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For vacuum wavelength endpoints ${\lambda }_{\min }<{\lambda }_{\max }$, the corresponding frequency span is $\Delta \nu =c(1/{\lambda }_{\min }-1/{\lambda }_{\max })$. The reciprocal-bandwidth estimate of optical coherence time is ${\tau }_c\sim 1/\Delta \nu$, giving ${\ell }_c\sim c/\Delta \nu$. This endpoint conversion is valid even for a broad band, whereas ${\ell }_c\sim {\lambda }_0^2/\Delta \lambda$ requires a relatively narrow optical spectrum.

For constant spectral density per unit frequency in the band, the normalized first-order optical coherence is $g^{(1)}(\tau )=e^{-i2\pi {\nu }_0\tau }\sin (\pi \Delta \nu \tau )/(\pi \Delta \nu \tau )$. Its first zero occurs at $\tau =1/\Delta \nu$. A different spectral shape or a different contrast criterion changes the numerical optical coherence length; endpoints alone do not define an exact universal value.

In-line holography uses nearly the same direction for the holographic reference wave and holographic object wave. In a transmission geometry, the undiffracted illumination supplies $R$ while the object supplies the scattered component $O$. This makes small optical path differences possible, but leaves the desired image, its twin image in holography and the bright background sharing an axis.

Dennis Gabor's electron-microscopy motivation led to optical tests: one line of a high-pressure mercury-vapor lamp illuminated a 3 μm pinhole, then approximately 1 mm microphotographs, recording approximately 1 cm holograms on sensitive photographic emulsions with minute-scale exposures. The roughly 0.1 mm optical coherence length favoured an in-line arrangement. Reillumination reconstructed images; focusing separated their depths incompletely. Distorted letters and random optical disturbances limited the results. These were optical demonstrations, not an operating electron microscope.^[ref]

The recorded microphotograph and its reconstructed lettering provide an object-to-image comparison in that account. The blocked-reference control described above is an explanatory control for the model; it is not a claim about an additional historical measurement.

A Michelson interferometer divides light at a beam splitter, reflects the two arms at mirrors and recombines the returning fields. For vacuum arm lengths ${\ell }_1,{\ell }_2$, the delay magnitude is $|\tau |=2|{\ell }_2-{\ell }_1|/c$. At each coarse delay, a small mirror scan produces maxima and minima of the detected light intensity. A photodiode and its electrical readout resolve these changes; the scan should cover several carrier cycles while changing $|g^{(1)}|$ negligibly. MIT's laboratory guide uses a mirror actuator, photodiode and electrical drive to scan interference fringes and extract interferometric visibility.^[ref]

A verified motivating experiment is Shaoul Ezekiel's MIT demonstration of source-spectrum control. A polarizer selects one or several lines from a laser whose adjacent lines have orthogonal linear polarizations. A sampled beam goes to a scanning Fabry–Pérot interferometer to check the optical spectrum; the main beam enters a Michelson interferometer, and a lens projects its interference fringes onto a screen. With one selected line the fringes retain good contrast through the demonstrated delay range. With several lines their contrast falls and returns as the arm length changes. At a fixed delay with poor contrast, selecting one line restores it; at matched arms both selections give good contrast. This control isolates the role of spectral composition rather than simply the arm position. The presentation is a qualitative demonstration, not a calibrated linewidth measurement. ^[ref].

A deliberately simpler two-line model explains the return of contrast. Take two stationary, mutually uncorrelated monochromatic contributions of equal power at ${\nu }_0\pm \delta \nu /2$. Their normalized wave-field correlation isfor equal detected single-arm light intensities. At $\tau =1/(2\delta \nu )$ the two contributions' interference fringes cancel on averaging; at $\tau =1/\delta \nu$ they agree again. This is a deduction from the stipulated spectrum, not a fit to Ezekiel's two- or three-line source. Finite individual linewidths eventually damp these revivals.

