Angle Tuning of Coatings
Changing the angle of incidence shifts the spectral response of thin-film coatings toward shorter wavelengths (blue shift). TE (s) and TM (p) respond differently: the p stop band narrows as the angle grows. The centre follows λ(θ) ≈ λ₀(cos θ_H + cos θ_L)/2, where θ_H and θ_L are the angles inside the layers.
Total layers = 10
Solid lines = TE (s-pol), Dashed = TM (p-pol). Each color = different angle.
λ_c(θ) ≈ λ₀(cos θ_H + cos θ_L)/2, cos θ_i = √(1 − (n₀ sin θ/n_i)²) (first order)
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
Exact s and p transfer-matrix spectra of a lossless (HL)^N quarter-wave stack with constant indices. The centre-wavelength curve is the first-order rule λ₀(cos θ_H + cos θ_L)/2, not a fit to the spectra.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 (phase thickness, tilted admittances) and ch. 9 (tilted coatings). CRC Press. doi:10.1201/9781420073034