
1.IntroductionIn recent years, integrated optical technology has been a research focus in optical communication networks.^{1} However, compared with electronics, the integration scale and integration level of optical devices are still fairly low.^{2}^{,}^{3} As an advanced technique, the photonic crystal waveguide (PCW) may eventually provide a way to achieve onchip photonic integration with roomtemperature operation. PCW has many advantages, such as a high degree of control, great potential bandwidth, and fine optical operation properties such as realizing slow light property.^{4}^{,}^{5} To date, slow light based on PCW has been a hot topic due to its broad range of applications, including optical delay lines and data synchronization in optical communications and information processing systems.^{6}^{–}^{8} Meanwhile, as an important nonlinear phenomenon and shapeunchanged optical pulse, optical soliton in photonic crystal fiber and PCW has been researched extensively. For example, Lefrançois et al.^{9} researched the scaling of dissipative soliton fiber lasers to megawatt peak powers by use of largearea photonic crystal fiber. Stark et al.^{10} studied soliton blue shift in tapered photonic crystal fibers, and Colman et al.^{11} researched temporal solitons and pulse compression in PCW. Using soliton pulses as information carriers in PCW not only can obtain pulse propagation without waveform distortion, but also can realize slow light property.^{12}^{–}^{14} This has opened a new avenue toward exploiting the low group velocity that PCW has exhibited. However, the previous studies of soliton propagation in PCWs^{12}^{–}^{14} have focused only on soliton propagation in the conventional PCW, and the bandwidth of soliton propagation at the given group index is extremely narrow.^{12}^{–}^{15} This is because the group index near the band edges has a large change, so the bandwidth at the given group index is quite narrow. To the best of our knowledge, the bandwidth improvement of soliton propagation in PCW has not been analyzed. For practical applications, it is quite necessary to extend the bandwidth for the soliton propagation at a given group index. Moreover, the required peak power must be optimized to adapt to the application criterion, since the peak power obtained in previous works on soliton propagation in PCW^{12}^{–}^{14} is a little large. Our research work indicates that the bandwidth and peak power can be further improved by optimizing the PCW structure parameters. In this paper, we focus on the bandwidth improvement and peak power reduction of the bright soliton propagation in linedefect PCW. First, we propose a method to improve the bandwidth of bright soliton propagation near the right band edge of the guided mode in PCW. Second, we focus on optimizing the bandwidth and required peak power by appropriately adjusting the PCW parameters, and the optimized PCW structures are introduced. Finally, the soliton propagation performances of the optimized PCW structures are numerically analyzed. 2.Theoretical Model2.1.PCW ModelThe basic PCW model in this paper consists of a single line defect and triangular lattice rods as shown in Fig. 1. The lattice constant is $a$. The radius of the first two rows of rods adjacent to the defect is denoted by ${r}_{1}$, the radius of the second two rows is ${r}_{2}$, and the radius of the remaining rows is $r=0.215a$. The figures $\mathrm{\Delta}{x}_{1}$ and $\mathrm{\Delta}{x}_{2}$ represent the shift of the first and second two rows adjacent to the defect along waveguide axis, respectively. $\mathrm{D}y$ is the conventional distance between the two adjacent rows, which becomes $\mathrm{d}y$ by changing the distance between the two adjacent rows. The refractive indexes of the background material polystyrene and the Sirods are 1.59 and 3.5, respectively. The nonlinear refractive indexes^{12}^{,}^{16} of Si and polystyrene are ${n}_{2\mathrm{Si}}=1.5\text{\hspace{0.17em}}\times \phantom{\rule{0ex}{0ex}}{10}^{16}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$ and ${n}_{2\text{polystyrene}}=9.3\times {10}^{13}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{m}}^{2}/\mathrm{W}$. Based on this model, by adjusting the structure parameters, we can optimize the bandwidth and peak power of soliton propagation and get broadband and low power bright soliton propagation. The soliton propagation mode is discussed below. 