We present a design agent which is a Large Language Model (LLM) that orchestrates, but does not perform, the numerical simulations to design a silicon-on-insulator (SOI) $2\times2$ directional coupler. We choose a symmetric phase-matched coupler where a lot of analytical results are available that help the design strategy. The LLM proposes candidate gap values (a geometrical dimension size) and judges convergence, while all physics is owned by deterministic solvers: a frequency-domain eigenmode solver estimates the coupling coefficient~$κ$ for the current design, and an independent Finite-Difference Time-Domain (FDTD) stage validates it. Both solvers operate on a common slab-projected two-dimensional (2D) effective-index reduction of the silicon film, so the design~$κ$ and the FDTD response are consistent by problem design; the residual between them is shown to be a single constant phase offset~$φ$, attributable to a fixed excess coupling length $L_{\mathrm{extra}}=\SI{2.837(11)}{\micro\meter}$ that we find invariant across a factor-of-two range in~$κ$. Folding this offset into a closed-loop length correction, the agent delivers a $50/50$ splitter whose FDTD-measured cross fraction is $0.498$ (target $0.500$), a residual of $0.0017$. Results are made self-consistent within the 2D effective-index model; and the LLM succeeds in delivering a suitable design over a number of attempts.

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