Fringe Visibility and Contrast

Point a camera at an interference pattern and you are really asking one question of the light: how dark do the dark fringes get? A pattern whose minima fall all the way to black is telling you the two beams are perfectly matched — same brightness, same polarization, and, above all, a phase relationship that holds steady for as long as the detector watches. A pattern whose "dark" fringes are merely a dim grey is confessing that something is wrong: the beams are unbalanced, or partly depolarized, or — most tellingly — their phase wanders. Fringe visibility, $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ , is the single number that reads this confession off the pattern. Its power is that it needs no calibration. You never measure an absolute intensity or a phase; you measure only the relative depth of the brightest and darkest points, a ratio robust against gain drift, exposure, and units. And that ratio is not merely cosmetic. Coherence theory shows that visibility is the modulus of the complex degree of coherence, weighted by how well the beam intensities balance: $V=\dfrac{2\sqrt{I_1 I_2}}{I_1+I_2}\,|\gamma_{12}|$ . For equal beams the prefactor is one and $V=|\gamma_{12}|$ exactly — so raw fringe contrast becomes a direct meter for coherence itself. That is why visibility is the master observable of interferometry. Scan the path difference and $V$ traces the temporal-coherence envelope, handing you the source's spectrum by Fourier transform. Widen the source and $V$ collapses along the spatial-coherence function, handing you the source's angular size. Michelson built a stellar interferometer on exactly this idea and weighed a star with it. One dimensionless number, read from the light, carries the coherence, the spectrum, and the geometry all at once.

Key takeaways

Fringe visibility (contrast) is $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ , a number between 0 and 1 measuring how sharply fringes stand out from their background — the modulation depth of the intensity.
For two partially coherent beams $V=\dfrac{2\sqrt{I_1 I_2}}{I_1+I_2}\,|\gamma_{12}|$ : contrast is the product of an intensity-balance factor and the degree of coherence.
With equal intensities the balance factor is 1 and $V=|\gamma_{12}|$ — raw contrast becomes a direct measurement of coherence, needing no phase or absolute-intensity calibration.
Unequal beams cap visibility even for perfect coherence: an intensity ratio $r=I_1/I_2$ gives at best $V=2\sqrt{r}/(1+r)$ (e.g. $r=4\Rightarrow V=0.8$ ).
An added incoherent background — stray light, dark counts, the unpolarized share — dilutes the pedestal and lowers $V$ below the true $|\gamma_{12}|$ , so contrast must be corrected before coherence is inferred.
Key idea: fringe visibility $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ equals $\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|$ ; for balanced beams it is exactly the modulus of the complex degree of coherence, making contrast a direct meter of coherence.

The core insight

Fringe visibility $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ measures the modulation depth of an interference pattern. Written in terms of the interfering beams it is $V=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|$ — an intensity-balance factor times the degree of coherence. When the beams are equal, $V=|\gamma_{12}|$ , so the darkness of the dark fringes is a direct, calibration-free measurement of how coherent the light is.

Where this lives today

Every modern interferometer treats visibility as a live diagnostic. In gravitational-wave detectors such as LIGO, the fringe contrast at the output port is monitored continuously and driven toward its ideal value by matching arm powers, polarizations, and spatial modes — a falling $V$ is the first sign of misalignment or contamination. In Fourier-transform infrared spectrometers the recorded interferogram is literally $I(\tau)\propto 1+V(\tau)\cos(2\pi\nu_0\tau)$ , and the spectrum is recovered by transforming the visibility-modulated signal; the achievable resolution is set by the largest delay over which fringes remain visible. The same number weighs stars and images galaxies. Optical and infrared stellar interferometers — from the historic Michelson–Pease apparatus to the CHARA array and the VLTI — measure the squared visibility of fringes on multiple baselines and invert it, via van Cittert–Zernike, into angular sizes and even resolved surface maps. In quantum optics the two-photon analogue, the visibility of Hong–Ou–Mandel or Bell-test coincidence fringes, is the standard witness of indistinguishability and entanglement, with "visibility above 71%" a familiar threshold for violating a classical bound.

The physical picture

Picture the intensity along a row of fringes as a cosine riding on a flat pedestal. Visibility is the ratio of the cosine's amplitude to the pedestal's height. A big wave on a low pedestal — fringes plunging nearly to black — is $V\to 1$ . A shallow ripple on a tall flat background — barely-there fringes — is $V\to 0$ . Nothing else about the pattern matters: not its brightness, not its fringe spacing, only how deep the modulation cuts.

