave devices, i.e., bends, a Y-branch and a directional coupler are discussed. II. SCATTERING MATRIX METHOD Fig. 1 shows the general calculation model of 2-D photonic crystals and the coordinate system. Here, arbitrary size optical atoms are arranged arbitrarily without overlapping. Fields are calculated by solving the Helmholtz equation using the Fourier Bessel expansion of scattered fields from all atoms. A precise result is expected within a short computation time when circular atoms are assumed, for which the analytical expression has been derived for the scattering matrix [7]. In [8], columns having the refractive index and the infinite height in the -direction were regarded as circular atoms. The background was a vacuum with a refractive index of 1.0. We considered the same condition in this paper to make the verification of our results easier. Now, let us define the transverse electric (TE) as the polarization with the electric field vector inside the plane, while the transverse magnetic (TM) as that with the magnetic field vector inside the plane. They correspond to the -polarization and the -polarization, respectively, which were defined in [10]. The component of the field at point is limited to the magnetic field for the TE, while the electric field for the TM. When is where is the field of an incident wave from excitation points, and are the index and total number of columns, respectively, is the th-order Hankel function of the first kind, corresponding to the time-dependent function, is the wave number in a vacuum, which is related to the vacuum wavelength, the angular frequency, and the vacuum velocity of light as, is the distance from column to point, and (P) is the angle of line P against the -axis. Assuming the plane wave as an incident wave, shown in (2) (6) at the bottom of the page, have been derived for the matrix =[ ] where brackets denote matrices, is the angle of the incident wave against the -axis, is the angle of line against the -axis, is the distance between columns and, and are the Bessel function of the first kind and its derivative, respectively, and is the wave number in columns. is known as the scattering matrix which shows the boundary condition for the scattered field at column [11]. It has been indicated that the summation with respect to in (1) can be reduced to from 2 to 2 or from 3 to 3 without increasing the error larger than 1% at PBG frequencies [8]. We verified this fact and estimated the error to be 10% and 0.1% for to and to, respectively, at the highest frequency assumed in this paper. It was less than 3% for to at PBG frequencies. Thus, we employed this extent of to achieve enough accuracy and a short computation time. Once the matrix is obtained by solving (2) (6), the field distribution is immediately given by (1). When we used a standard personal computer with Pentium II 266 MHz processor, it took nearly 3 min to obtain fields at points around 5 10 columns. The components of fields are obtained by taking the gradient of. The Poynting power spectrum is obtained from these orthogonal fields with changing the frequency of the incident wave. For a spectrum, the computation time is simply increased by the number of frequencies assumed in the calculation. III. UNIFORM CRYSTALS Fig. 2 shows calculation models of crystals with the triangular lattice, i.e., densely-packed structures A 1 and A 2 and (2) (3) (4) (5) TE TM (6)

3 1502 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 8, AUGUST 1999 Fig. 2. Calculation model of densely-packed photonic crystals A 1 and A 2, honeycomb crystal A 3, and corresponding Brillouin zone. honeycomb structure A 3, and the corresponding Brillouin zone. Overall calculations in this paper, a monochromatic plane wave propagating from to was assumed as an incident wave. This direction of the incident wave corresponds to the X direction in the Brillouin zone for crystals A 1 and A 3, while the J direction for crystal A. The number of unit cells in the -direction was limited to ten for crystals A 1 and A 2 and 5 for A 3 in order to save the computation time. That in the -direction was changed in each calculation. The refractive index was fixed to 3.5, and the diameter of columns 2 and pitch were 0.15 and 0.45 m, respectively, for A 1 and A 2, and 0.15 and 0.58 m, respectively, for A 3. They are typical for semiconductor columns in a vacuum, which exhibit PBG s in the wavelength range of the order of 1 m. For the