<p>The actual math is a bit complicated, but there's a simple intuitive explanation.</p>

<p>Longitudinal chromatic aberration happens because, when you cut a convergent lens in two, and you look at the cross-section, the edge of the lens looks a bit like a prism, doesn't it? (look at the diagram that you've posted above, the top of the lens) And it does exactly what a prism does - it makes a "pretty" little rainbow. Not so pretty when you're trying to do astronomy, however.</p>

<p>Short, fat, triangular prisms make widespread rainbows. Thin, elongated, thin-wedge shaped prisms make tight, narrow rainbows. The spread of the colors increases as the two sides of the prism are separated by a wider angle. It decreases as the two sides get closer together at a narrow angle. At the extreme, if the two sides of a "prism" were parallel, there would be no rainbow - but then it's not a prism anymore, it's just a glass plate, completely flat.</p>

<p>All else being equal, a single-lens-objective refractor has a "fatter" lens when focal length gets shorter (the two sides of the lens are more strongly curved, and/or less parallel); it has a "thinner" lens when the focal length gets bigger (the two sides of the lens are less strongly curved, and/or closer to parallel). A "fat" lens at the edge is like a fat, blunt prism. A "thin" lens at the edge is like a sharp, slender prism. The former makes a big spread of color. The latter makes a much more tight rainbow.</p>

<p>So that's what the old telescope makers did, back in the day (Hevelius, late 1600s). They just pumped up the focal ratio, made these super-long refractors with weakly curved lenses to combat the rainbow edge images which are so annoying and ultimately can falsify your observations.</p>

<hr>

<p>Now, if you study the properties of lenses, you notice that divergent lenses (the kind that make the image smaller) have a chromatic aberration opposite to convergent lenses (lenses that magnify, and that were used as primary, or objective lenses in telescopes back then). It could be argued that by combining a convergent and a divergent lens, chromatic aberration might be greatly reduced; as long as the convergent lens is stronger than the divergent companion, the combination should remain convergent.</p>

<p>Also, different glasses need to be used, so that the convergent lens is strongly convergent, but weakly dispersing, while the divergent lens is weakly divergent, but strongly dispersing. That's how you get near zero dispersion, but still pretty decent convergence.</p>

<p><a href="https://i.sstatic.net/Ty6XV.png" rel="noreferrer"><img src="https://i.sstatic.net/Ty6XV.png" alt="enter image description here"></a></p>

<p>In the early 1700s this was accomplished by British opticians. Chester Moore Hall is credited with the making of the first achromatic objective lens using the scheme shown above. But the actual work of grinding/polishing the glass was contracted to George Bass, who noticed that the two lenses he was making, when put together, had near-zero chroma. Hall failed to recognize the practical applications of the achromats, neglected his own invention, and years later Bass mentioned the achromat scheme to John Dollond, who understood its value and started to popularize the new design.</p>

<p><a href="https://en.wikipedia.org/wiki/History_of_the_telescope#Achromatic_refracting_telescopes" rel="noreferrer">https://en.wikipedia.org/wiki/History_of_the_telescope#Achromatic_refracting_telescopes</a></p>

<p><a href="https://en.wikipedia.org/wiki/Achromatic_lens" rel="noreferrer">https://en.wikipedia.org/wiki/Achromatic_lens</a></p>

<p>The achromatic combo guarantees zero chromatic aberration at two wavelengths. So you basically correct the aberration in the red and blue, or something like that, and the rest is "good enough". Things can be improved further by adding another lens - the apochromatic objective, or the "apo", or the "triplet". This was invented by Peter Dollond, John's son, in the mid-1700s.</p>

<p><a href="https://i.sstatic.net/E6feo.png" rel="noreferrer"><img src="https://i.sstatic.net/E6feo.png" alt="enter image description here"></a></p>

<p>The apochromat guarantees zero chromatic aberration at three wavelengths - usually in the middle and then closer to the two extremes of the visual spectrum. So that's overall better than the previous scheme.</p>

<p><a href="https://en.wikipedia.org/wiki/Apochromat" rel="noreferrer">https://en.wikipedia.org/wiki/Apochromat</a></p>

