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Photonic crystal tapers have been designed for coupling of light from ridge waveguides into low group velocity photonic crystal channel guides. The coupling efficiency is increased from 3 % (case of butt-coupling) to 97 % for frequencies in the band-edge region, corresponding to a group index close to 100, as predicted using 2D finite-difference time-domain simulations.
©2007 Optical Society of America
1. Introduction
Slowing down of light has been an area of increasing research for a number of years (see e.g. ref. [1]) – and its possible applications include building devices for all-optical computing and telecommunication systems, e.g. optical buffers and devices that use enhanced non-linearity. This approach is particularly relevant to semiconductor chips, since such chips support the slow-light functionality in a practical way. Dispersive periodic systems, such as gratings, coupled cavities and photonic crystals (PhC) can be used to manipulate the properties of propagating light, with the possibility of making light slow down [2–7]. In particular, PhCs present a diversity of configurations that offer this possibility. A PhC channel guide, e.g. as obtained by removing one row of elements in a 2D lattice, can easily present a region of frequencies in which guided waves have very low group velocities. However, coupling of guided light that has ‘normal’ velocity behavior into such a slow-light channel guide remains a challenge, because of the mismatch between the two modes [8–10]. Here, we present the simulation of a coupling device made from a PhC taper [11–14], which increases significantly the transmission properties of the coupled guide system.
2. Butt-coupling
PhC channel guides can be realized by removing one row of holes in a 2D triangular lattice of holes in a semiconductor waveguide (along the PhC ΓK direction) – and are then called W1 channel guides. The dispersion curves of the guided modes within the photonic band-gap (PBG) can be modified by altering the properties of the PhC. In particular, bands can be shifted in frequency by modifying the PhC channel effective width [15,7]. Here, we bring the two PhC regions closer, leaving a channel of width x.a√3 (taken between the centre of holes, a being the period). It is referred to later as a Wx channel guide (and the earlier W1-width also matches this definition). We have carried out simulations, using a 2D finite-difference time-domain (FDTD) approach, on a W0.8 channel guide created in a 2D PhC triangular lattice of holes in a Si/SiO2 heterostructure with a slab effective index of 2.98 for TE polarization (E in plane), with a filling factor of 0.35 – and a period of a = 425 nm. This configuration shifts the band-edge region where low group velocity occurs to a spectral region very close to 1.52 μm, which is our target wavelength. Such a channel guide, 16 periods long, is accessed by a ridge waveguide of (variable) width d, in its fundamental mode (Fig. 1). Input and output detectors are located on the input and output ridge access waveguides, at a distance of 4a from the PhC edges [see Fig. 1(b)] or at the equivalent positions for the case of the ridge [Fig. 1(a)]. The detectors measure the overlap integral with respect to the source in order to see how much light will recouple in the output ridge fundamental mode. The spatial discretization step used in the FDTD simulations is 10 nm.
Fig. 1. Schematic of the PhC channel guide of reduced width W0.8 accessed by ridge waveguides (width d). (a): reference ridge waveguide, (b): butt-coupling, (c): smooth transition with added linear PhC tapers, (d): smooth transition with added exponential PhC tapers. (e): example of fabrication of linear tapers in silicon on insulator.
Figure 2 shows the transmission of this device structure as a function of wavelength, together with the group index of a pulse traversing it. This group index is calculated by measuring the time of flight of the pulse, considering its position in time as the one given by the centre of gravity of its power profile. The calculation of the group index associated with the W0.8 channel guide is performed as follows: firstly, the group index of the input and output ridge access waveguides is calculated, using a reference ridge waveguide [Fig. 1(a)]. Then the group index of the PhC W0.8 channel guide is extracted by using the output detector of Fig. 1(b) and the input detector of Fig. 1(a) (so that no reflection is collected at the input detector) and subtracting the time spent traveling in the ridge access waveguide parts (over a length of 4a+4a). It can be seen in Fig. 2 that the transmission decreases progressively towards the band-edge whereas, simultaneously, the group index increases. However, high group indices can only be obtained at very low transmission coefficient values [9,5]. Ripples are also visible in transmission close to the band edge due to Fabry-Pérot effects between the ridge-PhC interfaces. The difference between the estimates for group index for the two pulses lengths in Fig. 2 arises because of the strongly dispersive situation near the band edge. The shorter (187 fs) pulse both spreads in time and develops substantial asymmetry. The centre-of-gravity estimate for the group index is dependant on the initial pulse length. We believe that the estimate obtained from using pulses of greater length (1.49 ps) is better in the wavelength range above 1.49 μm.
Fig. 2. Transmission through a PhC W0.8 channel guide accessed by a 425 nm (0.577a√3) wide ridge waveguide, and group index of ridge waveguide and PhC channel guide obtained from two pulse durations (power FWHM): 1.49 ps and 187 fs (2D FDTD).
