Elimination of cross-talk in waveguide intersections of triangular lattice photonic crystals

We design an intersection for crossing waveguides in triangular lattice photonic crystals with cross-talk smaller than 10. The cross-talk to the transverse waveguides is suppressed by symmetry mismatch between the cavity mode and the waveguide mode or by the mode-gap effect induced by air hole radius modulation of the waveguides. The transmission behavior of the crossing waveguides are illustrated by numerical simulations through finite difference time domain method. ©2008 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (130.5296) Photonic crystal waveguides; (250.5300) Photonic integrated circuits References and links 1. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819 (1999). 2. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, Y. H. Lee, “Electrically driven Single-Cell Photonic Crystal Laser,” Science 305, 1444-1447 (2004). 3. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207-210 (2005). 4. M. Tokushima and H. Yamada, “Light propagation in a photonic-crystal-slab line-defect waveguide,” IEEE J. Quantum Electron. 38, 753-759 (2002). 5. A. Sugitatsu, T. Asano, and S. Noda, “Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs,” Appl. Phys. Lett. 84, 5395-5397 (2004). 6. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960-963 (1998). 7. Z. Zhang and M. Qiu, “Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slab,” Opt. Express 13, 2596-2604 (2005). 8. B. K. Min, J. E. Kim, and H. Y. Park, “High-efficiency surface-emitting channel drop filters in two dimensional photonic crystal slab,” Appl. Phys. Lett. 86, 111106 (2006). 9. A. Shinya, S. Mitsugi, E. Kuramochi, and M. Notomi, “Ultrasmall multi-channel resonant-tunneling filter using mode gap of width-tuned photonic crystal waveguide,” Opt. Express 13, 4202-4209 (2005). 10. H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient multi-channel drop filter in a twodimensional hetero photonic crystal,” Opt. Express 14, 3491-3496 (2006). 11. M. F. Yanik, S. Fan, and M. Soljacic, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. 83, 2739-2741 (2003). 12. X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonicbandgap microcavity,” Nat. Photonics 2, 185-189 (2008). 13. S. G. Johnson, C. Manolatou, S. Fan, P. R. Villeneouve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross-talk in waveguide intersections,” Opt. Lett. 23, 1855-1857 (1998). 14. S. H. Kim, and Y. H. Lee, “Symmetry relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. 39, 1081-1085 (2003). 15. G. H. Kim, Y. H. Lee, A. Shinya, and M. Notomi, “Coupling of small, low-loss hexapole mode with photonic crystal slab waveguide mode,” Opt. Express 12, 6624-6631 (2004). 16. S. K. Kim, G. H. Kim, S. H. Kim, Y. H. Lee, S. B. Kim, and I. Kim, “Loss management using parityselective barriers for single-mode, single-cell photonic crystal resonators,” Appl. Phys. Lett. 88, 161119 (2006). 17. S. H. Kwon, T. Sünner, M. Kamp, and A. Forchel, “Ultrahigh-Q photonic crystal cavity created by modulating air hole radius of a waveguide,” Opt. Express 16, 4605-4614 (2008). #97052 $15.00 USD Received 6 Jun 2008; revised 3 Jul 2008; accepted 4 Jul 2008; published 15 Jul 2008 (C) 2008 OSA 21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 11399


