Wave particle duality in the seventeenth century Published in Abstract Wave-particle duality in the 17th century Newton's Opticks Isaac Newton published his Opticks in 1704. Between then and 1830 its conclusions, about the nature of light as a stream of particles, were accepted as correct by just about everybody. Maybe the mood was captured in Alexander Pope's epitaph to Newton God said, "Let Newton be", and all was light. In the early decades of the nineteenth century, with enormous difficulty and against the determined opposition of the scientific authorities, Thomas Young and Augustin Fresnel established the superiority of the wave theory of light. They were hardly recognized in their lifetimes — Young died in 1829 and Fresnel in 1827 — but by 1830 their correctness was conceded, and as a result the Opticks was considered as of purely antiquarian interest from then up to about 1920. The attitude was then reversed as a result of the growing popularity of the photon, a concept which Albert Einstein had introduced in 1905 to explain the photoelectric effect, and which today is widely considered to be a particle of light. In 1952 a modern edition of the Opticks was published with a Preface by Einstein which hailed Newton's ideas as a forerunner of the Wave-particle duality of the twentieth century. However, Einstein had himself seriously challenged this concept in a critical article he published along with Boris Podolsky and Nathan Rosen in 1935, and in the very year he wrote the Preface he said, in a letter to Michel Besso, Nowadays every Tom Dick and Harry thinks he knows what a photon is, but he is wrong. Refraction is the bending of a light ray at an obtuse angle as it goes from a rarefied medium like air into a denser medium like water or glass. It results in the optical illusion that a straight stick appears bent at the point that it enters a pool of water (see Fig.2). The law of refraction, corresponding to the mirror-image law of reflection, was first determined by Willebrord Snell around 1620. It states that the ratio of the sine of the incident angle to the sine of the refracted angle is a constant n, known as the refractive index, of the dense material - more accurately the ratio of the latter to the refractive index of air, but for most purposes that is one. y=√(a²+x²)+n√(b²+(d-x)²) should be a minimum. Nowadays a reasonably gifted sixteen year old can do this sum; all he or she has to do is calculate the derivative dy/dx and make it equal zero — and that gives us Snell's Law. That Fermat was able to do it in the 1640s, round about the time Newton and Leibniz, the inventors of the Differential Calculus, were born, is remarkable, and qualifies him for a substantial part of the credit which Newton and Leibniz were later to argue over between themselves. t(x)=((AB)/c)+((BC)/(c/n)) Since this gives t(x)=y(x)/c, Fermat's principle is simply the minimization of t with respect to x. This was a daring hypothesis indeed; the speed of light in air was first measured in the 1670s, and in water not until the 1850s. Interference Hooke had proposed that the various colours of a thin film arise because light can reach the eye either by reflection from the top surface, or by refraction into the film followed by reflection from the lower surface (see Fig.3). [Interference or fits? Hooke's explanation for the colours of thin films is that the signals ABCDE and A?B?C? interfere constructively, leading to a bright fringe, if the optical path lengths differ by a whole number of wavelengths. Newton's is that refraction occurs at C, leading to a dark fringe, if BC is an even number of fit lengths.] Because light is a wave motion, the two waves are phase related and they may either reinforce each other or cancel each other out; that is the Principle of Interference, and Hooke was the first to propose it, albeit in a somewhat vague and incomplete form. To make it more precise would require knowledge of the wave length of the light; if t is the thickness of the film, n its refractive index and λ the wave length of the light, and if the angle of refraction is r, then the reflected and refracted waves reinforce if the difference in path lengths for the two interfering waves contains a whole number of waves, that is, for some whole number N, 2ntcos r=Nλ . This, we may reasonably suppose, would have been Hooke's condition for a bright interference fringe, the dark ones being obtained by replacement of N by (N+1/2). tsec r=(2N+1)f , a dark ring being obtained by replacing 2N+1 by 2N. tcos r=2Nf . Now Newton's and the modified Hooke formulae differ in two respects. Firstly their criteria for dark and light fringes are precisely the reverse of each other; in this respect Newton would have been able to show the superiority of his own formula, using his own data on the ring diameters; the full wave explanation for this part of the phenomenon would have to wait until after 1800. But the other difference is the r dependence of the two formulae. It would have been particularly interesting to see Newton's formidable experimental technique used to test these latter two formulae against each other for nonzero r; they agree for r=0 which is the only situation observed by Newton, but their behaviour is different for nonzero r. As it turned out, questions of personal animosity between these two very substantial scientists intervened, as well as Newton's health, so publication of the new material had to wait until 1704, which was a year after the death of Hooke. Diffraction Grimaldi found that a beam of light, falling on a rectangular slit, casts an image on a screen some metres away which is somewhat wider than the slit, and that the shadow has coloured light and dark fringes close to its edges. A modern repetition of the experiment confirms his findings. In Fig.4 we depict the intensity in the image of a slit of width 9mm on a screen placed 1.08m behind the slit. If we define the shadow as the region in which the intensity is less than one tenth of the intensity at the