1. Introduction

Integrated silicon photonics has a great potential for providing significantly improved computing performance. For instance, faster and lower loss interconnects for multi-core processors are envisioned by utilizing silicon-compatible on-chip optical waveguides and modulators [1,2]. While on-chip light sources have been developed in recent years [3,4], the most common excitation method for photonic platforms are off-chip lasers. However, there is a substantial mode mismatch between off-chip light sources and on-chip waveguides, preventing efficient coupling. For example, the smallest spot size achievable from commercially available off-chip lasers (using lensed fiber-coupled lasers) is typically ≈3 µm while silicon-compatible single-mode waveguides typically have dimensions of ≈500 × 250 nm [5].

A number of solutions for fiber-to-chip coupling have been developed and can be classified into in-plane and out-of-plane couplers [6]. For in-plane coupling, the most prominent solutions are based on the inverse taper geometry [7–10]. In this configuration, light is coupled from a fiber into the narrow end of the taper where the mode is initially delocalized before being smoothly converted in both mode field size and effective index along the taper to the high-index on-chip silicon waveguide. While certain taper designs achieve insertion losses as low as 0.5 dB, they require multi-step lithography and extend over a long distance (>200 µm), occupying a large area on the chip [8].Recently, improved tolerance to fiber misalignment was achieved by introducing a Luneburg lens to focus light into an inverse nanotaper; however, this method did not result in increased coupling efficiency and requires the use of a complicated greyscale lithography process [11]. The most common out-of-plane couplers are based on diffraction gratings, in which the fiber is placed above the chip and light is incident at normal incidence to the plane of the chip [12–15].The main advantage of grating couplers is that the light can be coupled in and out at an arbitrary location on the chip, and not only at the chip facet. However, grating couplers are typically designed for a single wavelength and angle of incidence, thus limiting their bandwidth of operation. Furthermore, there is a fundamental limit to grating coupler efficiency due to the fact that the light is scattered in both the forward and backward directions. While the forward scattered light can be coupled to a waveguide mode, the backward scattered light is lost. Edge-ray optics approaches have been applied for efficient non-imaging fiber-to-fiber couplers, but these methods do not address compact modal index conversion, which is essential for coupling into high-index, high-confinement integrated silicon waveguides [16].

In this work, we demonstrate an approach employing transformation optics that allows reduction in the coupler size and complexity of fabrication, while also maintaining a relatively high coupling efficiency. Transformation optics provides a means to control light propagation through spatial transformations in the permittivity and permeability tensors [17].The theory has led to the development of several complex optical devices, including invisibility cloaks [18–22], multifunctional optical devices [23], and photonic black holes [24,25]. Based on this methodology, we present simulated and experimental results of a transformation-optical (TO) fiber-to-chip coupler. The coupler is fabricated using a single-step lithography process and exhibits low coupling losses while maintaining a significantly smaller footprint compared to other coupler designs [7–15].

2. Transformation design

Transformation optics can, in theory, provide a three-dimensional (3D) transformation that perfectly converts a Gaussian-like fiber mode into a mode of a rectangular waveguide through mode size and modal index matching. However, this 3Dcoordinate transformation requires permittivity and permeability tensors that vary in all three dimensions and would make the physical implementation extremely challenging with modern fabrication technology. A more realistic approach is the design of a coupler that only transforms light in the in-plane direction and accordingly utilizes a two-dimensional (2D) transformation. The design approach of converting a 3 µm fiber mode into a 500 × 250 nm waveguide mode is shown in Fig. 1(a) . The transformation compresses physical space in the in-plane direction, squeezing the fiber mode profile into a ridge waveguide mode. The ridge waveguide mode is obtained from a standard single-mode waveguide on the silicon-on-insulator (SOI) platform. Similar 2D spot-size converters have been proposed but have not been implemented yet due to fabrication challenges associated with transformations resulting in complex anisotropic permittivity and permeability tensors [26–28]. In order to reduce anisotropy in the transformation to yield isotropic material properties, a quasi-conformal mapping technique proposed by Chang et al. was used [29].The technique utilizes the solution to the Laplace equation with sliding boundary conditions which results in minimization of the anisotropy in the transformed medium for TE polarization. The virtual space was designed to be a 3 µm wide rectangular waveguide with an exponentially varying permittivity. This permittivity profile is used to account for the increase in index upon compression of the space, allowing for impedance matching with the output waveguide, eliminating reflections [26].The real space was tapered from 3 µm to 450 n mover a distance of 10 μm, as shown in Fig. 1(a).COMSOL Multiphysics software was utilized to solve the Laplace equations which defined the distorted coordinate system (Fig. 1(a)) and the isotropic relative permittivity profile resulting from the transformation (Fig. 1(b)).Considering the range and the gradient of permittivity accessible for different coupling lengths, a 10 µm coupler length was selected as it offered the best balance between high coupling efficiency and low losses due to scattering. As a general rule, the real space must avoid discontinuities in the first derivative of the boundary outline, as they cause singularities in the deformation tensor profile. A smooth coupler shape consisting of 2nd and 3rd order Bezier curves was empirically determined to provide the most efficient coupling.

