An asymmetric directional coupler scheme for the efficient injection of light into slow light slot photonic crystal waveguide modes is proposed and investigated using finite-difference time-domain simulation. Coupling wavelengths can be flexibly controlled by the geometrical parameters of a side-coupled subwavelength corrugated strip waveguide. This approach leads to a ~1dB insertion loss level up to moderately high light group indices (nG≈30) in wavelength ranges of 5-10nm. This work brings new opportunities to inject light into the slow modes of slot photonic crystal waveguides for on-chip communications using hybrid silicon photonics or sensing based on hollow core waveguides.
© 2014 Optical Society of America
Silicon photonics  has received a strong interest in the last years, and the vision of a possible convergence of electronics and optics within the same chips is now close to lead to concrete applications. In the same time, the Si photonics field has also broadened its potential applications to other topics like bio-sensing/lab-on-chips, monolithic integration of III/V semiconductors on Si for telecommunication receivers/transceivers, and nonlinear optics for all-optical signal processing at ultra-high data bit rates . In this perspective, slot optical waveguides first introduced by Almeida and associates  have opened the way to the hybrid integration of various low index materials with Si waveguides, bringing the advantages of the Si waveguides strong confinement with the luminescence/gain, nonlinear, or bio-sensitization properties of the introduced materials. The particular feature of this approach is that an excellent overlap between the propagating modes and the active material is obtained due to the slot mode mechanism coming from the electric field normal component discontinuity at the interface between the core and the cladding layers of high/low dielectric permittivities, respectively. Beyond this approach, one more step to further reinforce the light-matter interactions is to add one degree to the spatial compression of the propagating optical waves by the use of a slow light mode propagation mechanism within a slot photonic crystal waveguide (SPCW) geometry [4–8]. With respect to a simple slot waveguide approach, the introduction of a two-dimensional (2D) lateral periodical lattice (see Fig. 1(a)) leads to a photonic dispersion diagram characterized by a couple of even symmetry propagating slow light modes (e.g. flat band) within the 2D photonic crystal (PhC) bandgap frequency range , as illustrated in Fig. 1(b)). The lowest frequency one arises from the fundamental dielectric mode of the simple slot waveguide that is pulled within the photonic bandgap with a positive slope in the first Brillouin zone (FBZ) and called hereafter the “slot mode”, while the highest frequency one (“W1-like mode”) comes from the W1 PhC waveguide geometry and has a negative slope. The intrinsic “slot mode” has a similar behavior as the standard “W1-like” line mode: it is index guided far from the edge of the FBZ, but becomes gap guided when it reaches the edge of the FBZ due to the anti-crossing with the air band.
Although interesting in many configurations for the development of a silicon-organic hybrid (SOH) integration platform [9–11] overcoming the silicon intrinsic limitations by combining the mature CMOS processing and strong nonlinear light-matter interactions without introducing significant two-photon absorption (TPA) , the use of SPCW faces the practical issue of their efficient coupling with an incident standard strip one due to the strong mismatch between the involved electromagnetic modes’ profiles and high/low group velocities, respectively. Light injection into slow modes from input strip waveguides has been identified as a difficult point in previous works even for non-slotted PhC waveguides. Butt-coupling schemes with tapering input/output regions have been primarily investigated and previous works have shown the positive contribution of evanescent modes at and close to the interfaces between the fast and slow propagating modes’ regions [12, 13]. Several architectures have been proposed in recent years to solve the problem of light coupling into SPCW. All of them consist in a butt coupling approach between the two waveguide modes, respectively, consisting either in the use of optimized tapers , multimode couplers [14, 15], or in the use of properly designed defects near the edge of the strip/SPCW interface .
The aim of the present paper is to propose an alternative simpler solution to these approaches, which are all based on butt-coupling strategies and that often require a design demanding the collaboration of several structural parameters, precisely controlling the size of the defect holes, the lattice parameter, or the distance of the slot to the edge. We propose an effective asymmetric directional coupler (ADC) scheme [17, 18] based on a subwavelength corrugated strip waveguide placed in close proximity to the SPCW. With the help of the subwavelength corrugated strip waveguide parameters, light coupling from the fast light strip waveguide to the slow light slot modes can be adapted either to the W1-like slot mode or to the fundamental PhC slot one. The required phase-matching (PM) condition can be flexibly adapted to the one or the other situations by simply changing the width and the duty cycle of the subwavelength corrugated strip waveguide.
