1. Introduction

Endeavoring to reduce the speed of light may seem to be against the high speed optical communication occurring at close proximity to the speed of light. However, in some circumstances it is imperative to manipulate the speed of light in a controlled manner. For example, nonlinear optics, optical memories, buffers, delay lines and electro- and magneto-optic effects may gain benefit from slow light phenomena due to enhanced light-matter interaction and spatial compression of optical pulses [1–6]. Consequently, the concept of tailoring the speed of light has been an intense research area. The approaches in terms of underlying fundamentals can be mainly grouped into two categories. The first group utilizes the material dispersion in atomic media or solid-state systems. The second category belongs to the structural dispersion of specially designed photonic devices [7–9]. The pioneering studies in the field have been exploited the first type of slow light mechanism which achieved a record breaking speed of 17 m/s [10]. The two major problems associated with these approaches are the limited bandwidth nature and the practical constraints due to the requirement of special environments [11]. Carefully engineered dielectric structures show great promise to tame and manipulate the speed of light by lowering speed many orders of magnitude even though that value is still far beyond the achievement of the approaches belonging the first group.

The beauty of reducing speed of light based upon structural dispersion comes from the relatively easy engineering of the complex structures at will. Photonic crystals (PC) are periodic dielectric structures that may exhibit band gap phenomena so that we can modify them by inducing defects of certain type to obtain cavities and waveguides [12]. As a result, light can be confined in the desired regions or guided through certain symmetry directions. Photonic crystal waveguides (PCW) show dispersive characteristics so that the group velocity varies enormously within a certain k-space. The two regions which have low group velocities are near the Brillouin zone edges, namely, $(k_x,k_y)$ = $(0,0)2\pi /a$ or $(0,0.5)2\pi /a$. But near these points, band shape resembles quadratic behavior which naturally accompanies large group velocity dispersion (GVD) and small bandwidth. As a result, the optical signal gets distorted after propagating a certain distance [13]. One solution to tackle with the GVD lies behind the idea of obtaining flat bands. So having high group index $(n_g)$ values with small GVD and large bandwidth are important research objectives [14–29] because of the diverse applications of slow light as mentioned above. One crucial aspect that should be kept in mind while obtaining flat bands with constant slopes is the relatively simple structures that can be easily realized. In this work, we obtain different families of dispersion diagrams which ensure large and constant group index in addition to high bandwidth operation. This aspect of the work consists of one part of the study.

The second part as explained later in the paper is to explore systematically the trade-off relation between bandwidth and group index. We emphasized the importance of side rows of holes to reduce the speed of light. The band shape changes dramatically when the innermost holes’ radii reach a certain value as the rest of the background holes’ radii are kept constant. An inverse relation persists between the bandwidth and average group index. One may have either a large group index value of 590 or a small value of 14 with the corresponding normalized bandwidths of $5.35\times {10}^{-5}$ and 0.011, respectively for a certain filling factor. As we can see later, while one parameter diminishes by more than one order of magnitude the other parameter increases by more than two orders of magnitude. In the previous studies, attention was given to the flat-band creation and a figure of merit is defined as the product of “average group index and bandwidth”. The trade-off relation between the two parameters was mentioned in previous studies but none of them reported an exponential dependency. Flat-band occurs at two different spectral regions, one is close to dielectric band and the second region is close to air band. In between the two regions, there is a slow light waveguide mode whose band slope is almost zero. This in turn yields the most dispersive mode but at the same time the largest group index. The optimization of group index and bandwidth product was carried out in one of the flat bands region and a value of approximately 0.30 is obtained. The finite-difference time-domain (FDTD) method is used to verify propagation of slow light pico-second optical pulse [30]. An appropriate bandwidth of optical pulse (12 nm) is at 1550 nm selected and relevant parameters are extracted by the time-of-flight method. Both index and band gap guided modes are excited and the pulse separation in time domain was observed.

2. Slow light photonic crystal waveguide structure and frequency domain analysis

The intact photonic crystal (PC) used in this paper is a two dimensional structure with triangular lattice air holes in dielectric background as shown in Fig. 1 . The air hole radius is $r_b=0.30a$, where a is the lattice constant and background refractive index is 3.46. The symmetry directions are also shown in the figure. The waveguide is constructed by filling the center holes with the same background dielectric material along $\mathrm{\Gamma }\mathrm{{\rm K}}$ direction. The dispersion diagram of the PCW as shown in Fig. 2 is calculated by plane wave expansion method [31] using MIT’s freely available software MPB. The waveguide mode of even type is represented by a solid line. The odd mode is also included in the figure as a dashed-line. The group index value that corresponds to the waveguide mode is demonstrated in Fig. 3 . The group index is defined as $n_g=c/v_g$, where c is the speed of light. The group velocity as represented by $v_g$ is the first order derivative of waveguide mode, i.e., $v_g=\left|\partial \omega /\partial k\right|$. Due to the polarization sensitivity of 2D PCs only TE modes (electric fields are in the plane) are considered in the study.