 

Fig. 1 (a) Fiber mode field profile is shown on the left and ridge waveguide mode profile on the right; darker color indicates higher intensity. Transformed coordinate system isolines, obtained by quasi-conformal mapping, are shown in the middle.(b) Material permittivity resulting from the transformation.

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In order to realize the isotropic permittivity profile, a hexagonal lattice of fixed diameter sub-wavelength air holes with varying filling fraction was employed [20]. The hole array is placed in the device layer of an SOI wafer and the spatially varying filling fraction was computed in order to most closely match the effective waveguide mode index to the permittivity profile dictated by the transformation. We chose to implement uniform-size holes with variable spacing owing to the more forgiving fabrication tolerances that this approach offers compared to fixed-spacing and arbitrary hole size [22]. Other methods of obtaining variable permittivity that use random hole placement either based on a probability function or a grayscale image have a lower resolution and therefore require a much smaller hole size in order to be effective [21,30].

The Maxwell-Garnett effective medium approximation was used to determine the desired filling fraction profile. An algorithm was developed to determine the proper hole placement in the distorted lattice. The performance of the algorithm was evaluated by extracting the effective permittivity of the distorted lattice. The hexagonal lattice arrangement allows for maximum packing density and therefore can achieve a greater range of permittivity values. For our design, the hole diameter was chosen to be 110 nm to ensure consistent hole-to-hole uniformity using standard electron-beam lithography processing. Reducing the hole diameter would lead to a more accurate permittivity profile, resulting in improved coupler performance.

3. Fabrication

The coupler design was fabricated on an SOI wafer with a 220 nm p-type, 14–22 Ω-cm resistivity, Si(100) device layer and 1 μm buried oxide layer (SOITEC). Electron-beam lithography (JEOL JBX-9300–100kV) was performed using poly(methyl methacrylate) (PMMA) resist. After pattern exposure and development in methyl isobutyl ketone (MIBK) and isopropyl alcohol in a ratio of 1:3 (v/v), anisotropic reactive ion etching was performed (Oxford PlasmaLab 100) using C₄F₈/SF₆/Ar process gases to etch the exposed portion of the 220 nm Si layer. The 110 nm diameter holes with variable spacing were fabricated with relatively low size deviation due to proximity effects. In addition to the TO coupler (Fig. 2(a) ), several other geometries were fabricated for performance comparison: waveguides without a coupler, TO coupler outline without holes, and a TO coupler with holes only in the wider (x = 0-5 µm) region (“Restricted placement”). After etching, the chips were masked with a thick PMMA protective layer and then cleaved and polished down to the couplers on one side and cleaved through the waveguides on the other. The PMMA protective layer was removed with ultrasonication in acetone before measurements were performed. We note that the 3 μm wide end of the coupler could be extended to eliminate the need for polishing, although additional transmission losses through that section may be incurred. Experimental data was obtained using a lensed polarization-preserving fiber to couple light into and out of the waveguides. Measurements were performed using a broadband TE-polarized LED source (Agilent 83437A, 1500-1700 nm) and an optical spectrum analyzer (Agilent 86140B). For higher resolution measurements, a tunable TE-polarized laser source (Velocity 6328, 1520-1570 nm) and a broadband photodiode (Newport 1835-C) were utilized.

 

Fig. 2 (a) Scanning-electron microscope image of TO coupler, (b) Electric field (E_y) snapshot of the transformation-optical coupler with 110 nm holes demonstrating efficient in-plane mode conversion (top view), (c) Mode profiles at various points along the transformation-optical coupler with continuously-varying permittivity in 3D mode conversion, (d) Field (E_y)snapshot of the transformation-optical coupler with continuously-varying permittivity in 3D mode conversion (side view).

