Sporadic-Slot Photonic-Crystal Waveguide for All-Optical Buffers With Low-Dispersion, Distortion, and Insertion Loss

A novel structure is studied with implanting sporadic slots inside a photonic crystal waveguide (PCW) to form sporadic-slot PCW (SSPCW) for realizing compact, all-optical buffers with low-dispersion, distortion, and attenuation (DDA). We implement the first demonstration that, to the best of our knowledge, the SSPCW works for both TE- and TM-modes all-optical buffers and slow-light waveguides. High buffer performance and wider transmission bandwidth in telecommunication band are obtained, which are guaranteed through bit rate optimization that exceeded 2 Tb/s for both TE and TM modes in ultra-low dispersion region which is 20 times greater than that required for 5G mobile communications and it is also useful for all-optical signal processing. The storage length of 1 bit is obtained as <inline-formula> <tex-math notation="LaTeX">$4.5202~\mu \text {m}$ </tex-math></inline-formula> for TE mode and <inline-formula> <tex-math notation="LaTeX">$6.0525~\mu \text {m}$ </tex-math></inline-formula> for TM mode. Moreover, a low relative pulse distortion of 0.0801% <inline-formula> <tex-math notation="LaTeX">$\mu \text{m}^{-1}$ </tex-math></inline-formula> along the propagation path per unit length is acquired for a 0.63 ps pulse. Besides, the attenuation coefficient of the optical power pulse between the input and output is <inline-formula> <tex-math notation="LaTeX">$0.0170~\text {dB}/\mu \text {m}$ </tex-math></inline-formula>. Additionally, the deigned SSPCW is insensitive for small variations of waveguide parameters and the deviation is virtually negligible between the simulated results and that of the fabricated structures with totally acceptable tolerance below ±3 nm, which is the challenge in the fabrication process of conventional PCW and PCW-cavity.


I. INTRODUCTION
Photonic crystals (PhCs) have been scrutinized for miscellaneous applications, as a highly successful and unique platform for manipulating light in microscopic dimensions and designing their dispersions for particular needs [1]. PhCs have the potential in realizing optical processors to meet the need for higher CPU speeds. The common key for all-optical processors is the optical buffer, which stores the optical packets temporarily and fine-tunes CPU timing. Slow-light property in PhC waveguide (PCW) provides the basis for efficient buffering of optical packets in flexible onchip optical signal processing [2]. Moreover, they can operate The associate editor coordinating the review of this manuscript and approving it for publication was Nagendra Prasad Pathak. at room temperature with wide bandwidths [3] and possess low dispersion and distortion for light propagation [4]. With technologically valuable developmentsof silicon-based PCWs, slotted photonic crystal waveguide (SPCW)attracted the attention of many researchersdue to the combination of two important features in numerous applications: it can merge the abilities to confine light in the low-index material in slots and enhancing the slow-light property of PCWs [5]. In PCWs, the optical confinements are due to the deeper spreading of optical mode profile inside the PhC lattice with reducing group velocity [6], and generally the propagation loss upsurges for the slow-light modes. Further, some of the benefits derived from the slow-light effect can be debilitated. Contrariwise, in the SPCW due to the high index contrast of the silicon platform, the optical confinement consolidates VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ with the reduction of group velocity. At the interface with the high contrast of index, the guided modes are required to be highly intensive in the low index region (slots). Accordingly, the concentration percentage of energy in this region will increase instead of decrease by approaching the photonic bandgap edge [7]. This advantage can be utilized in applications mainly for compact optical communications [8], various types of sensor devices [9], [10] and probable applications in 5G communication systems.
