We propose a new method for mode conversion and coupling between an optical fiber and a sub-micrometer waveguide using a subwavelength grating (SWG) with a period less than the 1st order Bragg period. The coupler principle is based on gradual modification of the waveguide mode effective index by the SWG effect that at the same time frustrates diffraction and minimizes reflection loss. We demonstrate the proposed principle by two-dimensional Finite Difference Time Domain (FDTD) calculations of various SWG structures designed for the silicon-on-insulator (SOI) platform with a Si core thickness of 0.3 µm. We found a coupling loss as small as 0.9 dB for a 50 µm-long SWG device and low excess loss due to fiber misalignment, namely 0.07 dB for a transverse misalignment of ±1 µm, and 0.24 dB for an angular misalignment of ±2 degrees. Scaling of the SWG coupler length down to 10 µm is also reported on an example of a 2D slab waveguide coupling structure including aspect ratio dependent etching and micro-loading effects. Finally, advantages of the proposed coupling principle for fabricating 3D coupling structures are discussed.
©2006 Optical Society of America
A major problem in the design and fabrication of integrated microphotonic devices is the efficient coupling between compact planar waveguides and the outside macroscopic world. This problem has been identified from the earliest years of integrated optics . Various original solutions have been found since, but the coupling still remains a challenge particularly for waveguides of sub-micrometer dimensions made in high index contrast (HIC) materials such as III-V semiconductors, silicon oxynitride, and silicon-on-insulator (SOI). Very compact planar waveguide devices can be made in these platforms. In SOI waveguides, light is highly confined in the silicon core which can have cross-sections on the order of 200 nm x 200 nm or less, and bending radii can be reduced to a few micrometers. Beside the potential for chip size reduction, the benefit of integration of the mainstream microelectronic technology with photonics has been the main driving force in the emerging field of silicon photonics [2–7] with significant recent improvements in fabrication technology and many novel structures and devices reported, including modulators [9,10], lasers [11,12], arrayed waveguide gratings (AWGs) [7,13–22] and others [23–28].
Due to the large mode effective index and mode size disparities, the optical coupling between an optical fiber and a high index contrast waveguide with a small cross-section is largely inefficient. In order to match a large optical fiber mode to a HIC waveguide mode with an area typically two orders of magnitude smaller, mode size transforming structures in both the in-plane and out-of-plane directions need to be used. Such mode transformers are conceptually simple, but the out-of-plane tapering requires gray-scale lithography, which is not yet a standard technique in the industry. Photonic crystal and grating  couplers have been demonstrated, but their fabrication is demanding. An interesting approach is to use an inversely tapered waveguide [30–32] that adiabatically narrows down to a width of about 100 nm or less as the waveguide approaches the facet facing the fiber. The waveguide effective index is reduced by narrowing the waveguide width, which causes the mode to expand and to eventually match that of the fiber. An alternative approach using a coupler with a planar graded-index (GRIN) lens has been reported by Delâge et al. [33, 34]. The structure acts as an asymmetric GRIN lens that is the planar analogue of the conventional cylindrical GRIN lens. The GRIN coupler can be made very compact, about 15 µm in length, hence it requires precise placement with respect to the chip edge, with a tolerance of approximately ±1 µm. This can be guaranteed if the facet position is defined lithographically with etching rather than cleaving used for die separation .
2. Waveguide effective index modification by SWG effect
Here we propose a new method to couple light between an optical fiber and a planar waveguide circuit using a sub-wavelength waveguide grating. According to the homogenization theory or effective-medium theory , a composite medium comprising different materials combined at sub-wavelength scale (Λ<λ) can be approximated as a homogeneous media and its effective index can be expressed as a power series of the homogenization parameter χ=Λ/λ, where Λ is the grating period (pitch) and λ is the wavelength of light. The coupler principle is based on gradual modification of the waveguide mode effective index by the SWG effect. A general schematic of the proposed coupling method is illustrated in Fig. 1, with panel a showing the cross-sectional view (perpendicular to the chip plane), and panels b and c showing the in-plane views of SWG structures without and with waveguide width tapering, respectively.
