We propose a line defect waveguide structure along the Γ-? direction in two-dimensional triangular lattice silicon photonic crystal slabs. The modal dispersion relation and the transmission spectra of this waveguide are studied. The results show that by perturbing the width of the line defect and the diameter of the air holes adjacent to the waveguide core, one can control the width of the single mode transmission window and make it far broader than the original one. The proposed Γ-? waveguide will help to build a more flexible network of interconnection channel of light in two-dimensional photonic crystal slabs.
© 2008 Optical Society of America
Photonic crystals (PCs) have attracted great attention for many years due to their strong power in controlling the motion of photons. The periodical structures in these materials enable the creation of photonic band gaps (PBGs) to insulate light transmission [1–3]. Among various kinds of PCs, two-dimensional (2D) PC slabs have been studied most extensively as a possible candidate for future photonic integrated circuits due to relative ease for current micro-fabrication technologies to build these structures in the near infrared wavelengths. 2D triangular-lattice PC slab structures are known to possess a large PBG for the TE-like polarization mode and it is technologically convenient to incorporate various defects into them [4–8]. Usually, point defects work as micro-cavities with high-Q factor [9–11] and line defects serve as waveguide channels for light to propagate efficiently and freely. In previous works, single-mode or multi-mode optical waveguides are usually made by removing one or more rows of air holes along the Γ–M direction in the triangular lattice PC. It has been proved in plenty of works that the number of waveguide modes as well as the width of the transmission windows (which is equal to the total band width of all guided modes together) can be controlled by tuning the core width of the line defects [12–18]. However, waveguides along other directions in the triangular lattice PC, at least to our knowledge, were rarely discussed. Just like the Γ–M direction, waveguides along the Γ–M direction should also be able to guide confined mode due to the existence of a complete 2D PBG in the 2D triangular lattice PC. A large degree of freedom can be further explored to incorporate line defects in this crystalline direction. In this paper, we report a method for making line defects along the Γ–M direction and analyze the optical properties of such waveguides. Simulations and experiments show that the waveguide modes in these line defects can be properly tuned by changing the geometrical parameters such as the width of the waveguide core and radius of the air holes around the core. Our work for the first time shows that high-performance Γ–M waveguides can be successfully realized in the 2D triangular lattice PC slab.
2. Original Γ–M waveguides
The two directions in the triangular lattice for creating waveguides are illustrated in Fig. 1 and labeled with Γ–M and Γ–M. The waveguides along the Γ–M direction can be naturally formed by removing one or several lines of lattice points along this direction , but this method does not work well for the Γ–M direction because there is no natural pathway along this direction. Fortunately, after careful observation we find that the four neighboring lattice points along this direction compose a diamond region, as illustrated in Fig. 1(a). By removing a line of diamond areas, we can obtain a cluster-like waveguide along the Γ–M direction as depicted in Fig. 1(b). This waveguide is called the original Γ–M waveguide.
Before going on with our analysis of this line defect, it is important to first have a clear knowledge of the PBG property of the background PC structure. We choose the typical parameters for silicon 2D PC slab working at near infrared wavelengths: the lattice constant is a=430 nm, the radius of air holes is r 0=120 nm (0.28a) and the thickness of the slab is 220 nm (0.51a). We have employed the three-dimensional (3D) finite-difference time-domain (FDTD) method to solve the band diagram for both the background crystal and the waveguide. Since the period of the crystal in the Γ–M direction is a’=√3a, we use normalized frequency (a’/λ) and wave vector k in unit of (2π/a’) to describe the band diagram. The calculated modal dispersion relation and modal field profile of guided modes in the original Γ–M waveguide are displayed in Fig. 2(a) and Fig. 3(a). The figure indicates that a 2D PBG opens in the 0.45–0.54 (a’/λ) frequency window. The solid red line represents the edge of the light cone for the 2D PC slab. Since the period of the Γ–M waveguide turns out to be 3 times of the lattice constant, the slope of the light line is much more close to the Brillouin zone edge of the k vector than the conventional Γ–M waveguide. Figure 2(a) shows that the original Γ–M waveguide has two guided modes within the PBG, which are defined as even and odd modes based on the symmetry of their Ey field component with respect to the mirror reflection plane passing through the waveguide central axis. Since the light line is so close to the Brillouin zone edge, the bands of the guided modes become flat and limited under the light cone, which theoretically leads to very narrow transmission window of several nanometers (below 5 nm) and make the waveguide susceptible to structural disorders.
