Chromatic Aberration and the Achromatic Doublet
Why one piece of glass cannot focus blue and red at the same place, what the Abbe number measures, and how two glasses cancel each other’s color, computed live on a real lens.
Key Takeaways
- A single piece of glass focuses blue light closer than red, and no amount of reshaping it will help. Color is an error of material, not of shape.
- The Abbe number is one number per glass for how much its index spreads across the visible band. Higher means less color. Crowns are high, flints are low.
- Put a positive crown against a negative flint and their color errors cancel at two wavelengths. What is left over in between is the secondary spectrum.
Second of three. Color is separate from the five errors of shape, and it is the one you cannot fix by reshaping glass. Correcting the other five comes after.
The one aberration you cannot design away with shape
Here is a lens anyone could grind: a biconvex singlet of N-BK7 glass, radii 60 mm and −60 mm, 8 mm thick, working at f/3.0 with a 20 mm aperture. The sign on a radius says which way that surface bows, so this one is fat in the middle. Its focal length is 59.399 mm. Send yellow light through it, the helium d line at 587.6 nm, and the focus lands 56.700 mm behind the back surface.
The singlet, and its three foci
The same lens, three colors, drawn from the tool’s own ideal-ray (paraxial) trace. Below: the focal region magnified 60 times. Ideal rays, with this lens’s own spherical aberration set aside so that color is the only thing in the picture.
Now change nothing except the color. In blue light at 486.1 nm the focus moves to 56.085 mm. In red at 656.3 nm it moves to 56.978 mm. The lens’s own tolerance for being out of focus, meaning how far the sensor can move before the image visibly softens, is ±0.0104 mm. The gap between the blue focus and the red focus is 86 times that tolerance. Put the sensor at the yellow focus and the blue light arrives as a disk 105 µm in radius, sixty times the sharpest spot this aperture allows.
What “depth of focus” means here
The tolerance quoted throughout this page is the Rayleigh quarter-wave criterion, δz = ±2 λ (f/#)²: move the sensor further than that from best focus and the wavefront error passes a quarter of a wave. At f/3.0 in yellow light it is ±0.0104 mm; at f/4.4, where the doublets below work, it is ±0.0227 mm. It is a tolerance, not a measurement, and it is the fairest yardstick for a focus shift because it scales with the lens’s own aperture. The 105 µm disk is the geometric blur radius, |BFL(F) − BFL(d)| times the blue cone’s own exit slope from the paraxial trace: 0.614946 mm × 0.170142. The sharpest spot the aperture allows is the Airy radius 1.22 λ (f/#), which is 1.76 µm here, what a perfect f/3.0 lens would make in this color.
That is chromatic aberration, and it is different in kind from the five monochromatic aberrations. Spherical aberration, coma, astigmatism, field curvature and distortion are errors of shape: bend the surfaces, move the stop, add an element, and they move. Color is an error of material. Glass has a different refractive index at every wavelength, and no amount of reshaping one piece of glass will remove it. The fix is two glasses that disagree about color in a way you can exploit.
Two words you need before anything else. A crown is a glass whose index changes slowly across the visible spectrum. A flint is one whose index changes quickly. That is the whole distinction, and it is about dispersion, not density and not index. A common three-element camera lens, the one in our triplet aberrations piece, puts a flint in the middle at index 1.5679 and crowns outside at 1.6114. There the flint is the lower-index glass. What makes it a flint is that its index moves more as the color changes.
Axial color: the focus slides along the axis
Axial color is the shift of the focus along the optical axis with wavelength: every color forms its own image at its own distance, and no single sensor position is in focus for all of them.
The singlet above is the clean case. Its focal length, measured from a plane inside the lens rather than from its back face, which is why these numbers are larger than the focus distances above, is 58.774 mm in blue, 59.399 mm in yellow and 59.680 mm in red. The lens focuses more strongly in blue because the glass is stronger there: N-BK7’s index rises from 1.5143 at the red C line, 656.3 nm, to 1.5224 at the blue F line, 486.1 nm, and more index means more bending at both surfaces.
Two things follow. Refocusing does not fix it, it only picks a winner: move the sensor to the blue focus and the red is out by the same gap. And the error is there on axis, where a lens is otherwise at its best.
