The Physics
Opus in profectus


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"Nothing is perfect" is a content-free statement. It's an excuse used over and over again to explain why things don't work out as intended. It's an explanation that explains nothing. There's no room in science for palliative blanket statements like this. Science is not the pursuit of perfection. Perfection is a dumb concept to begin with.

In optics, the deviation from perfection is called aberration. More precisely, an aberration is a deviation of a ray from the behavior predicted by the simplified rules of geometric optics. The primary rule referred to here is the one that states that rays of light parallel to the principal axis of a lens or curved mirror meet at a point called the focus. If your only options for a statement are that it is either true or false, then this statement is definitely false — as are many physical laws. If you can think beyond the law of the excluded middle (which itself isn't a law, it's a logical fallacy) then you can appreciate a real answer with more nuance.

For an ideal image-forming optical system there are two basic expectations.

  1. There is a one-to-one correspondence between points in the object space and points in the image space — that is, points map to points not circles, ellipses or blobs. Aberrations of this sort result in images that are described as blurry, fuzzy, or soft and edge details accompanied by a glow or halo.
  2. Straight lines in the object space correspond to straight lines in the image space. Aberrations of this sort result in images that look distorted.

Aberrations arise for one of two basic reasons.

  1. Chromatic aberrations are caused by dispersion (the variation in the index of refraction of a medium with frequency). Images with noticeable chromatic aberration are typified by edge details with noticeable colored halos or fringes.
  2. Geometric aberrations are caused by geometry (the shape of the lens or mirror). They are sometimes called monochromatic aberrations because they occur even for images formed with light of a single frequency. Images with noticeable geometric aberration are typified by poor focus (the image looks fuzzy) or distortion (the image turns straight lines into curves).

chromatic aberration

Chromatic aberration is a kind of defect commonly found in simple lens systems caused by a variation in the index of refraction with wavelength. Different frequencies (or wavelengths or colors) originating from the same object point follow different paths after passing through a lens. The result is an out of focus image that cannot be corrected by merely changing the placement of the lens (focusing).

Chromatic aberration comes in two types: axial (or longitudinal) and lateral (or transverse).

Axial chromatic aberration (or longitudinal chromatic aberration) occurs when a lens cannot focus different wavelengths of light in the same focal plane. The foci of the different colors lie at different points along the principal axis in the longitudinal direction. The result is a blurring of the out of focus colors that is pretty much equally annoying all across an image. For a single converging lens made out of a typical transparent material, the focal lengths are shorter for shorter wavelengths and longer for longer wavelengths. Moving away from the lens: the blues focus toward the front, the greens focus in the middle, and the reds focus toward the back. In the human eye, the spread in focal lengths is about 0.7 mm or more than twice the thickness of the retina.

Axial chromatic aberration

The images below show a simulation of axial chromatic aberration applied to a simple object, a black grid on a white background. This is what a camera would see if it only experienced axial chromatic aberration and the camera lens was adjusted so that the three primary colors are in focus one at a time.

  1. The circular region on the left has the screen placed at the blue focal point. The blues are in focus, the greens are slightly blurry, and the reds are very blurry. This results a blue halo inside the white squares and a yellow halo inside the black lines. Yellow in this case being a combination of green and red light (basically, everything that isn't blue).
  2. The circular region in the center has the greens in focus. The blues and reds bleed into the adjacent black, leaving a green halo in the white areas. The black lines get a halo of magenta (the opposite of green).
  3. The circular region on the right has the reds in focus, but the greens are slightly blurry and the blues are very blurry. The white squares have a halo of red and the black lines a halo of cyan (white light minus red).


The apparent three dimensional appearance of the album cover reproduced below reveals an optical illusion caused by axial chromatic aberration in the human eye. The extreme juxtaposition of the hot pink background against the neon green cutout of the band name forces the visual system to make a decision. If the brain decides the eye muscles should focus on the pink background, then the green cutout is out of focus. If the brain decides the eye muscles should focus on the green cutout, then the pink background is out of focus. That gives the artwork an apparent 3D appearance. For some people that makes the green appear to pop out, for others it makes the pink appear to pop out. If your brain can't decide which of the two colors to focus on, the eyes will then shift focus between them and the artwork will appear to shimmer with a frequency on the order of ten times a second (~10 Hz). The frequency of the shimmer gives you a sense of how long it takes the brain make to make a decision — the reciprocal of ten times a second, or one tenth of a second (~0.1 s = ~100 ms).

