Temporal coherence depends on the whole spectrum

Represent one polarization of the light by a complex analytic field. For statistically stationary light, its average correlation with a delayed copy depends on the delay rather than on the starting time. Normalize by the average intensity to obtain the first-order coherence function. Its magnitude lies between zero and one. A perfectly monochromatic field has a rotating correlation phase but a constant unit magnitude, illustrating why phase evolution alone is not coherence loss.

A detector averages over many optical periods. Combining two copies gives the sum of their separate intensities plus a correlation-dependent cross term. When their intensities are equal and their polarizations match, fringe visibility equals the correlation magnitude. Unequal intensities or a polarization mismatch can reduce visibility even when the field is fully coherent. Therefore a low measured contrast does not identify a broad spectrum until instrumental causes have been checked.

The Wiener–Khinchin relation connects stationary field correlation to the power spectrum. Each frequency component accumulates a different phase during a delay. At zero delay these contributions align; later their phases spread and partially cancel in the spectral integral. This is a statistical statement about first-order correlation. It does not require a picture of sharply bounded little wave trains whose edges could be read directly from a spectrum.

For a narrow Lorentzian line, use detuning from its optical carrier and take the detuning integral over the real line. Its spectral full width at half maximum is Δν in hertz. The normalized transform has an exponential magnitude with decay rate πΔν. Extending detuning to negative infinity is the usual narrowband idealization, valid when the line width is much smaller than its carrier frequency. It is not a statement that physical light has negative photon energies.

Define coherence time here as the positive delay at which the correlation magnitude falls to one over e. The corresponding vacuum coherence length is c times that delay. Some authors instead use an integral of squared coherence, an envelope FWHM, or a rough inverse-bandwidth estimate. Those definitions can all be useful but are not numerically interchangeable. A Gaussian spectral line also has a Gaussian correlation envelope, so replacing its width by the Lorentzian formula introduces another avoidable error.

The waves instrument isolates the interference step rather than the statistical spectral average. For a fresh run, Reset and set both controls to phase difference zero degrees and amplitude ratio one; predict a doubled field amplitude. Reset again and set both to 180 degrees and one; predict cancellation. These are two fixed-frequency coherent sinusoids. Their changing phase does not simulate linewidth, random phase diffusion, or the decay envelope derived above; those require averaging a spectrum or a stationary random process.