Figure

An intensity fringe trace: the modulation depth between I_max and I_min sets the visibility V = (I_max − I_min)/(I_max + I_min)

Two entirely different faults can flatten that ripple, and visibility multiplies them together. First, the beams may be poorly correlated: if their relative phase drifts randomly during the exposure, the cross term averages toward zero and the fringes smear — a loss of coherence, carried by $|\gamma_{12}|$ . Second, even two perfectly correlated beams cannot cancel at the minima if one is much brighter than the other, because the weaker field simply has not enough amplitude to subtract the whole of the stronger. That is the balance factor $2\sqrt{I_1I_2}/(I_1+I_2)$ , and it is a purely geometric ceiling on contrast that has nothing to do with coherence. The next figure makes the coherence-versus-contrast link literal by drawing two patterns with different visibilities and reading the number straight off the extrema.

Figure

Two fringe patterns with different contrast: V = 1.0 (minima reach zero) versus V = 0.5, read directly from I_max and I_min

Read as a "signal-to-background on a 0–1 scale," visibility is exactly the interferometer's health gauge. Scan a knob — the path delay, the slit separation, the source size — and watch $V$ shrink: the shape of $V$ against that knob is the Fourier partner of the spectrum (when you scan delay) or of the source's brightness distribution (when you scan geometry). That is the whole engine of Fourier-transform spectroscopy and stellar interferometry, and it is why the darkness of the dark fringes is worth measuring carefully.

The core equations

Visibility is defined purely from the pattern, with no reference to what caused it. This is deliberate: it makes $V$ measurable from a single image without knowing the beams' intensities, phases, or the instrument's absolute gain.

V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} \qquad (1)

where: $V$ — fringe visibility (contrast), dimensionless, $0\le V\le 1$ ; $I_{\max}$ — intensity at a bright fringe; $I_{\min}$ — intensity at the adjacent dark fringe. Because both extrema scale with the overall gain, their ratio cancels it: $V$ is a pure modulation depth. To connect $V$ to physics we need the intensity pattern that two partially coherent beams actually produce. Detectors read the time-averaged intensity, and the cross term of the superposition carries a factor $|\gamma_{12}|$ — the modulus of the complex degree of coherence — plus a running phase set by the optical path difference.

I=I_1+I_2+2\sqrt{I_1 I_2}\,|\gamma_{12}|\cos(\alpha_{12}+\delta) \qquad (2)

where: $I_1,I_2$ — time-averaged intensities of the two beams; $|\gamma_{12}|$ — degree of coherence, $0\le|\gamma_{12}|\le 1$ ; $\alpha_{12}$ — phase of $\gamma_{12}=|\gamma_{12}|e^{i\alpha_{12}}$ (an offset that merely shifts the fringes); $\delta=2\pi\Lambda/\lambda_0$ — deterministic phase from the optical path difference $\Lambda$ . As $\delta$ sweeps, the cosine runs over $[-1,+1]$ , painting the fringes. Feeding the extrema of that cosine into the definition collapses everything to one compact statement: contrast is the product of how well the beams balance in intensity and how coherent they are.

V=\frac{2\sqrt{I_1 I_2}}{I_1+I_2}\,|\gamma_{12}| \qquad (3)

where: $\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}$ — intensity-balance factor, equal to 1 only when $I_1=I_2$ ; $|\gamma_{12}|$ — degree of coherence between the two field points or times. When the beams are equal this reduces to $V=|\gamma_{12}|$ , the relation that turns an interferometer into a coherence meter. When a source has spectral bandwidth, its self-coherence sets how $V$ decays with delay; the Wiener–Khinchin theorem makes that decay the Fourier transform of the spectrum, and its width defines the coherence length.

V(\tau)=|\gamma(\tau)|=\left|\frac{\int_0^\infty S(\nu)\,e^{-i2\pi\nu\tau}\,d\nu}{\int_0^\infty S(\nu)\,d\nu}\right|,\qquad \ell_c\approx c\,\tau_c\approx\frac{\lambda^2}{\Delta\lambda} \qquad (4)

where: $S(\nu)$ — power spectral density of the source; $\tau=\Lambda/c$ — time delay set by the OPD; $\ell_c$ — coherence length (the OPD at which $V$ has fallen to $\sim 1/e$ ); $\Delta\lambda$ — spectral width; $\lambda$ — mean wavelength. Visibility scanned against delay is the spectrometer.

