High-efficiency coupling between external and photonic crystal waveguides by longitudinally shifting waveguide junctions

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High-eﬃciency coupling between external and photonic crystal

waveguides by longitudinally shifting waveguide junctions

Chun-Wen Chang

a

, Szu-Cheng Cheng

b

, Wen-Feng Hsieh

a,*

a

Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC

b

Department of Physics, Chinese Culture University, Yang Ming Shan, Taipei, Taiwan, ROC Received 31 March 2004; received in revised form 1 August 2004; accepted 6 September 2004

Abstract

Longitudinally shifting waveguide junctions can achieve high efficiency light coupling from external silica wave-guides into photonic crystal wavewave-guides (PCWs) without requiring taper PCW structures in the waveguide junctions. Due to the periodic field distribution of PCWs along the propagation direction, maximum and minimum coupling effi-ciency occur when waveguide junctions are located near the peak and valley of the longitudinal mode profile, and cou-pling efficiency depends on both the localization of mode fields in PCWs and the group velocities of eigenmodes. Coupling efficiency of 86% between silicon/silica PCWs and external silica waveguides can be achieved without the need for special design in the junction region for the reduced-rod PCW. Due to large Bragg reflection, coupling of the band edge modes at the flat dispersion regions that possess small group velocities is inefficient.

Keywords: Photonic crystal waveguide; Coupling eﬃciency; Optical waveguide; Periodic structures; Bragg reﬂection; Integrated optics

1. Introduction

Photonic crystal waveguides (PCWs), con-structed by introducing line defects in photonic bandgap (PBG) structures, provide a potential ap-proach for achieving ultra-compact photonic inte-grated circuits (PICs). Because of their superior

ability to control and confine light propagation within the PBG in photonic crystals (PhCs), PCWs can confine the propagation of light effectively even with a sharp bend[1]. In this manner, various integrated photonic devices have been demon-strated in use of PCWs [2–4]. However, a key obstacle limiting the applications of PCWs in PICs is to develop a method of efficiently coupling light between PCWs and external lightwave circuits.

To achieve high coupling eﬃciency between external waveguides and PCWs, various analyses

*

Corresponding author. Tel.: +886 3 5712121/56316; fax: +886 3 5716631.

E-mail address:[email protected](W.-F. Hsieh).

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have been performed to improve the transverse mode-profile matching [5–12] at the junction of the external waveguide and the PCW. Among these structures, the external waveguide tapers and the PCW tapers with the localized defects have been shown capable to convert the transverse mode profiles of the external waveguides to match those of the PCWs[7]. These methods merely con-sider the transverse mode matching and ignore the mode-profile variation in the longitudinal direc-tion. However, due to the periodicity of a PCW structure, the modal field in a PCW is a Bloch state [10]. That is, the transverse mode profiles of the PCWs vary periodically along the propagation (or longitudinal) direction. This periodic variation characteristic of the modal field implies that the coupling efficiency of an external dielectric wave-guide with a PCW should depend on the position of junction between these two waveguides. To demonstrate the need to consider the mode match-ing longitudinally, in this investigation we analyze the coupling efficiency as a function of the longitu-dinal position of the junction and find the opti-mum coupling position.

The PhC analyzed in this study is a rod struc-ture PhC, where the PhC is formed by high

refractive index rods surrounded by a lower index medium. The single-line-defect rod structure PCW with single guided mode inside the PBG signifi-cantly improves the performance of PIC devices [13]. As seen inFig. 1, we analyzed the coupling efficiency of two general rod structure PCWs, i.e., the removed-rod and the reduced-rod PCWs, both of which are butt coupled to the external waveguides. This study analyzed the band struc-tures and the stationary mode profiles of the PCWs using the plane-wave expansion method and calculated the transmittances as a function of the longitudinal position of the waveguide junc-tion by the two-dimensional (2-D) finite-difference time-domain (FDTD) method. The butt coupling of the removed-rod PCWs is analyzed and the cou-pling efficiency is shown to be less than 30% with-out any special design at the waveguide junction [7]. However, by properly shifting the junction position along the longitudinal direction, the cou-pling efficiency of the removed-rod PCWs can be increased to up to 70% readily. The same tech-nique can be applied to the reduced-rod waveguide and the coupling efficiency of up to 86% can be achieved. Additionally, the transmission spectra of the reduced-rod PCWs are also analyzed to

Fig. 1. Schematic of the structures analyzed for coupling from an external waveguide to the PCWs including: (a) the removed-row PCW and (b) the reduced-row PCW. The incident waves are excited on the incident surface and the transmission ﬂux is integrated along the plane D while the waveguide junctions successively move from z = 0 toward the right.

