TMM Validation Results

This page documents actual simulation results from COMPASS, executed on a standard 1.0 um BSI pixel stack using the TMM (Transfer Matrix Method) solver. TMM treats the stack as a 1D planar thin-film structure, making it ideal for rapid stack design, BARL optimization, and baseline validation before full RCWA simulation.

Solver Note

TMM is a 1D solver — it ignores lateral patterning (microlens shape, Bayer CF pattern, DTI, metal grid). For full 2D/3D effects, use an RCWA solver (torcwa, grcwa, meent, fmmax). TMM results serve as a fast baseline and anti-reflection coating design tool.

Pixel Stack Structure

The simulated 1.0 um pitch BSI (Back-Side Illumination) pixel stack consists of 9 layers. Light propagates from air (top) through the stack to silicon (bottom).

Layer	Material	Thickness (um)	Patterned	n @ 550nm
air	air	1.000	-	1.000
microlens	polymer_n1p56	0.600	Yes	1.573
planarization	SiO₂	0.300	-	1.460
color_filter	cf_green	0.600	Yes	1.55+0.02i
barl_3	Si₃N₄	0.030	-	2.023
barl_2	SiO₂	0.015	-	1.460
barl_1	HfO₂	0.025	-	1.966
barl_0	SiO₂	0.010	-	1.460
silicon	silicon	3.000	Yes	4.08+0.03i

• Total stack height: 5.580 um
• BARL: 4-layer SiO₂/HfO₂/SiO₂/Si₃N₄ anti-reflection stack (80 nm total)
• Bayer pattern: RGGB 2×2 unit cell

Wavelength Sweep: 380–780 nm

Full visible spectrum sweep at normal incidence, unpolarized light.

Solver: TMM | Pixel: 1.0um BSI | Angle: 0° | Polarization: unpolarized
Runtime: 2.8 ms (41 wavelengths)

λ (nm)	R	T	A	R+T+A
380	0.0401	0.0517	0.9081	1.000000
400	0.0529	0.0722	0.8750	1.000000
420	0.0961	0.0830	0.8210	1.000000
440	0.0348	0.1085	0.8566	1.000000
460	0.0633	0.1443	0.7923	1.000000
480	0.0324	0.2549	0.7127	1.000000
500	0.1461	0.4549	0.3990	1.000000
520	0.0088	0.9135	0.0777	1.000000
540	0.2405	0.7023	0.0572	1.000000
560	0.1697	0.4775	0.3528	1.000000
580	0.0280	0.3331	0.6389	1.000000
600	0.0387	0.2423	0.7190	1.000000
620	0.0477	0.2175	0.7349	1.000000
640	0.0180	0.2258	0.7563	1.000000
660	0.0209	0.2336	0.7455	1.000000
680	0.0781	0.2284	0.6935	1.000000
700	0.1199	0.2252	0.6549	1.000000
720	0.1091	0.2345	0.6564	1.000000
740	0.0664	0.2522	0.6815	1.000000
760	0.0372	0.2667	0.6961	1.000000
780	0.0430	0.2721	0.6849	1.000000

Key observations:

• Peak absorption (90.8%) at 380 nm — silicon's large imaginary part of the refractive index (k ≈ 0.34) at short wavelengths leads to near-complete absorption.
• Transmission dip at 520 nm — the BARL and thin-film interference create a transmission window at 520 nm (T = 0.91), reducing silicon absorption to just 7.8%. This is a Fabry-Pérot resonance in the stack.
• Long wavelengths (> 600 nm) — absorption stabilizes around 65–75% as silicon's extinction coefficient decreases.

Energy Conservation Validation

Wavelength range : 380–780 nm (1 nm step, 41 points)
Max |R+T+A − 1| : 1.11 × 10⁻¹⁶
Mean |R+T+A − 1| : 1.08 × 10⁻¹⁷
Status           : PASS ✓

TMM achieves machine-precision energy conservation, as expected for an analytical method with no discretization error.

Polarization-Resolved Angle Sweep

Reflection, transmission, and absorption at 550 nm for TE, TM, and unpolarized incidence across 0–80°.

Angle	R_TE	T_TE	A_TE	R_TM	T_TM	A_TM	R_unpol
0°	0.2526	0.5618	0.1856	0.2526	0.5618	0.1856	0.2526
10°	0.2509	0.5615	0.1875	0.2410	0.5694	0.1896	0.2460
20°	0.1886	0.6036	0.2079	0.1558	0.6298	0.2145	0.1722
30°	0.0143	0.7252	0.2606	0.0089	0.7340	0.2570	0.0116
40°	0.1398	0.6263	0.2339	0.0664	0.6869	0.2467	0.1031
50°	0.1866	0.5875	0.2258	0.0458	0.6968	0.2574	0.1162
60°	0.0740	0.6626	0.2634	0.0597	0.6790	0.2613	0.0669
70°	0.5079	0.3471	0.1450	0.0283	0.6926	0.2792	0.2681
80°	0.7312	0.1870	0.0818	0.1200	0.6210	0.2590	0.4256

Key observations:

• Brewster-like minimum near 30° — R drops to ~1% for both TE and TM. This is a destructive interference minimum in the multilayer stack, not a simple Brewster angle (which only exists for TM at single interfaces).
• TE vs TM divergence above 40° — TM maintains low reflection while TE reflection increases sharply, as expected from Fresnel equations.
• Grazing incidence (80°) — TE reflectance reaches 73%, while TM remains at 12%, showing the strong polarization splitting.

