Abstract
An achromatic lens is called a lens with the chromatic aberration removed. This paper asks how far it is removed——the answer is at two wavelengths only, with a third colour remaining. No new mathematical theorem and no new law is claimed.
First, a single lens shifts focus with colour by the focal length divided by the Abbe number, $\mathbf{1.558361}$ mm (Section 2).
Second, the achromatic condition gives $f_1=43.322425$ mm and $f_2=-76.436624$ mm (Section 3).
Third, the second element must be negative, that is, diverging (Section 3).
Fourth, this is the core of the paper. Even with two glasses, $\mathbf{0.044604}$ mm remains (Section 4).
Fifth, the improvement is $\mathbf{34.9374}$-fold, and the residue’s cause is named by one number: the difference in partial dispersion (Section 4).
Sixth, the separator is first-order against second-order dispersion, not the number of elements (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the Abbe number, the achromatic condition, and the secondary spectrum are all standard. We do not build lens design——apochromat design and the selection of anomalous-dispersion glasses are not entered. They are named and no more. Other aberrations are not treated——spherical aberration, coma, and astigmatism are not the subject. The computation is thin-lens and paraxial. We measure no glass——Abbe numbers and partial dispersions are taken as representative figures from the literature. They are not the values of particular products. Manufacture is not treated——cementing and tolerances are not the subject. Diffraction is not treated——comparison with the diffraction limit is not the subject. Relation to earlier papers: Paper 370 showed a single anti-reflection layer cancelling at one wavelength only——there one layer for one wavelength; here two glasses for two. The same counting. Paper 363 showed a bat to have three sweet spots——there too the number of conditions matched the number of points. Paper 300 showed whether two things share a root is decidable——by that test, first-order and second-order dispersion have distinct roots. Paper 368 showed algebraically equal forms returning different digits——there too the residue’s cause could be named by one number. What is added is giving the single-lens shift as a number, solving the achromatic condition for the two focal lengths, confirming the combined focal length returns, showing the secondary spectrum falling by $34.9374$ without reaching zero, and putting the separator on first against second order.
1. Notation and the Question
Setting: Build an achromat of combined focal length $f=100$ mm from two glasses of Abbe number $V$.
$$\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2},\qquad \frac{1}{f_1V_1}+\frac{1}{f_2V_2}=0$$Take crown $V_1=64.17$ and flint $V_2=36.37$.
Observation 1.1 The Abbe number acts like an inverse dispersion. The larger it is, the less the colours separate——a single lens’s chromatic focal shift is $f/V$.
Observation 1.2 Two equations fix two unknowns. $f_1$ and $f_2$——the number of equations is the number of wavelengths that can be brought together.
Observation 1.3 There is no equation for a third colour. Only two conditions have been imposed——this paper gives the remainder as a number.
Question: How far does an achromat remove chromatic aberration?
2. A Single Lens Shifts by $1.558361$ mm
Calculation 2.1: Chromatic focal shift of a single crown lens (computed in this paper, $f=100$ mm).
| Quantity | Value |
|---|---|
| Abbe number $V$ | $64.17$ |
| Focal shift $f/V$ | $\mathbf{1.558361}$ mm |
| As a fraction of the focal length | $\mathbf{1.5584}$ per cent |
Observation 2.2 $1.5584$ per cent is not small. $1.5$ mm on a $100$ mm lens——blue and red focus $1.5$ mm apart, which in a photograph is a clear coloured fringe.
Observation 2.3 The shift scales with focal length. Because it is $f/V$——the longer the lens, the larger in absolute terms. That is why telescopes met the problem first, historically.
3. The Second Element Must Diverge
Calculation 3.1: Focal lengths from the achromatic condition (computed in this paper).
| Quantity | Value |
|---|---|
| Crown $f_1$ | $\mathbf{43.322425}$ mm |
| Flint $f_2$ | $\mathbf{-76.436624}$ mm |
| Check $1/f_1+1/f_2$ | $0.0100000000$ ($1/f=0.0100000000$) |
| Residual of the achromatic condition | $0.000\times10^{0}$ |
Observation 3.2 The second element is negative. $-76.436624$ mm——without adding a diverging lens the colour does not cancel.
