Dielectric Stack Theory
Quarter-wave dielectric stack reflectance. Alternating high/low index layers create high-reflectance mirrors — the basis of dielectric mirrors and VCSELs.
Peak Reflectance
98.72%
nH/nL Ratio
1.703
Total Layers
10
Stop band (infinite-stack edges)
471–661 nm, width 190 nm
Quarter-Wave Stack Theory
Structure: n₀ | (H L)^N | n_sub
d_H = λ₀/(4n_H), d_L = λ₀/(4n_L)
R_peak = {[(nH/nL)^(2N) − q] / [(nH/nL)^(2N) + q]}²
q = n₀ / n_sub
Transfer Matrix: M = ∏ M_j, M_j = [[cos δ, −i sin δ/η], [−i η sin δ, cos δ]]
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 an (HL)^N quarter-wave stack at normal incidence with constant, lossless indices; back surface ignored. Peak R at λ₀ and the stop-band edges (of an infinite stack) are closed forms.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 and ch. 6 (quarter-wave stack, high-reflectance zone). CRC Press. doi:10.1201/9781420073034
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §1.6.5 (periodically stratified media). Cambridge University Press.