We have firstly carried out an optimization process on the butt-coupling, obtained simply by adjusting the access ridge waveguide width. The results appear in Fig. 3 (only part of the simulated widths are shown) – and show an optimal width of d = 0.4a√3 (= 294 nm). Yet, the transmission remains low at the band-edge.
Fig. 3. Transmission through a PhC W0.8 channel guide for different widths of ridge access waveguide (2D FDTD) – case of butt-coupling.
3. Tapered transition
The key problem for obtaining high transmission arises from the difference, at this abrupt transition, between the mode profiles, and the effective indices, of the access ridge waveguide and the slow-light PhC channel guide. Here we propose, as already suggested in our previous paper [12], to operate a smooth transition by progressively narrowing a PhC channel guide, which will shift slowly the dispersion curve in the PBG region, allowing a progressive change of group velocity and effective index for a given frequency. This approach forms a PhC taper where rows of holes perpendicular to the direction of propagation are progressively shifted laterally. Our tapers begin at the W1 channel size and finish at W0.8, which is the size of the slow-light PhC channel guide (Fig. 1). W1 was chosen because it is a ‘natural’ size (i.e. corresponding to an integer value shift of the lattice) and has a group velocity close to that of the ridge waveguide. The mismatch between the access ridge guide and the W1 PhC guide at its entry is quite small, and can be optimized on its own [16,17,9]. Several kinds of taper were investigated: namely linear and exponential ones, for different lengths: 8a, 16a, 32a. The linear taper has a simple and ‘natural’ shape. The exponential taper has a width reduction that becomes smaller and smaller in going from its entry to its exit, thus providing a smoother transition where it is most needed, i.e. when reaching the slowest mode regime. The exponential shape is defined as the normalized function
with k = 7, in the x-y coordinate reference system, where the points A and B have the coordinates (0,0) and (1,1) respectively [see Fig. 1(d)]. The comparison of the coupling performances is shown in Fig. 4. These tapers show closely similar characteristics, with no obvious difference at this stage. Further refinements would be needed to support a properly based discussion, such as further increased resolution in frequency – and individual optimization of the access ridge waveguides. In the band edge region, estimated transmission values can exceed 1. This is an artefact that we attribute to the finite simulation time (although already long), which makes it impractical to render the exact transmission value – because a fractional amount of power is still trapped in the slow light waveguide section. This only occurs very close to the band-edge. For other wavelengths the simulation is long enough.
Fig. 4. Transmission through a PhC W0.8 channel guide for different types (linear, exponential) and lengths (8a, 16a, 32a) of input and output PhC tapers (from W1 to W0.8), themselves accessed by a 472 nm (0.641a√3) wide ridge waveguide (2D FDTD).
The short linear taper case (8a) was nevertheless optimized relative to the access ridge waveguide width. The results of Fig. 5 (where only part of the simulated widths are shown) show an optimized width of d = a√3 (= 736 nm). Eventually, we compare the (optimized) butt-coupling to the (optimized) PhC taper coupling (Fig. 6). The PhC taper shows a much sharper transition at the band-edge than the butt-coupling. Transmission levels of 96 % are reached up to just before the band-edge, where the transmission for butt-coupling has dropped down to as low as 3 %. This situation corresponds to a group index value of approximately 100. This result shows the clear improvement that the PhC taper brings in the mode-matching process.
Fig. 5. Transmission through a PhC W0.8 channel guide with input and output PhC tapers (linear type, from W1 to W0.8, length: 8a) for different widths of ridge access waveguide (2D FDTD).
Fig. 6. Transmission through a PhC W0.8 channel guide with butt-coupling from ridge access waveguide (green curve) and through additional input and output PhC tapers (linear type, from W1 to W0.8, length: 8a – red curve). The width of ridge access waveguides has been optimized in both cases (2D FDTD).
An animation of a pulse traversing a slow-light PhC channel guide, in combination with the designed PhC tapers is presented in Fig. 7. The pulse can clearly be seen to slow down when entering the region of the W0.8 PhC channel guide at the end of the input PhC taper. The field strength becomes higher and the color-coding for intensity then saturates in black and white – because the pulse is compressed in space, and the lateral extent of the mode becomes substantially wider, which is characteristic of slow-light modal behavior. At the end of the W0.8 PhC channel guide, the light re-couples to the ridge waveguide via the output PhC taper – and then propagates as a ‘normal’ pulse.
Fig. 7. Animation of a 291 fs (power FWHM) pulse (at λ = 1.48 μm) entering and leaving a W0.8 PhC channel guide with input and output PhC tapers (linear type, from W1 to W0.8, length: 8a) and accessed by ridge waveguides of width a√3 (= 736 nm). In the W0.8 section, the group index is 8.4. [Media 1]
4. Conclusions
We have shown that a PhC continuous taper can increase radically the coupling of light into slow modes in a PhC channel guide, with no noticeable reflection. Even a simple (linear) and short taper (a few micrometers long) can, with suitable choice of parameters, perform this task.
Acknowledgments
This work was supported by the EPSRC (UK) as part of the Ultrafast Photonics Collaboration.
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