Diffraction and Periodic Structures

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As pixel pitches shrink below 1 um, the pixel structures become comparable in size to the wavelength of light. At this point, light doesn't just travel in straight lines -- it bends around edges and through openings. This is diffraction, and it's the fundamental reason why we need wave-optics simulation rather than simple ray tracing.

Why Diffraction Matters for Image Sensors

When a CMOS pixel pitch (typically 0.7--1.4 um) approaches the wavelength of visible light (0.38--0.78 um), simple ray optics breaks down. Light diffracts around the pixel structures -- microlens edges, deep trench isolation (DTI) walls, metal grids -- and interference effects dominate the optical behavior. This is precisely why we need wave-optics solvers like RCWA and FDTD.

Diffraction Basics

Huygens' Principle

Every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the envelope of these wavelets. When a wave encounters an obstacle or aperture comparable to its wavelength, the wavelets from the edges spread into the shadow region -- this is diffraction.

Single Slit Analogy

Think of a single pixel aperture of width $a$. The angular spread of the first diffraction minimum is:

$$\sin \theta =\frac{\lambda }{a}$$

For a 1 um pixel at $\lambda =0.55$ um: $\theta \approx 33°$. This wide angular spread means light entering one pixel can easily reach a neighboring pixel's photodiode -- a major source of optical crosstalk.

Diffraction Gratings

Periodic Structures

A CMOS image sensor with its repeating pixel pattern is essentially a 2D diffraction grating. The Bayer color filter array, DTI grid, and metal wiring all form periodic structures.

For a grating with period $\mathrm{\Lambda }$ (the pixel pitch), incident light at angle ${\theta }_i$ is diffracted into discrete orders $m$:

$$\mathrm{\Lambda }(\sin {\theta }_m-\sin {\theta }_i)=m\lambda ,m=0,\pm 1,\pm 2,\dots$$

Each diffraction order carries a fraction of the incident power. The distribution among orders depends on the detailed structure within each period -- which is exactly what RCWA computes.

Grating Equation in 2D

For a 2D periodic structure with periods ${\mathrm{\Lambda }}_x$ and ${\mathrm{\Lambda }}_y$:

$$k_{x,mn}=k_{x,\text{inc}}+m\frac{2\pi }{{\mathrm{\Lambda }}_x},k_{y,mn}=k_{y,\text{inc}}+n\frac{2\pi }{{\mathrm{\Lambda }}_y}$$

where $k_{x,\text{inc}}=k_0\sin \theta \cos \varphi$ and $k_{y,\text{inc}}=k_0\sin \theta \sin \varphi$.

Floquet's Theorem (Bloch's Theorem)

The Key Insight

For a periodic structure with period $\mathrm{\Lambda }$, any solution to Maxwell's equations can be written as:

$$\mathbf{E}(x+\mathrm{\Lambda },y,z)=\mathbf{E}(x,y,z)\cdot e^{i{k}_{x,\text{inc}}\mathrm{\Lambda }}$$

In words: the field at $x+\mathrm{\Lambda }$ is the same as at $x$, multiplied by a phase factor determined by the incident angle. This is Floquet's theorem (or Bloch's theorem in solid-state physics).

Why This is Powerful

Floquet's theorem means we only need to simulate one unit cell of the periodic structure. The fields everywhere else are determined by phase shifts. This is why RCWA is so efficient for periodic structures like pixel arrays -- we simulate a single 2x2 Bayer unit cell, and the result applies to the entire sensor.

Fourier Expansion

Inside one period, we expand the fields in a Fourier series:

$$E_x(x,y,z)=\sum _{m,n}{a}_{mn}(z)e^{i(k_{x,m}x+k_{y,n}y)}$$

The number of Fourier terms retained (the Fourier order $N$) determines the accuracy. Higher orders capture finer structural details but increase computation as $O(N^3)$.

Pixel Pitch vs Wavelength

The ratio $\mathrm{\Lambda }/\lambda$ determines the diffraction regime:

Ratio $\mathrm{\Lambda }/\lambda$	Regime	Behavior	Example
$\gg 1$	Ray optics	Geometric shadows, minimal diffraction	10 um pixel, visible light
$\approx 1$--3	Resonance	Strong diffraction, interference dominates	0.7--1.4 um pixels
$\ll 1$	Sub-wavelength	Effective medium behavior	Nanophotonic structures

Modern BSI pixels (0.7--1.4 um pitch) fall squarely in the resonance regime, where wave-optics simulation is essential.

Connection to RCWA

RCWA leverages Floquet's theorem directly:

1. Periodic boundary conditions: The unit cell is repeated infinitely via Floquet phase shifts
2. Fourier expansion: Permittivity and fields are expanded in Fourier harmonics
3. Layer-by-layer: Each z-slice is solved as a uniform grating layer
4. S-matrix: Layers are cascaded using the scattering matrix algorithm

The Fourier order $N$ controls the trade-off between accuracy and speed. For a 2x2 Bayer cell at 1 um pitch, typical values are $N=5$--$15$ per direction, giving $(2N+1{)}^2=121$--$961$ harmonics.

Convergence and Fourier Order

Gibbs Phenomenon

Sharp permittivity boundaries (e.g., Si / SiO2 interface, DTI walls) cause the Fourier series to exhibit ringing near discontinuities. This is the Gibbs phenomenon and is a key source of slow convergence in RCWA.

Li's factorization rules (implemented in COMPASS's stability module) address this by correctly handling the Fourier factorization of products of discontinuous functions, dramatically improving convergence for TM polarization.

Fourier Order Approximation Demo

See how increasing the number of Fourier harmonics improves the approximation of a square wave (representing a DTI trench or metal grid cross-section). Notice the Gibbs phenomenon ringing at the edges.

Fourier Order N: 5

Duty Cycle: 50%

Total Harmonics:11

Matrix Size (RCWA):11×11

Gibbs Overshoot:18.8%

Gibbs phenomenon: Even with many harmonics, the Fourier series overshoots by ~9% at discontinuities. In RCWA, this affects convergence at sharp material boundaries (e.g., Si/SiO₂ DTI walls). Li's factorization rules mitigate this for TM polarization.

Convergence Guidelines

Structure	Minimum Order	Recommended
Uniform layers only	$N=1$	$N=3$
Color filter pattern	$N=5$	$N=7$--$9$
Metal grid (high contrast)	$N=9$	$N=11$--$15$
Fine DTI features	$N=7$	$N=11$--$13$

Always run a convergence test: increase $N$ until the QE change is below your tolerance (typically < 0.5%).

Further Reading

• Petit, R. (ed.), Electromagnetic Theory of Gratings, Springer (1980)
• Li, L., "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," JOSA A 13(5), 1996
• Moharam, M.G. et al., "Formulation for stable and efficient implementation of RCWA," JOSA A 12(5), 1995