The Fringes Die and Come Back ── The Visibility of Interference Fringes Is the Fourier Transform of the Spectrum, and the Sodium Doublet Kills the Fringes Every 0.577 mm: a Length Measured on a Bench Tells the Splitting in the Spectrum ── [Paper 218]
Yuuki Yamagishi
They went out not because information was lost but because the information had moved over to the side of length. 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. Splitting a beam in two, delaying one by a path difference and recombining, the visibility of the fringes is the difference of the greatest and least intensities over their sum; for two beams of equal intensity it equals the modulus of the degree of coherence, which is the Fourier transform of the spectrum, by the theorem of Wiener and Khinchin. The path difference over which the visibility dies is the coherence length. 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, the results being confirmed by numerical transform; quantum optics is not treated, neither photon correlations nor second-order coherence appearing; no measurement of a real source is claimed, the widths being representative values; and the general theory of partial coherence is not developed, only first-order coherence being treated. A note 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. And on the relation to Paper 214: there the radiance could not be raised by any means, whereas the coherence here can be raised, and what pays for it is light thrown away. First, the visibility is the Fourier transform of the spectrum. Transforming a Gaussian spectrum on a million points gives a visibility of one at zero path difference, 0.6413811693 at twenty micrometres, 0.0890953204 at fifty and 0.0002388201 at a hundred, differing from the closed form by 2.22 times ten to the minus sixteen, 3.95 times ten to the minus ten, 2.98 times ten to the minus ten and 2.76 times ten to the minus twelve. For a rectangular spectrum the values are 0.9354900075, 0.5045558077, 0.2162356772 and 0.0850403377 at ten, twenty, fifty and eighty micrometres, agreeing with the sinc function to about one part in a million. A Gaussian gives a Gaussian and a rectangle gives a sinc. Nothing was assumed; every number came out of the transform. The differences of order a millionth on the rectangular side come from putting a discontinuous spectrum on a finite grid. Second, real sources differ by eight orders. Setting the width in frequency against the coherence length, a stabilised laser at a megahertz gives three hundred metres, a helium-neon at one and a half gigahertz gives a fifth of a metre, a sodium lamp gives 588 micrometres, a filter ten nanometres wide gives 31.9 micrometres and white light gives 0.999 micrometres. 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. Third, this is the core. Putting in the two sodium lines, at 589.0 and 589.6 nanometres, a separation of 5.18 times ten to the eleventh hertz, and running the same transform, the visibility falls to nothing at path differences of 0.292297, 0.869397, 1.446497 and 2.023597 millimetres, against theoretical values of 0.289395, 0.868186, 1.446977 and 2.025767. 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 came back. Fourth, that spacing tells the splitting in the spectrum. The spacing between the deaths measures 0.577100 millimetres, differing by three parts in a thousand from the speed of light divided by the separation, which is 0.578791. That order can be read backwards: measure the spacing and the separation in frequency is the speed of light divided by it, and in wavelength the square of the wavelength divided by it. So the fine structure of a spectrum can be measured with a ruler and no spectrograph, which is what Michelson actually did. One single fact is lodged in two places, on the frequency side as the gap between two lines and 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. Fifth, there is a second coherence. For a round source the degree of coherence across the beam falls as twice the first Bessel function over its argument, and from its first zero at 3.83170597 the coherence radius is 0.61 wavelengths over the angular radius. At 550 nanometres this gives 72.1 micrometres for the Sun, 74.2 for the Moon, 671 for a millimetre lamp at a metre and 67100 for the same lamp at a hundred metres. This is a different thing from the coherence above: 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. Even sunlight is in step within seventy micrometres, 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 being the stellar interferometer. Sixth, coherence can be bought and light pays. Narrowing a filter about 550 nanometres, a width of three hundred nanometres gives a coherence length of 1.01 micrometres with all the light through; a hundred nanometres gives 3.02 micrometres with a third; ten nanometres gives 30.2 micrometres with a thirtieth; and a tenth of a nanometre gives 3020 micrometres with a three-thousandth. The product is 1.008 times ten to the minus six in every case and does not move. 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, and here the coherence can. Being forbidden and having a price are two different things. The separator and the fence. What separates the senses of coherent is what the spread is conjugate to: temporal coherence is set by the width in frequency and measured along the path difference, spatial coherence is set by the angular size of the source and measured across the beam. Both are the same Fourier relation, differing only in which quantities are paired. 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 according to the shape of the spectrum, on the space side according to the shape of the source. And what the third check 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. Important (scope note): a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem or physical law, none is claimed; any proof of a theorem, the results being confirmed by numerical transform; quantum optics, neither photon correlations nor second-order coherence appearing; any measurement of a real source, the widths being representative values; the general theory of partial coherence, only first-order coherence being treated. What is added: 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. What remains: draw two mirrors slowly apart and the fringes fade and go out. Stop there and the story ends at no longer visible. Draw them further and the fringes come back, and the spacing of their returns tells at once that the source has two lines in its spectrum and how far apart those lines are. They went out not because information was lost but because the information had moved over to the side of length. And the length over which light stays in step differs between sources by eight orders: the phrase coherent light says nothing at all until one settles over what length. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 見えなくなったのは情報が失われたからではなく、情報が長さの側へ移っていたからである。 論文207 は「ジョーンズ計算を囲っている柵はコヒーレンスである」と書き、その理論には触れなかった。本稿はその柵を測れる量にする。二つに分けた光を路程差 d だけずらして重ねたときの縞の可視度を V=(I_max−I_min)/(I_max+I_min) とすると、強度の等しい二本では V はコヒーレンス度の絶対値に等しく、それはスペクトルのフーリエ変換である(ウィーナー=ヒンチン)。V が落ちきるまでの路程差をコヒーレンス長という。新しい数学定理も新しい物理法則も主張しない:ウィーナー=ヒンチンの定理、ファン・シッタート=ゼルニケの定理、マイケルソンによる可視度からの分光は、いずれも標準的である。⚠️ 定理を証明しない——数値変換で確かめるのであって、定理は引用する。⚠️ 量子光学を扱わない(光子相関も二次のコヒーレンスも出てこない)。⚠️ 実在の光源の測定値を主張しない——幅は代表的な値であり、測定ではない。⚠️ 部分コヒーレンスの一般論を展開しない(扱うのは一次のコヒーレンスだけ)。 ⭐ 名前について一つ。体系には「ウィーナー」が 13 編に現れるが、その全部がウィーナー・フィルタかウィーナー=ホップ法であり、ウィーナー=ヒンチンの定理は本稿が初出である。⭐⭐ また論文214 との関係を書いておくと、あちらでは輝度がどうやっても上げられなかった——本稿のコヒーレンスは上げられる。ただし光を捨てることで、である。 第一に、可視度はスペクトルのフーリエ変換である。中心 5×10¹⁴ Hz・標準偏差 2×10¹² Hz のガウス型スペクトルを 2²⁰ 点で数値変換すると、路程差 0 で 1.0000000000、20 μm で 0.6413811693、50 μm で 0.0890953204、100 μm で 0.0002388201 であり、閉じた形との差はそれぞれ 2.22×10⁻¹⁶、3.95×10⁻¹⁰、2.98×10
•Aug 27, 2026•0 citations