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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 20 — Oct. 2, 2006
  • pp: 9444–9450
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Photonic crystal waveguides with semi-slow light and tailored dispersion properties

Lars H. Frandsen, Andrei V. Lavrinenko, Jacob Fage-Pedersen, and Peter I. Borel  »View Author Affiliations


Optics Express, Vol. 14, Issue 20, pp. 9444-9450 (2006)
http://dx.doi.org/10.1364/OE.14.009444


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Abstract

We demonstrate a concept for tailoring the group velocity and dispersion properties for light propagating in a planar photonic crystal waveguide. By perturbing the holes adjacent to the waveguide core it is possible to increase the useful bandwidth below the light-line and obtain a photonic crystal waveguide with either vanishing, positive, or negative group velocity dispersion and semi-slow light. We realize experimentally a silicon-on-insulator photonic crystal waveguide having nearly constant group velocity ~c0/34 in an 11-nm bandwidth below the silica-line.

© 2006 Optical Society of America

1. Introduction

The intricate confinement of light in a photonic crystal waveguide (PhCW) [1

1. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]

] and its resulting dispersion properties offer sophisticated possibilities for realizing complex nanophotonic circuits. Potentially, PhCWs may facilitate delay lines for package synchronization, dispersion compensation, and enhanced light-matter interactions [2

2. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanesku, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]

, 3

3. R. S. Jacobsen, K. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006). [CrossRef] [PubMed]

] in nanophotonic circuits by exploiting slow-light phenomena [4

4. 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, 253902 (2001). [CrossRef] [PubMed]

]. However, the practical utilization of ultra-slow light reaching group velocities below ~c 0/200 [3

3. R. S. Jacobsen, K. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006). [CrossRef] [PubMed]

,5

5. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real space observations of ultraslow light in photonic crystal waveguides,” Phys. Rev Lett. 94, 073903 (2005). [CrossRef] [PubMed]

] in PhCWs may be limited due to an inherent small bandwidth [6

6. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on Optical Filters,” IEEE J. Quantum Electron. 37, 525–532 (2001). [CrossRef]

], impedance mismatch [7

7. Y. A. Vlasov and S. J. McNab, “Coupling into the slow light mode in slab-type photonic crystal waveguides,” Opt. Lett. 31, 50–52 (2006). [CrossRef] [PubMed]

, 8

8. J. García, P. Sanchis, and J. Martí, “Detailed analysis of the influence of structure length on pulse propagation through finite-size photonic crystal waveguides,” Opt. Express 14, 6879–6893 (2006). [CrossRef] [PubMed]

], intensified loss mechanisms at scattering centers [9

9. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in Photonic Crystal Waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]

], and extreme dispersive pulse broadening [4

4. 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, 253902 (2001). [CrossRef] [PubMed]

, 10

10. 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, 1658–1672 (2006). [CrossRef] [PubMed]

]. Previously, it has been demonstrated that the dispersion properties of PhCWs can be altered via a structural tuning of the waveguide geometry, typically, by changing the waveguide width or by introducing bi-periodicity [11

11. K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic-crystal waveguides with high velocity,” Opt. Commun. 198, 395–402 (2001). [CrossRef]

, 12

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

, 13

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

]. However, such dramatic changes of the PhCW may lead to multimode operation, decreased coupling efficiency to a photonic wire, and structural continuity problems in, e.g., bend and splitting regions. Here, we show how the knowledge of the field distributions in a single-line defect (W1) PhCW can be exploited to tailor the dispersion properties of the fundamental even photonic bandgap (PBG) mode. In this way, one can realize a silicon-on-insulator (SOI) W1 PhCW with semi-slow light having a group velocity in the range ~(c 0/15–c 0/100); vanishing, positive, or negative group velocity dispersion (GVD); and low-loss propagation in a practical ~5–15 nm bandwidth.

2. Design aspects & modeling

The group velocity v g of light with frequency ω in an optical waveguide is given as [14

14. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley-Interscience, 1997).