Derivation AVisibility as the modulus of the degree of coherence

The goal here is to turn the qualitative statement "poor contrast means poor coherence" into the exact working formula $V=\tfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|$ , so that a number read off an image becomes a number about the light's correlation. We start from the time-averaged intensity of two overlapping quasi-monochromatic fields, isolate the interference cross term, normalise it into the degree of coherence, and then evaluate the pattern's extrema.

Step 1. Write the detected intensity as the time average of the squared total field, $I=\langle|E_1+E_2|^2\rangle=I_1+I_2+2\,\mathrm{Re}\,\langle E_1^*E_2\rangle$ . Why: a photodetector responds to intensity, the time-averaged square of the field, so the observable is the mean of $|E_1+E_2|^2$ , and the only new piece beyond $I_1+I_2$ is the cross term $\langle E_1^*E_2\rangle$ .

Step 2. Define the mutual coherence $\Gamma_{12}=\langle E_1^*E_2\rangle$ and normalise it: $\gamma_{12}=\Gamma_{12}/\sqrt{I_1I_2}=|\gamma_{12}|e^{i\alpha_{12}}$ , with $0\le|\gamma_{12}|\le 1$ by the Cauchy–Schwarz inequality. Why: normalising by the geometric mean of the intensities strips out beam brightness and leaves a pure correlation coefficient — the same idea as a statistical correlation, bounded by one.

Step 3. Insert the running phase from the optical path difference, $\delta=2\pi\Lambda/\lambda_0$ , so $I=I_1+I_2+2\sqrt{I_1I_2}\,|\gamma_{12}|\cos(\alpha_{12}+\delta)$ ; the extrema occur where the cosine is $\pm 1$ : $I_{\max/\min}=I_1+I_2\pm 2\sqrt{I_1I_2}\,|\gamma_{12}|$ . Why: sweeping the path difference just walks the cosine through its full range, and the brightest and darkest fringes are exactly where it hits its turning points.

Step 4. Apply the definition: $V=\dfrac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=\dfrac{4\sqrt{I_1I_2}\,|\gamma_{12}|}{2(I_1+I_2)}=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}\,|\gamma_{12}|$ . Why: the constant pedestal $I_1+I_2$ cancels out of the numerator and survives in the denominator, leaving contrast as the balance factor times the degree of coherence. *Verified: the reduction to $V=|\gamma_{12}|$ at $I_1=I_2$ and to $V=2\sqrt r/(1+r)$ at $|\gamma_{12}|=1$ is checked symbolically and numerically in verify_fringe-visibility-contrast.py.*

Go deeper · Derivation B · 1 minWhy intensity imbalance alone caps the contrast

Derivation BWhy intensity imbalance alone caps the contrast

This derivation isolates the second, purely geometric way to lose contrast — beam imbalance — and proves it is bounded, peaking at perfect balance. The aim is to show that the factor $2\sqrt{I_1I_2}/(I_1+I_2)$ can never exceed one and reaches one only when $I_1=I_2$ , so that even flawlessly coherent light of unequal brightness has a hard ceiling on its visibility.

Step 1. Write the balance factor in terms of the intensity ratio $r=I_1/I_2$ : $b(r)=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}=\dfrac{2\sqrt r}{1+r}$ . Why: contrast cannot depend on absolute brightness, only on the ratio, so folding both intensities into $r$ reduces the question to a single-variable one.

Step 2. Recognise this as the ratio of geometric to arithmetic mean of $\{I_1,I_2\}$ , which the AM–GM inequality bounds: $\sqrt{I_1I_2}\le\tfrac12(I_1+I_2)$ , so $b(r)\le 1$ . Why: the geometric mean of two positive numbers never exceeds their arithmetic mean, and the two are equal only when the numbers are equal — a classical inequality doing physical work.

Step 3. Find the maximum directly: $db/dr=0$ gives $\dfrac{(1+r)/\sqrt r - 2\sqrt r}{(1+r)^2}=0\Rightarrow 1+r=2r\Rightarrow r=1$ , where $b=1$ . Why: setting the derivative to zero locates the balance point, confirming the ceiling is reached precisely at equal intensities and nowhere else.