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show that the proposed method is very intuitive and suitable for designing high-eﬃciency and wide-bandwidth coupling structures.

2. Analysis method

The PhC structure analyzed in this work is composed of a 2-D triangular lattice of silicon rods surrounded by silica. This rod structure PhC has been investigated in previous studies [2,7,13]and demonstrated experimentally in a sili-con-based PIC[14]. The radii of the silicon rods of this photonic crystal are assumed to be r = 0.25a, where a denotes the lattice constant. The refractive indices of the silicon rods and the ambient silica are taken to be 3.46 and 1.45, respectively, at the wavelength k = 1.55 lm. With these parameters, this triangular lattice exhibits a bandgap of trans-verse magnetic (TM) modes in the normalized fre-quency range f = 0.2359–0.3258c/a, where c represents the speed of light in vacuum. From Fig. 1, the PCWs investigated in this study were created by removing a row of rods [type A, Fig. 1(a)] and reducing the radii of the whole row of rods to rd= 0.1a [type B, Fig. 1(b)] along the C– K direction of the PhC.Fig. 2 displays the band structures of the PCWs obtained using a 2-D plane-wave expansion method [15], which com-putes the Bloch wave vector k using the super cell enclosed in the boxes inFig. 1. From Fig. 2, the gray areas denote regions of the extended modes of the perfect PhC. Moreover, the solid and dashed lines within the bandgap represent the dis-persion curves of the type-A (removed-rod) and type-B (reduced-rod) PCWs, respectively. Both types of waveguides have a single guided mode in-side the PBG.

To investigate the longitudinal mode proﬁles of the PCWs, the stationary intensity distributions of the four modes A1, A2, B1, and B2, labeled on the dispersion curves in Fig. 2, were calculated by means of the plane-wave method and presented inFigs. 3(a)–(d), respectively. These modes include two type-A PCW modes: A1 for f = 0.290 c/a and A2 for f = 0.314c/a located near the band center and band edge, respectively, and both these two modes are situated in the slope region of the

dis-persion curve. The B1 mode, corresponding to the type-B PCW mode for f = 0.287c/a situated near the band center. Besides, to investigate the coupling issues of the modes with strong interac-tion between forward and backward fundamental Bloch modes, we also study the B2 mode (f = 0.3c/a), which is located near the ﬂat region of the dispersion curves with very small group velocity. The modes A2, B1, and B2 are located be-low the light line of vacuum. Thus all of these modes are expected to be conﬁned in PCW slabs. From Fig. 3, the electric energy distributions (jEj2

) of these four modes vary not only laterally but also periodically along the longitudinal direc-tion in the PCWs. Addidirec-tionally, the periodic vari-ations of the longitudinal modal fields have the same periods as the lattice of the original PhCs. The optical energy is localized around the original positions of the silicon rods. The localization of the modal fields depends both on the types of PCWs and on the frequencies of the waveguide modes. The modal field of the type-B PCW (re-duced-rod PCW) is more localized than that of the type-A (removed-rod) PCW, and the near-band-edge mode (A2 mode) displays more signifi-cant field localization than the band center mode (A1 mode). Especially, the light field of the type-B PCW is confined as a standing wave within the

Fig. 2. Band structure of the A PCW (solid line) and type-B PCW (dash line). The A1, A2, type-B1, and type-B2 are four modes analyzed in this study. The gray areas represent the extended modes of the bulk PhC.

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defect rods [see Fig. 3(c)], which covers less than one-tenth of wavelength.

If we ignore the coupling to the backward com-ponents of fundamental Bloch modes, the

depend-ence of the coupling eﬃciency g on the

longitudinal positions of waveguide junctions be-tween PCWs and conventional waveguides can be conceptually explained by the following overlap integral: g¼ Z EexðxÞEpcðxÞ dx 2, Z EexðxÞEexðxÞ dx Z EpcðxÞEpcðxÞ dx

[16], which is successfully employed to study cou-pling issues between z-invariant waveguides. Here Eex and E pc represent the mode fields of the external and the PhC waveguides, respectively, and x is the transverse coordinate. According to the above relation, since Epc varies periodically along the longitudinal direction, the coupling effi-ciency should depend on the junction location, and will significantly improve as the waveguide

junction approaches the ﬁeld-localized point of the PCWs.

