## Abstract

We demonstrate a concept for tailoring the group velocity and dispersion properties for light propagating in a planar photonic crystal waveguide. By perturbing the holes adjacent to the waveguide core it is possible to increase the useful bandwidth below the light-line and obtain a photonic crystal waveguide with either vanishing, positive, or negative group velocity dispersion and semi-slow light. We realize experimentally a silicon-on-insulator photonic crystal waveguide having nearly constant group velocity ~*c*
_{0}/34 in an 11-nm bandwidth below the silica-line.

©2006 Optical Society of America

## 1. Introduction

The intricate confinement of light in a photonic crystal waveguide (PhCW) [1] and its resulting dispersion properties offer sophisticated possibilities for realizing complex nanophotonic circuits. Potentially, PhCWs may facilitate delay lines for package synchronization, dispersion compensation, and enhanced light-matter interactions [2, 3] in nanophotonic circuits by exploiting slow-light phenomena [4]. However, the practical utilization of *ultra*-slow light reaching group velocities below ~*c*
_{0}/200 [3,5] in PhCWs may be limited due to an inherent small bandwidth [6], impedance mismatch [7, 8], intensified loss mechanisms at scattering centers [9], and extreme dispersive pulse broadening [4, 10]. Previously, it has been demonstrated that the dispersion properties of PhCWs can be altered via a structural tuning of the waveguide geometry, typically, by changing the waveguide width or by introducing bi-periodicity [11, 12, 13]. However, such dramatic changes of the PhCW may lead to multimode operation, decreased coupling efficiency to a photonic wire, and structural continuity problems in, e.g., bend and splitting regions. Here, we show how the knowledge of the field distributions in a single-line defect (W1) PhCW can be exploited to tailor the dispersion properties of the fundamental even photonic bandgap (PBG) mode. In this way, one can realize a silicon-on-insulator (SOI) W1 PhCW with semi-slow light having a group velocity in the range ~(*c*
_{0}/15–*c*
_{0}/100); vanishing, positive, or negative group velocity dispersion (GVD); and low-loss propagation in a practical ~5–15 nm bandwidth.

## 2. Design aspects & modeling

The group velocity *v*
_{g} of light with frequency *ω* in an optical waveguide is given as [14]:

where *k* is the wavevector along the waveguide and *n*
_{g} the group index. The dispersion relations *ω*(*k*) for a given waveguide system can easily be obtained numerically by using e.g. the plane wave expansion (PWE) method. Hereafter, it is straightforward to obtain *v*
_{g} by numerical differentiation according to Eq. (1). The group velocity in a PhCW is strongly dependent on the frequency [6] as quantified by the GVD parameter *β*
_{2}, given by the second order derivative of the dispersion relation [14]:

Figure 1(a) shows a typical band diagram for transverse-electric (TE) polarized light in a 2D W1 PhCW which is realized in silicon by arranging air holes of diameter *D*=0.6Λ in a triangular lattice with pitch *Λ*. The normalized dispersion relations are found for wavevectors *k _{z}* along the direction of the waveguide core [15], which is formed by introducing a single line-defect in the nearest-neighbor direction of the lattice. The inset of Fig. 1(a) shows the supercell used in the PWE calculation. As seen, the W1 PhCW supports one even (solid black) and one odd (dashed black) mode in the PBG which is located at frequencies ~(0.20–0.28) Λ/λ. The parities of the modes are defined by their in-plane symmetry with respect to the waveguide core. The even PBG mode is seen to flatten for

*k*≲0.3 and, eventually, obtains a zero slope at

_{z}*k*

_{z}=0.5. This is due to the folding at the Brillouin zone-end [1]. Figure 1(b) shows the resulting group velocity

*v*

_{g}for the even PBG mode (black) calculated by using Eq. (1). It reveals a dramatic reduction in

*v*

_{g}for increasing wavevectors (lower frequencies). As expected, the group velocity

*v*

_{g}≈

*c*

_{0}/4 is close to that of light propagating in a conventional silicon waveguide (with

*n*

_{Si}≈3.5) for

*k*

_{z}≳0.3. For

*k*

_{z}≳0.3,

*v*

_{g}decreases monotonically below

*c*

_{0}/20. Near

*k*

_{z}=0.5,

*v*

_{g}approaches zero, hence, addressed as the slow-light regime. Therefore, different frequencies will travel at very different speeds in a W1 PhCW. Correspondingly, the GVD parameter

*β*

_{2}(red) plotted in Fig. 1(b) increases by several orders of magnitudes from -10

^{4}ps

^{2}/km to an extreme negative GVD below -10

^{9}ps

^{2}/km in the slow-light regime. Such extreme GVD will distort any pulse-train propagating through the W1 PhCW. Hence, the monotonically changing group velocity along with the extreme GVD in the W1 PhCW will be devastating for the utilization of slow-light in e.g. a wavelength-division multiplexed system.

