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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 6 — Mar. 15, 2010
  • pp: 5707–5712
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Wideband slow light and dispersion control in oblique lattice photonic crystal waveguides

Feng-Chun Leng, Wen-Yao Liang, Bin Liu, Tong-Biao Wang, and He-Zhou Wang  »View Author Affiliations


Optics Express, Vol. 18, Issue 6, pp. 5707-5712 (2010)
http://dx.doi.org/10.1364/OE.18.005707


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Abstract

We find that the angle between elementary lattice vectors obviously affects the bandwidth and dispersion of slow light in photonic crystal line-defect waveguides. When the fluctuation of group index is strictly limited in a ±1% range, the oblique lattice structures with the angle between elementary lattice vectors slightly larger than 60° have broader available bandwidth of flat band slow light than triangular lattice structures. For example, for the angle 66°, there are increases of the available bandwidth from 20% to 68% for several different structures. For the same angle and a ±10% variation in group velocity, when group indices are nearly constants of 30, 48.5, 80 and 130, their corresponding bandwidths of flat band reach 20 nm, 11.8 nm, 7.3 nm and 3.9 nm around 1550 nm, respectively. The increasing of bandwidth is related to the shift of the anticrossing point towards smaller wave numbers.

© 2010 OSA

1. Introduction

Phenomenon of slow light has long been a focus due to its many potential applications, such as ultrafast all-optical signal processing [1

1. Y. Okawachi, M. Foster, J. Sharping, A. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14(6), 2317–2322 (2006). [CrossRef] [PubMed]

, 2

2. S.-M. Ma, H. Xu, and B. S. Ham, “Electromagnetically-induced transparency and slow light in GaAs/AlGaAs multiple quantum wells in a transient regime,” Opt. Express 17(17), 14902–14908 (2009). [CrossRef] [PubMed]

], quantum computing [3

3. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]

, 4

4. M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84(7), 1419–1422 (2000). [CrossRef] [PubMed]

], and enhancement of light-matter interactions [5

5. K. Kiyota, T. Kise, N. Yokouchi, T. Ide, and T. Baba, “Various low group velocity effects in photonic crystal line defect waveguides and their demonstration by laser oscillation,” Appl. Phys. Lett. 88(20), 201904 (2006). [CrossRef]

, 6

6. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009). [CrossRef]

]. Slow light propagation had been achieved by a variety of method, including electromagnetically induced transparency [2

2. S.-M. Ma, H. Xu, and B. S. Ham, “Electromagnetically-induced transparency and slow light in GaAs/AlGaAs multiple quantum wells in a transient regime,” Opt. Express 17(17), 14902–14908 (2009). [CrossRef] [PubMed]

,7

7. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reductionto 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

], coherent population oscillation [8

8. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]

], stimulated Brillouin scattering [9

9. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94(15), 153902 (2005). [CrossRef] [PubMed]

] and photonic crystals (PCs) [10

10. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]

]. As one kind of method to slow down the light speed, PC waveguides have attracted much attention recently because they have many merits, such as ability of being integrated on optical chip, operating at room temperature. Moreover, bandwidth and dispersion of slow light are crucial in telecommunications using short pulses and high-speed modulated slow-light signals, PC waveguides provide the ability of potential wide-bandwidth and dispersion-free propagation.

2. Design and numerical simulations

The studied structure is shown in Fig. 1
Fig. 1 Schematic diagram of the PC waveguide. The basic structure consists of a W0.85 waveguide in an oblique lattice. The waveguide width is W0.85=1.7acos(θ/2). Based on the basic structure, the first two rows of air-hole are shifted horizontally with distances of d1 and d2 respectively; and we define d1 and d2 to be positive if the air-holes are shifted along the directions denoted by the blue arrows.
. There is a line defect in a Si air-bridge slab PC consisting of the oblique lattice. The lengths of two elementary lattice vectors of the oblique lattice are a1= a2= a, where a is the lattice constant. The angle θ between two elementary lattice vectors (ABELV) is 66°. The radius of air-hole is r = 0.32a. The thickness of the slab is h = 0.5328a, and n = 3.5 is the refractive index of Si. The entire structure is symmetrical along X and Z directions.

Due to the fact that flat band can be systematically designed by changing the positions of the first two rows of air-hole of PC waveguides [17

17. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008). [CrossRef] [PubMed]

], we modify the positions of the first two rows of air-hole adjacent to the line-defect based on the W0.85 waveguide whose width is W0.85=1.7acos(θ/2). d1 and d2 denote the shift distances of the first two rows of air-hole from their original positions of an unmodified structure, as shown in Fig. 1.

