Wideband slow light with low dispersion in asymmetric slotted photonic crystal waveguides Bo Liu, Tao Wang,* Jian Tang, Xiaoming Li, Chuanbo Dong, and Yu He Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received 6 September 2013; revised 23 October 2013; accepted 5 November 2013; posted 7 November 2013 (Doc. ID 197284); published 25 November 2013
A new procedure of designing slotted photonic crystal waveguides is proposed to achieve slow light with improved normalized delay-bandwidth product and low group velocity dispersion that is suitable for both the W1 defect mode and the slot mode. The lateral symmetry of the waveguide in our study is broken by shifting the air holes periodically along the slot axis. The conversion of the “flat band” from band-up slow light to band-down slow light is achieved for the W1 defect mode. The group index curves of the W1 mode change from U-like to step-like and the group indices of 47, 67 and 130 are obtained with the bandwidth over 7.2, 4.8, and 2.3 nm around 1550 nm, respectively. We also obtain the group indices of 42, 55, and 108 for the slot mode with the bandwidth over 6.2, 5.6, and 2.2 nm, respectively. Then the low dispersion slow light propagation is numerically demonstrated by the finite-difference time-domain method. © 2013 Optical Society of America OCIS codes: (230.1150) All-optical devices; (230.5298) Photonic crystals; (230.7390) Waveguides, planar; (260.2030) Dispersion. http://dx.doi.org/10.1364/AO.52.008394
1. Introduction
Slow light has recently regained vitality for its extensive potential applications, such as ultrafast alloptical signal processing [1], quantum computing [2,3], and enhancement of light–matter interactions [4]. A photonic crystal, particularly constructed in a silicon-on-insulator (SOI) slab, is among the most optimal structures for achieving slow light in a waveguide formation [5]. 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 [6]. However, light is strongly confined to the high-index guiding layer in traditional PCWs, which would be a drawback in the interaction of light with low-index materials. After the slot photonic 1559-128X/13/348394-08$15.00/0 © 2013 Optical Society of America 8394
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crystal waveguide (SPCW) was first proposed in 2008 [7], researchers realized that a SPCW could combine the property to confine light in the nanometer-sized low-index slot with slow light enhancement available from PCW, which could be advantageously applied in practice [8]. However, as the group velocity decreases, the pulse shape envelop becomes broader and more asymmetric. This effect is called group velocity dispersion (GVD). In order to reduce the effect of GVD and obtain a wider bandwidth, many structures have been studied recently, such as chirping the waveguide property [9], infiltrating dielectric material [10], changing the hole sizes [11], and adjusting the hole positions [12–14]. From the aspect of fabrication processes, adjusting the hole positions is the best choice. First, it is important to emphasize the concept of the normalized delay-bandwidth product (NDBP), which characterizes the compromise between the light slowing down factor and bandwidth. The NDBP is given by
NDBP hnG i × Δω∕ω0 ;
(1)
where hnG i stands for the average group index over a bandwidth, Δω, and ω0 is the central normalized frequency (ω a∕λ, with a is the lattice constant). Previous studies of asymmetric structures mostly have the topic of a PCW [15,16], while the asymmetric SPCW is obtained here. The slow light performance of our structures is improved for both the W1 defect mode and the slot mode, whereas previous publications only optimized for one of the modes [14,17]. By using a 2D plane wave expansion (PWE) [18] with a resolution of 36 pixels∕a (a is the lattice constant), we engineer the dispersion curves of both the W1 defect mode and the slot mode among which the former has received more interests for its easier coupling from/to a photonic nanowire [19–22]. The slow light performances are shown here comparing with the previous published results in terms of the NDBP and bandwidth values. The conversion of the “flat band” from band-up slow light to band-down slow light is achieved for the W1 defect mode. The group index curves of the W1 defect mode are changed from U-like to step-like and high NDBP values of 0.223, 0.21, and 0.192 are obtained, respectively. For a special configuration, the NDBP value has been improved by 163% compared to previous results of the W1 mode. Then, we engineer the band curves of the slot mode and also obtain a high NDBP of 0.168, 0.163, and 0.133. A group index of 67, with a 4.8 nm bandwidth for the W1 defect mode, is demonstrated using the 2D finite-difference time-domain (FDTD) method simulation. Our asymmetric structures might have powerful potential for practical applications.
Fig. 1. Basic structure of a SPCW, where a is the lattice constant, r is the radius of the air holes, and ωs is the slot width; p1 and p2 denote the shift distance of the first two rows of holes along the x axis and are set to be optimized.
of the energy in the slot due to the symmetric constraint [23]. The group index of light can be obtained from the slope of the dispersion curve, which is given by ng c · dk∕dω;
where ω is the frequency, k is the wave vector, and c is the light velocity in a vacuum. The GVD parameter, β, is given by the second-order derivative of the dispersion relation as
2. Structure and Numerical Analysis
The basic structure of a SPCW is shown in Fig. 1, which is constructed in a SOI substrate (n 3.46) by replacing the central row of air holes (radii r 0.34a) with a narrow air slot (slot width ωs 0.28a) in a triangular lattice photonic crystal. We employ the 2D PWE method to study the structure, and the effective index of 2.9 is used for the 210-nm-thick slab [12]. In the numerical simulation, the supercell size p is set to 2a × 11a and the waveguide width is 3a, where a is the lattice constant (a 430 nm). The band diagram of the initial case for the transverse electric-like (TE)-polarized mode is shown in Fig. 2(a). There exists two available modes in the band gap, the slot mode and the W1 defect mode, that are indicated by different color lines. The red solid line in the top left corner of Fig. 2(a) indicates the light line, and the slab mode region is shaded in gray. The points M and N in the two curves are marked as the connection points between the index-guided mode and gap-guided mode, respectively. The electric field (Ey ) distributions of the two modes at kxv 0.42π∕a are shown in Fig. 2(b). It is obvious that the two even modes of interest concentrate most
(2)
β
∂2 k : ∂ω2
(3)
Fig. 2. (a) Band curves for the basic SPCW. Points M and N are marked as the connection points between the index-guided mode and the gap-guided mode; (b) the Ey electric field component distribution of the slot mode and the W1 defect mode at kx 0.42π∕a. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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(b) 150
p1=0a p1=0.01a p1=0.02a p1=0.03a p1=0.04a p1=0.05a
0.307 0.306 0.305
flat band region
0.304
p1=0a p1=0.01a p1=0.02a p1=0.03a p1=0.04a p1=0.05a
120
Group Index ng
Normalized Frequency (ωa/2πc)
(a) 0.308
0.303 0.302 0.301
90
60
0.300 30 0.299 0.298 0 0.297
0.297 0.30
0.35
0.40
0.45
0.50
0.298
0.299
0.300
0.301
0.302
0.303
0.304
Normalized Frequency (ω a/2πc)
Wave Vector (ka/2π)
Fig. 3. (a) Dispersion diagrams (band-up slow light) for different shifts of p1 from 0a to 0.05a with p2 0a and (b) the corresponding U-like group index curves for different shifts of p1 from 0a to 0.05a with p2 0a.
