Glossary entry

Bragg condition

The wavelength selection rule for a periodic refractive-index structure. Determines the wavelength of constructive backward reflection or out-of-plane diffraction in DFB lasers, fiber Bragg gratings, and grating couplers.

For a first-order periodic grating of period $\Lambda$ in a waveguide of effective index $n_\text{eff}$ , the Bragg condition for backward-reflected light is

\lambda_B \;=\; 2 \, n_\text{eff} \, \Lambda.

For $m$ th-order operation:

m \lambda_B \;=\; 2 \, n_\text{eff} \, \Lambda.

Light at $\lambda_B$ couples between the forward and backward propagating waveguide modes, producing strong reflection in a narrow band around $\lambda_B$ . Off-Bragg wavelengths pass through. Bandwidth of the reflected band scales with grating strength and length.

Applied configurations:

Distributed feedback (DFB) lasers — in-cavity Bragg grating provides wavelength-selective feedback, locking lasing to $\lambda_B$ (see DFB laser)
Distributed Bragg reflector (DBR) lasers — Bragg grating outside the gain region serves as a wavelength-selective mirror
Fiber Bragg gratings (FBG) — periodic refractive-index modulation along fiber, used as filters, dispersion compensators, and strain/temperature sensors
Surface grating couplers — the modified Bragg condition includes a free-space wavevector component (see grating coupler)

Temperature dependence of $\lambda_B$ includes both index and period contributions:

\frac{1}{\lambda_B}\frac{d\lambda_B}{dT} \;=\; \frac{1}{n_\text{eff}}\frac{d n_\text{eff}}{dT} + \frac{1}{\Lambda}\frac{d\Lambda}{dT}.

Typical values:

System	$d\lambda_B/dT$
InP-based DFB laser	0.08 – 0.10 nm/°C
Silicon photonic Bragg grating	0.07 – 0.08 nm/°C
Silica fiber Bragg grating	0.011 nm/°C
Silicon nitride Bragg grating	0.025 nm/°C

The silica FBG temperature coefficient is dominated by the thermo-optic coefficient of silica (much smaller than that of silicon).