When optical pulses pass through the PC waveguide, they will be broadened and distorted as the result of nonzero dispersion [21

21. Y. Tanaka, H. Kuwatsuka, H. Kawashima, N. Ikeda, Y. Sugimoto, T. Hasama, and H. Ishikawa, “Effect of third-order dispersion on subpicosecond pulse propagation in photonic-crystal waveguides,” Appl. Phys. Lett. 89(13), 131101 (2006). [CrossRef]

,22

22. R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, K. Asakawa, and L. Kuipers, “The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides,” Opt. Express 14(4), 1658–1672 (2006). [CrossRef] [PubMed]

]. Therefore, for the applications of small group velocities, the group velocity dispersion of the PC waveguides should be as small as possible. So we consider a stricter limit for group index that the fluctuation range of the group index is only ±1% in later discussions.

Figure 3
Fig. 3 Group index for θ = 66° and θ = 60°. The curves on the left of figure correspond to θ = 66°, while the curves on the right correspond to θ = 60°.
shows three sets group index curves. The left part and the right part correspond to θ = 66° and θ = 60°, respectively. There exists a constant group index for each curve within certain frequency range. And the constant group index is almost equal for the left part and the right part of each set curve. It is obvious that the bandwidth of flat band is increased for different group indices when θ varies from 60° to 66°. Δω1 and Δω2 are the bandwidths of flat band for θ = 60° and θ = 66°, respectively. For ng are about 32, 70, and 204, the ratios of Δω2/Δω1 are 1.37, 1.2, and 1.68, respectively.

3. Analysis

Figure 4 show the dispersion and the group velocity curves calculated by using the two-dimensional model with the effective index approximation. It should be noted that the width of the first Brillouin zone along the waveguide direction will become smaller with increasing θ, as shown in Fig. 4(a). In order to compare slow-light-zone of waveguides with different θ, we have translated the group velocity curves along the horizontal axis to make the right endpoints of them coinciding at wavevector k=0.52π/a in Fig. 4(b). In Fig. 4(b), the group velocity curves show that the anticrossing point shifts to smaller wave numbers with increasing θ, which result in an extended slow-light-zone. In this paper, we do not consider larger θ, because flat band likely come into the region where energies in waveguide are lost by coupling with air, with the anticrossing point shifts to smaller wave numbers. In fact, we can get wider flat band from our result calculated for θ = 70°.

To show the influence of slow-light-zone on bandwidth of flat band, group velocity curves of two waveguide structures with θ = 60° and θ = 66° are shown in Fig. 5
Fig. 5 The group velocity curves of two waveguides with different θ. For θ = 60°, d1 = 0.152, d2 = 0.06; while for θ = 66°, d1 = 0.128a, d2 = −0.002a. Other parameters used are as follows: r = 0.32a, h = 0.5328a, n = 3.5, effective refractive indices are 2.746, 2.775 for θ = 66° and 60° respectively.
. During the flat band range, they have the same group velocity (vg = c/27). It can be seen that the extension of slow-light-zone makes the flat band become wider.

4. Conclusion

In conclusion, we have demonstrated that near constant group index bandwidth can be increased by adopting oblique lattice structure with a proper ABELV for a PC waveguide. For ng = 30, ng = 48.5 and ng = 80, the bandwidth is 20 nm, 11.8 nm and 7.3 nm, respectively. When the fluctuation of the group index within flat band is in a ± 1% range, the calculated results suggest that the ratio of available bandwidth of flat band can reach 1.68 between an oblique lattice structure (θ = 66°) and a triangular one (θ = 60°). The method of adjusting the ABELV can be combined with other design approaches to tailor group velocity and dispersion, which enhance the possibilities of practical applications for PC waveguide.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (10874250, 10674183,10804131), National 973 Project of China (2004CB719804), and Ph. D Degrees Foundation of Education Ministry of China (20060558068).

References and links

1.

Y. Okawachi, M. Foster, J. Sharping, A. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14(6), 2317–2322 (2006). [CrossRef] [PubMed]

2.

S.-M. Ma, H. Xu, and B. S. Ham, “Electromagnetically-induced transparency and slow light in GaAs/AlGaAs multiple quantum wells in a transient regime,” Opt. Express 17(17), 14902–14908 (2009). [CrossRef] [PubMed]

3.

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]

4.

M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84(7), 1419–1422 (2000). [CrossRef] [PubMed]

5.

K. Kiyota, T. Kise, N. Yokouchi, T. Ide, and T. Baba, “Various low group velocity effects in photonic crystal line defect waveguides and their demonstration by laser oscillation,” Appl. Phys. Lett. 88(20), 201904 (2006). [CrossRef]

6.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009). [CrossRef]

7.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reductionto 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

8.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]

9.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94(15), 153902 (2005). [CrossRef] [PubMed]

10.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]

11.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef] [PubMed]

12.

