Phase Shift Coatings
Phase shift accumulated in thin film coatings. Explore how film thickness and refractive index affect the optical phase of reflected and transmitted light.
Film thickness
49.8 nm
Round trip 2β = 90.0° at λ₀
Reflected phase arg r at λ₀
-163.62°
R = 2.782 %
Transmitted phase arg t at λ₀
44.56°
T = 97.218 %
Interface coefficients
r₀₁ = -0.1597
r₁₂ = -0.0483
Phase relations
2β = 4π·n·d / λ (round trip in the film); quarter wave: 2β = π at λ₀, half wave: 2π
r = (r₀₁ + r₁₂·e^(2iβ)) / (1 + r₀₁·r₁₂·e^(2iβ))
t = t₀₁·t₁₂·e^(iβ) / (1 + r₀₁·r₁₂·e^(2iβ)), rij = (ni − nj)/(ni + nj), tij = 2ni/(ni + nj)
Born & Wolf §1.6.4, time dependence e^(−iωt); with e^(+iωt) (Macleod, Hecht) every phase changes sign. Normal incidence, lossless film. arg r is referred to the front face, arg t to the field at the substrate face over the incident field at the front face. The phase curves are unwrapped across the spectrum and placed in (−180°, 180°] at λ₀.
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
One lossless film between semi-infinite media at normal incidence, coherent. r and t come from the exact characteristic matrix; phases use e^(−iωt), as Born & Wolf do (Macleod's signs are opposite).
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §1.6.4 (a homogeneous dielectric film). Cambridge University Press.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 (characteristic matrix). CRC Press. doi:10.1201/9781420073034