Slow light with large group index–bandwidth product in ellipse-hole photonic crystal waveguides Xu Han, Tao Wang,* Jian Tang, Bo Liu, BoYun Wang, Yu He, and Youjiang Zhu Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received 11 November 2014; revised 10 January 2015; accepted 14 January 2015; posted 16 January 2015 (Doc. ID 226711); published 20 February 2015
In this study, we propose a new type of slow light photonic crystal waveguide structure to achieve wideband slow light with low dispersion. The waveguide is based on a triangular lattice ellipse-hole photonic crystal imposed simply by a selective altering of the locations of the holes adjacent to the line defect. Under a constant group index criterion of 10% variation, when group indices are nearly constants of 54, 69, and 80, their corresponding bandwidths of the flat band reach 12.7, 10.0, and 8.6 nm around 1550 nm, respectively. A nearly constant large group index–bandwidth product of 0.44 is achieved for all cases. Low dispersion slow light propagation is confirmed by studying the relative temporal pulse-width spreading with the two-dimensional finite-difference time-domain method. © 2015 Optical Society of America OCIS codes: (130.5296) Photonic crystal waveguides; (230.5298) Photonic crystals. http://dx.doi.org/10.1364/AO.54.001543
1. Introduction
Recently, slow light has attracted considerable interest for its great advantages such as room-temperature operation and easy high-density on-chip integration. It has many potential applications in the areas of optical delay lines [1–3], ultrafast all-optical signal processing [4–6], quantum computing [7,8], and nonlinear optical devices [9–11]. Photonic crystal, particularly constructed in a silicon-on-insulator slab, is among the most optimal structures for achieving slow light in a waveguide formation [12]. The slow light in a photonic crystal waveguide (PCW) enhances the light–matter interaction and reduces either the active length or the optical energy required for obtaining the same linear and nonlinear effect compared with the fast light regime [13]. The key parameters to evaluate the performance of slow light PCWs include the group index, bandwidth, propagation loss, and group velocity dispersion (GVD). The first two parameters are already 1559-128X/15/061543-05$15.00/0 © 2015 Optical Society of America
encapsulated by the commonly used figure of merit, namely the group index–bandwidth product (GBP), which indicates the obtained compromise between a large operating bandwidth and a large slow light group index. The propagation loss is mainly due to fabrication imperfections, and slow light propagation amplifies the disorder-induced scattering. The high backscattering loss with an n2g dependence is reported in [14], which has been experimentally confirmed [15]. But this is not the end of the road, and more recently, it has been demonstrated the propagation loss issue varies in different designs due to the different field intensity on the interfaces between the dielectric material and air [16]. That means the propagation loss issue can also be addressed by specially designing the dispersion of the waveguides. A large GVD may also accompany the slow light propagation, which distorts the pulse shape and thereby compromises the benefit of slow light [17]. So far, various approaches have been proposed to tailor the dispersion. The methods include the modification of the hole radius [8], altering the position of the air holes [18–22], merging coupled cavities [23], and employing annular holes [24–26], 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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crescent-shaped air holes [27] or eye-shaped holes [28]. Among these methods, shifting the air holes is considered to be the simplest and can be controlled much more accurately. While other methods may also perform well in theory, they have more challenging fabrication issues. An ellipse hole may be the easiest noncircular air hole shape to fabricate, and PCWs with ellipse-hole structures have been designed recently [29–33]. In this paper, a novel ellipse-hole photonic crystal slab waveguide is proposed, and ultralow GVD values in large slow light bandwidth can be achieved. Photonic band structures are calculated by the plane wave expansion (PWE) method with the MIT photonic bands package [34]. Numerical results showed the GBP around of 0.44 can be achieved for a group index ranging from 54 to 80. Low dispersion slow light propagation is numerically demonstrated by a two-dimensional (2D) finite difference time-domain (FDTD) method simulation. Therefore, our structures may have high potential for practical applications. 2. Structure and Numerical Analysis
Figure 1 shows the proposed PCW structure. The PCW comprises a triangular lattice with a lattice constant a. The x axis is parallel to the line defect, and the y axis is perpendicular to the line defect. The PCW is a modified W1 waveguide, which is base on an alternative row of ellipse-hole PCW (AEPCW) proposed in [32], where a high GBP of 0.399 with a slow light group index of 37 and bandwidth of 16.6 nm are obtained under the criterion of 10% variation. In the AEPCW, the major axis is along the x axis for the air holes in the even rows, while in the odd rows the major axis of the holes is along