In order to make clear of this issue, we investigate the performance of waveguides by experiment. The designed waveguide pattern was drawn on the top layer of silicon-on-insulator (SOI) wafer by electron-beam lithography, and then transferred to the silicon slab by inductively-coupled-plasma dry-etching. The insulator layer (SiO2) underneath the silicon pattern region was removed by wet-etching to form an air-bridged structure, which facilitates the tightest vertical confinement for photons. Each sample had a tapered silicon ridge waveguide as an input port to improve the coupling efficiency of the signal from the fiber into the PC waveguide. The length of the PC waveguide sample was deliberately set to be 20 µm in order to show the effect of the “leaky” modes and avoid the problem of mechanic fragility caused by air-bridge structure. The scanning electron microscope (SEM) image of the fabricated sample is shown in the upper panel of Fig. 4. The input optical signal came from a continuous wave tunable semiconductor laser with the wavelength ranging from 1500 to 1640 nm, launched into one facet of the ridge waveguide via a single-mode lensed fiber. Power meter was used to detect the optical signals transmitted through the waveguide and emitted from the output side. The measurement was made with TE polarization and normalized by a ridge waveguide on silicon with same length and width. Figure 2(b) shows the measured transmission spectrum of the Γ–M waveguides. There is only one transmission band in the whole tested region, which is highlighted by the yellow gray shadow area. We also use the gray yellow area to denote the corresponding pass band in the theoretical band diagram in Fig. 2(a). The simulated center and the width of the mode region are 1533nm and 22nm, while these parameters are 1524nm and 25nm for the practical sample. Despite a frequency deviation caused by the uncertainties in the fabrication process, the observed high transmission band coincides well with theory. Notice that this high transmission band consists of a large fraction of “leaky” modes that lie above the light line of the waveguide in addition to those guided modes that lie below the light line which should be truly lossless in principle. In other words, the leaky modes have a larger bandwidth than the lossless guided modes. It seems that the “leaky” mode above the light cone doesn’t “leak” for the 20 um long Γ–M waveguide, or at least, these “leaky” modes are low loss modes and still can propagate for a long path.
3. Optimized Γ–M waveguides
With the contribution of the seemingly “leaky” but actually low loss mode, the width of the transmission window is greatly broadened compared with the very limited guided mode band lying below the light line. However it is still not broad enough to accomplish practical applications. The narrow band features of the original Γ–M waveguide can be understood when we take a closer look at the geometric configuration of the waveguide. The waveguide has very rough walls and in some sense it can be assumed to be a coupled cavity waveguide made from a series of cavity-like diamond regions in Fig. 1(a). The distance between adjacent cavities is a’. The transportation of light through the waveguide is obstructed by the two protruding air holes of each diamond region and this results in the slow group velocity and flat band of the waveguide. In comparison, the usual Γ–M waveguide [made by removing one row of air holes along the Γ–M direction, as can be visualized in Fig. 1(a)] has much smoother walls and the light propagation pathway has little obstruction, leading to a much wider guided-mode band and much larger group velocity of light through the waveguide than the Γ–M waveguide. Similar ideas have been adopted in analyzing and designing optimal Γ–M waveguides with large group velocities and wide transmission windows . The above analysis in fact has suggested an appropriate way to modify the original Γ–M waveguide so that its transportation properties, in particular the guided-mode band width can be improved as much as possible: The walls of the waveguide should be made as smooth as possible. In the following we will show that such an intuition is efficient and effective in designing high-performance Γ–M waveguides.
To realize effective PBG waveguides, it is desirable to have confined waveguide modes with sufficiently broad transmission windows within the PBG region and a wide-band single mode operation is most preferred. To have a better idea on how a particular structural perturbation on the original Γ–M waveguide will affect the guided mode, we look more closely at the modal field profile of the two guided modes as displayed in Fig. 3(a). The corresponding k vector for these two modes are both 0.4 (2π/a’). The modal field of the two eigenmodes penetrates into the photonic crystal cladding and concentrates at the first and second row of air holes, thus it suffices to only consider some perturbations on the geometric parameters of air holes in these two rows in order to optimize the transmission band width of the guided modes.
As illustrated in Fig. 1(b), we define the radius of air holes in the first and second rows as r 1 and r 2. The width of the waveguide w d is defined by the spacing between the centers of the nearest air holes on the two sides of the waveguides, w 0=2a=860nm represents the width of the original Γ–M waveguide. Since the waveguide can be assumed as a connection of a periodic array of little microcavities, it is necessary to break the cavities and make the channel a smooth pathway. A natural way to achieve this aim is to shrink the size of the air holes in the first row and enlarge the size of the air holes in the second row. At the same time the width of the waveguide w d can be changed from the value of w 0.