]:

vg=dωdk=c0ng,
(1)

where k is the wavevector along the waveguide and n g the group index. The dispersion relations ω(k) for a given waveguide system can easily be obtained numerically by using e.g. the plane wave expansion (PWE) method. Hereafter, it is straightforward to obtain v g by numerical differentiation according to Eq. (1). The group velocity in a PhCW is strongly dependent on the frequency [6

6. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on Optical Filters,” IEEE J. Quantum Electron. 37, 525–532 (2001). [CrossRef]

] as quantified by the GVD parameter β 2, given by the second order derivative of the dispersion relation [14

14. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley-Interscience, 1997).

]:

β2=d2kdω2=dngdω1c0.
(2)

Figure 1(a) shows a typical band diagram for transverse-electric (TE) polarized light in a 2D W1 PhCW which is realized in silicon by arranging air holes of diameter D=0.6Λ in a triangular lattice with pitch Λ. The normalized dispersion relations are found for wavevectors kz along the direction of the waveguide core [15

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

], which is formed by introducing a single line-defect in the nearest-neighbor direction of the lattice. The inset of Fig. 1(a) shows the supercell used in the PWE calculation. As seen, the W1 PhCW supports one even (solid black) and one odd (dashed black) mode in the PBG which is located at frequencies ~(0.20–0.28) Λ/λ. The parities of the modes are defined by their in-plane symmetry with respect to the waveguide core. The even PBG mode is seen to flatten for kz≲0.3 and, eventually, obtains a zero slope at k z=0.5. This is due to the folding at the Brillouin zone-end [1

1. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]

]. Figure 1(b) shows the resulting group velocity v g for the even PBG mode (black) calculated by using Eq. (1). It reveals a dramatic reduction in v g for increasing wavevectors (lower frequencies). As expected, the group velocity v gc 0/4 is close to that of light propagating in a conventional silicon waveguide (with n Si≈3.5) for k z≳0.3. For k z≳0.3, v g decreases monotonically below c 0/20. Near k z=0.5, v g approaches zero, hence, addressed as the slow-light regime. Therefore, different frequencies will travel at very different speeds in a W1 PhCW. Correspondingly, the GVD parameter β 2 (red) plotted in Fig. 1(b) increases by several orders of magnitudes from -104 ps2/km to an extreme negative GVD below -109 ps2/km in the slow-light regime. Such extreme GVD will distort any pulse-train propagating through the W1 PhCW. Hence, the monotonically changing group velocity along with the extreme GVD in the W1 PhCW will be devastating for the utilization of slow-light in e.g. a wavelength-division multiplexed system.

Fig. 1. (a) Typical band diagram showing the normalized frequencies versus normalized wavevectors for a single-line defect 2D photonic crystal waveguide supporting an even (solid) and odd (dashed) mode in the bandgap. The inset of the graph sketches the supercell used in the plane wave expansion calculation. (b) Group velocity v g in units of the speed of light in vacuum, c 0, versus wavevector k z (black), calculated by using Eq. (1). The group velocity dispersion parameter β 2 obtained from Eq. (2) (red). (c) Modal field distributions in a W1 PhCW for the wavevectors marked by a red, yellow, and green square in (a) and (b).

Fig. 2. Movement of the even PBG mode when changing the diameter (a) D 1 of the first row and (b) D 2 of the second row of holes in a W1 PhCW. Bulk holes have diameter D=0.60Λ.

One design goal for a practical utilization of slow-light in a W1 PhCW could be to obtain semi-slow light with a group velocity in the range ~(c 0/15–c 0/100) and a vanishing GVD (i.e., a linear dispersion relation) in a ~5–15 nm bandwidth. This can be achieved by combining the effects illustrated in Figs. 2(a) and 2(b). By decreasing D 1, we perform a coarse tuning of v g, raising it to the desired range and increasing the bandwidth with linear dispersion for large wavevectors kz. By changing D 2, we can fine-tune v g for the tail of the mode, whereby we arrive at the desired group velocity and/or dispersion. Figure 3(a) shows a 3D calculation of the band diagram of the even PBG mode for three examples of perturbed W1 PhCWs. The diameter D 1 has been decreased by 30 nm (red), 40 nm (yellow), and 60 nm (green), whereas the diameter D 2 has been increased by 60 nm (red), 30 nm (yellow), and 10 nm (green) to obtain different group velocities and GVDs in different bandwidths.