The Ray-Optics Designer, our browser lens-design tool, reports axial color as a single number, the back focal length at the blue F line minus the back focal length at the red C line. For this singlet the panel reads −0.89248 mm. The sign convention is worth holding on to: negative means blue focuses nearer the lens, which is what an uncorrected positive lens always does.
Focal length or back focal length?
Two different differences get called axial color. The tool reports BFL(F) − BFL(C), the distance between the two focal planes, which is what you would measure by racking a sensor: −0.892485 mm here. The thin-lens theory in the next section is about EFL(F) − EFL(C), the difference in focal length: −0.906142 mm here. They differ because the principal planes also move with wavelength. On this singlet the rear principal plane sits 2.698 mm inside the back face at the d line, 2.689 mm at F and 2.703 mm at C, and that 0.014 mm of drift is exactly the 0.014 mm by which the two numbers above disagree. Neither is wrong; do not compare one against the other and call the gap an error.
Focus against wavelength
Start at the yellow d line and drag toward blue. The marker leaves the gray tolerance band before you reach 550 nm.
The spot this color makes on a sensor at the yellow focus. Dashed: the sharpest spot this aperture allows, same magnification.
back focal length
56.702 mm
focus shift
+0.0020 mm
inside the ±0.0104 mm band
spot radius
0.3 µm
sharpest possible
2.13 µm
The sensor is at this color’s own focus, so only the diffraction disk is left.
The Abbe number
Every glass catalog in the world describes dispersion with one number.
The Abbe number Vd is the ratio of a glass’s refracting power to the amount its index spreads across the visible: Vd = (nd − 1) / (nF − nC), where nd, nF and nC are the indices at 587.6, 486.1 and 656.3 nm.
The three wavelengths are historical spectral lines: d is the yellow helium line, F and C the blue and red hydrogen lines. The numerator is what the glass does to light at all, since a medium of index 1 does no refracting; the denominator is how much that varies from blue to red. Vd is large for a glass that bends a lot and disperses little.
For our two glasses, computed from the Sellmeier fits that give each glass its index at any wavelength:
| nC | nd | nF | nF − nC | Vd | |
|---|---|---|---|---|---|
| N-BK7 (crown) | 1.514322 | 1.516800 | 1.522376 | 0.008054 | 64.167 |
| N-SF5 (flint) | 1.666638 | 1.672707 | 1.687496 | 0.020858 | 32.251 |
The flint spreads its index 2.59 times as far between the F and C lines, and 2.77 times as far across the wider 400 to 700 nm range.
The reason Vd matters to a lens and not just to a glass is a single thin-lens result. A thin lens of focal length f in a glass of Abbe number V has
between the F and C lines. The Abbe number is, to a good approximation, the reciprocal of the fractional focal spread of any simple lens made from that glass. Our singlet is a real thick lens, and the tool traces its focal spread at 1/65.55 of its focal length. The rule says 1/64.17. Two percent apart, and the rule cost nothing.
That also explains the direction that reads backwards to everyone at first: higher Vd means less color, because dispersion sits in the denominator.
Why these three lines, and one letter to watch
nF − nC is called the principal dispersion, and Vd is fixed to it by convention rather than by physics, so two glasses with identical Vd can still behave differently outside the F to C interval; the g-line index used for the secondary spectrum further down is exactly that effect. Some catalogs quote Ve instead, on the mercury e line at 546.1 nm with F′ and C′, and the two are not interchangeable. Our catalog and the tool use Vd throughout.
Watch the case of the letter: lowercase d is the helium line at 587.5618 nm, which defines Vd and is the tool’s default wavelength; uppercase D is the sodium doublet at 589.2938 nm, 1.7 nm away, which is what the 1896 and 1902 patents on this page measured their own glasses at. Different lines, nearly the same color, and the tables below keep each patent’s own letter. The same trap sits at the violet end: G′ is the hydrogen line at 434.0 nm, the one the 1902 patent published, and g is the mercury line at 435.8 nm, 1.8 nm away, which is the one the secondary-spectrum section uses. Two letters, two lines, and this page uses both.
Two glasses, one axis
Swap the second glass to N-FK58 and watch V_d climb as the curve flattens.
N-BK7 is the fixed reference curve, drawn dashed. Each curve is that glass’s index minus its own value at the d line.