Introduction
Owing to the ability of photonic crystals (PhCs) to manipulate photons in the wavelengthdimension, various PhC devices such as microresonators [1][2][3], waveguides [4][5], channeldrop filters [6][7][8][9][10], and optical switches [11,12] have been developed as building blocks for integrated photonic systems.In order to integrate multiple devices in a small region, it is necessary to intersect waveguides connecting devices without cross-talk.In perpendicularly crossing waveguides, it was reported that the excited mode from the input waveguides can be prevented by symmetry mismatch from decaying into the transverse ports, which was illustrated in a square lattice PhC [13].However, although most of the two-dimensional (2D) PhC devices have been demonstrated in the triangular lattice because of the wider photonic bandgap (PBG) than in the square lattice, such a symmetry-based method can not be directly applied in the triangular lattice since the waveguides cross each other with 60 or 120 degrees and the cavity has C 6v symmetry [14].
On the other hand, it was demonstrated that symmetry mismatching between a waveguide mode and a cavity mode could prevent photons propagating from the cavity to the waveguide [15,16].In addition, spectral filters, which can eliminate transmission to the unwanted waveguides, were realized by using the mode-gap effect resulting from modulation of the width [9], the lattice constant [10], the air hole radius [17] of the waveguides.In this paper, we propose a new design of an intersection for crossing waveguides without cross-talk in triangular lattice PhCs by combing both: the effects of the symmetry mismatch and the modegap, which can be generally applied to any waveguide-cavity coupled system.We also illustrate the transmission behavior in the intersections by three-dimensional (3D) finitedifference time-domain (FDTD) simulations.Figure 1(a) shows the functionality of a proposed intersection schematically.Here the waveguide along the x-axis is WG1 and the 60 degree angled waveguide is WG2.A single cell cavity is introduced at the intersection of the waveguides.Signal 1 with the wavelength λ1 is transferred to the WG1 but not WG2 through the proposed intersection.In the case of signal 2 (λ2), it propagates only through WG2.However, when two waveguides cross in a triangular lattice, photons injected from one waveguide usually propagate into both waveguides, as long as the frequency is within the PBG, as shown in Fig. 1(b).Hence in order to prevent propagating into the transverse waveguides (cross-talk), it is necessary to design a special filter at the intersection, which selects a signal depending on the wavelength and transfers it to the wanted waveguide.
In this paper, we use a single line defect waveguide formed by filling in one row of air holes and enlarging the innermost holes (r wg ), as shown in Fig. 1(c).Here the enlarged innermost holes shift the frequency of the dispersion curve of the waveguide mode.Since the mode has a relatively broad single mode region, we focus on the fundamental waveguide mode.The magnetic field pattern of the fundamental mode has even mirror symmetry to the axis along the waveguide, as shown in Fig. 1(c).

Symmetries of quadrupole modes in a single cell cavity
We use a PhC cavity at the intersection for wavelength-selectivity.Here a single cell cavity is modified to have high quality factor (Q), as shown in Fig. 2(a).Six nearest neighbor holes are reduced to r n =0.18a and pushed away from the cavity centers.Here a is the lattice constant.The distance (c n ) from the cavity center to the nearest holes center is 1.1a.The slab thickness and the radii of the air holes (r PC ) constituting the PhC are 0.5a and 0.25a, respectively.The refractive index used for the PhC slab in our simulations is 3.4.In the single cell cavity, we choose two degenerated quadrupole modes among the observed six modes [14].Quadrupole modes have a relatively high Q and orthogonal symmetry to each other.The calculated Q and the normalized frequency (ω n ≡ a/λ) are 4.7×10 5 and 0.29, which are same for both modes.
The magnetic field patterns of the two quadrupole modes are shown in Figs.2(b) and 2(c).One mode (even mode) in Fig. 2(b) has even mirror symmetries for both x-and y-axis.The other mode (odd mode) has odd symmetries for both axes, as shown in Fig. 2(c).We investigated the coupling behavior of two quadrupole modes in the cavity connected with six waveguide arms, as shown in Figs.3(a) and 3(b).The frequencies of the quadrupole modes indicated by green line is placed in the multimode region of the waveguides with r wg ~0.25a,where there are the fundamental mode and the 2 nd mode, as shown in the black lines of Fig. 3(c).Therefore the innermost holes of the waveguides are enlarged to 0.32a so that the frequencies of the quadrupole modes can be placed in the single mode region of the fundamental waveguide mode, as shown in the red lines of Fig. 3(c).In addition, in contrast to the waveguide mode with r wg ~0.25a, the fundamental waveguide mode with r wg ~0.32a corresponding to the frequency of the cavity mode is confined in the slab by the total internal reflection (TIR) because the mode is placed below the light line (blue line).In order to describe the relatively weak amplitude of the magnetic fields in the waveguide arms, saturated color coding is used.The red and the blue colors represent positive and negative amplitudes of the magnetic field.An amplitude larger than 1/10 of the maximum is represented with saturated colors.In the case of the even mode, the excited mode can couple to all waveguides.Though the coupling strength to WG1 is three times stronger than to WG2 or to WG3 due to the same mirror symmetry between the cavity mode and the waveguide mode of WG1, considerable amounts of the fields still couple to WG2 and to WG3 from the cavity.In the case of the odd mode, the odd mode can couple to WG2 and WG3 but not WG1 due to the mirror symmetry.Because the odd mode has odd parity to the axis of the WG1 and the fundamental waveguide mode has even parity to the axis.Thus the estimated energy rate propagating to WG2 is 1000 times larger than that to WG1 in the odd mode.In a structure as shown in Fig. 3(c), when a signal is injected from WG1 left or WG2 down and excites the even mode, then the excited even mode emits photons to all waveguides arms.In contrast to the case of the even mode, when a signal is injected from WG2 down exciting the odd mode, then the odd mode emits photons only to WG2, not to WG1 due to the previously described coupling behavior.We already have succeeded in designing an intersection without cross-talk for the odd mode by using only the mirror symmetry.1 2

