Sum interface phases and the complete reflected series

1. Establish the optical mechanism

At a lossless normal-incidence interface the electric-field reflection coefficient is r_ij=(n_i−n_j)/(n_i+n_j). A negative coefficient is a π phase reversal; a positive coefficient is not. This interface sign is separate from the propagation phase accumulated while a wave traverses a film.

The original stack has incident index 1, film index 1.5, substrate index n2 and vacuum wavelength 600 nm. Therefore r01=−0.2 and r12=(1.5−n2)/(1.5+n2). A round trip through thickness d contributes φ=4π(1.5)d/600. Two interface signs can change relative phase without changing the optical path.

Transmitted light can reflect inside the film repeatedly. Summing that convergent geometric series gives total reflected field r=(r01+r12 exp(iφ))/(1+r01 r12 exp(iφ)). The transmitted field is t01 t12 exp(iφ/2)/(1+r01 r12 exp(iφ)), where t_ij=2n_i/(n_i+n_j). Power transmission must include the impedance factor n2/n0: T=(n2/n0)|t|², not merely |t|².

For real lossless indices the complete result obeys R+T=1. A two-reflection approximation may explain phase qualitatively but does not replace this exact series when a quantitative power balance is required. At zero thickness the stack reduces to a single incident/substrate interface; when n2 equals the film index, the lower interface disappears. Absorption, oblique polarization effects and roughness are outside this model.

2. Work the original evidence

For d=100 nm and substrate index 1, r12=+0.2 and φ=π. The complete reflectance is about 0.147929 and transmittance 0.852071. At d=0, the same surrounding media give R=0, not the 0.04 reflectance of an isolated first interface.

3. Predict and investigate

Can a reflection-phase reversal be replaced by adding an arbitrary geometric path length in every surrounding medium?

Track each interface coefficient and round-trip phase separately, then verify the full power sum and both zero-thickness and matched-index limits.

Primary mechanism context. Checked 31 October 2026. Original prose, records and diagrams; no copied source exercises or physical light-source operation. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.

Multiple-reflection derivation context: the lesson independently derives the original unequal-index stack and checks its lossless power balance.