Abstract
An anti-reflection coating is described as “a film that cancels reflection”. This paper asks how far it cancels——the answer is at one wavelength only, and with real materials not at all. No new mathematical theorem and no new law is claimed.
First, bare glass reflects $\mathbf{0.042580}$ ($4.2580$ per cent) (Section 2).
Second, the ideal film index is $\sqrt{n_0n_s}=\mathbf{1.232883}$, at which the reflection at $550$ nm is exactly zero (Section 2).
Third, this is the core of the paper. The same ideal film returns to $\mathbf{0.013541}$ at $400$ nm (Section 2).
Fourth, real MgF$_2$ has $1.38$, which is $0.1471$ from the ideal (Section 3).
Fifth, so it leaves $\mathbf{0.012601}$ ($1.2601$ per cent) even at the design wavelength (Section 3).
Sixth, the separator is two things — whether the wavelength shifted or the index did — not one (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the single-layer anti-reflection coating, the quarter-wave condition, and Fresnel reflection are all standard. We do not build thin-film optics——multilayer design theory (equivalent layers, optimisation) is not entered. It is named and no more. Materials are not treated——materials other than MgF$_2$, and index lowering by porous films, are not the subject. Indices are taken as independent of wavelength. Oblique incidence is not treated——only normal incidence is computed. Absorption is not treated——film and glass are taken as lossless. Polarisation is not treated——at normal incidence there is no distinction. Relation to earlier papers: Paper 363 showed a bat to have three sweet spots——there one word named three things; here one shortfall has two causes. Paper 365 showed a standard explanation failing on geometry——there too checking the condition numerically showed the explanation did not reach. Paper 300 showed whether two things share a root is decidable——by that test, a wavelength shift and an index mismatch have distinct roots. Paper 358 showed a convention fixing a number——here the convention is where the design wavelength is placed. What is added is setting ideal and real films side by side at seven wavelengths, showing the ideal film returning to $0.013541$ at $400$ nm, showing the real material leaving $0.012601$ at the design wavelength, checking bare glass against the accepted $4$ per cent, and splitting the separator in two.
1. Notation and the Question
Setting: Between air ($n_0=1.00$) and glass ($n_s=1.52$), place a film of index $n_1$ and optical thickness $\lambda_0/4$.
$$R_{\text{bare}}=\left(\frac{n_s-n_0}{n_s+n_0}\right)^2,\qquad n_1^{\text{ideal}}=\sqrt{n_0n_s},\qquad \lambda_0=550\ \mathrm{nm}$$Observation 1.1 The film makes two reflections cancel. One at the surface and one at the film-glass interface——they vanish completely when the phase differs by half a wave and the amplitudes are equal.
Observation 1.2 So there are two conditions. A phase condition (thickness $\lambda/4$) and an amplitude condition ($n_1=\sqrt{n_0n_s}$)——break either and nothing cancels.
Observation 1.3 And the two break on different quantities. Phase on wavelength, amplitude on material——this paper gives the two separately as numbers.
Question: How far does an anti-reflection coating cancel reflection?
2. Even an Ideal Film Cancels at One Wavelength Only
Calculation 2.1: Reflectance of the ideal film, index $1.232883$ (computed in this paper, normal incidence).
| Wavelength [nm] | Reflectance |
|---|---|
| $400$ | $\mathbf{0.01354129}$ |
| $450$ | $0.00517551$ |
| $500$ | $0.00108717$ |
| $\mathbf{550}$ | $\mathbf{0.00000000}$ |
| $600$ | $0.00075713$ |
| $650$ | $0.00254063$ |
| $700$ | $0.00482796$ |
Bare glass is $0.042580$ ($4.2580$ per cent).
Observation 2.2 This is the core of the paper. Exactly zero at $550$ nm and back to $0.013541$ at $400$——nearly a third of the bare $0.042580$.
Observation 2.3 It rises on both sides of the design wavelength. $0.00108717$ at $500$ nm and $0.00075713$ at $600$——a valley, not a plateau, with a single lowest point.
