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Optimizing Silicon-on-Oxide 2D-Grating Couplers
Lee Carroll,1 Dario Gerace,1 Ilaria Cristiani,2 Lucio Claudio Andreani1 Department of Physics, University of Pavia, via Bassi 6, 27100 Pavia, Italy 2 Department of Industrial and Information Engineering, University of Pavia, via Ferrata, 27100 Pavia, Italy Tel: (39) 0382 987491, Fax: (39) 0382 987563, e-mail: lee.carroll@unipv.it
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ABSTRACT 3D finite difference time domain calculations are used to optimize the coupling between a single-mode telecom fibre and a silicon-on-oxide 2D-grating coupler. By simultaneously varying the hole-radius and grating-pitch of the photonic crystal array that makes up the active region of the coupler, the maximum in coupling efficiency is tuned to 1550nm and optimized. Guided mode expansion indicates that an optically-active resonant guided mode in the photonic crystal array is responsible for the coupling. The results of the mode expansion calculation closely correspond to those of the significantly more computationally intensive finite difference time domain calculations. Using guided mode expansion to provide initial parameters that are later fine-tuned by 3D finite difference time domain calculations may open the way for truly globally optimized 2D-grating couplers that offer significantly improved coupling efficiencies and performance over current designs. Keywords: 2D-grating coupler, Si photonics, finite difference time domain, guided mode expansion. 1. INTRODUCTION In the last decade there has been a considerable push towards developing integrated silicon photonic circuits to support low-cost information and communication technology applications (ICT). These systems combine low cost complementary metal oxide semiconductor (CMOS) fabrication technology with the high optical transmission and index contrast of the silicon–on-insulator (SOI) platform that enables small footprints and strong optical confinement. One on-going challenge in the field is finding a low-cost scalable approach of coupling light from a telecom fibre into a Si-photonic circuit with high efficiency. The problem is two-fold – (i) the single-mode mean-field diameter of a fibre is ≈ 10 µm, while the SOI waveguide cross-section is ≈ 500 nm × 220 nm, and (ii) the polarization-state of the fibre is generally unknown and unstable, while many Si-photonic circuits contain elements that are optimized for (or only work with) TE-polarized light One promising solution to this coupling problem is offered by lithographically-etched grating couplers. Calculations indicate that an optimized one-dimensional (1D) grating coupler (using a non-uniform design and a distributed Bragg reflector) can couple up to 92% (-0.35 dB) of the input fibre mode into the SOI slab [1]. Unfortunately, these 1D-grating coupler designs are usually strongly polarisation sensitive, which makes them unsuitable for many coupling applications. A simple solution is the notional superimposition of two such 1Dgrating couplers, aligned in orthogonal directions, to form a 2D-grating coupler [2]. With this 2D-grating geometry, all input polarization states are separable into TE-polarized projections onto the pair of 1D-grating couplers, and so high efficiency coupling is possible for all polarization states of the input fibre mode. This notional superposition of two 1D-gratings leads to a photonic crystal array (PCA) of cylinders etched into the SOI platform, see Fig. 1, that couples light into the two SOI waveguide arms of the structure. The 2D-grating coupler performance is determined by the SOI slab thickness, and the etch-depth, hole-radius, and grating-pitch of the PCA. Measured coupling efficiencies of up to 37% (-4.3 dB) have been reported from CMOS-compatible 2D-grating couplers using a 70 nm etch depth in a 220 nm SOI slab [3]. However, until now, the design of these 2D-grating couplers have been based on approximations and extensions of the 2D finite difference time domain (2D-FDTD) calculations used to optimize 1D-grating couplers (or through lithographic tuning [2]). In this article, we report how 2D-grating couplers can be more fully designed and optimized using 3D-FDTD calculations. For a given SOI slab thickness (220 nm) and etch-depth (70 nm), a series of 3D-FDTD calculations are used to tune the coupler to the 1550 nm telecom band, and to determine what combination of hole-radius and grating pitch provides the highest coupling efficiency. Guided mode expansion (GME) calculations indicate that the coupling originates from a resonance between an optically active guided mode in the PCA and the input fibre mode. With very low computational cost, these GME results closely correspond to those of the 3D-FDTD calculations, and so offer a means of rapidly generating good initial parameter estimates that need only fine-tuning by the substantially more computationally intensive 3D-FDTD calculations. 2. DESCRIPTION OF 3D-FDTD SIMULATION A schematic of light from a telecom fibre incident on a 2D-grating coupler is shown in Fig. 1. The 10.4 µm mean field diameter fibre mode is near-normally incident on the 10 μm × 10 μm PCA of the coupler structure. The angle-of-incidence (AOI) in our simulation is 15° (in the plane that bisects the PCA at 45°, in order to create a symmetric coupling condition between the two arms of the structure). The total coupling efficiency is defined as the normalized sum of light intensity coupled into the two arms of the system. As the polarization state of the