This paper is organized as follows. We first introduce the principle of operation and the structure of the strip waveguide-SPCW ADC. Next, the band diagrams of the subwavelength corrugated strip waveguide and the SPCW are obtained by using the three-dimensional (3D) plane-wave expansion method (PWE)  and the physics inside the coupling scheme is discussed. Following that, three steps are implemented for the confirmation of the coupling behavior by using two-dimensional (2D) finite-difference time-domain method (FDTD) simulation .
2. Proposed coupling approach
2.1 Asymmetric directional coupler geometry
As mentioned in , efficient coupling into the slow light modes of slot PhC waveguides must rely on the consideration of the dispersive properties of the slot waveguides. A key difference between the fundamental slot and W1-like modes of such SPCW is indeed that the first and second ones have slopes of opposite signs in the FBZ. For this reason, the PM crossing condition between a standard strip waveguide mode having a positive slope in the FBZ with the fundamental slot mode should be prepared indeed in the FBZ, while it should be done in the second Brillouin zone for the W1-like one due to the band folding of both modes. In addition to this, the required PM effective indices at the crossing points are usually low due to the weak effective index values of empty-core slot waveguide modes, typically ranging between 1.7 and 1.9. Side coupling with a standard strip waveguide then requires an extremely narrow strip guide, raising difficulties both in the fabrication issues and in the precise control of the waveguide width with a reasonably small uncertainty. This difficulty is circumvented here by removing the strip waveguide and considering instead a subwavelength corrugated strip waveguide .
A schematic of the whole structure, a subwavelength corrugated strip waveguide placed in close proximity to a SPCW, is depicted in Fig. 2(a)). Hereafter, a possible implementation of the structure in silicon photonics is considered for illustration, starting from a suspended silicon membrane with a thickness of 220nm embedded in a low index material. The bulk silicon refractive index of 3.48 for TE polarization is adopted. And the low index material with a refractive index of 1.6 fills the holes and the slot itself.
We introduce now the main waveguide parameters: the lattice constant a of the triangular lattice SPCW and a’ the subwavelength corrugated strip waveguide one, respectively, with a’ = 0.5a. The width and the duty cycle of the subwavelength corrugated strip waveguide are labeled as Wsub and t, respectively, which are used hereafter to facilitate the phase-matching condition between the involved subwavelength corrugated strip waveguide mode and the SPCW one. The separation gap size between the SPCW and the subwavelength corrugated strip waveguide is fixed to 0.5a. With a lattice parameter a = 400nm, the separation gap is then of 200nm, which is thus feasible with actual existing nano-fabrication technologies. For the SPCW, a normalized hole radius of 0.3a and central width of 1.2*√3a are considered here. The distance between the hole and the edge is fixed to 0.2a. The widths of the slot (Wslot) are 0.4a and 0.8a for the W1-like mode and the fundamental SPCW slot modes, respectively.
2.2 Dispersion curves and phase matching wavelengths
In order to investigate the proposed coupling approach, a three-dimensional analysis is adopted here with the index values mentioned in section 2.1. Dispersion curves have been calculated using the Plane Wave Expansion (PWE) method with the MIT Photonic Bands software. The dispersion curves of the subwavelength corrugated strip waveguides and the SPCW are displayed in Fig. 3(a) and 3(b), for the W1-like and the fundamental slot modes, respectively. The dash lines represent the project lines of the low index material. And the dark blue shaded regions below the light lines are extended modes in the photonic crystal.
As shown in Fig. 3(a), the phase-matching frequency and thus the light group index for light coupling to the W1-like SPCW mode can be altered by adjusting the width and the duty cycle of the subwavelength corrugated strip waveguide. With the parameters chosen here for illustration of the proposed coupling method, group index values of 21, 31 and 38 with phase matching frequencies of 0.2683(2πc/a), 0.268(2πc/a) and 0.2678(2πc/a) are obtained, respectively. Similarly, one can get from Fig. 3(b), the group index values of the fundamental SPCW mode and the related frequencies as 25, 30 and 40 at 0.2771(2πc/a), 0.2773(2πc/a) and 0.2774(2πc/a), respectively.
2.3 Focus on the fundamental slot mode
Due to its better mode confinement properties, we focus hereafter on the light coupling into the fundamental SPCW mode. We emphasize first here that according to Fig. 3(b), the required PM effective index is around 1.8 with group index values ranging from 25 to 40, showing the intrinsic strong difficulty of efficient light coupling into such an unusual waveguide mode. Before estimating the coupling yield using transient FDTD simulation, we first plot the ADC supermodes profiles of the two coupled waveguides in order to get insight into the mode coupling mechanism for the interpretation of the FDTD results.
