I’ve been hard at work in recent months on a new textbook on electromagnetic waves, and that has led me to dig deep into understanding a number of subjects, and their history, that I have only really casually considered in the past. One topic that jumped out at me is the phenomenon of so-called “anomalous” dispersion. This name indicates that it is different from “normal” dispersion, which is the phenomenon that causes a prism to break up white light into a rainbow, as illustrated on Pink Floyd’s Dark Side of the Moon album cover:

In a prism, the angle at which light is refracted by the glass depends on the frequency of light, i.e. its color. In normal dispersion, the higher frequency light (blue and violet) is refracted more strongly than the lower frequency light (red and yellow). In anomalous dispersion, roughly the opposite happens: lower frequencies are refracted more than higher frequencies!

I thought it would be fun to talk a bit about the history of anomalous dispersion, which gets at some of the physics of matter, and also allows me to share lots of pretty pictures taken by Robert Williams Wood, who studied the phenomenon in the early 1900s!

To begin, let’s share some basics. When light passes from one medium to another, it changes its direction of propagation, in the phenomenon known as refraction. This is illustrated below.

Refraction can be loosely interpreted as arising from the change of speed of the wave as it passes into the new medium. To characterize this, scientists introduced the index of refraction, usually labeled n, which is the amount by which the speed of light is reduced.

For example, if we write the vacuum speed of light as c, then the speed of light in a medium is c/n, where n is a property of the specific medium. For water, n = 1.33, while for most glasses, n is around 1.5. In glass, then, light travels at roughly 2/3rds the vacuum speed of light.

This picture is too simple, however, because as we have noted the refractive index depends on the frequency (color) of the light in question. This means, as we have noted above, that different colors refract at different angles — are “dispersed” — when passing through a glass prism. Isaac Newton used this phenomenon of dispersion to argue, correctly, that white light is in fact a combination of all colors of light.

From the time of Newton (1700) until the mid-1800s, the dispersion properties of many materials were studied, and it was generally found that the refractive index increases with increasing frequency (or, equivalently, the refractive index decreases with increasing wavelength). We can see this in the Pink Floyd album cover: violet (high frequency) is more deflected (refracted) than red (lower frequency). The increase of index with wavelength is very small, usually a few percent over the visible light frequencies, which is why we often simply refer to a material like water as having a single refractive index whose value is somewhere around 1.33.

In 1862, however, the French physicist F.-P. Leroux was studying the refractive properties of iodine vapor, and he noticed something peculiar¹. Iodine is for the most part strongly absorbing throughout the visible light spectrum, with the exception of the extreme reds and violets. In his own words, Leroux found,

Iodine vapour disperses light in different direction to any substance yet studied; that is, a prism full of iodine vapour refracts red rays to a greater extent than blue rays.

It was not lost on Leroux, or others, that this “abnormal” dispersion (as he called it) occurred in the vicinity of wavelengths where light is absorbed by the material. This led other researchers to investigate the wavelength dependence of the refractive index near absorption lines; the next result was found by the Danish physicist Christian Christiansen, who in 1870 found anomalous dispersion in the magenta dye fuschine². (Christiansen would go on to have a small role in the history of invisibility physics, based on his dispersion research.)

It is worth saying a little bit about the nature of absorption before we continue. Due to their quantum nature, atoms emit and absorb radiation at discrete wavelengths, or “spectral lines.” This was first recognized by Joseph Fraunhofer in 1814 in observing the spectrum of light from the sun, as illustrated below.

Light gets emitted by the sun over almost all of the wavelengths (colors) of visible light, but Fraunhofer noted that there are individual wavelengths where the sun emits no light whatsoever; these dark “lines” were later found to represent the wavelengths at which atoms in the sun itself absorb the radiation that the sun is generating. In fact, when light is passed through any relatively dilute gas, one finds that the gas absorbs at a set of spectral lines characteristic of the material; these lines form a sort of “fingerprint” that can be used to identify the material itself. When atoms coalesce into liquids and solids, their interactions become very different and the lines are replaced with more complicated absorption spectra.

Iodine has a significant number of absorption lines throughout the visible spectrum, effectively cutting off most light transmission except for the reds and violets. Ordinary glass has no absorption lines in the visible spectrum, which is why it is transparent and clear.

Speaking of “clear,” it was clear to scientists of the time that anomalous dispersion was telling them something profound about the nature of the atom, and they dedicated significant effort to understand it. The next significant work was done by the German physicist August Kundt, who released a series of papers in 1871 announcing the discovery of anomalous dispersion near the absorption lines of a variety of substances³. In the third of his series, Kundt presented:

the conjecture that the gases also, which sometimes possess so energetic an absorption
for certain kinds of rays must exhibit anomalies of dispersion in the vicinity of these rays.

Kundt also created an absolutely gorgeous way to experimentally study the phenomenon of anomalous dispersion, which also was the inspiration of this post! His method is known as the method of crossed prisms, and it is, in essence, exactly what the name says.

