Chromatic Aberration and Achromats

A single lens cannot focus white light to a single point. Its focal length is set by the refractive index through the lensmaker equation, and the index of every ordinary glass falls with wavelength — blue light is bent more than red. So a simple lens has a shorter focal length for blue than for red, wrapping a white star in a coloured halo: the violet image floats in front of the red one, and no screen catches all colours sharp at once. This is chromatic aberration, and unlike the five monochromatic Seidel defects it comes not from geometry but from dispersion, $n=n(\lambda)$ . Its size is captured by one number per glass, the Abbe number $V=(n_d-1)/(n_F-n_C)$ , which measures refracting power divided by dispersion using three standard Fraunhofer wavelengths. High- $V$ "crown" glasses disperse gently ( $V\approx60$ ); low- $V$ "flint" glasses, loaded with heavy elements, disperse strongly ( $V\approx36$ ). A lens's fractional focal spread between blue and red is just $1/V$ — so a crown singlet of focal length $100$ mm shifts focus by about $1.7$ mm across the visible, plainly visible as colour fringing. The cure is the achromatic doublet: cement a strong positive crown lens to a weaker negative flint lens. Each bends light, but the flint's fierce dispersion, run in reverse, cancels the crown's colour error while leaving net converging power. The condition is exact and simple — $\varphi_1/V_1+\varphi_2/V_2=0$ — and it brings two chosen wavelengths to a common focus, collapsing the colour error to a small residual. That an achromat is possible at all was, notoriously, denied by Newton, who wrongly held dispersion to be inseparable from refraction; Hall and Dollond built working achromats within decades of his death. This article derives the singlet error, the Abbe number, and the doublet condition, and computes a real achromat.

Key takeaways

A lens's focal length depends on wavelength through $n(\lambda)$ : ordinary glass bends blue more than red, so blue focuses closer — chromatic (axial) aberration.
Each glass carries an Abbe number $V=(n_d-1)/(n_F-n_C)$ : high $V$ (crown, gentle dispersion) vs low $V$ (flint, strong dispersion).
A singlet's fractional focal spread between the blue (F) and red (C) lines is $1/V$ : longitudinal chromatic aberration $\Delta f\approx -f/V$ .
An achromatic doublet (crown $+$ flint) obeys $\varphi_1/V_1+\varphi_2/V_2=0$ with total power $\varphi=\varphi_1+\varphi_2$ ; it brings two wavelengths to one focus.
Solving those two equations gives a positive crown element and a negative flint element — the flint's dispersion cancels the crown's while net power remains.
A residual "secondary spectrum" survives because dispersion is not perfectly linear in wavelength; killing it needs special (anomalous-dispersion) glasses — an apochromat.
Key idea: a lens's colour error is refracting power divided by dispersion; combine a positive crown and a negative flint so their dispersions cancel, $\varphi_1/V_1+\varphi_2/V_2=0$ , and two wavelengths share a focus while the pair still converges.

The core insight

Chromatic aberration is dispersion made visible: because $n$ falls with wavelength, a lens focuses blue nearer than red, with a fractional focal spread of $1/V$ set by the glass's Abbe number $V=(n_d-1)/(n_F-n_C)$ . Pairing a positive crown lens with a negative flint lens so that $\varphi_1/V_1+\varphi_2/V_2=0$ cancels the colour error while leaving net converging power — the achromatic doublet.

Where this lives today

Almost every lens you use is at least an achromat. Camera lenses stack crown and flint (and, increasingly, low-dispersion "ED" and fluorite-like glasses) to hold colour correction across a zoom range; a "super-achromat" or apochromat brings three or four wavelengths to a common focus using anomalous-dispersion glasses, prized in astrophotography and microscopy where colour fringing on a bright star or a stained cell is unacceptable. The Abbe number remains the working currency of the optical-glass catalogue: Schott, Ohara, and Hoya publish thousands of glasses on the "glass map" of index versus $V$ , and a designer picks pairs by their separation on that map. Where colour correction must be perfect and weight is no object, the answer is still Newton's: a mirror. Every large telescope, from ground-based giants to the James Webb Space Telescope, images with reflectors precisely because a mirror focuses all wavelengths alike — dispersion is a property of refraction, and reflection escapes it entirely.

