Anti-Reflection Coating
A single transparent film on glass can reduce reflection at a design wavelength by destructive interference between waves reflected at the air–film and film–substrate interfaces. At normal incidence the reflectance follows the two-beam formula R = |(r₀₁ + r₁₂e^{2iβ})/(1 + r₀₁r₁₂e^{2iβ})|² with β = 2π n_f d / λ and Fresnel amplitude coefficients r_ij from the indices. Choosing d = λ₀/(4n_f) gives β = π/2 at λ₀ so the reflections can cancel partially if the indices are chosen well; the ideal quarter-wave index for a single layer on glass in air is n_f ≈ √(n_air n_glass). The graph scans R(λ) across the visible band. Absorption, oblique incidence, and bandwidth limits of real coatings are omitted — this is the textbook single-layer AR model.
Who it's for: Undergraduate electromagnetism and optical coating courses introducing thin-film matrix methods later.
Key terms
- Anti-reflection
- Quarter-wave plate
- Thin film
- Fresnel coefficients
- Interference
- Reflectance
- Refractive index matching
- Spectral tuning
Live graphs
How it works
A thin dielectric film on glass uses path interference between reflections at the two interfaces; at quarter-wave optical thickness reflection can nearly vanish at one wavelength.
Key equations
Frequently asked questions
- Why is reflectance not zero at all wavelengths?
- β depends on λ, so the phase slip 2β only equals π at the design condition. Away from that wavelength the two reflections no longer cancel.
- Why show √(n₀n_s)?
- For a single low-index film on a higher-index substrate, that geometric mean is the index that minimizes reflectance at the design wavelength when thickness is λ/(4n_f).
- Do real camera lenses use one layer?
- Modern systems stack many layers to widen the low-reflection band and handle oblique rays; the simulator isolates the physics of the simplest case.
- Is normal incidence a big limitation?
- Yes. At large incidence polarization splits s and p reflectance and the effective optical thickness changes; AR coatings for windows and camera lenses are optimized with those effects.
More from Optics & Light
Other simulators in this category — or see all 44.
Michelson Interferometer
I(Δ) = V cos²(πΔ/λ); tilt fringes; coherence length envelope.
Mach–Zehnder Interferometer
Two-beam recombination; I ∝ cos²(πΔ/λ) vs one-arm OPD; schematic + fringe plot.
Sagnac (Ring) Interferometer
Δφ ∝ Ω·A/λ for counter-propagating beams in a rotating loop — optical gyro idea.
Brewster Angle
tan θ_B = n₂/n₁; R_p→0; θᵢ+θₜ=90°; Fresnel R_s, R_p vs θᵢ.
Fermat's Principle
OPL = n₁AP+n₂PB vs hit point; minimum = Snell path.
Chromatic Aberration
Cauchy n(λ); thin-lens f(λ); paraxial rays R/G/B.