Mutual Coherence Function and Degree of Coherence

When two beams of light overlap, the intensity you measure is not simply the sum of their separate intensities — nor is it always the full-contrast fringe pattern of an idealized textbook. Laser-like beams interfere with maximal contrast; two independent light bulbs add with no fringes at all; real light lives everywhere in between. One complex function pins down exactly where a given pair of beams sits on that scale: the mutual coherence function $\Gamma_{12}(\tau)=\langle E_1^*(t)\,E_2(t+\tau)\rangle$ , the time-averaged cross-correlation of the field at one point and time with the field at another point a delay $\tau$ later. This single object contains all of the second-order coherence information there is, and it is the quantity every interferometer secretly measures. Its two limits are the two coherences you already know. Set the two points equal and $\Gamma_{11}(\tau)$ is the self-coherence whose width is the coherence time — the domain of temporal coherence and the Wiener–Khinchin theorem. Set the delay to zero and $\Gamma_{12}(0)$ is the mutual intensity, whose falloff with separation is spatial coherence and connects, through the Van Cittert–Zernike theorem, to the angular size of the source. Normalize $\Gamma_{12}$ by the geometric mean of the two intensities and you get the complex degree of coherence $\gamma_{12}(\tau)$ , bounded in modulus by unity, whose modulus is precisely the fringe visibility and whose phase shifts the fringe positions. The general interference law $I=I_1+I_2+2\sqrt{I_1 I_2}\,\mathrm{Re}\,\gamma_{12}$ then subsumes the ideal two-wave interference term as the fully coherent limit $|\gamma_{12}|=1$ . This article is the hub that unifies visibility, temporal coherence, and spatial coherence into one framework.

Key takeaways

The mutual coherence function $\Gamma_{12}(\tau)=\langle E_1^*(t)E_2(t+\tau)\rangle$ is the field–field cross-correlation between two spacetime points; it is the master quantity of classical partial coherence.
Its normalized form, the complex degree of coherence $\gamma_{12}(\tau)$ , obeys $0\le|\gamma_{12}|\le 1$ and equals the fringe visibility for equal-intensity beams.
The general interference law is $I=I_1+I_2+2\sqrt{I_1 I_2}\,\mathrm{Re}\,\gamma_{12}(\tau)$ , generalizing the interference term to partially coherent light.
Temporal coherence is the $\tau$ -dependence of the self-coherence $\Gamma_{11}(\tau)$ ; spatial coherence is the two-point dependence of the equal-time $\Gamma_{12}(0)$ ; both are faces of $\Gamma_{12}$ .
The bound $|\gamma_{12}|\le 1$ is a Cauchy–Schwarz inequality — it guarantees fringe visibilities never exceed one and intensities never go negative.
Key idea: a single complex function $\Gamma_{12}(\tau)=\langle E_1^*(t)E_2(t+\tau)\rangle$ completely determines the outcome of any two-beam interference experiment; its normalized modulus is the fringe visibility, $V=\frac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|$ .

The core insight

All second-order coherence of a light field is carried by one complex function, $\Gamma_{12}(\tau)=\langle E_1^*(t)E_2(t+\tau)\rangle$ . Its normalized modulus $|\gamma_{12}|\in[0,1]$ is the fringe visibility; the equal-point limit gives temporal coherence, the equal-time limit gives spatial coherence, and the general interference law $I=I_1+I_2+2\sqrt{I_1I_2}\,\mathrm{Re}\,\gamma_{12}$ predicts every two-beam experiment.

Where this lives today

Because $\Gamma_{12}$ propagates like a field, coherence theory is the working language of any instrument whose performance is a coherence budget. A Michelson stellar interferometer measures $|j_{12}|$ — the spatial degree of coherence at two apertures — and reads the star's angular diameter off the separation at which fringes vanish, via Van Cittert–Zernike. A Fourier-transform spectrometer scans the self-coherence $\Gamma_{11}(\tau)$ and inverts it (Wiener–Khinchin) to recover the source spectrum. Optical coherence tomography engineers a short self-coherence so that fringes localize a micron-thin slice of tissue. In each case the designer is propagating and shaping one function. The same object runs partially coherent imaging. In projection lithography and microscopy the illumination is described by the mutual intensity $J_{12}=\Gamma_{12}(0)$ across the source and pupil, and the Hopkins theory of image formation is built entirely on transmitting $J_{12}$ through the optical system — the degree of spatial coherence controls which spatial frequencies transfer and how sharp the image is. And at the frontier, the distinction the warning box draws — that $\Gamma_{12}$ is first-order while photon statistics live in $g^{(2)}$ — is exactly what Hanbury Brown and Twiss exploited to measure stellar diameters by intensity correlations, and what modern quantum optics uses to certify single-photon sources. One classical function, and the precise boundary of where classical stops.