calculation of transmission spectra, the Poynting power was obtained by averaging the -directed power observed on line Q behind the columns against the incident wave, as shown in Fig. 2. Line Q has the length and lies apart from the crystal end by the distance. For the calculation shown in Sections III and IV, we assumed m and m. The transmittance was defined as the ratio of the so-obtained Poynting power to that of the incident wave. Fig. 3 shows the field distribution for the TE polarization around crystal A 1. The incident wave is strongly reflected and the transmission is suppressed by the crystal when the frequency lies inside the PBG. Due to the finite width of the crystal, the incident wave is scattered so that radial patterns appear at side edges. When the frequency is outside the PBG, the incident wave is almost transmitted through the crystal. It suffers a phase delay in the crystal, which causes the focusing of light near line Q. Fig. 4 shows transmission spectra for the TM polarization. The transmittance takes values over 0 db at some frequencies due to the focusing effect. A strong attenuation of the transmitted wave is seen in the frequency range that accords well with the PBG obtained by the PWE method [12]. As shown in Fig. 4, this attenuation is simply strengthened with the increase of. The attenuation over 30 db is observed when. The value 30 db is equivalent to the power reflectivity over 99.9%, if the lateral scattering of the incident wave is negligible. This reflectivity is enough for a high quality laser cavity and a low loss waveguide. The factor for a laser cavity is of the order of, where and are the refractive index and length of the cavity, Fig. 3. Magnetic field distributions of TE-polarized wave around crystal A 1. For and,!a=2c = 0:96 and 0:40, respectively. respectively. Let us consider a semiconductor microcavity that satisfies. If the reflectivity is over 99.9%, the factor will be higher than This value corresponds to a mirror loss much lower than the standard internal absorption loss of semiconductors. The propagation loss of a waveguide with the width is of the order of [db/ ]. Let us assume m. If the reflectivity is over 99.9%, the loss will be several db/cm. This value is low enough for small lightwave devices, as discussed in Section VI. Thus, photonic crystals are not necessarily required to be infinite, but the performance of PBG s is enough for these applications, even though the number of cells in the crystal is limited. Fig. 4 also compares spectra for crystals A 1 and A 2, assuming. Almost the same frequency range and attenuation are observed for these crystals. In the following, we limit the discussion to crystals A 1 and A 3, which are both in the X direction. Fig. 4 compares spectra of crystal A 1 for the TE and TM polarizations. It is wellknown that crystal A 1 exhibits different PBG s for different polarizations [12]. However, attenuations at these PBG s are almost the same. Fig. 5 shows the field distribution for the TE polarization around honeycomb crystal A 3. Similar to that seen in

4 YONEKURA et al.: ANALYSIS OF FINITE 2-D PHOTONIC CRYSTALS 1503 Fig. 4. Transmission spectra. Compares those for TM polarizations with different M y. Solid and dotted curves indicate crystal A 1 and A 2, respectively. Compares those of crystal A 1 with M y =5for different polarizations. Solid and dashed curves indicate TM and TE polarizations, respectively. Not only in these figures but also in other figures, shadowed range indicates PBG obtained by PWE method. Fig. 5. Magnetic field distributions of TE-polarized wave around crystal A 3. For and,!a=2c = 0:96 and 0:40, respectively. Fig. 3, the light transmission is suppressed by the crystal when the frequency is inside the PBG. The focusing effect is also evident for this crystal when the frequency is outside the PBG. Fig. 6 compares transmission spectra for the two polarizations, assuming. The maximum attenuation from 0.95 to 1.1 is 36 db for the TE and 38 db for the TM. This frequency range also accords with the absolute PBG obtained by the PWE method [12]. IV. NONUNIFORM CRYSTALS Let us suppose the fabrication of columns using some lithographic technique. Normally, the nonuniformity is not brought to the pitch but to the diameter of each column 2. We evaluated the influence of the nonuniformity to the transmission spectra. In this calculation, the radius of each column was denoted as, where is the