Moth-Eye Graded-Index Anti-Reflection Calculator

Removing the Interface Instead of Coating It

A quarter-wave coating cancels one reflection at one wavelength. A sub-wavelength taper does something different: it removes the step in refractive index altogether, so there is no interface left to reflect from — across the whole visible band and over a wide range of angles.

A Taper Is a Stack of Effective Media

While the period is small enough that no order but the zeroth propagates, light cannot resolve the individual features and sees only their average. Slicing the taper and giving each slice the Bruggeman effective index of its local glass fraction turns the structure into an ordinary graded multilayer, which the same transfer-matrix engine solves exactly. Height is what matters first: below roughly half a wavelength no profile works, and a linear cone — which spends the most of its height in the middle of the index range — does best of a bad set. Past that the ordering inverts, and the smooth profiles whose index gradient vanishes at both ends pull ahead; a quintic taper near a micron leaves a few parts in \(10^5\), two orders of magnitude below a quarter-wave coating. What the height cannot fix is the period: below \(\lambda = \Lambda n_s\) the structure diffracts and this model stops being true.

The Bruggeman condition treats neither phase as the host, so it stays well behaved across the whole taper and returns the pure material at both ends. For two real, lossless phases it is a quadratic with a closed-form root, which is what each slice uses. Sixteen slices are already within about two percent of the converged answer — the slice count is a discretization control, not a design parameter. The same effective-medium reasoning is what the calculator's Bruggeman roughness layer applies to a rough interface.

Interactive design tool

This page includes a live transfer-matrix calculator for this design. Adjust the parameters to recompute spectra, layer stack, and performance figures in the browser, then open the design in the full calculator.