Fig. 3. (a). Band diagram for different even PBG modes in a 3D W1 PhCW where the diameter D 1/D 2 of the first/second row of holes have been changed according to the legend, relative to the bulk diameter D=222 nm. (b) Corresponding calculated group indices (bottom) and group velocity dispersion parameter β 2 (top, left and right) for the modes plotted in (a).

In the PWE calculation, the pitch Λ=370 nm and the diameter of the bulk holes D=0.60Λ=222 nm. The structure is set to penetrate a 340-nm layer of silicon placed on a buffer layer of silica and having air above. For comparison, the even mode for the generic W1 PhCW is also plotted (black). As seen, the perturbations represented by the yellow and green curves result in regions with nearly linear dispersion relations situated below the silica-line (violet) and are, thus, vertically confined to the silicon slab. Figure 3(b, bottom) shows the corresponding group indices, n g [see Eq. (1)], calculated by numerical differentiation of the bands in Fig. 3(a). As seen, the two perturbed PhCWs have plateaus where the group indices are constant within 5% around ~40 (green) and ~60 (yellow) over bandwidths of roughly ~13 nm and ~9 nm, respectively. These group-index plateaus are in sharp contrast to the group-index behavior for the generic W1 PhCW (black) that rises monotonically to extreme values. Inspecting the calculated GVD β 2 values in Fig. 3(b, top), we see that the group-index plateaus lead to relatively low and positive GVDs on the order of 105–106 ps2/km (yellow and green curves), which is not much larger in magnitude than in the index-like regime. A finer tuning of the geometry may lead to vanishing GVDs. Furthermore, we see that a perturbation of the W1 PhCW geometry also allows us to realize a PhCW having an extreme positive GVD (red curve), i.e. having a negative dispersion [14

14. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley-Interscience, 1997).

].

3. Fabrication and experimental results

Figure 4 shows a typical scanning electron micrograph of a fabricated 10-µm long perturbed W1 PhCW, where the diameter D 1/D 2 has been decreased/increased relative to the bulk holes with diameter D=234 nm. The triangular photonic crystal lattice with pitch Λ=370 nm penetrates the top 338-nm silicon slab of an SOI wafer having a 1-µm thick buffer layer of silica. The photonic crystal patterns were defined by using electron-beam lithography and transferred to the silicon by inductively coupled plasma reactive ions etching.

Fig. 4. Scanning electron micrograph of a perturbed photonic crystal waveguide. The diameter D 1/D 2 has been decreased/increased compared to the diameter D of the bulk holes.

Figure 5(a, right) shows the transmission spectrum (gray) measured for TE-polarized light in a 500-µm long perturbed PhCW where the diameters D 1/D 2 have been decreased/increased by ~54 nm/~9 nm. The band diagram for the fabricated structure is plotted in Fig. 5(a, left) with a ~1% blue-shift (solid black). The two transmission peaks of the measured spectrum can easily be correlated to the two linear dispersion regions for the even PBG mode located above (dotted red) and below (dotted green) the silica-line. The group index n g for light propagating through the 500-µm long perturbed PhCW has been measured experimentally and calculated numerically by using the time-of-flight methods as described in Ref. [16

16. R. Jacobsen, A. Lavrinenko, L. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express 13, 7861–7871 (2005). [CrossRef] [PubMed]

]. Figure 5(b) plots the group index obtained from measurement (black), 2D Finite-Difference Time-Domain calculation (FDTD, blue), and 3D PWE-calculation (red). The 2D FDTD group-index spectrum has been blue-shifted ~12.5% to match the experimental wavelength scale. The group index for the ~11-nm broad transmission peak corresponding to the linear part of the even PBG mode below the silica-line (indicated by the dotted green lines in Fig. 5) has been measured to n g=34±3, which is in close agreement with the values predicted by FDTD and PWE. The obtained uncertainty below 10% of the measured group-index is better compared to the uncertainty obtained in Ref. [16

16. R. Jacobsen, A. Lavrinenko, L. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express 13, 7861–7871 (2005). [CrossRef] [PubMed]