Observation 3.3 So the first is stronger than the target. $43.322425$ mm is shorter than $100$ mm——it must bend extra to survive the diverging element.
Observation 3.4 The formulae close. The combination gives $0.0100000000$ and the achromatic residual is exactly $0$——not an arithmetic error.
Observation 3.5 The difference of Abbe numbers sits in the denominator. $64.17-36.37=27.80$——a small difference makes both $f_1$ and $f_2$ extreme. The two must be glasses of different dispersion.
4. The Secondary Spectrum Leaves $0.044604$ mm
Calculation 4.1: Residual from the difference in partial dispersion (computed in this paper).
| Quantity | Value |
|---|---|
| Crown partial dispersion $P_1$ | $0.3076$ |
| Flint partial dispersion $P_2$ | $0.2952$ |
| Secondary spectrum $f(P_1-P_2)/(V_1-V_2)$ | $\mathbf{0.044604}$ mm |
| Relative to the single lens | $\mathbf{1/34.9374}$ |
| With $P_1=P_2$ | $0.000\times10^{0}$ mm |
Observation 4.2 This is the core of the paper. It falls by $34.9374$ and does not reach zero——$0.044604$ mm remains.
Observation 4.3 The cause is named by one number. The difference in partial dispersion, $P_1-P_2=0.0124$——set it to zero and the secondary spectrum is exactly zero. The residue’s origin is identified.
Observation 4.4 So it is not that “two elements are not enough”. Two elements satisfy the two conditions——what is missing is a pair of glasses whose partial dispersions agree.
Observation 4.5 The same shape as Paper 370. There a single layer demanded a material of the ideal index——here achromatism demands a pair of matched partial dispersions. Both reduce to whether a material meeting the condition exists.
Observation 4.6 To be stated honestly——this paper does not treat anomalous-dispersion glasses. With fluorite and the like the secondary spectrum falls further——cited and no more; what is computed here is the ordinary two-glass case.
5. The Separator: First Order, or Second?
Observation 5.1 The separator——whether the difference in Abbe number (first order) or in partial dispersion (second order) has been cancelled; not the number of elements.
| Cancelled term | Residue | Relative to a single lens | |
|---|---|---|---|
| Single lens | none | $\mathbf{1.558361}$ mm | $1$ |
| Achromat (two) | first order (Abbe) | $\mathbf{0.044604}$ mm | $\mathbf{1/34.9374}$ |
| Matched partial dispersions | first and second | $\mathbf{0.000000}$ mm | $0$ |
Observation 5.2 The third row is a condition on materials, not on design. Add elements and the residue survives if the partial dispersions differ——how far it can be removed is settled on the side of the available glass.
Observation 5.3 Distinct roots by the criterion of Paper 300. First-order dispersion comes from the Abbe number, second-order from partial dispersion——both produce the same symptom, “the colours miss”, from different roots.
Observation 5.4 The separator can be moved. Anomalous-dispersion glasses shrink $P_1-P_2$——but they are expensive and hard to work. It can be moved, at a price.
Observation 5.5 To be stated honestly——$0.044604$ mm and $34.9374$ are values under stated Abbe numbers and partial dispersions, thin-lens and paraxial. Another glass gives other numbers——what does not move is the structure: as many wavelengths meet as conditions imposed, and beyond that the material decides.
Observation 5.6 One last thing——one may say “removed” only for as many wavelengths as conditions were imposed. Impose two equations and two wavelengths meet, with no guarantee whatever for a third colour——to count what has been removed, count how many conditions were imposed.
6. The Census and the Close
- Single lens shifts $1.558361$ mm (Calculation 2.1): $1.5584$ per cent of $f$.
- The shift scales with $f$ (Observation 2.3): longer lenses are worse.
- $f_1=43.322425$ mm (Calculation 3.1): shorter than $100$ mm.
- $f_2=-76.436624$ mm (Calculation 3.1): a diverging element is required.