Album cover for illustrating chromatic aberration in the eye

Lateral chromatic aberration (or transverse chromatic aberration) occurs when image points formed by off axis rays are spread out away from the center of the image plane (in the lateral or transverse direction). This spreading is least apparent for rays parallel to the principal axis and increases with increasing angle. The result is a magnification that varies with color or, in technical language, a chromatic difference of magnification. Again, using the primary colors as examples: the blue frequencies produce the largest image, the green frequencies the mediumest, and the red frequencies the smallest.

Lateral chromatic aberration

The image below shows a simulation of lateral chromatic aberration applied to the same simple object as before, a black grid on a white background. This is what a camera would see if it only experienced lateral chromatic aberration. The result is fringes on high contrast areas — blue fringes blending into cyan on the inside edges and red fringes blending into yellow on the outside edges. The effect increases with distance from the center.


Lateral chromatic aberration does not occur in the human eye or, more accurately, if it does we don't notice it. I know from firsthand experience that it does occur in eyeglasses, however, but the effect varies with the material used. Price seems to be a determining factor. Since contact lenses rest directly on the cornea, any chromatic aberration is hard to detect.

To reduce chromatic aberration, a higher quality optical device would use a combination of lenses. The simplest such system consists of two lenses made of two different kinds of glass called an achromatic lens or an achromat. The most common acromat consists of a converging lens made of crown glass (the kind commonly used for drinking glasses and food jars) and a diverging lens made of flint glass (the slightly fancier kind of glass used in chandeliers and crystal decanters). The converging lens disperses the focal lengths one way and the diverging lens disperses them the other way canceling out some of the chromatic aberration. Flint glass has twice the dispersion of crown glass, so a converging crown glass lens of power +P paired with a diverging flint glass lens of power −½P will result in a reasonably good achromat with power P.


Technically speaking, an achromatic lens only aligns the focal points of the red and the blue source light waves to one another. It does not align the focal points of the red and the blue and the green to one another. To do that a third lens is needed. Such a system is called an apochromatic lens or an apochromat for short or, if you're in a real hurry, an APO. The lenses tend to be made of specialty materials. No more melted down jam jars and chandeliers. This one reason why expensive cameras are expensive.


Chromatic aberration can also be corrected for in digital cameras by computation. In color film cameras (and in the human eye) the locations of each of the three color images are fixed relative to one another. If the green image is in focus while the red and blue images are shifted and blurred in a film camera, well, too bad. You're stuck with it that way after you open the shutter and expose the film. If the same thing happens in a digital camera, well, it's all just numbers. Compute your way out of it. Not an easy problem to solve, but not an intractable one.

Some notes on chromatic aberration and vision I picked up along the way, mostly about the duochrome vision test — to be organized later.

Red-green duochrome eye test

spherical aberration


Spherical aberration

history or his story

The 17th century English scientist, mathematician, and theologian Isaac Newton was interested in the history of optical illusions. Is what we see there really there? To this end, he experimented on himself in a way that should never be repeated. When he was 24 years old, he inserted a bodkin (a blunt needle used to thread ribbon through lace) deep into the socket between his nose and eyeball.

Entry 58 from Newton's lab notebook described the one of these experiments. Spelling, capitalization, and punctuation rules were not well established in the 17th century, so some of this may look a bit odd to contemporary readers. Pen, ink, and paper were all difficult to come by (Newton had his own recipe for ink), so abbreviations were common as well. The letter "y" was often substituted for "th" so that "the" is written ye , "that" is written yt, and "them" is written ym.

Recreation of a page from Isaac Newton's experimental notebook

58 I tooke a bodkine gh & put it betwixt my eye & ye bone as neare to ye backside of my eye as I could: & pressing my eye with ye end of it (soe as to make ye curvature a,bcdef in my eye) there appeared severall white darke & coloured circles r,s,t, &c. Which circles were plainest when I continued to rub my eye with ye point of ye bodkine, but if I held my eye & ye bodkin still, though I continued to presse my eye with it yet ye circles would grow faint & often disappeare untill I removed ym by moving my eye or ye bodkin.

Pressing the side of the needle against his eyeball made colored circles appear in his field of vision at a point opposite that of the needle. These circles, which can be colored solid or take on animated geometric patterns, are an example of a visual phenomenon known as a phosphene — the sensation of light when there is no light — a mechanical phosphene in this case. Under normal circumstances, when the eye is being used for its intended purpose, light falls on the photoreceptor cells in the retina which causes them to become excited (formally) or fire (colloquially). In Newton's bodkin experiment, the photoreceptor cells were firing because they were being squeezed from behind. (Newton really wedged that thing deep into his eye socket according to his account.)

To confirm that the visions he was seeing were not formed by light, Newton repeated the experiment in a darkened room.

