It is well-known that light of two different frequencies illuminating a detector will produce an output with a component at the difference frequency. While such considerations are eminently useful (various heterodyne measurement techniques) it has not been practical until lasers came along. Regardless of practicality, the effect has been known for a long time.

The question is: how long? And, specifically, can you provide a reference to an analysis which establishes it? A web-accessible reference, for preference. I'd expect it was determined shortly after Maxwell's equations were established, but I could be wrong. Ideally, this would be an acknowledged seminal paper, comparable to Skutt's explanation of the color of the sky.

  • Please note that I'd like a pre-1900 reference for the bounty. – WhatRoughBeast Aug 1 '14 at 12:48
  • Are you specifically asking about visible light? – Floris Aug 5 '14 at 2:33
  • @Floris Yes. Young, for instance, knew about beat frequencies in sound, and was aware that interference fringes require both sources to be the same color (monochromatic, in his terms), so it's not unreasonable to think he realized that different colors would show a beat frequency too high to be distinguished, but that doesn't mean that he understood sum and difference production. Unless, of course, someone can find a quote which clearly states that he (or Fresnel, for instance - another likely candidate) actually did. Implication is not enough. – WhatRoughBeast Aug 5 '14 at 3:20
  • Poisson might be another candidate. I am sure you are aware of the origins of Poisson's spot… I love that story. But yeah - it is unlikely that a Poisson reference will win this bounty. – Floris Aug 5 '14 at 3:26
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    @WhatRoughBeast found a quote from a Rayleigh paper in 1889 – alemi Aug 9 '14 at 5:45
up vote 16 down vote

I've found some sources.


To start with, as for the mathematical notion of "beats", it seems that one Ibn Yunus (c. 950-1009) was responsible for first demonstrating the trigonometric identity

$$ \cos a \cos b = \frac 12 \left( \cos ( a + b) + \cos (a - b) \right ) $$

quoting A History of Mathematics By Carl B. Boyer, Uta C. Merzbach

At least one of these, that converting a product of cosines to a sum of cosines has been known to the Arabs in the time of ibn-Yunus [...]

and from Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1 By Ari Ben-Menahem

Ibn Yunus of Cairo (d. 1008), Alhazen's contemporary and countryman (they both lived in Egypt), introduced the formula $$ 2 \cos x \cos y = \cos (x + y ) + \cos (x - y) $$

but you don't seem to be interested in math, though one could say this lays the foundation for everything to follow.

Galileo and the frequency of sound

Jumping ahead, people knew about dissonances in the acoustical realm for a long time, but the first quantitative discussion of the phenomenon of beats seems to be due to Galileo, late in the first day of his Dialogues Concerning the Two World Systems in which we have a discussion suggesting that sound is an oscillatory wave, and the harmonies and dissonances people hear can be understood by their commensurate or discommensurate frequencies.

Salviati: [...] I assert that the ratio of a musical interval is not immediately determined either by the length, size, or tension of the strings but rather by the ratio of their frequencies, that is, by the number of pulses of air waves which strike the tympanum of the ear, causing it also to vibrate with the same frequency. This fact established, we may possible explain why certain pairs of notes, differing in pitch produce a pleasing sensation, others a less pleasant effect, and still others a disagreeable sensation. Such an explanation would be tantamount to an explanation of the more or less perfect consonances and of dissonances. The unpleasant sensation produced by the latter arises, I think, from the discordant vibrations of two different tones which strike the ear out of time [sproporzionatamente].

[... long fairly confusing description (though quantitative) follows ...]

Sagredo: I can no longer remain silent; for I must express to you the great pleasure I have in hearing such a complete explanation of phenomena withe regard to which I have so long been in darkness. Now I understand why unison does not differ from a single tone; I understand why the octave is the principal harmony, but so like unison as often to be mistaken for it and also why it occurs with the other harmonies. It resembles unison because the pulsations of strings in unison always occur simultaneously, and those of the lower string of the octave are always accompanied by those of the upper string; and among the latter is interposed a solitary pulse at equal intervals and in such a manner as to produce no disturbance; the result is that such a harmony is rather too much softened and lacks fire. But the fifth is characterized by its displaced beats and by the interposition of two solitary beats of the upper string and one solitary beat of the lower string between each pair of simultaneous pulses; these three solitary pulses are separated by intervals of time equal to half the interval which separates each pari of simultaneous beats from the solitary beats of the upper string. Thus the effect of the fifth is to produce a tickling of the ear drum such that its softness is modified with sprightliness, giving at the same moment the impression of a gentle kiss and of a bite.