The physical picture

Hold a strong magnifier to a distant streetlight and rack it toward and away from a wall. There is no single sharp image: a purple-fringed disc becomes a green-white spot becomes a red-fringed disc as you move. The lens is a very weak prism at every point of its surface, and like a prism it splits white light — blue deflected most, red least — so the blue rays converge to a nearer focus and the red rays to a farther one. Between them lies a smear of overlapping coloured discs.

Figure

A singlet focuses blue nearer than red: the blue focus f_F sits closer to the lens than the red focus f_C, so every colour has its own image plane and white light never sharpens to a point

The remedy is counter-intuitive: add a diverging lens. Take a positive crown element — gentle dispersion — and cement behind it a negative flint element whose dispersion is fierce. The flint subtracts some converging power, so the pair is weaker than the crown alone, but because the flint disperses so much more per unit power, running it in reverse it can undo all of the crown's colour error while sacrificing only part of the power. Choose the two powers so the colour errors cancel exactly at two wavelengths, and the blue and red foci merge.

Figure

Punchline — focal length versus wavelength: a crown singlet (blue curve) spreads by Δf ≈ f/V = 1.67 mm across 486–656 nm, while the achromat (green) touches the same focus at the F and C lines, leaving only a ~0.05 mm secondary-spectrum residual

The punchline is the pair of curves. The crown singlet's focal length climbs almost linearly from blue to red by $\Delta f=f/V\approx1.67$ mm on a $100$ mm lens — a defect you can see. The achromat's curve is bent into a shallow "U" that touches the same focus at both the F (blue) and C (red) lines; the leftover sag between them, the secondary spectrum, is smaller by a factor of thirty or so. Two glasses, two equations, and the colour error nearly disappears.

The core equations

Chromatic aberration is dispersion propagated through the lensmaker equation. A thin lens has power (reciprocal focal length)

\varphi=\frac1f=(n-1)\left(\frac1{R_1}-\frac1{R_2}\right)=(n-1)K \qquad (1)

where: $\varphi$ — lens power (dioptres if $f$ in metres); $n$ — refractive index at the wavelength of interest; $R_1,R_2$ — surface radii; $K=1/R_1-1/R_2$ — the fixed shape factor. Because $n$ depends on $\lambda$ and $K$ does not, the entire wavelength dependence of the power lives in the factor $(n-1)$ . The dispersion of a glass is graded by the Abbe number, refracting power over dispersion at three standard Fraunhofer lines — $C$ (red, $656.3$ nm), $d$ (yellow, $587.6$ nm), $F$ (blue, $486.1$ nm):

V=\frac{n_d-1}{n_F-n_C} \qquad (2)

where: $n_d$ — index at the yellow $d$ line (the "central" index); $n_F,n_C$ — indices at the blue and red lines; the denominator $n_F-n_C$ is the principal dispersion. Crown glasses (light, e.g. BK7) have $V\approx64$ ; flint glasses (dense, lead- or heavy-element-loaded, e.g. F2) have $V\approx36$ . A large $V$ means much bending for little colour spread. Differentiating $(1)$ across the visible gives the singlet's longitudinal chromatic aberration — the focal spread from blue to red:

\frac{\Delta\varphi}{\varphi}=\frac{\Delta n}{n-1}=\frac{n_F-n_C}{n_d-1}=\frac1V\quad\Rightarrow\quad \Delta f\approx-\frac{f}{V} \qquad (3)

where: $\Delta\varphi=\varphi_F-\varphi_C$ — power difference between the blue and red lines; $\Delta f=f_F-f_C$ — the (negative) focal spread; the sign says blue focuses shorter. The colour error is simply the focal length divided by the Abbe number — a $100$ mm crown singlet ( $V\approx60$ ) blurs by $1.7$ mm. For two thin lenses in contact, powers add, and demanding that the combined power be equal at the F and C lines gives the achromatic condition:

