The Debate on the "Polarity of Light" during the Optical Revolution

XIANG CHEN

Communicated by JED Z. BUCHWALD

Introduction

In the 1830s physical optics underwent a major transformation as the principles and traditions of emission (or particle) theory were displaced by those of the new undulatory (or wave) theory. Heated debates accompanied the change, during which the undulatory theory's supporters argued for its explana- tory superiority, whereas the theory's opponents presented evidence for its incompetence) Many of these debates concerned the respective physical models of the two traditions -- particles of light vs. waves in the ether -- and their abilities to account for known phenomena. Because the quantitative power of the undulatory theory seems, in retrospect, extraordinarily superior to that of its alternative, little attention has been paid either to what the alternative theory could do, or to what the undulatory theory had difficulties with. Recent work has however shown that the rival to the undulatory theory was hardly powerless, for its assumption that light consists of countable objects (rays) that gather in sets (beams) produced a great deal of quantitative work. 2 Indeed, this understanding was so powerful that many undulationists who were not for some time fully able to embody their understanding of light in a coherent apparatus frequently turned back to the apparatus of ray theory when they encountered novel phenomena, a This paper concentrates on a debate that concerned primarily the ability of undulatory principles to account for a new optical phenomenon. The debate was initiated in 1837 by DAVID BREWSTER, a stubborn opponent of the

1 For examples of the arguments for the undulatory theory, see LLOYD, 1834; for examples of the arguments against the undulatory theory, see BREWSTER, 1832. 2 For an example of a possible quantitative account of elliptic polarization from the emission tradition, see BUCHWALD, 1992, 67-74. 3 For example, even JOHN HERSCHEL had trouble in understanding the differences between ray and wave analysis. See BUCHWALD, 1989, 291-96. 360 X. CHEN undulatory theory, on the basis of a novel experimental result. Undulationists failed to explain the anomaly exposed by BREWSTER for more than a decade, and they accordingly suffered a temporary setback. The predicament of undula- tionists came not only from their failures in explanation, but from the fact that they inconsistently revised the theory's analytical apparatus, importing into it incompatible elements of ray optics. Thus, the controversy's closure was not easy to achieve: in the process undulatory techniques and tools themselves underwent further development. We will see that many of the salient issues in this debate remained below the surface, hidden beneath the explicit points regarding physical models or ex- planatory abilities. Indeed, it is precisely for this reason that the debate lasted as long as it did. We will see, in particular, that BREWSTER succeeded for so long because he based his reasoning tacitly on ray conceptions, which enabled him to characterize his discovery as a new property of light, in effect of a new optical kind. Undulationists did not recognize BREWSTER'S tacit apparatus, and they attempted to deal with his discovery by applying new tools that they developed specifically to deal with this special situation. Yet undulationists themselves reasoned initially in terms of rays, (albeit not as BREWSTER did) and not in terms of wave-fronts. It took years of experience, as well as people who trained from the beginning in the undulatory tradition (and who therefore had not been influenced by the rival form of optics), before the anomaly was removed. We will also see that the early undulatory analyses in this debate were significantly influenced by the specific experimental design, which appeared to be strikingly similar to interference experiments, and these could often be dealt with by ray analysis. The anomaly however resisted such a treatment, and it was in fact successfully dealt with only when interference methods, where rays may work, were replaced by the methods of diffraction, which require integra- tion over wave fronts. The successful classification and explanation of the anomaly were finally achieved on the basis of a new experimental device that produced the same result but in a different way, quickly and decisively remov- ing the irritant. This episode vividly shows that a full historical understanding of scientific debates requires examinations of not only articulated arguments but also of the many tacit factors that influence scientific practice, such as classifica- tion, the analytical apparatus deployed, and the specifics of experimental design.

1. The discovery

Fox TALBOt, an undulationist, discovered an interesting phenomenon in prismatic spectra in 1837. In this experiment, TALBOT used a prism with moder- ate dispersive power to produce a spectrum. Instead of observing the spectrum directly, he inserted a circular aperture the size of the pupil just in front of the eye, and he covered one half of the aperture with an extremely thin plate of glass (Fig. 1). When he viewed the prismatic spectrum in this way, TALBOT observed a group of parallel dark bands crossing throughout the spectrum, similar to those produced by absorption. He attributed the bands to interference. "Polarity of Light" during the Optical Revolution 361

Thin plate ~] The eye

Aperture ]~ Prism

Fig. 1. TALBOT'Sexperimental design.