Recently, sundry SPCW structures proposed for realizing slow-light property [4], [11], [12] have been designed based on two main frameworks: changing the periodic arrays or/and shape of the scatters. Examples of changing the periodic arrays of scatters are, adjusting the size of the hole [13], fine-tuning the holes' positions adjacent to the slot position [14], and changing the slot size [15], [16]. It turned out that it was difficult to precisely control the size of holes for a photonic lattice and reproduction in the fabrication processes [1]. Examples of changing the shape of the scatters are, ring-shaped [17], ellipse-shaped [11], eye-shaped [18], and crescent-shaped designs [19]. Nevertheless, novel structures are to be inspected based on square photonic crystal slab with a large and fixed size of all air holes together with SSPCW for realizing compact, all-optical buffers with low dispersion, distortion, and attenuation (DDA). In this paper, we proposed a unique platform from SSPCW for both TE-and TM-modes of all-optical buffers and slow-light waveguides. High buffer performance and ultra-wider transmission bandwidth in the telecommunication band are obtained.

II. GENERAL EQUATIONS OF SLOW LIGHT AND OPTICAL BUFFERS
To investigate and heighten slow-light properties, some formulae and definitions are requisite to be explained. The slowlight properties can be evaluated by the group index n g , which is defined as the ratio of light speed in vacuum c to group velocity v g , so [18] where U = ωa/2πc = a/λ and K = ka/2π are the normalized frequency and normalized wave vector, respectively. Considering the refraction effect of material to light, the group velocity v g of the guided modes is defined as the derived number of angular frequency ω to wave vector k, i.e., v g = dω/dk. According to Eq. (1), the slop of a dispersion curve or the first derivative of dispersion relation between U and K vector is n g . Furthermore, for all-inclusive valuation of slow-light property, normalizeddelay-bandwidth-product (NDBP) is more frequently considered for devices at different operating frequencies and/or different lengths. NDBP is defined as: where U 0 is the normalized central frequency of the linear slope region over normalized bandwidth U of a linear region within ±7.5% of n g variation about n g . To avoid optical signal distortion, U should be inside the linear region. Moreover, for ultra-low dispersion U 0 should be in the flattest dispersion curve and region. For all calculations in this study, n g variation is less than all other reported previously which restricted by ±10% [20]. The ±7.5% variation of the group index is a sufficient guarantor to achieve ultra-low dispersion below 5 × 10 4 (a/2πc 2 ). Alongside the transmission function of data connection, the SSPCW is superior for comprehending all-optical buffers in practical applications for its compact physical size in addition to its admirable performance of data transmission through slow-light modes. Consequently, the number of parameters to characterize the full buffering performance should be mentioned. The entering data to the buffer (slowlight region) contains a sequence of packets (i.e., a series of bits) with an optical central wavelength λ 0 . Each packet has a data bit period of τ bit with a base-bandwidth B packet = 1/τ bit . The optical data bandwidth in rad/s is ὼ = 4π B packet [21], which links with the normalized bandwidth by U = (a/2πc) ὼ. The first key figure of merit (FOM) for buffering is the bit rate R b which is well-defined by the number of bits per second. From the view of bit flow in a return-to-zero (RZ) format, R b can be calculated by [22]: For the constant value of lattice constant a, R b has a direct dependence on the bandwidth. The second key FOM parameter is the storage time T s for the buffers in the slow-light region of length L, calculated by Equation (4) shows that high n g is required for delay-time enhancement, which, however, leads to the reduction of bandwidth or the bit rate, and vice versa. The third key FOM parameter is the storage capacity (delay-bandwidth-product) C, which is defined by the maximum number of stored bits in the buffer. From Eqs. (3) and (4), C can be written as: It's obvious from Eq. 5 that C is directly proportional to the product of n g and U . The fourth key FOM parameter for all delay-line buffering is the minimum physical size of each stored bit in the buffer L bit , and by noticing Eq. (5) L bit can be written as: [21] L bit = v g ×τ bit = 2a (n g U ) (6) which is eventually limited by the maximum storage density (SD, the fifth key FOM parameter) of data bits in the buffer [23]. SD is defined as the number of stored bits per unit length, and from Eq. (6) SD can be written as: Subsequently, the guarantee for high capability all-optical buffering and slow light is the high compatible value between n g and U , hence the product of n g and U is the duty-bound quantity to be optimized. In real applications, for a fixed delay line of length L, the recommended structure has to be considered in line with the request of the delay time, bit length and storage density.