average radius and is the random value distributing around zero as the Gaussian function with the deviation. Fig. 7 shows spectra for densely-packed crystal A with. The frequent oscillation in spectral curves and indistinct band edges are caused by the random value. It is seen from the comparison of PBG s for the two polarizations that the attenuation for the TE polarization is more weakened by the nonuniformity than Fig. 6. Transmission spectra for structure A 3. Solid and dashed curves indicate TM and TE polarizations, respectively. that for the TM. The reason considered is the PBG for the TE lying in the higher frequency range, which suffers a larger phase disordering from nonuniform columns. Let us define the degree of the distinctness of PBG,, as the ratio of the reduced attenuation to the ideal attenuation, both of which are measured in the unit of db. This definition allows the general evaluation of the influence of the nonuniformity

5 1504 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 8, AUGUST 1999 Fig. 9. Calculation models of crystals with defects, i.e., one defect is brought to crystal A 1, and a series of defects are brought to crystal A 2. The former models a resonator, while the latter models waveguides B 1 and B 2 with different shape of input end. Fig. 7. Transmission spectra for crystal A 1 with =0:1r 0. Solid and dashed curves indicate TM and TE polarizations, respectively. Fig. 10. Electric field distributions of TM-polarized wave in crystal A 1 with one defect. and polarizations is opposite for honeycomb crystal A 3 ; from Fig. 8, and for the TE and TM, respectively. Thus, the influence of the nonuniformity is different for different structures and polarizations. One target that should be satisfied in experiments is. Fig. 8. Degree of distinctness of PBG, D, calculated with deviation. and are for crystals A 1 and A 3, respectively. Solid and dashed curves indicate TM and TE polarizations, receptively. in crystals with different number. Fig. 8 shows the degree as a function of deviation. Some irregular changes in the plotted points are caused by the manual reading of the attenuation from oscillating spectral curves. Curves in Fig. 8 indicate Gaussian distributions fitting to the points. It is seen in Fig. 8 that is more reduced for the TE than for the TM in crystal A 1. Let us use the deviation that gives = 0.5 as a criterion. From Fig. 8, and 0.3 for the TE and TM, respectively. In contrast to this, the relation between V. CRYSTALS WITH DEFECTS Fig. 9 shows calculation models of densely-packed crystals with defects. The polarization was fixed to the TM, since the wide PBG for this polarization, as seen in Fig. 4, is suitable for the clear demonstration of resonator and waveguide characteristics. Fig. 10 shows the field distribution around crystal A 1 with, which has one defect at the center. Here, the resonant frequency at the defect was assumed for the incident wave. Some amount of the incident wave couples with the defect and passes through the crystal, as in a Fabry Perot cavity. The energy of the resonant mode is confined around the defect by the reflection from two rows of columns before and behind the defect. Fig. 11 shows the transmission spectrum, where m and m are assumed. The resonant frequency lying at the center of the PBG exhibits a high transmission close to 0 db. These are almost the same results as in [8]. The maximum amplitude of the localized field is 35 times that of the incident field. For Fabry Perot cavities, the enhancement factor of the localized field is given by. From Fig. 4, the power reflectivity of two rows of columns

6 YONEKURA et al.: ANALYSIS OF FINITE 2-D PHOTONIC CRYSTALS 1505 Fig. 11. Transmission spectrum that corresponds to Fig. 10. at the resonant frequency is estimated to be 96%. Applying this reflectivity, the enhancement factor is calculated to be 49. The slight difference of the observed value from this value is considered to be due to the imperfect coupling of the incident wave to the localized field. For the series of defects, we expected the waveguide characteristics similar to those demonstrated in a square lattice crystal [9]. As shown in Fig. 9, we considered two waveguide patterns B 1 and B 2, for which shapes of the input end were different. Fig. 12 shows the field distribution in waveguide B 1 against different frequencies both in the PBG. Here, was reduced to nine in order to save the computer resource, while