Relying on HUYGENS' principle, TALBOT reasoned that every point in the spec- trum acted as a new source, emitting homogeneous rays that were focused by the crystalline lens of the eye onto the retina. Since the rays passing through the upper half of the lens experienced retardation caused by the plate, they could interfere with the unretarded rays that passed through the other half of the lens. When the retardation, which varied according to the color of a ray, was an odd number of half wavelengths, the light would be extinguished by interference. The spectrum would thereby be interrupted by dark bands. 4 TALBOT did not, however, attempt to produce a quantitative account. TALBOT'S discovery was published in the Philosophical Magazine in 1837, and it drew the attention of DAVID BREWSTER, who was a vociferous opponent of the undulatory theory. BREWSTER had begun to study optics as early as 1799, and he had an established reputation in experimental optics by the mid 1810s. Although his early optical researches were deeply influenced by the emission tradition, BREwsTER never publicly admitted that he was an emissionist, nor was he willing to give a straightforward answer to the question of whether light is particles or waves, s BREWSTER'S optical researches, however, always contained an implicit element that was thoroughly consonant with the emission tradition: he always thought of light as rays, and he analyzed optical phenomena in terms of their properties, occasionally thinking in terms of the deflection of rays by forces emanating from material bodies. The late 1830s was a difficult period for opponents of the undulatory theory in Britain. Wave optics dominated the journals after mid-decade, and the theory had strong advocates among the most prominent "gentlemen of science" of the

4 TALBOT, 1837, 364. TALBOT's explanation was problematic, because it also predicted the formation of bright bands (due to constructive interference) that had not been observed. TALBOT probably based his explanation on ARAGO's account of stellar scintillation, which attributed the momentary disappearance of starlight to interference between rays that passed the two halves of the eye's lens or the telescope's objective. s See BREWSTER, 1835, 1-2. 362 X. CHEN day, including GEORGE AIRY, the Astronomer Royal, BADENPOWELL, the Savilian Professor of geometry at Oxford, and HUMPHREY LLOYD, professor of natural and experimental science at Trinity College, Dublin. Its opponents in Britain did not however immediately surrender to the new system, and several of them continued to fight a rearguard action. BREWSTER, for one, worked hard to collect experimental results that were inconsistent with, or at least unexplained by, the undulatory theory. Foremost among these anomalies were dispersion and absorption, because undulationists themselves had difficulties here. 6 Thus, when TALBOT reported that his prismatic experiment produced an absorption- like effect, BREWSTER immediately thought to use the experiment in a further attack. BREWSTER repeated TALBOT'S experiment, but with two significant revisions. First, instead of using the naked eye, he examined the spectrum formed at the focus of an achromatic telescope, thereby producing a distinct and sharp spec- tral image. The existence of dark bands in prismatic spectra was well-known in the 1830s. Using a telescope to view the spectrum, FRAUNHOFERin the 1820s has reported the existence of more than 600 dark lines in the light from the sun. Since these dark lines could not be seen by the naked eye, the telescope was taken to be a standard device in line-observing experiments. BREWSTER thus added an achromatic telescope to TALBOT'S design. Because the object tens of the telescope in itself functioned as an aperture limiting the amount of light allowed to pass, BREWSTER did not use TALBOT'S circular aperture. He kept the thin plate of glass directly in front of his eye, covering one half of the pupil (Fig. 2). BREWSTER thereby made TALBOT'S dark bands more intense and distinct. BREWSTER'S second change to TALBOT'S design was to rotate the thin plate during the experiment. In his early experiments on polarization by refraction, conducted in the mid 1810s, BREWSTER found that observational results varied when the analyzer -- usually a piece of agate -- was rotated: some images disappeared altogether when the agate was turned to a particular angle with the plane of refraction. 7 Given the experience gleaned from this much-earlier work on polarization, BREWSTER decided to try rotating the thin plate in the modified TALBOT experiment. This produced an entirely novel phenomenon, one that had not been found by TALBOT. When he held the thin plate in its original position (covering the violet end of the spectrum) intensely dark bands appeared. How- ever, when BREWSTER rotated the plate in its own plane, the dark bands became less and less distinct as the angle between the edge of the plate and the lines of the spectrum increased. When the angle became 180 degrees, that is, when the thin plate covered the red end of the spectrum, the dark bands completely disappeared, s

6 HERSCHEL, 1827, 449-50; 1833, 401-12. 7 BREWSTER,1814, 220. 8 BREWSTER, 1837, 12-3; 1838, 13-4. "Polarity of Light" during the Optical Revolution 363

~ t~ eye

Prism Telescope The lens

Fig. 2. BREWSTER'Sexperimental design.