III. PHYSICAL MODEL AND SIMULATION RESULTS
The inspected structure is based on a square photonic crystal slab of n = 3.48 in silicon-on-insulator substrate (SOI) with large, fixed-size air holes of radius r = 0.44a, where a is the lattice constant. The proposed SSPCW of W2 PCW is constructed by switching the central two lines of air holes with one line of sporadic slots. Each slot has a width w x and length l z , and the shifting between every two sequentialslots in the Z direction is s z as shown in Fig. 1. For the minimization of scattering loss in the propagation direction, the lattice constant is kept as a constant between every two sequential slots to ensure optical-phase-matching between adjacent unit cells [24]. By appropriately modifying the sporadic slot dimensions, high buffer performance and wider transmission bandwidth are investigated. In the subsequent process, firstly, photonic band gaps (PBGs) were calculated for transverse electric/magnetic modes (TE/TM) by the spectral features of the basic square PhC structure for silicon slab of n = 3.48 and air holes of radius r = 0.44a. The output in time domain is transformed into the frequency domain through Fast-Fourier-Transform (FFT) algorithm. Figure 2 shows the transmission spectrum of the basic PhC along the propagation direction -X. Suppose, for instance, that the dielectric constant is independent on the Y-axis. Then the solutions all take the form of either TE modes with nonzero (H x , H z , E y ) or TM modes with nonzero (H y , E x , E z ). In all-purpose, conventional PhC structures are: either, slab on insulator substrate arranged by a triangular lattice of air holes supporting TM polarized mode, or square-lattice from dielectric rods arranged in air background supporting TE polarized mode. Fortunately, the square photonic crystal slab of n = 3.48 with large and fixed size of all air holes can supporting TE and TM polarized modes in our proposed case as shown in Fig. 2. Transmission spectrum has one TE-PBG of normalized frequency from ωa/2πc = 0.1775 to 0.2517, and two large TM-PBGs of normalized frequencies from ωa/2πc = 0.2015 to 0.3224 and from ωa/2πc = 0.3985 to 0.4967. For all-purpose extent, high bit rate and a low DDA can be acquired via optimization of the structure parameters (e.g., changing the radius, position, shape and refractive index and so on [4], [25]- [27]). However, in this paper, we introduce a unique method for both TE/TM polarized modes to get ultra-high bit rate and ultra-low DDA.
We first consider the linear slot with slot width w x = 1.0a and the shifting s z = 0.0a but with different slot length l z . This corresponds to a straight air slot in the center of the structure.
The dispersion curves in Fig. 3(a) are elaborated by planewave-expansion (PWE) method using the BandSOLVE module of software Rsoft for the SSPCW with w x = 1.0a, s z = 0.0a, and l z varying from 0.55a to 0.65a. It supports TE and TM modes as shown in Fig. 2(a). Moreover, as shown from the inset of Fig. 2, the supercell periodic is selected for eigenmode calculations with a lateral length of two rows of air holes in the x-direction and eleven rows in the z-direction. It is large enough that the coupling among adjacent parallel waveguides, which are created due to the super-cell model, can be ignored. Furthermore, high resolutions are used for both x and z directions with 65536 plane waves which equal to the spatial grid points (64 × 256) in the calculations. The dash-dot line signifies the light line. Figure 3(b) shows the contour map of group velocity v g /c for both TE and TM modes with l z varying from 0.55a to 0.65a via normalized wavevector K . For TE mode, we can see perceptibly that the linearity for constant slope in the long K range decreases with increasing l z and the smallest linearity of K range is around l z = 0.60a, for which we concern on the area below the light line. For the realization of slow-wave devices, the dispersion curves of waveguide modes should be under the light line, because the modes located above the light line are extremely lossy and the modes below the light line are intrinsically lossless [26]. Interestingly, TM modes have an acceptable wideband at l z = 0.60a. To make the simulation data in Fig.3 visible, we transferred it into Table 1. As is TABLE 1. Slow-light and all-optical buffer parameters for different l z at w x = 1.0a and s z = 0.0a. known, the PWE method considers infinite-area structures, however, the calculations are good for practical applications when the structure length is much longer than the operating wavelength which has the order of the lattice constant. In our simulations, the delay line length L is taken as 1 mm, which is much longer than the wavelength, so our calculations are applicable [4], [22].
