1 Introduction

Slowing light has attracted increasing attention from scientists and engineers and becomes an important area of optics. It is believed that slow light can provide an effective means to deeply understand the fundamental physics of light-matter interaction [1, 2]. Besides, slow light shows the promising prospective of potential applications, such as optical delay, optical storage, signal processing, and enhanced nonlinearity [3–5]. The current techniques to slow light are mainly based on either dispersive materials or dispersive structures. For the dispersive materials, the electromagnetically induced transparency (EIT) effect in atomic systems [6, 7] and the coherent population oscillation (CPO) effect in solid-state systems [8, 9] are employed. The wavelengths and bandwidths of the devices based on these effects are limited assigned to the material absorption lines. Meanwhile, it is impossible for the on-chip integration due to the large volume of the setups. Considering these, slow light based on dispersive structures is advantageous. Slow light effect within dispersive microstructures such as photonic crystal (PC) waveguides [1, 4, 10], and microring resonator [11, 12] and coupled cavity [13] optical waveguides has been demonstrated experimentally and theoretically.

Since the pioneering work done by Yablonovitch and John in 1987 [14, 15], the physics and fabrication technologies of PCs are gradually understood and well mastered. Consequently, particular and intensive attention has been paid to the PC structures to realize the slow light phenomena [1, 4, 10, 16–21]. These structures are supposed to easily enable the on-chip photonic circuits and favorable for photonic integration, which are critical to the next-generation optical communication systems. The key issues of the PC-based slow-light devices for the high-capacity optical communication systems are the possible narrow bandwidth (though wider than that of the dispersive materials) and large group velocity dispersion (GVD). These will limit the bit rates or distort the signals after propagating through the slow-light region. Several methods have been proposed and implemented experimentally to engineer the PC band structures or compensate the dispersion mutually in different sections to widen the bandwidth or lower the GVD. These include changing the hole/rod sizes, positions or shapes (e.g., annular rings [22, 23], sandglass-shaped holes [24]) near the waveguide region to obtain a wide flat band [17, 18, 25, 26], or chirping the structures [27, 28]. Some of these structures may raise the challenge from the fabrication point of view.

Recently, Casas-Bedoya et al. [29–31] have accomplished improving and tuning the slow light properties within the PC waveguides by selectively and controllably microfluidic infiltration. They have successfully demonstrated changing the central wavelengths, group indices and bandwidths of slow light effect by choosing different kinds of liquids (i.e., different refractive indices) infiltrated into the holes near the line-defect waveguide, which is also a promising way to realize the reconfigurable PC circuits [31, 32]. This method needs the precise positioning of the micropipette to the designed hole area (usually submicron-scale) with sophisticated technique. If the liquids with refractive indices responsive to the external stimuli are utilized as the background of the PC waveguides, the corresponding slow light characteristics can be tuned online by the external fields. This will greatly benefit the pragmatic applications as compared with the current techniques (all of them are offline tuning, viz. changing the structures/liquids). It will also avoid needing precise/repeated fabrication techniques and elaborate positioning means.

Magnetic fluids (MFs) are colloidal systems of surface-treated magnetic nanoparticles dispersed in suitable carrier liquids and have excellent properties of magnetic-field-dependent refractive indices. The interest in the novel optical properties and applications of MFs is increasing recently [33], which include optical negative refraction [34], terahertz Faraday rotation [35], fiber modulators [36], MF-infiltrated microstructured optical fiber [37], novel magnetic-field sensing [38–40], and light transmission [41, 42]. To this end, MFs are proposed to be the background of the silicon-rod PCs in this work. The external magnetic field is used to finely tune the average group index and the normalized bandwidth in online mode. On the other hand, almost all the published works about PC low light is based on the triangular lattice. This work will study the slow light effect with a rarely investigated but very simple PC structure, 45o-rotated square lattice, which shows extremely large bandwidth and ultralow-GVD slow light while has relatively high and almost constant value of normalized delay-bandwidth product (NDBP).

2 Model and waveguide design

Our proposed slow-light structure is omitting one row of rods within the two-dimensional PC with 45o-rotated square lattice to form the line-defect waveguide as shown in Fig. 1. The radius and period of the arrayed rods are \( r \) and \( a \), respectively. As an example, the rods are considered to be silicon with fixed refractive index \( n_{\text{Si}} \) = 3.48, which is the widely used material in integrated photonics. The background is a kind of MF with magnetic-field-tunable refractive index \( n_{\text{MF}} (H) \). Therefore, the external magnetic field \( H \) can tune the band structures of the PC and the corresponding modes of the line-defect waveguide, which may lead to the online tunability of the slow light.