BREWSTER'S persistent search for experimental facts unfavorable to the un- dulatory theory made him especially sensitive to every anomaly. He immediately realized that the undulatory theory might not be able to explain why the dark bands disappeared when the thin plate covered the red end of the spec- trum. The account given by TALBOT attributed the bands to interference. But, according to the undulatory theory, interference in this experiment should occur no matter what the orientation of the thin plate might be, and, therefore, the dark bands should never disappear on this account. BREWSTER did not want to miss this opportunity to embarrass the undulatory theory. He immediately reported his discoveries at the 1837 meeting of the British Association, charac- terizing the peculiar phenomenon he had found as "a very curious and entirely inexplicable property of light. ''9 BREWSTER'S report did not produce the kind of reaction from the audience that he expected. WHEWELL and LLOYD were present when BREWSTER read his report, but these two undulationists tried to reduce the impact of his discovery. WHEWELL simply denied that there was any new property of light involved here, although he was not quite sure about the details of the experiment. LLOYD admitted that he could not at the present imagine any probable way of explaining the fact, but he insisted that the as-yet unexplained phenomenon should not compel people to adopt the conclusion (which might otherwise account for the phenomenon) that "the time of an undulation of light could, under certain circumstances, be altered.''1~ These undulationists, secure in their new-found dominance of optics, argued that, though BREWSTER'S discovery might not be yet explained, nevertheless it was trivial and should not affect confidence in the theory.

9 BREWSTER,1837, 12-3. lo The Athenaeum, 1837, 518: 719. LLOYD's confidence in the undulatory theory came both from its various explanatory successes and from its impressive quantitative ability; see LLOYD, 1834, 295-413. A few other undulationists, such as AIRYand WHEWELL,also shared this opinion. 364 X. CHEN

2. Classification

BREWSTER did not give up despite the disappointing response at the 1837 meeting. He continued to make changes in the experiment, and he brought up the topic again at the 1838 meeting, where he did not simply describe his empirical findings; he also provided theoretical analyses. BREWSTER first told his audience about a new observation. Instead of using just one plate, he had let one half of the light beam pass through a series of thin plates, each the same thickness but having different widths, piled up so that different parts of light beam suffered different degrees of retardation. On look- ing through this pile of thin plates, BREWSTER said that he was surprised to observe a multitude of splendid bands and dark lines crossing the entire spectrum, which looked "as if it had been acted upon by absorbing media." The different degrees of retardation caused by the plate suggested, BREWSTER reason- ed, that "we have here dark lines and the effects of local absorptions produced by the interference of an unretarded pencil with different other pencils, proceed- ing in the same path with different degrees of retardation. ''11 BREWSTER con- tinued to agree with TALBOT that the formation of bands was caused by interference. BREWSTER next turned back to the major point he had made the year before, now emphasized by the multitude of new bands, and he made it the centerpiece of a new assertion. He reminded his audience that this "local absorption produced by interference" had a very peculiar feature. The bands appear only when the thin plate covers the violet end of the spectrum; they disappear when the-plate covers the red end. Interference alone cannot explain this asymmetric phenomenon, and BREWSTER now argued that the phenomenon might be caused by a new kind of luminous asymmetry or polarity that works to alter the conditions under which interference can occur. In the early 1830s polarization was the only asymmetry recognized by the optical community, and it was accordingly possible that the new phenomenon was somehow the result of the known interference properties of polarized light, with Which BREWSTER was familiar, most likely having read about them in HERscHEL'S "Light".Iz He knew that rays of light polarized in the same plane would interfere with each other in the same way as common light, but that rays polarized at right angles to each other produced no fringes. 13 These properties could be used in explaining BREWSTER'S new phenomenon only if the beams of light were in fact polarized. He had to counter this possibility in order to argue for an altogether new property. BREWSTER already knew that a light beam's polarization could not be ro- tated through a right angle by passage through thin plates, which spoke