\varphi=\varphi_1+\varphi_2,\qquad \frac{\varphi_1}{V_1}+\frac{\varphi_2}{V_2}=0 \qquad (4)

where: $\varphi_1,\varphi_2$ — powers of the two elements; $V_1,V_2$ — their Abbe numbers. The second equation is $\Delta\varphi_{\text{total}}=\varphi_1/V_1+\varphi_2/V_2=0$ : the two colour errors cancel. Solving the pair, $\varphi_1=\varphi\,V_1/(V_1-V_2)$ and $\varphi_2=-\varphi\,V_2/(V_1-V_2)$ — with $V_1>V_2$ (crown first) the crown element is positive and the flint is negative.

Derivation AThe achromatic-doublet condition

This is the owned result of the article: we prove that two thin lenses in contact focus blue and red light identically precisely when their powers divide their Abbe numbers to opposite, cancelling values — and that the crown must be the positive element and the flint the negative one. The strategy is to write the combined power, demand that it not change between the two Fraunhofer wavelengths that define the Abbe number, and solve the resulting two linear equations for the two element powers given a target total power.

Step 1. Two thin lenses in contact add their powers: $\varphi(\lambda)=\varphi_1(\lambda)+\varphi_2(\lambda)$ . Each element's power varies with wavelength as $(n_i(\lambda)-1)K_i$ , so its blue-to-red change is $\Delta\varphi_i=\varphi_i\,(n_{Fi}-n_{Ci})/(n_{di}-1)=\varphi_i/V_i$ . Why powers add: for thin lenses in contact the image of the first is the object of the second at zero separation, and the conjugate equation makes the reciprocal focal lengths sum.

Step 2. Achromatism between the F and C lines means the total power is the same at both: $\Delta\varphi=\Delta\varphi_1+\Delta\varphi_2=\dfrac{\varphi_1}{V_1}+\dfrac{\varphi_2}{V_2}=0$ . This is the achromatic condition. Why the F and C lines: choosing the two ends of the visible to coincide leaves the smallest residual across the band; the $d$ line then sits at the extremum of the secondary spectrum.

Step 3. Combine with the target power $\varphi_1+\varphi_2=\varphi$ . Solving the two linear equations: $\varphi_1=\varphi\dfrac{V_1}{V_1-V_2}$ , $\varphi_2=-\varphi\dfrac{V_2}{V_1-V_2}$ . With crown first ( $V_1>V_2$ ) the denominator is positive, so $\varphi_1>0$ (converging crown) and $\varphi_2<0$ (diverging flint). Why one must be negative: the colour errors have the same sign as their powers, so cancellation demands opposite-signed powers — you cannot achromatize two positive lenses of the same glass.

Step 4. For $\varphi=1/100\ \mathrm{mm}^{-1}$ , crown BK7 ( $V_1=64.12$ ) and flint F2 ( $V_2=36.37$ ): $\varphi_1=0.01\times64.12/27.75=0.02311\ \mathrm{mm}^{-1}$ so $f_1=+43.28$ mm, and $\varphi_2=-0.01\times36.37/27.75=-0.01311\ \mathrm{mm}^{-1}$ so $f_2=-76.32$ mm. The pair has $f=100$ mm and $\varphi_1/V_1+\varphi_2/V_2=0$ to machine precision. Why it works despite the negative element: the flint's dispersion per unit power is nearly twice the crown's, so a modest negative flint cancels the crown's colour while stealing only a third of its power. *Verified in verify_chromatic-aberration-achromats.py: the Abbe number $V=(n_d-1)/(n_F-n_C)$ for BK7 and F2; the doublet solution $f_1=+43.28$ mm, $f_2=-76.32$ mm for a $100$ mm target; the achromatic sum $\varphi_1/V_1+\varphi_2/V_2=0$ ; and $\varphi_1+\varphi_2=0.01\ \mathrm{mm}^{-1}$ .*

Go deeper · Derivation B · 1 minLongitudinal chromatic aberration of a singlet

Derivation BLongitudinal chromatic aberration of a singlet

Here we turn the lensmaker equation into the fringe you actually see, showing that the blue-to-red focal spread of any simple lens is just its focal length divided by its Abbe number, with the blue image nearer. The strategy is to differentiate the power with respect to index, recognise the ratio of dispersion to refractivity as the reciprocal Abbe number, and convert the power spread into a focal-length spread.