For the SPCW, the extreme negative or positive GVD values will seriously influence the width of the light pulse. Considering the practical application requirement, we should take the GVD below the order of 106 a∕2πc2 [24]. The mechanism for the slow light effect can be explained from a simple 1D grating perspective. The holes on each side of the line defect can be understood as period constrictions; the waveguide is narrow where there is a hole and is wider where there is not [19]. A standing wave form at the Brillouin zone boundary where the Bragg condition (i.e., λ∕2 a, where a is the period) occurs. In our simulations, the asymmetry is obtained by shifting the two rows of air holes and results in a redistribution of the effective index of the waveguide and the electromagnetic field. The modification of the first row of air holes has a strong effect on the index-guided mode, while the displacement of the second row has a stronger effect on the gap-guided mode. Thus, the modifications influence the position and strength of the anticrossing [25] between the index-guided mode and gap-guided mode. Before we used asymmetry, we also studied some symmetric cases in our simulations. The influence of the symmetric case on the distribution of the effective index along the waveguide is not as efficient as the asymmetric case, and cannot satisfy our needs of both the W1 and slot modes. So, we fix the asymmetric procedure to achieve slow light of both the W1 mode and slot mode.
Table 1.
A. Tuning the W1 Defect Mode
In order to eliminate these problems of narrow bandwidth and high GVD, the dispersion diagram should be linearized and flattened. As is seen in Fig. 2(d), the W1 defect mode can interact with several air holes adjacent to the slot, which means that adjusting these air holes can greatly influence the light– matter interaction. Compared with the slot mode, the W1 defect mode has received more interest because of its easier coupling from/to a strip waveguide. So, we first engineer the W1 defect mode in detail. In our structure, the parameters p1 and p2 denote the shift distance of each row from their original positions. We can clearly see that the lateral symmetry is broken. Figure 3(a) shows the dependence of the dispersion curves for the different shifted distances of p1, while other structure parameters remain unchanged. The solid black denotes the initial curve of the basic structure. It is obvious that a higher p1 moves the dispersion curves to a lower frequency and the slope increases. Figure 3(b) shows the corresponding group index curves for different p1 values. The ng , Δλ, and NDBP values of the W1 defect mode for different parameters p1 is shown in Table 1. ng is considered as constant when the fluctuation of the group index is within 10% range. We can see that the group index reduces as p1 increases from 0a to 0.05a, while the corresponding NDBP value increases from 0.084 to 0.170.
n g , Δλ, and NDBP of W1 Defect Mode for Different Parameters p1
Structure
p1 0a
p1 0.01a
p1 0.02a
p1 0.03a
p1 0.04a
p1 0.05a
ng Δλ nm NDBP Frequency interval
84 1.6 0.084 [0.30325, 0.30359]
73 2 0.095 [0.30313, 0.30352]
52 3.6 0.120 [0.30255, 0.30323]
37 6.2 0.144 [0.30165, 0.30283]
25 8.9 0.150 [0.30047, 0.30219]
17 15.2 0.170 [0.2984, 0.30156]
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(b) 250
0.306 0.305 0.304 0.303 0.302 0.301 0.300 0.299 0.298 0.297 0.296 0.295 0.294 0.293 0.292
p2=0a p2=0.05a p2=0.10a p2=0.15a p2=0.18a p2=0.19a p2=0.20a p2=0.21a
200
p2=0a p2=0.05a p2=0.10a p2=0.15a p2=0.18a p2=0.19a p2=0.20a p2=0.21a
0.30
Group Index ng
Normalized Frequency ( ωa/2πc)
(a) 0.307
flat band region
150
100
50
0.35
0.40
0.45
0 0.292
0.50
0.294
Wave Vector (ka/2π)
0.296
0.298
0.300
0.302
0.304
Normalized Frequency (ω a/2πc)
Fig. 4. (a) Dispersion diagrams for different shift of p2 from 0a to 0.21a with p1 0.03a. We change the flat band of the W1 defect mode from band-up slow light (p2 0a–0.15a) to band-down slow light (p2 0.18a–0.21a) and (b) the corresponding group index curves for different shifts of p2 from 0a to 0.21a with p1 0.03a. The group index curves change from U-like (p2 0a–0.15a) to step-like (p2 0.18a–0.21a).
Then, we keep p1 0.03a and study the dependence of the dispersion curves and group index curves on the different shifts of p2. The reason for not choosing p1 0.04a, 0.05a, or other values here is to avoid the W1 defect mode into the slab mode region when we sweep p2. Figure 4(a) shows the dispersion curves for different shifted distances of p2 with p1 0.03a. It is clear that the slope of the flat bands change from positive to negative. In other words, we change the flat band of the W1 defect mode from band-up slow light to band-down slow light. Figure 4(b) shows the corresponding changes of the group index curves for different shift distances of p2 with p1 0.03a. The group index curves change from U-like to step-like as the slope of flat band change. Table 2 shows ng , Δλ, and NDBP values of the W1 defect mode for different parameters p2 with the same p1 0.03a. As parameter p2 increases from 0a to 0.21a, the corresponding NDBP value first reduces from 0.144 to 0.08, and then increases to a high value of 0.22. Last, we change p1 and p2 simultaneously to study the comprehensive dependence of group index curves on the different shifts of p1 and p2. Three different asymmetric structures with improved NDBP values and low GVD are obtained, which are named as A1, A2, and A3, respectively. Figure 5(b) shows the group index curves for the three different structures with the group index of 47, 67, 130 and the NDBP values of 0.223, 0.21, and 0.192, respectively. The GVD curves for the three different sets of p1 and p2 are shown in Fig. 5(c). It is clear that the GVD values Table 2.
