Slow light in ellipse-hole photonic crystal line-defect ... - OSA Publishing

Varmazyari et al.

Vol. 31, No. 4 / April 2014 / J. Opt. Soc. Am. B

771

Slow light in ellipse-hole photonic crystal line-defect waveguide with high normalized delay bandwidth product Vali Varmazyari,1 Hamidreza Habibiyan,1,* and Hassan Ghafoorifard1,2 1

2

Photonics Engineering Group, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran Department of Electrical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran *Corresponding author: [email protected] Received December 9, 2013; revised January 29, 2014; accepted January 29, 2014; posted January 29, 2014 (Doc. ID 202740); published March 12, 2014

In this paper, a novel flatband slow light device with low group velocity dispersion (GVD) is presented in an ellipse-hole photonic crystal (PC) line-defect waveguide. Utilizing dispersion engineering in the proposed structure, normalized delay-bandwidth product (NDBP) under a constant group index criterion is significantly improved. A step-by-step optimization process is done on the adjacent rows to the waveguide, which are filled by silica. For optimum case a high NDBP of 0.461 with a group index of 41.86 and a bandwidth of 17.06 nm is obtained by three-dimensional plane-wave expansion method. To the best of our knowledge, this NDBP is one of the highest values in PC waveguides reported to date, in which the group index value is relatively high. The numerical results show that GVD is negligible over a broad wavelength range. Also, optical pulse propagation through the waveguide is performed based on the finite-difference time-domain method. The results indicate that the shape of output pulse experiences a broadening of 2.1% compared with the incoming pulse after traveling a distance of 30a. © 2014 Optical Society of America OCIS codes: (350.4238) Nanophotonics and photonic crystals; (130.5296) Photonic crystal waveguides; (130.2790) Guided waves; (260.2030) Dispersion. http://dx.doi.org/10.1364/JOSAB.31.000771

1. INTRODUCTION All-optical integrated devices are key components for future telecommunication networks. A category of such devices is optical delay lines based on slow light, which have various applications, such as phase shifters, modulators, and also are used to design controllable optical buffers, optical switches, biosensors, quantum all-optical data storage, and data processing devices. So the slow light, which can dramatically reduce the group velocity of propagating optical pulse, has attracted much attention. In early stages, many researchers used the quantum mechanical effect of atomic levels, such as electromagnetically induced transparency [1,2] and coherent population oscillation [3–5], to attain the slow light. But these methods have clear disadvantages in the practical applications at room temperature and suffer from generating slow light at any desired wavelength because the carrier frequency of the propagated pulse depends on material resonance frequency. Also an important criterion for evaluating the slow light performance is the delay-bandwidth product (DBP), which is a trade-off between the achieved group index and slow light bandwidth. In most studies reported based on quantum mechanical methods, the DBP value was lower than 10 because they mainly focused on the long delay, ignoring the bandwidth [6]. On the contrary, slow light in photonic crystal (PC) has attracted much attention, because it is compatible with on-chip integration, room-temperature operation, and ability to attain desired communication wavelength. Also a PC-based slow light device can offer wide-bandwidth and dispersion-free propagation 0740-3224/14/040771-09$15.00/0

(a DBP of over 100 [7]) by engineering of structural parameters. The two basic kinds of PC structures, including line-defect waveguides [8,9] and coupled-cavity waveguides [10–15], have been used to achieve slow light. Within the past years, PC slow light structures mostly have been presented based on linedefect waveguides, which are suitable for enhanced light– matter interactions. Several novel methods have been proposed to improve the slow light performance by tuning the structural parameters of line-defect waveguides and their neighborhood rows. Improvements include changing the width of the line defect [16,17], altering the position or radii of the air holes [18–20], employing ellipse air holes [21–23], eye-shape holes [24], PC with ring-shape holes [25–27], liquid infiltration into the air holes [28,29]. In this paper, a novel modified line-defect PC slab waveguide is proposed. Dispersion engineering in the presented structure is performed, in order to attain the slow wave mode with high bandwidth and relatively large group index. Important geometric parameters for improvement of slow light characteristics, including the major and minor radii of silica rows and their position in x and y directions, are tuned. The remainder of this paper is organized as follows. In Section 2, the proposed slow light structure is introduced and its optimization process is presented. Also normalized field distribution of the slow wave mode in Si slab is investigated and the slow light performance of an optimized PC waveguide (PCW) is compared with the previous published results. In Section 3, in order to verify the band structure analysis, the © 2014 Optical Society of America

772

J. Opt. Soc. Am. B / Vol. 31, No. 4 / April 2014

Varmazyari et al.