2.2.Model of the Nonlinear Pulse Propagation in PCWThe nonlinear propagation model of optical pulses inside a PCW satisfies the equation^{12}^{,}^{17} Eq. (1)$$j(\frac{\delta A}{\delta z}+\frac{\mathrm{\Gamma}}{2A})+\sum _{i\ge 2}{j}^{m(l)}\frac{{\beta}_{l}{\delta}^{l}A}{l!\delta {T}^{l}}+\gamma {A}^{2}\phantom{\rule[0.0ex]{2em}{0.0ex}}A=0,$$From Eq. (1), the wellknown nonlinear Schroedinger equation, we know that, if ${\beta}_{2}<0$, bright soliton can be supported. In common PCW, near the left band edge, ${\beta}_{2}$ can be negative and support bright solitons.^{12} The initial condition for the bright soliton solution can be given^{12}^{,}^{17} as Eq. (2)$$A(0,T)=\sqrt{{P}_{0}}\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}h(T/{T}_{0}),$$Eq. (4)$${\beta}_{2}=\frac{\mathrm{d}({v}_{g}^{1})}{\mathrm{d}\omega}=\frac{1}{c}\times \frac{\mathrm{d}{n}_{g}}{\mathrm{d}\omega}=\frac{{\mathrm{d}}^{2}k}{{\mathrm{d}\omega}^{2}},$$3.Simulation Results3.1.Bandwidth Improvement of Soliton Propagation in PCWThe group index and GVD curves for TM polarized mode in PCW are numerically calculated by the 2D PWE method. The defect mode inside the band gap is studied with a super cell that is 1 unit in the $x$direction and 10 units in the $z$direction in the PWE calculation. Figure 2 shows the group index and group velocity dispersion of the initial PCW structure (shown in Fig. 1) without any structure parameter adjusted. The minimum and maximum values of $K$ represent the left and right band edges of the guided mode, respectively.^{12} The GVD curve in the inset figure indicates that the dispersion relation is ${\beta}_{2}<0$ near the left band edge, so the bright soliton can be considered in this case.^{12} The bright soliton propagation near the left band edge of the guided mode in PCW has been studied previously,^{12}^{–}^{15} but the bandwidth for bright soliton propagation was extremely narrow. Moreover, near the right band edge of the mode, ${\beta}_{2}>0$, the bright soliton cannot be supported, which is the same situation as in Thomas’s work.^{12}^{–}^{14} Here, the bandwidth improvement of slow light based on the soliton propagation is obtained by adjusting PCW structure parameters $\mathrm{\Delta}{x}_{1}$ and $\mathrm{\Delta}{x}_{2}$. In our work, better bandwidth performance is obtained when $\mathrm{\Delta}{x}_{1}=0.2a$, $\mathrm{\Delta}{x}_{2}=0.344a$, and the PCW structure is defined as PCWI. Figure 3 shows the group index ${n}_{g}$ and GVD ${\beta}_{2}$ of PCWI. The inset picture indicates ${n}_{g}$ and ${\beta}_{2}$ near the right band edge. By adjusting $\mathrm{\Delta}{x}_{1}$ and $\mathrm{\Delta}{x}_{2}$, near the right band edge of the guided mode, a region appears where the group index ${n}_{g}$ can be considered as constant^{3} with a range of $\pm 10\%$. In addition, as shown in the figure, within the region of constant ${n}_{g}$, part of the GVD is ${\beta}_{2}<0$, so the bright soliton can be considered. This is distinguished from the results of previous PCW structure research^{12}^{–}^{15} in that, near the right band edge, ${\beta}_{2}>0$, and bright soliton cannot be considered in the initial PCW structure. In this paper, for bright soliton propagation, the bandwidth is defined as the frequency (wavelength) range where ${\beta}_{2}<0$. Figure 4 shows the relationships between the GVD coefficient ${\beta}_{2}$ and the wavelength for the bandwidth range of ${\beta}_{2}<0$ for PCWI. The soliddotted curve is ${\beta}_{2}$ versus the wavelength. In this PCW, the bandwidth for ${\beta}_{2}<0$ is 1.96 nm, which is defined as BWAll. Therefore, the bright soliton of Eq. (2) can be supported in BWAll. According to Eq. (3), it can be easily deduced that the required soliton peak power ${P}_{0}$ is different at different values of ${\beta}_{2}$, so it is different at different wavelengths. In practical applications, it is hard to have precise control of the peak power ${P}_{0}$ for soliton propagation. This problem can be solved by using perturbation methods developed for soliton stability research. When the peak power does not exactly match the value of $N=1$, mathematically, if $0.5\le N\le 1.5$, solitons can be formed, though the peak power ${P}_{0}$ varies within a large range.