of the three optimized structures are all far below the order of 106 a∕2πc2 , which is required for practical application. Furthermore, as each GVD curve presents both a positive region and a negative region, it reveals the applications of dispersion compensation. B. Tuning the Slot Mode
Similar dispersion engineering presented in the previous section can be applied to tune the slot mode. Compared with the W1 defect mode, the slot mode is in a lower frequency region and has a stronger optical confinement in the slot. We first study the dependence of dispersion diagrams on the shift distance of p1. Figure 6(a) shows the dispersion curves of different parameters of p1 0.03a, 0.04a, 0.05a, 0.06a, and 0.07a, which are compared with the basic structure. The solid black curve denotes the initial curve of the basic structure. It is clear that the higher p1 moves the dispersion curves to the lower frequency. The corresponding group index curves are presented in Fig. 6(b). As the parameter p1 increases, the group index of the flat band decreases. Meanwhile, the bandwidth and NDBP value moves to a higher level as p1 increases. For p1 0.03a, 0.04a, 0.05a, 0.06a, and 0.07a, the corresponding NDBP values are 0.11, 0.112, 0.139, 0.145, and 0.152, respectively. Then, we shift the first and second row of air holes simultaneously. Three asymmetric structures with improved NDBP values and low GVD are also
n g , Δλ, and NDBP of W1 Defect Mode for p1 0.03a and Different Parameters p2
Structure
p2 0a
p2 0.05a
p2 0.10a
p2 0.15a
p2 0.18a
p2 0.19a
p2 0.20a
p2 0.21a
ng Δλ nm NDBP Frequency interval
37 6.2 0.144 [0.30165, 0.30283]
39 4.9 0.130 [0.30135, 0.30235]
56 3.3 0.120 [0.30028, 0.30093]
219 0.6 0.08 [0.29821, 0.2983]
140 1.7 0.154 [0.29588, 0.29632]
84 2.9 0.160 [0.2949, 0.29566]
70 4.7 0.213 [0.29402, 0.29491]
59 5.6 0.220 [0.29309, 0.29413]
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p1=0a p2=0a p1=0.03a p2=0.215a p1=0.03a p2=0.202a p1=0.215a p2=0.16a
0.307 0.306 0.305 0.304 0.303 0.302 0.301 0.300 0.299 0.298 0.297 0.296 0.295 0.294 0.293 0.292 0.291
(b) 150
A3
A3
120
Group Index ng
Normalized Frequency (ω a/2πc)
(a) 0.308
90
A2 60
A1 p1=0a p2=0a p1=0.03a p2=0.215a p1=0.03a p2=0.202a p1=0.025a p2=0.16a
A2
A1
30
0 0.30
0.35
0.40
0.45
0.50
0.292
0.296
0.298
0.300
0.302
0.304
0.306
x 105
(c)
p1=0.03a p2=0.215a p1=0.03a p2=0.202a p1=0.025a p2=0.16a p1=0a p2=0a
8
Group Velocity Dispersion (a/2 π c2)
0.294
Normalized Frequency (ω a/2πc)
Wave Vector (ka/2 π)
6
A3
4
2
A1
A2
0
-2
-4 0.292
0.294
0.296
0.298
0.3
0.302
0.304
0.306
Normalized Frequency (wa/2 π c)
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(a)
0.254 0.252 0.250 0.248 0.246 0.244 0.242 0.240
p1=0a p1=0.03a p1=0.04a p1=0.05a p1=0.06a p1=0.07a
0.238 0.236 0.234 0.232 0.30
0.35
0.40
0.45
0.50
Wave Vector (ka/2 π)
(b)
300
250
Group Index ng
presented here, that have different parameters p1 and p2 and are named B1, B2, and B3, respectively. The dispersion curves are presented in Fig. 7(a). The corresponding group index curves and dispersion curves are depicted in Figs. 7(b) and 7(c). We obtain the group indices of 42, 54, and 108; and the NDBP values of 0.168, 0.163, and 0.133 for the three different structures B1, B2, and B3, respectively. It is obvious that the low GVD reduces down to the order of 106 a∕2πc2 and the GVD curves are extremely flat at the slow light region. We note that the GVD changes with positive and negative parts, which also has applications for dispersion compensation. After works of tuning the W1 defect mode and the slot mode, higher NDBP values of 0.223, 0.21, 0.192 for the W1 defect mode and 0.168, 0.163, 0.133 for the slot mode are obtained. We also compare the results with previous reference papers [14,17,26–29]. We have a principle of comparison and NDBP values with a similar group index are compared here. As Table 3 shows, compared to the previous papers, the NDBP values of the W1 defect mode are improved by 85%, 163%, and 60%, respectively; and the NDBP values of the slot mode are also improved by 40%, 75%, and 28% for the three optimized structures B1, B2, and B3, which is illustrated in Table 4. This means that our procedure for designing the “flat
Normalized Frequency (ωa/2πc)
Fig. 5. (a) Dispersion diagrams, (b) group index curves, and (c) the GVD curves for three different sets of p1 and p2, which are named A1, A2, and A3. For the three asymmetric structures A1, A2, and A3, we obtain the higher NDBP values of 0.223, 0.21, and 0.192, respectively. All of them are compared with the basic structure.
200
150
100
p1=0a p1=0.03a p1=0.04a p1=0.05a p1=0.06a p1=0.07a
50
0 0.245
0.246
0.247
0.248
0.249
0.250
0.251
0.252
Normalized Frequency (ω a/2π c)
Fig. 6. (a) Dispersion diagrams and (b) group index curves with different sets of p1 for the slot mode.