where a is the lattice constant. In the proposed structure the two adjacent rows of line-defect waveguide are filled by SiO2 with a refractive index of 1.44. Also, the major and minor radii of ellipse silica regions in the basic proposed structure are r a and r b , respectively. The radii and position of these regions are tuned by the optimization process. A convenient method to evaluate the properties of slow light devices is the plane-wave expansion (PWE) method. Accurate modeling of dispersion relation for the proposed structure is performed by three-dimensional (3D) PWE method. The rectangle with the dashed line in Fig. 1 shows the supercell used in numerical calculation for the presented structure. For the sufficient calculation precision, the number of plane waves in each axis and the eigenvalue tolerance are selected by 32 and 10−8 , respectively. Before any change in the structural parameters, the dispersion diagram is calculated for r a Ra 0.4a and r b Rb 0.33a. Figure 2(a) shows the band diagram of basic structure for the transverse electric-like (TE-like) polarization mode. The main propagation mode is shown with a solid curve. This slow mode of the line defect PCW is confined vertically by index-guiding and horizontally by gap-guiding, which is located below the light-line and close to the lower edge of the bandgap zone. The normalized H-field distribution of the considered band at kx 0.52π∕a is shown in Fig. 2(b). The even-type profile of the selected mode is highly concentrated in Si slab between the first rows adjacent to the line-defect waveguide. With the dispersion diagram of the relevant mode shown in Fig. 2(a), the important parameters of slow light can be calculated. The most significant issues in this context are group velocity vg , group index ng , and group velocity dispersion (GVD). The group index indicates deceleration of propagating pulse in medium [30]. The group velocity of a propagating pulse is calculated as the derivative of the angular frequency over the wave vector [30]. The group velocity and group index are given by

optimized PCW is studied in time domain by the finitedifference time-domain (FDTD) method. Also, the effects of fabrication faults on device performance are modeled and the fabrication process of the suggested structure is presented. A brief summary of the work is given in Section 4.

2. PROPOSED STRUCTURE AND NUMERICAL ANALYSIS Figure 1 shows the basic proposed structure, which is a silicon air-bridge slab (nsi 3.46) with thickness of 300 nm, perforated by a triangular lattice of elliptical holes. In this platform light is confined in the in-plane direction by the photonic bandgap (PBG) effect and in the out-of-plane (vertically) direction by total internal reflection mechanism. Our reason for using the air-bridge platform is the reduction of scattering loss in the out-of-plane direction. As shown in Fig. 1, the line-defect waveguide is constructed by removing one line of elliptical holes in the x direction. In odd rows, the major radii of elliptical holes are placed along the x axis, while in even rows, the major radii are placed along the y axis. By considering the trade-off between fabrication technology and access to the wide PBG, the major and minor radii of elliptical holes are selected Ra 0.4a and Rb 0.33a, respectively,

a Y X

Dy Dy

Z

Dx Dx

300 nm 2Ra

vg

2ra

Air Hole

2Rb

Sio2 Rod

(1)

dn ; dω

(2)

2rb

ng n ω

Fig. 1. Schematic representation of the basic proposed structure. The dashed lines show the supercell used in numerical calculation. The white ellipse shapes correspond to air holes while the blue ones filled by SiO2 ; also Dx and Dy are waveguide displacement in horizontal and vertical direction.

where ω is the light frequency, c is the velocity of light in vacuum, and k is the propagation constant. Beside the group

X 0

0.5

0.28

- 0.5

Normalized Frequency (ωa/2 c)

dω c ; dk ng

0.26 0.24 0.22

Y

0.2 0.18

0.15

0.2

0.25

0.3

0.35

Wave vector (ka/2 )

(a)

0.4

0.45

0.5 0.00

0.25

0.50

0.75

1.00

(b)

Fig. 2. (a) Dispersion curve of line defect PC waveguide for parameters of r a Ra 0.4a and r b Rb 0.33a. The solid curve is considered as the guided mode. (b) 3D view of normalized H-field distribution of the relevant mode in Si slab at kx 0.52π∕a.

Varmazyari et al.


773

index, the GVD effect needs to be concerned because it can cause significant distortion in short optical pulses. GVD parameter β2 is expressed by the second-order derivative of the dispersion relation as [13] β2

3 2 d2 k 1 d ω 1 dng − : 2 v c dω dω dk2 g

(3)

Another important factor in a slow light device is the DBP, which is defined as the product of the time group delay Δt and bandwidth Δf in slow light. In order to evaluate the capacity of slow light PCWs with different lengths, the expression of the DBP is modified to the normalized DBP (NDBP), which is more useful when devices that have different lengths and different operating frequencies are compared. The NDBP is given by [25] NDBP ng ×

Δω ; ω0

(4)