The waveguide mode effective index is altered by chirping the SWG duty ratio r(z)=a(z)/Λ(z), where a(z) is the length of the waveguide core segment. In a general case both Λ(z) and a(z) may vary along the propagation direction z, but in this paper we consider a simple case of constant Λ, as varying a(z) suffices to explain the SWG effect. The duty ratio and hence the volume fraction of the waveguide core material is modified such that at one end of the coupler the effective index is matched to the HIC waveguide while near the chip facet it matches that of the optical fiber. The effective index of the mode in our SWG coupling structures increases with the grating duty ratio. The SWG effect can be advantageously combined with waveguide width tapering (Fig. 1, panel c) and also with SWG segment height or etch depth tapering arising from the reactive ion etching lag effect near the ends of the coupler (Fig. 1, panel a), the latter is discussed at the end of section 3. Unlike in the waveguide grating couplers based on diffraction, the SWG mechanism is non-resonant, hence intrinsically wavelength insensitive. This is a consequence of the fine scale of the segmentation compared to the wavelength of light, as it is predicted from the effective medium theory. Diffraction by the grating is frustrated provided the grating period Λ is less than the 1st order Bragg period ΛBragg=λ/(2n eff), unlike in long-period tapers and mode converter with Λ>λ/(2n eff) [37–39]. The latter have been proposed for mode size transformation in low index contrast waveguides such as those made in a silica-on-silicon platform, but their application in HIC waveguides is hindered by the reflection and diffraction losses incurred at the boundaries of different segments. Applying the SWG approach in an SOI waveguide with Si core and SiO2 cladding, an increasing portion of the Si core is removed and replaced with SiO2 when approaching the coupler end facing the fiber, and an effective index close to that of silica glass is obtained at the fiber-chip interface. Reflection loss at the coupling interface can be very small since index matching can be achieved as in photonic circuits fabricated using conventional silica-on-silicon technology.
3. Calculated examples of SWG coupling structures
We demonstrate the proposed coupling principle on examples of various two-dimensional coupling structures using 2D Finite Difference Time Domain (FDTD) calculations . In our couplers, the duty ratio is chirped linearly from r min=0.1 at the coupler end facing the fiber to r max=1 at the opposite end of the coupler, except for structure (H) with r min~0.33, see Table 1. The coupling structures were simulated for an SOI waveguide with Si core thickness 0.3 µm and SiO2 cladding thickness of 6 µm, with the corresponding refractive indices of nSi=3.467 and nSiO2=1.5.
In our FDTD calculations, at the input of the coupler (z=z 1) a continuous-wave (cw) Gaussian field with a width equivalent to the mode filed diameter (MFD) of the optical fiber mode at a wavelength λ=1.55 µm was assumed. MFD=10.4 µm of an SMF-28 fiber was used, for some structures also compared with MFD=5.9 µm of a C-type high numerical aperture fiber. The SWG waveguide is positioned along the z axis. Typical simulation window dimensions used were 50 µm (propagation direction) by 13 µm (transverse direction). The mesh size was 10 nm in both dimensions and the simulations ran for a total of 20,000 time steps each of Δt=2.2·10-17 s. The time step was chosen according to the Courant limit to ensure numerical stability of the algorithm. The coupler efficiency was calculated as η=ΓP 2/P 1, where P 1 is the input power injected at the right edge of the computation window (z=z 1), and P 2 is the output power crossing the output plane obtained by integrating the S z component of the Poynting vector along the left edge of the computation window (z=z 2) where the coupler joins the silicon waveguide, and Γ is the power overlap integral of the calculated field at the output plane z=z 2 with the fundamental mode of the Si waveguide.