(b) 300
0.252
B3
0.250
250
B2
0.248
B1
Group Index ng
Normalized Frequency (ωa/2πc)
(a) 0.254
0.246 0.244 0.242
200
150
B3 100
0.240 p1=0a p2=0a p1=0.05a p2=0.2a p1=0.045a p2=0.15a p1=0.025a p2=0.13a
0.238 0.236
B2 50
p1=0a p2=0a p1=0.05a p2=0.2a p1=0.045a p2=0.15a p1=0.025a p2=0.13a
B1
0 0.30
0.35
0.40
0.45
0.50
0.248
Wave Vector (ka/2 π)
(c)
8
0.250
0.251
0.252
x 106 p1=0.05a p1=0.045a p1=0.025a p1=0a
7
Group Velocity Dispersion (a/2 π c2)
0.249
Normalized Frequency (ω a/2 πc)
6
p2=0.2a p2=0.15a p2=0.13a p2=0a
5 4 3 2 1
B3
B2
B1 0 -1 -2 0.2475
0.248
0.2485
0.249
0.2495
0.25
0.2505
0.251
0.2515
0.252
Normalized Frequency (ωa/2π c)
Fig. 7. (a) Dispersion diagrams for the three optimized structures, B1, B2, and B3; (b), (c) the corresponding group index curves and GVD curves for B1, B2, and B3, respectively.
band” of the SPCW is suitable for both the W1 defect mode and the slot mode. Furthermore, this type of procedure [12–14,17] is simpler from the aspect of the fabrication process since we did not change the radius of the air holes. 3. Time-Domain Analysis
In order to confirm the results obtained from dispersion curve calculations based on the PWE Table 3.
Comparison among the Optimized W1 Defect Mode and References
Structure The present asymmetric structures Reference symmetric structures
method, the transmission characteristics of optical pulses in structure SPCW-A2 are simulated for comparison by the 2D FDTD method [30]. The schematic of the FDTD simulation system is shown in Fig. 8(a). A resolution of 40 pixels∕a (a is the lattice constant) is adopted in this simulation. Perfectly matched absorbing boundary layers are applied to the surroundings of the structure. The asymmetric case with group index ng 67 for structure A2 is
A1 A2 A3 Ref. [26] Ref. [27] Ref. [28] Ref. [29]
p1
p2
0.03a 0.215a 0.03a 0.202a 0.025a 0.16a A comb SPCW A comb slot and change the radius of air holes Ring-shape-hole SPCW Bragg-like slot
ng
Δλ nm
NDBP
47 67 130 47 48
7.2 4.8 2.3 4 4.5
0.223 0.21 0.192 0.121 0.139
67 95
1.8 2
0.08 0.122
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Table 4.
Comparison among the Optimized Slot Mode and References
Structure The present asymmetric structures
p1
Reference symmetric structures
p2
0.05a 0.2a 0.045a 0.15a 0.025a 0.13a A comb SPCW Transversal shift The oblique structure
B1 B2 B3 Ref. [26] Ref. [17] Ref. [14]
considered in our simulation. The transmission length between the input time monitor and the output time monitor is 50a. The result for SPCW-A2 is shown in Fig. 8(b). The normalized center frequency of a Gaussian pulse source is centered at 0.29432πc∕a with a frequency width of 0.000892πc∕a. The peaks of the fluxes are located at ∼5531a∕c and ∼8815a∕c in the input and output detecting points, respectively. The total time delay between the peaks in Fig. 8(b) is Δt 3284a∕c, and the average group index in the SPCW-A2 is calculated as ng c∕L∕Δt 65.6, which agrees with the calculated result from the PWE simulation. The full width at half-maximum (FWHM) of the input pulse is 2133a∕c, while the FWHW of thse output pulse is 2195a∕c. Thus, the relative pulse distortion is only 3.0%, which indicates that the GVD is adequately low for optical pulse propagation. (a)
ng
Δλ nm
NDBP
42 55 108 35 54 110
6.2 5.6 2.2 6 3.3 1.8
0.182 0.20 0.153 0.135 0.114 0.127
4. Conclusion
In short, asymmetric slotted PCWs, formed by shifting holes along the slot axis, are designed here. By careful and detailed optimizing, we achieve the conversion of the “flat band” from band-up slow light to band-down slow light for the W1 defect mode. A large group index, ng 130, with a wide bandwidth, Δλ 2.3 nm, is obtained for the W1 defect mode in the asymmetric structure SPCW-A3. A large index, ng 108, with a wide bandwidth, Δλ 2.2 nm, is also obtained for the slot mode in the structure SPCW-B3. The FDTD simulation is carried out to observe the light propagation in the optimal waveguide. The results agree well with our prediction in the PWE calculations. Compared with previous publications, our asymmetric procedure is suitable for improving the slow light performance of both the W1 defect mode and the slot mode, which would be of powerful potential for further practical applications. This work is supported by the National Natural Science Foundation of China (Grant No. 61376055), and National Basic Research Program of China (Grant No. 2010CB923204). References
(b)
input
Normalized Field Amplitude
1.0
output 0.8
0.6
2133a/c
0.4
2195a/c 0.2
0.0 2000
4000
6000
8000
10000
12000
Time delay (a/c)
Fig. 8. (a) Schematic of FDTD simulation system of the proposed slow light waveguides. (b) Temporal pulses detected at the input and output detecting points in the proposed SPCW-A2. 8400
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10. L. Dai, T. Li, and C. Jiang, “Wideband ultralow high-orderdispersion photonic crystal slow-light waveguide,” J. Opt. Soc. Am. B 28, 1622–1626 (2011). 11. H. Kurt, K. Üstün, and L. Ayas, “Study of different spectral regions and delay bandwidth relation in slow light photonic crystal waveguides,” Opt. Express 18, 26965–26977 (2010). 12. J. Liang, L. Y. Ren, M. J. Yun, and X. J. Wang, “Wideband slow light with ultralow dispersion in a W1 photonic crystal waveguide,” Appl. Opt. 50, G98–G103 (2011). 13. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal wave-guide,” Opt. Lett. 34, 1072–1074 (2009). 14. Y. Zhao, Y. N. Zhang, and Q. Wang, “Wideband slow light with large group index and low dispersion in slotted photonic crystal waveguide,” J. Lightwave Technol. 30, 2812–2817 (2012). 15. J. Ma and C. Jiang, “Flatband slow light in asymmetric linedefect photonic crystal waveguide featuring low group velocity and dispersion,” IEEE J. Quantum Electron. 44, 763–769 (2008). 16. J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 20, 1237–1239 (2008). 17. J. Wu, Y. P. Li, C. Peng, and Z. Y. Wang, “Wideband and low dispersion slow light in slotted photonic crystal waveguide,” Opt. Commun. 283, 2815–2819 (2010). 18. 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). 19. J. H. Wülbern, S. Prorok, J. Hampe, A. Petrov, M. Eich, J. Luo, A. K. Y. Jen, M. Jenett, and A. Jacob, “40 GHz electro-optic modulation in hybrid silicon–organic slotted photonic crystal waveguides,” Opt. Lett. 35, 2753–2755 (2010). 20. X. Wang, C. Y. Lin, S. Chakravarty, J. Luo, A. K. Y. Jen, and R. T. Chen, “Effective in-device r33 of 735 pm/V on electrooptic polymer infiltrated silicon photonic crystal slot waveguides,” Opt. Lett. 36, 882–884 (2011).