Fig. 1. Schematic structure of the proposed PCW: the major axis of the blue holes in the first row adjacent to the waveguide is along the x axis, and the blue holes are shifted along the arrows. The displacement relative to the ideal lattice is given by p, where shifts toward the positive direction of the x axis is positive. The supercell periodicity has taken to be a0 2a. 1544
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the y axis. In our structure, we selectively changed the orientation of the holes in the first row. As shown in Fig. 1, the blue holes are located with its major axis along the x axis. Furthermore, the blue holes are shifted toward the positive direction of the x axis with a displacement denoted by p in order to get wideband slow light with larger group index. The lengths of the long and short shafts of the ellipse are denoted by A and B, respectively. The optimized parameters of A 0.81a and B 0.73a reported in [32] are used in this study for simplicity. 2D analysis with an equivalent slab effective index of 2.9 is adopted in this paper. The supercell is selected with 2 units in the x direction and 11 × 1.732 units in the y direction, and a resolution of 60 grids per spatial period was adopted. The group index can be increased from 54 to 80, whereas the GPB can be maintained at approximately 0.44. Clearly shown in Fig. 2(a), as we increase the magnitude of the parameter p, the lower frequency of the band moves down, while the higher frequency of the band nearly unchanged. This phenomenon demonstrates that only the mode in the slow light region “feels” the increase in the shifting parameter p. The calculation of the GBP is performed by means of the expression GBP n¯ g ×
Δω : ω0
(1)
Fig. 2. (a) Dispersion curves for the optimized PCWs. (b) Calculated group index as a function of normalized frequency.
Here n¯ g denotes an average group index, which is defined as [35] Z n¯ g
ω0 Δω∕2 ω0 −Δω∕2
ng ω ×
dω ; Δω
(2)
and Δω∕ω0 is defined as the normalized bandwidth of a slow light region. Here we calculate the GBP with the constant group index criterion, where the group index ng remains constant within 10% [19]. This criterion may seem too simplistic and lead to the appearance of high order dispersion. However, in practice it is sufficient for most cases because most experimental situations tend to be limited by propagation loss rather than by dispersion broadening. Figure 2(b) shows the calculated group index as a function of the normalized frequency. The group index increased with p, and the slow light bands still remain rather flat for those higher group indices. The simulation results for different optimized parameters are listed in Table 1. Bandwidths of 12.7, 10.0, and 8.6 nm were obtained for nominal group indices of 54, 69, and 80, respectively. A nearly constant GBP of 0.44 was achieved for all cases. Compared with previous works based on both circular air holes and elliptical air holes [28,29,31,32], it is observed that much larger GBP values were obtained in our PCW geometry. Figure 3 shows the modal field distributions for the even mode for the AEPCW and our three optimized PCWs when the wavevector is fixed at k 0.44 (2π∕a0 ). Figure 3(a) shows the Ey electric field component distributions of the AEPCW. All the waveguides are in the slow light regime when the wavevector is fixed at k 0.44 (2π∕a0 ). The electric field is highly concentrated in the first and second row of holes, Compared to the AEPCW, the electric field of our structures penetrated more toward the air holes adjacent to the line defect. The magnitude of the electric field in the line defect of our structure is smaller than the AEPCW and becomes even smaller as we increase the shifting parameter p.
Fig. 3. Ey electric field component distributions of (a) the AEPCW and our three optimized PCWs: (b) p 0, (c) p 0.04a, (d) p 0.05a. The wavevector is fixed at k 0.442π∕a0 .
The GVD parameters (β2 d2 k∕dω2 ) of our structures are shown in Fig. 4. The dispersion parameters β2 for our three optimized PCWs are lower than 2 × 105 a∕2πc2 . If we take β2 < 106 a∕2πc2 as low GVD [33], then all our proposed structures are low GVD devices.
Table 1. Group Index, GBP, and Bandwidth under Different Optimized Shifting Parameters and Comparison Between This Paper and Reference Papers
Current work p0 p 0.05a [32] [18] [19]
[31]
Optimized shifting parameters 54 p 0.04a 80 AEPCW Two hole diameter variations Shift of two rows
n¯ g
Bandwidth centered at GBP 1550 nm (nm)
0.442 12.7 69 0.443 0.445 8.6 37 0.399 34 0.240
32 50 93 Adjust the position and the 41 hole size of ellipse-holes 73
0.304 0.305 0.260 0.214 0.207
10.0 16.6 11.0 14.7 9.5 4.3 8.1 4.4
Fig. 4. GVD relation of the slow light PCWs when the group indices are nominally constant at 54, 69, and 80. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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index variation within a 10% range, we respectively, attain available bandwidths of 12.7, 10.0, and 8.6 nm around 1550 nm, which are much improved compared with that of the previous work based on other methods. The results are confirmed by using the FDTD simulation, and the results essentially coincide with the prediction performed by PWE method. This work is supported by the National Natural Science Foundation of China (Grants No. 61376055 and 61006045), and National Basic Research Program of China (Grant No. 2010CB923204). Fig. 5. Time-domain optical pulse propagation in the proposed slow light PCW with the group index ng 54.
