How this calculator works
The two film interfaces reflect amplitudes that acquire a round-trip phase inside the film. Summing all internal reflections gives the coherent single-layer reflectance, evaluated separately for s and p polarization. Interface amplitude signs include reflection phase reversals. The film is lossless, uniform and fully coherent; indices are real and constant over the plotted wavelength range. Absorption, evanescent layers, total internal reflection and multilayer stacks require a more general complex-index model.
Formula
OPD = 2n2 d cosθ2; phase φ = 2π OPD/λ. R = [r12²+r23²+2r12r23 cosφ]/[1+(r12r23)²+2r12r23 cosφ]. Quarter-wave d = λ/(4n2 cosθ2). At normal incidence a zero-reflection quarter-wave design also requires n2² = n1n3.
Worked example
At normal incidence with n1 = 1, n2 = 2 and n3 = 4, a 62.5 nm film is one quarter wave at 500 nm and has zero ideal s and p reflectance. Equal indices everywhere also give zero reflectance.
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Frequently asked questions
Does quarter-wave thickness always remove reflection?
No. At normal incidence, the film index must also be the geometric mean of the surrounding indices for complete cancellation.
Why are s and p different?
Their electric-field orientations produce different Fresnel amplitude coefficients at oblique incidence.
Can I model an absorbing metal film?
This page assumes real indices and no absorption. A metal coating requires complex refractive indices and a broader transfer-matrix model.