A. Yu. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85(21), 4866–4868 (2004). [CrossRef]

13.

M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express 15(1), 219–226 (2007). [CrossRef] [PubMed]

14.

A. Säynätjoki, M. Mulot, J. Ahopelto, and H. Lipsanen, “Dispersion engineering of photonic crystal waveguides with ring-shaped holes,” Opt. Express 15(13), 8323–8328 (2007). [CrossRef] [PubMed]

15.

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14(20), 9444–9450 (2006). [CrossRef] [PubMed]

16.

A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, “Mode dispersion in biperiodic photonic crystal waveguides,” Phys. Rev. B 68(23), 233102 (2003). [CrossRef]

17.

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008). [CrossRef] [PubMed]

18.

D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Lett. 15, 5264–5270 (2007).

19.

T. Baba, T. Kawaaski, H. Sasaki, J. Adachi, and D. Mori, “Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide,” Opt. Express 16(12), 9245–9253 (2008). [CrossRef] [PubMed]

20.

S. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]

21.

Y. Tanaka, H. Kuwatsuka, H. Kawashima, N. Ikeda, Y. Sugimoto, T. Hasama, and H. Ishikawa, “Effect of third-order dispersion on subpicosecond pulse propagation in photonic-crystal waveguides,” Appl. Phys. Lett. 89(13), 131101 (2006). [CrossRef]

22.

R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, K. Asakawa, and L. Kuipers, “The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides,” Opt. Express 14(4), 1658–1672 (2006). [CrossRef] [PubMed]

OCIS Codes
(260.2030) Physical optics : Dispersion
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Photonic Crystals

History
Original Manuscript: December 14, 2009
Revised Manuscript: January 17, 2010
Manuscript Accepted: January 30, 2010
Published: March 5, 2010

Citation
Feng-Chun Leng, Wen-Yao Liang, Bin Liu, Tong-Biao Wang, and He-Zhou Wang, "Wideband slow light and dispersion control in oblique lattice photonic crystal waveguides," Opt. Express 18, 5707-5712 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5707


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References

  1. Y. Okawachi, M. Foster, J. Sharping, A. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14(6), 2317–2322 (2006). [CrossRef] [PubMed]
  2. S.-M. Ma, H. Xu, and B. S. Ham, “Electromagnetically-induced transparency and slow light in GaAs/AlGaAs multiple quantum wells in a transient regime,” Opt. Express 17(17), 14902–14908 (2009). [CrossRef] [PubMed]
  3. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]
  4. M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84(7), 1419–1422 (2000). [CrossRef] [PubMed]
  5. K. Kiyota, T. Kise, N. Yokouchi, T. Ide, and T. Baba, “Various low group velocity effects in photonic crystal line defect waveguides and their demonstration by laser oscillation,” Appl. Phys. Lett. 88(20), 201904 (2006). [CrossRef]
  6. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic crystal waveguides,” Nat. Photonics 3(4), 206–210 (2009). [CrossRef]
  7. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reductionto 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
  8. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]
  9. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94(15), 153902 (2005). [CrossRef] [PubMed]
  10. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]
  11. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. 87(25), 253902 (2001). [CrossRef] [PubMed]
  12. A. Yu. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85(21), 4866–4868 (2004). [CrossRef]
  13. M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express 15(1), 219–226 (2007). [CrossRef] [PubMed]
  14. A. Säynätjoki, M. Mulot, J. Ahopelto, and H. Lipsanen, “Dispersion engineering of photonic crystal waveguides with ring-shaped holes,” Opt. Express 15(13), 8323–8328 (2007). [CrossRef] [PubMed]
  15. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14(20), 9444–9450 (2006). [CrossRef] [PubMed]
  16. A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, “Mode dispersion in biperiodic photonic crystal waveguides,” Phys. Rev. B 68(23), 233102 (2003). [CrossRef]
  17. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16(9), 6227–6232 (2008). [CrossRef] [PubMed]
  18. D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Lett. 15, 5264–5270 (2007).
  19. T. Baba, T. Kawaaski, H. Sasaki, J. Adachi, and D. Mori, “Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide,” Opt. Express 16(12), 9245–9253 (2008). [CrossRef] [PubMed]
  20. S. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef] [PubMed]
  21. Y. Tanaka, H. Kuwatsuka, H. Kawashima, N. Ikeda, Y. Sugimoto, T. Hasama, and H. Ishikawa, “Effect of third-order dispersion on subpicosecond pulse propagation in photonic-crystal waveguides,” Appl. Phys. Lett. 89(13), 131101 (2006). [CrossRef]
  22. R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. van Hulst, K. Asakawa, and L. Kuipers, “The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides,” Opt. Express 14(4), 1658–1672 (2006). [CrossRef] [PubMed]

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