 

Fig. 1 The representative picture of the investigated slow light photonic crystal waveguide. The holes inside the rectangles with dashed lines are subject to size variation. Their radii are represented by$r_d$. The radii of the remaining holes in background are $r_b=0.30a$ and the relative permittivity of dielectric material is 12. The two insets on the right and left hand sides show the two defined variables, $r_b$and$r_d$, respectively.

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Fig. 2 The defect bands of the dispersion diagram of PCW. Solid- and dashed-lines correspond to even and odd modes, respectively. The radii of holes in the first rows and background are $r_b$ = $r_d$ = 0.30a.

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Fig. 3 The group index spectrum of photonic crystal waveguide with respect to normalized frequency.

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Considering the out-of-plane losses, we focus on only region of interest in the reciprocal space where mode stays under the light line. This part of the region is apparent if we only present dispersion diagrams within the interval from $k=0.25(2\pi /a)$ to band-edge. Figure 3 shows group index variation of a regular PCW. As evident from this plot, it diverges very fast at the close proximity of cut-off frequency. The two main problems of such a waveguide mode are the narrow bandwidth operation and high GVD. The way of eliminating these problems lies behind the idea of obtaining flat-bands (non-zero, constant slope). Flat bands with small slopes provide constant group index and have very low GVD.

As the field distribution of the guided mode is highest in the center of the waveguide and decreases along transverse to propagation direction, the first rows of holes affect the field distribution and the dispersion diagram more than the second and third rows. Therefore, in this study we modified the side rows holes’ size and changed symmetrically their radii along the waveguide.

The dispersion diagram of PCW contains various curves as shown in Fig. 4 when we change the first row of holes’ radii from $r_d=0.20a$ to $0.4750a$ with an incremental step of 0.0125a. The background holes radii are kept constant at $r_b=0.30a$. Within the resolution of our parameter scan, we realized that when hole radius is around 0.36a and beyond that first flat-band region emerges in the dispersion diagram. For the values of hole radii between 0.30a and 0.36a, the waveguide mode is very dispersive and there is no signature of flat-band region. The second flat-band regime occurs when $r_d$ takes values around 0.20a. Most of the previous studies targeted to tune geometric parameters within a small interval. Hence, some of the general trends that may be captured were unrevealed. As a result, large structural variations reveal important observations. The first observation in the dispersion diagram which is also a trivial one is the bands movement to higher/lower frequencies as we increase/decrease the side row of holes’ radii. The band flattens as the waveguide mode comes close to continuum of dielectric band. The flatness disappears and mode becomes highly dispersive at intermediate values of$r_d$ due to hyperbolic forms of bands. We will demonstrate later in the paper that, there is always a turning point ($r_d=0.35a$, designated with a marker in the figure) as we change the filling factor of the PC by altering $r_d$. This is the second important observation that can be drawn from the dispersion diagram. The slope of the band is almost zero at the turning point and below/above this value the bands have negative/positive slopes. The upward movement speed increases even though the amount of increment is kept the same. The three spectral intervals are roughly separated by double-arrows in Fig. 4. While the lower and upper parts contain flat-band regions suitable for slow light application the middle part is highly dispersive.

 

Fig. 4 Evolution of dispersion curves of photonic crystal waveguide with changing boundary holes’ size. The background holes have radii of $r_b=0.30a$ and first row of holes’ radii are varied from $r_d=0.20a$ to$0.4750a$.

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To gather group index variation of the bands in the three regions, we prepared Fig. 5 . The band which has the smallest slope is also indicated with a marker. There are two interesting regimes in the figure in terms of slow light perspective. Bands with constant group indices appear both at lower and upper parts of the spectrum. The first one is close to dielectric band continuum states. On the contrary, the second part is close to continuum states of the air band. The same types of lines correspond to same modes in the plots of $\omega (k)$ and $n_g(\omega )$ in Figs. 4 and 5, respectively. Hence, the two figures complement each other. The useful low dispersion area in Fig. 4 can be easily detected by inspecting Fig. 5. The constant group index regions plotted with red-dotted lines imitate a U-shaped behavior which is the region of low GVD. We selected three representative points (shown with arrows) in the group index spectrum where the corresponding group indices are 27.9, 43.4 and 10.4, respectively to investigate e-field distributions. The result is shown in Fig. 6 . The e-field penetrates more towards transverse to propagation direction in Fig. 6(a) than the rest. The field components reach up to the 4th row. This is because of the relevant mode closeness to dielectric continuum states. On the other hand, it decays very quickly over the 1st and 2nd rows for the other two cases in Figs. 6(b) and 6(c). The unique behavior of e-field distributions for different bands tells us that one may play with the rows up to 3rd or 4th row to engineer slow light in lower part of the spectrum. However, 1st and 2nd row have the most pronouncing role to manipulate mode shape for slow light in the upper part of the group index spectrum.