<p>The best apo refractors have extremely low chromatic aberration, basically invisible to the eye in most situations. But they are expensive.</p>

<p>Further improvements can be made. The most expensive amateur refractors are "quad" schemes, whereby using 4 lenses zeroes out chroma at 4 wavelengths. Takahashi is a brand nowadays known for excellent quality quad refractors, used in top-level astrophotography rigs.</p>

<hr>

<p>BTW, not just chroma, but basically all aberrations can be reduced by increasing the focal ratios of single lens objectives. Pretty much every single aberration gets better if you put a piece of paper in front of it, with a hole smaller than the diameter of the lens. But the aperture gets smaller, resolving power gets worse, there's less light captured by the instrument, etc. Tradeoffs...</p>

<hr>

<p>W.r.t detailed math of single lens chromatic aberrations, see chapter 5.3.1. Chromatic Aberration of a Singlet in 'Telescopes, Eyepieces, Astrographs' by G.H. Smith, R. Ceragioli, R. Berry. I believe that chapter is exactly the answer to your question, complete with working out the math for the aberration as a function of lens parameters.</p>

<p><a href="http://www.willbell.com/tm/TelescopesEyepiecesAstrographs.html" rel="noreferrer">http://www.willbell.com/tm/TelescopesEyepiecesAstrographs.html</a></p>

<p>Given your interest in the theory of optical systems, you should read that book.</p>

<p>Also worth mentioning:</p>

<p>Introduction to Lens Design, by J.M. Geary</p>

<p><a href="http://www.willbell.com/tm/lens_design/index.htm" rel="noreferrer">http://www.willbell.com/tm/lens_design/index.htm</a></p>

<p>Practical Computer-Aided Lens Design, by G.H. Smith:</p>

<p><a href="http://www.willbell.com/tm/lensdesn.htm" rel="noreferrer">http://www.willbell.com/tm/lensdesn.htm</a></p>

<p>Some good theoretical info, as always, can be found on telescope-optics.net</p>

<p><a href="https://www.telescope-optics.net/chromatic.htm" rel="noreferrer">https://www.telescope-optics.net/chromatic.htm</a></p>

CC BY-SA 4.0
2019-02-14T22:47:18.600Z
This is a beautifully written answer, thank you! But I've asked for a mathematical understanding. Since chromatic aberration results in different focal lengths for different wavelengths, perhaps you could develop one from a depth-of-field argument?
uhoh
CC BY-SA 4.0
2019-02-14T23:42:11.440Z
Or perhaps for strong lens figures it's no longer reasonable to approximate $\sin(\theta) \approx \theta$ when applying Snell's law at each interface?
uhoh
CC BY-SA 4.0
2019-02-15T01:03:36.767Z
@uhoh When dealing with aberrations you throw out all approximations. I mean, that's why aberrations exist, because things are not exactly the way they are described by those pretty and simple equations.
Florin Andrei
CC BY-SA 4.0
2019-02-15T01:17:11.160Z
@uhoh I made an edit and added links to reading material that contains the math you're looking for.
Florin Andrei
CC BY-SA 4.0
2019-02-16T10:33:20.817Z
`-1` Thanks, but I'm looking for an answer to my question as asked. As beautiful as this post is, in terms of my question it's still a *link-only answer*.
uhoh
CC BY-SA 4.0
2019-02-14T21:08:08.457Z
2019-02-15T01:24:54.877Z
2019-02-15T01:24:54.877Z
Florin Andrei
<p>The answer lies in your first image. You see how the different colours are refracted differently? This is for a very-short-focus lens. The cone of light between the lens and the image is very wide—the angle between the top and bottom rays is very wide.</p>
<p>Now, when you have a long-focus lens, this cone of light becomes narrower and narrower. The relative space between each “colour-focus” becomes smaller. So you end up with red and blue rays focused almost in the same place, at least proportionally speaking.</p>
<p>I don’t have an image to illustrate this, sorry, but I hope my explanation is clear enough to be understood.</p>