^{17} The two dashed lines in Fig. 4 show the maximum and minimum of ${\beta}_{2}$ and the peak power. For PCWI, as shown in Fig. 4, within BWAll, the maximum GVD is ${\beta}_{\mathrm{max}}=2.24\times {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ps}}^{2}/\mathrm{km}$ (the lower dashed line), and the calculated required soliton peak power is ${P}_{\text{max}}=79.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}/\mathrm{m}$. To get a bandwidth in a continuous wavelength range, ${P}_{\text{max}}=79.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}/\mathrm{m}$ corresponds to $N=1.5$. Then, according to Eq. (3), the minimum peak power is ${P}_{\text{min}}=12.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}/\mathrm{m}$, which corresponds to $N=0.5$, and the corresponding GVD is ${\beta}_{\mathrm{min}}=3.171\times {10}^{6}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ps}}^{2}/\mathrm{km}$ (the higher dashed line). If the input soliton peak power is ${P}_{0}=35.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}/\mathrm{m}$, which is calculated by ${P}_{0}=0.5\times {P}_{\text{max}}=\phantom{\rule{0ex}{0ex}}1.5\times {P}_{\text{min}}$, this satisfies the condition that $0.5\le N\le 1.5$. Hence, the soliton can form and propagate at the constant input peak power of ${P}_{0}=35.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}/\mathrm{m}$ within the bandwidth, and the bandwidth for this is about 1.8 nm, which is defined as BWP. So far, BWAll and BWP have been defined, and their calculations have been given. Within BWAll, the required soliton peak power ${P}_{0}$ is different at different wavelengths, but within BWP, only one peak power ${P}_{0}$ is needed. As to PCWI, BWAll is 1.96 nm, and BWP is 1.8 nm. The required power $P$ can be estimated by multiplying the power density ${P}_{0}$ by ${\omega}_{\mathrm{eff}}$. After calculating the field in the supercell, the effective mode aperture of PCWs in this paper is about ${\omega}_{\mathrm{eff}}=0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu \mathrm{m}$, which is the same as that in Theocharidis et al.^{12} Here, it can be calculated that bright soliton can propagate during BWP only for one peak power of $P=\phantom{\rule{0ex}{0ex}}{P}_{0}\times {\omega}_{\mathrm{eff}}=35.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}/\mathrm{m}\times 0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu \mathrm{m}=17.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu \mathrm{W}$. Meanwhile, ${n}_{g}$ approaches a constant of about 51. 3.2.Optimizing of Bandwidth and Peak PowerIn this section, the bandwidth and required soliton peak power are optimized by adjusting the PCW structure parameters ${r}_{1}$, ${r}_{2}$, and $\mathrm{d}y$ step by step. The group index and group velocity dispersion of the two final optimized PCW structures, which are defined as PCWII and PCWIII, are shown in Fig. 5(a) and 5(b). The PCW parameters are given in Table 1. Figure 5(a) indicates that, after being optimized, a much larger flat band appears around the work wavelength of 1,550 μm in the group velocity curve for the two cases. This means that, physically, the bandwidth is improved. Figure 5(b) clearly shows that, among the flat band ranges, a region exists where ${\beta}_{2}<0$ for the two cases. In physics, the bright soliton can be supported. The concreted performance parameters such as bandwidth and power are also shown in Table 1. This table gives a summary and comparison of the structure parameters and soliton performance (when ${R}_{b}=100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{Gb}/\mathrm{s}$) of the two optimized PCWs we have found. Detailed descriptions are given below. Table 1Structure parameters and soliton performance of the PCWs.