21. J. Gao, J. F. McMillan, M. C. Wu, J. Zheng, S. Assefa, and C. W. Wong, “Demonstration of an air-slot mode-gap confined photonic crystal slab nanocavity with ultrasmall mode volumes,” Appl. Phys. Lett. 96, 051123 (2010). 22. J. M. Brosi, C. Koos, L. C. Andreani, M. Waldow, J. Leuthold, and W. Freude, “High-speed-low voltage electro-optic modulator with a polymer infiltrated silicon photonic crystal waveguide,” Opt. Express 16, 4177–4191 (2008). 23. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). 24. R. J. P. Engelen, Y. Sugimoto, Y. Watanabe, J. P. Korterik, N. Ikeda, N. F. 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). 25. 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). 26. C. Caer, X. L. Roux, L. Vivien, and E. Cassan, “Enhanced localization of light in slow wave slot photonic crystal waveguides,” Opt. Lett. 37, 3660–3662 (2012). 27. C. Caer, X. L. Roux, L. Vivien, and E. Cassan, “Dispersion engineered slot photonic crystal waveguides for slow light operation,” Appl. Phys. A 109, 895–899 (2012). 28. K. T. Zhu, T. S. Deng, Y. Sun, Q. F. Zhang, and J. L. Wu, “Slow light property in ring-shape-hole slotted photonic crystal waveguide,” Opt. Commun. 290, 87–91 (2013). 29. C. Caer, X. L. Roux, V. K. Do, D. M. Morini, N. Izard, L. Vivien, D. S. Gao, and E. Cassan, “Dispersion engineering of wide slot photonic crystal waveguides by Bragg-like corrugation of the slot,” IEEE Photon. Technol. Lett. 23, 1298–1300 (2011). 30. 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).
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A new procedure of designing slotted photonic crystal waveguides is proposed to achieve slow light with improved normalized delay-bandwidth product and low group velocity dispersion that is suitable for both the W1 defect mode and the slot mode. The lateral symmetry of the waveguide in our study is broken by shifting the air holes periodically along the slot axis. The conversion of the “flat band” from band-up slow light to band-down slow light is achieved for the W1 defect mode. The group index curves of the W1 mode change from U-like to step-like and the group indices of 47, 67 and 130 are obtained with the bandwidth over 7.2, 4.8, and 2.3 nm around 1550 nm, respectively. We also obtain the group indices of 42, 55, and 108 for the slot mode with the bandwidth over 6.2, 5.6, and 2.2 nm, respectively. Then the low dispersion slow light propagation is numerically demonstrated by the finite-difference time-domain method. © 2013 Optical Society of America OCIS codes: (230.1150) All-optical devices; (230.5298) Photonic crystals; (230.7390) Waveguides, planar; (260.2030) Dispersion. http://dx.doi.org/10.1364/AO.52.008394
1. Introduction
Slow light has recently regained vitality for its extensive potential applications, such as ultrafast alloptical signal processing [1], quantum computing [2,3], and enhancement of light–matter interactions [4]. A photonic crystal, particularly constructed in a silicon-on-insulator (SOI) slab, is among the most optimal structures for achieving slow light in a waveguide formation [5]. 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 [6]. However, light is strongly confined to the high-index guiding layer in traditional PCWs, which would be a drawback in the interaction of light with low-index materials. After the slot photonic 1559-128X/13/348394-08$15.00/0 © 2013 Optical Society of America 8394
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crystal waveguide (SPCW) was first proposed in 2008 [7], researchers realized that a SPCW could combine the property to confine light in the nanometer-sized low-index slot with slow light enhancement available from PCW, which could be advantageously applied in practice [8]. However, as the group velocity decreases, the pulse shape envelop becomes broader and more asymmetric. This effect is called group velocity dispersion (GVD). In order to reduce the effect of GVD and obtain a wider bandwidth, many structures have been studied recently, such as chirping the waveguide property [9], infiltrating dielectric material [10], changing the hole sizes [11], and adjusting the hole positions [12–14]. From the aspect of fabrication processes, adjusting the hole positions is the best choice. First, it is important to emphasize the concept of the normalized delay-bandwidth product (NDBP), which characterizes the compromise between the light slowing down factor and bandwidth. The NDBP is given by
NDBP hnG i × Δω∕ω0 ;
(1)
where hnG i stands for the average group index over a bandwidth, Δω, and ω0 is the central normalized frequency (ω a∕λ, with a is the lattice constant). Previous studies of asymmetric structures mostly have the topic of a PCW [15,16], while the asymmetric SPCW is obtained here. The slow light performance of our structures is improved for both the W1 defect mode and the slot mode, whereas previous publications only optimized for one of the modes [14,17]. By using a 2D plane wave expansion (PWE) [18] with a resolution of 36 pixels∕a (a is the lattice constant), we engineer the dispersion curves of both the W1 defect mode and the slot mode among which the former has received more interests for its easier coupling from/to a photonic nanowire [19–22]. The slow light performances are shown here comparing with the previous published results in terms of the NDBP and bandwidth values. The conversion of the “flat band” from band-up slow light to band-down slow light is achieved for the W1 defect mode. The group index curves of the W1 defect mode are changed from U-like to step-like and high NDBP values of 0.223, 0.21, and 0.192 are obtained, respectively. For a special configuration, the NDBP value has been improved by 163% compared to previous results of the W1 mode. Then, we engineer the band curves of the slot mode and also obtain a high NDBP of 0.168, 0.163, and 0.133. A group index of 67, with a 4.8 nm bandwidth for the W1 defect mode, is demonstrated using the 2D finite-difference time-domain (FDTD) method simulation. Our asymmetric structures might have powerful potential for practical applications.