The physical picture

Picture two detectors sampling one optical field. Multiply the field at detector 1 (at time $t$ ) by the field at detector 2 (a delay $\tau$ later) and average over the light's random fluctuations. If the product averages to something large, the two samples keep a fixed relative phase across that separation and delay — they are coherent. If it averages to zero, their random phases have washed out — incoherent. That multiply-and-average operation is $\Gamma_{12}(\tau)$ . It is complex because it encodes two things at once: the modulus sets how sharp the fringes are, and the argument sets where the fringes sit. Interference fringes are, quite literally, a physical plot of $\mathrm{Re}\,\Gamma_{12}$ scanned across delay or across slit separation. The payoff of a single function is that a star, a discharge lamp, and a laser are not three different physics problems — they are three values of the same $\gamma_{12}$ , differing only in how fast it decays with delay (temporal) or with separation (spatial). Coherence stops being an all-or-nothing label and becomes a continuous number on $[0,1]$ , and the fringe contrast reads it out.

Figure

Two detectors sample the field at (r₁,t) and (r₂,t+τ); their cross-correlation ⟨E₁*(t)E₂(t+τ)⟩ = Γ₁₂(τ) is the one function that fixes both fringe contrast and position

The most compact way to see the whole scale is to hold the two beams equal in intensity and vary only their correlation. When $|\gamma_{12}|=1$ the fringes swing all the way from bright to fully dark; when $|\gamma_{12}|=0$ the field is flat with no fringes; and partial coherence sits between, with a contrast equal to $|\gamma_{12}|$ itself. The visibility is a direct, calibration-free readout of the coherence.

Figure

Fringe contrast is the degree of coherence: for equal beams V = |γ₁₂| — |γ|=1 gives V=1 (Imin=0), |γ|=0.6 gives V=0.6 (Imax:Imin = 4:1), |γ|=0 gives a flat field

The core equations

Fix two points in a wavefield and correlate the field at the first with the field at the second, delayed by $\tau$ . Averaging over the light's fluctuations defines the master quantity of coherence theory; the analytic-signal (complex) representation is used so the correlation keeps its phase.

\Gamma_{12}(\tau)=\langle E_1^*(t)\,E_2(t+\tau)\rangle=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}E_1^*(t)\,E_2(t+\tau)\,\mathrm{d}t \qquad (1)

where: $E_i(t)$ — complex analytic signal at point $\mathbf r_i$ ; $\tau$ — time delay between the two field samples; $\langle\cdot\rangle$ — time (= ensemble, by ergodicity) average; $\Gamma_{12}$ — mutual coherence function, with the dimensions of intensity. Dividing by the geometric mean of the two intensities removes the dimensions and the beam-brightness dependence, leaving a pure correlation whose modulus never exceeds one — the number that actually appears in fringe contrast.

\gamma_{12}(\tau)=\frac{\Gamma_{12}(\tau)}{\sqrt{\Gamma_{11}(0)\,\Gamma_{22}(0)}}=\frac{\Gamma_{12}(\tau)}{\sqrt{I_1 I_2}},\qquad 0\le|\gamma_{12}|\le 1 \qquad (2)

where: $\Gamma_{ii}(0)=I_i$ — mean intensity at point $\mathbf r_i$ ; $\gamma_{12}$ — complex degree of coherence, dimensionless. Its equal-point limit $\gamma_{11}(\tau)$ is the temporal-coherence factor; its equal-time limit $\gamma_{12}(0)$ is the spatial-coherence factor. Superpose the two contributions at a screen and time-average the squared field. The self-terms give the two intensities; the cross-term survives only through the correlation, so it is carried entirely by $\Gamma_{12}$ . This is the law behind every fringe pattern.