CC BY-SA 4.0
2021-12-30T22:37:03.493Z
"smaller" based on what metric exactly? Angles inside the telescope, or angular resolution projected back to the celestial sphere? At this point I really feel that a math-based answer is needed, one that convinces the reader that an image of a given object would be clearer; that chromatic aberration would have a lower impact on image clarity when the focal length is very long. I have a hunch that a few lines of algebra is all that's needed, small angle approximation where $\sin(\theta) \approx \theta$, Taylor expansion of Snells' law to first order in $\lambda$ etc.
uhoh
CC BY-SA 4.0
2021-12-30T23:15:06.740Z
Hmm… I tried using the *lensmaker’s formula*, but the results I get would make it for a *short*-focus lens to have less chromatic aberration… I’m puzzled…
Pierre Paquette
CC BY-SA 4.0
2021-12-30T23:18:03.333Z
Yes, in my  bounty message I threaten to answer this but so far I have not attempted to solve what I claimed there (without evidence) to be a "pretty simple" problem. :-)
uhoh
CC BY-SA 4.0
2021-12-30T22:31:01.243Z
2021-12-30T22:31:01.243Z
Pierre Paquette
<p>Below are two cropped views of &quot;<a href="https://en.wikipedia.org/wiki/Aerial_telescope" rel="nofollow noreferrer">Johannes Hevelius's 8 inch telescope with an open work wood and wire &quot;tube&quot; that had a focal length of 150 feet to limit chromatic aberration.</a>&quot; from <a href="https://iiif.lib.harvard.edu/manifests/view/ext:aHR0cHM6Ly9paWlmLmxpYi5oYXJ2YXJkLmVkdS9tYW5pZmVzdHMvaWRzOjE2NDcwMzE1$1i" rel="nofollow noreferrer">Harvard University, Houghton Library, pga_typ_620_73_451_fig_aa</a> (found <a href="https://hollis.harvard.edu/primo-explore/fulldisplay?context=L&amp;vid=HVD2&amp;search_scope=everything&amp;tab=everything&amp;lang=en_US&amp;docid=01HVD_ALMA211764767520003941" rel="nofollow noreferrer">here</a>) the first of which I've zoomed and enhanced the contrast.</p>
<p>According to Wikipedia's <a href="https://en.wikipedia.org/wiki/Aerial_telescope#Very_long_%22tubed%22_telescopes" rel="nofollow noreferrer">Aerial telescope; Very long &quot;tubed&quot; telescopes</a>:</p>
<blockquote>
<p><strong>Very long &quot;tubed&quot; telescopes</strong></p>
<p>Telescopes built in the 17th and early 18th century used single element non-achromatic objective lenses that suffered from interfering rainbow halos (chromatic aberration) introduced by the non-uniform refractive properties of single glass lenses. This degraded the quality of the images they produced. Telescope makers from that era found that very long focal length objectives had no appreciable chromatic aberration (the uncorrected chromatic aberration fell within the large diffraction pattern at focus). They also realized that when they doubled the diameter of their objectives they had to make the objective's focal length 4 times as long (focal length had to be squared) to achieve the same amount of minimal chromatic aberration. As the objective diameter of these refracting telescopes was increased to gather more light and resolve finer detail they began to have focal lengths as long as 150 feet. Besides having very long tubes, these telescopes needed scaffolding or long masts and cranes to hold them up. Their value as research tools was minimal since the telescope's support frame and tube flexed and vibrated in the slightest breeze and sometimes collapsed altogether.</p>
</blockquote>
<p>A refractor's primary lens produces an image because the angle it deflects is proportional to the distance <span class="math-container">$x$</span> from the center of the lens of focal length <span class="math-container">$f$</span> as</p>
<p><span class="math-container">$$\theta \approx x/f$$</span>.</p>
<p>For a dispersive material the deflection at a given point on the lens varies with wavelength, and therefore the focal length of the lens varies with wavelength.</p>
<p><a href="https://i.sstatic.net/KBEYI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KBEYI.png" alt="enter image description here" /></a> <a href="https://en.wikipedia.org/wiki/File:Chromatic_aberration_lens_diagram.svg" rel="nofollow noreferrer">Source</a></p>
<p>This is called <a href="https://en.wikipedia.org/wiki/Chromatic_aberration" rel="nofollow noreferrer">chromatic aberration</a> and its severity for a given type of glass is often expressed by its <a href="https://en.wikipedia.org/wiki/Abbe_number" rel="nofollow noreferrer">abbe number</a>.</p>
<p><strong>Question:</strong> Is there a simple mathematical way to explain how making the focal length longer could possibly <em>improve</em> a telescope's performance by decreasing the impact of chromatic aberration on its performance?</p>
<p>Or do I misunderstand what the benefits are to making these very long telescopes?</p>
<hr />
<p><a href="https://i.sstatic.net/iFTbX.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iFTbX.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/1I0eh.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1I0eh.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2019-02-14T11:06:33.853Z
related but different: [How did Johannes Hevelius' long telescope work? Why all the round holes?](https://astronomy.stackexchange.com/q/29317/7982)
uhoh
CC BY-SA 4.0
2019-02-14T16:21:22.520Z
I don't have my copy of Smith Modern Optical Engineering handy so just as a comment:  for a dispersive material, the chromatic aberration is a nonlinear function of the lens power,  Weaker lens means significantly less chromatic aberration in the final image.
Carl Witthoft
<telescope><history><optics><angular-resolution>
How does making a refracting telescope very long reduce the chromatic aberration of an uncorrected lens?
uhoh
# How does making a refracting telescope very long reduce the chromatic aberration of an uncorrected lens?