where Δω∕ω0 is the normalized bandwidth of a slow light region and ng is the average group index. The main challenge in this context is to achieve slow light with large ng and small β2 in a wide wavelength region, corresponding to the large NDBP. In recent years, efforts to achieve slow light using ellipsehole structures have been reported, which is probably the easiest noncircular air hole shape appropriate to fabricate. Here, we explain the advantages of the suggested structure. To this end, we compare the basic proposed structure shown in Fig. 1, with three other W1 waveguides. The first PCW consists of a silicon air-bridge slab perforated by a triangular lattice of circular air holes with radii r 0.363a. The filling factor of both the circular-holes PCW and the proposed device are kept the same. The major and minor radii of ellipse holes in the basic structure are fixed, respectively, as 0.4a and 0.33a, and then the radii of circular air holes in the first waveguide should be 0.363a. By deforming the whole circular air holes of the first PCW into the elliptical air holes with the major axis along the x direction and keeping the filling factor constant (Ra 0.4a and Rb 0.33a), the second waveguide is obtained. Finally, the third waveguide is formed by alternative 90° rotation of elliptical air holes in the even rows of the second structure. Indeed, our basic suggested platform is the same alternative rotated ellipse air hole PCW (ARE-PCW) with the innermost rows filled by SiO2 . Figure 3(a) shows the dispersion relation of guided modes for these structures. The dotted region shows the slow light part of guided modes close to the Brillouin zone edge. Comparison of slow light bandwidth for the above-mentioned waveguides is shown in Figs. 3(b)–3(d). It can be seen that by deforming the circular air holes into elliptical shapes, the bandwidth of the slow light guided mode (Δω) is increased from 0.00226 to 0.00243, which corresponds to ∼8% improvement [Fig. 3(b)]. By changing the major and minor axis of ellipse holes in every other row, Δω is increased by 90% [Fig. 3(c)]. Finally, filling the innermost rows with high-index material, such as SiO2 , results in ∼36% improvement of Δω. In other words, the flatband region of the guided mode for the basic proposed structure is broadened significantly compared with the conventional circle-hole PCW (approximately 2.8 times). In the slow light region, ng

Fig. 3. (a) Dispersion relation of guided modes for different PCWs. Comparison of slow light frequency window between (b) conventional circle-hole PCW and ellipse-hole PCW, (c) ellipse-hole PCW and ARE-PCW, and (d) ARE-PCW and basic proposed PCW.

has not significantly changed and then due to improvement of Δω, the NDBP parameter is increased substantially. From the viewpoint of physical origin, the guided modes in PCWs can be classified with respect to their field distribution as index-guided and gap-guided modes [31], which have different natures. The energy of the index-guided mode is concentrated inside the defect by the total internal reflection due to index contrast between the waveguide and its surroundings.


Varmazyari et al. 300

0.226 ra= 0.36a

0.224

ra = 0.37a

0.222

ra = 0.38a

0.22

ra = 0.39a ra = 0.40a

0.218 0.216

ra = 0.41a

0.35

0.4

0.45

0.5

Group Index - ng

Normalized Frequency (ωa/2πc)

774

ra= 0.36a

250

ra = 0.37a

200

ra = 0.38a

150

ra = 0.39a

100

ra = 0.40a

50

ra = 0.41a

0

0.216

Wave vector (ka/2 π)

0.218

0.22

0.224

(b)

(a)

(a) Dispersion relation and (b) group index of the guided mode for different values of r a .

The gap-guided mode is confined by distributed Bragg reflection due to the periodic structure within the waveguide. An intrinsic anticrossing between these modes determines local shape of the guided mode dispersion curve, which its slope determines the group velocity of the mode [32]. For low value of the wave vector (near k 0), the index-guided part has dominated effect to form the guided mode having the low ng . In contrast, for the value of the wave vector close to k 0.52π∕a, the gap-guided part has dominated effect and large value of ng is obtained [23]. The GVD has a maximum at the anticrossing point (close to the Brillouin zone edge), because of the fast change of group velocity there. The anticrossing vitiates the flat region of the gap-guided mode and should be avoided to achieve constant group velocity [32]. One of the possibilities is to shift the anticrossing point to the left side of band diagram. This is achieved by deforming the shape of PCW holes. For example, as described previously, by deforming the whole circular air holes of a PCW into the elliptical air holes, both index-guided and gap-guided modes moves upward, but the gap-guided mode moves faster. Therefore, anticrossing near the Brillouin zone edge is weakened, the group velocity increases, and the corresponding bandwidth broadens [Fig. 3(b)]. On the other hand, the holes nearest to the line-defect waveguide are the physics conjunction of index-guided and gap-guided modes [33]. In recent years, various researches have reported the improvement of slow light properties by parameters engineering in PCs. In this work, due to important effect of SiO2 regions in the value of guided-mode’s group velocity, we change the major and minor radii and position of these regions to enhance the NDBP. In the following subsections, we study the influences of these parameters on the slow light properties. A. Changing the Major Radii of Silica Regions To study the effect of major radii of silica regions located adjacent to the waveguide on slow light properties, the ratio of r a ∕a is modified. The band diagram and the group index curves are calculated for relevant guided mode. The band diagram variation for r a , ranging from 0.36a to 0.41a is shown in Fig. 4(a), while r b , Dx , and Dy are fixed on previous values (r b 0.33a, Dx 0 and Dy 0). Also the group indices corresponding to these modes are shown in Fig. 4(b). As can be seen, when r a increases (i.e., making holes more elliptical), effective refractive index in the ΓK direction is decreased. Therefore, both index-guided and gap-guided modes are pulled upward, but the gap-guided mode moves faster and the anticrossing point shifts to the left side of band diagram. So, the group velocity increases and the corresponding bandwidth broadens. The bandwidth is a frequency region in which

