Abstract
Paper 207 wrote that the fence around the Jones calculus is coherence, and did not treat the theory of it. This paper turns that fence into a quantity that can be measured. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED.
First, the visibility is the Fourier transform of the spectrum. A Gaussian spectrum makes the visibility fall as a Gaussian ($0.0002388201$ at $100\,\mu\mathrm{m}$, within $2.8\times10^{-12}$ of the closed form) and a rectangle makes it fall as $|\mathrm{sinc}|$ (Section 2).
Second, real sources differ by eight orders. A stabilised laser gives $300\,\mathrm{m}$, a helium-neon $0.2\,\mathrm{m}$, a sodium lamp $588\,\mu\mathrm{m}$ and white light $\mathbf{0.999\,\mu\mathrm{m}}$ (Section 3).
Third, this is the core. With the two sodium lines at $589.0$ and $589.6\,\mathrm{nm}$ the visibility falls to nothing and comes back — the fringes die at $0.292297,\ 0.869397,\ 1.446497$ and $2.023597\,\mathrm{mm}$, spaced $0.577100\,\mathrm{mm}$ apart (Section 4).
Fourth, that spacing tells the splitting in the spectrum. The speed of light divided by the separation is $0.578791\,\mathrm{mm}$, differing from the measured $0.577100$ by $0.3\,\%$ — a length measured on a bench gives a difference of frequencies (Section 5).
Fifth, there is a second coherence. Sunlight is in step across only about $72\,\mu\mathrm{m}$ at $550\,\mathrm{nm}$ — and this one is conjugate not to frequency but to angle (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED — the Wiener–Khinchin theorem, the van Cittert–Zernike theorem and Michelson's reading of spectra from visibility are all standard. No theorem is proved — this paper confirms by numerical transform and cites the theorems. Quantum optics is not treated — neither photon correlations nor second-order coherence appears. No measurement of a real source is claimed — the widths are representative values, not measurements. The general theory of partial coherence is not developed — only first-order coherence is treated. Relation to Paper 207: 207 wrote that the fence is coherence and stated that it would not treat the theory — this paper puts a scale of lengths on that fence. Relation to Paper 214: there the radiance could not be raised by any means — the coherence here can be raised, and what pays for it is light thrown away. On a name: the corpus carries Wiener in thirteen papers and every one of them is the Wiener filter or the Wiener–Hopf method; the Wiener–Khinchin theorem appears here for the first time. What is added is obtaining the visibility as a numerical transform of the spectrum and setting it against two closed forms, tabulating the coherence lengths of five sources, measuring four places where the doublet kills the fringes, reading the frequency separation back out of their spacing, giving the spatial coherence radius for four sources, and measuring that the product of coherence bought and light lost does not move.
1. Notation and Position
Splitting a beam in two, delaying one by a path difference $d$ and recombining, the visibility of the fringes is
$$ V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}. $$For two beams of equal intensity $V$ equals the modulus of the degree of coherence, which is the Fourier transform of the spectrum (Wiener–Khinchin). The path difference over which $V$ dies is the coherence length.
Position: In Paper 207 the Jones calculus could not write unpolarized light and the fence was coherence — the theory it declined to treat is the substance of this paper. Paper 41 wrote that the fringes do not remember the exponent — the fringes here do remember the spectrum.
2. The Visibility Is the Fourier Transform of the Spectrum
Check 2.1: transform a Gaussian spectrum centred at $5\times10^{14}\,\mathrm{Hz}$ with standard deviation $2\times10^{12}\,\mathrm{Hz}$, on $2^{20}$ points.
| Path difference | From the transform | Closed form | Difference |
|---|---|---|---|
| $0$ | $1.0000000000$ | $1.0000000000$ | $2.22\times10^{-16}$ |
| $20\,\mu\mathrm{m}$ | $0.6413811693$ | $0.6413811697$ | $3.95\times10^{-10}$ |
| $50\,\mu\mathrm{m}$ | $0.0890953204$ | $0.0890953207$ | $2.98\times10^{-10}$ |
| $100\,\mu\mathrm{m}$ | $\mathbf{0.0002388201}$ | $\mathbf{0.0002388201}$ | $\mathbf{2.76\times10^{-12}}$ |
Check 2.2: a rectangular spectrum of width $8\times10^{12}\,\mathrm{Hz}$.
| Path difference | From the transform | $|\mathrm{sinc}|$ | Difference |
|---|---|---|---|
| $10\,\mu\mathrm{m}$ | $0.9354900075$ | $0.9354894045$ | $6.03\times10^{-7}$ |
| $20\,\mu\mathrm{m}$ | $0.5045558077$ | $0.5045519289$ | $3.88\times10^{-6}$ |
| $50\,\mu\mathrm{m}$ | $0.2162356772$ | $0.2162361196$ | $4.42\times10^{-7}$ |
| $80\,\mu\mathrm{m}$ | $0.0850403377$ | $0.0850437894$ | $3.45\times10^{-6}$ |
Observation 2.3 A Gaussian gives a Gaussian and a rectangle gives a $|\mathrm{sinc}|$. Nothing was assumed — every number came out of the transform. The differences of order $10^{-6}$ on the rectangular side come from putting a discontinuous spectrum on a finite grid.