(Asked by: username_0 on 2019-02-14)

Below are two cropped views of "[Johannes Hevelius's 8 inch telescope with an open work wood and wire "tube" that had a focal length of 150 feet to limit chromatic aberration.](https://en.wikipedia.org/wiki/Aerial_telescope)" from [Harvard University, Houghton Library, pga\_typ\_620\_73\_451\_fig\_aa](https://iiif.lib.harvard.edu/manifests/view/ext:aHR0cHM6Ly9paWlmLmxpYi5oYXJ2YXJkLmVkdS9tYW5pZmVzdHMvaWRzOjE2NDcwMzE1$1i) (found [here](https://hollis.harvard.edu/primo-explore/fulldisplay?context=L&vid=HVD2&search_scope=everything&tab=everything&lang=en_US&docid=01HVD_ALMA211764767520003941)) the first of which I've zoomed and enhanced the contrast.

According to Wikipedia's [Aerial telescope; Very long "tubed" telescopes](https://en.wikipedia.org/wiki/Aerial_telescope#Very_long_%22tubed%22_telescopes):

> **Very long "tubed" telescopes**
>
> Telescopes built in the 17th and early 18th century used single element non-achromatic objective lenses that suffered from interfering rainbow halos (chromatic aberration) introduced by the non-uniform refractive properties of single glass lenses. This degraded the quality of the images they produced. Telescope makers from that era found that very long focal length objectives had no appreciable chromatic aberration (the uncorrected chromatic aberration fell within the large diffraction pattern at focus). They also realized that when they doubled the diameter of their objectives they had to make the objective's focal length 4 times as long (focal length had to be squared) to achieve the same amount of minimal chromatic aberration. As the objective diameter of these refracting telescopes was increased to gather more light and resolve finer detail they began to have focal lengths as long as 150 feet. Besides having very long tubes, these telescopes needed scaffolding or long masts and cranes to hold them up. Their value as research tools was minimal since the telescope's support frame and tube flexed and vibrated in the slightest breeze and sometimes collapsed altogether.

A refractor's primary lens produces an image because the angle it deflects is proportional to the distance $x$ from the center of the lens of focal length $f$ as

$$\theta \approx x/f$$.

For a dispersive material the deflection at a given point on the lens varies with wavelength, and therefore the focal length of the lens varies with wavelength.

[![enter image description here](https://i.sstatic.net/KBEYI.png)](https://i.sstatic.net/KBEYI.png) [Source](https://en.wikipedia.org/wiki/File:Chromatic_aberration_lens_diagram.svg)

This is called [chromatic aberration](https://en.wikipedia.org/wiki/Chromatic_aberration) and its severity for a given type of glass is often expressed by its [abbe number](https://en.wikipedia.org/wiki/Abbe_number).