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4. Simulation and experiment

In order to verify proper operation of the TO coupler, in-plane mode conversion was studied by restricting the size of the out-of-plane input mode profile to match the height of the waveguide and coupler. To evaluate coupling efficiency, 3D finite-difference time-domain (FDTD) simulations were carried out for the coupler with the designed spatially varying hole profile using freely available software, MEEP [31]. A 3000 nm × 250 nm input rectangular Gaussian pulse was launched into the wide end of the coupler and the transmission was determined by accumulating flux through two planes, one at each end of the coupler. A Fourier transform was then performed on the resulting fluxes; transmitted power was determined by the ratio of the transformed output plane flux to the transformed input plane flux [31]. The mode conversion efficiency was nearly 100% for some wavelengths within the evaluated bandwidth (1500-1700 nm) (field snapshot shown in Fig. 2(b), transmission plot shown in Fig. 3 ). The origin of the oscillations in the transmission spectrum is discussed below. In comparison, the peak efficiency of the TO coupler outline without holes was less than 85% (Fig. 3), confirming that the transformation optics approach to in plane mode conversion works as designed. The TO coupler outline design provides mode size conversion but does not achieve modal index matching. Several mode profiles at different positions along the TO coupler are shown in Fig. 2(c).Bending of the field can be observed at x = 5 µm, illustrating how the light is compressed as it propagates inside the transformed coordinate system.

 

Fig. 3 Simulated transmission plot for in-plane mode conversion for TO coupler and TO coupler outline without holes.

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To analyze the efficiency of the TO coupler for fiber-to-waveguide mode conversion (3D mode conversion), the input mode was approximated by a rectangular 3000 nm × 3000 nm plane wave centered in the middle of the coupler. Instead of the Gaussian-like field profile of a real fiber, the plane wave has a uniform field profile contributing to additional loss (illustrated in Fig. 2(d)) in the simulation which is not present in experimental measurements. Despite this additional loss, the simulation results (Fig. 4(a) ) provide a good comparison between different coupler designs and match reasonably well with the experimental measurements (Fig. 4(b)). The slight differences between simulations and experiments are likely caused by fabrication errors, namely non-uniformity in the hole size and distortion in the hole shape. For the butt-coupled waveguide without any included transition region between the fiber and waveguide, an average transmission of 1% was simulated and very low power was measured at the output of the waveguide. The fabricated TO coupler showed a peak 5-fold improvement over butt coupling in experiment for the design with 110 nm holes. A maximum transmission of 17% was simulated. In comparison, the TO coupler outline without holes showed an average simulated transmission of 10% and a 3-fold improvement over butt-coupling in measurement. Due to the out-of-plane losses and inaccuracies in approximating the fiber mode in simulation, the maximum achievable transmission for our in-plane mode conversion TO coupler was found to be approximately 21% by simulating a continuously varying index profile (“Perfect index”) for the TO coupler.

 

Fig. 4 Plot of the (a) simulated power transmission and (b) experimental transmission through the couplers: TO coupler, TO coupler with holes present only in wider half of coupler (Restricted placement), and TO coupler outline without holes. For comparison, a coupler with a perfect refractive index gradient transformation was simulated and a butt-coupled waveguide without a coupler was simulated and experimentally investigated.

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Besides the lack of an out-of-plane transformation, the TO coupler efficiency is also affected by inaccuracies in replicating the designed permittivity profile. As shown in Fig. 4,the TO coupler efficiency varies with wavelength and does not match the efficiency of the continuously-varying index obtained directly from the transformation (perfect index). This inconsistency is due to the discretization in the refractive index profile caused by use of110 nm diameter air holes. The discretization becomes more drastic in the narrow end of the coupler where the spacing between air holes is large enough to cause interference effects. Namely, the two holes at the end of the coupler, separated by about 1 µm, form a Fabry-Perot type structure, which exhibits similar oscillations to the ones seen in the TO coupler transmission. Furthermore, due to the strong field confinement in the narrow end of the coupler, small deviations in the lateral positioning of the holes in this section lead to large differences in the effective index of the mode, making it challenging to match the perfect index profile.

In order to demonstrate that the transmission curve for the TO coupler can be flattened to increase the wavelength tolerance of the coupler performance, two approaches could be taken: (1)eliminating the transformation in the narrow end of the coupler at the expense of peak coupling efficiency, and (2)reducing the size of the holes. The second approach will minimize the transformation errors due to the permittivity discretization but, because of the fabrication difficulties, it was not investigated. The first approach was simulated and fabricated by removing all holes in the narrow half of the coupler (x = 5-10µm, “Restricted placement”), thus truncating the transformation. As shown in Fig. 4(b), this approach indeed demonstrated flattening of the transmission spectrum. The experimental transmission curve shows an increase with wavelength, which does not appear in simulation. One possible reason for this small discrepancy between experiment and simulation is the slight inhomogeneity in the hole profile, which causes stronger scattering from holes at shorter wavelengths.

5. Conclusion

An ultra-compact fiber-to-chip coupler was designed and fabricated employing the transformation optics approach. The coupler showed a 5fold improvement over butt coupling while being only 10 µm in length, representing a major improvement compared with conventional coupler designs. Due to the compact design, ease of alignment, and CMOS compatible fabrication techniques, the TO coupler is a promising candidate for future use in the development of integrated silicon photonic components.