PCWII is obtained by adjusting ${r}_{1}$ and ${r}_{2}$ for PCWI. As shown in Table 1 and in the dotted lines of Fig. 5(a) and 5(b), compared with PCWI, PCWII has an improved bandwidth, BWAll increases from 1.96 nm to 2.47 nm, and BWP increases from 1.8 nm to 2.35 nm. Also, ${n}_{g}$ has decreased from 51 to 38. At the same time, it can be found that the peak power $P$ is only about 8.1 μW when ${n}_{g}$ is 38 — a $7.53\times {10}^{7}$ times reduction^{9} compared with $P=6.1\times {10}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}$ when ${n}_{g}=30$. Considering the DWDM system with 0.2 nm of channel spacing in optical networks, about 11 channels can be supported within the 2.35nm BWP wavelength range. PCWIII is obtained by adjusting $\mathrm{d}y$ in PCWII. As Table 1 and the solid lines of Fig. 5(a) and 5(b) show, compared with PCWII, BWAll increases to 4.08 nm (a 1.61nm improvement), and BWP increases to 3.61 nm (a 1.26nm improvement). It can be seen that the bandwidth can be further improved, and it can be deduced that $\mathrm{d}y$ has a larger impact on the bandwidth improvement than ${r}_{1}$ and ${r}_{2}$. Therefore, about 18 channels can be supported within the 3.61nm BWP wavelength range for DWDM application with 0.2 nm of channel spacing. In PCWIII, the soliton peak power $P$ is about 35.7 μW when ${n}_{g}$ is 44, of a $1.71\times {10}^{7}$ times reduction^{12} from $P=6.1\times {10}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}$ when ${n}_{g}=30$. To summarize, for the two final optimized structures PCWII and PCWIII, PCWII has a peak power $P$ as low as 8.1 μW. However, PCWIII has a larger bandwidth, where BWAll is 4.08 nm and BWP is 3.61 nm. For a DWDM system with 0.2 nm of channel spacing in optical networks, about 11 and 18 channels can be supported in the 2.35 and 3.61nm bandwidths of PCWII and PCWIII, respectively. Therefore, DWDM applications with 8 and 16 channels can be realized in PCWII and PCWIII, respectively. These results indicate that the bandwidth and required peak power for soliton propagation in PCW have been significantly improved in our proposed structures, and the delay based on soliton propagation in PCW can be applied in alloptical networks. The bright soliton propagation and the field pattern of PCWIII are numerically investigated to verify the above results. Figure 6(a) shows the temporal domain bright soliton propagation in PCWIII with ${\beta}_{\mathrm{max}}=3.82\times {10}^{7}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ps}}^{2}/\mathrm{km}$ and $\gamma =117.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{W}}^{1}$, which is obtained by employing the numerical splitstep Fourier method to solve Eq. (1). The curves along the normalized time axis and the amplitude axis in Fig. 6(a) denote the shape of the bright soliton. The initial soliton shape is given from Eq. (2), which is a hyperbolic secant function of time. The distance axis $L$ in Fig. 6(a) is in units of the dispersion length ${L}_{D}$. The PCW length is $L=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}$, so the distance is $L=190\times {L}_{D}$. As shown in Fig. 6(a), the soliton pulse propagates in a stable manner in the PCW along the distance axis without waveform distortion. This means physically that the dispersion and the nonlinearity are balanced. Figure 6(b) shows the spacial field pattern of the optical wave in PCWIII, where the $x$axis denotes the longitudinal dimension of PCWIII that corresponds with the distance axis of Fig. 6(a). The $z$axis shows the transverse dimension of PCWIII. From Fig. 6(b), it can be seen that the light signal is well confined spatially in PCWIII, and the spatial diffusion and scattering are small. At last, we can combine Fig. 6(a) and 6(b) to determine that the optical wave signal can propagate effectively in PCWIII without temporal dispersion and spatial scattering. As a result, it is especially appealing for optical communication. 