I=I_1+I_2+2\sqrt{I_1 I_2}\;\mathrm{Re}\,\gamma_{12}(\tau) \qquad (3)

where: $I$ — time-averaged intensity at the superposition point; $\mathrm{Re}\,\gamma_{12}=|\gamma_{12}|\cos(2\pi\bar\nu\tau-\alpha_{12})$ — the oscillating fringe term, a fast cosine modulated by the slow envelope $|\gamma_{12}|$ ; $\bar\nu$ — mean optical frequency. Extremizing the interference law over the fringe phase gives the contrast. For unequal beams the amplitude mismatch enters as a separate factor; balanced beams isolate the coherence itself.

V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=\frac{2\sqrt{I_1 I_2}}{I_1+I_2}\,|\gamma_{12}(\tau)|\ \xrightarrow{I_1=I_2}\ |\gamma_{12}(\tau)| \qquad (4)

where: $V$ — fringe visibility; $I_{\max},I_{\min}$ — fringe extrema as the phase is scanned. The prefactor $2\sqrt{I_1I_2}/(I_1+I_2)\le 1$ is the amplitude-balance penalty; the coherence $|\gamma_{12}|\le 1$ is the correlation penalty, and both must be near one for crisp fringes. Finally, the bound that makes the whole theory physical: the correlation of two fields cannot exceed the geometric mean of their self-correlations. This is a Cauchy–Schwarz inequality and it guarantees $I_{\min}\ge 0$ .

|\gamma_{12}(\tau)|^2\le\gamma_{11}(0)\,\gamma_{22}(0)=1 \qquad (5)

where: $\gamma_{ii}(0)=1$ — a field is perfectly correlated with itself at the same point and instant, which fixes the normalization of the entire framework.

Derivation AThe general interference law and the unification of visibility

The goal of this derivation is to show that one line of algebra — averaging the squared sum of two fields — produces the interference law for any state of coherence, and that extremizing it yields a visibility that unifies three quantities usually taught separately: fringe contrast, temporal coherence, and spatial coherence. This is the article's owned structural result: visibility, coherence time, and coherence area are all readings of the single function $\gamma_{12}$ .

Step 1. Superpose the two contributions reaching the screen point, $E=E_1(t)+E_2(t+\tau)$ , and time-average the squared modulus: $I=\langle|E|^2\rangle=\langle|E_1|^2\rangle+\langle|E_2|^2\rangle+2\,\mathrm{Re}\langle E_1^*(t)E_2(t+\tau)\rangle$ . Why: intensity is the time-averaged square of the field, and the cross-term is the only place the two beams "talk" — it is exactly the mutual coherence function.

Step 2. Identify the pieces, $I=I_1+I_2+2\,\mathrm{Re}\,\Gamma_{12}(\tau)=I_1+I_2+2\sqrt{I_1I_2}\,\mathrm{Re}\,\gamma_{12}(\tau)$ . Why: the self-terms are the intensities and the cross-term is $\Gamma_{12}$ ; normalizing by $\sqrt{I_1I_2}$ turns the dimensioned correlation into the bounded $\gamma_{12}$ .

Step 3. Write $\mathrm{Re}\,\gamma_{12}=|\gamma_{12}|\cos\phi$ with $\phi$ the total fringe phase, and read off the extrema $I_{\max/\min}=I_1+I_2\pm 2\sqrt{I_1I_2}\,|\gamma_{12}|$ . Hence $V=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|$ , which for $I_1=I_2$ collapses to $V=|\gamma_{12}|$ . Why: the fringe contrast is a product of two independent penalties — amplitude balance and field correlation — and only the second is intrinsic to the light's coherence.

Step 4. Specialize the arguments of $\gamma_{12}$ : with the two points equal, $\gamma_{11}(\tau)$ decays over the coherence time (temporal coherence, article 1419); with the delay set to zero, $\gamma_{12}(0)$ decays over the coherence area (spatial coherence, article 1420); the fringe contrast article (1417) simply reports $V=|\gamma_{12}|$ . Why: three chapters, one function — this is why $\Gamma_{12}$ is called the master quantity of coherence theory. *Verified: the expansion $\langle|E_1+E_2|^2\rangle$ , the extrema $I_{\max/\min}=I_1+I_2\pm2\sqrt{I_1I_2}|\gamma|$ , and $V=2\sqrt{I_1I_2}/(I_1+I_2)|\gamma|\to|\gamma|$ are checked symbolically in verify_mutual-coherence-function.py.*