The 3D FDTD method is used again to investigate the underlying principle of optimization by looking at the band diagram of different perturbed Γ–M waveguides. Firstly, we vary r 1 from 50 nm to 90 nm (0.12a to 0.2a) with a step of 20 nm (0.04a). The other structural parameters are fixed as r 2=r 0=120 nm (0.28a) and w d=0.75 w 0. The calculation result of dispersion curves is basically consistent with our hypothesis. Figure 3(b) clearly shows that the width and position of the entire waveguide mode strongly depend on r 1. The odd mode increases its band width from 6 to 45 nm, but the even mode just exhibits a small increase in width (below 3 nm). The decrease in r 1 has resulted in an apparent increase in the transmission band width. Similar operation is made on the air holes in the second row. We fix r 1=50 nm (the optimal value) and w d=0.75 w 0 and increase r 2 from 150 nm to 190 nm (0.35a to 0.44a) with a step of 20 nm. As can be found from Fig. 3(c), the width of the transmission window reaches the maximum quantity at r 2=170 nm (0.4a). On the contrary to variation of r 1, the change of r 2 has a much larger influence on the even mode than on the odd mode: The band width is broadened by about 50% for the even mode but just 18% for the odd mode. The difference effects on the even and odd mode can be understood from the modal field profile of these two modes. As can be found in Fig. 3(a), the first side lobe of the modal field is located at the first row of air holes for the odd mode while it is concentrated on the second row of air holes for the even mode. A perturbation on the geometric configuration of a region where the local electromagnetic field energy concentrates can lead to a much more remarkable change to the mode feature than on the other regions. On the other hand, our numerical simulations show that the broadening of the transmission band width exhibits a saturation behavior upon the variation of r 1 and r 2. For r 1<50 nm or r 2>190 nm, the width of the transmission window barely increases when we further decrease r 1 or increase r 2. It seems that the modulation on the guided-mode band have their limitations by the current optimization scheme.
In the above we keep the apparent width of the waveguide w d unchanged. Now we further explore this additional structural freedom for optimizing the width of the transmission windows. We have investigated a series of waveguides with w d varying from 0.65 to 0.75 w 0. The calculation results of guided-mode band diagram are displayed in Fig. 3(d). The effect of modulation is clearly seen in the dispersion curves. As w d decreases while r 1=50nm and r 2=170nm, the width of the transmission window increase in the PBG region and finally reaches the maximum quantity of 50 nm at w d=0.65 w 0. The frequency bandwidth is far broader than that in original Γ–M waveguide. Furthermore, it can be seen from Fig. 3(d) that at this parameter the even and odd band is almost free from overlap in frequency with the band width of the odd mode much broader than the even mode and as a result the single-mode operation is realized. This wide-band single mode feature is favorable for practical applications, e.g., in the wave division multiplexer.
Having had a clear picture of how to optimize the Γ–M waveguide through systematic numerical simulations, we proceed to realize the designed waveguide in experiment on the SOI slabs. The length of the modified Γ–M waveguides is 20µm (the same value as the original waveguide). The SEM picture of a typical optimized Γ–M waveguide sample is shown in the lower panel of Fig. 4. The subtle difference in structure from the original Γ–M waveguide sample [upper panel of Fig. 4] can be seen clearly. Figure 5 shows the measured transmission spectra of several optimized Γ–M waveguides. We measured the transmission intensity of the same sample more than five times. The measured intensity fluctuation was less than 0.5dB. Distinctive transmission windows and cutoffs appear in the spectra. To facilitate direct theory-experiment comparison, we plot the theoretical result of transmission windows, including the “leaky” mode areas by gray lines. Compared with the calculated dispersion curve in Figs. 3(b), 3(c), and 3(d), the observed propagating band coincides well with our expectations, despite slight frequency deviations caused by the uncertainties in the fabrication process. Obviously, the width and position of the guided modes strongly depend on the parameters of r 1 and r 2. The “leaky” modes in Fig. 3 are verified again to be the low loss mode for their contributions in the observed transmission spectra. The optimized waveguides all have a high transmission band that is much broader than the original waveguide. Besides, the intensities of the transmission spectra are much higher than the original one. The maximal values almost increase more than 10 times: from −15dB to −5dB.
