Quarter-Wave Stack Calculator

Design single-layer AR coatings or calculate HR dielectric stack reflectance. Computes film thickness, residual reflectance, and HR bandwidth.

Dielectric thin-film coatings exploit interference between reflections at each layer interface. The simplest design is the quarter-wave layer: a film of thickness d = λ/(4n_f) deposited on a substrate. For a single-layer AR coating, minimum reflectance occurs when n_f = √(n₀ · n_s); MgF₂ (n ≈ 1.38) approximates this ideal for glass substrates in the visible. For high-reflector (HR) stacks, alternating high- and low-index quarter-wave layers build constructive interference with each added pair. Reflectance for N pairs follows R = ((1 − (n_H/n_L)²ᴺ · n₀/n_s) / (1 + (n_H/n_L)²ᴺ · n₀/n_s))² and can reach 99.999% for ten or more pairs of TiO₂/SiO₂. The tool computes film thickness, residual reflectance compared to the uncoated substrate, and — in HR mode — the fractional high-reflectance bandwidth and physical layer thicknesses.

AR Coating Design
AR Coating Design
Film thickness (d)
99.6nm
Ideal film index √(n₀·n_s)
1.232
Residual reflectance (R)
1.28%
Uncoated reflectance
4.22%

Note: Zero residual reflectance requires the film index to equal the ideal value (1.232). MgF₂ (n = 1.38) is the standard choice for glass substrates despite being above ideal.

Abridged Optics — Quarter-Wave Stack Calculator v1.0AR: d = λ/(4n_f), R = ((n_f²−n₀n_s)/(n_f²+n₀n_s))². HR: R = ((1−(n_H/n_L)^(2N)·n₀/n_s)/(1+(n_H/n_L)^(2N)·n₀/n_s))².

All information, equations, and calculations have been compiled and verified to the best of our ability. For mission-critical applications, we recommend independent verification of all values. If you find an error, please let us know.