Amplitude Splitting
Multiple-beam interference from amplitude splitting at a thin film. Shows how partial reflections from each interface combine to form interference fringes.
r₀₁
-0.1597
r₁₂
-0.0483
Coefficient F
0.0313
Finesse (FSR/FWHM)
— (F < 1: T never falls to half)
Amplitude Splitting Theory
Eᵣ = r₀₁ + t₀₁t₁₀r₁₂e^(−iδ) + t₀₁t₁₀r₁₂²r₁₀e^(−2iδ) + ...
Sum: r = (r₀₁ + r₁₂e^(−iδ)) / (1 + r₀₁r₁₂e^(−iδ)), T = (n₂/n₁)·|t|²
(e^(+iωt) convention; R and T don't depend on it)
Airy function: T = T_max / (1 + F·sin²((δ − δ_max)/2))
F = 4√(R₁R₂) / (1 − √(R₁R₂))², R₁ = r₀₁², R₂ = r₁₂²
Finesse ℱ = FSR/FWHM = 2π / (4 asin(1/√F)) ≈ π√F / 2 for F ≫ 1 (needs F ≥ 1)
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
A lossless film between semi-infinite media at normal incidence, coherent multiple beams, constant indices. R and T come from the exact characteristic matrix; the Airy finesse needs F ≥ 1.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 (characteristic matrix). CRC Press. doi:10.1201/9781420073034
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §7.6.1 (Airy formulae, finesse). Cambridge University Press.
- Hecht E. (2016). Optics, 5th ed., §9.4.