 

Fig. 5 Group index spectrum of bands in Fig. 4. The three arrows show the spectral points whose electric field distributions are presented in Fig. 6. The double arrow indicates the interval of a group of modes that have large GVD.

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Fig. 6 Electric field distributions of three cases are prepared. (a) $(r_b,r_d)=(0.30a,0.20a)$, (b) $(r_b,r_d)=(0.30a,0.35a)$ and (c) $(r_b,r_d)=(0.30a,0.4625a)$. The wave vector is fixed at $k=0.40(2\pi /a)$for the three cases.

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The nice behavioral change between group index and bandwidth in the upper part of the group index spectrum (Fig. 5) directed us to extract quantitative values. We re-produced group index variation for $r_d$ changing from 0.3625a to 0.45a keeping $r_b$ fixed at 0.30a. Different y-scales are used in Figs. 7(a) and 7(b). From these plots, constant-group index and the corresponding bandwidth values are extracted. We can see that there are constant group index regions with different bandwidths. The $n_g$ values are U-shaped and the bottom of this U shape is very appropriate for obtaining constant group index with small GVD and high bandwidth. We extracted the value of minimum group index and bandwidth from Fig. 7. The bandwidth is defined as $\Delta \omega /{\omega }_0$, where $\Delta \omega$ is the difference between maximum and minimum frequencies and ${\omega }_0$ is the center frequency of the U-shape. The maximum and minimum frequencies are determined so that corresponding $n_g$ values are within the 10% of the minimum group index. The corresponding variations of bandwidth and group index are shown in Figs. 8(a) and 8(b), respectively. As we can see, while bandwidth increases exponentially, the group index decreases also exponentially but with a different decay rate.

 

Fig. 7 (a) Group index vs. frequency plots for $r_d$values ranging from 0.3625a to 0.40a. (b) Group index vs. frequency plots for $r_d$values from 0.4125a to 0.45a.

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Fig. 8 (a) The bandwidth variation corresponding to an interval such that group index does not exceed 10% of the local minima. (b) The constant group index dependency with respect to holes’ radii.

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Since the group index and bandwidth values are inversely proportional, we had to determine a figure of merit (FOM). We first evaluated average group index and bandwidth values and then defined the product of them as a FOM. The result is shown in Fig. 9 . The FOM increases almost five times reaching a value of 0.16 as we increase air holes radii of the first rows from 0.3625a to 0.45a.

 

Fig. 9 The average group index and bandwidth product dependency to boundary hole size variation.

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The GVD graph calculated by evaluating ${\partial }^{\text{2}}k/\partial {\omega }^2$ is depicted in Figs. 10(a) and 10(b) for radii of holes 0.3875a and 0.45a, respectively. The low GVD reduces down to the order of 10⁵ ${\text{ps}}^{\text{2}}\text{/km}$. We should note that GVD changes symmetrically with positive and negative parts which can be used for dispersion compensation application.

 

Fig. 10 The group velocity dispersion relation of the slow light PCW for (a) $r_d=0.3875a$ and (b) $r_d=0.45a$.

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3. Time domain analysis of slow light propagation in photonic crystal waveguide

In this section, the slow light results are reinforced by the time domain study using FDTD method. It allows us to visualize slow light propagation inside PCW. The structure length is taken to be 300a. The computational domain is surrounded by perfectly matched layer boundary condition. Different detection points are placed throughout the structure and the distance between each detection point is 32a. We made simulations on the structure with side row radii equal to 0.4375a. The central frequency of Gaussian source is fixed at $\omega a/2\pi c=0.2452$with a normalized bandwidth of 0.001. We excited the slow light mode by sending a tilted input source. After a certain distance which is proportional to the difference between the slopes of the waveguide mode at two different k points, the separation of the input pulse into fast (index-guided) and slow modes (band-gap guided) is easily observable.