So, at the very least, we have some quantitative discussion of the phenomenon of beats in the acoustic realm as early as 1632.

Newton: Frequency of Light

The first person to hypothesize a frequency type relationship with light was Newton

May not the harmony and discord of colours arise from the proportions of the vibrations propagated through the fibres of the optic nerve into the brain, as the harmony and discord of sound sarise from the proportions of the vibrations of the air?

(From Newton's Opticks Qu. 14. [gutenberg] )

and he also made some early observations of interference phenomenon as well. That Young will reference to build his case.

Young and Light

Skipping forward a bit, we come to 1802 and Young's (hattip to Trimok) experiments on the interference of light. He gave a long lecture to the philosophical society of london in which he gives a nice historical overview of light experiments up to that time, and proposes that all known phenomenon can be explained with a wave theory of light.

A year later in 1803, and in particular his report on his experiments on the interference of light to the royal society where we have some reports on the interference of light, and an analogy made to acoustic phenomenon. His experiment was the original double split experiment, in which he observed interference fringes causes by white sunlight. Upon closer examination, he concluded that the white bands were actually a mixtures of all of the colors, and could see slight color variations across the bright bands themselves.

His sketch of interference of water waves, to illustrate the phenomenon: Young's sketch of interference in water waves (From wikipedia)

His full lectures shows that he had a good understanding on the phenomenon, which is ultimately a result of the difference phenomena, but in the spatial domain instead of the frequency domain. Just as the difference phenomenon in the frequency domain can be used to take two nearby signals and through their difference create a low frequency signal that is easier to detect, Young's real genius was to use two small slits as sources of spherical waves, and use their small spatial separation to enable him to make a quantitative measure of the very small wavelength of light. In modern notation, on the screen, he was observing

$$ 2 \cos k\ell_1 \cos k\ell_2 = \cos\left( k (\ell_1 + \ell_2) \right) + \cos \left( k ( \ell_1 - \ell_2 ) \right) $$ Where $\ell_1$ is the distance the light travelled from one hole to a particular point on the screen and $\ell_2$ is the distance the light travelled from the other hole. This enabled him to make a precise measurement of $k$, since $\ell_1 - \ell_2$ was very tiny, you could hope to see results of the very tiny wavelength of light (~500 nm). By simply looking at different positions on the screen, up and down, you vary the $\ell_1 - \ell_2$ and so for a distant screen you see fringes where $$ \theta \sim \frac{\lambda}{D} $$ where $\theta$ is the angle the part of the screen makes with the point half way between your two slits.

Its really worth appreciating how genius the apparatus is. You use two interacting sources of light (from the slits), so that by the difference phenomenon you can create an even smaller distance than you can build directly. Then you take advantage of geometry to take small angles and amplify them to measureable distances by just moving your observing plate backwards.

Young understood all of this and his lecture has the details all worked out. From his Backerian lecture in 1803:

On the supposition that the dark line is produced by the first interference of the light reflected from the edges of the knives, with the light passing in a straight line between them, we may assign by calculating the difference of the two paths, the interval for the first disappearence of the brightest light [...] the second bright line being supposed to correspond to a double interval, the second dark line to a tripler interval, and the succeeding lines to depend on a continuation of the progression.

Here I reproduce his table:

Young's tables of calculated values

In which he calculates the the value of 0.0000149 inches, or 378 nm of light. Which is pretty good for 1803 if you ask me.

He goes on to draw a strong analogy with the difference phenomenon in sound:

... The advocates for the projectile hypothesis of light, must consider which link in this chain of reasoning they may judge to be the most feeble; for, hitherto, I have advanced in this Paper no general hypothesis whatever. But, since we know that sound diverges in concentric superficies, and that musical sounds consist of opposite qualities, capable of neutralizing each other, and succeeding at certain intervals, which are the different according to the difference of note, we are fully authorized to conclude, that there must be some strong resemblance between the nature of sound and that of light.

So, by 1803, Young is directly demonstrating his knowledge of the difference phenomenon in acoustics, and attempting to apply it to help explain his observations in interference. He also suggests it is well known by this time in the acoustic realm.