For a quantitative version of the experiment, the visibility law suggests explicit controls. Measure each arm separately at every coarse delay, subtract the detector's dark signal and any measured background, then normalize by $2\sqrt{I_1{I}_2}/(I_1+I_2)$. Keep the returning beams aligned on the same detector region with parallel linear polarizations; keep detector gain fixed and avoid saturation. Resolve the small mirror scan electrically, rather than averaging over its moving interference fringes. Repeat scans to identify vibration or drift. These controls follow from the averaging and mode-matching assumptions in first-order optical coherence; loss of contrast alone cannot distinguish source decorrelation from apparatus motion, misalignment or insufficient spatial resolution. The finite travel range gives access only to a finite range of delays.

The practical gain is that wave interference measures field correlations even when the detector cannot follow the optical oscillation itself. A long optical coherence length allows unequal-arm measurements; a short one localizes high contrast near matched delays. The beautiful feature is that both fading and reappearance of interference fringes are sums of the same rotating complex amplitudes.

The distance over which a constant phase mismatch accumulates phase difference $\pi$. In unseeded, undepleted sum-frequency generation, the generated intensity first reaches a maximum at this length and returns to zero at $2{\ell }_c$. This length concerns relative phases of interacting waves, rather than the temporal optical coherence length of one source.

In this temporal sense, the optical coherence length is $L_c=c{\tau }_c$, where $c$ is the vacuum speed of light and ${\tau }_c$ is an explicitly chosen optical coherence time. It measures a delay as a vacuum-equivalent path difference; it does not measure transverse spatial optical coherence. ^[ref].

For the stipulated $1\mathrm{GHz}$ profiles, the $1/e$ convention gives $L_c\approx 9.54269\mathrm{cm}$ for the Lorentzian and $15.8896\mathrm{cm}$ for the Gaussian. In a vacuum Michelson interferometer, changing one mirror position by $x$ changes the round-trip delay by $2x/c$, so the mirror travel to either of these delays is $L_c/2$. In a medium, convert physical travel using the actual group delay, which can depend on frequency. For a narrow line centered at vacuum wavelength ${\lambda }_0$, differentiating $\nu =c/\lambda$ gives $\Delta \nu \approx c\Delta \lambda /{\lambda }_0^2$; this small-width approximation must use consistently defined spectral widths.

The coherence length from a finite spectral band is most safely estimated by converting its endpoints to frequency, because the quoted band is broad:The reciprocal-bandwidth optical coherence time and corresponding longitudinal optical coherence length areThe given temperature is consistent with the spectral peak: Wien's displacement law gives ${\lambda }_{\mathrm{peak}}=2.898\times 10^{-3}\mathrm{m}\mathrm{K}/5778\mathrm{K}\simeq 502\mathrm{nm}$. Using the common narrow-band estimate ${\lambda }_{\mathrm{peak}}^2/\Delta \lambda$, with $\Delta \lambda =600\mathrm{nm}$, gives about $0.42\mu \mathrm{m}$. Its closeness here does not justify the narrow-band approximation for arbitrary broad spectra.

The $0.40\mu \mathrm{m}$ result is a bandwidth-scale estimate. A flat spectral density per unit frequency would make it the first-zero optical coherence length; a truncated Planck law spectrum is not flat and would have a different correlation function. Neither the stated endpoints nor the word “coherence” specifies an exact visibility threshold. Propagation to Earth sets the angular size and transverse scale, but does not by itself narrow the solar optical spectrum or lengthen this longitudinal scale.

A Michelson interferometer splits a collimated beam at a partially reflecting beam splitter. Two mutually perpendicular arms reflect from mirrors and recombine at the same beam splitter; the superposed returning beams produce interference fringes at the detector. Translate one mirror along its arm while holding the other fixed.