Fig. 1. Basic structure of a SPCW, where a is the lattice constant, r is the radius of the air holes, and ωs is the slot width; p1 and p2 denote the shift distance of the first two rows of holes along the x axis and are set to be optimized.
of the energy in the slot due to the symmetric constraint [23]. The group index of light can be obtained from the slope of the dispersion curve, which is given by ng c · dk∕dω;
where ω is the frequency, k is the wave vector, and c is the light velocity in a vacuum. The GVD parameter, β, is given by the second-order derivative of the dispersion relation as
2. Structure and Numerical Analysis
The basic structure of a SPCW is shown in Fig. 1, which is constructed in a SOI substrate (n 3.46) by replacing the central row of air holes (radii r 0.34a) with a narrow air slot (slot width ωs 0.28a) in a triangular lattice photonic crystal. We employ the 2D PWE method to study the structure, and the effective index of 2.9 is used for the 210-nm-thick slab [12]. In the numerical simulation, the supercell size p is set to 2a × 11a and the waveguide width is 3a, where a is the lattice constant (a 430 nm). The band diagram of the initial case for the transverse electric-like (TE)-polarized mode is shown in Fig. 2(a). There exists two available modes in the band gap, the slot mode and the W1 defect mode, that are indicated by different color lines. The red solid line in the top left corner of Fig. 2(a) indicates the light line, and the slab mode region is shaded in gray. The points M and N in the two curves are marked as the connection points between the index-guided mode and gap-guided mode, respectively. The electric field (Ey ) distributions of the two modes at kxv 0.42π∕a are shown in Fig. 2(b). It is obvious that the two even modes of interest concentrate most
(2)
β
∂2 k : ∂ω2
(3)
Fig. 2. (a) Band curves for the basic SPCW. Points M and N are marked as the connection points between the index-guided mode and the gap-guided mode; (b) the Ey electric field component distribution of the slot mode and the W1 defect mode at kx 0.42π∕a. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS
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(b) 150
p1=0a p1=0.01a p1=0.02a p1=0.03a p1=0.04a p1=0.05a
0.307 0.306 0.305
flat band region
0.304
p1=0a p1=0.01a p1=0.02a p1=0.03a p1=0.04a p1=0.05a
120
Group Index ng
Normalized Frequency (ωa/2πc)
(a) 0.308
0.303 0.302 0.301
90
60
0.300 30 0.299 0.298 0 0.297
0.297 0.30
0.35
0.40
0.45
0.50
0.298
0.299
0.300
0.301
0.302
0.303
0.304
Normalized Frequency (ω a/2πc)
Wave Vector (ka/2π)
Fig. 3. (a) Dispersion diagrams (band-up slow light) for different shifts of p1 from 0a to 0.05a with p2 0a and (b) the corresponding U-like group index curves for different shifts of p1 from 0a to 0.05a with p2 0a.
For the SPCW, the extreme negative or positive GVD values will seriously influence the width of the light pulse. Considering the practical application requirement, we should take the GVD below the order of 106 a∕2πc2 [24]. The mechanism for the slow light effect can be explained from a simple 1D grating perspective. The holes on each side of the line defect can be understood as period constrictions; the waveguide is narrow where there is a hole and is wider where there is not [19]. A standing wave form at the Brillouin zone boundary where the Bragg condition (i.e., λ∕2 a, where a is the period) occurs. In our simulations, the asymmetry is obtained by shifting the two rows of air holes and results in a redistribution of the effective index of the waveguide and the electromagnetic field. The modification of the first row of air holes has a strong effect on the index-guided mode, while the displacement of the second row has a stronger effect on the gap-guided mode. Thus, the modifications influence the position and strength of the anticrossing [25] between the index-guided mode and gap-guided mode. Before we used asymmetry, we also studied some symmetric cases in our simulations. The influence of the symmetric case on the distribution of the effective index along the waveguide is not as efficient as the asymmetric case, and cannot satisfy our needs of both the W1 and slot modes. So, we fix the asymmetric procedure to achieve slow light of both the W1 mode and slot mode.
Table 1.
A. Tuning the W1 Defect Mode
In order to eliminate these problems of narrow bandwidth and high GVD, the dispersion diagram should be linearized and flattened. As is seen in Fig. 2(d), the W1 defect mode can interact with several air holes adjacent to the slot, which means that adjusting these air holes can greatly influence the light– matter interaction. Compared with the slot mode, the W1 defect mode has received more interest because of its easier coupling from/to a strip waveguide. So, we first engineer the W1 defect mode in detail. In our structure, the parameters p1 and p2 denote the shift distance of each row from their original positions. We can clearly see that the lateral symmetry is broken. Figure 3(a) shows the dependence of the dispersion curves for the different shifted distances of p1, while other structure parameters remain unchanged. The solid black denotes the initial curve of the basic structure. It is obvious that a higher p1 moves the dispersion curves to a lower frequency and the slope increases. Figure 3(b) shows the corresponding group index curves for different p1 values. The ng , Δλ, and NDBP values of the W1 defect mode for different parameters p1 is shown in Table 1. ng is considered as constant when the fluctuation of the group index is within 10% range. We can see that the group index reduces as p1 increases from 0a to 0.05a, while the corresponding NDBP value increases from 0.084 to 0.170.
n g , Δλ, and NDBP of W1 Defect Mode for Different Parameters p1
Structure
p1 0a
p1 0.01a
p1 0.02a
p1 0.03a
p1 0.04a
p1 0.05a
ng Δλ nm NDBP Frequency interval
84 1.6 0.084 [0.30325, 0.30359]
73 2 0.095 [0.30313, 0.30352]
52 3.6 0.120 [0.30255, 0.30323]
37 6.2 0.144 [0.30165, 0.30283]
25 8.9 0.150 [0.30047, 0.30219]
17 15.2 0.170 [0.2984, 0.30156]
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(b) 250
0.306 0.305 0.304 0.303 0.302 0.301 0.300 0.299 0.298 0.297 0.296 0.295 0.294 0.293 0.292
p2=0a p2=0.05a p2=0.10a p2=0.15a p2=0.18a p2=0.19a p2=0.20a p2=0.21a
200
p2=0a p2=0.05a p2=0.10a p2=0.15a p2=0.18a p2=0.19a p2=0.20a p2=0.21a
0.30
Group Index ng
Normalized Frequency ( ωa/2πc)
(a) 0.307
flat band region
150
100
50
0.35
0.40
0.45
0 0.292
0.50
0.294
Wave Vector (ka/2π)
0.296
0.298
0.300
0.302
0.304
Normalized Frequency (ω a/2πc)
Fig. 4. (a) Dispersion diagrams for different shift of p2 from 0a to 0.21a with p1 0.03a. We change the flat band of the W1 defect mode from band-up slow light (p2 0a–0.15a) to band-down slow light (p2 0.18a–0.21a) and (b) the corresponding group index curves for different shifts of p2 from 0a to 0.21a with p1 0.03a. The group index curves change from U-like (p2 0a–0.15a) to step-like (p2 0.18a–0.21a).