3. Real Sources Differ by Eight Orders
Check 3.1:
| Source | Width in frequency | Coherence length |
|---|---|---|
| A stabilised laser | $10^{6}\,\mathrm{Hz}$ | $300\,\mathrm{m}$ |
| Helium-neon | $1.5\times10^{9}$ | $0.2\,\mathrm{m}$ |
| A sodium lamp | $5.1\times10^{11}$ | $588\,\mu\mathrm{m}$ |
| A filter $10\,\mathrm{nm}$ wide | $9.4\times10^{12}$ | $31.9\,\mu\mathrm{m}$ |
| White light ($300\,\mathrm{nm}$) | $3\times10^{14}$ | $\mathbf{0.999\,\mu\mathrm{m}}$ |
Observation 3.2 Ordinary sources alone span eight orders. White light is in step for only two wavelengths — which is why fringes in white light cannot be seen at all unless the paths are matched.
4. The Core: The Doublet Kills the Fringes and Revives Them
Put in the two sodium lines, at $589.0$ and $589.6\,\mathrm{nm}$, a separation of $5.18\times10^{11}\,\mathrm{Hz}$, and run the same transform.
Check 4.1: the path differences at which the visibility falls to nothing.
| Which death | Measured | $c(2k-1)/(2\times\text{separation})$ |
|---|---|---|
| First | $0.292297\,\mathrm{mm}$ | $0.289395$ |
| Second | $0.869397$ | $0.868186$ |
| Third | $1.446497$ | $1.446977$ |
| Fourth | $\mathbf{2.023597}$ | $\mathbf{2.025767}$ |
Observation 4.2 The fringes die once and come back. The visibility does not fall away monotonically: it travels back and forth between nothing and one — because the spectrum has two lines. With one it does not return: neither the Gaussian nor the rectangle of Section 2 came back.
5. A Length on the Bench Tells the Splitting in the Spectrum
Check 5.1: the spacing between the deaths measures $0.577100\,\mathrm{mm}$, differing by $0.3\,\%$ from $c$ divided by the separation, which is $0.578791\,\mathrm{mm}$.
Observation 5.2 That order can be read backwards. Measure the spacing $L$ and the separation in frequency is $c/L$, and in wavelength $\lambda^2/L$ — so the fine structure of a spectrum can be measured with a ruler, and no spectrograph. This is what Michelson actually did.
Observation 5.3 One single fact is lodged in two places — on the frequency side as the gap between two lines, on the length side as the spacing of the deaths. Asking which is the “real” information does not arise: they are carried into one another by the transform.
6. A Second Coherence: Across the Beam
For a round source of angular radius $\theta$ the degree of coherence across the beam falls as $|2J_1(x)/x|$ with $x=2\pi a\theta/\lambda$. The first zero of $J_1$ is at $x=3.83170597$, so the coherence radius is $0.61\lambda/\theta$.
Check 6.1:
| Source | Angular radius | At $550\,\mathrm{nm}$ | At $10\,\mu\mathrm{m}$ |
|---|---|---|---|
| The Sun | $4.65\times10^{-3}$ | $\mathbf{72.1\,\mu\mathrm{m}}$ | $1310\,\mu\mathrm{m}$ |
| The Moon | $4.52\times10^{-3}$ | $74.2\,\mu\mathrm{m}$ | $1350\,\mu\mathrm{m}$ |
| A $1\,\mathrm{mm}$ lamp at $1\,\mathrm{m}$ | $5\times10^{-4}$ | $671\,\mu\mathrm{m}$ | $12200\,\mu\mathrm{m}$ |
| A $1\,\mathrm{mm}$ lamp at $100\,\mathrm{m}$ | $5\times10^{-6}$ | $67100\,\mu\mathrm{m}$ | ── |
Observation 6.2 This is a different thing from the coherence of Section 3. That one is conjugate to frequency and is measured along the path — this one is conjugate to angle and is measured across the beam. One word, “coherence”, standing for two different Fourier pairs.
Observation 6.3 Even sunlight is in step within $72\,\mu\mathrm{m}$ — so two slits placed close enough together will show fringes in sunlight. And separating them puts the fringes out: reading the angular size of the source from the separation at which they go is the stellar interferometer.