Step 1. From $\varphi=(n-1)K$ with fixed shape $K$ , a change in index gives $\Delta\varphi=K\,\Delta n=\varphi\,\dfrac{\Delta n}{n-1}$ . Take $\Delta n=n_F-n_C$ (blue minus red). Why only $n$ varies: the radii are ground fixed, so the whole colour dependence rides on $(n-1)$ .

Step 2. The bracket is the reciprocal Abbe number: $\dfrac{n_F-n_C}{n_d-1}=\dfrac1V$ , so $\Delta\varphi=\varphi/V$ . Since blue has the larger index, $\varphi_F>\varphi_C$ : the lens is stronger for blue. Why blue is stronger: normal dispersion means $n$ rises toward the blue, and more index means more bending and more power.

Step 3. Convert to focal length: $\Delta f=f_F-f_C=-f\,\Delta\varphi/\varphi=-f/V$ . For $f=100$ mm and $V=60$ , $\Delta f=-1.67$ mm — the blue focus sits $1.67$ mm in front of the red one. Why the minus: raising power lowers focal length, so the stronger blue lens focuses nearer — the negative sign is the near-blue/far-red ordering. *Verified in verify_chromatic-aberration-achromats.py: $\Delta f=-f/V=-1.667$ mm for $f=100$ mm, $V=60$ ; the fractional spread $1/V=0.0167$ ; and the ordering $f_F<f_C$ .*

Limiting cases

Case	Result	Meaning
Single glass, one lens	$\Delta f=-f/V$	irreducible colour error set by the glass; blue focuses nearer
Two identical glasses	$\varphi_1/V+\varphi_2/V=(\varphi_1+\varphi_2)/V\neq0$	cannot achromatize — you need two different $V$
Crown $+$ flint, $\varphi_1/V_1+\varphi_2/V_2=0$	$f_F=f_C$	achromat: blue and red share a focus
Achromat at the $d$ line	small residual (secondary spectrum)	dispersion is not linear in $\lambda$ ; a leftover $\sim f/2000$ remains
Anomalous-dispersion glasses (e.g. fluorite)	three wavelengths coincide	apochromat — secondary spectrum largely removed
Mirror (reflector)	$\Delta f=0$	reflection has no dispersion — telescopes went to mirrors for this reason

Worked examples

1 · Designing a cemented achromat (the owned computation)

Design an achromatic doublet of focal length $f=100$ mm from crown BK7 ( $V_1=64.12$ ) and flint F2 ( $V_2=36.37$ ). Find the focal length of each element and confirm the colour error cancels.

Step 1. The target power is $\varphi=1/100=0.0100\ \mathrm{mm}^{-1}$ . Split it by the achromatic solution: $\varphi_1=\varphi\,V_1/(V_1-V_2)=0.0100\times64.12/27.75=\boxed{0.02311\ \mathrm{mm}^{-1}}$ , so the crown element has $f_1=+43.28$ mm (strongly converging).

Step 2. The flint takes the rest: $\varphi_2=\varphi-\varphi_1=-0.01311\ \mathrm{mm}^{-1}$ , so $f_2=\boxed{-76.32\ \mathrm{mm}}$ (diverging). Check the ratio: $\varphi_2/\varphi_1=-V_2/V_1=-0.567$ , exactly the Abbe-number ratio.

Step 3. Confirm achromatism: $\varphi_1/V_1+\varphi_2/V_2=0.02311/64.12-0.01311/36.37=3.60\times10^{-4}-3.60\times10^{-4}=\boxed{0}$ . The doublet focuses the F and C lines identically; a $\sim0.05$ mm secondary spectrum remains at intermediate colours — thirty times smaller than the $1.7$ mm error of a crown singlet of the same power.