The group index extraction is performed by tracing arrival times of the central peaks of the slow light. The lattice constant a is fixed at 380 nm targeting the optical regime of 1550 nm. The bandwidths of the input pulse and constant group index region are approximately 6.3 nm and 12.6 nm, respectively. The pulse is allowed to propagate for more than 10000 a/c time. By measuring delay time of the pulses at different locations, we obtain Fig. 11(a) . The slope of the curve which approximates FDTD data gives group index value of $n_g=22.85$. The delay time information of the first detection point is scaled to zero time to take it as a reference. This number is very close to the frequency domain result of $n_g=20.59$. It takes 8.69 ps for the input pulse to travel 300a length PCW. The delay time can be increased either by using longer PCW or using different rod radii value. By using different rod radii for innermost holes, larger group index value can be attained at the expense of narrowing the bandwidth.

 

Fig. 11 (a) A part of the computational domain employed in the FDTD study is shown as an inset in the figure. The side rows of holes radii are 0.4375a. The length of the PCW is 300a and the distance between two detection points is 32a. Time domain data are stored at equally placed detection points along waveguide centerline. Lines with circular and square markers indicate slow light’s and fast light’s delay time data, respectively. (b)-(e) Time domain optical pulse evolutions of slow light at four selected propagation distances.

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Figure 11(a) also indicates time delay information for fast light. As can be seen from Fig. 7(b), fast light has a smaller group index (lower tail part of dispersion line). As a result, the slope of the curve corresponding to fast light is almost three times smaller than slow light. The temporal evolutions of slow light optical pulses while propagating down the waveguide are presented in Figs. 11(b)–11(e) at four selected detection points, 32a, 96a, 160a and 224a, respectively. The absence of broadening and distortion in the pulse waveforms are an indication of negligible GVD. The fast light reaches the end of the waveguide earlier than slow light. Hence, the reflected portion of it interferes with the traveling slow light. As a result, a small broadening in the pulse form appears only in Figs. 11(d) and 11(e).

The pulse propagation through PCW is presented in Fig. 12 . There are two detection windows used for the observation of light propagation, one is placed in the middle of the waveguide while the other one is located at the end of the waveguide. First fast light appears at the observation plane as expected due to small group index as compared with the slow light which has almost three times larger group index. As can be seen from Fig. 12(a), e-field distribution resembles well the index guided mode pattern. After certain time delay, the slow light arrives with distinct field patterns such as diffusing more towards the transverse direction and appearing beating patterns along the longitudinal direction as indicated in Fig. 12(b). To prove further the dispersion free propagation of the slow light, we provided another snap-shot of the e-field of slow light at the end of the waveguide. That means the field in Fig. 12(b) is allowed to travel almost a distance of 300a to reach the end of the waveguide. It can be seen from the plot in Fig. 12(c) that the pulse propagation does not suffer from any group velocity dispersion. Media 1 shows the optical pulse propagation along the waveguide.

 

Fig. 12 (Media 1) Light propagation through PCW. (a) The time snap-shots of the fast and slow light are shown in (a) and (b), respectively. (c) The spatial field distribution of slow light in (b) after propagating until the end of the waveguide.

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4. Optimization of figure of merit and loss analysis

The previous analyses were performed by keeping background holes’ radii at $r_b=0.30a$. It is worth considering the other cases where $r_b$ takes values smaller and larger than 0.30a. As a result, we consider the change in the figure of merit when we scan different combinations of $r_b$ and $r_d$.

Figure 13(a) shows the color-map plot of average group index and bandwidth product with respect to $r_b$ and $r_d$. The FOM value reaches up to 0.30 at the upper right corner of the figure where $r_b$ and $r_d$ take larger values, 0.42a and 0.48a, respectively. To underpin the constituent parameters variation clearly, we show group index and bandwidth variation along the dotted line which corresponds to $r_b=0.36a$in Figs. 13(b) and c. The insets show the enlarged view of the selected portions. When group index takes the largest value for certain $r_d$, the corresponding bandwidth is at the smallest value. Hence, defined FOM is a signature designating the inverse relation between the two parameters. The optimizations were performed only considering the first two rows of holes. It may be possible to enhance FOM further if one considers also the role of second row of holes. If higher FOM is sought for, one should use larger $r_b$ and $r_d$ values. On the other hand, larger group index with small bandwidth is available for various $r_b$ values when $r_d$ is kept relatively smaller. FOM values in the previous studies can be as low as on the order of 0.01 [28] and may increase up to 0.85 by judicious waveguide design [26]. The FOM of the present work as compared to them stays within an intermediate value.The parametric scan of radii allowed us to identify one additional feature of slow light PCW. For each radius of background holes, there is a certain $r_d$ value of first row of holes where the bands have the smallest slope (close to zero within the numerical resolution). The U-shaped group index region starts to appear above this $r_d$ value. Table 1 shows the specific $r_d$ values which are transition regions between lower flat bands and upper flat bands. The modes are highly dispersive at and around these $r_d$ values. When we increase $r_b$ from 0.28a to 0.38a, the turning points first decrease and then start to increase. Table 1 informs us which region to search for slow light with flat band if we fix the filing factor of PCW. The turning point is reminiscent of the existence of lower flat band. As we increase $r_b$ above 0.38a, continuum states of dielectric band intervene strongly with the waveguide mode and lower flat band disappears. Hence the trend in Table 1 can be presented up to a certain $r_b$ value.

