 First-order optical coherence

Consider one selected polarization of waves, represented by complex fields $E_1(t)$ and $E_2(t)$ at two sampling positions. Suppose their second moments are finite and stationary: $J_j=⟨|E_j(t){|}^2⟩$ and the following uncentered wave-field correlation do not depend on the absolute time $t$. DefineThe star denotes the complex conjugate; brackets denote an ensemble average. A long detector average estimates this quantity only if it adequately samples the fluctuations. The superscript $1$ identifies a field correlation: it does not mean the autocorrelation of the measured light intensity. The one-position version is $g^{(1)}(\tau )=⟨E^{\ast }(t)E(t+\tau )⟩/⟨|E{|}^2⟩$, so $g^{(1)}(0)=1$ for every nonzero field covered by this definition; this alone says nothing about its correlation at nonzero delay. ^[ref].

The Cauchy-Schwarz inequality gives $0\le |g_{12}^{(1)}(\tau )|\le 1$. At a specified delay, magnitude one means $E_2(t+\tau )=a{E}_1(t)$ in mean square for a fixed nonzero complex $a$: the samples have a reproducible relation in both wave amplitude and wave phase. Magnitude zero means the averaged field cross term vanishes. It need not mean statistical independence. A deterministic phase difference changes the argument of $g^{(1)}$, not its complex modulus; a distorted but fixed wavefront can therefore remain fully coherent. ^[ref].

To see the observable, recombine the samples in the same detected spatial mode with matched wave polarizations. With a common positive conversion factor $C$, let $I_j=C{J}_j$ be the single-arm light intensities reaching that detector. Adding $E_1(t)+e^{i\varphi }{E}_2(t+\tau )$ givesThis follows simply by expanding the squared complex modulus and averaging, then taking the maximum and minimum as $\varphi$ spans a full cycle. The extra wave phase scan must leave $I_1,I_2$ and the magnitude of the correlation effectively unchanged. Thus interferometric visibility equals $|g^{(1)}|$ only for equal nonzero light intensities and ideal mode matching. For $I_2/I_1=1/4$, perfectly coherent beams have interferometric visibility $0.8$; a measured value $0.4$ would imply $|g^{(1)}|=0.5$ under these assumptions. Orthogonal linear polarizations, background light, unresolved interference fringes or vibrations can reduce measured contrast further. One absent arm gives zero contrast and makes the normalized two-sample correlation undefined.