From Table 1, in the case of a straight-line slot with w x = 1.0a, the optimum slot length is l z = 0.625a and 0.60a for TE and TM modes, respectively. These bit rates do not exceed 1.5 Tb/s and 1.0 Tb/s for TE and TM modes, respectively. The central wavelength can be scaled to be the optical communication wavelength of 1.55 µm by reducing the lattice constant a from 1 µm according to Table 1. Therefore, in this case to storage 1 bit, we need 6.9743 µm for TE (SD = 1 bit/6.9743 µm = 0.1434 bit/µm) and 13.1249 µm for TM (SD = 0.0762 bit/µm). It is not so satisfied as yet for applications, particularly for all-optical signal processing and new-generation communication systems in which high bit rates are needed with a small physical size of each bit which requires higher storage density. In the following, we will see that the SSPCW is the potential candidate for all-optical signal processing requirements. Our idea is physically based on the following: the sporadic slots in the proposed SSPCW act as microcavities of low index material (air slot). Meanwhile, each slot represents a defect and each defect can be considered as a cavity. Consequently, a series of slot cavities can be coupled together to form a line of defect cavities as a waveguide. The zigzag-arranged cavities provide more delay to the waves. The operating transmission bandwidth is enlarged based on the concept of impedance matching [26]. With the suitable design (suitable w x and l z values) and impedance matching, the transmission bandwidth can be wider on account of the group index.
In the following, w x = 1 and will be varied from 0.85a to 0.95a with an increment of w x = 0.05a, and the length l z will be varied from 0.50a to 0.70a with an increment of l z = 0.025a. In this case, the central region is a channel with a broken straight slot, i.e., having periodic air slots arranged in a silicon background. Fig. 4 shows the effect of changing w x and l z on the group velocity and normalized frequency. The inset of Fig. 4(a) shows the supercell for PWE calculations. For TE at w x = 0.85a and l z changing from 0.50a to 0.70a, it can be seen that most flat-band areas are located above the light line which has high losses and the ellipse-marked regions around K = 0.4 are the only small flat area for slow-light modes. Whereas, for w x = 0.90a and 0.95a, linear dispersion regions in the long K range are available below the light line with wide bandwidth around l z = 0.55a as shown in Fig. 4(a). Moreover, for observing more clearly the transmission bandwidth, Fig. 4(b) shows the variation of n g with normalized frequency for TE mode as l z changing from 0.525a to 0.60a with w x = 0.90a and 0.95a. For TM mode at w x = 0.95a and l z changing from 0.50a to 0.70a, it's clear that the flat-band area located below the light line is small, as shown from the vertical ellipse-marked region around K = 0.4. Whereas for w x = 0.85a and 0.90a, linear-dispersion regions in the long K range are available below the light line with wide bandwidth above l z = 0.625a as shown in Fig. 4(c). Moreover, for more clear observation of the transmission bandwidth, Fig. 4(d) shows the variation of n g with normalized frequency for TM modes as l z changing from 0.60a to 0.70a with w x = 0.85a and 0.90a. The operating frequency bands within ±7.5 % variation of n g are colored by dark gray solid balls in Figs. 4(b, d). To compare the simulation results of Figs. 4(b, d) clearly, we transferred them into Table 2. Table 2 shows the slow-light and all-optical buffer parameters at w x = 0.90a, 0.95a combined with l z changing from 0.525a to 0.60a by an increment of l z = 0.025a with s z = 0.0a for TE modes. Whereas for TM modes, l z changes from 0.65a to 0.70a by an increment of l z = 0.025a combined with w x = 0.85a, 0.90a. The lattice constant a is ranged for the operating wavelength in the optical communication band around 1550 nm. Meanwhile, for TE modes, at w x = 0.90a and l z = 0.55a lead to the highest compatible value of NDBP = 0.6858 for the operating bandwidth and group index. By varying w x and l z , R b exceeds 2 Tb/s in the ultra-low dispersion region which is 20 times the speed demonstrated for all-optical signal processing and 200 times the speed of 5G mobile communication, and it can also be used for 6G mobile communication network due to its high bit rate with the smallest physical size of each stored bit around 4.5202 µm. Furthermore, to