To demonstrate the results based on the PWE method, the time domain pulse propagation was investigated through the optimized structures using the 2D FDTD method [34–37]. A resolution of 60 grids per spatial period was adopted in this simulation. Perfectly matched absorbing boundary layers were applied to the surroundings of the structures. The PCW with group index nominally constant at ng 54 was considered in the simulation. The transmission length is 80a. Given the terminal coupling loss between the PCW and waveguide, we positioned the input time monitor 10a to the left of the computational window; the output time monitor was positioned 10a to its right. Hence, the distance between the monitors is L 60a. Figure 5 shows the normalized field amplitude detected by the input and output time monitors placed at the two sides of the PCW. A Gaussian pulse source centered at 0.2802 (2πc∕a) with a frequency width of 0.0022 (2πc∕a) was used. A lattice constant of 434 nm was chosen for the central wavelength of λ 1550 nm. The peaks of the normalized field amplitude were located at approximately 3065 a∕c and approximately 6380 a∕c in the input and output detecting points, respectively. The pulse has been delayed in the waveguide, and the delay time of Δt 3315 a∕c was obtained between the two pulse peaks. The average group index was calculated as ng c∕L∕Δt 55.3, which was in agreement with the PWE calculation. Meanwhile, the full width at halfmaximum of the pulse only expanded minimally: from 980 (a∕c) to 1030 (a∕c). Thus, the relative pulse distortion was only 5.10%, which indicated that the dispersion was adequately low for optical pulse propagation [38]. 3. Conclusion
In this work, simply by a selective altering of the locations of the holes adjacent to the line defect of the ellipse-hole photonic crystal, we can easily acquire wideband and low dispersion slow light waveguides. Through the optimization of the structures by the PWE method, PCWs with the group indices at 54, 69, and 80 are obtained, while restricting the group 1546
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References 1. 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, 65–69 (2005). 2. Y. J. Zhou and B. J. Yang, “A 4-way wavelength demultiplexer based on the plasmonic broadband slow wave system,” Opt. Express 22, 21589–21599 (2014). 3. K. Kiyota, T. Kise, N. Yokouchi, T. Ide, and T. Baba, “Various low group velocity effects in photonic crystal ine defect waveguides and their demonstration by laser oscillation,” Appl. Phys. Lett. 88, 201904 (2006). 4. Y. Okawachi, M. Foster, J. Sharping, A. Gaeta, Q. Xu, and M. Lipson, “All-optical slow-light on a photonic chip,” Opt. Express 14, 2317–2322 (2006). 5. S. M. Mirjalili, “SoMIR framework for designing high-NDBP photonic crystal waveguide,” Appl. Opt. 53, 3945–3953 (2014). 6. M. Janfaza and M. A. Mansouri-Birjandi, “Wideband slow light in photonic crystal slab waveguide based on geometry adjustment and optofluidic infiltration,” Appl. Opt. 52, 8184–8189 (2013). 7. D. L. Wang, Z. Y. Yu, Y. M. Liu, X. T. Guo, C. G. Shu, and S. Zhou, “Transverse-electric and transverse-magnetic mode slow light propagation in a two-dimensional photonic crystal waveguide,” Appl. Opt. 52, 6523–6528 (2013). 8. 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, 233903 (2004). 9. M. Christelle, E. H. Majid, and J. E. Benjamin, “Slow light enhancement nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344– 356 (2010). 10. B. Corcoran, C. Monat, D. Pudo, B. J. Eggleton, T. F. Krauss, D. J. Moss, L. O’Faolain, M. Pelusi, and T. P. White, “Nonlinear loss dynamics in a silicon slow-light photonic crystal waveguide,” Opt. Express 35, 1073–1075 (2010). 11. C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. J. Eggleton, T. P. White, L. O’Faolain, J. Li, and T. F. Krauss, “Four-wave mixing in slow light silicon crystal waveguide,” Opt. Express 18, 22915–22927 (2010). 12. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008). 13. Y. Zhai, H. P. Tian, and Y. F. Ji, “Slow light property improvement and optical buffer capability in ring-shape-hole photonic crystal waveguide,” J. Lightwave Technol. 29, 3083–3090 (2011). 14. 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). 15. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and L. Ramunno, “Disorder-induced scattering loss of linedefect waveguides in photonic crystal slabs,” Phys. Rev. B 72, 161318 (2005). 16. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010). 17. E. Dulkeith, F. N. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-