Step 4. Read off representative values: $r=4\Rightarrow b=0.8$ , $r=9\Rightarrow b=0.6$ , $r=100\Rightarrow b\approx 0.198$ . Why: contrast degrades gently near balance but collapses for lopsided beams, which is why interferometers are aligned to equalise arm powers before anyone worries about coherence. *Verified: $b(r)=2\sqrt r/(1+r)\le 1$ with maximum at $r=1$ , and the values $b(4)=0.8$ , $b(9)=0.6$ , are checked in verify_fringe-visibility-contrast.py.*

Go deeper · Derivation C · 1 minWhy measured visibility falls below the true coherence

Derivation CWhy measured visibility falls below the true coherence

Real instruments always report a visibility smaller than $|\gamma_{12}|$ , and this derivation accounts for the two commonest culprits — an additive incoherent background and finite detector averaging — so that a measured contrast can be corrected back to the physical coherence. We add a constant background $I_b$ to the pedestal and average the cosine over a detector spanning a fraction of a fringe.

Step 1. Add an incoherent background $I_b$ (stray light, dark counts, the unpolarized share) that raises both extrema equally: $I_{\max/\min}=I_1+I_2+I_b\pm 2\sqrt{I_1I_2}|\gamma_{12}|$ . Why: incoherent light carries no fringe modulation, so it lifts the whole pedestal without touching the cosine amplitude — pure dilution.

Step 2. Recompute the contrast: $V=\dfrac{2\sqrt{I_1I_2}\,|\gamma_{12}|}{I_1+I_2+I_b}$ , strictly below the background-free value. Why: the background enlarges the denominator only, so any $I_b>0$ pushes $V$ down — a low reading can mean glare, not decoherence.

Step 3. Add finite detector width $w$ relative to fringe spacing $\Lambda_f$ : averaging $\cos(2\pi x/\Lambda_f)$ over the face multiplies the modulation by $\mathrm{sinc}(w/\Lambda_f)=\sin(\pi w/\Lambda_f)/(\pi w/\Lambda_f)$ . Why: a pixel wider than a fringe blurs the cosine across its own face, and the average of a cosine over a window is the sinc of that window — the same envelope as single-slit diffraction.

Step 4. Collect the instrumental factors: $V_{\text{meas}}=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2+I_b}\,|\mathrm{sinc}(w/\Lambda_f)|\,|\gamma_{12}|$ ; invert them to recover $|\gamma_{12}|$ . Why: only after dividing out balance, background, and averaging does the residual number report the physics — a discipline that separates instrument health from source coherence. *Verified: the background-diluted $V$ and the $\mathrm{sinc}(w/\Lambda_f)$ washout (with $V\to 0$ at $w=\Lambda_f$ ) are checked numerically in verify_fringe-visibility-contrast.py.*

Limiting cases

Case	Result	Meaning
balanced, fully coherent	$I_1=I_2,\ \|\gamma_{12}\|=1\Rightarrow V=1$	fringes reach zero at the minima — the ideal Young or Michelson pattern
balanced, partially coherent	$I_1=I_2\Rightarrow V=\|\gamma_{12}\|$	visibility equals the degree of coherence — the basis of every coherence measurement
unbalanced, fully coherent	$\|\gamma_{12}\|=1,\ r=I_1/I_2\Rightarrow V=2\sqrt r/(1+r)$	$r=4\Rightarrow V=0.8$ : contrast lost to imbalance, not coherence
incoherent limit	$\|\gamma_{12}\|=0\Rightarrow V=0$	intensities simply add; a uniformly lit field with no fringes
with incoherent background	$V=\dfrac{2\sqrt{I_1I_2}\|\gamma_{12}\|}{I_1+I_2+I_b}$	glare or dark counts dilute the pedestal and lower $V$
detector wider than a fringe	$V\to V_0\,\|\mathrm{sinc}(w/\Lambda_f)\|$	fringe averaging washes contrast away, hitting zero at $w=\Lambda_f$

Worked examples

1 · Reading coherence off the fringes (confidence builder)

A Young's fringe pattern fed by two equal-intensity beams is measured to have $I_{\max}=9$ and $I_{\min}=1$ in arbitrary units. Find the visibility, and the degree of coherence $|\gamma_{12}|$ .

Step 1. Apply the definition: $V=\dfrac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}=\dfrac{9-1}{9+1}=\dfrac{8}{10}=\mathbf{0.80}$ .

Step 2. Because the beams are equal, the balance factor is 1 and $V=|\gamma_{12}|$ , so $|\gamma_{12}|=\mathbf{0.80}$ — the two fields are 80% correlated.