This study uses silica as the material of the feeding waveguide, which can more easily couple with optical fibers through a waveguide taper than other high-index materials. We feed optical power into the PCW through the air-clad silica waveguide, with the waveguide width wg= 1.0a. The dispersion curve of the feeding waveguide is also shown inFig. 2. In this case, the external waveguide is excited by the fundamental mode of k= 1.55 lm from the incident plane in Fig. 1. The length L of the PCW is 10a, and the PCW is surrounded by seven rows of rods on each side. The perfect match layer conditions are consid-ered in the FDTD calculation to avoid the reflec-tion of radiareflec-tion waves in the boundaries of the calculation domain. To identify the optimum coupling point of the waveguide junction, as illustrated in Fig. 1, the transmittances from the external waveguides to the PCWs are calculated using the FDTD method by shifting the wave-guide junction with a step of Dz = 0.1a. During moving the junction forward, the remainder of

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the PhC cladding behind the junction (the PhC cladding in the Dz region) was removed and be-came air, which could be achieved by dry etching technology. A resolution of 40 grid points per lattice is employed to ensure accurate modeling of the 0.1a longitudinal shift. The transmittance T is calculated by integrating the Poynting vector over the integration plane D inFig. 1and then is normalized to the input power while the wave-guide junctions successively move from z = 0 to-ward the right. Since the contribution of radiation fields in the PCW will affect the accu-racy of the integrated transmission power, the longitudinal position of the integration plane D should be far enough to ensure that the station-ary mode field has been reached. Therefore, we calculate the transmittance as a function of the longitudinal position of the plane D, and find that the calculated result converges to a constant as long as the integration plane D is farther than z = 5.5a. As can be seen inFig. 1, the integration plane D is located at a distance of z = 8.5a, and is extended to the second row of rod in PhC claddings to ensure that it is wide enough to cov-er most of the PCW mode. Furthcov-ermore, for examining the coupling bandwidth, the transmis-sion spectrum of the reduced-rod PCW, which has the highest coupling efficiency, is also analyzed.

3. Results and discussion

Fig. 4 shows the transmittances (T) and the reflectances (R) as a function of the longitudinal position of waveguide junctions, which originates from the valley point of the PCW where the modal field intensity of the PCW is minimal. Since the radiation scattering losses from the modal mis-match is not collected by the detector, which col-lect the reflected guided waves only within the silica waveguides, the sum of the transmittance and the reflectance is not equal to unity. The trans-mittances of these four modes vary considerably as the waveguide junction shifts longitudinally, and all of them reach their maxima when the wave-guide junctions are located near the maximal intensity of the modal fields of the PCWs. On the

other hand, the transmittance is very low when the waveguide junction is near the valley position of the PCW. In addition to the positions of the waveguide junctions, the group velocities of eigen-modes and the PCW types also significantly affect coupling efficiency.

As can be seen in the B1–PCW mode with the stronger field localization than the type-A PCW mode and the highest coupling efficiency, the transmittance reaches up to 86% when the wave-guide junction is near the maximum intensity posi-tion, and is below 5% when the waveguide junction is near the valley position. In the B2 mode, the mode field is even more localized than that of the

Fig. 4. Transmittances and reﬂectances of the A1, A2, B1, and B2 modes as a function of the longitudinal positions of the waveguide junction between external waveguides and PhC waveguides.

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B1 PCW mode [Fig. 3(d)]. We would have ex-pected that the transmittance of the B2 mode were higher than that of the B1 mode. However, inFig. 4, we found that the B2 mode has much less trans-mittances (maximum efficiency < 40%) but very high reflectances. Besides, the modulation depths of both the transmittance and reflectance along the longitudinal position for the B2 mode are much less than those of the B1 mode. Notice that the summations of the transmittance and reflect-ance for both modes almost reach 90%. Most of the energy in the B2 mode is reflected with the maximum reflectance of 82% (z = 0.1a), and R = 50% even when the waveguide junction is near the maximum intensity position (z = 0.4a), where has the maximum transmittance T = 39%. This may be partly because of the large group velocity mismatch between the feeding silica waveguide mode and the B2 mode which has a far smaller group velocity, and partly because of Bragg reflec-tion when the operareflec-tion frequency of the PCW mode such as B2 is close to the band edge where the PCW mode is resemble to the standing wave not just within the defect rods. On the other hand, since both the A1 and A2 modes locate in the slope region of the dispersion curve, the group velocities of these two modes are similar and the difference of the coupling efficiency between these two modes is unapparent. If we shift the A2 mode to the flat region of the dispersion curve (e.g., k = 0.42p/a, f = 0.32c/a), the reflectance at the optima junction position (z = 0.6a) grows up 40% due to the back-ward coupling from large Bragg reflection and group velocity mismatches. However, the summa-tion of the R and T is only 67% at the optima junc-tion posijunc-tion since this flat-dispersion mode strongly couples with leaky modes to cause serious radiation losses. From the FDTD simulation (not shown here) we can see that the field extends over the bulk PhC regions. Furthermore, the calculated transmittances as a function of the longitudinal position do show asymmetry however the Epc is symmetric within the lattice period. Thus the clas-sical approximate formula of the overlap integral, which considers only fundamental forward Bloch modes, is not suitable to predict the coupling of rod structure PCW modes with considerable back-ward components.