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28. Y. Wan, K. Fu, C. H. Li, and M. J. Yun, “Improving slow light effect in photonic crystal line defect waveguide by using eye-shaped scatterers,” Opt. Commun. 286, 192–196 (2013). 29. F. Wang, J. Ma, and C. Jiang, “Dispersionless slow wave in novel 2-D photonic crystal line defect waveguides,” J. Lightwave Technol. 26, 1381–1386 (2008). 30. S. Rawal, R. Sinha, and R. M. De La Rue, “Slow light miniature devices with ultra-flattened dispersion in silicon-oninsulator photonic crystal,” Opt. Express 17, 13315–13325 (2009). 31. F. Long, H. Tian, and Y. Ji, “Buffering capability and limitations in low dispersion photonic crystal waveguides with elliptical airholes,” Appl. Opt. 49, 4808–4813 (2010). 32. Y. M. Xu, L. Xiang, E. Cassan, D. S. Gao, and X. L. Zhang, “Slow light in an alternative row of ellipse-hole photonic crystal waveguide,” Appl. Opt. 52, 1155–1160 (2013). 33. J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008). 34. J. Tang, T. Wang, X. M. Li, B. Liu, B. Y. Wang, and Y. He, “Systematic design of wideband slow light in ellipse-hole photonic crystal waveguides,” J. Opt. Soc. Am. B 31, 1011–1017 (2014). 35. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). 36. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible freesoftware package for electro-magnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010). 37. B. Liu, T. Wang, J. Tang, X. M. Li, C. B. Dong, and Y. He, “Wideband slow light with low dispersion in asymmetric slotted photonic crystal waveguides,” Appl. Opt. 52, 8394–8401 (2013). 38. Y. Zhao, Y. N. Zhang, and H. F. Hu, “Dispersion engineering of slow light in ellipse-shaped-hole slotted photonic crystal,” J. Lightwave Technol. 32, 2144–2151 (2014).
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In this study, we propose a new type of slow light photonic crystal waveguide structure to achieve wideband slow light with low dispersion. The waveguide is based on a triangular lattice ellipse-hole photonic crystal imposed simply by a selective altering of the locations of the holes adjacent to the line defect. Under a constant group index criterion of 10% variation, when group indices are nearly constants of 54, 69, and 80, their corresponding bandwidths of the flat band reach 12.7, 10.0, and 8.6 nm around 1550 nm, respectively. A nearly constant large group index–bandwidth product of 0.44 is achieved for all cases. Low dispersion slow light propagation is confirmed by studying the relative temporal pulse-width spreading with the two-dimensional finite-difference time-domain method. © 2015 Optical Society of America OCIS codes: (130.5296) Photonic crystal waveguides; (230.5298) Photonic crystals. http://dx.doi.org/10.1364/AO.54.001543
1. Introduction
Recently, slow light has attracted considerable interest for its great advantages such as room-temperature operation and easy high-density on-chip integration. It has many potential applications in the areas of optical delay lines [1–3], ultrafast all-optical signal processing [4–6], quantum computing [7,8], and nonlinear optical devices [9–11]. Photonic crystal, particularly constructed in a silicon-on-insulator slab, is among the most optimal structures for achieving slow light in a waveguide formation [12]. The slow light in a photonic crystal waveguide (PCW) enhances the light–matter interaction and reduces either the active length or the optical energy required for obtaining the same linear and nonlinear effect compared with the fast light regime [13]. The key parameters to evaluate the performance of slow light PCWs include the group index, bandwidth, propagation loss, and group velocity dispersion (GVD). The first two parameters are already 1559-128X/15/061543-05$15.00/0 © 2015 Optical Society of America
encapsulated by the commonly used figure of merit, namely the group index–bandwidth product (GBP), which indicates the obtained compromise between a large operating bandwidth and a large slow light group index. The propagation loss is mainly due to fabrication imperfections, and slow light propagation amplifies the disorder-induced scattering. The high backscattering loss with an n2g dependence is reported in [14], which has been experimentally confirmed [15]. But this is not the end of the road, and more recently, it has been demonstrated the propagation loss issue varies in different designs due to the different field intensity on the interfaces between the dielectric material and air [16]. That means the propagation loss issue can also be addressed by specially designing the dispersion of the waveguides. A large GVD may also accompany the slow light propagation, which distorts the pulse shape and thereby compromises the benefit of slow light [17]. So far, various approaches have been proposed to tailor the dispersion. The methods include the modification of the hole radius [8], altering the position of the air holes [18–22], merging coupled cavities [23], and employing annular holes [24–26], 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
1543