Fig. 1
figure 1

Schematic of line-defect waveguide within the two-dimensional photonic crystal with 45o-rotated square lattice. The cylinders and background are silicon and magnetic fluid, respectively

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To study the slow light properties of the designed structures numerically, the information about the magnetic-field-dependent refractive indices of MFs must be known (the magnetic permeability μ is taken as 1 at optical frequency [43]). Ref. [44] only gave the experimental refractive data of MFs at several discrete strengths of magnetic field. Besides, the actual values of refractive indices may deviate from the measured ones due to the experimental errors. So we use Chen’s model to acquire the MFs’ refractive indices at any desired magnetic field [45]. Chen et al. [45] have pointed out that the magnetic-field-dependent refractive index of a MF can be described by the modified Langevin function.

$$ n_{\text{MF}} (H) = \left\{ \begin{gathered} n_{0} , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;H < H_{c, n} \hfill \\ \left( {n_{s} - n_{0} } \right)\left[ {\coth \left( {\alpha \frac{{H - H_{c, n} }}{T}} \right) - \frac{T}{{\alpha \left( {H - H_{c, n} } \right)}}} \right] + n_{0} ,\;H > H_{c, n} \hfill \\ \end{gathered} \right., $$
(1)

where \( n_{0} \) is the refractive index of MF under magnetic field lower than the critical value \( H_{c, n} \) (depending on specific MF), \( n_{s} \) is the saturated refractive index of MF at sufficiently high field, \( T \) is the temperature in Kelvin, \( H \) is the field strength in Oe and \( \alpha \) is the fitting parameter. We take the experimental refractive data of a MF from Ref. [44] as an example (sample thickness L = 200 μm, concentration of MF ϕ = 1.52 % or M s  = 0.85 emu/g, sweep rate of externally applied magnetic field dH/dt = 10 Oe/s, and ambient temperature T = 24.3 °C), which are depicted in Fig. 2 as solid symbols. For this MF, \( n_{0} = 1.46193 \), \( H_{c, n} = 12.98701 \) Oe, and T = 24.3 °C. Therefore, the magnetic-field-dependent refractive index at \( H > H_{c, n} \) can be written explicitly as

Fig. 2
figure 2

Refractive index of magnetic fluid as a function of magnetic field. The solid symbols and red solid line are the discrete data from Ref. [44] and modified Langevin function Eq. 2, respectively

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$$ n_{\text{MF}} (H) = \left( {n_{s} - 1.46193} \right)\left[ {{ \coth }\left( {\alpha \frac{H - 12.98701}{24.3 + 273.15}} \right) - \frac{24.3 + 273.15}{{\alpha \left( {H - 12.98701} \right)}}} \right] + 1.46193 $$
(2)

Fitting the experimental data to Eq. 2, we can get the red solid line in Fig. 2 and the following fitting parameters are obtained: \( n_{s} = \)1.49583 and \( \alpha \) = 6.97151. Figure 2 displays that the modified Langevin function can fairly describe the magnetic-field-dependent refractive index of the MF. By substituting the values of \( n_{s} \) and \( \alpha \) into Eq. 2, the refractive index of the MF at any magnetic field can be easily achieved. Table 1 lists the refractive indices of the MF we used at nine fixed magnetic fields according to Eq. 2.

Table 1 Refractive indices of the magnetic fluid in this work at several fixed magnetic fields calculated by the modified Langevin function Eq. 2
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To avoid the waveguide modes approaching the band edges and then easily coupling to the bulk modes of the PC (for instance, the actual modes may deviate from the designed ones due to the inaccuracy of the fabrication), it is desirable to design a PC structure with a relatively large bandgap. We find that when \( r = 0.24a \), the PC with square lattice has a considerable bandgap for TE modes (electrical field parallels the silicon rods). Figure 3a shows the typical band diagram for the TE modes when \( r = 0.24a \) and \( n_{\text{MF}} \) set at a moderate value (1.4854 at \( H \) = 150 Oe). The ordinate in Fig. 3a denotes frequency, which has been normalized to \( 2\pi c/a \) (c is the light speed in vacuum) and equals \( a/\lambda \), i.e., \( \omega_{N} = \omega a/( 2\pi c) = a/\lambda \). The solid lines in Fig. 3a are the bands of guided modes in the PC and the shaded region is the bandgap. It is clear from Fig. 3a that the designed PC exhibits a large TE bandgap in the range 0.2485 < ω N  < 0.2975 with a gap ratio (defined as the width of the full forbidden band to its central frequency) of 17.9 %.

