Backscattering design for a focusing grating coupler with fully etched slots for transverse magnetic modes

Grating couplers are a fundamental building block of integrated optics as they allow light to be coupled from free-space to on-chip components and vice versa. A challenging task in designing any grating coupler is represented by the need for reducing back reflections at the waveguide-grating interface, which introduce additional losses and undesirable interference fringes. Here, we present a design approach for focusing TM grating couplers that minimizes these unwanted reflections by introducing a modified slot that fulfills an anti-reflection condition. We show that this antireflection condition can be met only for the Bloch mode of the grating that concentrates in the dielectric. As a consequence the light is scattered from the grating coupler with a negative angle, referred to as “backscattering design”. Our analytic model shows that the anti-reflection condition is transferrable to grating couplers on different waveguide platforms and that it applies for both TE and TM polarizations. Our experimentally realized focusing grating coupler for TM-modes on the silicon photonics platform has a coupling loss of (3.95 ± 0.15) dB at a wavelength of 1.55 µm. It has feature sizes above 200 nm and fully etched slots. The reflectivity between the grating coupler and the connected waveguide is suppressed to below 0.16%.

One dimensional photonic crystal with anti-reflection boundary. In the following, we show that a 1D photonic crystal anti-reflection boundary can be designed to suppress back reflections when coupling from homogenous space into a photonic crystal. We show that the anti-reflection condition is met when half of the first slot in the grating is filled (i.e. 1 / = 0.5 ). A necessary condition to suppress reflections is that light is coupled into the dielectric mode of the photonic crystal slab, that is, the mode below the band gap with energy mainly concentrating in the dielectric material. This mode scatters light with a negative angle into free space as shown in Fig. 1(b) of the main text.  The Bloch mode in every layer of the stack can be described by a superposition of forward and backward running plane waves denoted by their amplitudes as and . Back reflections into the homogenous medium, that shall be mitigated by our approach, are denoted as .
When coupling from the homogenous medium into the photonic crystal, certain modes can be excited that obey the photonic crystal dispersion relation schematically shown in Fig. S1(b).
Generally, bands located above the light line can scatter light into free space. The relevant bands are the air and dielectric band at the second photonic crystal bandgap (PhC BG 2) as indicated in Fig. S1(b). Modes located at the first band gap do not have to be considered since they lie below the light line and do not scatter light to free space. In Fig. S1(a) we show schematically the electric field energy density distribution of the modes close to the second bandgap. We note that compared to the air mode, the dielectric mode concentrates a larger percentage of electric field energy in the silicon. We also remark that these modes accumulate a phase of 2 within the lattice constant Λ since they are located at the second bandgap.
We now show that reflections can be mitigated when coupling to the dielectric band of the photonic crystal. Following a transfer matrix approach, a system of linear equations can be formulated to describe the coupling from homogenous space into a photonic crystal 1,2 with Δ Si,Air being the transition matrix at the silicon/air boundary defined as Δ Si,Air = and Si,Air = Si− Air where 1+ Si,Air = Si,Air . Note that for normal incidence these coefficients are polarization independent. is the transmission coefficient of the Bloch mode described by a vector ( , ).
Solving Eq. (1) for yields In the case of suppressed back reflections, = 0, and it follows that Si,Air + = 0 since where Λ is the product of transition Δ and propagation matrices Π given by Λ = Thus, we will now consider the case that − / is real. As we have shown, this condition is fulfilled if the first air layer is terminated at half of the width of all other air layers (i.e. 1 / = 0.5 ) since it satisfies mirror symmetry at this position. Si,Air > 0 according to Eq. (3) since we couple from a high refractive index medium into a multilayer stack that starts with a low refractive index medium as shown in Fig. S1(a). The case Si,Air < 0 does not have to be considered since it corresponds to the case where we couple from a low refractive index medium into the grating that starts with a high refractive index medium. This case is not given when coupling from the waveguide into the grating.
For Si,Air > 0 it follows that / < 0 according to Eq. (5) meaning that and are out of phase in the air layer. This situation is only obtained when coupling to the dielectric mode of the photonic crystal which has out-of-phase components in air and consequently a low intensity in air. We remark that it cannot be fulfilled for the air mode since this mode has inphase components in air and thus / > 0. As shown in the band diagram in Fig. S1 (b), coupling to the dielectric mode results in a negative scattering angle since we can only couple to modes with a positive group velocity g > 0 when exciting from the direction of the waveguide (i.e. homogenous space).
In the last step it is important to match / and Si,Air in order to suppress according to Eq. (5). This matching happens automatically close to the band edge. At the band edge = 1 since modes are standing waves. However, / reduces from a value of 1 away from the band edge. Thus, there exists a frequency where the reflection is exactly zero.
In conclusion, we emphasize that the anti-reflection boundary approach only works when coupling to the dielectric mode of the photonic crystal slab. Therefore, designs in the antireflection configuration necessarily result in a negative scattering angle . The approach equally works for TE-and TM-polarizations and any filling fraction of the grating ((Λ − )/ Λ) since no assumption on these parameters is made. The refractive index contrast in the photonic crystal slab is also arbitrary making the approach transferrable to other waveguide platforms. Irrespective of these parameters, the condition to suppress reflections is always given when half of the first slot in the grating is filled , i.e. 1 / n = 0.5, representing a conceptually simple design rule.