the group index remained almost constant. The results extracted from Fig. 4 are listed in Table 1. It can be seen that, by changing the r a the group index and bandwidth centered at the wavelength of 1550 nm are varied from 186.3 to 29.1 and 1.64 to 18.69 nm, respectively. Also NDBP parameter is changed from 0.198 to 0.359. In this step, the maximum NDBP 0.359 is selected as the best value for r a 0.40a. B. Changing the Minor Radii of Silica Regions In the second step, the influence of minor radii of silica regions is studied on the NDBP for previously tuned value of r a 0.40a. The r b is swept in the interval from 0.3a to 0.50a. The behavior of the guided mode and group index are investigated for each case of r b . Figure 5 shows the dispersion relation of the relevant mode for different values of r b . Similar to Fig. 4(a), when r b increases, the effective refractive index in the ΓK direction is a little decreased. The guided modes are moved to the higher frequencies and the anticrossing point experiences a little shift to the left side of the band diagram. Therefore, slopes of these curves do not significantly change and almost all cases have the same slopes corresponding to the approximately equal ng . The simulation results are listed in Table 2. It can be seen that the Table 1. Slow Light Properties for Various ra r a a

ng

Bandwidth Centered at 1550 nm (nm)

NDBP

0.36 0.37 0.38 0.39 0.40 0.41

186.3 110.3 71.4 51.5 38.2 29.1

1.64 3.35 6.01 9.93 14.56 18.69

0.198 0.239 0.277 0.33 0.359 0.351


Fig. 4.

0.222

Normalized Frequency (ωa/2πc=a/λ)

0.226

rb= 0.45a

0.224

rb = 0.46a rb = 0.47a

0.222

rb = 0.48a

0.22

rb = 0.49a

0.218

rb = 0.50a

0.35

0.4

0.45

0.5

Wave vector (ka/2 ) Fig. 5. Dispersion relation of the guided mode for different values of r b (with previously tuned value r a 0.4a).

Varmazyari et al.


775

Table 2. Slow Light Properties for Various rb (with Previously Tuned Value ra 0.4a)

Table 3. Slow Light Properties for Various Dx (with Previously Tuned Values ra 0.4a and rb 0.48a)

r b a

ng


NDBP

Dx a

0.45 0.46 0.47 0.48 0.49 0.50

39.09 38.43 38.26 38.1 37.71 37.46

14.74 16.21 16.48 16.68 16.77 16.75

0.397 0.402 0.407 0.410 0.408 0.405

0.01 0.02 0.03 0.04 0.05 0.06

group index varies from 39.09 to 37.46. In this step, the maximum NDBP 0.41 is selected as the best value for r b 0.48a, which corresponds to the group index of 38.1 over the bandwidth of 16.68 nm. C. Changing the Position of Silica Regions in the x Direction The influence of the position of silica regions in the x direction on slow light performance is investigated in the third step of the optimization. The radii of silica regions are set on the previously tuned values, i.e., r a 0.40a and r b 0.48a. The displacement of the silica regions (Dx ) in the x direction is varied from 0.01a to 0.06a with the interval of 0.01a. The dispersion curves of the guided modes are shown in Fig. 6(a). By changing Dx , the geometric symmetry becomes weak and both index-guided and gap-guided modes are slightly moved to the higher frequencies. Therefore, the anticrossing point experiences a little shift to the larger values of k. The important parameters of slow light for this case are given in Table 3. It illustrates that, although displacement of the silica regions in the x direction can enlarge the slow light bandwidth, the reduction rate of ng is faster and then the NDBP gradually drops from the previously optimized value of 0.41. Thus, the best case is Dx 0 (i.e., the silica regions do not shift in the x direction).


NDBP

37.8 36.1 34.2 32.6 30.5 26.7

16.48 16.78 16.89 17.21 18.14 20.26

0.402 0.391 0.375 0.362 0.357 0.349

spectra for all states are shown in Figs. 7(a) and 7(b), respectively. We can observe that by increasing Dy , the band diagram curves move to the upper frequencies and the anticrossing point experiences a shift to the left side of the band diagram. The obtained results are listed in Table 4 and show that group index and NDBP values have upward trend for Dy from 0.075a to 0.15a, and have downward trend after Dy 0.15a. Therefore, the maximum NDBP 0.461 is selected as the optimum result for Dy 0.15a, which corresponds to the group index of 41.86 over the bandwidth of 17.06 nm. In suggested structure the maximum NDBP of 0.461 was obtained by using a four-step tuning process on the important parameters of waveguide. The GVD is investigated for optimum values of parameters. In Fig. 8 the GVD parameter β2 is plotted as a function of normalized frequency, which varies from negative value less than −5 × 105 ps2 ∕km to positive value more than 5 × 105 ps2 ∕km. A flat region with small GVD in the frequency range between 0.2255 (a∕λ) and 0.228 (a∕λ) corresponds to the bandwidth of slow light. It can be observed that, for this range of frequency, the order of GVD is 105 ps2 ∕km, which is an acceptable GVD. Thus, it is suitable for pulse propagation with negligible dispersion in optical buffering applications. Also, the GVD is changed symmetrically with positive and negative values, which can be used for dispersion compensation application [30]. In order to demonstrate the superiority of the optimized device, we compare our structure with previously published results in various contexts of slow light in PCs, such as [18,22,23], and [34–37]. Comparison is done on three main parameters, including group index ng , bandwidth Δλ, and NDBP, which are given in Table 5. We should notice that the comparison of the NDBP in different devices should be performed at the approximately same ng . Based on Table 5, the NDBP value for our proposed structure has been improved by 110% compared to [34], 143% compared to [27], 48% compared to [18], 155% compared to [35], 25% compared 200