If that mirror moves a measured distance $\Delta x$, the round-trip optical path difference changes by $2\Delta x$ in air, approximated as vacuum. Every change of one wavelength moves one complete bright-dark-bright fringe cycle past a fixed detector point. Count $N$ such cycles to obtainUse a spectral line selected with a suitable optical filter, and count many interference fringes to reduce the fractional uncertainty of the distance and count. Keep the two arms within the source's optical coherence length and maintain overlap and alignment. If the vacuum wavelength is required for an arm filled with air of known refractive index $n$, the corrected relation is ${\lambda }_0=2n\Delta x/N$ at stable $n$.

The length over which the harmonic and its quadratic source accumulate a relative wave phase of $\pi$. For nonzero second-harmonic phase mismatch, $L_c=\pi /|\Delta k|$; the first output zero of the uniform undepleted model occurs at $2L_c$. This is a propagation mismatch length, distinct from the pump's optical coherence length. It diverges as perfect phase matching is approached.

Spatial optical coherence compares field samples at different positions, usually across a transverse plane at equal time. Two small openings separated by $d$ sample this correlation: recombine their light with nearly matched travel times and measure interferometric visibility near the central interference fringes. A narrow opening in front of a broad-spectrum lamp can improve transverse spatial optical coherence while leaving its short temporal optical coherence. These are separate controls: narrow spectral selection by itself need not make an extended source spatially coherent. MIT's lecture notes illustrate the two-opening test and these contrasting source preparations. ^[ref].

To isolate source width, stipulate a distant, narrowband source with uniform angular brightness on $-\Theta /2\le \alpha \le \Theta /2$, and mutually uncorrelated emission from distinct source directions. Here $\Theta \ll 1$ is the full angular width in radians, $\lambda$ the common wavelength, and the two openings are separated by $d$ along the same transverse direction as the source extent. Neglect amplitude differences across the openings. Each source direction contributes a phase difference approximately $2\pi d\alpha /\lambda$. Averaging its complex cross term givesThe sinc function is extended continuously to one at $q=0$. The sign convention in the exponential depends on which opening is numbered first; the symmetric model gives the same real result either way. Equal detected single-opening light intensities make the magnitude equal to interferometric visibility. A displaced source center adds a deterministic wave phase, shifting the interference fringes without changing their contrast.

The strips show $I/(I_1+I_2)=1+g_{12}^{(1)}(0)\cos \varphi$ with equal single-opening light intensities, a common phase difference coordinate $\varphi$ and no single-slit diffraction envelope. At $q=1/2$, contrast is $2/\pi \approx 0.636620$; at $q=1$, it vanishes. At $q=3/2$, the correlation is $-2/(3\pi )\approx -0.212207$: weak fringes return with bright and dark positions exchanged. For $\lambda =500\mathrm{nm}$ and $d=0.5\mathrm{mm}$, the first zero occurs at $\Theta =1\mathrm{mrad}$. This is a calculated example, not an experimental measurement or a universal source-width law.

A proposed source-width control follows directly from this model: illuminate two fixed openings with a distant adjustable luminous slit of full width $w$ at distance $R$, so $\Theta \approx w/R$ under the small-angle approximation, and hold $d$ and the spectral selection fixed. Increasing $w$ overlays more mutually incoherent, displaced interference fringe patterns. Record profiles over several near-central fringes, measure each opening separately and correct intensity imbalance as in first-order optical coherence. Restrict the recorded region so the path difference stays small compared with the optical coherence length, and resolve the fringes with the detector. Real source nonuniformity, finite opening widths and spectral spread modify the stipulated sinc function law. This protocol is a model-based extension of the verified two-opening method, not a claim that the plotted curves were measured.

Thus an extended incoherent source can give locally excellent wave interference when the sampled separation is small, and almost none when it is larger. The useful scale $d\Theta \sim \lambda$ expresses how source extent limits the aperture over which coherent addition is available. The same angular averaging explains why narrowing a source opening can recover visible interference fringes, even without narrowing its optical spectrum.