Optical Simulation Chapter Overview

This chapter explains the numerical methods COMPASS uses to solve Maxwell's equations for an image sensor pixel. It covers how a solver actually computes reflection, transmission, and absorption — not the optics or sensor architecture, which are covered in the previous chapters.

Why we need numerical methods

Maxwell's equations have closed-form solutions only for trivial geometries (a planar multilayer, a single sphere). A real BSI pixel has 3D periodic structures — color filter grids, sub-wavelength microlenses, DTI walls — for which no analytical solution exists. We discretize the problem and solve it on a computer.

There are three families of methods worth distinguishing:

Family	Domain	What it discretizes	COMPASS solver(s)
Analytical / TMM	1D planar	Closed-form transfer matrices per layer	tmm
Modal / RCWA	Frequency domain, periodic	Permittivity in Fourier basis, fields as Bloch modes	torcwa, grcwa, meent, fmmax
Time domain / FDTD	Time domain, arbitrary geometry	Yee-grid finite differences in space and time	fdtd_flaport, fdtdz, fdtdx, meep

A note on permittivity

All solvers internally represent materials by their complex permittivity $\epsilon ={\tilde{n}}^2=(n+ik{)}^2$. COMPASS stores material data as $(n,k)$ pairs in MaterialDB and converts to $\epsilon$ when building the simulation geometry. RCWA solvers expand $\epsilon (x,y)$ in a 2D Fourier series; FDTD solvers sample $\epsilon$ on the Yee grid; TMM uses $\epsilon$ per layer directly.

Choosing a solver

A rough decision tree:

• 1D planar stack (no lateral structure): use TMM. It is ~1000× faster than RCWA and exact for this geometry.
• 2D / 3D periodic, frequency-domain QE sweep: use RCWA. Fast for moderate Fourier order, well-suited to sub-wavelength gratings.
• Strongly resonant or large lateral extent, broadband transient effects: use FDTD. Slower per wavelength but handles any geometry.
• Cross-validation: run two methods of different families and compare — see RCWA vs FDTD.

What this chapter covers

Page	Topic	Key takeaway
RCWA Explained	Fourier expansion, eigenmodes, S-matrix recursion	Why RCWA is fast for periodic stacks
FDTD Explained	Yee lattice, update equations, Courant condition	Why FDTD is general but more expensive
RCWA vs FDTD	Side-by-side comparison and use cases	When to use which
Numerical Stability	Eigenvalue conditioning, S-matrix vs T-matrix, mixed precision	How COMPASS keeps RCWA stable at high Fourier order

Boundary with nearby pages

Nearby page	Difference
Optics	Defines the electromagnetic equations and wave phenomena; this chapter explains discretization and solver behavior.
Image Sensor	Defines the physical target and metrics; this chapter explains how those metrics are computed.
Reports	Publishes generated validation evidence; this chapter explains the methods behind those reports.

Prerequisites

This chapter assumes the Optics chapter — especially Electromagnetic Waves and Diffraction. If terms like Bloch mode or S-matrix are new, read those first.