Integrate a spectrum to predict a fringe envelope

1. Establish the optical mechanism

Two delayed copies of one source can retain coherence at each frequency while different frequency components average incoherently in the detector. The relevant calculation averages the phase-dependent cross term over the power spectrum; it does not first add unrelated spectral fields as though all frequencies had a fixed common phase.

The original source has normalized constant spectral density 1/B on ν0−B/2 to ν0+B/2, with ν0=10 inverse time units. For delay τ, integrate μ(τ)=(1/B)∫exp(i2πντ)dν. Factoring out the central phase gives μ=exp(i2πν0τ) sin(πBτ)/(πBτ), with limit μ(0)=1.

For equal unit-intensity beams, the recorded intensity at that delay is 2[1+Re μ]. If an additional phase is scanned while holding the delay envelope fixed, contrast is |μ|. The signed sinc can become negative, shifting fringe phase; its absolute value is the contrast envelope. These are different quantities.

The first envelope zero occurs at τ=1/B, corresponding to path difference c/B in a nondispersive medium with speed c. Calling that the coherence scale is a stated first-zero convention, not a universal cutoff. A rectangular band has secondary lobes; Gaussian and Lorentzian spectra give other envelopes and width conventions. Numerical integration and the analytic limit should agree before an instrumental interpretation.

2. Work the original evidence

With B=2 and τ=0.25, the envelope is 2/π≈0.636620 and the central phase is 5π. Thus μ is negative real and the equal-beam intensity is about 0.726760, while contrast is positive 0.636620. First-zero delay is 0.5.

3. Predict and investigate

Does a zero at one delay mean that the two copies can never interfere at another delay?

Integrate the real and imaginary spectral terms independently, compare them with the analytic sinc, and distinguish intensity at a delay from a phase-scan visibility.

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.