Abbe number, N-BK7
64.167
Abbe number, N-SF5
32.251
blue-to-red spread, N-BK7
0.008054
blue-to-red spread, N-SF5
0.020858
the three indices behind each Abbe number
N-BK7, yellow line
1.516800
N-BK7, blue line
1.522376
N-BK7, red line
1.514322
N-SF5, yellow line
1.672707
N-SF5, blue line
1.687496
N-SF5, red line
1.666638
The glass map
Put refractive index on one axis and Abbe number on the other and every optical glass in the world becomes a point. The map is conventionally drawn with Vd decreasing to the right, so the low-dispersion crowns sit on the left and the flints on the right.
The Ray-Optics Designer ships 399 glasses from 4 manufacturers, running from index 1.437 at one corner, a fluorophosphate crown, to 2.1042 at the other, a dense flint. Both corners are labeled on the map below.
The crown/flint line is convention, not nature. SCHOTT’s technical note TIE-29 puts it at Vd = 55 for glasses below index 1.60 and Vd = 50 above; under that rule the catalog splits into 151 crowns and 248 flints.
Two things about the map surprise people. It is not full: the upper left, high index with low dispersion, is where every designer wants to be and where very little glass exists. And it is lumpy. 11 of the 399 glasses sit within 0.02 in index and 3 in Vd of N-BK7, because the same workhorse glass is sold by several manufacturers under several names.
The map is also where the tool’s optimizer works. Given permission it treats index and Abbe number as continuous variables, slides a fictitious “model glass” to whatever point minimizes the design’s errors, then snaps it to the nearest real catalog glass. The snap is the honest step: the fictitious glass always wins, and what counts is what the design does after it.
How the snap measures distance, and one cautionary run
Nearest is measured on a scaled plane, (Δnd / 0.01)² + (ΔVd / 5)², which says that 0.01 in index and 5 in Abbe number are the same size of mistake. A model glass landing at (1.72, 50.0) snaps to LAC10 at (1.72, 50.34), a distance of 0.068, which is close enough that the design barely notices. One at (1.90, 70.0), in the empty corner, has nothing nearer than TAFD30 at a distance of 6.1, and a design that depended on it does not survive the snap. The manual has the mechanics, and our camera-triplet guide walks a run where the greedy version lands all three elements on the same glass and one airspace at −3.0 mm.
Every optical glass, as one point
Drag the open marker into the empty upper-left corner, high index and low dispersion, and see what the catalog offers instead.
model glass
1.720 · 50.0
nearest real glass
LAC10
its index and Abbe number
1.72 · 50.34
distance
0.068
Distance: how far the nearest real glass is, counting 0.01 in index and 5 in Abbe number as the same size of step. All 399 glasses the tool ships are on the map.
The achromatic doublet
An achromatic doublet is two elements of different glasses, one positive and one negative, whose powers are chosen so the pair has the same focal length at two chosen wavelengths.
Optical power is one over focal length, so a 100 mm lens has a power of 0.01 per millimeter. It is the useful variable here because the powers of two thin lenses in contact simply add, which is what makes two equations enough. The pair must add up to the power you want, and its color errors must cancel:
where 1 is the crown and 2 the flint. The second equation is the achromatic condition, and it says something physical: each element contributes color in proportion to its own power divided by its own Abbe number, and the two contributions must cancel. Since V₁ and V₂ are both positive, φ₁ and φ₂ must have opposite signs. An achromat is always a positive element fighting a negative one.
Solve the pair for N-BK7 with N-SF5 and the crown comes out at 2.0105 times the power of the whole doublet, the flint at −1.0105 times. The crown has to be more than twice as strong as the doublet it is part of, because the flint is throwing most of that power away again.
The gap between the two Abbe numbers is what sets that ratio, and it sits in the denominator of both element powers. The wider apart the glasses are on the map, the gentler both elements can be. Halve the gap and you double both element powers, with all the spherical aberration and manufacturing sensitivity that steeper surfaces bring. With the same glass twice the gap is zero and there is no solution at all. That is why an achromat is a crown and a flint, not two crowns.