crescent-shaped air holes [27] or eye-shaped holes [28]. Among these methods, shifting the air holes is considered to be the simplest and can be controlled much more accurately. While other methods may also perform well in theory, they have more challenging fabrication issues. An ellipse hole may be the easiest noncircular air hole shape to fabricate, and PCWs with ellipse-hole structures have been designed recently [29–33]. In this paper, a novel ellipse-hole photonic crystal slab waveguide is proposed, and ultralow GVD values in large slow light bandwidth can be achieved. Photonic band structures are calculated by the plane wave expansion (PWE) method with the MIT photonic bands package [34]. Numerical results showed the GBP around of 0.44 can be achieved for a group index ranging from 54 to 80. Low dispersion slow light propagation is numerically demonstrated by a two-dimensional (2D) finite difference time-domain (FDTD) method simulation. Therefore, our structures may have high potential for practical applications. 2. Structure and Numerical Analysis
Figure 1 shows the proposed PCW structure. The PCW comprises a triangular lattice with a lattice constant a. The x axis is parallel to the line defect, and the y axis is perpendicular to the line defect. The PCW is a modified W1 waveguide, which is base on an alternative row of ellipse-hole PCW (AEPCW) proposed in [32], where a high GBP of 0.399 with a slow light group index of 37 and bandwidth of 16.6 nm are obtained under the criterion of 10% variation. In the AEPCW, the major axis is along the x axis for the air holes in the even rows, while in the odd rows the major axis of the holes is along
Fig. 1. Schematic structure of the proposed PCW: the major axis of the blue holes in the first row adjacent to the waveguide is along the x axis, and the blue holes are shifted along the arrows. The displacement relative to the ideal lattice is given by p, where shifts toward the positive direction of the x axis is positive. The supercell periodicity has taken to be a0 2a. 1544
APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015
the y axis. In our structure, we selectively changed the orientation of the holes in the first row. As shown in Fig. 1, the blue holes are located with its major axis along the x axis. Furthermore, the blue holes are shifted toward the positive direction of the x axis with a displacement denoted by p in order to get wideband slow light with larger group index. The lengths of the long and short shafts of the ellipse are denoted by A and B, respectively. The optimized parameters of A 0.81a and B 0.73a reported in [32] are used in this study for simplicity. 2D analysis with an equivalent slab effective index of 2.9 is adopted in this paper. The supercell is selected with 2 units in the x direction and 11 × 1.732 units in the y direction, and a resolution of 60 grids per spatial period was adopted. The group index can be increased from 54 to 80, whereas the GPB can be maintained at approximately 0.44. Clearly shown in Fig. 2(a), as we increase the magnitude of the parameter p, the lower frequency of the band moves down, while the higher frequency of the band nearly unchanged. This phenomenon demonstrates that only the mode in the slow light region “feels” the increase in the shifting parameter p. The calculation of the GBP is performed by means of the expression GBP n¯ g ×
Δω : ω0
(1)
Fig. 2. (a) Dispersion curves for the optimized PCWs. (b) Calculated group index as a function of normalized frequency.
Here n¯ g denotes an average group index, which is defined as [35] Z n¯ g
ω0 Δω∕2 ω0 −Δω∕2
ng ω ×
dω ; Δω
(2)
and Δω∕ω0 is defined as the normalized bandwidth of a slow light region. Here we calculate the GBP with the constant group index criterion, where the group index ng remains constant within 10% [19]. This criterion may seem too simplistic and lead to the appearance of high order dispersion. However, in practice it is sufficient for most cases because most experimental situations tend to be limited by propagation loss rather than by dispersion broadening. Figure 2(b) shows the calculated group index as a function of the normalized frequency. The group index increased with p, and the slow light bands still remain rather flat for those higher group indices. The simulation results for different optimized parameters are listed in Table 1. Bandwidths of 12.7, 10.0, and 8.6 nm were obtained for nominal group indices of 54, 69, and 80, respectively. A nearly constant GBP of 0.44 was achieved for all cases. Compared with previous works based on both circular air holes and elliptical air holes [28,29,31,32], it is observed that much larger GBP values were obtained in our PCW geometry. Figure 3 shows the modal field distributions for the even mode for the AEPCW and our three optimized PCWs when the wavevector is fixed at k 0.44 (2π∕a0 ). Figure 3(a) shows the Ey electric field component distributions of the AEPCW. All the waveguides are in the slow light regime when the wavevector is fixed at k 0.44 (2π∕a0 ). The electric field is highly concentrated in the first and second row of holes, Compared to the AEPCW, the electric field of our structures penetrated more toward the air holes adjacent to the line defect. The magnitude of the electric field in the line defect of our structure is smaller than the AEPCW and becomes even smaller as we increase the shifting parameter p.