Then, we keep p1 0.03a and study the dependence of the dispersion curves and group index curves on the different shifts of p2. The reason for not choosing p1 0.04a, 0.05a, or other values here is to avoid the W1 defect mode into the slab mode region when we sweep p2. Figure 4(a) shows the dispersion curves for different shifted distances of p2 with p1 0.03a. It is clear that the slope of the flat bands change from positive to negative. In other words, we change the flat band of the W1 defect mode from band-up slow light to band-down slow light. Figure 4(b) shows the corresponding changes of the group index curves for different shift distances of p2 with p1 0.03a. The group index curves change from U-like to step-like as the slope of flat band change. Table 2 shows ng , Δλ, and NDBP values of the W1 defect mode for different parameters p2 with the same p1 0.03a. As parameter p2 increases from 0a to 0.21a, the corresponding NDBP value first reduces from 0.144 to 0.08, and then increases to a high value of 0.22. Last, we change p1 and p2 simultaneously to study the comprehensive dependence of group index curves on the different shifts of p1 and p2. Three different asymmetric structures with improved NDBP values and low GVD are obtained, which are named as A1, A2, and A3, respectively. Figure 5(b) shows the group index curves for the three different structures with the group index of 47, 67, 130 and the NDBP values of 0.223, 0.21, and 0.192, respectively. The GVD curves for the three different sets of p1 and p2 are shown in Fig. 5(c). It is clear that the GVD values Table 2.
of the three optimized structures are all far below the order of 106 a∕2πc2 , which is required for practical application. Furthermore, as each GVD curve presents both a positive region and a negative region, it reveals the applications of dispersion compensation. B. Tuning the Slot Mode
Similar dispersion engineering presented in the previous section can be applied to tune the slot mode. Compared with the W1 defect mode, the slot mode is in a lower frequency region and has a stronger optical confinement in the slot. We first study the dependence of dispersion diagrams on the shift distance of p1. Figure 6(a) shows the dispersion curves of different parameters of p1 0.03a, 0.04a, 0.05a, 0.06a, and 0.07a, which are compared with the basic structure. The solid black curve denotes the initial curve of the basic structure. It is clear that the higher p1 moves the dispersion curves to the lower frequency. The corresponding group index curves are presented in Fig. 6(b). As the parameter p1 increases, the group index of the flat band decreases. Meanwhile, the bandwidth and NDBP value moves to a higher level as p1 increases. For p1 0.03a, 0.04a, 0.05a, 0.06a, and 0.07a, the corresponding NDBP values are 0.11, 0.112, 0.139, 0.145, and 0.152, respectively. Then, we shift the first and second row of air holes simultaneously. Three asymmetric structures with improved NDBP values and low GVD are also
n g , Δλ, and NDBP of W1 Defect Mode for p1 0.03a and Different Parameters p2
Structure
p2 0a
p2 0.05a
p2 0.10a
p2 0.15a
p2 0.18a
p2 0.19a
p2 0.20a
p2 0.21a
ng Δλ nm NDBP Frequency interval
37 6.2 0.144 [0.30165, 0.30283]
39 4.9 0.130 [0.30135, 0.30235]
56 3.3 0.120 [0.30028, 0.30093]
219 0.6 0.08 [0.29821, 0.2983]
140 1.7 0.154 [0.29588, 0.29632]
84 2.9 0.160 [0.2949, 0.29566]
70 4.7 0.213 [0.29402, 0.29491]
59 5.6 0.220 [0.29309, 0.29413]
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p1=0a p2=0a p1=0.03a p2=0.215a p1=0.03a p2=0.202a p1=0.215a p2=0.16a
0.307 0.306 0.305 0.304 0.303 0.302 0.301 0.300 0.299 0.298 0.297 0.296 0.295 0.294 0.293 0.292 0.291
(b) 150
A3
A3
120
Group Index ng
Normalized Frequency (ω a/2πc)
(a) 0.308
90
A2 60
A1 p1=0a p2=0a p1=0.03a p2=0.215a p1=0.03a p2=0.202a p1=0.025a p2=0.16a
A2
A1
30
0 0.30
0.35
0.40
0.45
0.50
0.292
0.296
0.298
0.300
0.302
0.304
0.306
x 105
(c)
p1=0.03a p2=0.215a p1=0.03a p2=0.202a p1=0.025a p2=0.16a p1=0a p2=0a
8
Group Velocity Dispersion (a/2 π c2)
0.294
Normalized Frequency (ω a/2πc)
Wave Vector (ka/2 π)
6
A3
4
2
A1
A2
0
-2
-4 0.292
0.294
0.296
0.298
0.3
0.302
0.304
0.306
Normalized Frequency (wa/2 π c)
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(a)
0.254 0.252 0.250 0.248 0.246 0.244 0.242 0.240
p1=0a p1=0.03a p1=0.04a p1=0.05a p1=0.06a p1=0.07a
0.238 0.236 0.234 0.232 0.30
0.35
0.40
0.45
0.50
Wave Vector (ka/2 π)
(b)
300
250
Group Index ng
presented here, that have different parameters p1 and p2 and are named B1, B2, and B3, respectively. The dispersion curves are presented in Fig. 7(a). The corresponding group index curves and dispersion curves are depicted in Figs. 7(b) and 7(c). We obtain the group indices of 42, 54, and 108; and the NDBP values of 0.168, 0.163, and 0.133 for the three different structures B1, B2, and B3, respectively. It is obvious that the low GVD reduces down to the order of 106 a∕2πc2 and the GVD curves are extremely flat at the slow light region. We note that the GVD changes with positive and negative parts, which also has applications for dispersion compensation. After works of tuning the W1 defect mode and the slot mode, higher NDBP values of 0.223, 0.21, 0.192 for the W1 defect mode and 0.168, 0.163, 0.133 for the slot mode are obtained. We also compare the results with previous reference papers [14,17,26–29]. We have a principle of comparison and NDBP values with a similar group index are compared here. As Table 3 shows, compared to the previous papers, the NDBP values of the W1 defect mode are improved by 85%, 163%, and 60%, respectively; and the NDBP values of the slot mode are also improved by 40%, 75%, and 28% for the three optimized structures B1, B2, and B3, which is illustrated in Table 4. This means that our procedure for designing the “flat
Normalized Frequency (ωa/2πc)
Fig. 5. (a) Dispersion diagrams, (b) group index curves, and (c) the GVD curves for three different sets of p1 and p2, which are named A1, A2, and A3. For the three asymmetric structures A1, A2, and A3, we obtain the higher NDBP values of 0.223, 0.21, and 0.192, respectively. All of them are compared with the basic structure.
