Bragg Wavelength Calculator

A periodic perturbation of period $\mathrm{\Lambda }$ couples a guided mode to its counter-propagating copy when the phase gained over one period matches an integer multiple $m$ of half a wavelength in the guide. The tool uses the resulting Bragg condition in both directions:

Here $n_{\text{eff}}$ is the effective index of the mode averaged over one period of the corrugated waveguide, and it must be evaluated at ${\lambda }_B$ itself; an index taken from a mode solve at another wavelength, or from the unperturbed waveguide, shifts the answer. The relation gives the center of the reflection band of a uniform grating and nothing else: the width of the stop band, set by the coupling coefficient and the grating length, is not computed. A uniform DFB grating without a phase shift lases at the two edges of the stop band rather than at ${\lambda }_B$. Higher-order gratings satisfy the same condition with a longer period, but also diffract light out of the waveguide, which a second-order grating sends normal to the surface.

Worked example, with the SOI 220 nm strip, TE preset ($n_{\text{eff}}=2.4$, 1550 nm) in the compute-period mode. First order gives $\mathrm{\Lambda }=1550\mathrm{/}4.8=322.917$ nm, and second order gives 645.833 nm. Switching to the compute-wavelength mode with the default period of 322.9 nm gives 2 × 2.4 × 322.9 = 1549.92 nm. The same relation sets the tolerances: at first order a 1 nm error in period moves ${\lambda }_B$ by $2n_{\text{eff}}=4.8$ nm, and an error of 0.01 in $n_{\text{eff}}$ moves it by 1550 × 0.01/2.4 = 6.46 nm, which is why the index from a mode solver matters more than the lithographic rounding of the period. The result panel reports both sensitivities for the current inputs, $0.01{\lambda }_B\mathrm{/}{n}_{\text{eff}}$ and $2n_{\text{eff}}\mathrm{/}m$.

The SOI TM preset, 2.05, is the fundamental TM solution of a 220 nm silicon slab clad in oxide at 1550 nm, taking 3.476 for silicon and 1.444 for silica; it comes from the slab dispersion relation $\tan (\kappa d\mathrm{/}2)=(n_{\text{Si}}^2\mathrm{/}{n}_{{\text{SiO}}_2}^2)\gamma \mathrm{/}\kappa$ with $d$ the thickness and $\kappa$, $\gamma$ the transverse wavenumbers in the core and cladding. It is an upper limit, since a strip of finite width confines the mode less. The other presets are representative strip values and stand in for a mode solve of the actual grating.

References: H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327 (1972). T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277 (1997).