store 1 bit, a space of 4.5202 µm length is needed, so the storage density SD is 0.2212 bit/µm. For TM modes, w x = 0.90a and l z = 0.65a lead to the As the present optical modulators [28], [29] has the modulation speed of 10 − 100 Gb/s, the proposed SSPCW can be utilized for developing high-speed optical modulator. Besides, SD of slow light in practical PhC delay line devices is 1 bit/20 µm [30]. Whereas our SD for TE SSPCW is 0.2212 bit/µm, and 0.1652 bit/µm for TM SSPCW. Finally, to obtain further optimum results for TM mode, we considerchanging the displacement between each adjacent slots form s z = 0.06a to 0.1a by an increment of s z = 0.01a, while keeping w x = 0.90a and l z = 0.65a at the optimum values as shown in Table 2. The influence of s z on n g is shown in Fig. 5 and Table 3. It is comprehended from Table 3 that for s z = 0.07a, it has an n g of 9.73, and there are the largest R b and wider bandwidth λ about 1.9142 Tb/s and 61.3192 nm respectively, with the lowest 1-bit storage size of 8.0534 µm. Therefore, ultra-high R b was achieved for both TE and TM modes. This is significant as in future telecommunication networks, high R b , all-optical signal processing exceeding 40 Gb/s will be desirable [31] and 1 Tb/s for metro transportation systems [32]. Also, the SSPCW can provide promising applications for random access memories in all-optical signal processors and quantum information processing.

IV. TIME-DOMAIN ANALYSES
To validate our results founded by PWE method, the timedomain propagation of optical pulse is studied over the proposed SSPCW with the two-dimensional finite-difference time-domain (2D-FDTD) method through FullWAVE module of software Rsoft [33] with 25 grid points in resolution for each period and perfectly matched absorbing boundary layers (PMLs) surrounding the structure. The transmission length is elongated to be 50a. n g is calculated from [26] n g = c T /L, where T is delay time by ps unit between the input and output pulses and L is transmission length. Throughout the simulation, two monitors are positioned to measure the field intensity. To grantee being inside the slow-light region, the input monitor is located inside at a position of 10a to the leftmost air-hole row and the output monitor is located at a position of 10a to the rightmost air-hole row of the structure, so the transmission length monitored is L = 30a.
Three cases are considered for 2D-FDTD simulations. The first case is the optimized instance for TE mode at w x = 0.90a, l z = 0.55a and s z = 0.0a with n g = 14.35 and λ = 74.0776 nm at 1550 nm. According to Table 2 and clear comparison between PWE and FDTD methods, the Gaussian pulse source is launched with central normalized frequency U 0 = 0.2405 and to operate at optical communication wavelength of 1550 nm, the lattice constant a is 372 nm. Fig. 6(a) reveals the intensity of the normalized field of input and output monitors with the time delay by ps. As clarified, the normalized field intensities detected by the input and output monitors are at 2.384 ps and 2.873 ps respectively, thus the delay time T between the input and output is 0.489 ps over the length L = 30a. Consequently, n g = 13.11. The slight difference between the two values of n g from PWE and FDTD is principally from the number of plane waves for calculation in the PWE method and the limited discretization of FDTD [34]. The full widths at half maximum (FWHM) detected by the input and output monitors are 0.445 ps and 0.592 ps, respectively, viewing a pulse distortion. This displays that the relative pulse distortion (D p = FWHM out /FWHM in − 1) is 33.0337% and the relative pulse distortion per unit length (D PR = D P /L) is 2.9536 % µm −1 , which is a little high. The light pulse transmission for TE inside the designed SSPCW has a benefit to realizing the physical mechanism as shown in Fig. 6(b). The Electric field (E-field) profile shows that the pulse transmission is slow and completely different from the counterpart in conventional PCWs. The E-field profile reflects a wider bandwidth. The second case is the optimized instance for TM mode at w x = 0.90a, l z = 0.65a and s z = 0.0a with n g = 30.96 and λ = 25.6425 nm at 1550 nm. According to Table 2, the Gaussian pulse source is launched with U 0 = 0.2983 and a = 462 nm. From Fig. 6(c), the normalized field intensities detected by the input and output monitors are at 4.298 ps and 2.912 ps respectively, thus