2 · The colour error of a simple magnifier (exam level)

A single crown lens has $f=100$ mm and $V=60$ . How far apart are its blue and red foci, and how does this compare to the achromat above?

Step 1. Longitudinal chromatic aberration is $\Delta f=-f/V=-100/60=\boxed{-1.67\ \mathrm{mm}}$ : the blue (F) focus lies $1.67$ mm in front of the red (C) focus. As a fraction of focal length that is $1/V=1.67\%$ .

Step 2. The achromatic doublet of Example 1 reduces this to its secondary spectrum, roughly $f/2000\approx0.05$ mm — a thirty-fold improvement — which is why every quality objective, from a microscope to a camera, is at minimum a cemented achromat rather than a singlet.

Physics of the portrait

Chromatic aberration is the defect that split refractors from reflectors, embarrassed Newton, and made the achromatic doublet the most-copied optical invention in history.

Origins · 1 minchromatic aberration and how this was discovered

chromatic aberration

the colour defect of refraction · Newton's error, Dollond's cure 1758 · why glass cannot focus white light to a point

Chromatic aberration is dispersion caught in the act of imaging, and its history is a cautionary tale about a great physicist's overreach. Newton, having discovered that white light is a mixture of colours refracted by different amounts, concluded — from too few glasses — that dispersion was rigidly proportional to refraction, so any lens that bent light must smear its colours and no combination of glasses could ever undo it. On that mistaken generalization he declared achromatic refractors impossible and turned to the reflecting telescope, whose mirror has no dispersion at all. He was wrong about the premise: different glasses disperse differently for the same bending, exactly the freedom an achromat exploits. Chester Moore Hall quietly built cemented crown-and-flint doublets around 1733, and John Dollond, after systematic experiments, patented the achromatic doublet in 1758 — within a generation of Newton's Opticks. Joseph von Fraunhofer later mapped the dark solar lines (the C, d, F wavelengths that define the very Abbe number used to design achromats) and built the finest achromatic objectives of his age. The lesson survives in every lens: colour error is not destiny but a number, $1/V$ , that a second glass can cancel.

History

1666

— Newton's prism experiments reveal that white light is a mixture of colours, each refracted differently

1704

— Newton's Opticks argues (wrongly) that dispersion is proportional to refraction, so achromatic lenses are impossible

1733

— Chester Moore Hall builds the first crown-and-flint achromatic doublets, keeping them largely private

1758

— John Dollond patents the achromatic doublet, disproving Newton's claim

1814

— Joseph von Fraunhofer catalogues the dark solar lines, later used to standardize the Abbe number

1830

— Joseph Jackson Lister's paper on achromatic object-glasses opens the era of corrected microscope objectives

1886

— Abbe and Schott develop new glasses enabling apochromats that remove the secondary spectrum

Going further

Texts. Hecht, Optics, 5th ed., §6.3 (chromatic aberration, achromatic doublets) · Born & Wolf, Principles of Optics, §4.7 · Kingslake & Johnson, Lens Design Fundamentals · Smith, Modern Optical Engineering · Schott/Ohara optical-glass catalogues (the glass map).

Provenance. The Abbe number, singlet chromatic aberration, and the achromatic-doublet condition follow Hecht §6.3 and Born & Wolf §4.7. verify_chromatic-aberration-achromats.py reproduces: $V=(n_d-1)/(n_F-n_C)$ for BK7 ( $64.1$ ) and F2 ( $36.4$ ) from catalogue indices; the doublet split $f_1=+43.28$ mm, $f_2=-76.32$ mm for a $100$ mm target with $\varphi_1/V_1+\varphi_2/V_2=0$ and $\varphi_1+\varphi_2=0.01\ \mathrm{mm}^{-1}$ ; the singlet $\Delta f=-f/V=-1.67$ mm ( $f=100$ , $V=60$ ); and the power ratio $\varphi_2/\varphi_1=-V_2/V_1$ . The Newton achromat-impossibility episode is the classic base-error trap (consistent with the sibling article Dispersion and Prisms); dates web/knowledge-verified (Newton 1704, Hall ~1733, Dollond 1758, Fraunhofer lines 1814).

