Dual-Band AR Coating
Three-layer anti-reflection coating fitted for two wavelengths: thicknesses, residual reflectance and the band below 1 % around each.
R at λ₁ = 450 nm
0.037 %
R < 1 %: 419 – 474 nm
R at λ₂ = 1064 nm
0.234 %
R < 1 %: 983 – 1366 nm
Fitted thicknesses (outer → substrate)
251.5 / 16.3 / 71.4 nm
Uncoated face: R = 4.26 %
The three thicknesses are fitted to minimise R(λ₁) + R(λ₂): a grid search with each layer between 0 and a half wave at the longer wavelength, refined by Nelder–Mead. Three layers have enough freedom for two zeros only for some index sets; otherwise the fit returns the best compromise it finds, which need not be the global one. Quarter waves at each wavelength (the usual first guess) don't work: with the defaults they leave 12.9 % at 1064 nm. Exact transfer matrix at normal incidence, lossless layers with constant indices (real coating materials disperse, which shifts a dual-band design), substrate back face ignored.
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Exact transfer matrix of three lossless layers with constant indices at normal incidence. Thicknesses fitted (grid + Nelder–Mead, each ≤ a half wave at the longer λ) to minimise R(λ₁) + R(λ₂).
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 4 (multilayer antireflection coatings). CRC Press. doi:10.1201/9781420073034
- Nelder J. A., Mead R. (1965). A simplex method for function minimization. Comput. J. 7, 308–313. doi:10.1093/comjnl/7.4.308