The knowledge of the modal field distribution can be exploited to tailor the dispersion characteristics of the W1 PhCW. Figure 1(c) shows the modal field distributions for the even PBG mode for three different wavevectors marked by the green, yellow, and red squares in Figs. 1(a) and 1(b). For *k*
_{z}≲0.3 with *v*
_{g}≈*c*
_{0}/4 (green square), the mode is seen to be well-confined in the waveguide core and the mode profile looks similar to that of a fundamental mode in a ridge waveguide. Hence, in this index-like regime the mode is usually referred to as being index-guided. Entering the slow-light regime (yellow square), the mode starts to penetrate into the photonic crystal cladding and, eventually, has its field highly concentrated in the first and second row of holes (red square). Thus, the properties of the even PBG mode in the index-like regime depend mainly on the parameters of the first row of holes, whereas the slow-light part of the mode is strongly dependent on the parameters of the photonic crystal cladding, especially, the first two rows of holes. A consequence of this is illustrated in Fig. 2, where the even PBG mode is plotted for a W1 PhCW with bulk hole diameter *D*=0.60*Λ* and various diameters (a) *D*
_{1} of the first row of holes, and (b) *D*
_{2} of the second row of holes. Clearly, the mode moves down in frequency in *both* the index-like and slow-light regimes when the diameter *D*
_{1} is decreased, as the effective index is increased locally. Moreover, the slope at large wavevectors increases in magnitude and the bandwidth, in which the dispersion relation is linear, increases in the slow-light regime. On the contrary, the mode is only affected at large wavevectors when the diameter *D*
_{2} is changed, as only the part of the mode in the slow-light regime ‘feels’ the second row of holes. By decreasing *D*
_{2} the slope of the tail increases in magnitude. Thus, the field distributions in Fig. 1(c) nicely document a frequency dependent behavior of the even PBG mode on the waveguide geometry and can be used as guidelines to selectively tailor the dispersion properties of a W1 PhCW in the slow-light regime.

One design goal for a practical utilization of slow-light in a W1 PhCW could be to obtain *semi*-slow light with a group velocity in the range ~(*c*
_{0}/15–*c*
_{0}/100) and a vanishing GVD (i.e., a linear dispersion relation) in a ~5–15 nm bandwidth. This can be achieved by combining the effects illustrated in Figs. 2(a) and 2(b). By decreasing *D*
_{1}, we perform a coarse tuning of *v*
_{g}, raising it to the desired range and increasing the bandwidth with linear dispersion for large wavevectors *k _{z}*. By changing

*D*

_{2}, we can fine-tune

*v*

_{g}for the tail of the mode, whereby we arrive at the desired group velocity and/or dispersion. Figure 3(a) shows a 3D calculation of the band diagram of the even PBG mode for three examples of perturbed W1 PhCWs. The diameter

*D*

_{1}has been

*decreased*by 30 nm (red), 40 nm (yellow), and 60 nm (green), whereas the diameter

*D*

_{2}has been

*increased*by 60 nm (red), 30 nm (yellow), and 10 nm (green) to obtain different group velocities and GVDs in different bandwidths.

In the PWE calculation, the pitch *Λ*=370 nm and the diameter of the bulk holes *D*=0.60*Λ*=222 nm. The structure is set to penetrate a 340-nm layer of silicon placed on a buffer layer of silica and having air above. For comparison, the even mode for the generic W1 PhCW is also plotted (black). As seen, the perturbations represented by the yellow and green curves result in regions with nearly linear dispersion relations situated below the silica-line (violet) and are, thus, vertically confined to the silicon slab. Figure 3(b, bottom) shows the corresponding group indices, *n*
_{g} [see Eq. (1)], calculated by numerical differentiation of the bands in Fig. 3(a). As seen, the two perturbed PhCWs have plateaus where the group indices are constant within 5% around ~40 (green) and ~60 (yellow) over bandwidths of roughly ~13 nm and ~9 nm, respectively. These group-index plateaus are in sharp contrast to the group-index behavior for the generic W1 PhCW (black) that rises monotonically to extreme values. Inspecting the calculated GVD *β*
_{2} values in Fig. 3(b, top), we see that the group-index plateaus lead to relatively low and positive GVDs on the order of 10^{5}–10^{6} ps^{2}/km (yellow and green curves), which is not much larger in magnitude than in the index-like regime. A finer tuning of the geometry may lead to vanishing GVDs. Furthermore, we see that a perturbation of the W1 PhCW geometry also allows us to realize a PhCW having an extreme *positive* GVD (red curve), i.e. having a negative dispersion [14].

