Derive local intensity from complex field addition

1. Establish the optical mechanism

A complex amplitude records two real pieces of information: magnitude and phase relative to a reference. For equally polarized waves of the same frequency in the same medium, the detector intensity is proportional to the squared magnitude of the summed field. The common proportionality constant is absorbed into the original normalized units here.

Write E1=1 and E2=sqrt(r) exp(iφ). Multiplying E1+E2 by its complex conjugate gives I=1+r+2sqrt(r)cosφ. The last term is not extra emitted energy: interference redistributes local flux, and a complete energy account must include the other positions or output channels of the optical system.

As phase varies, Imax=(1+sqrt(r))² and Imin=(1−sqrt(r))². Their difference divided by their sum yields visibility V=2sqrt(r)/(1+r). Equal intensity gives complete ideal contrast, whereas unequal amplitudes leave a nonzero minimum even with perfect coherence.

If the relative phase samples a full uniform cycle during observation, the average cosine is zero and the intensity becomes 1+r. This is an incoherent or phase-averaged comparison, not the field at a particular phase. Orthogonal polarizations also require vector treatment; the scalar formula cannot silently be used for arbitrary polarization.

2. Work the original evidence

For r=4 and φ=90°, E=1+2i and I=5. The phase scan ranges from 1 to 9, so V=8/10=0.8. The intensity at this one phase happens to equal the incoherent average, but the remaining phase scan distinguishes them.

3. Predict and investigate

Can the same local intensity arise from a coherent and an incoherent record?

Reconstruct real and imaginary components before intensity. Compare a complete phase scan with a fixed reading, then change the beam ratio and derive visibility from the observed extrema.

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.