Go deeper · Derivation B · 1 minWhy the degree of coherence cannot exceed one

Derivation BWhy the degree of coherence cannot exceed one

This derivation proves the bound $|\gamma_{12}|\le 1$ , the inequality that keeps coherence theory physically consistent: it forbids visibilities greater than one and guarantees that the interference law never predicts a negative intensity. The proof is a direct Cauchy–Schwarz inequality applied to the field cross-correlation viewed as an inner product, with equality only when the two fields are perfectly proportional — the fully coherent case.

Step 1. Define an inner product on the fluctuating signals, $\langle f,g\rangle\equiv\langle f^*(t)\,g(t)\rangle$ , with $f(t)=E_1(t)$ and $g(t)=E_2(t+\tau)$ . Why: the time average of a product of a conjugate and a field is a genuine positive-definite inner product, so the machinery of inner-product spaces applies to optical fields.

Step 3. Translate to coherence functions, $|\Gamma_{12}(\tau)|^2\le\Gamma_{11}(0)\,\Gamma_{22}(0)=I_1 I_2$ , and divide: $|\gamma_{12}(\tau)|=|\Gamma_{12}|/\sqrt{I_1I_2}\le 1$ . Why: substituting the definitions turns the abstract inequality into the physical bound on the degree of coherence.

Step 4. Check consistency with the interference law: $I_{\min}=I_1+I_2-2\sqrt{I_1I_2}|\gamma_{12}|\ge(\sqrt{I_1}-\sqrt{I_2})^2\ge 0$ , so intensity is never negative, and equality $|\gamma_{12}|=1$ occurs only when $E_2\propto E_1$ . Why: a bound that allowed $|\gamma_{12}|>1$ would predict dark fringes with negative energy — the inequality is what makes the theory obey conservation of energy. *Verified: Cauchy–Schwarz gives $|\gamma_{12}|\le1$ ; the resulting $I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2\ge0$ for $|\gamma|=1$ is checked in verify_mutual-coherence-function.py.*

Go deeper · Derivation C · 1 minFactorizing coherence: temporal times spatial

Derivation CFactorizing coherence: temporal times spatial

For quasi-monochromatic light from an extended source, the degree of coherence between two beams often factorizes into a temporal part (set by the path-difference delay) and a spatial part (set by the transverse separation): $|\gamma_{12}|\approx\mu_{\text{temporal}}\,\mu_{\text{spatial}}$ . This is how a real interferometer's contrast is budgeted in practice — poor coherence in either channel drags the whole visibility down multiplicatively.

Step 1. Write the field at each pinhole as a slow envelope times a common carrier, and suppose the temporal fluctuations (frequency spread) and the spatial fluctuations (finite source size) are statistically independent. Why: for a quasi-monochromatic extended thermal source the delay-dependence and the separation-dependence come from unrelated physical causes — bandwidth and source geometry — so their correlations multiply.

Step 2. The correlation then separates, $\gamma_{12}(\tau)\approx\gamma_{11}(\tau)\times j_{12}$ , whose moduli are the temporal factor $\mu_{\text{temporal}}=|\gamma_{11}(\tau)|$ (from article 1419) and the spatial factor $\mu_{\text{spatial}}=|j_{12}|$ (from the Van Cittert–Zernike theorem, article 1420). Why: the self-coherence carries the delay information and the equal-time mutual intensity carries the separation information; independence lets them multiply.

Step 3. Combine numerically: a source with $\mu_{\text{temporal}}=0.90$ (small path imbalance within the coherence length) and $\mu_{\text{spatial}}=0.70$ (pinholes near the edge of the coherence area) gives $|\gamma_{12}|=0.90\times0.70=0.63$ , hence $V=0.63$ for balanced beams. Why: this shows that "good enough" in each channel is not good enough overall — 0.9 and 0.7 still cost you a third of the contrast.