0.229 0.2285 0.228

Dx= 0.01a Dx = 0.02a

0.2275 0.227 0.2265

Dx = 0.03a Dx = 0.04a Dx = 0.05a

0.226 0.2255

Dx = 0.06a

0.35

0.4

0.45

Wave vector (ka/2 )

(a)

0.5

Group Index - ng


D. Changing the Position of Silica Regions in the y Direction In the final step of the optimization process, the effect of the position of silica regions in the y direction on slow light performance is studied. The radii and position of silica regions in the x direction are set on the previously optimized values, i.e., r a 0.40a, r b 0.48a, and Dx 0. The displacement of the silica regions (Dy ) in the y direction is swept in the interval from 0.075a to 0.20a. The band diagrams and the group index

ng

Dx= 0.01a

150

Dx = 0.02a Dx = 0.03a

100

Dx = 0.04a Dx = 0.05a

50

Dx = 0.06a

0

0.2255

0.2265

0.2275

0.2285

Normalized Frequency (ωa/2 c=a/ )

(b)

Fig. 6. (a) Dispersion relation and (b) group index of the guided mode for different values of Dx (with previously tuned values r a 0.4a and r b 0.48a).


0.23 0.229 0.228 0.227 0.226 0.225 0.224 0.223

Varmazyari et al. 200 Dy= 0.075a Dy = 0.10a Dy = 0.125a Dy = 0.150a Dy = 0.175a

Group Index - ng


776

Dy= 0.075a

150

Dy = 0.10a Dy = 0.125a

100

Dy = 0.150a Dy = 0.175a

50

Dy = 0.20a

Dy = 0.20a

0.35

0.4

0.45

0 0.223

0.5

0.225

0.227

0.229

0.231

Normalized Frequency (ωa/2 c=a/ )

Wave vector (ka/2 )

(a)

(b)

Fig. 7. (a) Dispersion relation and (b) group index of the guided mode for different values of Dy (with previously tuned values r a 0.4a, r b 0.48a, and Dx 0.).

Table 4. Slow Light Properties for Various Dy (with Previously Tuned Values ra 0.4a, rb 0.48a, and Dx 0) Dy a

ng


NDBP

0.075 0.10 0.125 0.15 0.175 0.20

39.2 39.95 40.92 41.86 38.5 31.7

17.12 17.07 16.85 17.06 17.79 19.90

0.433 0.440 0.445 0.461 0.442 0.407

to [23], 105% compared to [37], and 4% compared to [22]. One of the latest studies in this context was presented by Üstün and Kurt that proposed the triangular lattice PC line-defect waveguide with highest value of NDBP 0.511 [36]. Although their structure has high NDBP, but the group index is less than 20. The obtained NDBP for our proposed structure provides high buffering capacity, which means that large number of data bits can be stored in the buffer. Also, by choosing appropriate values for r a , r b , Dx , and Dy required delay at delay lines and desirable buffering capacity can be provided.

3. TIME DOMAIN ANALYSIS x

10

5

6

2 2 (ps /km)

4 2 0 -2 -4 -6 0.2255 0.226 0.2265 0.227 0.2275 0.228

Normalized Frequency (ωa/2 c=a/ ) Fig. 8. GVD parameter β2 of the optimized PCW as a function of normalized frequency. The inset of figure shows schematic of the optimized structure.

Table 5. Comparison between the Optimized PCW and Reference Papers Reference [34] [27]a [18] [35]a [36] [23]a [37] [22]a The present work

ng

Δλ(nm)

NDBP

Published Year

34 37 44 42 17.78 31 35 42 41.86

11 8 11 6.7 44 18.54 10 16.4 17.06

0.22 0.19 0.312 0.181 0.511 0.37 0.225 0.446 0.461

2006 2007 2008 2010 2012 2012 2013 2013 —

a These items are 2D simulation based on effective index method; Other items are 3D simulation.

In order to verify obtained results from frequency-domain calculations, optical pulse propagation through the optimized PCW is investigated by using FDTD simulation. The perfectly matched layer as absorbing boundary conditions at the spatial edges of the computational domain is considered, which eliminates any outward propagating energy that impinges on the domain boundaries. We choose the relevant parameters in the presented structure by previously optimized values of r a 0.4a, r b 0.48a, Dx 0, and Dy 0.15a. For operating at the wavelength of 1550 nm, the lattice constant is selected a 351 nm. The schematic of the simulated device is shown in Fig. 9(a). In this structure, the input and output monitors are placed at a position of 5a and 35a, respectively. So the distance between input and output monitors is 30a and whole length of the structure is 40a. The central frequency of the Gaussian source is fixed at 1550 nm. The group index value can be determined based on the relationship between the propagation time Δt and the pulse traveling distance between two monitors l, i.e., Δt l∕vg ng ∕c∕l. Figure 9(b) shows the normalized pulse shapes corresponding to the propagation time through the input and output monitors. The peak of pulse at the input detecting point is located at 0.6 ps, while it is at 2 ps at the output detecting point. The total delay and distance between the two peaks are approximately 1.4 ps and 10.53 μm, respectively. According to the above equation, the group index is obtained by ng Δt · c∕l 40.55, which is highly close to the obtained result of the PWE calculation. The slight discrepancy between the two values mainly results from the limited discretization of FDTD and certain number of plane waves used in the supercell calculation of the PWE. The full width at half-maximum (FWHM) of the incident optical pulse in the input detecting point is 0.38 ps, while the corresponding value in the output detecting point is 0.388 ps. Thus, the relative pulse distortion

Varmazyari et al.