To obtain a further insight into the high-effi-ciency coupling of the type-B (reduced-rod) PCW, the electric fields of the input coupling be-tween the external waveguide and the type-B PCW at the maximal and minimal transmittances were calculated using the FDTD method. Strong coupling (86%) clearly occurs [seeFig. 5(a)] when the waveguide junction is located near the center of the defect (z = 0.4a) where the light field is strongly localized. On the other hand, weak cou-pling (4.5%) occurs [seeFig. 5(b)] when the wave-guide junction is situated at the point of minimum modal field intensity (z = 0). In this case, the inci-dent optical field is mostly reflected at the wave-guide junction owing to the significant modal

Fig. 5. Snapshots of the electric ﬁeld distributions for: (a) maximum (z = 0.4a) and (b) minimum (z = 0) transmittance of the B1 mode.

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mismatch. Besides, since the time-domain electric field distribution of a Bloch mode depends on both PhC periodicities and wave vectors, the field pat-terns shown inFig. 5also display the wave vector induced periodicity. To investigate the effect of wave vectors, we calculate the transmittance of the B1 mode (k = 0.3212p/a) as a function of the longitudinal position for two periods of the PhC and find that the difference between these two cy-cles is less than 1%. Therefore, the wave vector in-duced periodicity would not affect the physical explanation of the optimum junction location.

To examine the coupling bandwidth of the pro-posed scheme, we analyzed the transmission spec-trum of the B1–PCW mode with the external waveguide at the maximum coupling efficiency. FromFig. 6, the transmittance exceeds 77% within the frequency range f = 0.247–0.289c/a (corre-sponding to k = 1539 to 1801 nm). On both ex-treme ends of the transmission spectrum, the coupling efficiency abruptly drops to almost zero. That is because of large Bragg reflection for the band edge modes as mentioned earlier. Conse-quently, coupling between external and PhC wave-guides varies with the longitudinal position of the junction, and high efficiency and wide bandwidth coupling of the reduced-rod PCW can be achieved by locating the junction near the peak of the longi-tudinal mode profile.

4. Conclusion

To summarize, this study presented a conven-ient method for high-efficiency coupling between external and PhC waveguides by shifting the wave-guide junction longitudinally. The distribution of the PCW mode field was calculated using the plane-wave expansion approach, and was found to vary periodically along the PCW. The localiza-tion of the modal field in the PCW depends on both the waveguide structure and the eigenmode fre-quency. High-efficiency coupling can be achieved by conveniently shifting the waveguide junction near the peak position of the longitudinal mode profile. Additionally, the coupling efficiency de-pends on both the types of PCWs and the group velocity of Bloch modes. The B1 mode of the type-B (reduced-rod) waveguide located near the band center with a strongly localized field and a high group velocity has the best coupling with the silica waveguide. In practical PCW slabs, out-of-plane scattering losses in cladding layers due to modal mismatches should be significant [10,17], and the dispersion curves will be shifted for some frequency range because of the index confinement in the perpendicular direction in a real PCW slab [5]. Using 2-D calculations one might be able to identify the function of the coupling structure qual-itatively in the PCW slabs before performing three-dimensional (3-D) exact electromagnetic analyses, which can be used to achieve more quantitative re-sults. Therefore, the proposed concept can be com-bined with planar tapered waveguides [18] to achieve high efficiency and wide bandwidth cou-pling between external silica and silicon/silica PhC waveguides in PICs. Furthermore, since these flat dispersion PCW modes with small group veloc-ities are very useful to design novel PIC devices, the efficient coupling of the band edge modes should be a challenging work that must be investigated.

Acknowledgment

This work is partially supported by the National Science Council and the Ministry of Eco-nomics Taiwan under Grants NSC-92-2112-M009-037 and 92C240.

Fig. 6. Transmission spectrum for the B1–PCW mode with the external waveguide at the maximum coupling eﬃciency.

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