Edge Filter Design
Long-pass and short-pass quarter-wave edge filters, (H/2 L H/2)^N and (L/2 H L/2)^N: design wavelength for an edge, 50 % point and R/T spectrum.
50 % point of this stack
609.4 nm
Transmits above this wavelength
Design wavelength λ₀
508.7 nm
Quarter-wave layers at λ₀; R(λ₀) = 99.983 %
Reflection zone of the infinite stack: 441.5 nm – 600.0 nm (Δg = 0.1522)
dH = 54.1 nm, dL = 87.7 nm; the two outer layers are halves
Layers: 21
Long-pass λ₀ = λedge(1 − Δg), short-pass λ₀ = λedge(1 + Δg), Δg = (2/π) asin((nH − nL)/(nH + nL)). A finite stack's 50 % point lies a little past the zone edge, toward the pass band. The outer half layers keep the ripple low on the pass side; the other side of the zone ripples more.
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
Transfer matrix of (H/2 L H/2)^N or (L/2 H L/2)^N, lossless constant indices, normal incidence, semi-infinite substrate. λ₀ puts the infinite stack's zone edge at the edge wavelength.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 6 (zone width) and ch. 7 (edge filters, symmetric periods). CRC Press. doi:10.1201/9781420073034
- Epstein L. I. (1952). The design of optical filters. J. Opt. Soc. Am. 42, 806. doi:10.1364/JOSA.42.000806