The power split in closed form
Solving φ₁ + φ₂ = φ with φ₁/V₁ + φ₂/V₂ = 0 gives φ₁ = φ · V₁/(V₁ − V₂) and φ₂ = −φ · V₂/(V₁ − V₂), which is where the gap in the denominator comes from and why the same glass twice is a division by zero rather than a hard design.
How the power splits between the two elements
Pick N-KZFS4, the flint closest to the crown, and watch both bars grow.
N-BK7 power
+2.0105 φ
f = 49.739 mm
N-SF5 power
−1.0105 φ
f = −98.961 mm
Abbe gap
31.916
Abbe numbers
64.17 · 32.25
The crown stays N-BK7; f is for a 100 mm doublet.
Now do it on a real lens. The Ray-Optics Designer opens on a preset called “Cemented Achromat Doublet (aberration anchor)”, named for its kind. It is a cemented doublet of exactly these two glasses, radii 50, −35 and −120 mm, 6 mm of crown and 3 mm of flint, focal length 87.878 mm, working at f/4.4 because it is the same 20 mm aperture on a longer lens. Its glasses are the achromat pair, but its radii were never solved for the achromatic condition, and the tool measures its axial color at 0.17146 mm, 7.6 times the depth of focus. Fixing that is the next two paragraphs.
The doublet, element by element
A cemented doublet is two elements ground to a common radius and glued together, so the pair has three optical surfaces rather than four and no air gap in the middle. This is the tool’s own “Cemented Achromat Doublet (aberration anchor)” preset, drawn from its own layout and the tool’s own ideal-ray (paraxial) trace: a positive N-BK7 crown in front, a negative N-SF5 flint behind, cemented on surface 2, with the aperture stop on surface 1. Below: the focal region magnified 250 times, with this lens’s own quarter-wave depth of focus drawn as a band about the yellow focus. Ideal rays, with this lens’s own spherical aberration set aside so that color is the only thing in the picture.
So solve it. Keep the front radius and both thicknesses, ask for a focal length of 87.878 mm and for the F and C foci to coincide, and let the cemented interface and the back surface move. Two conditions, two free radii, one answer:
| radius 1 | radius 2 (cemented) | radius 3 | axial color, as the tool prints it | |
|---|---|---|---|---|
| as the tool ships it | 50 | −35 | −120 | 0.17146 mm |
| solved, to the four decimals below | 50 | −39.1630 | −131.0973 | 1.6453e−6 mm |
| solved, at a shop’s rounding | 50 | −39.16 | −131.10 | 0.00020058 mm |
Every cell in that last column is what the tool’s own panel prints when you type the radii on its row, so none of it is a solver reporting a flattering zero. The middle row leaves the blue and red foci 1.6 nanometers apart, and it leaves them apart at all only because the radii on that row are themselves rounded to four decimals. Round them further, to the 0.01 mm any grinder can hold, and the residual is 0.2 µm, under one percent of the depth of focus. Achromatism is not a delicate condition.
What the thin-lens formula got right, and by how much
Measure the two elements of the solved design directly from their radii with the thin-element formula φ = (nd − 1)(1/R_a − 1/R_b): the crown comes out at 42.495 mm and the flint at −83.017 mm, against the thin-lens targets 43.710 and −86.965. The thin-lens condition is off by 2.9 % on the crown and 4.8 % on the flint, because the elements have real thickness and the second surface is not at the first one’s position. That is the normal division of labor: the thin-lens condition tells you the shape of the answer and lands you close enough for a solver to finish. In the tool the finishing step is the optimizer, given the focal length and the axial color as its two targets and the two radii as its two variables.
One honest warning about the spot size
The solved doublet’s on-axis spot also improves, from 40.0 µm RMS to 5.9 µm at the d line. Almost none of that is the color correction. Both numbers are measured in one color, so the improvement is a monochromatic one: solving for achromatism moved the radii, and the new bending happens to have much less spherical aberration. Two different things improved at once, and it would be wrong to credit the color condition for both.
Lateral color: the fringe that grows toward the corner
Lateral color is the change of image height with wavelength: each color forms an image of a different size, so off-axis points break into radial colored fringes that vanish on the axis and grow toward the corner of the field.