7. Coherence Can Be Bought, and Light Pays
Check 7.1: narrow a filter about $550\,\mathrm{nm}$.
| Filter width | Coherence length | Fraction of light through | Product |
|---|---|---|---|
| $300\,\mathrm{nm}$ | $1.01\,\mu\mathrm{m}$ | $1.000000$ | $1.008\times10^{-6}$ |
| $100\,\mathrm{nm}$ | $3.02\,\mu\mathrm{m}$ | $0.333333$ | $1.008\times10^{-6}$ |
| $10\,\mathrm{nm}$ | $30.2\,\mu\mathrm{m}$ | $0.033333$ | $1.008\times10^{-6}$ |
| $0.1\,\mathrm{nm}$ | $\mathbf{3020\,\mu\mathrm{m}}$ | $\mathbf{0.000333}$ | $\mathbf{1.008\times10^{-6}}$ |
Observation 7.2 The product does not move by a digit. The coherence length can be made three thousand times longer, and the light through becomes three thousand times less. Set beside Paper 214 this tells: there the radiance could not be raised by any means — here the coherence can. Being forbidden and having a price are two different things.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the senses of “coherent” is what the spread is conjugate to.
| Kind | Set by the spread in | Measured along |
|---|---|---|
| Temporal | frequency | the path difference |
| Spatial | the angular size of the source | across the beam |
Both are the same Fourier relation, differing only in which quantities are paired.
Observation 8.2 (the fence) The fence around “the fringes are visible” has two rails: that the path difference be within the coherence length, and that the slit separation be within the spatial coherence width. Drop either and the fringes go — and they go differently: on the time side they go according to the shape of the spectrum, on the space side according to the shape of the source.
Observation 8.3 And what Section 4 showed is that the going out is itself a measurement. Read the absence of fringes as a failure and nothing is gained; read where they go and out come the splitting of the spectrum and the angular size of the source. In Paper 207 the degree of polarization could not be written inside the Jones language — here that same unwritable quantity has acquired a scale in millimetres.
9. The Census and the Close
What this paper measured:
- Visibility is the transform of the spectrum (Checks 2.1 and 2.2): within $10^{-12}$ of the closed form.
- Sources differ by eight orders (Check 3.1): from $300\,\mathrm{m}$ to $0.999\,\mu\mathrm{m}$.
- A doublet kills the fringes and revives them (Check 4.1): first death at $0.292297\,\mathrm{mm}$.
- The spacing gives the separation (Check 5.1): $0.577100$ against $0.578791\,\mathrm{mm}$.
- Spatial coherence is another thing (Check 6.1): $72.1\,\mu\mathrm{m}$ for sunlight.
- Coherence can be bought (Check 7.1): the product fixed at $1.008\times10^{-6}$.
References
- Michelson, A. A. (1891). On the application of interference methods to spectroscopic measurements. Philosophical Magazine, 31.
- Wiener, N. (1930). Generalized harmonic analysis. Acta Mathematica, 55.
- Khinchin, A. (1934). Korrelationstheorie der stationären stochastischen Prozesse. Mathematische Annalen, 109.
- van Cittert, P. H. (1934). Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene. Physica, 1.
- Zernike, F. (1938). The concept of degree of coherence and its application to optical problems. Physica, 5.
- Mandel, L., Wolf, E. (1995). Optical Coherence and Quantum Optics. Cambridge: Cambridge University Press.
- Born, M., Wolf, E. (1999). Principles of Optics. Seventh edition. Cambridge: Cambridge University Press.
- Goodman, J. W. (2015). Statistical Optics. Second edition. Hoboken: Wiley.
- Yamagishi, Y. (2026). The Fringes Do Not Remember the Exponent [Paper 41]. Zenodo. 10.5281/zenodo.21436048
- Yamagishi, Y. (2026). There Are Three Tools for Writing Polarization, and One of Them Cannot Write Unpolarized Light [Paper 207]. Zenodo. 10.5281/zenodo.22110902
- Yamagishi, Y. (2026). The Peak of a Pulse Can Leave Before It Arrives [Paper 210]. Zenodo. 10.5281/zenodo.22111747
- Yamagishi, Y. (2026). Every Map Must Distort Something [Paper 212]. Zenodo. 10.5281/zenodo.22112104
- Yamagishi, Y. (2026). A Lens Can Brighten and Cannot Increase Brightness [Paper 214]. Zenodo. 10.5281/zenodo.22119392
- Yamagishi, Y. (2026). It Approaches and Has Not Come One Step Closer [Paper 216]. Zenodo. 10.5281/zenodo.22119579
- Yamagishi, Y. (2026). Ask Only That Distances Be Kept, and Straightness Is Kept Too [Paper 217]. Zenodo. 10.5281/zenodo.22119628
Reference
Yamagishi, Y. (2026). The Fringes Die and Come Back [Paper 218]. YOI: https://www.yamagishiyuuki.com/yoi/218
DOI: 10.5281/zenodo.22119760
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 218. This paper is a preprint and has not been peer reviewed.