TMM QE Calculator

Compute the quantum efficiency spectrum of a BSI CMOS pixel using the Transfer Matrix Method. Adjust silicon thickness, BARL layers, and angle of incidence to see real-time results.

TMM Quantum Efficiency Calculator

Configure a BSI pixel layer stack and see real-time R/T/A/QE spectra computed via the transfer matrix method.

Preset:

Silicon thickness: 3.0 µm

Angle of incidence: 0°

Polarization: s p unpolarized

Blue peak QE:68.2%

Green peak QE:85.1%

Red peak QE:65.5%

Average QE:72.9%

Model scope

Use this browser tool for intuition, relative trends, and design-space exploration. Its local simplified model is not a substitute for RCWA/FDTD sign-off, silicon calibration, or vendor process data.

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Quantum Efficiency · TMM Validation Results recipe

Physics Notes

Transfer-Matrix QE Model

Plain-English Intuition

Think of the pixel stack as many transparent plates. Each interface reflects a small wave; each layer adds phase delay and absorption. The transfer matrix keeps the amplitude and phase of all forward and backward waves coherent, so a thin BARL layer can increase QE by canceling reflection at one wavelength while hurting another wavelength.

The calculator treats the BSI pixel stack as a coherent one-dimensional thin-film system: air, polymer microlens proxy, planarization, color filter, BARL, and silicon. It computes wavelength- and angle-dependent reflection, transmission, and layer absorption; the plotted QE is the silicon-layer absorption proxy.

Assumptions

Every layer is laterally infinite, planar, homogeneous inside the layer, and coherent over the simulated optical path.
The plotted QE is silicon optical absorption, not collected charge after carrier transport.
Microlens, metal grid, DTI, and color-filter relief are collapsed into planar effective films.

Outputs

Wavelength-dependent $R(\lambda)$, $T(\lambda)$, parasitic layer absorption, and silicon absorption by color channel.
Angle and polarization trends for a flat stack, useful before switching to RCWA/FDTD for lateral geometry.

Validation Example

With all coating layers removed, compare the normal-incidence air/Si reflection against $R=|(1-n_{Si})/(1+n_{Si})|^2$ at the selected wavelength.
For a passive stack, verify $R+T+\sum_j A_j\approx1$ before interpreting QE peaks.

Core Equations

Complex refractive index

$$\tilde{n}_j(\lambda)=n_j(\lambda)+i k_j(\lambda)$$

$\tilde{n}_j$: Complex refractive index of layer $j$
$n_j$: Real refractive index controlling phase velocity
$k_j$: Extinction coefficient controlling absorption
$\lambda$: Vacuum wavelength

The browser model uses tabulated, Sellmeier, Cauchy, or constant material data depending on the layer.

Snell relation in each film

$$\tilde{n}_0\sin\theta_0=\tilde{n}_j\sin\theta_j, \quad \cos\theta_j=\sqrt{1-\left(\frac{\tilde{n}_0\sin\theta_0}{\tilde{n}_j}\right)^2}$$

$\theta_0$: External angle of incidence selected in the simulator
$\theta_j$: Internal propagation angle in layer $j$
$\tilde{n}_0$: Incident medium index, air in this simulator

At oblique incidence, every layer gets a different optical path length and different s/p polarization response.

Layer phase thickness

$$\delta_j = \frac{2\pi}{\lambda}\tilde{n}_j d_j\cos\theta_j$$

$\delta_j$: Complex phase thickness of layer $j$
$d_j$: Physical thickness of layer $j$
$\cos\theta_j$: Obliquity factor inside the layer

Interference fringes shift when layer thickness, wavelength, or angle changes because this phase term changes.

Characteristic matrix

$$M_j=\begin{bmatrix}\cos\delta_j & -i\sin\delta_j/\eta_j \\ -i\eta_j\sin\delta_j & \cos\delta_j\end{bmatrix}, \quad M=\prod_j M_j$$

$M_j$: Transfer matrix of layer $j$
$M$: Total stack transfer matrix
$\eta_j$: Optical admittance; $\eta_j=\tilde{n}_j\cos\theta_j$ for s polarization and $\eta_j=\tilde{n}_j/\cos\theta_j$ for p polarization in this implementation

The total matrix maps the boundary fields at the incident side to those at the substrate side.