11 BREWSTER,1838, 13-4. 12 HERSCHEL, 1827, 529-33. Although HERSCHEL'S"Light" was not published until 1845, it was widely circulated among the optical community since it was completed in 1827. la BREWSTER, 1835, 179--181. "Polarity of Light" during the Optical Revolution 365 powerfully against the property being involved here. To prove directly that polarization was not responsible, BREWSTE• later replaced the thin plate of glass in the original experiment with a piece of doubly-refracting crystal. He now saw two systems of dark bands in the spectrum when the crystal covered the violet end, one produced by the ordinary rays and the other by the extraordinary rays. However, all these bands disappeared when the crystal covered the red end. The observation of double refraction proved that all the light (retarded as well as unretarded) in his original experiment was in fact unpolarized. Further- more, the disappearance of dark bands produced by both the ordinary and extraordinary rays implied that the polarity phenomenon also existed in polar- ized light. BRBWST~ accordingly claimed that "I have no hesitation in consider- ing this property of light as indicating a new species of polarity in the simple elements of light, whether polarized or unpolarized. "14 According to BREWST~IL as well as to others who deployed ray conceptions, polarization as an observable phenomenon was caused by a special property of the rays of light, namely that a ray has sides. 15 On this basis BREWSTER was able to provide explanations for many phenomena related to polarization, and this confirmed for him the exemplary power of ray analysis. BREWSTBrt often attributed novel optical phenomena to correspondingly novel properties of rays. Although this might seem to be a clear case of introducing ad hoc assumptions, the procedure was in fact quite powerful since it could, and did, have broader empirical implications than those for which it had been designed. Furthermore, assigning new properties to rays had the signal virtue of leaving intact their identity as individual objects, and it was precisely this identity that permitted BREWSa'ER and those who thought like him to consider rays collectively. For example, when BkEWSa'ER in the early 1830s discovered a new species of polar- ization -- elliptic and circular -- in experiments of metallic reflection, he was able to provide impressive quantitative detail by ascribing to rays a new property linked to a specific notion of "phase. "16 BREWSrE~ thought that the newly discovered asymmetry was tightly bound to a ray's refrangibility. The bands appeared when the plate covered the more refrangible end of the spectrum (the violet end), or "when the least refrangible side of the retarded ray is towards the most refrangible side of the spectrum, or towards the most refrangible side of the unretarded ray." The bands disap- peared altogether when the plate covered the less refrangible end (the red end), or "when the most refrangible side of the retarded ray is towards the least refrangible side of the unretarded ray" (Fig. 3). 17 Hence BREWSTER concluded that "the different sides of the rays of homogeneous light have different proper- ties when they are separated by prismatic refraction ..., -- that is, these rays have polarity. ''is

1,* BREWSTER, 1838, 13; original emphases. 15 See BREWSTER, 1815, 149-151; 1830, 176-77. 16 For detail of BREWSTER'sexplanation, see BUCHWALD, 1992, 50-54, 67-74. 17 BR~WST~R,1838, 13-4; original emphases. 18 BeEWSTER,1838, 14; original emphases. 366 X. CHZN

High refrangibility Violet

Retarded ray Unretarded ray

Unretarded ray Retarded ray

(A) When the least refrangible side of (B) When the most refragible the retarded ray is towords the most side of the retarded ray is refrangible side of the unretarded ray. Red towords tile lest refrangible side of the unretarded ray. Low refrangibility

Fig. 3. The Relation between polarity and refrangibility.

More specifically, BREWSTZR thought that his polar refrangibility was pro- duced by a process of ray-sorting that occurs when rays interact with material particles. "When", he wrote, "light is rendered as homogeneous as possible by absorption, or when it is emitted in the most homogeneous state by certain coloured flames, it exhibits none of the indications of polarity above mentioned. The reason of this is, that the more or less refrangible sides of the rays lie in every direction, but as soon as these sides are arranged in the same direction by prismatic refraction ..... the light displays the same properties as if it had originally formed part of a spectrum. ''19 This process of ray-sorting was similar to the one that occurs in polarization, which was, obviously, B~zwszz~'s model (although the latter has nothing to do with refrangibility). BI~ZWSTER might have developed a quantitative explanation of his experi- mental findings on this basis. Given his spatially-directed conception of polar refrangibility, he would have needed two additional assumptions to provide a mathematics for his system: one assumption would specify how the thin plate alters the direction of a ray's polar refrangibitity, while the other would specify the conditions for interference between rays with different polar refrangibili- ties. 2~ But BRZWSTER did not work out such a scheme. Indeed, he never openly described precisely how his assumptions of polar refrangibility could be used to explain the experiment. Given the situation that BREWSTER faced in the late 1830s, it is hardly surprising that he cautiously decided not to provide a theor- etical account for the experiment. With the undulatory theory's powerful domi- nance, no theoretical account with traces of the emission theory, which the notion of a ray-linked refrangibility inevitably raises, could possibly be accepted

