Hard Coating Design
Single hard coating on a lens or window: reflectance and the absentee (half-wave) thickness, substrate bow from film stress (Stoney), and the thickness below which the film cannot delaminate.
Optics
Stress and adhesion
R at λ₀ (coated face)
23.64 %
Uncoated: 4.26 %
Mean R, 400–700 nm
16.56 %
Range 4.26 % – 23.75 %
Nearest absentee thickness
261.9 nm
Multiples of λ₀/(2n) = 131.0 nm keep R(λ₀) at the uncoated value
Substrate bow over the diameter
0.508 µm
Radius 153.8 m, film side convex (compressive)
Stored elastic energy U
0.250 J/m²
(1 − νf)σ²t/Ef; Γ = 5 J/m²
Thickness where U = Γ
4.00 µm
Thinner: it cannot delaminate, whatever its flaws
Optics: one lossless layer on the substrate (exact transfer matrix, normal incidence, back face ignored). A quarter wave gives R = ((ns − n²)/(ns + n²))² = 23.75 % at λ₀, which is a reflectance maximum (n > √n_s): a high-index hard coat is not an AR coat; a half wave is absentee. Thick hard coats ripple, so their mean R over the band is what counts. Stress: Stoney's equation gives the substrate curvature 6σt(1 − νs)/(Ests²) (film ≪ substrate, small bow); the bow is the sagitta over the radius. A debond can't release more energy per area than the film stores, so a film thinner than tc = ΓEf/((1 − νf)σ²) can't delaminate (Hutchinson & Suo 1991); above it, compressive films buckle and tensile films crack from flaws, at thicknesses that depend on flaw size. Hardness and scratch resistance aren't modelled.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Optics: exact one-layer transfer matrix, lossless. Stress: Stoney (film ≪ substrate, small bow) and the energy bound (1 − ν)σ²t/E < Γ against delamination. Hardness and scratch resistance not modelled.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 4 (single-layer coatings). CRC Press. doi:10.1201/9781420073034
- Stoney G. G. (1909). The tension of metallic films deposited by electrolysis. Proc. R. Soc. Lond. A 82, 172–175. doi:10.1098/rspa.1909.0021
- Hutchinson J. W., Suo Z. (1991). Mixed mode cracking in layered materials. Adv. Appl. Mech. 29, 63–191. doi:10.1016/S0065-2156(08)70164-9