Fig. 3
figure 3

Band structures for the TE modes of the designed photonic crystal without defect (a) and the projected dispersion diagram along the waveguide propagation direction of the corresponding line-defect photonic crystal (b) when \( r = 0.24a \) and \( n_{MF} \) = 1.4854 (\( H \) = 150 Oe). In (a), the shadow region denotes the forbidden band and the insets show schematically the structure in the real space and the first Brillouin zone in the wave-vector space. In (b), the insets show schematically the structure in the real space and the typical electrical field intensity \( \left| {E_{y} } \right|^{2} \) distribution at k N  = 0.15

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Removing a row of silicon rods along the propagation direction will form the line-defect PC waveguide (also called W1 waveguide) as shown in Fig. 1. The guided modes will be created within the bandgap of the PC, which is shown in Fig. 3b. The wave vector in Fig. 3b has been normalized to \( \sqrt 2 \pi /a \), i.e., \( k_{N} = ka/({\sqrt 2 \pi}) \). It is obvious from Fig. 3b that the defect-state mode with ω N in the range of 0.265–0.287 is well within the PC bandgap and simultaneously far away from the band edges. Also, there is a wide region of nearly constant slope of the waveguide mode dispersion curve, which is called the flat band as indicated in Fig. 3b. In this region (0.225 < k N  < 0.386 or 0.269 < ω N  < 0.281), the GVD of the waveguide mode will be very low and then gives rise to the wideband operation.

3 Results and discussion

To investigate the influence of externally applied magnetic field on the guided mode dispersion of the PC waveguide, the projected band diagrams of the PC waveguide at several magnetic fields are calculated using the data in Table 1. The results are depicted in Fig. 4a. Figure 4b, c shows the electrical field intensity \( \left| {E_{y} } \right|^{2} \) distributions of the guided modes at given points indicated in Fig. 4a.

Fig. 4
figure 4

Projected band diagrams of the photonic crystal waveguide at several magnetic fields (a) and the corresponding electrical field intensity \( \left| {E_{y} } \right|^{2} \) distributions of the guided modes at given points (b) and (c)

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Figure 4a implies that the dispersion curves shift to low frequency with increasing magnetic field. This is contributed to the increased refractive indices of MF with magnetic field. In addition, the shift in the low field regime is greater than that in the high field regime, which can be well explained by the saturated trend of the refractive indices of MF at relatively high magnetic field (see Fig. 2). For all cases, there is a wide flat band region faring away from their band edges, which are favorable for the wideband-low-dispersion slow light. The electrical field intensity \( \left| {E_{y} } \right|^{2} \) distributions in Fig. 4b, c reveal clearly that the mode energy is concentrated in the defect regions and propagates along z direction. Figure 4b displays that the electrical field intensity distributions have different patterns at different wave-vector points although the magnetic field is fixed at \( H \) = 75 Oe, which may imply the different propagation properties, while the same electrical field intensity patterns at a fixed wave-vector point (k N  = 0.26, point D, E and F within the flat band) but different magnetic fields in Fig. 4c may state the same propagation properties. Actually, all the dispersion curves around point D, E and F have the same properties of flat band. Similarly, the electrical field intensity pattern at point B (also within the flat band) is similar to those shown in Fig. 4c.

According to the projected band diagrams, the group index \( n_{g} = \frac{c}{{v_{g} }} = \frac{c}{{{\text{d}}\omega /{\text{d}}k}} = \frac{1}{{\sqrt 2 \cdot \left( {{\text{d}}\omega_{N} /{\text{d}}k_{N} } \right)}} \) can be calculated out. Figure 5 illustrates the group index of the flat band as a function of frequency at different magnetic field strengths. It is obvious from Fig. 5 that the \( n_{g} \sim \omega_{N} \) curves are U-shaped, which result in the wide range of frequency with almost constant value of \( n_{g} \)(around 10). This means the wideband-low-dispersion slow light may happen. Besides, the central frequencies \( \omega_{N0} \) of the U-shaped \( n_{g} \sim \omega_{N} \) curves decrease with the magnetic field strength, but their widths \( \Updelta \omega_{N} \) increase slightly with the magnetic field strength. Then, the normalized bandwidth \( \Updelta \omega_{N} /\omega_{N0} = \Updelta \omega /\omega_{0} = \Updelta f/f_{0} = - \Updelta \lambda /\lambda_{0} \) will increase with the magnetic field strength (also see Fig. 7).