As the total loss for the optimized Γ–M waveguide is 5dB, the propagation loss for the 20µm long waveguide must be well below 5dB. To have a more quantitative value about the propagation loss level, we employed the 3D FDTD method to simulate the propagation loss of the pure Γ-M waveguide. Three waveguides with their length changed from 10µm to 15µm and to 30µm are tested, and the calculated transmission spectra are shown in Fig. 6. The boundaries of the gap and the mode region coincide with the dispersion relations in Fig. 2 and Fig. 3 and the measured transmission spectra in Fig. 5. The peak transmissivity values of the waveguide modes are -0.1, -1.2, -1.3dB and the minimum transmissivity values are -12.9, -13.0 and -14.4 dB, which also coincide approximately to the measurement data in Fig. 5. This result indicates that as the length of the pure Γ–M waveguide increases the intensities of the transmission barely change. The propagation loss for the leaky modes is estimated to be below 1.0 dB per 20µm. Overall, our results confirm that by fine tuning the geometric parameters of the structure we can realize wide single-mode transmission bands in the Γ–M waveguide in 2D triangular lattice PC slabs.
One may notice from Fig. 3 that in the optimized Γ–M waveguide the high transmission band still consists of a large fraction of leaky modes that lie above the light line and the lossless guided modes that lie below the light line still have a limited bandwidth. The reason is that the light line of the waveguide is very close to the Brillouin zone edge in the k-space due to the relatively large lattice constant (√3a) of the waveguide compared with the conventional Γ–M waveguide whose lattice constant is a. This leaves a very limited k-space for lossless guided modes to greatly expand the bandwidth. However, as has been shown both experimentally and theoretically in the above, the optimized waveguide has a relatively low loss in the whole wide transmission band, so such an optimized waveguide structure can still find applications in areas that do not require a very long propagation path of light signal.
In summary, we have investigated theoretically and experimentally the optical properties of Γ–M waveguides made in 2D triangular-lattice PC slabs. We have found that the original waveguide has a very narrow band width of guided modes. The band width can be significantly enlarged by modifying the radius of the air holes in the first and second rows adjacent to the waveguide core as well as the width of the waveguide core. Optical characterization of the fabricated waveguides shows good agreement with the theoretical prediction. As the Γ–M waveguide is perpendicular to the usual Γ–M waveguide, it offers an alternative to construct a waveguide interconnection beyond the usual scheme of Γ–M with Γ–M waveguides. A high-performance wide-band Γ–M waveguide should be of great help to build integrated-optical devices such as interconnection networks, channel-drop filters, and wave division multiplexers with more flexible geometrical configurations in 2D PC slabs.
The authors would like to acknowledge the financial support from the National Natural Science Foundation of China at Grant No. 10525419 and the National Key Basic Research Special Foundation of China at Grant Nos. 2006CB921702 and 2007CB613205
References and links
1. E. Yablonovitch and Science289, 557–559 (2000).
2. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in Photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
4. H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient in-plane channel drop filter in a two-dimensional heterophotonic crystal,” Appl. Phys. Lett. 86, 241101 (2005). [CrossRef]
5. 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, 4202–4208 (2005). [CrossRef]
6. B. S. Song, T. Asano, Y. Akahane, Y. Tanaka, and S. Noda, “Multichannel add/drop filter based on in-plane hetero photonic crystals,” IEEE J. Lightwave Technol. 23, 1449–1455 (2005). [CrossRef]
8. C. Ren, J. Tian, S. Feng, H. H. Tao, Y. Z. Liu, K. Ren, Z. Y. Li, B. Y. Cheng, and D. Z. Zhang, “High resolution three-port filter in two dimensional photonic crystal slabs,” Opt. Express 14, 10014–10020 (2006). [CrossRef]
10. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]
12. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001). [CrossRef]
13. M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, “Singlemode transmission within photonic bandgap of width-varied single-line-defect photonic crystal waveguides on SOI substrates,” Electron. Lett. 37, 293–295 (2001). [CrossRef]
14. K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic crystal waveguides with high group velocity,” Opt. Commun. 198, 395–402 (2001). [CrossRef]
15. M. Notomi, A. Shinya, K. Yamada, J. I. Takahashi, C. Takahashi, and I. Yokohama, “Structural tuning of guiding modes of line-defect waveguides of silicon-on-insulator photonic crystal slabs,” IEEE J. Quantum Electron 38, 736–742 (2002). [CrossRef]
16. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and I. Borel,” Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006). [CrossRef]
17. B.-S. Song, T. Asano, and S. Noda, “Heterostructures in two-dimensional photonic-crystal slabs and their application to nanocavities,” J. Phys. D 40, 2629–2634 (2007). [CrossRef]
18. Y. Z. Liu, S. Feng, J. Tian, C. Ren, H. H. Tao, Z. Y. Li, B.Y. Cheng, and D. Z. Zhang, “Mulitchannel filters with shape designing in two-dimensional photonic crystal slabs,” J. Appl. Phys. 102, 043102 (2007). [CrossRef]
19. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, (1995).