Savuer and Smith: Young's inspiration in sound

I then tried to find more discussion of the phenomenon of beats in acoustics, and the earliest reference I could find dicussing the phenomenon in modern algebraic terms is in Harmonics, or The philosophy of musical sounds. By Robert Smith published in 1759. In which in Section VI, he gives a proper mathematical treatments of beats, and dicusses how if two sounds are nearby in frequency, it causes an apparent sound with frequency given by the difference in their frequencies. Its a bit hard to parse as the language is very particular, but just a snippet to show he's onto the right thing:

Coroll. 4. Hence as musical intervals are proportional to the logarithms of the ratios of the single vibrations of the terminating sounds ($k$), if any part or parts of the comma $c$ denoted by $ \frac q p c $, be the interval of imperfect unisons, the ratio of the times of their single vibrations will be $161 p + q $ to $ 161 p - q $.

It would also appear that Lagrange in 1758 in his Miscellanea Taurinensia discusses acoustics and in particular

The article concludes with a masterly discussion of echoes, beats, and compound sounds.

Though I haven't managed to find English translations, so I can't confirm, the french seems to be available here.

Another guy active in acoustic beats was Joseph Sauveur, who presented his work to the French Academy in 1701, though I can't find much of the primary sources in english, historical accounts confirm that he worked on this as well. Smith's Harmonics mentions Sauveur's work, but thinks he was confused about some detail about beats. There is also evidence that Young had read Sauveur's, as well as Smith's work, so it could explain his familiarity with the phenomenon.

To quote from Sounds of Our Times: Two Hundred Years of Acoustics By Robert T. Beyer

Sauveur combined a knowledge of the ratios of musical tones with the recently observed phenomenon of beats to make it possible to determine the actual frequencies of the tones. Sauveur considered two organ pipes whose rather low tones differed by a half-tone, standing in the ratio of 15 to 16. Sauveur was able to count six beats when the two pipes where sounded simultaneously and therefore assigned them the frequencies of 90 and 96 oscillations per second.

Demonstrating that he had mastery of the difference phenomenon.

Historic Summary

So, to summarize a bit. The optical phenomenon was first suggested by Young in 1801 (and observed in the spatial domain in his double slit experiments), in analogy to beats in acoustics. Acoustical beats seem to have their first quantitative treatment that I could find in Galileo's Dialogues in 1632, though the discussion there is very old timey in terms of ratios and the like, and the first modern algebraic treatment was probably Sauveur in 1701, though the most recent in english I can find is Smith's in 1759. Of course, the primary mathematical identity responsible was first worked out sometime near the year 1000 by Ibn Yunus.

Jumping forward in time past Maxwell and Hertz and the like to near the turn of the century (~1900), a few sources make it clear that scientists of the time completely understand difference frequencies.


In On the Limit to Interference when Light is radiated from Moving Molecules By Lord Rayleigh in 1889 [doi], Lord Rayleigh calculates what the title suggests.

We find the following:

A section from Rayleigh's paper


I just found the book Light Waves and Their Uses by Albert Abraham Michelson, published in 1903 although in the preface it says it is based on lectures given in the spring of 1899 at Lowell Institute. It is available for free on google books as well as

The book itself is a popular introduction to light and its many uses, as the title suggests.

In chapter 1: Wave Motion and Interference. Amazingly, the chapter follows much of the progression of this post. Michelson introduces the audience to a wave theory of light by analogy to sound and water waves. He discusses interference of all forms, again by analogy to sound, and offers this picture:

Michelson I

As he discusses interference phenomenon and the difference phenomenon in words.

Later in Chapter IV: Application of Interference Methods to Spectroscopy, he discusses using an interferometer to determine the wavelength of monochromatic light generated from a sodium lamp (or any other element) sent through a prism.

Michelson 2: Sodium lamp

He goes on to give a description of how one can use interference to measure the difference in frequency of the two lines in the sodium doublet:

Michelson and the sodium doublet

He goes on to give an introduction to the use of a harmonic analyzer (a crazy mechanical fourier analysis computer thing) (also the subject of a recent set of youtube video lectures) and presents results generated for various spectral line combinations:

Michelson Harmonic Analyzer Results

There is plenty more, but I think I've made the point. Notice that this is all from lectures Michelson gave in 1899, and presented to a popular audience, suggesting this is all well known by the time. It additionally features results generated from experiments on monochromatic light.

The book is actually a very entertaining read and I highly recommend it after looking at it. I will also point out that the descriptions Michelson gives to explain how the difference frequency can be used in experiment relies heavily on references to historical accounts (most of the ones I mention above) and draws heavily on analogies with sound. It is clear that he and his contemporaries fully understood the implications the difference frequency and suggests that science as a whole came to understand it much in the way I've described, as most things in science it was an incremental process building upon the works of many great contributors, spread across centuries, all unified in their desire for understanding.