]. This is due to a lower loss and an increased optical path length, i.e. a longer absolute time-of-flight, for the PhCW in the present investigation. The fringes in the FDTD-calculated spectrum are caused by the finite length [8

8. J. García, P. Sanchis, and J. Martí, “Detailed analysis of the influence of structure length on pulse propagation through finite-size photonic crystal waveguides,” Opt. Express 14, 6879–6893 (2006). [CrossRef] [PubMed]

] of the 20-µm long PhCW used in the simulation. The PWE group-index plateau in Fig. 5(b) reveals a positive GVD parameter β 2 on the order of 105–106 ps2/km, which is of opposite sign and several orders of magnitudes lower than β 2 in the slow-light regime of a W1 PhCW [see Fig. 1(b)]. The propagation loss for the perturbed PhCW has been measured with a typical error of 2–3 dB/mm using the cut-back method with four PhCWs of lengths 50, 100, 500, and 1000 µm and is shown in Fig. 5(b) (green curve). As seen, the propagation loss in the ~11-nm wavelength range is less than ~20 dB/mm and drops below 5 dB/mm in a ~2-nm bandwidth from ~(0.2385–0.2388) Λ/λ. We have fabricated a different set of perturbed PhCWs having a ~7-nm group-index plateau with n g=17±2 and showing a propagation loss of 4.2±1.2 dB/mm. The advantage of the perturbed PhCWs is that the losses induced by the silica-line and extreme dispersion are diminished in comparison with W1 PhCWs [17

17. M. Settle, M. Salib, A. Michaeli, and T. F. Krauss, “Low loss silicon on insulator photonic crystal waveguides made by 193nm optical lithography,” Opt. Express 14, 2440–2445 (2006). [CrossRef] [PubMed]

]. Thus, perturbed PhCWs are very promising for the SOI configuration as they offer bandwidth, slow-light, and low losses.

Fig. 5. (a). 3D band diagram (left) and transmission spectrum (right) for a perturbed 500-µm PhCW with ΔD1=-60 nm and ΔD2=+10 nm. The even PBG mode (solid black) gives rise to two transmission peaks: one located above (dotted red) and one located below (dotted green) the silica-line (violet). (b) Measured (black), 2D FDTD (blue) and 3D PWE (red) calculated group index for the perturbed PhCW. The measured propagation loss is also plotted (green).

4. Conclusion

The field distributions in a single-line defect photonic crystal waveguide have shown to be useful guidelines for tailoring the dispersion relation of the fundamental, even photonic bandgap mode. The tailoring is obtained by doing a simple perturbation to the diameters of the first and second row of holes adjacent to the waveguide core. This general concept can be applied to photonic crystal waveguides of any design. In this way, it is possible to realize waveguides having low-loss bandwidths with semi-low group velocity, and vanishing, negative, or positive group velocity dispersion. We have realized a W1 photonic crystal waveguide in silicon-on-insulator material having an ~11-nm bandwidth below the silica-line with a nearly constant group velocity ~c 0/34 and relatively low and positive group velocity dispersion with β 2 on the order of 105–106 ps2/km. Within this bandwidth, the measured propagation loss is less than 20 dB/mm and drops below 5 dB/mm in a ~2-nm bandwidth.

With tailored group velocity and dispersion, photonic crystal waveguides become much more attractive for practical applications. The perturbed waveguides offer novel possibilities for realizing compact integrated components used for e.g. pulse shaping, dispersion compensation, and enhanced non-linear effects. They may also help to understand how propagation losses in photonic crystal waveguides scale with group velocity and dispersion.

Acknowledgment

This work was supported in parts by the Danish Technical Research Council via the PIPE (Planar Integrated PBG Elements) project and by the New Energy and Industrial Technology Development Organization (NEDO) via the Japanese Industrial Technology Research Area.

References and links

1.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000). [CrossRef]

2.

M. Soljacic, S. G. Johnson, S. Fan, M. Ibanesku, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]

3.

R. S. Jacobsen, K. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006). [CrossRef] [PubMed]

4.

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, 253902 (2001). [CrossRef] [PubMed]

5.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real space observations of ultraslow light in photonic crystal waveguides,” Phys. Rev Lett. 94, 073903 (2005). [CrossRef] [PubMed]

6.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on Optical Filters,” IEEE J. Quantum Electron. 37, 525–532 (2001). [CrossRef]

7.