My rough illustration of the idealized experiment is shown in the figure below. Imagine sending a beam of white light through an ordinary glass prism that is oriented vertically. This will disperse the light, like on the Pink Floyd cover, with all the colors spread out along the horizontal axis. Now let that second prism be filled with the substance whose dispersion you wish to study, and oriented so that light hits the surface at an angle. The light of a given frequency will be deflected upwards or downwards depending on the refractive index at that frequency, and the resulting image projected on a screen will effectively be a plot of the refractive index as a function of frequency!

The first actual illustration of the output of such a system was done by Kundt for sodium vapor in 1880⁴, and his figure is below.

In this image, you can see the anomaly in refractive index: though the line is mostly horizontal from red (low frequency) to blue (high frequency), there is an abrupt change around “D”, where the index increases up to an absorption line (vertical axis is reversed) and then seems to change quickly, dropping dramatically in value and then increasing again past the “D”. The dark break represents the absorption of light, so we can’t directly see what the refractive index is doing in that region.

Get used to this shape — we’ll be seeing it a lot!

Why study sodium in particular? I have noted that substances like iodine have a large number of absorption lines in the visible spectrum, making it exceedingly difficult to study the refractive index near them. Kundt himself wondered,

But whether we shall ever succeed in demonstrating the refraction-anomalies in each single absorption-band of the gases and incandescent vapours, some of which show so great a number of thin absorption-lines, must be left undecided.

Sodium, however, is a remarkable substance that has what appears to be an isolated absorption line in the orange part of the spectrum; in fact, there are two closely-spaced absorption lines present, that were labeled the “D” lines by Fraunhofer. With no other apparent absorption lines nearby to confound or complicate measurements, they evidently became the ideal laboratory test subjects for anomalous dispersion.

Kundt had an interesting challenge at the time in preparing his experiment. At room temperature and pressure, sodium is a solid; to get sodium gas, it must be burned, but in Kundt’s time there was no way to do this in a prism. Kundt’s solution was simple and ingenious:

By means of electric light a horizontal intensely bright spectrum was projected, through a prism with the edge vertical, upon the screen. In the path of the rays a Bunsen burner was placed, and with a small iron spoon a piece of sodium introduced into it. If the spoon is brought exactly into the middle of the flame of the Bunsen burner, it is easy to maintain the flame above it as a cone of intensely yellow brightness. Now this cone acts like a prism with horizontal refracting angle turned upwards.

In short: burning sodium created a vertical cone of gas that worked approximately as a prism inverted with respect to the diagram I drew above!

Experiments were improved by others. In 1898, none other than Henri Becquerel, discoverer of radioactivity, studied refraction around the D lines himself⁵. I have yet to translate his full paper, but he appears to have measured the refractive index anomaly between the two sodium D lines, and also measured the variations in the effect based on what part of the sodium flame was analyzed.

This brings us, more or less, to Robert Williams Wood, whose book inspired me to write this whole post! I’ve talked a lot about Wood before, who was a phenomenal and influential physicist and more. Most recently, I discussed his early contribution to the physics of invisibility, which of course fascinated me because of my whole book and stuff. He also coauthored a pair of science fiction novels, The Man Who Rocked the Earth (1915) and The Moon-Maker (1916), both of which were ahead of their time. He was also an avid lecture demonstrator, and I’ve posted before about his published lecture demos. I have yet to write about his most famous discovery, known as Wood’s anomalies, which are curious phenomena associated with diffraction gratings (though I need to understand them myself before blogging about them).

Wood took up the analysis of sodium vapor_⁶ in 1902. He appears to have been motivated by two factors. First, he was interested in verifying the so-called Sellmeier equation that was introduced in 1872. Sellmeier introduced an empirical formula to model the refractive index of transparent substances, based on the assumption that the refractive index of a material is largely determined by the nature of its absorption lines. The equation had been tested for solids, but solids were expected to be an imperfect test of Sellmeier’s formula due to the strong interaction between atoms of the solid. Sodium, with its gaseous form and simple and strong absorption structure, appeared to be an ideal way to test Sellmeier’s result.

Second, Sellmeier’s formula could not yet be tested using existing sodium measurements because all researchers to date had only studied the refractive index very close to the D lines; Wood wanted to measure the index over a broader range of frequencies.

I don’t want to dwell on the details of Wood’s results, but rather the way he depicted them. Wood had a tools available that were relatively new to the scientific community: photographs and color plates. He also introduced a modified experimental setup to produce cleaner results; his illustration is given below.

I am a theorist, and not as well-versed in experimental techniques as I would like, but my understanding of Wood’s experiment is that he, in essence, reversed the order of the prisms in the crossed prisms experiment. Light from an electric arc lamp (on the right) was collimated by a lens and a horizontal slit, to produce a flat horizontally-propagating beam of white light. This beam passed through a tube with sodium inside, and each of the five Bunsen burners in the system heated up the sodium and basically produced a prism-like shape of sodium gas! Wood’s illustration of this, from his later Physical Optics book, gives the idea:

So, after passing through the tube, each color of the spectrum is inclined upwards or downwards, depending upon the refractive index it experienced. Then, after leaving the tube, this separated spectrum could either be viewed directly, or it could be put into a spectroscope and separated out to give the same sort of colored plot of refractive index that the crossed prisms method did.