Input Monitor

Output Monitor

30a

is only 2.1%. As we expected due to the low dispersion characteristics, the pulse can be transmitted along the presented PCW without obvious broadening. One of the important issues after designing of PC devices is production of them in large scales. Usually, due to imperfect operation of equipment during the fabrication process, there are some differences between the real operation of the device and its simulation results. Several sources of errors, including electron scattering in e-beam lithography process, stitching errors and etching anisotropies, lead to surface roughness and the radii variation of holes that should be analyzed [38]. The size of silica regions has important effect on the value of the NDBP. Therefore, we model the effects of fabrication errors on the device performance by changing the major and minor radii of silica regions. In Table 6, eight cases of errors are modeled for the presented structure by using the PWE method. In the first two cases, the r b is changed by 3 nm and the r a remains constant. For the next two cases, the r a is changed by 3 nm, while the r b is constant. In the case 5, both r a and r b are decreased by 3 nm to 137 nm and 165 nm, respectively. On the contrary, these radii in the sixth case are increased by 3 nm to 143 nm and 171 nm. In the case 7, the r a is decreased by 3 nm and the r b is increased by 3 nm. Whereas, in the case 8, the r b is decreased by 3 nm and the r a is increased by 3 nm. The results show that the group index is varied from 37.2 to 47.7 over the bandwidth of 19.08–14.94 nm, respectively. In all cases, the NDBP value has no significant changes. At the worst case, for r a 143 nm and r b 168 nm, the NDBP is equal to 0.444. Also, it can be seen that the variation of ng and Δλ are such that the overall NDBP parameters are decreased from optimized value. For example, in case 6 despite the increasing of Δλ, the group index is decreased to 37.2 and the NDBP is decreased to 0.458. For modeling of fabrication errors by the PWE method, we introduce disorders within a supercell and assume the structure remains periodic. This is a simplistic view to the fabrication errors, because other supercells may have different disorders. Therefore, to investigate the variation of air holes and silica regions in the whole structure and their effects on the performance of the device, we used the FDTD method. In this modeling, the major and minor diameters of ∼20% of ellipse shapes are randomly changed by 6 nm, as shown in Fig. 10. By measuring the total delay between input and output peaks and FWHM of output pulse, the group index and the

…… …… …… …… …… 40a

(a)

Normalized field Amplitude

1 0.8 0.6 0.4

0.38 0.388

0.2 0 0.5 1 2 1.5 Propagation Time (ps)

2.5

(b) Fig. 9. (a) Schematic of the structure used in FDTD simulation. The relevant parameters are adapted by previously optimized values. Whole length of the structure is 40a and the distance between input and output monitors is 30a. (b) Temporal pulse propagation at input and output detectors placed at the points 5a and 35a, respectively.

Table 6. Effects of Fabrication Errors on Slow Light Parameters of Suggested Structure No. 1 2 3 4 5 6 7 8

r a (nm)

r b (nm)

ng

Δλ (nm)

NDBP

140 140 137 143 137 143 137 143

165 171 168 168 165 171 171 165

42.1 38.83 47.7 38.4 43.2 37.2 45.4 39.6

16.6 18.08 14.94 17.92 16.03 19.08 15.5 17.57

0.451 0.453 0.46 0.444 0.447 0.458 0.454 0.449

C

B

C

D C

2 (µm)

A

4 (µm)

C

6 (µm)

B C

B C

B

C

D A

A A

A

A A

C D

C

C D

C

C D

B B

B

D

B

A A

D

C

B D

B

A

D

B

A B

D

C B

C A

D

A

A C

B

B

C

D

D

B

A

D B

B

C

D

C A

D

A

0

A

B A

D C

A

D

C

A B

777

D

D

D B

8 (µm)

10 (µm)

Fig. 10. Proposed structure with randomly distributed ellipse-shape imperfections. Both minor and major radii of regions labeled by A are decreased by 3 nm. The minor radii of regions labeled by B are decreased by 3 nm and major radii are increased by 3 nm. The minor radii of regions labeled by C are increased by 3 nm and major radii are decreased by 3 nm. Both minor and major radii of regions labeled by D are increased by 3 nm.

778


(a)

Varmazyari et al.