4.ConclusionIn this paper, we studied the improvement of bandwidth and power performance for bright soliton propagation near the right band edge in linedefect PCW. The simulation results show that bandwidth improvement of bright soliton propagation can be obtained by properly adjusting the structure parameters $\mathrm{\Delta}{x}_{1}$, $\mathrm{\Delta}{x}_{2}$, ${r}_{1}$, ${r}_{2}$, and $\mathrm{d}y$. In the proposed structure PCWII, a 2.47nm BWAll bandwidth for bright soliton propagation with different peak powers is obtained. A 2.35nm BWP bandwidth is also obtained, within which a constant peak power as low as 8.1 μW is needed with a constant group velocity about 38. For the proposed structure PCWIII, BWAll is 4.08 nm, and BWP is 3.61 nm; both figures are larger than those of PCWII. The constant peak power is 35.7 μW within the 3.61nm BWP bandwidth, with a constant group velocity about 44. For a DWDM system with 0.2 nm of channel spacing in optical networks, 8 and 16 channels can be supported within the 2.35nm and 3.61nm BWP bandwidths of PCWII and PCWIII, respectively. The soliton pulse envelope propagation and the field pattern of the optical wave in the optimized PCW have also been numerically investigated. Our research shows the optimal optical wave propagation performance of the PCW, which can be applied in alloptical network. AcknowledgmentsThis research was supported in part by National 973 Program (No. 2012CB315705), National 863 Program (No. 2011AA010303), China. ReferencesT. F. Krauss,
“Slow light in photonic crystal waveguides,”
J. Phys. D: Appl. Phys., 40
(9), 2666
–2670
(2007). http://dx.doi.org/10.1088/00223727/40/9/S07 JPAPBE 00223727 Google Scholar
T. Baba,
“Slow light in photonic crystals,”
Nat. Photon., 2
(8), 465
–473
(2008). http://dx.doi.org/10.1038/nphoton.2008.146 17494885 Google Scholar
F. Long, H. P. Tian and Y. F. Ji,
“A study of dynamic modulation and buffer capability in low dispersion photonic crystal waveguides,”
J. Lightwave Technol., 28
(8), 1139
–1143
(2010). http://dx.doi.org/10.1109/JLT.2010.2040708 JLTEDG 07338724 Google Scholar
D. J. Gauthier, A. L. Gaeta and R. W. Boyd,
“Slow light: from basics to future prospects,”
Photon. Spectra, 40
(3), 44
–50
(2006). PHSAD3 07311230 Google Scholar
T. Baba and D. Mori,
“Potential of slow light in photonic crystal,”
Proc. SPIE, 6351 63511Z
(2006). http://dx.doi.org/10.1117/12.690817 PSISDG 0277786X Google Scholar
M. Notomi et al.,
“Extremely large group velocity dispersion of linedefect waveguides in photonic crystal slabs,”
Phys. Rev. Lett., 87
(25), 253902
(2001). http://dx.doi.org/10.1103/PhysRevLett.87.253902 PRLTAO 00319007 Google Scholar
X. Zhang, H. P. Tian and Y. F. Ji,
“Group index and dispersion properties of photonic crystal waveguides with circular and square airholes,”
Opt. Commun., 283
(9), 1768
–1772
(2010). http://dx.doi.org/10.1016/j.optcom.2009.12.022 OPCOB8 00304018 Google Scholar
Y. Hamachi, K. Shousaku and B. Toshihiko,
“Slow light with low dispersion and nonlinear enhancement in a latticeshifted photonic crystal waveguide,”
Opt. Lett., 34
(7), 1072
–1074
(2009). http://dx.doi.org/10.1364/OL.34.001072 OPLEDP 01469592 Google Scholar