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fibre mode varies, i.e. as the polarization angle of φ in Fig. 1 rotates, the proportion of light coupled into each arm changes (CEX and CEY), but the total coupling efficiency (CET = CEY + CEX) is very nearly constant [4]. The 15 µm × 15μm × 5 µm volume around the 2D-grating coupler is simulated using a commercial 3D-FDTD package from Lumerical Inc. Our simulation is based on an industrially standard SOI wafer that consists of a 220 nm Si slab on top of a 2.0 μm bottom-oxide layer (BOX) grown on a Si substrate. To simulate a realistic device that supports a process flow for other integrated Si-photonic circuit components, a 750 nm top-oxide layer (TOX) is included in the simulation. The refractive index of the SOI slab and the Si substrate is taken as nSi = 3.47, and the refractive index of the TOX and BOX as nOX = 1.44. The etched-cylinders making up the PCA are assumed to be fully and perfectly filled by the SiO2 of the TOX.

Figure 1. Schematic of light from a telecom fibre incident on a 2D-grating coupler, showing the photonic crystal array (PCA) and siliconon-oxide (SOI) waveguide arms of the structure. The inset shows a material cross-section, made up of a 750 nm top-oxide layer (TOX), a 220 nm silicon-on-oxide (SOI) layer, a 2.0 μm bottom-oxide layer (BOX), and semi-infinite substrate (SUB). The hole-radius (R), grating-pitch (P), and etch-depth (E) of the PCA are also indicated.

The 3D simulation mesh used in the 3D-FDTD calculations is optimized using a conformal algorithm that gives a spatial resolution of ≈ 20 nm in the region of the photonic crystal array, and approximately half that resolution in the BOX and TOX layers. Convergence testing of the mesh confirmed that this resolution was sufficient to describe the physics of the 2D-grating coupler. One full simulation of the 2D-grating coupler volume takes ≈ 90 minutes on a 3.6 GHz QuadCore Desktop PC with 32 GB of RAM.

Figure 2: (a) The variation of coupling with etch-depth for 2D-grating couplers with fixed radius and pitch. (b) The variation of coupling as a function of hole-radius for 2D-grating coupler with fixed etch-depth and grating-pitch. The SOI slab is 220nm in all cases.

3. 3D-FDTD OPTMIZATION OF COUPLER The efficiency of a 2D-grating coupler is largely determined by the performance of its PCA. As illustrated in Fig. 2, both the peak efficiency and central wavelength of coupling depend sensitively on the etch-depth (E), hole-radius (R), and grating-pitch (P) of the PCA. The variation in coupling as a function of etch-depth, for the 2D-grating coupler described in Fig. 1, with fixed hole-radius of 215 nm and a grating-pitch of 616 nm, is shown in Fig. 2(a). Deeper etching results in a blue-shift of the coupling, and the highest coupling efficiency is achieved with an etch-depth of ≈ 110 nm, but not at the target wavelength of 1550 nm. A similar trend is observed in Fig. 2(b), where the 2D-grating coupler is simulated with a constant etch-depth of 70 nm, a grating-pitch of 610 nm, and a variable hole-radius. As the hole-radius is increased, the coupling spectra blue-shift and the highest coupling efficiency is for R ≈ 225 nm at a wavelength of 1525 nm, which is again offset from the 1550 nm target.

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Figure 3: (a) Coupling spectra of a series of 3D-FDTD tuned radius and pitch pairs giving coupling at 1550 nm. The etch-depth is 70 nm and the SOI slab is 220 nm. (b) The peak coupling efficiency from (a) plotted a function of the ratio of the corresponding radius and pitch.

In order to tune the 2D-grating coupler to 1550 nm, it is necessary to simultaneously adjust the hole-radius and grating-pitch of the PCA. The coupling spectra of a number of these 3D-FDTD “tuned” radius and pitch pairs that provide coupling centred at 1550 nm for a 70 nm etch-depth, are shown in Fig. 3(a). While each of these radius-pitch pairs gives their maximum coupling at 1550 nm, they each do so with different coupling efficiencies. As shown in Fig. 3(b), when the coupling efficiency is plotted as a function of the ratio between the radius and pitch of these pairs, it is easy to identify the optimum parameters. For a 2D-grating coupler with an etch-depth of 70 nm in a 220 nm SOI slab, the best choice is a hole-radius of 215 nm, and grating-pitch of 616 nm. This gives a coupling efficiency of 37% (-4.3 dB) at 1550 nm, which is in excellent agreement with the measured efficiency (-4.2 dB) from a 2D-grating coupler using the same 70 nm etch in 220 nm SOI design, albeit with a non-uniform pattern [3].

Figure 4: (a) Energy band dispersion of the PCA with an etch-depth of 110 nm, a hole-radius of 215 nm, and a grating-pitch of 616 nm, as calculated by multi-layer guided-mode expansion (GME). The inset shows the principle symmetry directions in the Brillouin zone of the square lattice. Modes are classified as TM (dashed lines) or TE (full lines) with respect to the vertical plane of incidence, which depends on the symmetry line. (b-e) Details of the relevant variations of photonic mode dispersion close to the normal incidence in the Γ-M direction for a series of different etch-depths. The energy of the coupling resonance of the PCA, and so the 2D-grating coupler, corresponds to the crossing of the 10° light line with the TE-like dipole-active mode. The 10° light line in the GME corresponds to the 15° AOI in the 3D-FDTD calculations, after refraction by the TOX layer.