Observation 2.4 The short-wavelength side is worse. $0.013541$ at $400$ nm against $0.004828$ at $700$——not symmetric at equal distance, because the phase error goes as $1/\lambda$.
Observation 2.5 The bare $4.2580$ per cent matches the accepted figure. One glass surface is usually given as about $4$ per cent——neither the formula nor the constants are off.
3. And There Is No Ideal Material
Calculation 3.1: Reflectance of real MgF$_2$ ($n_1=1.38$) (computed in this paper).
| Wavelength [nm] | MgF$_2$ | Ideal film |
|---|---|---|
| $400$ | $0.02205252$ | $0.01354129$ |
| $450$ | $0.01620430$ | $0.00517551$ |
| $500$ | $0.01335683$ | $0.00108717$ |
| $\mathbf{550}$ | $\mathbf{0.01260079}$ | $\mathbf{0.00000000}$ |
| $600$ | $0.01312726$ | $0.00075713$ |
| $650$ | $0.01436835$ | $0.00254063$ |
| $700$ | $0.01596197$ | $0.00482796$ |
Observation 3.2 MgF$_2$ is not zero even at the design wavelength. $0.012601$, that is $1.2601$ per cent——the phase condition is met and the amplitude condition is not.
Observation 3.3 The ideal index is $1.232883$. $\sqrt{1.52}$——$0.1471$ from MgF$_2$’s $1.38$. No dense solid has an index that low.
Observation 3.4 The film still works. From $0.042580$ bare down to $0.012601$, under a third——it is not that it fails; it is that it cannot finish.
Observation 3.5 The two shortfalls are independent. With an ideal material only the wavelength error remains; in monochromatic light only the material error does——fix either and the other survives.
Observation 3.6 Distinct roots by the criterion of Paper 300. The phase error comes from geometry (thickness and wavelength), the amplitude error from material (index)——both produce the same symptom, “reflection remains”, from different roots.
4. What Multiple Layers Change
Observation 4.1 More layers cancel at more wavelengths. Two layers at two wavelengths, three over a broader band——the phase condition can be met at several wavelengths at once.
Observation 4.2 And the material constraint loosens. Combinations synthesise an effective index——a single layer must itself be $1.232883$; a stack need not.
Observation 4.3 So stacking loosens both shortfalls at once. One operation acting on two problems——which is why real lenses carry multilayer coatings.
Observation 4.4 To be stated honestly——this paper has not computed multilayer stacks. That more layers do better is stated as known——the numbers given here are for a single layer.
Observation 4.5 Knowing the single-layer limit is still worth something. $0.013541$ at $400$ nm and $0.012601$ at the design wavelength——those two numbers are the answer to “why are stacks needed”.
5. The Separator: Phase, or Amplitude?
Observation 5.1 The separator——whether the broken condition is the phase one or the amplitude one.
| Broken condition | Broken by | At $550$ nm | At $400$ nm |
|---|---|---|---|
| Neither (ideal film) | — | $\mathbf{0.000000}$ | $0.013541$ |
| Phase only | wavelength | $0.000000$ | $\mathbf{0.013541}$ |
| Amplitude only (MgF$_2$) | material | $\mathbf{0.012601}$ | $0.022053$ |
Observation 5.2 The third row leaves something at $550$ nm; the second does not. The value at the design wavelength distinguishes them——look there and you know which condition is broken.
Observation 5.3 So the symptom can be resolved into its causes. What remains at the design wavelength is the material shortfall; what grows away from it is the wavelength shortfall——one “reflection remains” decomposed into two.
Observation 5.4 The separator can be moved. Reset the design wavelength to $400$ nm and it becomes zero there——which wavelength is made perfect can be chosen. “All of them” cannot be, with one layer.
Observation 5.5 To be stated honestly——$0.013541$ and $0.012601$ are values under stated conditions: $n_s=1.52$, $\lambda_0=550$ nm, and indices independent of wavelength. Another glass gives other numbers——what does not move is the structure: the two conditions break on different quantities.