Step 4. Read the design lesson: to keep fringes crisp, keep the arm imbalance well inside $\ell_c$ (temporal) and the pinholes well inside the coherence width (spatial); either one near its limit throttles $V$ . Why: interferometer alignment is a two-channel coherence budget, and the mutual coherence function is the ledger. *Verified: the factorization $|\gamma_{12}|=\mu_t\mu_s=0.90\times0.70=0.63$ and $V=0.63$ are checked in verify_mutual-coherence-function.py.*

Limiting cases

Case	Result	Meaning
Complete coherence	$\|\gamma_{12}(\tau)\|=1$	fields perfectly correlated; unit-visibility fringes — the idealized monochromatic or single-mode-laser limit
Complete incoherence	$\gamma_{12}(\tau)=0$	no correlation; intensities simply add, $I=I_1+I_2$ , no fringes — independent sources, or beyond the coherence length/area
Partial coherence	$0<\|\gamma_{12}\|<1$	the generic case for real light; fringes with reduced contrast $V=\|\gamma_{12}\|$ (equal beams)
Equal points, $\tau\ne 0$	$\gamma_{11}(\tau)$	temporal coherence; width = coherence time $\tau_c$ , Wiener–Khinchin
Equal time, $\mathbf r_1\ne\mathbf r_2$	$\gamma_{12}(0)=j_{12}$	spatial coherence; falloff with separation ↔ source angular size (Van Cittert–Zernike)
Self-coherence at zero delay	$\gamma_{11}(0)=1$	a field always fully correlates with itself here; fixes the normalization

Common trap“Full fringe visibility (|γ₁₂| = 1) means the light is fully coherent in every sense”

The mutual coherence function is a first-order, field–field correlation, and fringe visibility reads only its modulus. Two beams with $|\gamma_{12}|=1$ produce identical, perfect fringes whether they come from a laser or from carefully filtered thermal light — the interferometer cannot tell them apart. What distinguishes them lives in the second-order, intensity–intensity correlation $g^{(2)}(\tau)$ , measured by coincidence counting (Hanbury Brown–Twiss), which sees photon bunching and antibunching. So " $|\gamma_{12}|=1$ " certifies coherence for interference but says nothing about photon statistics. A related trap: $\gamma_{12}$ is a normalized ratio bounded by one, while $\Gamma_{12}$ carries the dimensions of intensity — never quote a visibility straight from an unnormalized $\Gamma_{12}$ .

Worked examples

1 · Reading the degree of coherence off a fringe pattern (confidence builder)

Two equal-intensity beams produce fringes with $I_{\max}/I_{\min}=9$ . What is $|\gamma_{12}|$ at that delay, and what does it say about the light?

Step 1. Compute the visibility directly from the extrema: $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})=(9-1)/(9+1)=\mathbf{0.80}$ . Only the ratio is needed, so no absolute calibration of the detector is required.

Step 2. For equal beams the amplitude prefactor is one, so $V=|\gamma_{12}|$ exactly: $|\gamma_{12}|=\mathbf{0.80}$ . The light is partially coherent at this point and delay — well correlated but not perfectly.

Step 3. Turn it around: at $|\gamma_{12}|=0.80$ the interference law predicts $I_{\max}:I_{\min}=(1+0.8):(1-0.8)=1.8:0.2=9:1$ , reproducing the measurement. The map between contrast ratio and coherence is one-to-one.

Step 4. Interpretation: whether the shortfall from unity is temporal (path imbalance approaching $\ell_c$ ) or spatial (pinholes approaching the coherence width) is decided by which knob you turned — $\Gamma_{12}$ holds both. |γ₁₂| = 0.80, partial coherence.

2 · Unequal beams and a two-channel coherence budget (exam level)

Part A: two fully coherent beams have intensity ratio $I_1:I_2=1:0.25$ . Find the visibility. Part B: a quasi-monochromatic extended source has temporal factor $\mu_{\text{temporal}}=0.90$ and spatial factor $\mu_{\text{spatial}}=0.70$ ; find the overall degree of coherence and the balanced-beam visibility.

Step 1. Part A — even with perfect coherence the amplitude mismatch caps the contrast: $V=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|=\dfrac{2\sqrt{1\times0.25}}{1+0.25}\times 1=\dfrac{2\times0.5}{1.25}=\mathbf{0.80}$ . A 4:1 intensity mismatch (2:1 in amplitude) already costs 20% of the contrast.

