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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 26 — Dec. 22, 2008
  • pp: 21483–21491
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Γ–M waveguides in two-dimensionaltriangular-lattice photonic crystal slabs

Ya-Zhao Liu, Rong-Juan Liu, Chang-Zhu Zhou, Dao-Zhong Zhang, and Zhi-Yuan Li  »View Author Affiliations


Optics Express, Vol. 16, Issue 26, pp. 21483-21491 (2008)
http://dx.doi.org/10.1364/OE.16.021483


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Abstract

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

1. Introduction

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 [19], 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.

Fig. 1. Schematic image of a triangular lattice photonic crystal slab with waveguides. The original Γ–M waveguide as illustrated in panel (b) is formed by removing a line of diamond pattern of lattice points as depicted in panel (a) along the Γ–M crystalline direction.

Fig. 2. (a). Calculated dispersion relation and (b) measured transmission spectra of guided modes in the original Γ-M waveguides (with parameters of a=430nm, r 1=r 2=120nm, and w 0=2a). The width of the transmission window is 22nm. The red line in panel (a) represents the light line, and the yellow shadow region in panel (b) represents the theoretical mode band.

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 [14

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]

]. 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.

Fig. 3. (a). Calculated modal field profile for the even (upper plot) and odd (lower plot) guided mode and dispersion relation of guided modes in the modified waveguide structures by varying the parameters of (b) r 1, (c) r 2, and (d) w d. In panel (b), r 2=120nm, w d=0.75 w 0, r 1 varies from 0 to 90nm. In panel (c), r 1=50nm, w d=0.75 w 0, r 2 varies from 150 to 190nm. In panel (d), r 1=50nm, r 2=170nm, w d varies from 0.65 w 0 to 0.75 w 0. The black lines in panels represent the light line.

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.

Fig. 4. SEM pictures of the fabricated Γ–M waveguide sample before (upper panel) and after (lower panel) optimization. The structural parameters for the two waveguides are a=430nm, r 1=r 2=r 0=120nm, w d=w 0, and a=430nm, r 1=50nm, r 2=170nm, w d=0.65 w 0.

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.

Fig. 5. Measured transmission spectra of several optimized Γ–M waveguides. The theoretical results of transmission windows of the entire guide modes are represented by the horizontal gray lines. Shadow areas represent the pass-band regions. In all the structures a=430nm and w d=0.65w 0.
Fig. 6. Simulated transmission spectra of the optimized Γ–M waveguides with three different lengths. The structural parameters for the waveguide is a=430nm, r 1=50nm, r 2=170nm, w d=0.65 w 0. The transmission intensities of the waveguide mode have the peak values of -0.1, -1.2, -1.3dB and minimum values of -12.9, -13.0 and -14.4 dB respectively.

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.

4. Summary

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.

Acknowledgments

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]

3.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000). [CrossRef] [PubMed]

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] [PubMed]

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]

7.

H. Takano, B. S. Song, T. Asano, and S. Noda, “Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal,” Opt. Express 14, 3491–3496 (2006). [CrossRef] [PubMed]

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] [PubMed]

9.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express 13, 1202–1214 (2005). [CrossRef] [PubMed]

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]

11.

T. Asano, B. S. Song, and S. Noda, “Analysis of the experimental Q factors (~1 million) of photonic crystal nanocavities,” Opt. Express 14, 1996–2002 (2006). [CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

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).

20.

D. Gerace and L. C. Andreani, “Low-loss guided modes in photonic crystal waveguides,” Opt. Express 13, 4939 (2005). [CrossRef] [PubMed]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(230.3990) Optical devices : Micro-optical devices
(130.5296) Integrated optics : Photonic crystal waveguides
(230.5298) Optical devices : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: September 9, 2008
Revised Manuscript: October 10, 2008
Manuscript Accepted: November 25, 2008
Published: December 15, 2008

Citation
Ya-Zhao Liu, Rong-Juan Liu, Chang-Zhu Zhou, Dao-Zhong Zhang, and Zhi-Yuan Li, "Γ−Μ waveguides in two-dimensionaltriangular-lattice photonic crystal slabs," Opt. Express 16, 21483-21491 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-26-21483


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References

  1. E. Yablonovitch, "How to be truly photonic," Science 289, 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]
  3. S. Noda, A. Chutinan, and M. Imada, "Trapping and emission of photons by a single defect in a photonic bandgap structure," Nature 407, 608-610 (2000). [CrossRef] [PubMed]
  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] [PubMed]
  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]
  7. H. Takano, B. S. Song, T. Asano, S. Noda, "Highly efficient multi-channel drop filter in a two-dimensional hetero photonic crystal,"Opt. Express 14, 3491-3496 (2006). [CrossRef] [PubMed]
  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] [PubMed]
  9. Y. Akahane, T. Asano, B. S. Song and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005). [CrossRef] [PubMed]
  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]
  11. T. Asano, B. S. Song, and S. Noda, "Analysis of the experimental Q factors (~ 1 million) of photonic crystal nanocavities," Opt. Express 14, 1996-2002 (2006). [CrossRef] [PubMed]
  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] [PubMed]
  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] [PubMed]
  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).
  20. D. Gerace and L. C. Andreani, "Low-loss guided modes in photonic crystal waveguides," Opt. Express 13, 4939 (2005). [CrossRef] [PubMed]

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