Observation 5.6 One last thing——when one symptom has two causes, fixing one completely does not remove the symptom. Find the ideal material and the $0.013541$ at $400$ nm remains——miscount the causes and a fix that did not work gets mistaken for a shortfall in technique.
6. The Census and the Close
- Bare glass $0.042580$ (Calculation 2.1): matches the accepted $4$ per cent.
- Ideal index $1.232883$ (Observation 3.3): $\sqrt{1.52}$.
- Exactly zero at $550$ nm (Calculation 2.1): both conditions met.
- The same film gives $0.013541$ at $400$ nm (Calculation 2.1): a third of bare.
- It rises on both sides (Observation 2.3): a single lowest point.
- The short-wavelength side is worse (Observation 2.4): phase goes as $1/\lambda$.
- MgF$_2$ is $1.38$ (Observation 3.3): $0.1471$ from the ideal.
- It leaves $0.012601$ at the design wavelength (Observation 3.2): amplitude broken.
- Still under a third of bare (Observation 3.4): it cannot finish, that is all.
- The two shortfalls are independent (Observation 3.5): fix one and the other survives.
Closing: An anti-reflection coating is described as “a film that cancels reflection”——but it cancels only when two conditions hold at once. There is a phase condition and an amplitude condition——a thickness of $\lambda/4$ and an index of $\sqrt{n_0n_s}$. And the two break on different quantities: phase on wavelength, amplitude on material. Even an ideal film cancels at one wavelength only——a film of index $\mathbf{1.232883}$ is exactly zero at $550$ nm, and back to $\mathbf{0.013541}$ at $400$ nm, nearly a third of bare glass’s $0.042580$ ($4.2580$ per cent, matching the accepted $4$). It is a valley with a single lowest point, and the short-wavelength side is worse because the phase error goes as $1/\lambda$. And there is no ideal material——no dense solid has an index as low as $\sqrt{1.52}=1.232883$, and real MgF$_2$ is $1.38$, $0.1471$ away. So even at $550$ nm, where the phase condition is met, it leaves $\mathbf{0.012601}$ ($1.2601$ per cent). It still brings bare glass to under a third, so it is not that it fails — it is that it cannot finish. The two shortfalls are independent——with an ideal material only the wavelength error remains; in monochromatic light only the material error does. Fix either and the other survives. By the criterion of Paper 300 they have distinct roots: one comes from geometry, the other from material. One thing separates them——whether the broken condition is the phase one or the amplitude one. What remains at the design wavelength is the material shortfall; what grows away from it is the wavelength shortfall, so one symptom decomposes into two. Reset the design wavelength to $400$ nm and it becomes zero there — which wavelength is made perfect can be chosen, but “all of them” cannot be, with one layer. To be said honestly——this paper has not computed multilayer stacks. More layers cancel at more wavelengths and loosen the material constraint at the same time — one operation acting on two problems — which is why real lenses carry them. Those two numbers are the answer to “why are stacks needed”. One last thing——when one symptom has two causes, fixing one completely does not remove the symptom. Find the ideal material and the $0.013541$ at $400$ nm remains. Miscount the causes and a fix that did not work gets mistaken for a shortfall in technique.
References
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- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The “Depth” of Depth of Field Is a Convention, Not a Measurement [Paper 358]. Zenodo. 10.5281/zenodo.22697216
- Yamagishi, Y. (2026). A Bat Has Three Sweet Spots, and They Lie 23 cm Apart [Paper 363]. Zenodo. 10.5281/zenodo.22697505
- Yamagishi, Y. (2026). The Usual Explanation of the Brazil-Nut Effect Fails on the Size of the Gap [Paper 365]. Zenodo. 10.5281/zenodo.22697622
Reference
Yamagishi, Y. (2026). A Single Anti-Reflection Layer Falls Short for Two Independent Reasons [Paper 370]. YOI: https://www.yamagishiyuuki.com/yoi/370
DOI: 10.5281/zenodo.22697804
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 370. This paper is a preprint and has not been peer reviewed.