200
150
100
p1=0a p1=0.03a p1=0.04a p1=0.05a p1=0.06a p1=0.07a
50
0 0.245
0.246
0.247
0.248
0.249
0.250
0.251
0.252
Normalized Frequency (ω a/2π c)
Fig. 6. (a) Dispersion diagrams and (b) group index curves with different sets of p1 for the slot mode.
(b) 300
0.252
B3
0.250
250
B2
0.248
B1
Group Index ng
Normalized Frequency (ωa/2πc)
(a) 0.254
0.246 0.244 0.242
200
150
B3 100
0.240 p1=0a p2=0a p1=0.05a p2=0.2a p1=0.045a p2=0.15a p1=0.025a p2=0.13a
0.238 0.236
B2 50
p1=0a p2=0a p1=0.05a p2=0.2a p1=0.045a p2=0.15a p1=0.025a p2=0.13a
B1
0 0.30
0.35
0.40
0.45
0.50
0.248
Wave Vector (ka/2 π)
(c)
8
0.250
0.251
0.252
x 106 p1=0.05a p1=0.045a p1=0.025a p1=0a
7
Group Velocity Dispersion (a/2 π c2)
0.249
Normalized Frequency (ω a/2 πc)
6
p2=0.2a p2=0.15a p2=0.13a p2=0a
5 4 3 2 1
B3
B2
B1 0 -1 -2 0.2475
0.248
0.2485
0.249
0.2495
0.25
0.2505
0.251
0.2515
0.252
Normalized Frequency (ωa/2π c)
Fig. 7. (a) Dispersion diagrams for the three optimized structures, B1, B2, and B3; (b), (c) the corresponding group index curves and GVD curves for B1, B2, and B3, respectively.
band” of the SPCW is suitable for both the W1 defect mode and the slot mode. Furthermore, this type of procedure [12–14,17] is simpler from the aspect of the fabrication process since we did not change the radius of the air holes. 3. Time-Domain Analysis
In order to confirm the results obtained from dispersion curve calculations based on the PWE Table 3.
Comparison among the Optimized W1 Defect Mode and References
Structure The present asymmetric structures Reference symmetric structures
method, the transmission characteristics of optical pulses in structure SPCW-A2 are simulated for comparison by the 2D FDTD method [30]. The schematic of the FDTD simulation system is shown in Fig. 8(a). A resolution of 40 pixels∕a (a is the lattice constant) is adopted in this simulation. Perfectly matched absorbing boundary layers are applied to the surroundings of the structure. The asymmetric case with group index ng 67 for structure A2 is
A1 A2 A3 Ref. [26] Ref. [27] Ref. [28] Ref. [29]
p1
p2
0.03a 0.215a 0.03a 0.202a 0.025a 0.16a A comb SPCW A comb slot and change the radius of air holes Ring-shape-hole SPCW Bragg-like slot
ng
Δλ nm
NDBP
47 67 130 47 48
7.2 4.8 2.3 4 4.5
0.223 0.21 0.192 0.121 0.139
67 95
1.8 2
0.08 0.122
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Table 4.
Comparison among the Optimized Slot Mode and References
Structure The present asymmetric structures
p1
Reference symmetric structures
p2
0.05a 0.2a 0.045a 0.15a 0.025a 0.13a A comb SPCW Transversal shift The oblique structure
B1 B2 B3 Ref. [26] Ref. [17] Ref. [14]
considered in our simulation. The transmission length between the input time monitor and the output time monitor is 50a. The result for SPCW-A2 is shown in Fig. 8(b). The normalized center frequency of a Gaussian pulse source is centered at 0.29432πc∕a with a frequency width of 0.000892πc∕a. The peaks of the fluxes are located at ∼5531a∕c and ∼8815a∕c in the input and output detecting points, respectively. The total time delay between the peaks in Fig. 8(b) is Δt 3284a∕c, and the average group index in the SPCW-A2 is calculated as ng c∕L∕Δt 65.6, which agrees with the calculated result from the PWE simulation. The full width at half-maximum (FWHM) of the input pulse is 2133a∕c, while the FWHW of thse output pulse is 2195a∕c. Thus, the relative pulse distortion is only 3.0%, which indicates that the GVD is adequately low for optical pulse propagation. (a)
ng
Δλ nm
NDBP
42 55 108 35 54 110
6.2 5.6 2.2 6 3.3 1.8
0.182 0.20 0.153 0.135 0.114 0.127
4. Conclusion
In short, asymmetric slotted PCWs, formed by shifting holes along the slot axis, are designed here. By careful and detailed optimizing, we achieve the conversion of the “flat band” from band-up slow light to band-down slow light for the W1 defect mode. A large group index, ng 130, with a wide bandwidth, Δλ 2.3 nm, is obtained for the W1 defect mode in the asymmetric structure SPCW-A3. A large index, ng 108, with a wide bandwidth, Δλ 2.2 nm, is also obtained for the slot mode in the structure SPCW-B3. The FDTD simulation is carried out to observe the light propagation in the optimal waveguide. The results agree well with our prediction in the PWE calculations. Compared with previous publications, our asymmetric procedure is suitable for improving the slow light performance of both the W1 defect mode and the slot mode, which would be of powerful potential for further practical applications. This work is supported by the National Natural Science Foundation of China (Grant No. 61376055), and National Basic Research Program of China (Grant No. 2010CB923204). References
(b)
input
Normalized Field Amplitude
1.0
output 0.8
0.6
2133a/c
0.4
2195a/c 0.2
0.0 2000
4000
6000
8000
10000
12000
Time delay (a/c)
Fig. 8. (a) Schematic of FDTD simulation system of the proposed slow light waveguides. (b) Temporal pulses detected at the input and output detecting points in the proposed SPCW-A2. 8400
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