was 15. The coupling of the incident wave with the waveguide depends on whether the frequency is inside or outside the transmission band. The penetration of the guided mode into the crystal is remarkably suppressed. The transmission spectrum was evaluated from the Poynting power on line Q with m and m outside the waveguide. The length m is the same as the waveguide width. One may consider that it should be evaluated inside the waveguide to divide the propagation loss of the waveguide and other losses. It should be noted, however, that the scattering matrix method cannot divide the forward and backward propagating waves, since it gives static solutions for electromagnetic fields. The imperfect matching of fields between the waveguide and the free space causes some amount of reflection at both ends and accordingly the resonance inside the waveguide. It seriously confuses the evaluation of the transmittance. This is the reason we evaluated the pointing power outside the waveguide. Fig. 13 shows transmission spectra. In contrast to the case of Fig. 12, assumed here was 21, just as shown in Fig. 9, while was reduced to seven in order to save the computation time. A wide transmission band lies in the higher side of the PBG. In the transmission band, the transmittance slightly changes in the range from 0.7 to 3 db. We consider that it is not affected by the propagation loss of the waveguide but other losses and diffracted waves behind the output end. The propagation loss should be negligible, because of the short waveguide and the high reflectivity of the photonic crystal claddings. If anything, we found in the calculation that the spectrum in the transmission band is shifted above or far below 0 db only by changing the length and the distance of line Q. This is due to the strong diffraction of the output wave, as Fig. 12. Electric field distributions of TM-polarized light around waveguide B 1. For and,!a=2c =0:43 and 0:33, respectively. seen in Fig. 12, which rapidly changes the power density of the output wave. However, the slight change in transmittance was independent of and. This indicates that this change is caused by the frequency dependence of the reflection loss at waveguide ends and the resonance in the waveguide. We also found that the transmittance is strongly affected by the shape of the input end. Fig. 13 also compares spectra for waveguide B 1 and B 2. In contrast to the relatively high transmittance for waveguide B 1 with the tapered input end, the transmittance is decreased to less than 10 db for waveguide B 2 with the narrow input end. The incident wave is strongly reflected by the front two columns at the input end of waveguide B 2. This result implies that transmission characteristics in photonic crystal waveguides can be greatly changed by a small number of columns placed at changing points of waveguides. VI. LIGHTWAVE DEVICES Fig. 14 shows calculation models of waveguides with 60 bends, Y-branch, and two types of directional couplers. For the former two, and, while for the latter two, and. The tapered input end was assumed for each structure. The polarization was fixed to the

7 1506 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 8, AUGUST 1999 Fig. 13. Transmission spectra for defect waveguides. Solid and dotted curves indicate waveguide B 1 and B 2, respectively. Fig. 14. Calculation models of waveguides with 60 bends and Y-branch and directional couplers C 1 and C 2. TM. Fig. 15 shows field distributions in waveguide with 60 bends and in the Y-branch. Owing to the abrupt bends, the Y-branch is outstandingly shorter than conventional ones; the assumed length is only four times the crystal pitch, i.e., 1.8 m including bends. The guided mode smoothly passes through such abrupt bends and branch. The penetration of guided modes into crystals is still very small. Fig. 16 shows the corresponding transmission spectra, where and are the same as for Fig. 13. The transmission band lies in the same frequency range as for the straight waveguide. By comparing the spectral curve with that for the straight waveguides, the excess loss caused by the reflection at bends was estimated to be less than 1 db. We also found in a similar calculation that a pair of 120-degree-bends exhibit a narrow attenuation band in the transmission band, and that four series of 60 bends, almost similar to a pair of 120 bends, flatten the transmission band. For Y-branch, on the other hand, the