Wood’s figure of the separated spectrum, taken from his paper, is shown below. This is a drawing that attempts to capture the appearance of the colored spectrum.

This drawing is in essence a rough map of the refractive power of different colors. The center violet band matches the slit and indicates that the refractive index is nearly that of vacuum. Red, below the center, suggests a lower index, while green suggests a higher one. The drawing is only an approximation of what Wood was seeing in the lab; as he said, “it is quite impossible to represent by means of pigments the sparkling brilliancy of the colours.”

Wood took photographs of all his results, but these were in black and white and it is the colored drawings that he did that really stand out. Next, we show you the colored drawings of the refractive index as a function of frequency, from the 1905 edition of Wood’s Physical Optics:

The two different spectra shown are from different densities of sodium gas. In the lower figure, the temperature of the burners is higher, and one sees significant changes in the spectrum. The D lines are still present, but there are additional dark lines — absorption resonances — in the green at low temperatures, and also in the red at high temperatures.

Just to wrap up the images, here’s one of the black and white photographs of the D lines from Wood’s book.

For purposes of this blog post, one big question remains: what is the origin of this sharp change of refractive index, this anomalous dispersion, around the absorption lines? By Wood’s era, the question had already been at least partly answered by the Dutch scientist Hendrik Antoon Lorentz. In the late 1800s, he introduced a simple model of an atom to describe light-matter interactions. In short, he assumed that electrons are trapped in the neighborhood of atoms by a simple harmonic oscillator force, like the force a spring exerts. A simple harmonic oscillator oscillates at a fixed frequency that is determined by the mass of the electron and the force of the spring. Further assuming that the electron oscillations are damped, i.e. that they lose energy as they oscillate, he was able to produce a model for the refractive index of a material near an absorption line that matched the experimental observations of anomalous dispersion stunningly well. An image of the theoretical refractive index n_R and the strength of absorption n_I around an absorption line is shown below. Looking at the refractive index and comparing it to Wood’s photograph above, you can see the striking similarity between the images. In both cases, the refractive index shoots up as one approaches the resonance frequency, then it plummets across that frequency and then shoots back up on the other side. Wood and others could not see the “plummet” because it occurs at frequencies where light is strongly absorbed.

It should be noted that this plummet explains the early anomalous decrease of refractive index by frequency observed by Leroux and others. Far from an absorption line (to the left and right of the figure above), the refractive index is always increasing with frequency. However, because of the plummet, if one compares the refractive index on the immediately left and right of absorption, one can see that it appears to have decreased in value — and it has, but not in a smooth way like it increases.

Lorentz’s theoretical work demonstrated that the strange features observed in the refractive index could be explained by light waves interacting with electrons in atoms, and that those electrons have characteristic frequencies of vibration. Exactly why electrons have characteristic frequencies of vibration could not be explained until the birth of quantum mechanics… which is another story.

The sodium D lines would continue to play an important role in physics. It turns out that atoms will emit light at the same frequencies at which they absorb light. By exciting sodium electrically, one gets a light source with a very pure spectral content: the sodium radiates at the wavelength of the sodium D lines, at about 589 nanometers. Sodium lamps therefore became one standard source of light where the wavelength of emission is very precisely known, and this source has been used for all sorts of optical measurements. When one looks up old tables of refractive index values by wavelength, for example, one will almost always find measurements have been taken at 589 nanometers, because they were measured with a sodium lamp.

To conclude, I can’t help but share one really cool consequence of the fact that matter absorbs and emits light at the same characteristic frequencies. Suppose we burn some sodium in a fire. That sodium will emit light at the D line, which gives it a characteristic orange glow. But once it has emitted that light and cooled, it can reabsorb light at that same D line frequency. If you illuminate a room with a sodium lamp, and burn sodium in a fire, you will get a black flame!

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1. F.-P. Leroux, “Researches on the refractive indices of bodies which only assume the gaseous condition at high temperatures. Abnormal dispersion of iodine vapour,” Phil. Mag. 24:160 (1862), 245-247.
2. C. Christiansen, “Über die Brechungsverhältnisse einer weingeistigen Lösung des Fuchsins,” Ann. Phys. 217:11 (1870), 479-480.
3. A. Kundt, “Nachtrag zum Aufsatz: “Ueber die anomale Dispersion der Körper mit Oberflächenfarben,”” Ann. Phys. 219:5 (1871), 149-152.
4. A. Kundt, “On anomalous dispersion in incandescent sodium vapour,” Phil. Mag. 10:59 (1880), 53-57.
5. H. Becquerel, “Sur la dispersion anomale et le pouvoir rotatoire magnetique de certaines vapeurs incandescentes,” Comptes Rendus 127 (1898), 899-904.
6. R.W. Wood, “The anomalous dispersion of sodium vapor,” Proc. Roy. Soc. Lond. 69 (1902), 157-171.