Fig. 5
figure 5

Group index of the flat band as a function of frequency at different magnetic field strengths

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Using Fig. 5, the GVD \( \beta_{2} = \frac{{{\text{d}}^{2} k}}{{{\text{d}}\omega^{2} }} = \frac{1}{c}\frac{{{\text{d}}n_{g} }}{{{\text{d}}\omega }} = \frac{a}{{2\pi c^{2} }}\frac{{{\text{d}}n_{g} }}{{{\text{d}}\omega_{N} }} \) is easily obtained. The results are displayed in Fig. 6, which show the chair-shaped \( \beta_{2} \sim \omega_{N} \) curves. There is a wide range of very low GVD under every magnetic field. The magnetic-field-dependent properties of the \( \beta_{2} \sim \omega_{N} \) curves are very similar to those of \( n_{g} \sim \omega_{N} \) curves shown in Fig. 5.

Fig. 6
figure 6

Group velocity dispersion of the slow light mode as a function of frequency at different magnetic fields

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To further quantify the properties of the slow light and compare with other results in the literatures, the normalized delay-bandwidth product is employed, that is \( {\text{NDBP}} = \widetilde{n}_{g} \cdot \frac{\Updelta \omega }{{\omega_{0} }} = \widetilde{n}_{g} \cdot \frac{{\Updelta \omega_{N} }}{{\omega_{N0} }} \), where \( \widetilde{n}_{g} \) is the average group index within the bandwidth \( \Updelta \omega_{N} \). Therefore, it is critical to define the appropriate \( \Updelta \omega_{N} \) to obtain the correct NDBP and then do the comparison with the published work. We use the concept of flat ratio proposed by Hao et al. [17], which is given as \( \mu = \frac{{n_{g\hbox{max} } - n_{g\hbox{min} } }}{{\widetilde{n}_{g} }} \), where \( n_{g\hbox{max} } \) and \( n_{g\hbox{min} } \) are the maximum and minimum values of group index \( n_{g} \) within the bandwidth \( \Updelta \omega_{N} \). The flat ratio \( \mu \) is set at 0.2 corresponding to a maximum of 10 % \( n_{g} \) variation with respect to the average group index \( \widetilde{n}_{g} \) (consistent with Ref. [17]). For this small variation of \( n_{g} \) and considering the central-flat \( n_{g} \sim \omega_{N} \) curves, it is sufficient to use \( (n_{g\hbox{max} } + n_{g\hbox{min} } )/2 \) instead of \( \widetilde{n}_{g} \). Then, \( \mu = 0.2 \) leads to \( n_{g\hbox{max} } = \frac{11}{9}n_{g\hbox{min} } \) (\( n_{g\hbox{min} } \) is easily obtained from Fig. 5). The bandwidth is obtained as \( \Updelta \omega_{N} = \omega_{NH} - \omega_{NL} \), where \( \omega_{NH} \) and \( \omega_{NL} \) are the high and low frequencies corresponding to \( n_{g\hbox{max} } \). The central frequency of the bandwidth is expressed as \( \omega_{No} = (\omega_{NH} + \omega_{NL} )/2 \). Using this method, the NDBP of our designed structure at different magnetic fields are plotted in Fig. 7. The average group index \( \widetilde{n}_{g} \) and normalized bandwidth \( \Updelta \omega /\omega_{0} \) are also shown in Fig. 7.

Fig. 7
figure 7

Normalized delay-bandwidth product, average group index and normalized bandwidth as functions of magnetic field

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It is clear from Fig. 7 that the normalized delay-bandwidth product NDBP is almost constant and remains around 0.43. This value is very larger than that of other W1 waveguide with NDBP = 0.02 [18]. Even optimizing the structures by varying the size/position/shape of the holes/rods outside the defect region, NDBP is in the range of 0.14–0.35 [18, 46] and still smaller than this work. Moreover, optimizing the structures will incur the complexity and decrease the accuracy of fabrication. Recently, Üstün and Kurt have attained a relatively very large NDBP of 0.618 with optimization on a regular triangular lattice photonic crystal [47]. Figure 7 also indicates that the average group index \( \widetilde{n}_{g} \) decreases with the externally applied magnetic field, while the normalized bandwidth \( \Updelta \omega /\omega_{0} \) increases with the magnetic field. Actually, the down-shift of the flat band with magnetic field in Fig. 4 qualitatively implies that \( \omega_{0} \) decreases and then \( \Updelta \omega /\omega_{0} \) increases with the magnetic field. Both \( \widetilde{n}_{g} \) and \( \Updelta \omega /\omega_{0} \) tend to saturate at high field region, which is again due to the saturated trend of the refractive indices of MF at relatively high magnetic field. So, it is favorable to operate at low magnetic field to magnetically tune the slow light.