The parameters and calculated coupler efficiencies of different structures are summarized in Table 1. Panels a and b in Fig. 2 show the Poynting vector component S z=Re(E x H y*)/2 obtained for a 2D FDTD calculation of structures (A) and (D) representing designs without and with waveguide width tapering, respectively. The structure (A) has an overall length of 40 µm, SWG pitch of 0.2 µm, and the duty ratio r is linearly chirped from 0.1 to 1. The calculated coupling efficiency is 73.3%, hence the coupling loss is 1.35 dB. In Fig. 2a it is observed that the loss is primarily incurred along the first 10 µm of the coupler length. To easy the transition, parabolic rather than linear tapering can be used. Here we include linear waveguide width tapering in two steps (denominated as the waveguide width tapering type 2 in Table 1) because such approximated structure is easier to script that the ideal (parabolicaly tapered) SWG and still effectively easies the transition. The structure (D) has the waveguide width linearly tapered from w 1=30 nm (at z=z 1) to w=150 nm along the first 2/3 of the coupler length, and then to w 2=0.3 µm (at z=z 2). The simulated Poynting vector for this structure is shown in Fig. 2b. The calculated coupler efficiency is 76.1%, corresponding to a loss of 1.19 dB. Only 0.03% of power is reflected back by the SWG, yielding a negligible return loss of -35 dB. Using the same taper with a high-NA fiber, the calculated coupling efficiency is 81.4%, hence a loss of 0.89 dB (structure (E), Table 1). Further loss reduction can be expected by a judicious design, including parabolic rather then linear tapering of waveguide width and chirping the SWG pitch.
The results were obtained for an input mode with the electric field parallel to the simulation plane shown in Fig. 2. Because these 2D SWG structures are invariant (strips of infinite length) in direction orthogonal to the simulation plane with obviously no subwavelength segmentation effect existing in that direction, the 2D structures are not effective for electric field polarized along that direction. At this point we do not have computational tools capable of 3D FDTD simulations of these SWG coupling structures. Also notice that the coupling efficiency of the SWG structures does not vary significantly even for quite large variations in the grating parameters (see Table 1), indicating that the proposed method is robust and potentially tolerant to fabrication errors. For example, an increase in the SWG pitch from 0.2 µm to 0.3 µm results in a negligible excess loss of 0.03 dB, see Table 1, structures (D) and (F).
We calculated coupling tolerances to transverse and angular fiber misalignment for coupling from standard SMF-28 fiber. We found that transverse misalignments of ±1 µm and ±2 µm (along y axis, Fig. 2) result in an increased coupling loss by only 0.07 dB and 0.47 dB, respectively. The angular misalignment tolerance is also large, with only 0.24 dB loss penalty for angular misalignment of ±2 degrees. These results were obtained for the structure (D).
We also investigated a possibility to further reduce the SWG coupler size. We reduced the coupler length to 10 µm using the SWG pitch of 0.3 µm (structure (H) in Table 1), hence with as few as 33 grating periods. We also considered waveguide height tapering due to aspect ratio dependent etching, also known as reactive ion etching (RIE) lag that manifests itself as a variation in depth or height of an etched feature proportional to its width . The coupling effect of the sub-wavelength grating is shown in Fig. 3 with the spatial evolutions of the field components H y and E x visualized in the microscopic scale as the light propagates along the SWG structure. It can be observed that the input plane wavefront is rapidly transformed into a convergent spherical wavefont as a consequence of the SWG effect. The calculated coupling efficiency is 65.2% (1.85 dB loss) at λ=1550 nm, and 66.2% (1.79 dB loss) at λ=1570 nm, indicating a negligible wavelength dependence (0.05 dB loss variation for a 20 nm wavelength change), as expected from the nature of the SWG effect. We expect the loss can be further improved by optimizing the SWG structure, for example by chirping the grating period. This may be particularly beneficial in the first half of the coupler where the confinement is relatively weak and the loss is higher, as it can be inferred from the ‘wings’ of the fields in Fig. 3a. This compact SWG structure is among the smallest yet efficient couplers reported, and further reduction in length appears feasible.
We have proposed a new method for efficient coupling of light between an optical fiber and a high index contrast waveguide with sub-micrometer dimensions. Our FDTD calculations demonstrate that the effective index modification by a subwavelength waveguide grating is an effective method for making highly efficient and compact fiber-chip couplers and mode transformers. The method is obviously not limited to waveguides in high index contrast platforms such as SOI, III–V semiconductors, and silicon oxynitride, and it can also be used, for example, in conventional silica-on-silicon waveguides and silica photonic wires . The principle proposed here can be generalized to other types of waveguide modifications by SWG effect, opening new possibilities for engineering of waveguide properties at subwavelength scale.
We would like to thank Dr. Stoyan Tanev for many stimulating discussions on FDTD theory and simulations.
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