59 If ye experiment were done in a light roome so yt though my eyes were shut some light would get through their lidds There appeared a greate broade blewish darke circle outmost (as ts), & wthin that another light spot srs whose colour was much like yt in ye rest of ye eye as at k. Within wch spot appeared still another blew spot r espetially if I pressed my eye hard & wth a small pointed bodkin. & outmost at vt appeared a verge of light.

Then he did something really dumb (as if sticking a needle into your eye socket wasn't dumb enough). He stared at the Sun — maybe. He was hopefully more sensible and stared at a bright patch of the Sun's light projected onto a wall. Staring at a bright light source overstimulates the photoreceptor cells in the retina. This reduces their sensitivity, which is a response that allows our visual system to adapt to surroundings with different brightnesses. When the bright source is removed, the overstimulated photoreceptors are now under-sensitive (a word I just made up). The human visual system is complicated, so there's a bit more to it than that. Let's just say that staring at a bright light screws up your eyesight for a while.

63 Looking on a very light object as ye Sun or his image reflected; for a while after there would remaine an impression of colours in my eye: viz: white objects looked red & soe did all objects in the light but if I went into a dark roome ye Phantasma was blew.

We would call this thing that Newton saw an afterimage, but at the time that word did not exist and Newton was not the one to invent it. Instead he used the word phantasma (φαντασμα in Greek) which is a variation on the word phantasm or phantom — in other words, a ghost or at least something ghost-like. It's ingenious and imaginative, but also a bit otherwordly.

The reason Newton did these experiments on himself wasn't because he was some thick headed frat boy. Rather, he was fascinated by the difference between objective reality and illusion (or even delusion). One of the ways we can be fooled is in the perception of color. Newton showed through a series of now famous experiments using glass prisms that white light, which up to that point was thought to be the purest form of light, is actually a blended form of light with different colors.

Line drawing of Newton's prism experiment

7 Taking a Prisme, (whose angle fbd was about 60gr) into a Darke roome into wch ye sun shone only at one little round hole k, and laying it close to ye hole k in such manner yt ye rays, being equally refracted at (n & h) their going in & out of it, cast colours rstv on ye opposite wall. The colours should have beene in a round circle were all ye rays alike refracted, but their forme was oblong terminated at theire sides r & s wth straight lines; theire breadth rs being 2⅓inches, theire length to about 7 or eight inches, & ye centers of ye red & blew, (q & p) being distant about 2¾ or 3 inches. The distance of ye wall trsv from ye Prisme being 260inches.

What Newton saw projected on the wall of his darkened laboratory looked something like this.

A continuous spectrum running horizontally across the page

Near the end of Entry 6 in his notebook, Newton called it a "phantom".

And looking on it through the Prisme, it appeared broken in two twixt the colours, the blew parte being nearer the Prisme than the red parte. Soe that blew rays suffer a greater refraction than red ones. I call those blew or red rays &c, which make the Phantome of such colours.

Six years later, when he described the prism experiment in a public letter to the Royal Society, Newton had begun the transition from the Greek loanword "phantasm" to the Latin loanword "spectrum". This is the first written example of the word spectrum with its current meaning.

Comparing the length of this coloured Spectrum with its breadth, I found it about five times greater; a disproportion so extravagant, that it excited me to a more then ordinary curiosity of examining, from whence it might proceed….

He did not completely abandon the original word phantasm, however.

But, to determine more absolutely, what Light is, after what manner refracted, and by what modes or actions it produceth in our minds the Phantasms of Colours, is not so easie. And I shall not mingle conjectures with certainties.

Both words had similar meanings in the 17th century — something ghostly or not of this world. Much like spelling and punctuation, scientific terminology wasn't systematized in the 17th century. It may well have been seen as a mark of proficiency to mix up spellings, punctuation placements, and word choices. (This was about the time when the thesaurus was invented after all.) In the 21st century, however, scientific terminology is reasonably well organized and consistent and, for unrelated reasons, the word spectrum has lost all its supernatural connotations.

The spectrum that Newton first saw and then named is a colored band of light produced when a source of mixed light has been decomposed or broken up into components and sorted into a characteristic sequence — sorted by frequency, it was later determined. It is a real thing and is not an optical illusion or mental delusion.

Because Newton was a bit of a mystic and seven is a number with mystical connotations, he divided the spectrum up into seven named segments giving primary school children everywhere something to memorize. He identified these as the "primary colors" but later experiments have shown this notion to be wrong. (Sorry primary school children.) The preferred term now is spectral colors or prismatic colors for the things Newton was naming. (The primary colors of red, green, and blue are discussed elsewhere in this book.) There are also many more than seven distinguishable colors of light in the visible spectrum — a point Newton makes clear near the end of this quotation.

red orange yellow green blew indico violet-purple

There are therefore two sorts of colours. The one original and simple, the other compounded of these. The Original or primary colours are, Red, Yellow, Green, blew, and a Violet-purple, together with Orange, Indico, and an indefinite variety of Intermediate gradations.

