Thin-Film Optics, Camera Lenses, Solar PV Glass & Photonics
Quarter-Wave Anti-Reflection (AR) Coating Thickness & R% Formula
Calculate Ideal Zero-Reflection Index SQRT(n_0 * n_s), Physical Quarter-Wave Film Thickness d (nm), Bare-Substrate Fresnel Reflectance (%), and Coated Quarter-Wave Reflectance (%).
Destructive interference needs BOTH a quarter-wave optical path d = lambda / (4*n1) AND geometric-mean index SQRT(n0 * ns)!
Uncoated camera elements, silicon solar cells, and microscope objectives lose light at every air-glass interface via Fresnel normal reflectance R_bare = ((n_s - n_0) / (n_s + n_0))^2! Depositing a dielectric thin film of refractive index n_1 (with n_0 < n_1 < n_s so BOTH interfaces undergo a 180° phase flip!) and physical Quarter-Wave Thickness d = lambda_0 / (4 * n_1) forces the round-trip beam (2 * n_1 * d = lambda_0 / 2) to emerge 180° out of phase with the front-surface reflection! Exact Thin-Film Matrix Interference gives Quarter-Wave Normal Reflectance R_AR = (((n_0 * n_s - (n_1^2)) / (n_0 * n_s + (n_1^2)))^2) * 100—vanishing to 0.00% when n_1 equals the geometric mean SQRT(n_0 * n_s)!
Verify on a Magnesium Fluoride (MgF2) Coated BK7 Crown-Glass Lens at Photopic Green lambda_0 = 550.0 nm—Incident_Medium_Index n_0 = 1.00 (air, B2), Substrate_Index n_s = 1.52 (BK7 glass, B3 -> Ideal AR Index = SQRT(1.52) = 1.2329 in E2), Coating_Index n_1 = 1.38 (durable MgF2, B4), and Design_Wavelength lambda_0 = 550.0 nm (B5): Physical MgF2 Film Thickness d = 550 / (4 * 1.38) = 99.64 nm (E3), Bare Glass Loss of 4.26% (E4) drops more than 3x down to 1.26% per surface (E5)!
| Quarter-Wave AR Coating KPI (Air n0=1.0, BK7 ns=1.52, MgF2 n1=1.38, 550 nm) | Formula | Verified Output |
|---|---|---|
| Ideal Zero-Reflectance Index | SQRT(Incident_n0 * Substrate_ns) | 1.2329 (1.3784 @ Flint ns = 1.90) |
| Physical Quarter-Wave Film d (nm) | Wavelength_nm / (4 * Coating_n1) | 99.64 nm |
| Bare Uncoated Reflectance (%) | (((ns - n0) / (ns + n0))^2) * 100 | 4.26% (9.63% @ Flint ns = 1.90) |
| Coated Quarter-Wave R_AR (%) | (((n0*ns - n1^2)/(n0*ns + n1^2))^2)*100 | 1.26% (0.00% @ Flint ns = 1.90!) |
Exact Single-Layer Quarter-Wave Thin-Film Reflectance Formula
Validate 1.0 <= n_0 < n_1 < n_s <= 4.5 (phase-matching ordering!) and Wavelength lambda_0 in [150, 15000] nm!
=LET(n0_i, B2, ns_b, B3, n1_c, B4, lm_n, B5, IF(OR(n0_i < 1, n1_c <= n0_i, ns_b <= n1_c, ns_b > 4.5, lm_n < 150, lm_n > 15000), "Invalid AR Coating Input", ROUND((((n0_i * ns_b - (n1_c ^ 2)) / (n0_i * ns_b + (n1_c ^ 2))) ^ 2) * 100, 2)))High-Index Dense Flint Glass Perfect MgF2 Impedance Match Check (Set n_s = 1.90)
Switch Substrate_Index B3 from 1.52 (BK7 crown) to 1.90 (dense lanthanum flint glass): Ideal AR Index E2 rises to SQRT(1.90) = 1.3784—nearly identical to MgF2 (1.38)—slashing reflection from 9.63% bare (E4) all the way to 0.00% (E5)!
=IF(ISTEXT(E5), E5, ROUND((((B3 - B2) / (B3 + B2)) ^ 2) * 100, 2))Try the example
Use the original values or change them to test your understanding.