Axial color and lateral color are independent, and the second one turns on where the aperture stop sits. A cemented doublet with the stop, the opening that limits the ray bundle, right against it has essentially none of the second kind: at 5° off axis the solved achromat above puts the blue and red image points 0.074 µm apart on an image height of 7.686 mm. Nothing.
You need a lens whose elements sit far from the stop. Our Tessar preset, from Rudolph’s 1902 patent US 721,240 with the patent’s own indices at three lines, is one: four elements, stop buried in the middle, 30° half field. Its chief rays, the ones that run from an off-axis point through the middle of the stop, land at a common sensor plane to make the blue image consistently smaller than the yellow one. Rudolph measured his glasses at the sodium D line, 1.7 nm from the helium d line above and near enough to the same yellow, so the table keeps his letter:
| half-field | image height (D line, 589.3 nm) | blue F minus D | violet G′ (434.0 nm) minus D |
|---|---|---|---|
| 10° | 17.489 mm | −3.24 µm | −7.05 µm |
| 20° | 35.998 mm | −7.95 µm | −17.08 µm |
| 30° | 56.640 mm | −13.71 µm | −29.44 µm |
Zero on the axis, rising steadily to the corner, and worse the further into the violet you go. Refocusing cannot touch it, because it is not a focus error at all. On a modern sensor with 4 µm pixels a 29 µm fringe is seven pixels of color at the edge of the frame.
The fringe grows toward the corner
Drag the field back toward the axis and watch the three dots merge into one.
A strip 82 µm across, on an image height of 56.640 mm.
blue F minus D
−13.71 µm
violet G′ minus D
−29.44 µm
The 1902 patent published its glass indices at three wavelengths only, so this lens can be traced in exactly three colors: 589.3, 486.1, 434.0 nm. The lens is the tool’s own “Tessar f/5.5 (Rudolph 1902, US 721,240)” preset.
What is left over: the secondary spectrum
Two glasses buy two wavelengths. The rest of the spectrum is still free to miss.
That residual, the failure of the wavelengths between and beyond the corrected pair to join them, is the secondary spectrum.
Trace the solved achromat again, this time across the whole visible band. The F and C foci coincide, by construction. The yellow d line focuses 45 µm closer, and the deepest point of the curve sits near 555 nm. That is twice this lens’s own depth of focus, on a lens that is perfect at two wavelengths.
Toward the violet it gets much worse. At the mercury g line, 435.8 nm, the solved doublet focuses 0.214 mm beyond the yellow focus, which puts a 24 µm blur radius on the d-line sensor plane against an Airy radius of 2.3 µm. Ten times the diffraction limit, in the violet, on a lens that is perfect at two wavelengths. This is the residue behind the violet halo that a plain achromat leaves on a bright edge.
The full picture in one comparison, focal shift across 400 to 700 nm:
| lens | total spread of the focus | in units of its own depth of focus |
|---|---|---|
| N-BK7 singlet, f/3.0 | 1.9439 mm | 188 |
| doublet as the tool ships it, f/4.4 | 0.8845 mm | 39 |
| doublet solved for achromatism, f/4.4 | 0.5151 mm | 23 |
Why can’t a better glass pair fix it? Because of a stubborn regularity. Define the partial dispersion PgF, the share of the blue-to-red index spread that falls in the violet end alone. Plot PgF against Vd and almost every real glass lies close to one straight line, the normal line. A doublet’s secondary spectrum scales as the difference in P between its two glasses divided by the difference in their V, and for any two glasses on that line that ratio is just the line’s slope, the same for everyone. Pairing N-BK7 with five different flints, from the mildest to the densest, changes the residual by a factor of two, and the best of the five is already an off-line glass.
That is the escape: a glass off the line. Our catalog has them at both ends. The fluorophosphate crown at the left corner of the map, FCD100, sits 0.060 higher in PgF than the line predicts, and short flints, meaning flints whose violet end is unusually tame, sit below it: N-KZFS11 by 0.018. Pairing an abnormal crown with an ordinary flint is how an apochromat, corrected at three wavelengths rather than two, gets built, and it is why fluorite and fluorophosphate elements cost what they do.
Rudolph’s 1902 Tessar prints its own glass indices at three lines, so the tool can trace its focal curve without any modern data at all, and what comes out is a focal curve with a turning point inside the visible band, published in 1902: the blue focuses 0.102 mm nearer the lens than the yellow, and the violet comes back up to 0.058 mm nearer.