Step 2. Cross-check via extrema: $I_{\max}=(\sqrt{1}+\sqrt{0.25})^2=(1.5)^2=2.25$ , $I_{\min}=(1-0.5)^2=0.25$ , so $V=(2.25-0.25)/(2.25+0.25)=2.00/2.50=\mathbf{0.80}$ — the two routes agree.

Step 3. Part B — independent channels multiply: $|\gamma_{12}|=\mu_{\text{temporal}}\times\mu_{\text{spatial}}=0.90\times0.70=\mathbf{0.63}$ . For balanced beams $V=|\gamma_{12}|=0.63$ ; the corresponding fringe ratio is $I_{\max}:I_{\min}=1.63:0.37\approx 4.4:1$ .

Step 4. Design reading: neither channel is bad (0.90 and 0.70), yet the product loses more than a third of full contrast. Crisp fringes demand both the arm imbalance inside $\ell_c$ and the apertures inside the coherence area. Part A V = 0.80; Part B |γ₁₂| = 0.63, V = 0.63.

Physics of the portrait

The mutual coherence function is not a person but a unifying idea, and its "portrait" is really the mid-twentieth-century synthesis that turned scattered observations about fringes into one predictive theory.

Origins · 1 minmutual coherence function and how this was discovered

mutual coherence function

classical coherence theory · van Cittert to Wolf · the one function behind every fringe

Coherence had been an experimental fact since Young's fringes, but it became a theory only when the right object was identified. In 1934 Pieter van Cittert computed the correlation between the light at two points illuminated by an extended incoherent source, and in 1938 Frits Zernike distilled it into a measurable "degree of coherence" tied directly to fringe visibility — the normalized $\gamma_{12}$ we still use. The full synthesis came from Emil Wolf in the mid-1950s: he defined the mutual coherence function $\Gamma_{12}(\tau)$ , showed that it propagates through space obeying a pair of wave equations, and thereby made coherence a field you can track from source to detector rather than a property assigned by hand. Max Born and Emil Wolf codified the whole edifice in Principles of Optics (1959), the text that taught generations to treat visibility, coherence time, and coherence area as three faces of one function. The framework proved so robust that when Roy Glauber built the quantum theory of optical coherence in 1963 — for which he shared the 2005 Nobel Prize — the classical $\Gamma_{12}$ reappeared as the first-order correlation $g^{(1)}$ , with the genuinely quantum physics living one order higher, in $g^{(2)}$ .

History

1934

— van Cittert computes the correlation of the field at two points from an extended incoherent source

1938

— Zernike introduces the degree of coherence and its measurement through fringe visibility

1955

— Wolf formulates the mutual coherence function and its propagation equations

1959

— Born and Wolf's Principles of Optics codifies classical coherence theory

1963

— Glauber's quantum coherence functions extend the framework, recovering Γ₁₂ as the first-order g⁽¹⁾

Going further

Texts. Hecht, Optics, 5th ed., §12.1 (Coherence Theory) · Saleh & Teich, Fundamentals of Photonics, ch. 11 (Statistical Optics) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 3 · Born & Wolf, Principles of Optics, 7th ed., ch. 10 · Mandel & Wolf, Optical Coherence and Quantum Optics, ch. 4 · Goodman, Statistical Optics, 2nd ed., ch. 5.

Provenance. Definitions of $\Gamma_{12}$ , $\gamma_{12}$ , and the general interference law follow Hecht §12.1, Saleh & Teich ch. 11, Born & Wolf ch. 10, and Mandel & Wolf ch. 4. verify_mutual-coherence-function.py recomputes: the interference-law expansion $\langle|E_1+E_2|^2\rangle$ ; the visibility $V=2\sqrt{I_1I_2}/(I_1+I_2)|\gamma|\to|\gamma|$ ; the Cauchy–Schwarz bound $|\gamma_{12}|\le 1$ with $I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2\ge 0$ ; the worked values $V=(9-1)/(9+1)=0.80$ , the unequal-beam $V=0.80$ , and the factorization $|\gamma_{12}|=0.90\times0.70=0.63$ . The Betelgeuse angular-diameter and stellar-interferometer context is developed in the spatial-coherence article; no constants are fabricated here.

Prerequisites. Superposition Principle of Electromagnetic Waves · Two-Wave Interference and the Interference Term · Conditions for Observable Interference · Optical Path Difference and Phase Retardation. Leads to. Fringe Visibility and Contrast · Temporal Coherence and Coherence Length · Spatial Coherence and the Van Cittert–Zernike Theorem · Young's Double-Slit Experiment.