**Question:** Is there a simple mathematical way to explain how making the focal length longer could possibly *improve* a telescope's performance by decreasing the impact of chromatic aberration on its performance?

Or do I misunderstand what the benefits are to making these very long telescopes?

---

[![enter image description here](https://i.sstatic.net/iFTbX.png)](https://i.sstatic.net/iFTbX.png)

[![enter image description here](https://i.sstatic.net/1I0eh.jpg)](https://i.sstatic.net/1I0eh.jpg)

**Tags:** telescope, history, optics, angular-resolution
> **Comments:**
> **username_0**: related but different: [How did Johannes Hevelius' long telescope work? Why all the round holes?](https://astronomy.stackexchange.com/q/29317/7982)  
> **username_1**: I don't have my copy of Smith Modern Optical Engineering handy so just as a comment:  for a dispersive material, the chromatic aberration is a nonlinear function of the lens power,  Weaker lens means significantly less chromatic aberration in the final image.  

---

## 2 answers

### Answer by username_2 on 2019-02-14. Score: 9

The actual math is a bit complicated, but there's a simple intuitive explanation.

Longitudinal chromatic aberration happens because, when you cut a convergent lens in two, and you look at the cross-section, the edge of the lens looks a bit like a prism, doesn't it? (look at the diagram that you've posted above, the top of the lens) And it does exactly what a prism does - it makes a "pretty" little rainbow. Not so pretty when you're trying to do astronomy, however.

Short, fat, triangular prisms make widespread rainbows. Thin, elongated, thin-wedge shaped prisms make tight, narrow rainbows. The spread of the colors increases as the two sides of the prism are separated by a wider angle. It decreases as the two sides get closer together at a narrow angle. At the extreme, if the two sides of a "prism" were parallel, there would be no rainbow - but then it's not a prism anymore, it's just a glass plate, completely flat.

All else being equal, a single-lens-objective refractor has a "fatter" lens when focal length gets shorter (the two sides of the lens are more strongly curved, and/or less parallel); it has a "thinner" lens when the focal length gets bigger (the two sides of the lens are less strongly curved, and/or closer to parallel). A "fat" lens at the edge is like a fat, blunt prism. A "thin" lens at the edge is like a sharp, slender prism. The former makes a big spread of color. The latter makes a much more tight rainbow.

So that's what the old telescope makers did, back in the day (Hevelius, late 1600s). They just pumped up the focal ratio, made these super-long refractors with weakly curved lenses to combat the rainbow edge images which are so annoying and ultimately can falsify your observations.

---

Now, if you study the properties of lenses, you notice that divergent lenses (the kind that make the image smaller) have a chromatic aberration opposite to convergent lenses (lenses that magnify, and that were used as primary, or objective lenses in telescopes back then). It could be argued that by combining a convergent and a divergent lens, chromatic aberration might be greatly reduced; as long as the convergent lens is stronger than the divergent companion, the combination should remain convergent.

Also, different glasses need to be used, so that the convergent lens is strongly convergent, but weakly dispersing, while the divergent lens is weakly divergent, but strongly dispersing. That's how you get near zero dispersion, but still pretty decent convergence.

[![enter image description here](https://i.sstatic.net/Ty6XV.png)](https://i.sstatic.net/Ty6XV.png)

In the early 1700s this was accomplished by British opticians. Chester Moore Hall is credited with the making of the first achromatic objective lens using the scheme shown above. But the actual work of grinding/polishing the glass was contracted to George Bass, who noticed that the two lenses he was making, when put together, had near-zero chroma. Hall failed to recognize the practical applications of the achromats, neglected his own invention, and years later Bass mentioned the achromat scheme to John Dollond, who understood its value and started to popularize the new design.

<https://en.wikipedia.org/wiki/History_of_the_telescope#Achromatic_refracting_telescopes>

<https://en.wikipedia.org/wiki/Achromatic_lens>

The achromatic combo guarantees zero chromatic aberration at two wavelengths. So you basically correct the aberration in the red and blue, or something like that, and the rest is "good enough". Things can be improved further by adding another lens - the apochromatic objective, or the "apo", or the "triplet". This was invented by Peter Dollond, John's son, in the mid-1700s.