Y. A. Vlasov and S. J. McNab, “Coupling into the slow light mode in slab-type photonic crystal waveguides,” Opt. Lett. 31, 50–52 (2006). [CrossRef] [PubMed]

8.

J. García, P. Sanchis, and J. Martí, “Detailed analysis of the influence of structure length on pulse propagation through finite-size photonic crystal waveguides,” Opt. Express 14, 6879–6893 (2006). [CrossRef] [PubMed]

9.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in Photonic Crystal Waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]

10.

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, 1658–1672 (2006). [CrossRef] [PubMed]

11.

K. Yamada, H. Morita, A. Shinya, and M. Notomi, “Improved line-defect structures for photonic-crystal waveguides with high velocity,” Opt. Commun. 198, 395–402 (2001). [CrossRef]

12.

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

13.

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

14.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley-Interscience, 1997).

15.

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

16.

R. Jacobsen, A. Lavrinenko, L. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express 13, 7861–7871 (2005). [CrossRef] [PubMed]

17.

M. Settle, M. Salib, A. Michaeli, and T. F. Krauss, “Low loss silicon on insulator photonic crystal waveguides made by 193nm optical lithography,” Opt. Express 14, 2440–2445 (2006). [CrossRef] [PubMed]

OCIS Codes
(230.7390) Optical devices : Waveguides, planar
(260.2030) Physical optics : Dispersion

ToC Category:
Optical Devices

History
Original Manuscript: August 23, 2006
Revised Manuscript: September 19, 2006
Manuscript Accepted: September 20, 2006
Published: October 2, 2006

Citation
Lars H. Frandsen, Andrei V. Lavrinenko, Jacob Fage-Pedersen, and Peter I. Borel, "Photonic crystal waveguides with semi-slow light and tailored dispersion properties," Opt. Express 14, 9444-9450 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9444


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References

  1. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, "Linear waveguides in photonic crystal slabs," Phys. Rev. B 62, 8212-8222 (2000). [CrossRef]
  2. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanesku, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002). [CrossRef]
  3. R. S. Jacobsen, K. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri and A. Bjarklev, "Strained silicon as a new electro-optic material," Nature 441, 199-202 (2006). [CrossRef] [PubMed]
  4. 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, 253902 (2001). [CrossRef] [PubMed]
  5. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers,"Real space observations of ultraslow light in photonic crystal waveguides," Phys. Rev Lett. 94, 073903 (2005). [CrossRef] [PubMed]
  6. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on Optical Filters," IEEE J. Quantum Electron. 37, 525-532 (2001). [CrossRef]
  7. Y. A. Vlasov and S. J. McNab, "Coupling into the slow light mode in slab-type photonic crystal waveguides," Opt. Lett. 31, 50-52 (2006). [CrossRef] [PubMed]
  8. J. García, P. Sanchis, and J. Martí, "Detailed analysis of the influence of structure length on pulse propagation through finite-size photonic crystal waveguides," Opt. Express 14, 6879-6893 (2006). [CrossRef] [PubMed]
  9. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic optical scattering loss in Photonic Crystal Waveguides: Role of fabrication disorder and photon group velocity," Phys. Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]
  10. 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, 1658-1672 (2006). [CrossRef] [PubMed]
  11. K. Yamada, H. Morita, A. Shinya, and M. Notomi, "Improved line-defect structures for photonic-crystal waveguides with high velocity," Opt. Commun. 198, 395-402 (2001). [CrossRef]
  12. A. Jafarpour, A. Adibi, Y. Xu, and R. K. Lee, "Mode dispersion in biperiodic photonic crystal waveguides," Phys. Rev. B 68, 233102 (2003). [CrossRef]
  13. A. Y. Petrov, and M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004). [CrossRef]
  14. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley-Interscience, 1997).
  15. S. Johnson and J. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
  16. R. Jacobsen, A. Lavrinenko, L. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. Borel, "Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides," Opt. Express 13, 7861-7871 (2005). [CrossRef] [PubMed]
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