Newton produced his spectrum by refraction (the change in direction of a wave through a medium associated with changes in the wave's speed) or more precisely dispersion (the variation of a wave's speed in a medium with frequency). All transparent media are dispersive to some degree. Therefore any optical system that uses refraction to do what it needs to do will also experience dispersion. If the goal of your optical system is to produce a spectrum, then dispersion is a fine thing. If the goal of your optical system is to produce a reliable image, to "see" something for what it really is, then dispersion is a problem.

Maybe dispersion could be reversed. Newton tried a second prism as a part of an "error correction" experiment. Disperse the light with one prism then un-disperse it with a second to see if there were any distortions caused by impurities or irregularities in the glass.

Then I suspected, whether by any unevenness in the glass, or other contingent irregularity, these colours might be thus dilated. And to try this, I took another Prisme like the former, and so placed it, that the light, passing through them both, might be refracted contrary ways, and so by the latter returned into that course, from which the former had diverted it. For, by this means I thought, the regular effects of the first Prisme would be destroyed by the second Prisme, but the irregular ones more augmented, by the multiplicity of refractions. The event was, that the light, which by the first Prisme was diffused into an oblong form, was by the second reduced into an orbicular one with as much regularity, as when it did not at all pass through them. So that, what ever was the cause of that length, 'twas not any contingent irregularity.

Dispersion is a one way street. This realization caused Newton to rethink his work in optics. No optical device would ever be able to produce a "true" (for lack of a better word) image if it relied on refraction. It would suffer from what we now call chromatic aberration — initially collinear rays of light would follow different paths depending on their color. There would be no way for all the colored rays of an image to be in focus together. Newton was interested in astronomical telescopes at the time.

When I understood this, I left off my aforesaid Glass-works; for I saw, that the perfection of Telescopes was hitherto limited, not so much for want of glasses truly figured according to the prescriptions of Optick Authors, (which all men have hitherto imagined,) as because that Light it self is a Heterogeneous mixture of differently refrangible Rays. So that, were a glass so exactly figured, as to collect any one sort of rays into one point, it could not collect those also into the same point, which having the same Incidence upon the same Medium are apt to suffer a different refraction.

The way around this is to eliminate at least one of the lenses from the telescope (the bigger lens, the one that faces the stars, the objective lens) and replace it with a mirror.

[telescopes illustration]

All rays of light obey the law of reflection in the same way, regardless of their color. Problem solved. Newton even understood that the mirror needed to be ground and then polished with a parabolic curvature to eliminate spherical aberration — the inability of a spherical surface to bring rays far from its axis into proper focus. He most certainly didn't do this, however, as the method of grinding a parabola is much more complicated that that of grinding a sphere. (Optical devices with curved surfaces are usually ground into the desired shape instead of being cast or molded.)

This made me take Reflections into consideration, and finding them regular, so that the Angle of Reflection of all sorts of Rays was equal to their Angle of Incidence; I understood, that by their mediation Optick instruments might be brought to any degree of perfection imaginable, provided a Reflecting substance could be found, which would polish as finely as Glass, and reflect as much light, as glass transmits, and the art of communicating to it a Parabolick figure be also attained.

This was Newton at 30 reflecting back on thoughts he had when he was 24. It took that long for the reflecting telescope to go from concept to working prototype. (The bubonic plague didn't help things much.)

Amidst these thoughts I was forced from Cambridge by the Intervening Plague, and it was more then two years, before I proceeded further. But then having thought on a tender way of polishing, proper for metall, whereby, as I imagined, the figure also would be corrected to the last; I began to try, what might be effected in this kind, and by degrees so far perfected an Instrument (in the essential parts of it like that I sent to London,) by which I could discern Jupiters 4 Concomitants, and shewed them divers times to two others of my acquaintance. I could also discern the Moon-like phase of Venus, but not very distinctly, nor without some niceness in disposing the Instrument.

The reflecting telescope was a success. Not only did Newton exhibit great theoretical insight when it came to optics, but he also demonstrated that he could apply his theoretical knowledge to practical applications. He was accepted as a Fellow of the Royal Society that year. The prototype telescope he sent them is still in their archives. It is the telescope more than anything else that ushered Isaac Newton on to the public stage of 17th century science — more than his work on gravity, the laws of motion, or the invention of calculus.

geometric aberrations




field curvature