Builds on: Thin Lens and Lensmaker's Equation · Dispersion and Prisms · Refractive Index and Optical Path Length · Snell's Law. Leads to: Thick Lenses and Principal Planes · Image Formation and Magnification · Plane and Spherical Mirrors. See also: Normal and Anomalous Dispersion (the physics of why $n$ depends on $\lambda$ ).

What is the difference between chromatic aberration and the Seidel aberrations?

They have different causes and are budgeted separately. The five Seidel aberrations are monochromatic — they arise from the third-order geometry of spherical surfaces and would occur even if glass had no dispersion. Chromatic aberration comes entirely from dispersion, the wavelength dependence $n(\lambda)$ , so a lens has a different focal length for each colour. You correct them with different tools: the Seidel sums with shapes, stops, and aspheres; chromatic aberration by pairing glasses of complementary Abbe number (an achromat) or by using mirrors. A lens can be sharp in monochromatic light yet colour-fringed, or colour-corrected yet soft at the rim — the two errors are independent.

Why can't I just use one very low-dispersion glass to avoid an achromat?

Because dispersion never reaches zero for a refracting material, so any singlet still has $\Delta f=-f/V$ , only smaller for a higher- $V$ glass. Fluorite and modern "ED" glasses push $V$ toward $90-95$ , roughly halving the colour error of ordinary crown, but they do not eliminate it — and they are expensive and delicate. An achromat, by contrast, cancels the error at two wavelengths with two ordinary glasses, and an apochromat using an anomalous-dispersion glass brings a third wavelength in too. Combining dispersions is far more powerful than chasing a single low-dispersion material.

What is the "secondary spectrum" that remains after an achromat?

An achromat forces the blue (F) and red (C) foci to coincide, but dispersion is not a straight line in wavelength — the intermediate colours, especially near the yellow-green $d$ line, focus at a slightly different distance. That residual is the secondary spectrum, typically about $f/2000$ for a standard crown-flint pair, some thirty times smaller than the singlet's error. Because it comes from the curvature of $n(\lambda)$ , correcting it needs glasses whose partial dispersions match — anomalous-dispersion glasses like fluorite — which bring three wavelengths to a focus. A lens corrected that way is an apochromat.

Was Newton simply wrong, or was there something to his claim?

He was wrong about the crucial generalization but right about the phenomenon. Newton correctly showed that refraction disperses white light and that a single ordinary lens must therefore have chromatic aberration. His error was to assume, from a limited set of materials, that the ratio of dispersion to refraction was the same for all transparent media — which would indeed make achromats impossible. Different glasses in fact disperse differently for the same bending (that is exactly what the Abbe number quantifies), so Hall and Dollond could pair a gently dispersing crown with a fiercely dispersing flint and cancel the colour. It is a textbook case of a correct observation and an overreaching theory.

Does stopping down a lens reduce chromatic aberration?

Only weakly, and not in the way it helps the Seidel aberrations. Stopping down narrows the cone of rays, so the coloured blur discs at any given plane shrink somewhat — but the separate blue and red foci stay exactly where they are, set by the glass, so axial colour persists. In practice a stopped-down singlet still shows colour fringing on high-contrast edges. The real fix is the glass combination (an achromat) or, for perfect correction, a mirror. This is a key difference from spherical aberration and coma, which stopping down attacks directly.

Why are the wavelengths 486, 588, and 656 nm chosen for the Abbe number?

They are Fraunhofer's dark solar lines — sharp, universally available reference wavelengths. The $F$ line ( $486.1$ nm) is hydrogen-beta (blue), the $C$ line ( $656.3$ nm) is hydrogen-alpha (red), and the $d$ line ( $587.6$ nm) is helium yellow (older references use the sodium $D$ lines near $589$ nm). Bracketing the visible with $F$ and $C$ and centring on $d$ makes the Abbe number $V=(n_d-1)/(n_F-n_C)$ a fair single measure of a glass's visible dispersion, and it lets any two labs specify and compare glasses against the same fixed spectral marks — which is exactly why Fraunhofer's line catalogue mattered so much to lens making.