T between the input and output is 1.386 ps over the length L = 30a. Consequently, n g = 30. The FWHMs detected by the input and output monitors are 0.630 ps and 0.637 ps respectively, viewing a very small pulse distortion of D p = 1.1111% and D PR = 0.0801 % µm −1 . The magnetic field (M-field) profile of pulse transmission is shown in Fig. 6(d), which reflects a lower transmission bandwidth as compared with that of TE mode transmission in Fig. 6(b). Periodic oscillations of the field profile are observed through spatial light confinement along the propagation path. It displays that the pulse experiences periodic compression, extension, and compression, and the output of the pulse width is nearly the same as that of the input pulse. The third case is the optimized instance for TM mode at w x = 0.90a, l z = 0.65a and s z = 0.07a with n g = 9.73 and λ = 61.3192 nm at 1550 nm. According to Table 3, the Gaussian pulse source is launched with U 0 = 0.4454 and a = 690 nm. From Fig. 6(e), the normalized field intensities detected by the input and output monitors are at 2.387 ps and 3.058 ps respectively, thus T between input and output pulses is 0.671 ps over L = 30a.   Consequently, n g = 9.725 which agrees completely with the PWE calculations. The FWHMs for the input and output monitors are 0.48 ps and 0.489 ps respectively, viewing also a very small pulse distortion of D p = 1.8750% and D PR = 0.0905 % µm −1 . Thus, the pulses can be transmitted lengthways in the proposed SSPCW without explicit distortion. The magnetic field profile of pulse transmission is shown in Fig. 6(f), which shows that the distortion in the pulse transmission is ignorable. The proposed SSPCW provides an abundant higher value of slow-light property (NDBP) and all-optical buffer parameters compared with all other structures based on slotted PCW waveguides reported previously in literature as shown in Table 4. In comparison with previous works, we find that our work has achieved high NDBPs in the cases of both TE and TM. Ultra-low dispersion is achieved within ±7.5% of the group index in the proposed SSPCW, unlike previous literature that achieved results with a change in ±10% of the group's index.

V. IMPLEMENTATION CONSIDERATIONS
There are a few particular issues for the implementation of buffer devices, for instance, dispersion, losses and fabrication tolerance, to be considered.

A. GVD
The second derivative of dispersion relation gives the group velocity dispersion (GVD = d 2 k/dω 2 = (a/2πc 2 )dn g /dU ) which induces phase modulation and results in pulse width broadening. Subsequently, the carried bit data by the transmitted pulses become distorted along the propagation path. Therefore, compensation to the distortion due to the GVD along the propagation is required [1], [6], [36]. Naturally, less distortion (L-D) corresponds to less GVD (L-GVD) in transmission. So, we searched for low dispersion distortion (L-DD). Fig. 7 shows the GVD variation of optimum cases for TE/TM SSPCWs within ±7.5 % variation of n g color in green. The ±7.5 % variation of n g is a small variation for the light to have ultra-low dispersion, i.e., nearly zero dispersion. To see the masked range of green color in the GVD curve clearly, the corresponding data in Fig. 7 is transferred into Table 5 with a summary of distortion calculation in the previous section for the optimum cases. . GVD variation with normalized frequency a/λ of the SSPCW within the ±7.5% variation of n g colored in green for optimum cases. Table 5, ultra-low dispersion is achieved within the ±7.5% of the group index. For general application, we can consider GVD < 10 6 (a/2πc 2 ) as low GVD, and then all our proposed SSPCWs are ultra L-DD and they are proper for optical pulse transmission and all-optical buffer applications [37]. As can be seen from Table 5, within the ±7.5% variation of group index, it is sufficient to achieve ultra-low dispersion below 1 × 10 4 (a/2πc 2 ) and 1 × 10 5 (a/2πc 2 ) of GVD for TE and TM modes, respectively. Alternatively, an appropriate GVD can be used to compensate for the pulse distortion. For compensation purposes, the designs having positive and negative GVDs are shown in Fig. 7. Besides, slow light transmitted in a waveguide experience generally an intrinsic dispersion. Hence, designing a wide zero-dispersion band is required for slow-light systems and important for high fidelity in the transmission of light signals.

