Energy Budget Analyzer
Track where every photon's energy goes as it traverses the pixel stack. At each wavelength, the incident energy splits into reflection, per-layer absorption, and transmission.
Energy Budget Analyzer
Visualize where photon energy goes at each wavelength — how much is reflected, absorbed in each layer, or transmitted through the pixel stack.
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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Photon Energy Accounting
Think of every incoming photon as a coin. Some coins bounce off the stack, some pass through the active silicon without being used, some are spent heating filters or coatings, and only the coins absorbed in the photodiode silicon can become signal. The budget tells you which bucket is stealing photons.
The energy-budget view decomposes incident optical power into reflection, escape transmission, useful silicon absorption, and parasitic absorption in non-silicon layers. It is a diagnostic layer on top of the TMM result: it explains why QE changes, not just that QE changes.
Assumptions
- The stack is treated as passive, so incident energy must split into reflection, transmission, and layer absorption.
- Layer absorption is an optical power bookkeeping term; it does not guarantee charge collection in the photodiode.
- The browser model uses simplified layer spectra and therefore supports diagnosis rather than process sign-off.
Outputs
- Per-wavelength fractions for reflected, transmitted, useful silicon-absorbed, and parasitic absorbed energy.
- A diagnosis of whether QE loss is dominated by front-surface reflection, filter/coating absorption, or insufficient silicon absorption.
Validation Example
- For every wavelength, verify that $R+T+\sum A_j$ stays close to 1 before interpreting layer tradeoffs.
- Removing an absorbing filter should reduce parasitic absorption and move that energy into reflection, transmission, or silicon absorption.
Core Equations
- \(P_{\text{inc}}\): Incident optical power at a wavelength, normalized to unity
- \(\lambda\): Wavelength at which the stack is evaluated
All bars in the budget are fractions of the same incident power, so channels and wavelengths can be compared directly.
- \(\varepsilon\): Numerical energy-balance residual
- \(R,T\): Reflected and transmitted power fractions
- \(A_j\): Absorption fraction in layer $j$
A large residual means the optical model, interpolation, or layer accounting should be inspected before interpreting QE.
- \(P_{\text{top},j},P_{\text{bot},j}\): Normalized Poynting flux at the top and bottom of layer $j$
- \(E,H\): Electric and magnetic field amplitudes recovered from the transfer matrix
- \(H^*\): Complex conjugate of magnetic field amplitude
This assigns absorption to physical layers instead of only reporting total stack loss.
- \(A_{\text{use}}\): Absorption that can potentially contribute to signal
- \(A_{\text{parasitic}}\): Absorption in filters, coatings, metal, or other non-photodiode layers
- \(A_{\text{Si}}\): Absorption in the silicon photodiode layer
A design that reduces reflection but increases parasitic absorption may not improve sensor signal.
- \(QE_{\text{ext}}\): External quantum efficiency after carrier collection effects
- \(\eta_{\text{cc}}\): Carrier-collection probability for generated electron-hole pairs
The budget reports optical absorption; electrical QE can only be lower unless carrier collection is ideal.
- \(S_{x_k}\): Sensitivity of silicon absorption to design variable $x_k$
- \(x_k\): Layer thickness, material index, angle, or wavelength-dependent filter parameter
Sensitivity helps identify whether a QE loss is controlled by BARL thickness, silicon depth, filter absorption, or angle.
Model Interpretation
- Reflection-limited loss points to coating or BARL work; parasitic absorption points to material or thickness choices; escape transmission points to silicon thickness or light trapping.
- Energy accounting is the bridge between a spectral curve and an engineering action.
- The budget is most useful when compared across a before/after design change rather than read as a standalone number.
Diagnosis Workflow
- First check $\varepsilon$ to confirm power conservation, then inspect whether loss sits in $R$, $T$, $A_{\text{parasitic}}$, or missing $A_{\text{Si}}$.
- If $R$ dominates, tune BARL/coating; if $A_{\text{parasitic}}$ dominates, revisit color-filter or metal absorption; if $T$ dominates, adjust silicon thickness or backside trapping.
- Compare the same budget for R/G/B because a fix for one channel can move loss into another channel.
Calibration Targets
- Use measured reflectance/transmittance spectra to anchor $R$ and $T$ before trusting layer absorption allocation.
- Replace generic $n,k$ with process-specific ellipsometry for color filters, BARL layers, silicon, and passivation.
- Validate $A_{\text{Si}}$ against measured QE only after estimating carrier collection probability.
Known Missing Physics
- The budget does not include lateral redistribution: photons absorbed in silicon outside the photodiode ROI may become crosstalk.
- It does not model electrical collection, diffusion, recombination, or depletion-region geometry.
- Structured pixels need RCWA/FDTD field maps to decide where within silicon absorption occurs.
References
- Catrysse & Wandell, "Optical efficiency of image sensor pixels", JOSA A, 2002 — Image-sensor optical efficiency framing and connection between absorption and pixel-level efficiency.
- Green, "Self-consistent optical parameters of intrinsic silicon at 300 K", Solar Energy Materials and Solar Cells, 2008 — Silicon optical constants for wavelength-dependent absorption-depth and escape-loss analysis.
- Macleod, Thin-Film Optical Filters, 5th ed., CRC Press, 2017 — Thin-film energy balance and multilayer optical power accounting background.