(b)

(d)

(c)

(e)

Fig. 11. Schematic fabrication process of the proposed structure: (a) multilayer structure used in the fabrication of Si membrane waveguide, (b) transferring pattern of silica regions into Si slab, (c) deposition of SiO2 into the etched regions, (d) transferring pattern of air holes into Si slab, and (e) removing the BOX layer underneath the waveguide.

broadening of the output pulse are obtained 40.22 and 2.7%, respectively. In this case, no significant variation was found in the NDBP value. In our structure, it is possible to minimize the effects of imperfections by further correction methods. Various methods have been studied and used to dynamically tune the properties of silicon PC devices by an external control, such as electro-optic effect [13], electromagnetically induced transparency [15], microfluidic infiltration [29], and thermal tuning [39]. Using these tuning methods we can modify the properties of the silicon waveguide or silica region in the presented structure so that the NDBP parameter is close to the previous optimized value. As mentioned previously, our suggested device is a silicon air-bridge (membrane) platform. To date, most experimental studies of Si PCWs have focused on different structures, such as membrane or silicon-on-insulator (SOI) or silicon slab fully embedded in bulk silica [40]. Air-bridge geometry provides two important features: a high-index contrast for strong out-of-plane confinement, and a symmetric air cladding that ensures orthogonality of the TE-like and TM-like slab modes. The lowest PCW losses have been achieved for membrane structures. In SOI PCs, the silica cladding breaks the vertical symmetry of the structure, then coupling between the TE-like and TM-like modes of the silicon slab is permitted and propagation loss is increased. However, the well-established silicon slab embedded in silica has been reported with much improved accuracy and relatively low loss [40]. According to these explanations, we can also use other configurations. In such structures, only the refractive indices used in simulations will be altered, but the overall operation of the proposed device and concept will not change. Our proposed Si air-bridge slab has usual fabrication process. The main process flow is shown schematically in Fig. 11, which is compatible with standard silicon CMOS technologies [41,42]. To fabricate the proposed device, a multilayer platform is used. In this structure, layers from top to down consist of a silicon layer, which is used to create a line-defect waveguide; the buried oxide (BOX) layer with two thin layers of silica, which are used as holders of silicon membrane; and the latest layer is a silicon substrate. In the second layer, two thin layers of silica are obtained by combining the optical

lithography and chemical etching, and then the BOX layer is deposited. After preparation of the multilayer structure, the fabrication process is concisely divided into following steps. In the first step, the pattern of the silica regions by using e-beam lithography and standard reactive-ion etching (RIE) process is transferred on the Si layer. Then the etched regions are deposited with silicon dioxide by a plasma-enhanced chemical vapor deposition technique at low temperatures. In the next step, similar to the method used for silica regions, the pattern of the air holes is transferred on the Si slab. Finally, the BOX layer is removed by wet etching with selective BOX etchant. In order to minimize the effect of fabrication errors on the size of the designed parameters, the conditions of the fabrication steps should be carefully performed. Thus, using some high precision process, such as high quality pattern, high-resolution lithography, high-aspect-ratio etching and suitable sidewall passivation process in the RIE step, is necessary.

4. CONCLUSION In summary, a novel slow light device in a Si air-bridge PC with triangular lattice of ellipse hole was proposed, that the adjacent rows of line-defect waveguide were filled by silica. Dispersion engineering in the presented slow light structure was performed using a four-step optimization process on the waveguide parameters, such as major and minor radii and position of silica regions in the x and y directions. After the optimization, high NDBP under a constant group index criterion and wideband slow light was achieved. For the optimum case, the NDBP of 0.461 with a group index of 41.86 and a bandwidth of 17.06 nm was obtained by the 3D PWE method. Using this feature, a small and compact optical buffer with high capacity can be achieved that has many applications in optical telecommunications. In order to study the presented structure in time domain, the optical pulse propagating in the optimized waveguide was investigated by using the FDTD method. The simulation results indicated that a propagating optical pulse with FWHM of 0.38 ps, has a low relative broadening of 2.1% after traveling time of 1.4 ps.

Varmazyari et al.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9.

10. 11. 12.

13. 14. 15. 16. 17. 18. 19.