BARL Anti-Reflection Stack Impact

Effect of the 4-layer BARL (Bottom Anti-Reflection Layer) on optical performance at 550 nm.

Configuration	R	T	A
No BARL	0.0434	0.6877	0.2689
Standard 4-layer BARL	0.2526	0.5618	0.1856
2× thickness BARL	0.0240	0.6958	0.2802

Interpretation

At 550 nm, the standard BARL configuration is not at its optimal anti-reflection point — it actually increases reflection compared to no BARL. This is because the BARL is optimized for broadband performance across 380–780 nm, not for a single wavelength. The 2× thickness BARL achieves better anti-reflection at 550 nm (R = 2.4%) but may have different broadband characteristics. BARL optimization requires a full wavelength sweep analysis.

Material Refractive Indices

Complex refractive index (n + ik) for all stack materials at key wavelengths. Values from COMPASS MaterialDB using cubic spline interpolation of tabulated data.

Material	400 nm	450 nm	500 nm	550 nm	600 nm	650 nm	700 nm
air	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
polymer_n1p56	1.5850	1.5798	1.5760	1.5732	1.5711	1.5695	1.5682
SiO₂	1.4701	1.4656	1.4623	1.4599	1.4580	1.4565	1.4553
cf_green	1.55+0.12i	1.55+0.11i	1.55+0.04i	1.55+0.02i	1.55+0.10i	1.55+0.12i	1.55+0.12i
HfO₂	2.0250	1.9988	1.9800	1.9661	1.9556	1.9473	1.9408
Si₃N₄	2.0726	2.0500	2.0344	2.0232	2.0149	2.0085	2.0035
silicon	5.38+0.34i	4.64+0.21i	4.21+0.13i	4.08+0.03i	3.88+0.05i	3.80+0.04i	3.75+0.03i
tungsten	3.46+2.72i	3.55+2.86i	3.61+2.98i	3.65+3.08i	3.68+3.17i	3.70+3.25i	3.72+3.33i

Notes:

• Silicon has strong dispersion: n drops from 5.38 (400 nm) to 3.75 (700 nm). The extinction coefficient k peaks at short wavelengths (UV absorption edge).
• Color filter (green) has a passband centered near 500–550 nm where k is minimized (0.02–0.04), with blocking regions (k > 0.10) at 400 nm and > 600 nm.
• Tungsten (metal grid material) has high k across all wavelengths, as expected for a refractory metal.

Snell's Law CRA Shift vs Tangent Approximation

Comparison of the upgraded Snell's law ray tracing shift (traces through all intermediate layers) versus the old tangent approximation (Δx = tan(CRA) × h × 0.5).

CRA (deg)	Snell shift (um)	tan approx (um)	Difference (um)
0°	0.000000	0.000000	0.000000
5°	0.109021	0.026247	+0.082775
10°	0.217825	0.052898	+0.164927
15°	0.326181	0.080385	+0.245796
20°	0.433829	0.109191	+0.324638
25°	0.540466	0.139892	+0.400574
30°	0.645724	0.173205	+0.472518

Key observations:

• Snell shift is 3–4× larger than the old tangent approximation at all angles. This is because the tangent formula only considered the microlens height (0.6 um × 0.5), while Snell's law traces through the entire stack below the microlens: planarization (0.3 um, n=1.46), color filter (0.6 um, n=1.55), BARL (0.08 um, n=1.46–2.02), and silicon to photodiode center (0.5 um, n=4.08).
• The refraction reduces angles at each interface (Snell's law: n₁ sin θ₁ = n₂ sin θ₂), so the lateral displacement per layer is smaller than geometric tan(θ), but the total accumulated shift is larger because it accounts for all layers.
• At 30° CRA, the Snell shift is 0.646 um — more than half the pixel pitch (1.0 um), indicating that proper microlens alignment is critical for edge-of-sensor performance.

TMM vs RCWA for CRA Analysis

The TMM solver treats all layers as uniform slabs, so microlens shift has no effect on TMM's R/T/A values. The shift only matters when using RCWA (which models the actual microlens shape and lateral light distribution). To see the full QE impact of CRA shift compensation, use torcwa or grcwa.

Execution Environment

Solver      : TMM (Transfer Matrix Method)
Backend     : NumPy (CPU)
Python      : 3.11.6
PyTorch     : 2.5.0 (available but not used by TMM)
Platform    : macOS (Darwin 25.2.0, Apple Silicon)

Simulation	Wavelengths	Runtime
Single wavelength (550 nm)	1	0.8 ms
Full sweep (380–780 nm, 20 nm step)	21	2.8 ms
Full sweep (380–780 nm, 10 nm step)	41	~4 ms
CRA sweep (7 angles × 2 configs)	14 runs	~12 ms