Step 3. Sanity-check the physics: the dark fringes sit at $I_{\min}=1$ , one-ninth of the peak, not at zero, exactly because the correlation is imperfect. Full coherence would have driven $I_{\min}$ to 0 and $V$ to 1; complete incoherence would have flattened the pattern to $V=0$ . V = 0.80, and since the beams are balanced, |γ₁₂| = 0.80.

2 · The same visibility from imbalance instead of decoherence (exam level)

Two perfectly coherent beams ( $|\gamma_{12}|=1$ ) have intensities $I_1=1$ and $I_2=0.25$ . Find the visibility two ways — from the fringe extrema and from the balance factor — and comment.

Step 1. The fields add and subtract in amplitude, so with $\sqrt{I_1}=1$ and $\sqrt{I_2}=0.5$ : $I_{\max}=(\sqrt{I_1}+\sqrt{I_2})^2=(1+0.5)^2=2.25$ and $I_{\min}=(\sqrt{I_1}-\sqrt{I_2})^2=(1-0.5)^2=0.25$ .

Step 2. Definition route: $V=\dfrac{2.25-0.25}{2.25+0.25}=\dfrac{2.00}{2.50}=\mathbf{0.80}$ .

Step 3. Balance-factor route: $V=\dfrac{2\sqrt{I_1I_2}}{I_1+I_2}|\gamma_{12}|=\dfrac{2\sqrt{0.25}}{1.25}\cdot 1=\dfrac{1.0}{1.25}=\mathbf{0.80}$ — the two routes agree.

Step 4. The lesson: this $V=0.80$ comes entirely from a $4{:}1$ intensity imbalance, with the light perfectly coherent — numerically identical to Example 1, where $V=0.80$ came from $|\gamma_{12}|=0.80$ with balanced beams. Contrast alone cannot tell the two apart; only dividing out the balance factor does. V = 0.80 by both routes — but here it is imbalance, not decoherence, that limits the contrast.

Physics of the portrait

Fringe visibility is not owned by any one person; it is a definition — but a definition sharp enough to have started a field. The idea and the symbol trace to Albert A. Michelson, who in the 1890s needed a rigorous way to say how far he could push a path difference before his fringes faded, and who realised the very fading was a measurement.

Origins · 1 minfringe visibility and how this was discovered

fringe visibility

formalised by A. A. Michelson, 1890s · the interferometer's figure of merit

Fringe visibility, $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ , was introduced by Albert Abraham Michelson around 1890 as the quantitative "clearness" of interference fringes. Michelson's insight was that the decline of $V$ as he lengthened one arm of his interferometer was itself data: it traced the coherence of the light and, by what we now call the Wiener–Khinchin theorem, the shape of the source's spectral line. He used the collapse of visibility with wavelength to resolve the hyperfine structure of spectral lines a decade before quantum theory could explain them, and to redefine the metre in terms of the coherence length of the cadmium red line. In 1920, feeding a 6-metre beam onto the 100-inch telescope, he and Francis Pease watched the visibility of Betelgeuse's fringes fall to zero at a known baseline and so measured the first stellar diameter — about 0.047 arcseconds — turning a dimensionless ratio into a ruler for stars. The concept has since become the master observable of coherence theory, unifying the spectrum (via temporal coherence), the source size (via spatial coherence and the van Cittert–Zernike theorem), and instrument health into a single number between 0 and 1.

History

1801

— Young observes two-beam interference fringes, making contrast a visible phenomenon

1890

— Michelson introduces the visibility function V = (I_max − I_min)/(I_max + I_min)

1891

— Michelson resolves spectral-line fine structure from the periodic revival of fringe visibility

1920

— Michelson and Pease measure the diameter of Betelgeuse from the vanishing of fringe visibility

1938

— Zernike formalises the degree of coherence, of which visibility is the modulus

Going further

Texts. Hecht, Optics, 5th ed., §9.4 (fringe contrast and visibility) · Born & Wolf, Principles of Optics, 7th ed., Ch. 10 (partial coherence, Michelson visibility) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 3 (degree of coherence and visibility) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 11 (statistical optics) · Demtröder, Laser Spectroscopy, Vol. 1 (interferometer contrast, coherence length).