transmittance is decreased by the reflection loss at the branch. Since the Poynting power was evaluated for one output end, the ideal transmittance is 3 db. Taking account of this value and the difference from the spectral curve for the bent waveguide, the reflection loss at the branch was estimated to be db. The frequency dependence of the reflection loss arises from the resonance; we observed the standing wave of resonance between the branch and the input end in the animation of the field distribution, which was made by changing the phase of the input wave. For directional couplers, the input of the incident wave was limited to one waveguide by filling three columns at the input end of another waveguide. Besides, one waveguide was bent on the way and separated from another to avoid the mixing of output fields. Let us call the waveguide having the opened input end the straight, and that having Fig. 15. Electric field distributions of TM-polarized wave around bends and Y-branch, assuming!a=2c =0:43. the closed input end the cross. As shown in Fig. 14, the arrangement of the bent waveguide was changed according as the transmittance was evaluated for the straight or the cross. Fig. 17 shows field distributions against different frequencies both in the transmission band of the single waveguide. The output waveguide is switched by the nearly 20% change of the frequency. Since the defect waveguide assumed in this paper is an antiguide-type, the propagation constant of the guided mode can take discrete values from to zero. The ray of the mode orients as it almost crosses the waveguide near the cutoff. It is seen in Fig. 17 that is as small as The corresponding ray angle of the mode, which is taken from the waveguide orientation, is 65. Fig. 18 shows transmission spectra, where and are the same as for Fig. 13. The major output is observed in a wide frequency range from the straight. The ratio of outputs is maximally 27 db. At, the input light is equally divided to the two output ports, as in a 3-dB-coupler. An interesting property of this directional coupler is seen near the cutoff of the transmission band,. Considering the strong optical confinement in the waveguide, is approximated as. Near the cutoff,

8 YONEKURA et al.: ANALYSIS OF FINITE 2-D PHOTONIC CRYSTALS 1507 Fig. 16. Transmission spectra of TM-polarized wave for waveguide with bends and Y-branch. Solid, dashed, and dash-dotted curves indicate straight waveguide, bent waveguide, and Y-branch, respectively. Fig. 18. Transmission spectra for directional couplers. Solid and dotted curves indicate straight output from C 1 and cross output from C 2, respectively. outputs is limited to nearly 10 db due to the different profile of the even and odd modes caused by the different. Fig. 17. Electric field distributions of TM-polarized wave in directional couplers. For coupler C 1 with!a=2c =0:43 are assumed, for coupler C 2 with 0.375, respectively. changes rapidly so that orthogonal even and odd modes of the directional coupler, into which the guided mode of the single waveguide is expanded in the coupling region, have much different with each other. This results in the frequent switching of output waveguide within a short coupling length. We found that such a switching is possible within a coupling length as short as 2 times the pitch, i.e., 0.9 m. The ratio of VII. CONCLUSIONS The frequency range for the inhibition of light transmission in finite 2-D photonic crystals, which were calculated by the scattering matrix method, well accorded with PBG s obtained by the PWE method. Over 30 db attenuation of transmitted wave is expected by only four rows of unit cells of semiconductor columns for both the densely-packed crystal and the honeycomb crystal. This attenuation is weakened to half when the nonuniformity from 10 to 30% is brought to a diameter of columns. It was found in the analysis of defect waveguides that the transmittance is affected by the reflection loss at waveguide ends. The reflection loss at the input end is drastically changed by the arrangement of a few columns at the input end. For 60 bends, the minimum reflection loss was estimated to be less than 1 db. The transmission band for a 120 bend is flattened by replacing with the pair of 60 bends. The reflection loss from 0 to 4.6 db occurs at the Y-branch. The directional coupler with a submicron coupling length exhibits a demultiplexing and power dividing functions owing to an