## 3. Fabrication and experimental results

Figure 4 shows a typical scanning electron micrograph of a fabricated 10-µm long perturbed W1 PhCW, where the diameter *D*
_{1}/*D*
_{2} has been decreased/increased relative to the bulk holes with diameter *D*=234 nm. The triangular photonic crystal lattice with pitch *Λ*=370 nm penetrates the top 338-nm silicon slab of an SOI wafer having a 1-µm thick buffer layer of silica. The photonic crystal patterns were defined by using electron-beam lithography and transferred to the silicon by inductively coupled plasma reactive ions etching.

Figure 5(a, right) shows the transmission spectrum (gray) measured for TE-polarized light in a 500-µm long perturbed PhCW where the diameters *D*
_{1}/*D*
_{2} have been decreased/increased by ~54 nm/~9 nm. The band diagram for the fabricated structure is plotted in Fig. 5(a, left) with a ~1% blue-shift (solid black). The two transmission peaks of the measured spectrum can easily be correlated to the two linear dispersion regions for the even PBG mode located above (dotted red) and below (dotted green) the silica-line. The group index *n*
_{g} for light propagating through the 500-µm long perturbed PhCW has been measured experimentally and calculated numerically by using the time-of-flight methods as described in Ref. [16]. Figure 5(b) plots the group index obtained from measurement (black), 2D Finite-Difference Time-Domain calculation (FDTD, blue), and 3D PWE-calculation (red). The 2D FDTD group-index spectrum has been blue-shifted ~12.5% to match the experimental wavelength scale. The group index for the ~11-nm broad transmission peak corresponding to the linear part of the even PBG mode below the silica-line (indicated by the dotted green lines in Fig. 5) has been measured to *n*
_{g}=34±3, which is in close agreement with the values predicted by FDTD and PWE. The obtained uncertainty below 10% of the measured group-index is better compared to the uncertainty obtained in Ref. [16]. This is due to a lower loss and an increased optical path length, i.e. a longer absolute time-of-flight, for the PhCW in the present investigation. The fringes in the FDTD-calculated spectrum are caused by the finite length [8] of the 20-µm long PhCW used in the simulation. The PWE group-index plateau in Fig. 5(b) reveals a positive GVD parameter *β*
_{2} on the order of 10^{5}–10^{6} ps^{2}/km, which is of opposite sign and several orders of magnitudes lower than *β*
_{2} in the slow-light regime of a W1 PhCW [see Fig. 1(b)]. The propagation loss for the perturbed PhCW has been measured with a typical error of 2–3 dB/mm using the cut-back method with four PhCWs of lengths 50, 100, 500, and 1000 µm and is shown in Fig. 5(b) (green curve). As seen, the propagation loss in the ~11-nm wavelength range is less than ~20 dB/mm and drops below 5 dB/mm in a ~2-nm bandwidth from ~(0.2385–0.2388) Λ/λ. We have fabricated a different set of perturbed PhCWs having a ~7-nm group-index plateau with *n*
_{g}=17±2 and showing a propagation loss of 4.2±1.2 dB/mm. The advantage of the perturbed PhCWs is that the losses induced by the silica-line and extreme dispersion are diminished in comparison with W1 PhCWs [17]. Thus, perturbed PhCWs are very promising for the SOI configuration as they offer bandwidth, slow-light, and low losses.

## 4. Conclusion

The field distributions in a single-line defect photonic crystal waveguide have shown to be useful guidelines for tailoring the dispersion relation of the fundamental, even photonic bandgap mode. The tailoring is obtained by doing a simple perturbation to the diameters of the first and second row of holes adjacent to the waveguide core. This general concept can be applied to photonic crystal waveguides of any design. In this way, it is possible to realize waveguides having low-loss bandwidths with semi-low group velocity, and vanishing, negative, or positive group velocity dispersion. We have realized a W1 photonic crystal waveguide in silicon-on-insulator material having an ~11-nm bandwidth below the silica-line with a nearly constant group velocity ~*c*
_{0}/34 and relatively low and positive group velocity dispersion with *β*
_{2} on the order of 10^{5}–10^{6} ps^{2}/km. Within this bandwidth, the measured propagation loss is less than 20 dB/mm and drops below 5 dB/mm in a ~2-nm bandwidth.

With tailored group velocity and dispersion, photonic crystal waveguides become much more attractive for practical applications. The perturbed waveguides offer novel possibilities for realizing compact integrated components used for e.g. pulse shaping, dispersion compensation, and enhanced non-linear effects. They may also help to understand how propagation losses in photonic crystal waveguides scale with group velocity and dispersion.

## Acknowledgment

This work was supported in parts by the Danish Technical Research Council via the PIPE (Planar Integrated PBG Elements) project and by the New Energy and Industrial Technology Development Organization (NEDO) via the Japanese Industrial Technology Research Area.

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