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:

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

where $w_\text{tooth}$ is the width of the high-index tooth and $\Lambda$ 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:

$$\bar{n} \;=\; f \cdot n_\text{Si} + (1 - f) \cdot n_\text{etched}, \qquad \lambda_B \;=\; 2 \bar{n} \Lambda / m,$$

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 $\Delta n$ that drives coupling between modes is proportional to $\sin(2\pi f)$ — 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:

Application	Typical $f$
Silicon photonic surface grating coupler	0.35 – 0.55
Distributed Bragg reflector (DBR)	0.5 (maximum reflectivity)
DFB laser gain coupling	0.35 – 0.5
Apodized grating (varying $f$ along length)	0.1 – 0.9 across grating
Resonant grating filter	0.4 – 0.6

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
• Linear chirp ($f$ varies linearly while period also varies): broadband or wavelength-multiplexing applications
• Phase-shifted (abrupt change in $f$ at one point): 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 $\pm 2\%$ filling factor accuracy requires $\pm 6$ nm lithographic accuracy — within DUV stepper capability for silicon photonic foundry processes. Tighter tolerances (e.g., 0.1% $f$ accuracy) require EUV or e-beam lithography.

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 etch depth variation produces position-dependent $f$ (deeper etch → smaller remaining tooth → smaller $f$). This is one of the dominant sources of wavelength tolerance 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.