Mode-gap of waveguides with different radii of air holes
In the next step, it is necessary to prevent the coupling from the even mode to WG2 and WG3.
In the structure of Fig. 3(c), the cutoff frequencies (WG cutoff) of the fundamental waveguide modes are smaller than the resonant frequencies (ω even , ω odd ) of the even and the odd modes, as expressed in Eq. ( 1).Here the cutoff frequency corresponds to the frequency of the waveguide mode at the bandedge of wavevector 0.5(2π/a).In order to prevent the coupling from the even mode to WG2 and WG3, the frequency of the even mode is tuned to a frequency range where the photons can propagate through WG1 but not through WG2 and WG3 like in Eq. ( 2).In contrast to the even mode, the frequency of the odd mode should be tuned higher than the cutoff frequency of the WG2, and hence the odd mode still is able to couple to WG2.On the other hand, although the frequency of the odd mode is placed in the guided mode region of WG1, the coupling to WG1 is prevented by symmetry mismatch as shown in Fig. 3(b).
1 2 In order to arrange the frequencies of the cavity modes and the waveguide modes as Eq. ( 2), the frequencies of the cavity modes are firstly modified.We have carefully investigated the electric field intensity patterns of the modes to tune the frequencies of the even and the odd mode independently.Since most of electric field is concentrated at the nearest six air holes for both modes, the modulation of those holes induces the similar frequency shifts for both modes, which is not suitable for the cavity to satisfy Eq. ( 2).In the air holes indicated red arrows in Figs.4(a) and 4(b), considerable amount of the electric field is observed only in the even mode but not in the odd mode.On the other hand, in the air holes indicated yellow arrows, the electric field is observable in this scale only in the odd mode.Therefore reducing the red air holes and enlarging the yellow air holes result in lowering the frequency of the even mode and increasing the frequency of the odd mode by changing the effective refractive indexes of the both modes, independently.The red and yellow holes are modified to 0.10a and 0.30a so that the frequencies of the even and odd mode are tuned to 0.281(a/λ) and 0.291 (a/λ) from 0.290 (a/λ).Second, the cutoff frequency of WG2 shifts to a higher frequency by enlarging the radii of the innermost air holes to r wg ~0.37a.Several groups reported that the coupling to unwanted waveguides can be prevented by the mode-gap of the waveguide mode caused by waveguide modulations such as air hole radii or lattice constant [9,10,17].In this way, the frequencies of the waveguide modes and the cavity modes are arranged like in Eq. ( 2) and Fig. 4(c).Based on the cavity design of Fig. 4, we investigated the coupling behavior of the even and the odd modes in the cavity with the modified air holes to the modified waveguides, as shown in Figs.5(a) and 5(b).The radii of the innermost holes in the WG1, WG2, and WG3 are 0.32a, 0.37a, and 0.37a.The even mode couples only to WG1 because the mode-gap effect inhibits coupling to WG2 and WG3.The odd mode couples only to WG2 and WG3 because of the symmetry mismatch to WG1.
Finally the transmission behavior of the designed intersection is investigated and the results are shown in Figs.5(c) and 5(d).The field sources are located at the positions indicated by red arrows.When the signal with the resonant frequency of the even mode is injected from WG1 left , 73% of the input signal passes to WG1 right in Fig. 5(c) with cross-talk of 10 -5 to both arms of WG1.Here the cross-talk is defined as the ratio of the energy to transverse waveguide to the injected energy.When the signal with the frequency of the odd mode is injected from WG2 down , 67% of the input signal passes to WG2 up in Fig. 5(d) with cross-talk of 10 -6 to both arms of WG2.We successfully show that the signals transmit through the proposed intersection with negligible cross-talk.The throughput and the cross-talk depend on the distance between the cavity and the waveguides and the innermost air hole sizes.Such quantitative investigations on the various structure parameters will be further studied.