Reflection and transmission amplitudes

$$r=\frac{\eta_0M_{00}+\eta_0\eta_sM_{01}-M_{10}-\eta_sM_{11}}{\eta_0M_{00}+\eta_0\eta_sM_{01}+M_{10}+\eta_sM_{11}}, \quad t=\frac{2\eta_0}{\eta_0M_{00}+\eta_0\eta_sM_{01}+M_{10}+\eta_sM_{11}}$$

$r,t$: Complex reflection and transmission amplitudes
$\eta_0$: Incident-side optical admittance
$\eta_s$: Substrate-side optical admittance
$M_{mn}$: Elements of the total transfer matrix

The implementation averages separate s and p calculations for unpolarized light.

Power balance and layer absorption

$$R=|r|^2, \quad T=\frac{\operatorname{Re}(\eta_s)}{\operatorname{Re}(\eta_0)}|t|^2, \quad A_j=P_{\text{top},j}-P_{\text{bot},j}$$

$R,T$: Reflected and transmitted power fractions
$A_j$: Absorbed power fraction assigned to layer $j$
$P_{\text{top},j}, P_{\text{bot},j}$: Normalized Poynting flux at the top and bottom of layer $j$

For a consistent passive stack, $R+T+\sum_j A_j\approx1$; numerical clipping only prevents tiny negative absorption artifacts.

Displayed QE proxy

$$QE_{c,\text{opt}}(\lambda)=100\,A_{\text{Si},c}(\lambda)$$

$QE_{c,\text{opt}}$: Displayed optical QE proxy for color channel $c$
$A_{\text{Si},c}$: Absorption fraction in the silicon layer of the selected color-filter stack
$c$: Color channel: red, green, or blue

This is an optical upper bound: carrier collection efficiency, recombination, electrical conversion gain, and pixel aperture effects are outside the model.

Model Interpretation

Best for flat BSI stack trends, BARL thickness tuning, color-filter absorption intuition, and angle/polarization sensitivity screening.
The color-filter curves in this browser tool are compact pedagogical spectra, not proprietary process-verified pigment data.
The polymer microlens layer is treated as a planar film, so focusing, lens shift, fill factor, and CRA-dependent spot displacement are not solved.

How To Read The Plot

A QE peak moving with silicon thickness usually indicates interference and absorption-depth tradeoff, not a direct change in electrical quantum yield.
A BARL layer that improves green may reduce blue or red because the phase-cancellation condition is wavelength dependent.
Angle sweeps should be interpreted as planar-stack angular response; they are not a full chief-ray-angle pixel model.

Calibration Checklist

Replace simplified $n,k$ data with process-specific ellipsometry for BARL, color filter, polymer, and silicon if quantitative accuracy matters.
Compare $R(\lambda)$ and $T(\lambda)$ against wafer optical metrology before trusting silicon absorption trends.
Use RCWA/FDTD when the stack has lateral structures: metal grids, DTI, color-filter relief, sub-wavelength texture, or microlens curvature.

Known Missing Physics

No diffraction, scattering, lateral crosstalk, finite pixel aperture, or microlens focusing is included.
Silicon absorption is not the same as collected charge; depletion depth, recombination, and carrier collection probability are omitted.
The calculation assumes laterally infinite coherent films, so roughness, thickness non-uniformity, and incoherent thick-layer effects are approximated away.

References

Macleod, Thin-Film Optical Filters, 5th ed., CRC Press, 2017 — Reference for thin-film characteristic matrices, optical admittance, and multilayer coating interpretation.
Green, "Self-consistent optical parameters of intrinsic silicon at 300 K", Solar Energy Materials and Solar Cells, 2008 — Useful source for wavelength-dependent intrinsic silicon optical constants.
Catrysse & Wandell, "Optical efficiency of image sensor pixels", JOSA A, 2002 — Connects optical efficiency modeling to image-sensor pixel stacks and QE interpretation.
Born & Wolf, Principles of Optics, 7th ed., Cambridge University Press, 1999 — Background for electromagnetic boundary conditions, polarization, and coherent optics.