Beamsplitter Design
Dielectric beamsplitters split light into reflected and transmitted beams. A single quarter-wave layer of ZnS reflects only 32 % from air on glass; 50 % would need n ≈ 2.98. Quarter-wave stacks H, HLH, HLHLH step up to 32, 66 and 85 %. For 50 %, thin one layer of HLH (exact at λ₀ only) or use H M H with n_M ≈ 1.86 (flat at λ₀).
Thinned-layer design, 50 % at λ₀ (your nH, nL)
H 58.51 nm | L 48.12 nm | H 58.51 nm
HLH with the second (L) layer at 0.507 of a quarter wave
Single quarter-wave layer for 50 %
n = 2.976
n² = n₀ns(1 + √R)/(1 − √R). Visible coating materials reach about 2.4 (ZnS, TiO₂ films); higher indices are IR materials such as Si and Ge.
All quarter waves: H M H for 50 %
nM = 1.855
nM = nH²/n with n from the single-layer card. R is flat at λ₀ (a maximum when nM < nH).
Quarter-wave stacks at λ₀ (bare substrate: 4.26 %):
H: 32.30 %
HLH: 65.66 %
HLHLH: 85.24 %
HLHLHLH: 94.11 %
A quarter-wave layer maps the admittance Y → n²/Y, so R steps up in jumps as layers are added. The thinned design takes the smallest quarter-wave stack H(LH)m that reaches the target and thins one layer until R(λ₀) equals it (R rises monotonically with that thickness). It meets the target at λ₀ only: its R peak moves to shorter wavelengths, so λ₀ sits on a slope (chart). H M H keeps every layer a quarter wave and gets a flat top, but needs a middle index nM. Exact transfer matrix at normal incidence, lossless layers, back surface ignored. A plate splitter at 45° splits s and p differently; this page doesn't model that.
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 at normal incidence, constant lossless indices, back surface ignored. A plate splitter at 45° splits s and p differently; not modelled.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 and ch. 5 (beam splitters). CRC Press. doi:10.1201/9781420073034