Conclusion
In this paper, we present the principle of the operation of a waveguides-crossing with negligible cross-talk in a triangular lattice PhC by 3D FDTD simulations.The signal from either of two input ports mostly passes to the designated output port through the resonant tunneling of one of two modes in a single-cell cavity.The cross-talk to the transverse ports can be strongly suppressed by the symmetry mismatch between the cavity mode and the waveguide mode and the mode-gap resulting from the radius modulation of the innermost air holes of waveguide.The transmitted signals have 70% throughput and 10 -5 ~10 -6 cross-talk.The proposed device can be utilized to construct complex photonic crystal systems.The demonstrated symmetries and the mode-gap ideas can be also applied to analyze the coupling behaviors between the cavity and the waveguide modes in any PhC based systems.

Fig. 1 .
Fig. 1. (a).A schematic diagram of the proposed intersection.Signals with wavelengths λ1 and λ2 cross at the intersection.(b).The propagation of the magnetic field in the normal crossing waveguides (c).Magnetic field pattern of a fundamental waveguide mode with r wg ~0.28a.The radius (r PC ) of the air holes in the surrounding PhC is 0.25a.

Fig. 2 .
Fig. 2. (a).Modified single cell cavity structure.r pc , r n , and c n are 0.25a, 0.18a, and 1.1a, respectively.Magnetic field patterns of (b) even quadrupole mode and (c) odd quadrupole mode.

Fig. 3 .
Fig. 3. Magnetic field patterns in a cavity with six waveguide arms of (a) even mode and (b) odd mode.The r wg of the six arms is 0.32a.Saturation color coding is used.(c) Dispersion curves of the fundamental (1 st : solid lines) and the 2 nd (dotted lines) waveguide modes with r wg ~0.25a (black lines) and r wg ~0.32a (red lines).The green line represents the frequencies of the cavity modes.(d) The intersection consisting of the modified single cell cavity and the crossing waveguides with r wg ~0.32a.

Fig. 4 .
Fig. 4. Electric field intensity patterns of (a) even mode and (c) odd mode.(log scale) Gray circles indicate air holes.Air holes (indicated by red arrows) are 0.10a and the holes (indicated by yellow arrows) are 0.30a.(c) Band structures of the fundamental waveguide modes with r wg ~0.32a and r wg ~0.37a.The dashed lines represent the odd and the even cavity modes.

Fig. 5 .
Fig. 5. Electric field intensity patterns of (a) even mode and (b) odd mode in a cavity with six waveguide arms.(log scale).Movies of propagation patterns of electric field, in which signals are injected from the position of the red arrows and propagate along the waveguide and across the intersection for (c) (2.5MB) even mode and (d) (3.0MB) odd mode.(log scale).Red arrows indicate electric field source positions.