Thin Film Optics

Prerequisites

Optics Primer -> Light Basics -> Electromagnetic Waves -> this page

Thin films are everywhere in an image sensor pixel -- the anti-reflection coating (BARL), the color filter, even the microlens. When layers are only nanometers thick, interference effects become critical. Understanding thin film optics helps you design better BARL stacks and predict how layer thicknesses affect QE.

Thin film interference is one of the most important optical effects in image sensor design. Anti-reflection coatings, BARL layers, and even the planarization layer all rely on thin-film principles.

Single thin film

Consider a thin film of thickness $d$ and refractive index $n_f$ on a substrate. Light reflected from the top and bottom surfaces of the film interferes:

$$\text{Path difference}=2n_{f}d\cos {\theta }_f$$

where ${\theta }_f$ is the angle of propagation inside the film. The interference condition for minimum reflection (destructive interference of the two reflected beams) is:

$$2n_{f}d\cos {\theta }_f=(m+\frac{1}{2})\lambda ,m=0,1,2,\dots$$

For a quarter-wave anti-reflection (AR) coating at normal incidence ($m=0$):

$$d=\frac{\lambda }{4n_f}$$

And zero reflection occurs when the film index satisfies:

$$n_f=\sqrt{n_{\text{air}}\cdot n_{\text{substrate}}}$$

Interactive Fresnel Calculator

Explore how reflection and transmission depend on refractive indices and incidence angle.

n₁ (incident medium)Air=1.0, Glass=1.5, Water=1.33

n₂ (transmitted medium)SiO₂=1.46, Si₃N₄=2.0, Si=3.5

Angle of incidence: 0.0°

R_s (TE)

4.00%

R_p (TM)

4.00%

R_avg (unpolarized)

4.00%

Brewster Angle

56.31°

Multi-layer stacks

Real image sensors use multiple thin films. The BARL (Bottom Anti-Reflection Layer) is a multi-layer dielectric stack at the color-filter / silicon interface, but the exact materials, the number of layers (typically 2 to 5), and the stacking order vary substantially from vendor to vendor — these choices are part of each vendor's process recipe and are usually not disclosed. Common building blocks include SiO2, Si3N4, HfO2, Al2O3, TiO2, and Ta2O5, used in different combinations to hit a target broadband anti-reflection band.

The example below is one illustrative 4-layer stack that ships with the COMPASS sample configs. It is not a "correct" recipe — re-optimize the materials, layer count, and thicknesses for whatever silicon thickness, color filter, and target QE band you are actually modeling:

barl:
  layers:
    - thickness: 0.010   # example layer 1
      material: "sio2"
    - thickness: 0.025   # example layer 2
      material: "hfo2"
    - thickness: 0.015   # example layer 3
      material: "sio2"
    - thickness: 0.030   # example layer 4
      material: "si3n4"

For multi-layer stacks, the transfer matrix method (TMM) provides an exact solution. Each layer is represented by a 2x2 matrix:

$$M_j=\left(\begin{array}{cc}\cos {\delta }_j& -\frac{i}{{\eta }_j}\sin {\delta }_j\\ -i{\eta }_j\sin {\delta }_j& \cos {\delta }_j\end{array}\right)$$

where ${\delta }_j=k_0{n}_j{d}_j\cos {\theta }_j$ is the phase thickness and ${\eta }_j$ is the admittance (different for TE and TM). The total system matrix is the product:

$$M=M_1\cdot M_2\cdots M_N$$

The overall reflection and transmission coefficients follow from the matrix elements.

Interactive Thin Film Reflectance Calculator

Compute reflectance spectra using the transfer matrix method for common anti-reflection coating configurations.

Preset:Single layer ARCBARL (SiO2/HfO2)Quarter-wave stack

Layer 1Si3N4 (n=2.00)

Thickness: 69 nm

Min Reflectance

0.00%

At Wavelength

552 nm

Substrate

Si (n=4.0)

Incident Medium

Air (n=1.0)

Role in BSI pixels

In a backside-illuminated pixel, light enters through the silicon backside and must pass through several layers before reaching the photodiode:

Incident light
      |
      v
  [Air]
  [Microlens]         -- focuses light onto pixel
  [Planarization]     -- uniform dielectric
  [Color filter]      -- wavelength-selective absorption
  [BARL layers]       -- anti-reflection at color-filter/silicon interface
  [Silicon + DTI]     -- photodiode region

The BARL stack is designed to minimize reflection at the color-filter-to-silicon interface. Without it, the large refractive index mismatch (color filter $n\approx 1.55$, silicon $n\approx 4$) would cause roughly 30--40% reflection, severely reducing QE.

Spectral response effects

Thin film interference produces wavelength-dependent oscillations in the QE spectrum. These "Fabry-Perot" fringes are a common feature of image sensor simulations:

• Constructive interference at certain wavelengths boosts the QE.
• Destructive interference at other wavelengths creates dips.

The fringe spacing is approximately:

$$\mathrm{\Delta }\lambda \approx \frac{{\lambda }^2}{2nd}$$

For a 3 um silicon layer at 600 nm, $\mathrm{\Delta }\lambda \approx 15$ nm. This means you need a wavelength step of at most 5 nm to resolve the fringes (Nyquist criterion).

WARNING

If your QE spectrum looks jagged or shows unexpected oscillations, check that your wavelength step is fine enough to resolve thin-film fringes. A step of 10 nm or smaller is recommended.

Angle-induced peak shift

For a Fabry-Perot-like resonator (such as a color filter or a thin-film bandpass filter integrated on a sensor), the transmission peak at wavelength ${\lambda }_0$ shifts to shorter wavelengths under oblique incidence approximately as

$$\lambda (\theta )\approx {\lambda }_0\sqrt{1-{\left(\frac{\sin \theta }{n_{\text{eff}}}\right)}^2}$$

where $n_{\text{eff}}$ is the effective index of the cavity mode. When a real lens focuses light onto the pixel, the simulation must therefore integrate over the cone of incidence angles set by the f-number and the chief-ray angle (CRA); the measured peak both broadens and blue-shifts relative to the plane-wave result. Goossens et al. (2018) derive a closed-form convolution model and a compact correction formula for this shift -- see Key Papers § 6.4. In COMPASS this effect is captured by the cone_illumination source.

COMPASS implementation

COMPASS handles thin films differently depending on the solver:

• RCWA: Each uniform thin film layer is represented exactly as a single layer slice with thickness $d$ and permittivity $\epsilon (\lambda )$. No approximation is needed.
• FDTD: Thin films must be resolved by the spatial grid. A 10 nm BARL layer requires grid spacing $\le 5$ nm, which can increase memory and computation time.

The PixelStack class automatically constructs all layers from the YAML configuration and provides the appropriate representation to each solver type.