Figure 7 shows that the normalized bandwidth \( \Updelta \omega /\omega_{0} \) varies from 4.162 to 4.375 %. Then, the wavelength bandwidth centered at a wavelength of \( \lambda_{0} \) = 1,550 nm is in the range of 64.5110–67.8125 nm. This is about 179 times larger than that of the conventional W1 waveguide [18], 22 times larger than those of the microfluidic-infiltrated W0.8(W0.9) waveguides [29, 30] and 1.5 times larger than the best result through optimizing structures [17]. Meanwhile, the NDBP is still 21.5, 2.5(7.2) and 1.4 times larger than the compared structures, respectively.

According to our definition of bandwidth, the GVD \( \beta_{2} \) lies in the range from −1,191\( \frac{a}{{2\pi c^{2} }} \) to 855\( \frac{a}{{2\pi c^{2} }} \) and only slightly depends on the frequency and magnetic field strength within the bandwidth. This is about three orders of magnitude smaller than the previous work [46, 48], while the bandwidth and NDBP are still very large. Ebnali-Heidari et al. [30] have also achieved the low dispersion with the same order of magnitude using microfluidic infiltration technique, but the bandwidth is only several nanometers. On the other hand, Fig. 6 manifests that the GVD \( \beta_{2} \) is very sensitive to the frequency and magnetic field strength outside the bandwidth and its value can be either negative or positive. These characteristics may be exploited for dispersion-compensation or magnetically tunable dispersion-compensation applications.

We would like to point out that the group index of our system is not very high, though the bandwidth is extremely wide and the NDBP is comparably large. For instance, the group index of our currently designed structure is only around 1/9 of that of Hao et al.’s structure [18], 1/8 of that of Casas-Bedoya et al. [29] structure, and 1/2–1/11 of that of Ebnali-Heidari et al. [30] structure. For the cases needing high group index, one may engineer the band structures utilizing the methods proposed in the published work [17, 18]. Moreover, the properties and types of the MF, for example the concentrations and types of magnetic nanoparticles and carrier liquids, can be appropriately chosen to produce the desired band structures, which will result in high group index. As shown in Fig. 7, the magnetically tuning range of the slow light properties is limited. Therefore, it is suitable for finely tuning the slow light properties after finishing designing and optimizing the structures to set the slow light parameters in the rough range. Furthermore, we believe that use of more magnetically sensitive MFs (e.g., MFs with higher concentrations) can increase the tuning range of the slow light properties. Actually, there is much work concentrating on the dispersion compensation and tunability of slow light specifically, such as those of Baba et al. [49–52]. Finally, the concept of this work is easily extended to construct other structures consisting of external-stimulus-responsive (such as suitable electro-optical, thermo-optical, and photosensitive) materials to realize the online tunability of slow light.

4 Conclusions

In summary, wideband-low-dispersion slow light is proposed with a new kind of W1 waveguide. The designed structure is based on the 45o-rotaed square lattice photonic crystal made of silicon rods and MF background, which is simple and compatible with the photonic integration. The wide flat bands of the projected band diagrams lead to the U-shaped \( n_{g} \sim \omega_{N} \) curves and then the chair-shaped \( \beta_{2} \sim \omega_{N} \) curves. These create the extremely large bandwidth and ultralow-dispersion slow light. The achieved bandwidth is about 179 times larger than that of the conventional W1 waveguide, 22 times larger than those of the microfluidic-infiltrated W0.8/W0.9 waveguides, and 1.5 times larger than the best result through optimizing structures. The GVD is about three orders of magnitude smaller than the previous work in Refs. [45, 46]. The NDBP is comparably large and kept around 0.43. MF as an external-stimulus-responsive material provides the “in vivo” tunable trade-off between the \( \widetilde{n}_{g} \) around 10 and \( \Updelta \omega /\omega_{0} \) around 4.25 %. The structures and the parameters of the materials can be further adjusted to acquire the desirable slow light properties.