Figure 4 shows the 3D-PWE calculated supermodes dispersion curves in the configuration of Fig. 3(b) for Wslot = 0.8a, Wsub = 1.14a and t = 0.6, as well as the supermodes Hz-fields profiles at the A1-A3 and B1-B3 points. From the plotted field maps, one can clearly identify the subwavength dielectric (B1, A3) and slow light slot (A1, B3) modes far from the coupling region located around ω = 0.277, while the mode hybridization is clearly visible for example at the A2 point. Above a given frequency threshold, the flat band slot waveguide supermode vanishes, preventing any more the possible beating between the two supermodes for light power exchange between the subwavelength corrugated strip and the slot PhC waveguides.
Based on the discussion above, three steps have been implemented for the confirmation of the coupling behavior by using 2D FDTD simulation. An equivalent slab waveguide index of 2.872 for TE polarization was adopted for this. A low index material with a refractive index of 1.6 was considered for the filling of holes and slot regions. The width and the duty cycle of the subwavelength corrugated strip waveguide have been additionally modified to maintain the slow light configurations (i.e. nG = 25, 30, and 40) obtained from the 3D dispersion curve calculations previously performed. The structural parameters of the modified subwavelength corrugated strip waveguide are displayed in Table 1, along with the corresponding phase matching wavelengths obtained for a lattice parameter a = 400nm.
At first, the incident light is injected into the subwavelength corrugated strip waveguide and the spectral response in the port of the SPCW output is calculated. The wavelength of the maximum power transmission is taken as the phase matching wavelength. The related phase matching wavelengths are 1569nm, 1567nm and 1566nm, respectively. The obtained phase matching wavelengths are thus in close agreement with the ones in Table 1.
In a second step, a continuous wave excitation at each PM wavelength is considered through a long enough ADC (40μm) for getting the beating lengths from the magnetic field (Hz) steady-state maps by using the FDTD simulation software package developed at MIT (MEEP), as shown in Fig. 5.In the three slow light configurations considered above (e.g. nG = 25, 30, and 40, respectively), the obtained beating lengths are 12.4μm, 12μm and 13.2μm, respectively, showing that the proposed coupling approach into slow light modes of SPCW is not obtained at the expense of the device footprint compactness.
Lastly, a transient pulse is injected into the subwavelength corrugated strip waveguide, and the cross-channel transmission spectra are calculated. Results are plotted in Fig. 6, showing the coupling yields both in log and linear scales, respectively. The overall shape of each transmission curve is consistent with the above analysis from the supermodes dispersion curves: a clear coupling cut-off is observed.
Beyond this qualitative agreement, we also get the quantitative result that coupling losses are of 1.2dB, 0.8dB and 3.9dB for group index values of 25, 30 and 40 in the SPCW, respectively. We thus get the interesting result that a reasonable ~1dB loss level is obtained for coupling into slow light modes of the tightly confined and low equivalent effective index of the fundamental photonic crystal slot mode up to nG~30 in wavelength ranges of 5-10nm.
We have designed an asymmetric directional coupler scheme consisting of a laterally coupled subwavelength corrugated strip waveguide and a slot photonic crystal waveguide in the silicon photonics technology. The proposed approach circumvents the use of extremely narrow strip waveguides that would be necessary in order to get the low equivalent effective index of the excited slot modes (neff~1.8). Slow light with group index values ranging from 25 to 40 have been considered. This scheme has been analyzed by using Plane Wave Expansion method and FDTD simulations. The gathered results show that coupling efficiencies with a loss level of ~1dB can be achieved in the 5-10nm slow light bandwidth of the slot photonic crystal waveguide up to nG~30 even for the tightly confined fundamental slot photonic crystal mode. Additionally, the asymmetric directional coupler scheme offers a flexible method to tune the phase-condition condition between the fast and slow light coupled modes by choosing the subwavelength corrugated strip waveguide width and/or duty cycle. The proposed approach offers a new method for efficiently exciting slow light Bloch waves in hollow core slotted waveguides and may serve for applications in hybrid silicon photonics for all-optical signal processing using nonlinear effects or sensing.
The authors would like to acknowledge the NSFC Major International Joint Research Project (No. 61320106016), the NSFC/ANR Sino-French Joint Program (POSISLOT Project), the National Basic Research Program of China (Grant No. 2011CB301704), the National Science Fund for Distinguished Young Scholars (No. 61125501), and the Opened Fund of the State Key Laboratory on Integrated Optoelectronics (No. 2011KFJ002).
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