B. LOSSES
In general, there are two classes of losses: material loss and structure loss. By using appropriate materials, material loss can be neglected or avoided [2], [38]. Structure loss for PCWs or any waveguide possesses two originations. The first one is the coupling loss at the input and output of a structure or a device; the second one is the propagation loss through the buffer region. To conserve the bandwidth and storage advantages of buffer devices, it is adequate to employ classical Si waveguide as a transmission medium devoid of any loss, and use a buffer and slow-light devices as active elements [39]. Now, coupling loss in the buffer and slowlight devices can be considered as a mostly resolved issue [4], [39], [40]. Propagation losses in the slow light and buffer PCW can be classified into intrinsic and extrinsic losses. For the practical realization of slow-light devices, the dispersion curve of waveguide modes should be under the light line, because the modes located above the light line are extremely lossy, and the modes below the light line are intrinsically lossless [26] and all our calculations are done below light lines. Providentially, both issues of the dispersion and losses are controlled by particular designs of the proposed structures since the benefited property of the slow light is a joint result by the pulse broadening and loss [4], [41]. So, the design of compensating-device GVD with opposite signs shown in Fig. 7 is proposed for overcoming the pulse broadening. For the issue of propagation losses, the optimized case of TE SSPCW is considered for a transmission length L = 50a with w x = 0.90a, l z = 0.55a, s z = 0.0a, and a = 373 nm. A Gaussian pulse source is launched by using 2-D FDTD method with 4 monitors inserted every 10a where the first monitor is positioned at 10a from the leftmost row of the SSPCW for observing the attenuation in the slow-light region, and the last monitor is located at 10a from the rightmost row of the SSPCW. Hence, the distance is L = 30a between the first and last monitors.  0.728 a.u. and 0.714 a.u., respectively, showing a decay along the transmission path. The attenuation coefficient is 0.0189 dB/µm between the first and second monitors (L = 10a and a = 0.373 µm), 0.0095 dB/µm between the second and third monitors, and 0.0226 dB/µm between the third and fourth monitors. Thus, The average attenuation coefficient of 0.0170 dB/µm, is a lower attenuation (L-A) compared to that reported in the literature [42]- [44]. For example, in [42], the attenuation was 1.5 dB/µm and in [43] was 0.05 dB/µm. However, less attenuation is better for applications, and thus it is an interesting future work to acquire further less attenuation. Moreover, we achieved low DDA in one proposed structure and further, we will show that this structure is insensitive for a small variation of structure parameters, which represents a challenge in the fabrication of conventional PCWs.