middle of the image, then the width of the image is about 10mm, and the first bright fringe is about 1mm from the edge of the shadow. This figure represents the diffraction pattern with yellow sodium light at wave length 589nm. The fringes are in different positions if the colour of the light is changed, and with the white light used by Grimaldi it would not have been possible to observe more than the three fringes on each side that he reported. Grimaldi has informed us, that if a beam of the Sun's light be let into a dark room through a very small hole, the shadows of things in this light will be larger than they ought to be if the rays went on by the bodies in straight lines… We see from Fig.4 that it would be more accurate to say that the shadow is made smaller, which is the way Grimaldi described his finding; he went on to infer that this was evidence for the wave nature of light, since it had bent slightly around the edge of the slit and into the shadow. Newton, on the other hand, consistently fails to see such evidence in any of his Observations, insisting very strongly that the rays of light are always bent away from the edge. For Grimaldi's diffraction of waves Newton substitutes inflexion of rays. Grimaldi, just like Hooke, was able to give only a qualitative account of his theory, but in this case Newton is no different. In Fig.2 of Opticks, Book Three he draws a diagram of a stream of inflected particles going past a narrow rectangular object, but he does not attempt to calculate the pattern which such a stream would make on a distant screen. In Observations 1-11 he goes on to make accurate measurements of the positions of the light and dark diffraction fringes, but, in the absence of the kind of theory he has constructed for the interference fringes of Book Two, he is unable to draw anything other than qualitative conclusions, based on that figure. These conclusions were not only incorrect, as was revealed by experiments made after 1830; they also caused him to claim he saw shadows where in fact he should have observed illuminated areas. His most spectacular failure is in Observation 6, where he observes the diffraction by a narrow slit, whose width is .06mm. Such slits fall in the region of what we now call Fraunhofer diffraction, named after Joseph Fraunhofer (1787-1826), the inventor of the diffraction grating, for which the complicated behaviour of Fig.4 is replaced by the simple formula for the intensity I(x)=I(0)(((λd)/(2πax)))²sin²(((2πax)/(λd))) , which is depicted, for the case λ=589nm, d=1.08m, 2a=.06mm, in Fig.5. Newton mapped very accurately the position of the first three fringes on either side of the central one, but failed to see the central peak — this in spite of the area under the central peak being more than nine times the total area under all of the other peaks put together! When I made the foregoing observations, I designed to repeat most of them with more care and exactness, and to make some new ones for determining the manner how the rays of light are bent in their passage by bodies, for making the fringes of colours with the dark lines between them. But I was then interrupted, and cannot now think of taking these things into farther consideration. And since I have not finished this part of my design, I shall conclude with proposing only some Queries, in order to a farther search to be made by others. Double refraction Starting from the Fermat Principle of Least Time, Christian Huygens set out to provide a causal description of reflection and refraction. For Huygens it was the wave front of a light beam, rather than the rays considered by Newton, which were fundamental. There is a simple connection; the rays are perpendicular to the wave front (see Fig.6). The wave front travels through air at a speed c, and when it falls upon a reflecting surface it sends out spherical wavelets also travelling at speed c. But the wave front reaches the surface at different times, so at some later time the set of all these wavelets has varying radii. Huygens constructs the reflected wave front by drawing the tangent plane to all of these wavelets, and the reflected rays are simply the perpendiculars to this wave front. It is then a matter of simple geometry to prove that the reflected rays make the same angle as the incident rays with the reflecting surface. It is then a matter of simple geometry to prove that the direction of the refracted rays is in accord with Snell's Law. The optical lengths of the paths A?B?C?, A?B?C?, A?B?C? and A?B? are all equal, and in modern terminology we say that the Huygens construction interprets Fermat's Principle as stating that the wave front is a surface of constant phase. Now draw the vertical plane through the edge common to these two faces; it bisects the obtuse angle of the parallelogram on the top face of the crystal, and any plane parallel to it is known as a plane of perpendicular refraction (PPR). The line connecting the two images lies in a PPR, and in the ray description this means that a vertical incident ray (that is one with i=0), is split into an ordinary ray which is vertical (r=0) and an extraordinary ray with an angle r', measured to be between 6 and 7 degrees, lying in the PPR. Further observation had shown, by the time Huygens wrote his Treatise on Light in 1690, that, for any angle of incidence the ordinary ray satisfies Snell's Law. The extraordinary ray, however, completely violates it. Indeed, if the incident ray is not in the PPR, then the refracted ray is not even in the same vertical plane as the incident ray. The wave front of the refracted wave is, as before, the common tangent of all these wavelets, but now the ray direction, which is the vector from the centre of the spheroid to the point of contact with this wave front, is no longer perpendicular to it. In modern terminology we use the term walkoff angle for the angle between the ray vector of the extraordinary ray and its wave vector. By considerations of symmetry Huygens was able to infer that the optic