Angular Response Simulator
In real cameras, light hits pixels at oblique angles depending on sensor position and lens design. Chief ray angle (CRA) can reach 20–30° at the sensor edge, significantly affecting QE.
Angular Response Simulator
Explore how quantum efficiency changes with angle of incidence — critical for understanding Chief Ray Angle (CRA) effects in image sensors.
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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Chief-Ray Angle and Polarization Response
A tilted ray does not simply travel through the same stack sideways. It refracts at each interface, sees different effective thicknesses, and splits into s and p polarization responses. At the edge of a camera field, that planar-film angular rolloff is only one part of the story; microlens shift and finite-cone averaging decide whether the focused spot still lands on the photodiode.
Angular response explains how QE changes when light enters a planar pixel stack away from normal incidence. It combines Snell refraction, s/p Fresnel behavior, TMM phase changes, longer absorption path length, and normalization against the on-axis response.
Assumptions
- The page evaluates planar-stack angular response, not a complete off-axis pixel with shifted microlenses and finite aperture.
- s and p polarization are computed separately and can be averaged for unpolarized illumination.
- CRA is represented as incidence angle at the stack entrance; lens-pupil cone averaging is handled by separate cone-illumination tools.
Outputs
- Normalized QE or transmission versus angle, with separate s, p, and unpolarized trends.
- Screening information for angular rolloff, color-channel imbalance, and when CRA compensation needs microlens or stack redesign.
Validation Example
- At $\theta=0$, s and p responses should coincide for an isotropic planar stack.
- Increasing angle should generally lengthen optical path and shift thin-film features; if no feature moves, check whether the stack is too simplified.
Core Equations
- \(\tilde{n}_0,\tilde{n}_j\): Complex refractive indices of incident medium and layer $j$
- \(\theta_0\): External incidence angle or chief-ray angle in the planar-stack model
- \(\theta_j\): Internal angle inside layer $j$
High-index silicon strongly reduces the internal ray angle, but upper polymer/filter layers can still see meaningful obliquity.
- \(r_s,r_p\): Reflection amplitudes for s and p polarization
- \(n_i,n_t\): Incident and transmitted side refractive indices
- \(\theta_i,\theta_t\): Incident and transmitted angles at an interface
s and p responses can diverge strongly at high angles, especially near Brewster-like conditions.
- \(\eta_j^{(s)},\eta_j^{(p)}\): Optical admittance of layer $j$ for s and p polarization
- \(\tilde{n}_j\): Complex refractive index
- \(\cos\theta_j\): Obliquity factor inside the layer
The same physical stack has different effective boundary conditions for the two polarization states.
- \(\delta_j(\theta_0)\): Phase thickness induced by external incidence angle $\theta_0$
- \(d_j\): Layer physical thickness
- \(\lambda\): Vacuum wavelength
A phase shift with angle can move BARL minima and color-filter interference features across wavelength.
- \(AR_{p/s}\): Normalized angular response for each polarization
- \(QE_{p/s}\): Optical QE proxy for s or p polarization
- \(AR_{\text{unpol}}\): Unpolarized angular-response estimate
Normalization removes absolute QE scale and highlights angular rolloff or angular gain.
- \(\Omega\): Angular cone set by lens f-number and chief-ray direction
- \(W(\theta,\phi)\): Angular weighting function of the optical system
- \(\phi\): Azimuthal angle inside the cone
A real camera pixel receives a cone of rays, not a single plane wave, so edge-pixel behavior needs cone averaging.
Model Interpretation
- Planar angular response is a stack property; full CRA response is a pixel property that also includes microlens focusing and lateral collection.
- s/p separation is not optional at high angle because coating minima and Fresnel loss can split by polarization.
- A good edge-pixel design usually needs stack tuning, microlens shift, and finite-cone validation together.
How To Read The Plot
- If both s and p fall together, the loss is likely path-length or absorption driven; if they split, Fresnel/admittance effects are important.
- A response above 1 at some angle can occur when interference minima shift in a favorable direction; it is not automatically an error.
- Compare wavelength slices, not only broadband averages, because angular color shading is channel dependent.
CRA Design Implications
- At the sensor edge, the chief ray may enter at high angle even when the lens cone has many nearby angles.
- Microlens shift compensates the lateral focus displacement, while stack tuning controls planar-film throughput at that angle.
- Use this angular page before expensive ray/FDTD studies to identify angle bands and wavelengths at risk.
Known Missing Physics
- The model does not trace the focused spot to a finite photodiode or model microlens shift.
- It ignores lateral diffraction, metal-grid shadowing, color-filter relief, and DTI-induced waveguide effects.
- For finite aperture filters or Fabry-Perot stacks, integrate the full angular cone rather than relying on a single angle.
References
- Hwang & Kim, "A Numerical Method of Aligning the Optical Stacks for All Pixels", Sensors, 2023 — Practical CIS context for aligning optical stacks under pixel-dependent chief-ray angle.
- Goossens et al., "Finite aperture correction for spectral cameras with integrated thin-film Fabry-Perot filters", Applied Optics, 2018 — Finite-aperture angular averaging reference for spectral cameras with thin-film filters.
- Macleod, Thin-Film Optical Filters, 5th ed., CRC Press, 2017 — Thin-film angular and polarization response theory.