C. FABRICATION TOLERANCE
After designing any proposed structure, the fabrication of buffer devices on a large scale is one of the critical issues. Attributable to the inadequate process of equipment in the progression of fabrication, there may exist rough changes between the simulated results and that of the structure fabricated. The errors include, for example, that due to the electron scattering in electron beam lithography, etching anisotropies, and stitching [45]. These errors lead to the changing of air holes in the basic PhC and also the slot width and length in the structure. As the PBG is larger than the bandwidth of the transmitted slow-light pulse, thus the errors of air holes involving the PhC structure can be neglected. Consequently, acceptable tolerance of the slot width and length is modeled by changing w x and l z around the optimum values of 0.90a and 0.55a respectively for TE SSPCW. Table 6 displays 9 cases of fabrication errors of the optimized TE SSPCW. The width w x changes from 0.54a to 0.56a around the optimized value 0.55a by ±0.01a = ±3.73 nm at a = 373 nm. The length l z changes from 0.89a to 0.91a around the optimized value 0.90a by ±0.01a = ±3.73 nm. The results in Table 6 show that n g is scrimpy ranging from 14.23 to 14.51 over a narrow changing of bandwidth from 70.2515 nm to 74.9807 nm, with average values exposed in the last line. The averaged R b , L bit and SD of 2.2803 Tb/s, 4.58 µm and 0.2184 b/µm respectively are significantly close to the optimum values in No. 5. Moreover, the slow-light and all-optical buffer parameters have insignificant variation. Hence, the errors are totally acceptable in this range and thus the deviation is virtually negligible about the optimum values. Finally, we can conclude that below the variation of ±0.01a around ±3 nm there is no deviation between the simulated results and that of fabricated structure. So, the proposed PCW with sporadic slots in the waveguide is insensitive for small variations of waveguide parameters, which is the challenge in conventional PCW fabrication.

VI. CONCLUSION
In conclusion, a new paradigm structure (SSPCW) is inspected based on the SPCW. The results are numerically discussed for all-optical buffers and slow-light property of TE and TM polarized modes by PWE and confirmed with FDTD methods for the first time. Besides, low DDA is obtained. By appropriately modifying the sporadic-slot dimensions, higher buffer performance and wider transmission bandwidth in the telecommunication band are demonstrated. The bandwidth centered at 1550 nm fluctuates from 16.7 nm to 75 nm, corresponding to a fluctuation of the bit rate from 0.5209 Tb/s to 2.3125 Tb/s within ±7.5% variation of group index which is a sufficient guarantor to achieve ultra-low dispersion below 5 × 10 4 (a/2πc 2 ) with the highest value of NDBP about 0.6858 and 0.5122 for TE and TM modes, respectively. For low distortion, the incident optical pulse of width 0.63 ps is broadened to be solitary 0.637 ps, corresponding to a relative pulse distortion along the propagation path per unit length of 0.0801 % µm −1 . Moreover, the attenuation coefficient of the optical-pulse power between the input and output is obtained as 0.0170 dB/µm. ZIXIAN LIANG received the Ph.D. degree from the Chinese Academy of Sciences, in 2010. He is currently an Associate Professor with the Institute of Microscale Optoelectronics, Shenzhen University. His research interests primarily include photonics crystals, metamaterials, and transformation optics.
JIHONG PEI (Member, IEEE) was born in Gansu, China. He received the Ph.D. degree from the Xi'an University of Electronic Science and Technology, in 1998. From 1989 to 2002, he worked at the Xi'an University of Electronic Science and Technology. He has been with Shenzhen University, since 2002, where he is currently the Director of the Department of Electronics, School of Information Engineering, and also works with the ATR National Defense Science and Technology Key Laboratory. His main research interests include image and video content analysis, intelligent video surveillance, multicamera panoramic video technology, pattern recognition and machine learning, multispectral remote sensing image analysis, vector field signals, and image analysis. He is a member of the China Aeronautical Member Information Fusion Branch.
ZHENGBIAO OUYANG (Member, IEEE) received the B.S. degree in radio engineering from the Harbin Institute of Technology, Harbin, China, in 1983, and the Ph.D. degree in electron physics from the University of Electronic Science and Technology of China, in 1988. He is currently a Full Professor and the Director of the Solid-State Photonics Laboratory, Shenzhen University, the Vice Director of the THz Technical Research Center, Shenzhen University, the Vice Director of the Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology, a Trustee of the Chinese Society of Optical Engineering, a Committeeman of the Opto-Electronic Committee of the Chinese Space Aeronautics Association, a Senior Member of the Chinese Optical Society, and a member of the Optical Society of America. His main research interests in recent years include nano photonic materials, optical logic integrated circuits, nano photonic devices based on photonic crystals, metamaterials, and metasurfaces, high-speed optical communication technology, and THz technology.