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    You are very thorough. And all that for unicorn coins. A scholar. – Floris Aug 10 '14 at 18:00

It seems that Thomas Young (1773-1829) was one of the first who where interested in beats and interferences, and in fact, it seems that the concept of beats leads him to the concept of interferences (around $1801$), see page $92$ of this reference (in french only). For instance, it is said (page $92$), (traduced in English by google):

This is the phenomenon of beats which seems to have suggested to the first Young idea of the interference vibrations. Undulations where the beats are not the result of the same origin or period; but if the periods are slightly different, these vibrations are alternately in the same favorable conditions and their capacity to their mutual weakness, and these adverse effects are susceptible to ear

This phenomenon is not special to light or Maxwell's equations: it's a simple consequence of detector nonlinearity and I should think that this would have been pretty clear to any bright experimentalist who thought carefully about how his or her kit makes its measurements.

In the simplest case, the detector's response $y(t)$ as a function of time $t$ is instantaneous and is thus some one-to-one function of the input signal $x(t)$: $y=\mu(x)$. If you wish to use the output $y$ to measure your input, then it is simplest if $\mu$ is a simple proportionality relationship, but anything one-to-one will make it a workable detector. Now, for "reasonably behaved" $\mu$, postulate a Taylor series (i.e. assume that $\mu$ is analytic or $C^\omega$) so that

$$\mu(x) =\sum\limits_{k=0}^\infty \mu_k\,x^k;\;|x|<\epsilon$$

for some interval of convergence defined by $\epsilon>0$. So, given the output comprises power terms $x^k$, what happens when you put $x(t) = a_1 \cos(\omega_1 t+\delta_1) + a_2\cos(\omega_2 t+\delta_2)$? Expand by the integer binomial theorem and the $m^{th}$ term of $x(t)^k$ is

$$\left(\begin{array}{c}k\\m\end{array}\right)\, a_1^m \cos(\omega_1 t+\delta_1)^m a_2^{k-m} \cos(\omega_2 t+\delta_2)^{k-m}$$

With many nonlinearities, the square term is the dominant nonlinear term, so this gives rise to the cross term:

$$a_1\,a_2 \cos(\omega_1 t+\delta_1)\cos(\omega_2 t+\delta_2) = \frac{1}{2}\left(\cos((\omega_1 +\omega_2) t+\delta_1+\delta_2)+\cos((\omega_1 -\omega_2) t+\delta_1-\delta_2)\right)$$

whence the sum and difference frequencies. Note that some nonlinearities have purely odd symmetry, in which case there is no square term and you do not get sum and difference terms: then, the dominant nonlinearity is the cubic term and you get terms at frequencies $|2\,\omega_1 \pm\omega_2|$ and $|2\,\omega_1 \pm\omega_2|$.

The phenomenon you refer to in its most general form is called intermodulation and you should look this word up if you've not yet done son.

When was this noticed? It is impossible to pin it down precisely, but it would have been noticed by many people once the notion of analytic became known to and widely used in science, i.e. after Brooke Taylor's publication of the notion in 1715, so the best answer would be "sometime during the Leonard Euler's lifetime".

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    Look, I know the math. I spent more than a decade involved with system design for a series of heterodyne imaging laser radars. But the idea of intermodulation and nonlinearity seems unlikely in 1715, given the conceptual tools of the time, particularly as applies to light. For instance, in 1715 Newton's corpuscular theory was king, and that does not lend itself to the analysis you provide. Even so, the question I asked was specific - provide a reference to an early exposition on the subject. The phenomenon is obvious in light of current knowledge, but this has not always been so. – WhatRoughBeast Jul 30 '14 at 15:04
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    I'm sorry, I could not glean your background from either your question or your profile. And for the second time - the phenomenon does not arise from light - if you know the math, then you can clearly see from the above it is the detection process, not the signal itself, that begets intermodulation. So you could still explain a sum and difference term from a corpuscular theory in terms of sinusoidally varying with time fluxes fluxes. Nonlinearity and Taylor series were hot ideas at the time I cite: people were trying to apply them to all kinds of physics. – WetSavannaAnimal aka Rod Vance Jul 30 '14 at 21:21
  • Fine. And for the second time, can you provide a paper which discusses this? And particularly can you provide a reference which specifically connects the analysis with the behavior of visible light? Nowadays we know about the common framework of various phenomena which allow this approach, but this was not always so. The Hooke/Newton dispute is clear evidence of this. As you say, "people were trying to apply them to all kinds of physics." I'm dubious in this case, but I'm willing to be convinced. Did they do so to light, and if so, who did it and when? – WhatRoughBeast Jul 30 '14 at 21:27

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