Bragg Reflector
Dielectric distributed Bragg reflector: reflectance spectrum, stop band and period, and the band structure of the 1D photonic crystal with its gap/midgap ratio and Bloch decay per period.
Peak Reflectance (at λ₀)
97.9171%
Δn = 0.90, 5 pairs, λ₀ = 1550 nm
Stop band (infinite-stack edges)
1345.2–1828.3 nm, width 483.1 nm
gap/midgap Δω/ω₀ = 30.44 %
Period Λ = dH + dL
432.13 nm
dH = 164.89 nm, dL = 267.24 nm
Bloch decay at λ₀: κ = ln(nH/nL)
0.4829 Np per period
field falls to 1/e in 2.07 periods (895 nm)
Band structure of the infinite crystal (1D photonic crystal)
Bloch waves of the endless (HL) stack: cos KΛ = cos δH cos δL − ½(nH/nL + nL/nH) sin δH sin δL. Where |cos KΛ| > 1 the wave decays by Im KΛ per period: these are the stop bands, at odd multiples of the design frequency for quarter-wave layers. N pairs between equal media reflect tanh²(N·Im KΛ) at the band centre.
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, the stop-band edges, the gap/midgap ratio and the Bloch decay ln(n_H/n_L) are closed forms for the infinite stack; the band diagram is its exact Bloch dispersion.
- 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.
- Yeh P., Yariv A., Hong C.-S. (1977). Electromagnetic propagation in periodic stratified media. I. General theory. J. Opt. Soc. Am. 67, 423–438. doi:10.1364/JOSA.67.000423
- Joannopoulos J. D., Johnson S. G., Winn J. N., Meade R. D. (2008). Photonic Crystals: Molding the Flow of Light, 2nd ed., ch. 4. Princeton University Press.