antiguide dispersion characteristic of the defect waveguide near the cutoff of the transmission band. This paper discussed conventional lightwave devices transformed by photonic crystals. However, peculiar characteristics of defect waveguides suggested the possibility of a novel functional device. Extremely short Y-branch and directional coupler are very attractive, when one desires the size reduction of lightwave circuits. For this purpose, the reduction of the reflection loss and the improvement of the transmission efficiency are crucial problems for the actual use of such waveguides in large scale circuits. This paper demonstrated the high-speed calculation of the scattering matrix method. However, the discussion was limited to 2-D, while experiments now being done are 3-D; even though photonic crystals are 2-D, channel waveguides need some optical confinement structure for another direction. Very recently, we have observed the light propagation in a photonic crystal waveguide, for the first time, which was composed of a thin film slab waveguide and a 2-D photonic crystal of holes [13]. In that study, we observed the photonic bandgap that qualitatively agreed but quantitatively disagreed with that

9 1508 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 17, NO. 8, AUGUST 1999 expected by the 2-D calculation. The theoretical research will be developed by the combination of a qualitative discussion by high-speed 2-D calculations and quantitative design by precise 3-D ones. ACKNOWLEDGMENT The authors would like to thank Prof. Y. Kokubun and Prof. Y. Hirose, Yokohama National University, and Prof. K. Iga and Associate Prof. F. Koyama, Tokyo Institute of Technology, for encouragement and discussion. REFERENCES [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., vol. 58, pp , [2] K. M. Leung and Y. F. Liu, Full vector wave calculation of photonic band structure in face-centered-cubic dielectric media, Phys. Rev. Lett., vol. 65, pp , [3] Ze Zhang and S. Satpathy, Electromagnetic wave propagation in periodic structures: Bloch wave solution of Maxwell s equations, Phys. Rev. Lett., vol. 65, pp , [4] K. M. Ho, C. T. Chan, and C. M. Soukoulis, Existence of photonic gap in periodic dielectric structures, Phys. Rev. Lett., vol. 65, pp , [5] R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, Accurate theoretical analysis of photonic band-gap materials, Phys. Rev. B, vol. 48, pp , [6] J. B. Pendry and A. MacKinnon, Calculation of photon dispersion relations, Phys. Rev. Lett., vol. 69, pp , [7] D. Felbacq, G. Tayeb, and D. Maystre, Scattering by a random set of parallel cylinders, J. Opt. Soc. Amer. A, vol. 11, pp , [8] G. Tayeb and D. Maystre, Rigorous theoretical study of finite-size twodimensional photonic crystals doped by microcavities, J. Opt. Soc. Amer. A, vol. 14, pp , [9] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, High transmission through sharp bends in photonic crystal waveguides, Phys. Rev. Lett., vol. 77, pp , [10] M. Plihal and A. A. Maradudin, Photonic band structure of twodimensional systems: The triangular lattice, Phys. Rev., vol. 44, pp , [11] G. O. Olaofe, Scattering by two cylinders, Radio Sci., vol. 5, pp , [12] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals. Princeton, NJ: Princeton University Press, [13] T. Baba, N. Fukaya, and J. Yonekura, Observation of light transmission in photonic crystal waveguides with bends, Electron. Lett., vol. 35, pp , Jun Yonekura was born in Shizuoka Prefecture, Japan, on July 19, He received the B.E. and M.E. degrees from the Division of Electrical and Computer Engineering, Yokohama National University, Japan, in 1997 and 1999, respectively. He is currently with Canon, Inc., Tokyo, Japan. During his thesis work, he studied neural network optical systems and devices, and also studied scattering matrix analysis and the fabrication of photonic crystal lightwave circuits. Mr. Yonekura is a member of the Japan Society of Applied Physics. Mitsutaka Ikeda was born in Kanagawa Prefecture, Japan, on July 5, He received the B.E. and M.E. degrees from the Division of Electrical and Computer Engineering, Yokohama National University, Japan, in 1996 and 1998, respectively. He is currently with Fujitsu Ltd., Kawasaki, Japan. During his thesis work, he studied scattering matrix analysis and the fabrication of semiconductor two-dimensional photonic crystal. Toshihiko Baba (M 93), for photograph and biography, see this issue, p