- The combination gives $0.0100000000$ (Observation 3.4): the formulae close.
- The achromatic residual is exactly $0$ (Calculation 3.1): both conditions met.
- Secondary spectrum $0.044604$ mm (Calculation 4.1): a factor of $34.9374$.
- With $P_1=P_2$ it is exactly $0$ (Observation 4.3): the cause is named.
- What is missing is a pair of glasses (Observation 4.4): not more elements.
- The Abbe difference is in the denominator (Observation 3.5): different dispersions are needed.
Closing: An achromatic lens is called a lens with the chromatic aberration removed——but what is removed is as many wavelengths as conditions were imposed. A single lens shifts focus with colour by the focal length divided by the Abbe number——$\mathbf{1.558361}$ mm at $f=100$ mm and $V=64.17$, $1.5584$ per cent of the focal length. Blue and red focus $1.5$ mm apart on a $100$ mm lens, a clear coloured fringe in a photograph. Since the shift is $f/V$ it grows with focal length, which is why telescopes met the problem first. Two equations fix two focal lengths——$f_1=\mathbf{43.322425}$ mm and $f_2=\mathbf{-76.436624}$ mm. The second must be negative — without a diverging element the colour does not cancel. The first is therefore shorter than the target $100$ mm, bending extra to survive it. The combination gives $0.0100000000$ and the achromatic residual is exactly $0$. And still a third colour remains——the secondary spectrum from the difference in partial dispersion is $\mathbf{0.044604}$ mm, a fall by $\mathbf{34.9374}$ that does not reach zero. And its cause is named by one number: set the partial-dispersion difference $P_1-P_2=0.0124$ to zero and the secondary spectrum is exactly zero. So it is not that two elements are not enough — what is missing is a pair of glasses whose partial dispersions agree. The same shape as Paper 370——there a single layer demanded a material of the ideal index, and here achromatism demands a matched pair. Both reduce to whether a material meeting the condition exists. One thing separates them——whether the first-order (Abbe) or the second-order (partial dispersion) difference has been cancelled; not the number of elements. Add elements and the residue survives if the partial dispersions differ, so how far it can be removed is settled on the side of the available glass. Anomalous-dispersion glasses move it, at the price of cost and workability. To be said honestly——$0.044604$ mm and $34.9374$ hold for the stated Abbe numbers and partial dispersions, thin-lens and paraxial. This paper does not treat anomalous-dispersion glasses and computes the ordinary two-glass case. One last thing——one may say “removed” only for as many wavelengths as conditions were imposed. Impose two equations and two wavelengths meet, with no guarantee whatever for a third colour. To count what has been removed, count how many conditions were imposed.
References
- Abbe, E. (1873). Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung. Archiv für Mikroskopische Anatomie.
- Kingslake, R., Johnson, R. B. (2010). Lens Design Fundamentals (2nd ed.). Amsterdam: Academic Press.
- Smith, W. J. (2007). Modern Optical Engineering (4th ed.). New York: McGraw-Hill.
- Hecht, E. (2016). Optics (5th ed.). Boston: Pearson.
- Born, M., Wolf, E. (1999). Principles of Optics (7th ed.). Cambridge: Cambridge University Press.
- 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). A Bat Has Three Sweet Spots, and They Lie 23 cm Apart [Paper 363]. Zenodo. 10.5281/zenodo.22697505
- Yamagishi, Y. (2026). Two Algebraically Equal Forms Return Answers Differing by 15 Digits [Paper 368]. Zenodo. 10.5281/zenodo.22697726
- Yamagishi, Y. (2026). A Single Anti-Reflection Layer Falls Short for Two Independent Reasons [Paper 370]. Zenodo. 10.5281/zenodo.22697804
Reference
Yamagishi, Y. (2026). An Achromat Meets at Two Wavelengths Only [Paper 377]. YOI: https://www.yamagishiyuuki.com/yoi/377
DOI: 10.5281/zenodo.22698749
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 377. This paper is a preprint and has not been peer reviewed.