Notch Filter
A notch filter reflects a narrow band and transmits elsewhere. A quarter-wave stack of two close indices does this: the index contrast sets the notch width, the number of pairs its depth (optical density).
Blocking at λ₀
OD 3.41
R = 99.9608 %, T = 10−OD
Notch FWHM (R ≥ R(λ₀)/2)
42.83 nm
511.4 nm – 554.3 nm
Pairs needed
OD 3: 36, OD 4: 47, OD 6: 68
Layers now: 80
Zone of the infinite stack: 514.0 nm – 551.3 nm (37.3 nm)
dH = 81.6 nm, dL = 91.1 nm
A quarter-wave stack (HL)N reflects R = tanh²(N ln(nH/nL) + ½ ln(ns/n₀)) at λ₀. Its notch is about as wide as the zone λ₀/(1 ± Δg), Δg = (2/π) asin((nH − nL)/(nH + nL)), so the index contrast sets the width and the number of pairs the depth. It reflects again near λ₀/3, λ₀/5, … Real notch filters also suppress the side lobes (apodized or rugate profiles) and add AR layers; this page doesn't. Exact transfer matrix, normal incidence, lossless layers, back surface ignored.
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 a lossless quarter-wave stack (HL)^N at normal incidence, back surface ignored. No apodization or AR layers, so the side lobes and the outer-face ripple remain.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 and ch. 6 (quarter-wave stacks). CRC Press. doi:10.1201/9781420073034