Si₃N₄ → UHNA3 inverse-taper SSC @ 1550 nm

Platform & fiber parameters

UHNA3 (Nufern) is an ultra-high-NA fiber: NA 0.35, MFD ≈ 4.1 µm at 1550 nm. The last field is the mode-field diameter the inverse-taper tip presents to the fiber; the tip width is chosen so this ≈ the fiber MFD. Set it equal to the bare-core MFD to see the loss without a converter.

The two FD values are the authoritative mode-overlap results from the sparse finite-difference eigensolver (downloads below). The interactive row is a quick Gaussian–Gaussian overlap estimate as you vary the tip-mode MFD.

Mode-field cross-section — blue: UHNA3 fiber Gaussian, red dashed: bare Si₃N₄ core mode (tiny), green: expanded inverse-taper tip mode. Larger overlap with the fiber means lower loss.

• Si₃N₄ (n ≈ 1.996) in SiO₂ (n ≈ 1.444) is a high index-contrast platform. A 1.5 × 0.3 µm core tightly confines the mode to ≈ 1.5 × 0.9 µm — far smaller than the UHNA3 fiber (MFD 4.1 µm), so the bare butt-coupling loss is a huge ≈ 5.0 dB.
• Unlike the low-contrast silica SSCs (which widen the core), the fix here is to make the core narrower: an inverse taper shrinks the Si₃N₄ width toward a sub-wavelength tip. As the width drops, the effective index falls toward the SiO₂ cladding and the mode delocalizes and expands into the oxide.
• At the optimal tip width the expanded mode matches the fiber; the taper is made long enough (≈ 200 µm) to be adiabatic, so the device coupling loss equals the tip-mode / fiber mismatch.
• UHNA3's small 4.1 µm MFD is chosen deliberately: it sits between the on-chip mode and a standard SMF-28 (10.4 µm), so a short UHNA fiber pigtail bridges Si₃N₄ to SMF with low loss at both ends.

Butt-coupling loss is the mode-overlap integral between the fiber mode and the waveguide (tip) mode:

The 2-D waveguide mode E_wg is found by solving the scalar Helmholtz eigenproblem (∂²ₓ + ∂²ᵧ)E + k₀²n(x,y)²E = β²E on the real Si₃N₄/SiO₂ index cross-section with a sparse shift-invert eigensolver (largest n_eff); E_fiber is the UHNA3 field (Gaussian, MFD 4.1 µm).

Why not an imaginary-distance BPM mode solve. For a high-contrast Si₃N₄ wire near cutoff, the paraxial imaginary-distance iteration is unstable — it collapses onto a spurious tightly-bound mode. The direct finite-difference eigensolve is grid-convergent and robust, so it is used for the reported numbers.

Sweeping the Si₃N₄ tip width and overlapping each tip mode with the UHNA3 field gives a clear minimum. The mode expands monotonically as the tip narrows; the loss bottoms out where the tip mode best matches the 4.1 µm fiber.

The optimum is broad — a tip width in ≈ 0.22–0.24 µm keeps the loss under 0.45 dB, giving good lithography tolerance. The residual ≈ 9 % (0.42 dB) is the intrinsic shape mismatch between the near-cutoff waveguide mode (peaked centre, exponential tails) and the fiber's Gaussian; even when the D4σ matches, the overlap is capped near 91 %. The 3 µm thermal-oxide BOX is assumed thick enough that substrate leakage of the ≈ 4.6 µm tip mode is negligible.

Everything needed to reproduce the result above:

• ssc_optimized.gds — Si₃N₄ inverse-taper mask (1.5 µm chip → 0.23 µm tip over 200 µm, layer 1).
• ssc_reference.gds — the bare straight 1.5 µm Si₃N₄ waveguide (no converter).
• run_sin_uhna3_ssc_1550.py — driver: sweeps the tip width, finds the optimum, computes the UHNA3 mode-overlap loss, and writes the GDS + figures.
• fdmode.py — the sparse finite-difference scalar mode solver (Si₃N₄/SiO₂ index builder, Helmholtz eigensolve, overlap & D4σ helpers).
• ssc_result_summary.png — full result figure.
• ssc_compare.png — mode-matching comparison figure.

Run with python3 run_sin_uhna3_ssc_1550.py --out . (needs numpy, scipy, gdstk, matplotlib). Coupling loss is the mode-overlap integral between the eigensolver's 2-D waveguide mode and the UHNA3 field; pass --dx 0.018 --dy 0.013 for a finer convergence check.