Provenance. The definition $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ and the coherence relation $V=2\sqrt{I_1I_2}\,|\gamma_{12}|/(I_1+I_2)$ are standard (Born & Wolf Ch. 10; Loudon Ch. 3). The special cases $V=|\gamma_{12}|$ for equal beams and $V=2\sqrt r/(1+r)$ for coherent unbalanced beams, and the numeric checks $I_{\max}=9,I_{\min}=1\Rightarrow V=0.8$ and $I_1{:}I_2=1{:}0.25\Rightarrow V=0.8$ , are reproduced in verify_fringe-visibility-contrast.py. The Michelson visibility function (c. 1890), the 1920 Betelgeuse diameter (≈0.047″), and the Wiener–Khinchin link to the spectrum are historically standard; no non-standard numbers are asserted.

Prerequisites. Two-Wave Interference and the Interference Term · Superposition Principle of Electromagnetic Waves · Optical Path Difference and Phase Retardation · Mutual Coherence Function and Degree of Coherence. Leads to. Temporal Coherence and Coherence Length · Spatial Coherence and the Van Cittert–Zernike Theorem · Wavefront-Splitting and Amplitude-Splitting Interferometers · Conditions for Observable Interference.

Is visibility the same thing as contrast, or are they different?

For interference fringes they are the same number: "fringe visibility" and "fringe contrast" both name $V=(I_{\max}-I_{\min})/(I_{\max}+I_{\min})$ . The word "contrast" is also used more loosely in imaging (e.g. Michelson contrast versus Weber or RMS contrast), but in coherence optics the interferometric definition is universal. Its virtue is that it is bounded to $[0,1]$ and needs no absolute calibration — only the relative depth of the darkest and brightest points, which is why it survives gain drift and exposure changes that would ruin an absolute-intensity measurement.

Why does the degree of coherence enter as its modulus and not its full complex value?

The complex degree of coherence $\gamma_{12}=|\gamma_{12}|e^{i\alpha_{12}}$ carries two pieces of information. Its phase $\alpha_{12}$ shifts the position of the fringes — it slides the whole cosine sideways — but does not change how deep the modulation cuts. Visibility measures only the depth, so it depends on the magnitude $|\gamma_{12}|$ alone. To recover the phase you must track where the fringes sit, not merely how sharp they are, which is exactly what phase-referenced interferometry and closure-phase techniques in stellar interferometry are built to do.

If two beams are perfectly coherent, why can't I always get V = 1?

Because contrast also needs the beams to be balanced. At the dark fringes the two fields subtract, and the residual intensity is $(\sqrt{I_1}-\sqrt{I_2})^2$ ; this vanishes only when $I_1=I_2$ . If one beam is four times the other, the weaker field cannot fully cancel the stronger, leaving $I_{\min}=(2-1)^2/(\dots)$ finite and capping $V$ at $2\sqrt r/(1+r)=0.8$ . Perfect coherence guarantees the fields track each other; it does not guarantee they have equal amplitude, and visibility needs both.

How is visibility connected to the coherence length of a source?

Scan the optical path difference in an equal-arm interferometer and the visibility becomes the normalized autocorrelation of the field, $V(\tau)=|\gamma(\tau)|$ . By the Wiener–Khinchin theorem this is the Fourier transform of the power spectrum, so a broad spectral line (large $\Delta\nu$ ) gives a fast-decaying $V(\tau)$ and a short coherence length $\ell_c\approx c/\Delta\nu\approx\lambda^2/\Delta\lambda$ . White light shows fringes only within a micrometre or two of zero path difference; a single-mode laser keeps $V$ high over hundreds of metres. The width of the visibility envelope is the coherence length.

Can visibility ever exceed 1 or be negative?

No. By construction $I_{\max}\ge I_{\min}\ge 0$ , so the numerator $I_{\max}-I_{\min}$ is non-negative and never larger than the denominator $I_{\max}+I_{\min}$ ; hence $0\le V\le 1$ . A reported value above 1 or below 0 signals a processing error — a mis-identified baseline, a subtracted background that overshot, or negative counts from detector electronics. In the two-photon (quantum) case the analogous coincidence visibility is likewise bounded by 1, and crossing the classical threshold of $1/\sqrt2\approx 0.71$ is what certifies non-classical correlations.

Does the fringe spacing matter for visibility?

The spacing does not enter the definition — $V$ is about depth, not period — but it matters instrumentally. If your detector or pixel is comparable to the fringe spacing $\Lambda_f$ , it averages the cosine across its own width and reports $V_{\text{meas}}=V_0\,|\mathrm{sinc}(w/\Lambda_f)|$ , hitting zero when the pixel spans exactly one fringe. This is why interferograms are sampled with several detector elements per fringe (a Nyquist-like rule) and why demagnifying optics are used to spread fringes across many pixels before the contrast is read.