[![enter image description here](https://i.sstatic.net/E6feo.png)](https://i.sstatic.net/E6feo.png)

The apochromat guarantees zero chromatic aberration at three wavelengths - usually in the middle and then closer to the two extremes of the visual spectrum. So that's overall better than the previous scheme.

<https://en.wikipedia.org/wiki/Apochromat>

The best apo refractors have extremely low chromatic aberration, basically invisible to the eye in most situations. But they are expensive.

Further improvements can be made. The most expensive amateur refractors are "quad" schemes, whereby using 4 lenses zeroes out chroma at 4 wavelengths. Takahashi is a brand nowadays known for excellent quality quad refractors, used in top-level astrophotography rigs.

---

BTW, not just chroma, but basically all aberrations can be reduced by increasing the focal ratios of single lens objectives. Pretty much every single aberration gets better if you put a piece of paper in front of it, with a hole smaller than the diameter of the lens. But the aperture gets smaller, resolving power gets worse, there's less light captured by the instrument, etc. Tradeoffs...

---

W.r.t detailed math of single lens chromatic aberrations, see chapter 5.3.1. Chromatic Aberration of a Singlet in 'Telescopes, Eyepieces, Astrographs' by G.H. Smith, R. Ceragioli, R. Berry. I believe that chapter is exactly the answer to your question, complete with working out the math for the aberration as a function of lens parameters.

<http://www.willbell.com/tm/TelescopesEyepiecesAstrographs.html>

Given your interest in the theory of optical systems, you should read that book.

Also worth mentioning:

Introduction to Lens Design, by J.M. Geary

<http://www.willbell.com/tm/lens_design/index.htm>

Practical Computer-Aided Lens Design, by G.H. Smith:

<http://www.willbell.com/tm/lensdesn.htm>

Some good theoretical info, as always, can be found on telescope-optics.net

<https://www.telescope-optics.net/chromatic.htm>

> **Comments:**
> **username_0**: This is a beautifully written answer, thank you! But I've asked for a mathematical understanding. Since chromatic aberration results in different focal lengths for different wavelengths, perhaps you could develop one from a depth-of-field argument?  
> **username_0**: Or perhaps for strong lens figures it's no longer reasonable to approximate $\sin(\theta) \approx \theta$ when applying Snell's law at each interface?  
> **username_2**: @uhoh When dealing with aberrations you throw out all approximations. I mean, that's why aberrations exist, because things are not exactly the way they are described by those pretty and simple equations.  
> **username_2**: @uhoh I made an edit and added links to reading material that contains the math you're looking for.  
> **username_0**: `-1` Thanks, but I'm looking for an answer to my question as asked. As beautiful as this post is, in terms of my question it's still a *link-only answer*.  

---

### Answer by username_3 on 2021-12-30. Score: 0

The answer lies in your first image. You see how the different colours are refracted differently? This is for a very-short-focus lens. The cone of light between the lens and the image is very wide—the angle between the top and bottom rays is very wide.

Now, when you have a long-focus lens, this cone of light becomes narrower and narrower. The relative space between each “colour-focus” becomes smaller. So you end up with red and blue rays focused almost in the same place, at least proportionally speaking.

I don’t have an image to illustrate this, sorry, but I hope my explanation is clear enough to be understood.

> **Comments:**
> **username_0**: "smaller" based on what metric exactly? Angles inside the telescope, or angular resolution projected back to the celestial sphere? At this point I really feel that a math-based answer is needed, one that convinces the reader that an image of a given object would be clearer; that chromatic aberration would have a lower impact on image clarity when the focal length is very long. I have a hunch that a few lines of algebra is all that's needed, small angle approximation where $\sin(\theta) \approx \theta$, Taylor expansion of Snells' law to first order in $\lambda$ etc.  
> **username_3**: Hmm… I tried using the *lensmaker’s formula*, but the results I get would make it for a *short*-focus lens to have less chromatic aberration… I’m puzzled…  
> **username_0**: Yes, in my  bounty message I threaten to answer this but so far I have not attempted to solve what I claimed there (without evidence) to be a "pretty simple" problem. :-)  

---