4. GUIDED MODE EXPANSION 3D-FDTD calculations can generate the optimized parameters for a given 2D-grating coupler design, but they do not provide any insight into the coupling mechanism. Guided mode expansion (GME) calculations can give a more comprehensive picture of the physics at play by describing the photonic mode dispersion of the PCA. GME is based on solving the 2nd-order Maxwell equation, after expanding the electromagnetic field in terms of a separable basis of plane waves in the plane, and guided modes of the vertical dielectric stack. Within this formalism, the photonic modes of the PCA are radiatively coupled to propagating modes in the TOX and BOX by perturbation theory. The main assumptions of GME are that (i) guided modes are coupled to radiative modes of an effective waveguide with a refractive index averaged over a unit cell of the PCA, and (ii) 2nd-order corrections to the resonance energies of the guided modes (induced by coupling to the radiative modes) are not

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taken into account [5]. GME calculations require negligible computational effort, but assume an infinitely extended PCA with semi-infinite BOX and TOX layers. In contrast, 3D-FDTD is computationally intensive, but more adequately accounts for the finite dimensions of the simulated structure and waveguide coupling.

Figure 5: (a) Comparison of the coupling resonance from GME, corresponding to the crossing of the 10° light line with the dipole-active TE mode along the Γ-M direction, and the peak coupling from the 3D-FDTD calculations in Fig 2(a). (b) Comparison between GME and the 3D-FDTD calculations in Fig 2(b). The shaded areas correspond to the FWHM of the 3D-FDTD coupling peaks.

Figure 4 shows the variation in GME-calculated photonic band dispersion as a function of etch-depth for a PCA with the same fixed radius and pitch as that in the 3D-FDTD calculations of Fig. 2(a). The energy of the coupling resonance is given by the crossing of the 10° light line of the input mode with the TE-like dipole-active mode of the PCA (Note: the 10° light line in the GME corresponds to the 15° AOI in the 3D-FDTD calculations, after refraction by the TOX layer). As shown in Fig. 5(a) and 5(b), the coupling resonance predicted by GME is in excellent agreement with the peak of the 3D-FDTD calculated coupling spectra of Fig. 2(a) and 2(b). This agreement is better than 25 nm and always within the full-width half-maximum of the spectra line-width. This strong correspondence between the two numerical approaches is an important result, because it indicates that the fast GME calculations can be used to generate reliable initial values for the 2D-grating design that then need only be fine-tuned by 3D-FDTD calculations to account for the finite dimensions of the coupler structure. This opens the way for a global optimization of the 2D-grating coupler over a wide parameter-space that includes not only the hole-radius and grating-pitch, but also the etch-depth, SOI layer thickness, angle-ofincidence, etc. This level of parameter optimization is not practical with 3D-FDTD calculations alone, because they are too computationally intensive. The combination of joint GME and 3D-FDTD optimization may lead to significantly improved coupling performance by determining the absolute best combination of many different design parameters. 5. CONCLUSIONS We have shown that 3D-FDTD calculations can be used to determine what combination of hole-radius and grating-pitch in a SOI 2D-grating coupler that deliver the highest coupling efficiency at the desired wavelength. There is a strong correspondence between the results of 3D-FDTD simulations of the 2D-grating coupler region and the computationally light GME calculations of the corresponding PCA. Combining GME and 3D-FDTD calculations as a means of practical global optimization of 2D-grating coupler design parameters may lead to significantly improved performance in the future. ACKNOWLEDGEMENTS This work was supported by the EU through FP7-ICT-2011-8 Contract No. 318704 “FABULOUS” REFERENCES [1] D. Taillaert, et al.: A compact two-dimensional grating coupler used as a polarization splitter, IEEE Photon. Technol. Lett., vol 15, pp. 1249-1251, 2003. [2] D. Taillaert, et al.: Grating couplers for coupling between optical fibers and nanophotonic waveguides, Jpn. J. Appl. Phys., vol. 45. pp. 6071-6077, 2006. [3] S. Pathak, et al.: Compact SOI-based polarization diversity wavelength de-multiplexer circuit using two symmetric AWGs, Opt. Express, vol. 20, pp. B493-B500, 2012. [4] F. Van Laere, et al.: Efficient polarization diversity grating couplers in bonded InP-membrane, IEEE Photon. Technol. Lett., vol. 20, pp. 318-320, 2008. [5] L.C. Andreani, D. Gerace: Photonic crystal slabs with a triangular lattice of triangular holes investigated using a guided mode expansion method, Phys. Rev. B, vol. 73, pp. 235114-1-16, 2006.

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