Why is Γ₁₂ complex when the physical field is real?

Because coherence theory uses the analytic-signal representation: the real field is replaced by a complex $E(t)$ whose real part is the physical field and whose imaginary part is its Hilbert transform. The reward is that the single complex $\Gamma_{12}$ encodes two independent things — the correlation strength in its modulus (which sets fringe contrast) and the mean phase difference in its argument (which sets fringe position). A purely real correlation function could carry contrast but would lose the clean separation of envelope and carrier that makes $|\gamma_{12}|$ the visibility and $\arg\gamma_{12}$ the fringe shift. It is the same trick that makes phasors convenient in AC circuits, applied to fluctuating optical fields.

How is γ₁₂ different from the second-order correlation g⁽²⁾, and why can't visibility tell them apart?

$\gamma_{12}$ (equivalently $g^{(1)}$ ) is a field–field correlation, measured by interference; fringe visibility reads its modulus. $g^{(2)}$ is an intensity–intensity correlation, measured by coincidence counting (Hanbury Brown–Twiss). They are logically independent: a coherent-state laser has $|g^{(1)}|=1$ and $g^{(2)}=1$ , filtered thermal light can have $|g^{(1)}|=1$ locally but $g^{(2)}=2$ (bunching), and a single-photon source has $g^{(2)}(0)\to 0$ (antibunching) regardless of its $g^{(1)}$ . Since fringes depend only on $g^{(1)}$ , two sources with identical spectra give identical fringes even when their photon statistics are utterly different. That is why photon statistics require a separate, intensity-correlation experiment.

What has to be true for the time average in Γ₁₂ to equal an ensemble average?

Two properties. First, stationarity: the statistics of the field cannot depend on absolute time, so correlations depend only on the delay $\tau$ , not on when you start averaging. Second, ergodicity: a single long time record must explore the full statistical ensemble, so that averaging over time is equivalent to averaging over many independent realizations. Thermal sources and stabilized lasers satisfy both to excellent approximation on timescales long compared with the coherence time. When they fail — for pulsed, chirped, or transient light — one must keep the explicit two-time correlation and cannot collapse it to a function of $\tau$ alone.

If Γ₁₂ contains "all" coherence information, why do we still teach temporal and spatial coherence separately?

Because the two limits are governed by different physics and different theorems, and separating them is how you diagnose a real instrument. Temporal coherence lives in the delay-dependence $\gamma_{11}(\tau)$ and is fixed by the source spectrum through Wiener–Khinchin; spatial coherence lives in the separation-dependence $\gamma_{12}(0)$ and is fixed by the source geometry through Van Cittert–Zernike. When fringes fade, knowing which limit you are in tells you whether to narrow the bandwidth or shrink the source. $\Gamma_{12}$ unifies them conceptually and lets them multiply in a coherence budget, but the two knobs on the bench remain physically distinct.

Can the degree of coherence ever be negative, or is it always between 0 and 1?

The modulus $|\gamma_{12}|$ is always between 0 and 1 — that is the Cauchy–Schwarz bound. But $\gamma_{12}$ itself is complex, and its real part $\mathrm{Re}\,\gamma_{12}$ , which is what appears in the interference law, swings between $-|\gamma_{12}|$ and $+|\gamma_{12}|$ as the phase is scanned; that sign change is exactly the alternation between bright and dark fringes. So "negative coherence" is not a thing, but a negative real part of the coherence is just a dark fringe. When people quote a single number for the degree of coherence they mean the modulus, the envelope that caps the fringe contrast.

Does the mutual coherence function survive when I send the light through lenses and free space?

Yes — and that is Wolf's key result. $\Gamma_{12}$ is not a property frozen at the source; it propagates, obeying a pair of wave equations (one in each point), so you can compute the coherence at a detector from the coherence at the source the same way you propagate a field. This is what makes the Van Cittert–Zernike theorem possible: an incoherent source acquires spatial coherence simply by propagating far enough, because the mutual coherence function spreads according to those wave equations. In imaging systems the equal-time version, the mutual intensity $J_{12}$ , is propagated through the lenses (Hopkins theory) to predict partially coherent image formation. Coherence flows.