 

Fig. 13 Average group index and bandwidth product map of slow light photonic crystal waveguide. The filling factor changes with $r_b$, and two sides air holes are altered by parameter $r_d$. (b) and (c) indicate group index and bandwidth variation, respectively when $r_b$is fixed at 0.36a.

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Table 1. Turning point variation

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A detailed loss analysis is out of the scope of the present work. However, to have an idea on the performance of the proposed slow light structure under the loss, we performed some additional work. Among the various loss mechanisms that may occur in PCW, we focused on the loss due to backward scattering. For this purpose, we introduced randomly varying hole diameters within an interval of 0.4375a±0.01a. Such randomness may occur as a consequence of an imperfect structure fabrication. The background radii of holes are kept constant at $r_b=0.30a$. The expected group index value of the waveguide without inclusion of loss parameter is $n_g=22.85$. However, uniformly distributed and randomly varying hole diameters’ should modify the dispersion curve. As a result, group index may increase or decrease correspondingly. The length of the perturbed part within the PCW is varied up to 40a while keeping the rest of the structure unchanged. The result is presented in Fig. 14 . The estimated loss coefficient extracted within the limited resolution of FDTD discretization is approximately 0.05dB/a. At first glance, such a loss value may seem to be high if a comparison is made with the literature [14]. However, careful examination of the way of introducing fabrication loss component into the current analysis provides a reasonable explanation. The holes’ radii of first rows are artificially perturbed from the original value, 0.437a to 0.4275a and 0.4475a (extreme cases). In these situations, dispersion analyses yield constant group indices of the following values 26.92 and 14.56 with different band widths. As compared to the loss free case, group index change occurs within a large interval of 20.59 ± (~6). Besides, slow light waveguide mode moves to higher/lower frequencies and in some cases there may be no overlap between the constant group index regions. As a result, slow light mode excited for $n_g=20.59$ faces with a huge scattering loss due to group index mismatch and band movement. Uniformly distributed perturbation parameters also exaggerate the fabrication disorder loss. Instead, a more realistic distribution such as Gaussian distribution can be adapted. Finally, field distribution of the slow light mode plays a crucial role for the loss amount. Strongly concentrated fields especially around the perturbed area are more amenable to high loss. It is possible to tailor the disorder induced losses in slow light waveguides by manipulating appropriately the locations of the e-field concentration [32]. The electric field concentration should be distributed at regions where disordered places do not overlap with the high intensity of the field. The out of plane losses in PCWs will not be a major concern as long as the operating spectral region is selected to be below the light line [33,34]. In the present work, slow light region appears below the light as can be seen from Fig. 2.

 

Fig. 14 The logarithmic plot of the disorder induced losses versus length of slow light waveguide. The solid line is a fitting curve to discrete data points.

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5. Conclusion

In conclusion, we have studied tuning of group index and bandwidth values by means of obtaining flat dispersion curves in a simplistic manner. We performed detailed analysis to engineer the dispersion and show that complete scan of parameters is very beneficial for slow light concept to identify and categorize different spectral trends that appear in dispersion diagrams. The frequency domain calculations confirm that when the radii of the side rows of holes increase above certain value for each $r_b$ then unique properties in the waveguide mode appear due to the interaction with continuum states (from below/above: dielectric/air band). The inverse exponential relation between group index and bandwidth is explored. The defined figure of merit shows increment with respect to increase in the radii of the side rows of holes reaching a value of approximately 0.30. The group velocity of 0.0017c is achieved in the modified photonic crystal waveguide. The transmissions of pico-second optical pulses were realized by time domain study. The presented results obtained by simple geometrical modifications via detailed parameter scan guide us how to obtain large bandwidth, constant group index and low GVD for slow light applications.