The formulas, and which shift is which
Partial dispersion is PgF = (ng − nF)/(nF − nC), with ng the index at the mercury g line, 435.8 nm. PgF is 0.534930 for N-BK7 and 0.598359 for N-SF5. The normal line through those two runs P = 0.534930 −0.00198735 (V − 64.167).
Two different residuals are quoted in this section and they are not the same measurement. The 45 µm is the d line against the joined F/C focus, one part in 1971 of the focal length, the number a visual user notices. The five-flint comparison is the g line against that same focus, which is three to four times larger because the g line is far outside the corrected interval: it runs from one part in 414 (N-SF66) to one part in 818 (N-KZFS4) on a 100 mm thin-lens pair. For thin elements in contact that one is Δf ≈ −f (P₁ − P₂)/(V₁ − V₂). On our solved doublet the tool traces the g line 0.169822 mm beyond the F/C focus; the formula gives 0.174645 mm, 2.8 % apart, the same thin-versus-thick gap as before.
The Tessar’s three points, measured
Back focal length 90.710335 mm at the yellow D line, 90.608056 mm at blue F, and 90.652127 mm at violet G′. Those are three points, not a curve: the preset’s glasses are published at three named lines only, and the tool returns no index between them, by design.
Three lenses, one axis of focus shift
All three are on. Turn off the singlet and the doublet as the tool ships it, and the achromat’s own U-shape appears.
N-BK7 singlet
f/3.0 · 1.9439 mm · 188×
doublet as the tool ships it
f/4.4 · 0.8845 mm · 39×
doublet solved for achromatism
f/4.4 · 0.5151 mm · 23×
Each span is that lens’s own focus shift across the band, in millimeters and in its own quarter-wave depths of focus. That depth of focus is the faint band about zero, and it is a hairline until the singlet is switched off. The vertical scale fits whatever is switched on, so it changes when you toggle a lens; the horizontal axis never does.
Try it
Open the Ray-Optics Designer. It starts on “Cemented Achromat Doublet (aberration anchor)”: N-BK7 and N-SF5, radii 50, −35 and −120. The panel titled First-order data shows the axial color, 0.17146 mm, and it needs no account.
Then do the design. Type −39.16 into the second radius and −131.10 into the third, and watch the axial color fall from 0.17146 mm to 0.00020058 mm while the focal length moves only from 87.878 to 87.881 mm. Compare the spot diagrams at all three wavelengths before and after; those need a free account. Most of that improvement is spherical aberration, not color. Then load “Biconvex Singlet (teaching)”, the lens the first half of this page is built on. The last one, “Tessar f/5.5 (Rudolph 1902, US 721,240)”, is eight surfaces and needs a free account; load it to see what a stop buried between elements does at the corner of a 30° field.
All of this is the prerequisite for the three-glass triplet, where color correction and the five monochromatic aberrations have to be satisfied at once with eight free numbers. That is our camera-triplet design guide, which starts where this page stops. And the same three-element lens, taken aberration by aberration, is our triplet aberrations piece: the correction that needs three elements.
References
- SCHOTT, TIE-29: Refractive Index and Dispersion (technical information note): the definitions of nd, nF, nC, the Abbe number, the partial dispersion PgF and the normal line, and the crown/flint boundary used on the glass map here.
- R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic Press, 2010), chapters 6 to 11: the aberration-by-aberration treatment, including the achromatic doublet’s power split and the secondary spectrum.
- W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008), chapter 3: axial and lateral color as ray-level pictures, and the thin-lens achromat.
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), chapter 4: dispersion and the chromatic terms of the aberration expansion.
- P. Rudolph, “Photographic Objective”, US Patent 721,240 (1903), assigned to Carl Zeiss, Jena: the Tessar prescription and the three published glass indices used for the lateral-color and Tessar numbers here.
- H. D. Taylor, “Lens”, US Patent 568,052 (1896): the three-element camera lens quoted in the opening for its low-index flint.
- Glass data:
refractiveindex.infodatabase (CC0), release6f3b772, carrying the manufacturers’ own 2017-vintage Sellmeier and power-series coefficients. The tool’s copy and its provenance are described in the manual.