Fig. 3. Ey electric field component distributions of (a) the AEPCW and our three optimized PCWs: (b) p 0, (c) p 0.04a, (d) p 0.05a. The wavevector is fixed at k 0.442π∕a0 .
The GVD parameters (β2 d2 k∕dω2 ) of our structures are shown in Fig. 4. The dispersion parameters β2 for our three optimized PCWs are lower than 2 × 105 a∕2πc2 . If we take β2 < 106 a∕2πc2 as low GVD [33], then all our proposed structures are low GVD devices.
Table 1. Group Index, GBP, and Bandwidth under Different Optimized Shifting Parameters and Comparison Between This Paper and Reference Papers
Current work p0 p 0.05a [32] [18] [19]
[31]
Optimized shifting parameters 54 p 0.04a 80 AEPCW Two hole diameter variations Shift of two rows
n¯ g
Bandwidth centered at GBP 1550 nm (nm)
0.442 12.7 69 0.443 0.445 8.6 37 0.399 34 0.240
32 50 93 Adjust the position and the 41 hole size of ellipse-holes 73
0.304 0.305 0.260 0.214 0.207
10.0 16.6 11.0 14.7 9.5 4.3 8.1 4.4
Fig. 4. GVD relation of the slow light PCWs when the group indices are nominally constant at 54, 69, and 80. 20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS
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index variation within a 10% range, we respectively, attain available bandwidths of 12.7, 10.0, and 8.6 nm around 1550 nm, which are much improved compared with that of the previous work based on other methods. The results are confirmed by using the FDTD simulation, and the results essentially coincide with the prediction performed by PWE method. This work is supported by the National Natural Science Foundation of China (Grants No. 61376055 and 61006045), and National Basic Research Program of China (Grant No. 2010CB923204). Fig. 5. Time-domain optical pulse propagation in the proposed slow light PCW with the group index ng 54.
To demonstrate the results based on the PWE method, the time domain pulse propagation was investigated through the optimized structures using the 2D FDTD method [34–37]. A resolution of 60 grids per spatial period was adopted in this simulation. Perfectly matched absorbing boundary layers were applied to the surroundings of the structures. The PCW with group index nominally constant at ng 54 was considered in the simulation. The transmission length is 80a. Given the terminal coupling loss between the PCW and waveguide, we positioned the input time monitor 10a to the left of the computational window; the output time monitor was positioned 10a to its right. Hence, the distance between the monitors is L 60a. Figure 5 shows the normalized field amplitude detected by the input and output time monitors placed at the two sides of the PCW. A Gaussian pulse source centered at 0.2802 (2πc∕a) with a frequency width of 0.0022 (2πc∕a) was used. A lattice constant of 434 nm was chosen for the central wavelength of λ 1550 nm. The peaks of the normalized field amplitude were located at approximately 3065 a∕c and approximately 6380 a∕c in the input and output detecting points, respectively. The pulse has been delayed in the waveguide, and the delay time of Δt 3315 a∕c was obtained between the two pulse peaks. The average group index was calculated as ng c∕L∕Δt 55.3, which was in agreement with the PWE calculation. Meanwhile, the full width at halfmaximum of the pulse only expanded minimally: from 980 (a∕c) to 1030 (a∕c). Thus, the relative pulse distortion was only 5.10%, which indicated that the dispersion was adequately low for optical pulse propagation [38]. 3. Conclusion
In this work, simply by a selective altering of the locations of the holes adjacent to the line defect of the ellipse-hole photonic crystal, we can easily acquire wideband and low dispersion slow light waveguides. Through the optimization of the structures by the PWE method, PCWs with the group indices at 54, 69, and 80 are obtained, while restricting the group 1546
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