S. Lefrançois et al.,
“Scaling of dissipative soliton fiber lasers to megawatt peak powers by use of largearea photonic crystal fiber,”
Opt. Lett., 35
(10), 1569
–1571
(2010). http://dx.doi.org/10.1364/OL.35.001569 OPLEDP 01469592 Google Scholar
S. P. Stark, A. Podlipensky and P. St. J. Russell,
“Soliton Blueshift in Tapered Photonic Crystal Fibers,”
Phys. Rev. Lett., 106
(8), 083903
(2011). http://dx.doi.org/10.1103/PhysRevLett.106.083903 PRLTAO 00319007 Google Scholar
P. Colman et al.,
“Temporal solitons and pulse compression in photonic crystal waveguides,”
Nat. Photon., 4
(12), 862
–868
(2010). http://dx.doi.org/10.1038/nphoton.2010.261 17494885 Google Scholar
A. Theocharidis et al.,
“Linear and nonlinear optical pulse propagation in photonic crystal waveguides near the band edge,”
IEEE. J. Quantum Electron., 44
(11), 1020
–1027
(2008). http://dx.doi.org/10.1109/JQE.2008.2002106 IEJQA7 00189197 Google Scholar
T. Kamalakis and T. Sphicopoulos,
“A new formulation of coupled propagation equations in periodic nanophotonic waveguides for the treatment of kerrinduces nonlinearities,”
IEEE. J. Quantum Electron., 43
(10), 923
–933
(2007). http://dx.doi.org/10.1109/JQE.2007.902933 IEJQA7 00189197 Google Scholar
I. Neokosmidis, T. Kamalakis and T. Sphicopoulos,
“Optical delay lines based on soliton propagation in photonic crystal coupled resonator optical Waveguides,”
IEEE. J. Quantum Electron., 43
(7), 560
–567
(2007). http://dx.doi.org/10.1109/JQE.2007.898842 IEJQA7 00189197 Google Scholar
L. Y. Liu, H. P. Tian and Y. F. Ji,
“Soliton pulse propagation and optical delay properties in photonic crystal waveguide,”
Acta Phys. Sin., 60
(10), 104216
(2011). http://dx.doi.org/10.7498/aps.60.104216 WLHPAR 10003290 Google Scholar
C. R. Mendon et al.,
“Nonlinear refractive indices of polystyrene films doped with azobenzene dye Disperse Red 1,”
Elect. Lett., 34
(1), 116
–117
(1998). http://dx.doi.org/10.1049/el:19980010 ELLEAK 00135194 Google Scholar
G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego
(1989). Google Scholar
BiographyHuiping Tian received BS and PhD degrees from Shanxi University, China, in 1998 and 2003, respectively. She is an associate professor in the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT), China. Her research interests are focused on ultrashort and ultrafast processes in the transmission of optics, photonic crystals, and broadband information networking. Daquan Yang received a bachelor’s degree in electronic information science and technology from JiNan University, China, in 2009. He currently is a master’s candidate in the State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications (BUPT), China. His research focuses on photonic crystals and optical communication. Lingyu Liu received a master’s degree in information and communication systems from the State Key Laboratory of Information and Optical Communications, Beijing University of Posts and Telecommunications (BUPT), China, in 2012. Her research focused on nonlinear effects in photonic crystals. She currently works in Jilin province and majors in electric power communication technique. Yuefeng Ji received a PhD degree from Beijing University of Posts and Telecommunications (BUPT), China, where he is now a professor and executive dean of the Institute of Information Photonics and Optical Communications. His research interests are primarily broadband communication networks and optical communications, with emphasis on key theory, realization of technology, and applications. 