20. 21.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). I. Novikova, R. L. Walsworth, and Y. Xiao, “Electromagnetically induced transparency-based slow and stored light in warm atoms,” Laser Photon. Rev. 6, 333–353 (2012). M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90, 113903 (2003). P. Ch. Ku, F. Sedgwick, C. J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S. W. Chang, and S. L. Chuang, “Slow light in semiconductor quantum wells,” Opt. Lett. 29, 2291–2293 (2004). H. Su and Sh. L. Chuang, “Room-temperature slow light with semiconductor quantum-dot devices,” Opt. Lett. 31, 271–273 (2006). T. Baba, J. Adachi, N. Ishikura, Y. Hamachi, H. Sasaki, T. Kawasaki, and D. Mori, “Dispersion-controlled slow light in photonic crystal waveguides,” Proc. Jpn. Acad. 85, 443–453 (2009). T. Baba, T. Kawasaki, H. Sasaki, J. Adachi, and D. Mori, “Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide,” Opt. Express 16, 9245–9253 (2008). D. Wang, Z. Yu, Y. Liu, X. Guo, and S. Zhou, “Optimization of a two-dimensional photonic crystal waveguide for ultraslow light propagation,” J. Opt. 14, 125101 (2012). C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17, 2944–2953 (2009). K. Üstün and H. Kurt, “Ultra slow light achievement in photonic crystals by merging coupled cavities with waveguides,” Opt. Express 18, 21155–21161 (2010). D. O’Brien, M. D. Settle, T. Karle, A. Michaeli, M. Salib, and T. F. Krauss, “Coupled photonic crystal heterostructure nanocavities,” Opt. Express 15, 1228–1233 (2007). S. C. Huang, M. Kato, E. Kuramochi, C. P. Lee, and M. Notomi, “Time-domain and spectral-domain investigation of inflectionpoint slow-light modes in photonic crystal coupled waveguides,” Opt. Express 15, 3543–3549 (2007). H. Tian, F. Long, W. Liu, and Y. Ji, “Tunable slow light and buffer capability in photonic crystal coupled-cavity waveguides based on electro-optic effect,” Opt. Commun. 285, 2760–2764 (2012). M. S. Moreolo, V. Morra, and G. Cincotti, “Design of photonic crystal delay lines based on enhanced coupled-cavity waveguides,” J. Opt. A 10, 064002 (2008). V. Varmazyari, H. Habibiyan, and H. Ghafoorifard, “All-optical tunable slow light achievement in photonic crystal coupledcavity waveguides,” Appl. Opt. 52, 6497–6505 (2013). D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005). 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). J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). R. Hao, E. Cassan, X. Le Roux, D. Gao, V. Do Khanh, L. Vivien, D. Marris-Morini, and X. Zhang, “Improvement of delay-bandwidth product in photonic crystal slow-light waveguides,” Opt. Express 18, 16309–16319 (2010). 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). 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).


779

22. Y. Xu, L. Xiang, E. Cassan, D. Gao, and X. Zhang, “Slow light in an alternative row of ellipse-hole photonic crystal waveguide,” Appl. Opt. 52, 1155–1160 (2013). 23. N. Janrao, R. Zafar, and V. Janyani, “Improved design of photonic crystal waveguides with elliptical holes for enhanced slow light performance,” Opt. Eng. 51, 064001 (2012). 24. Y. Wan, K. Fu, C. Li, and M. Yun, “Improving slow light effect in photonic crystal line defect waveguide by using eye-shaped scatterers,” Opt. Commun. 286, 192–196 (2013). 25. Y. Zhai, H. Tian, and Y. Ji, “Slow light property improvement and optical buffer capability in ring-shape-hole photonic crystal waveguide,” J. Lightwave Technol. 29, 3083–3090 (2011). 26. M. Mulot, A. Saynatjoki, S. Arpiainen, H. Lipsanen, and J. Ahopelto, “Slow light propagation in photonic crystal waveguides with ring-shaped holes,” J. Opt. A 9, S415–S418 (2007). 27. A. Säynätjoki, M. Mulot, J. Ahopelto, and H. Lipsanen, “Dispersion engineering of photonic crystal waveguides with ring-shaped holes,” Opt. Express 15, 8323–8328 (2007). 28. S. Rawal, R. K. Sinha, and R. De La Rue, “Slow light propagation in liquid-crystal infiltrated silicon-on-insulator photonic crystal channel waveguides,” J. Lightwave Technol. 28, 2560–2571 (2010). 29. M. Ebnali-Heidari, C. Grillet, C. Monat, and B. J. Eggleton, “Dispersion engineering of slow light photonic crystal waveguides using microfluidic infiltration,” Opt. Express 17, 1628–1635 (2009). 30. T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D 40, 2659–2665 (2007). 31. 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). 32. A. Yu. Petrova and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). 33. J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009). 34. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006). 35. J. Wu, Y. Li, C. Peng, and Z. Wang, “Wideband and low dispersion slow light in slotted photonic crystal waveguide,” Opt. Commun. 283, 2815–2819 (2010). 36. K. Üstün and H. Kurt, “Slow light structure with enhanced delaybandwidth product,” J. Opt. Soc. Am. B 29, 2403–2409 (2012). 37. M. Hosseinpour, M. Ebnali-Heidari, M. Kamali, and H. Emami, “Optofluidic photonic crystal slow light coupler,” J. Opt. Soc. Am. B 30, 717–722 (2013). 38. M. Y. Tekeste and J. M. Yarrison-Rice, “High efficiency photonic crystal based wavelength demultiplexer,” Opt. Express 14, 7931–7942 (2006). 39. N. Ishikura, R. Hosoi, R. Hayakawa, T. Tamanuki, M. Shinkawa, and T. Baba, “Photonic crystal tunable slow light device integrated with multi-heaters,” Appl. Phys. Lett. 100, 221110 (2012). 40. T. P. White, L. O’Faolain, J. Li, L. C. Andreani, and T. F. Krauss, “Silica-embedded silicon photonic crystal waveguides,” Opt. Express 16, 17076–17081 (2008). 41. M. Loňcar, T. Doll, J. Vučković, and A. Scherer, “Design and fabrication of silicon photonic crystal optical waveguides,” J. Lightwave Technol. 18, 1402–1411 (2000). 42. J. Feng, Y. Chen, J. Blair, H. Kurt, R. Hao, D. S. Citrin, C. J. Summers, and Z. Zhou, “Fabrication of annular photonic crystals by atomic layer deposition and sacrificial etching,” J. Vac. Sci. Technol. B 27, 568–572 (2009).