Filling factor

The filling factor (sometimes called duty cycle) $f$ of a periodic refractive-index structure is the fraction of one period occupied by one of the two index regions. For a grating:

$$f=\frac{w_{\text{tooth}}}{\mathrm{\Lambda }},$$

where $w_{\text{tooth}}$ is the width of the high-index tooth and $\mathrm{\Lambda }$ is the grating period.

In a typical silicon photonic surface-grating coupler with a 600 nm period and 300 nm wide silicon teeth: $f=0.5$. With 200 nm teeth: $f=0.33$.

Why filling factor matters. Filling factor controls the average refractive index of the grating, which sets the effective index and therefore the Bragg wavelength:

$$\begin{array}{rl}{\displaystyle \bar{n}}& {\displaystyle =f\cdot n_{\text{Si}}+(1-f)\cdot n_{\text{etched}},}\\ {\displaystyle {\lambda }_B}& {\displaystyle =2\bar{n}\mathrm{\Lambda }\mathrm{/}m,}\end{array}$$

for a Bragg grating in order $m$. A grating coupler's center wavelength can be precisely set by tuning the filling factor (more silicon → higher index → longer Bragg wavelength).

Filling factor also controls the coupling strength: the perturbation amplitude $\mathrm{\Delta }n$ that drives coupling between modes is proportional to $\sin (\pi f)$ (the first Fourier component of a rectangular grating), maximum at $f=0.5$ (50% duty cycle). For weaker coupling (needed for narrow-linewidth gratings, narrow-band filters, etc.), departures from $f=0.5$ produce smaller coupling.

Standard ranges:

Design trade-offs.

• Maximum coupling at $f=0.5$: standard design choice for grating couplers requiring strong perturbation
• Reduced coupling at $f\to 0$ or $f\to 1$: useful for weak, narrow-band gratings or for grating-couplers requiring controlled apodization
• Bandwidth scaling: stronger grating coupling produces wider stopband (or wider coupling wavelength range); $f=0.5$ gives the widest band, $f\to 0$ or $f\to 1$ gives the narrowest
• Loss scaling: deeper or wider perturbations scatter more light out of guided modes into radiation; the highest-coupling $f=0.5$ designs may also have the highest excess loss

Apodization with filling factor variation. Many high-performance gratings use position-dependent $f$ to engineer the coupling profile along the grating length. Standard apodization profiles:

• Gaussian ($f$ varies smoothly from low to peak to low along length): produces Gaussian-shaped spectral response, minimizing sidelobes
• Chirped (the period varies along the grating, sometimes with $f$ adjusted alongside it): broadband or wavelength-multiplexing applications
• Phase-shifted (an abrupt shift in grating phase at one point, such as a quarter-wave shift): narrowband filter applications, defect-mode lasers

Manufacturing implications. Fabricating a grating with controlled filling factor requires lithographic patterning at the period scale. For a 600 nm period grating with $f=0.5$, the silicon tooth width is 300 nm. Achieving $\pm 2\mathrm{\%}$ filling factor accuracy requires $\pm 6$ nm lithographic accuracy, within DUV stepper capability for silicon photonic foundry processes. Tighter tolerances soon exceed what any lithography can hold: 0.1% in $f$, on the same basis, would mean 0.3 nm in tooth width.

Common errors.

1. Sidewall angle: most etched gratings have sloped sidewalls (typically 5 – 15° off vertical), which means $f$ measured at the top of the etched feature differs from $f$ at the bottom. The optical mode samples some weighted average; standard design tools use effective-index methods to compute the equivalent rectangular-grating $f$.

2. Etch depth nonuniformity: across-wafer variation in etch depth changes the strength of the grating and its effective index, which shifts the coupling and the center wavelength; with sloped sidewalls it also changes the effective $f$. It is one of the main sources of wavelength variation in silicon photonic grating couplers.

3. Misalignment of subsequent layers: if there are multiple gratings or grating-overlay structures, $f$ of the composite structure depends on alignment of each layer, which is limited by stepper overlay accuracy ($\sim 5$ nm at modern silicon photonic processes).

References: Halir et al., Waveguide grating coupler with subwavelength microstructures, Optics Letters 2009 (subwavelength grating engineering); Chrostowski & Hochberg, Silicon Photonics Design (2015), Ch. 4 for the grating coupler treatment.