axis makes an equal angle of p=45.38 deg. with each of the three crystal planes meeting at O in Fig.8. He, or anyone else taking his theory seriously, could therefore translate his Fig.9 into a mathematical relation between the ordinary and extraordinary angles r and r', namely, putting tanr=x, and tanr'=y, y=x(1+ttan p)√(((1-tcot p)/(1-tx²tan p)))+t . The only unknown quantity in this formula is the parameter t, which is the value of tanr' when r is zero. This quantity must be measured; Newton gave it as the tangent of 6.67 deg., which is quite close to the modern value of 6.23 deg. Since p is extremely close to 45 deg., we observe that the above formula may be adequately approximated by y=x(1+t)√(((1-t)/(1-tx²)))+t . Because of the existence of such a simple diagram, readily translated into a simple formula, the challenge posed by Huygens to Newton's corpuscular theory was more serious than had been posed by either Hooke's interference or Grimaldi's diffraction. Newton's response had to wait until the second edition of the Opticks in 1717, where he stated, in Query 28, Are not all hypotheses erroneous, in which light is supposed to consist in pression or motion, propagated through a fluid medium?… The waves on the surface of stagnating water, passing by the sides of a broad obstacle which stops part of them, bend afterwards and dilate themselves gradually into the quiet water behind the obstacle… But light is never known to follow crooked passages nor to bend into the shadow… To explain the unusual refraction of Iceland crystal by pression or motion has not hitherto been attempted… except by Huygens, who for that end supposed two several vibrating mediums within that crystal… But how two aethers can be diffused through all space, one of which acts upon the other, and by consequence is reacted upon, without retarding, shattering, dispersing and confounding one anothers motions, is inconceivable. Note the curious way Newton has of posing a "Query". This is not the response of the scientist, but rather of the ideologue. After restating what was, as we saw in the last section, an incorrect description of the diffraction phenomenon, he goes on to say it is "inconceivable" that two kinds of light wave coexist within the same medium. This seems a wholly inadequate reason for him, or anyone else, to refuse further consideration of Huygens's explicit proposal contained in Fig.9. His willful blindness in this matter was to hold up recognition of the correct wave description of light for more than a hundred years. y=x+t , for the extraordinary, or "unusually sided" particles. This, to be sure, is mathematically simple, even though the mechanical model leading up to it was extremely convoluted. What is amazing is that neither Newton nor any of his disciples during the next century thought to test this formula against Huygens's Fig.9. If they had done so, they would surely have found Huygens to be correct, because the difference between them (see Fig.10) is something the painstaking experimenter of Opticks Book Two would have easily detected.
There is a substantial difference between Newton's treatment of interference and his treatment of the other two phenomena we have considered here. Only his description of interference may be said to anticipate the wave-particle duality of the twentieth century, while his descriptions of both diffraction and double refraction are simply corpuscular. Both of the latter parts of his theory can be, and should have been, disproved using nothing more than the experimental technology already available during Newton's life time. But, as for how this occurs, I have found nothing until now which satisfies me. This modest statement is of the same quality as Newton's own, in respect of the possible causes of the gravitational interaction. Newton conceded, in a letter to Richard Bentley in the early 1690s, that something like the modern field theory of Einstein would be superior to his own hypothesis (Yes!) of instantaneous attraction, but indicated that the current state of knowledge did not allow him to make any further hypothesis. Information about this Article Published on Sunday 29th October, 2006 at 15:58:53.
added 2nd November, 2006 at 10:26:51 My first attempt at posting this article was unsuccessful, because I was not able to send the figures. The editors are helping me with the IT and I hope to submit an improved version shortly. I am now writing aditional material because I was told by the automatic editor that my Comment was too short! You will appreciate that my IT skills are rather limited! Peer review added 12th November, 2006 at 10:24:21 Despite saying that your IT skills are no good, I find that your article has many diagrams, which is good! Peer review added 19th December, 2006 at 22:48:16 We now know that Newton did not destroy Hooke’s records, when he succeeded Hooke as Secretary of the Royal Society. The handwritten minutes of the Royal Society as recorded by Robert Hooke were discovered in an attic and purchased early in 2006 by the Society. The ~520 yellowing and stained pages are currently being restored and recorded (including the numerous comments in the margins) to be released for general study. Peer review added 20th August, 2011 at 20:34:28 In spite of the good graphical presentation of the article its main point about Newton’s theory of light ‘corpuscles’ assimilation with photons, or the classical waves with quantum wave function is both superficial and incorrect. The classical wave theories of light have been much improved in quantum physics. Thus, using the classical wave approach one cannot readily explain the photoelectric effect, nor can one explain the well-known blackbody radiation as Planck has done. The author may benefit from reading the contributed, general article on the History of Quantum Mechanics http://en.wikipedia.org/wiki/History_of_quantum_mechanics , that also has many of the references missing from this preprint. |

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