19 BREWSTER, 1848, 14. 20 For example, BREWSTER'Sexperimental findings could be explained if he assumed that the retarding plate altered the direction of polar refrangibility by 180 degrees, and that interference between two rays occurred when their polar refrangibilities were heading at each other, but not the other way around. "Polarity of Light" during the Optical Revolution 367 by undulationists. Instead, presenting apparently neutral experimental results that could not be explained by either theory effectively weakened the dominant system, provided that nobody denied the value of the experiments. BREWSTER accordingly adopted a careful strategy: in the first stage, he presented new facts that neither the emission nor the undulatory theory could cover, and that accordingly challenged the latter's completeness. After these facts were widely accepted, a second campaign for the downfall of the undulatory theory proper could be launched. 21 Presenting unexplained observational facts raised problems. Undulationists could dismiss the anomalies by treating them as trivial, at least until they appeared in many different experimental settings. BREWSTER therefore needed to show that his findings were not artifactual, but that they reflected the existence of a new, general optical property. BREWSTER, thus, tried to convince others that what he had found was not simply a novel, possibly singular phenomenon, but a "new species of polarity.''zz He wanted to categorize the polarity phenomenon as a new kind of optical asymmetry similar to polarization. Indeed, there are parallels between polarization and BREWSTER'S polar refrangibility. First, both are caused by orderly arrangements of a particular property of rays, which is the side of rays in polarization and the refrangibility of rays in the other. Random arrangements of these properties in both kinds result in the disappear- ance of asymmetry. Second, both kinds of asymmetry can prevent two light beams from interfering with each other, though they require different specific conditions. Thus, BREWSTER'S "new species of polarity" tacitly implied a taxon- omy for the phenomena of optical asymmetry, which included, in addition to the existing category of polarization, a new one called "polar refrangibility" (Fig. 4). 23 The introduction of "polar refrangibility" as a major category of asymmetry similar to "polarization" was crucial to BREWSTER'S argument. If this taxonomy for optical asymmetry were accepted, then his experimental discovery would not be trivial, because it would imply a new classification for light itself, in which case the undulatory theory would fail not in a single case but in a whole range of phenomena. BREWSTER'S strategy was apparently successful. His discussions drew the attention of the audience at the 1838 British Association meeting. Both W~EWELL and LLOYD were again present when BREWSTER made his report. This time, however, these two undulationists no longer denied the significance of BREWSTER'S findings. They asked BREWSTERmany questions about the experimental

21 BREWSTERcontinued this strategy in his later fights with the undulatory theory. For how he applied this strategy in the late 1840s and the early 1850s, see CHEN and BARKER, 1992, 78-81. 22 BREWSTER,1838, 14. 23 In Fig. 4, polar refrangibility is a subcategory under polarization, because BREW- STER's experiment with a double-refractingcrystal showed that polar refrangibilityoccurred in both polarized and unpolarized light. On the other hand, it is also possible that polarization is a subcategory of polar refrangibilityin BREWSTER'Staxonomy, but he did not have experimental evidence to support this arrangement. 368 X. CHEN

Light Beam

Polarized Beam Unpolarized Beam (Orderly arrangement (Random arrangement of rays' side) of rays' sides)

Beam With Beam Without Beam With Beam Without Polar Polar Polar Polar Refrangibility RefrangibUity Refrangibility Re frangibility (Orderly arrangement (Randomarrangement (Orderly arrangement (Randomarrangement of rays' refranglbili~) of rays' refrangibiliZy) of rays' refrangib~y) of rays' refrangibility)

Fig. 4. BREWSTER'staxonomy of optical asymmetry. design, but did not raise any substantial objections. JOHN HERSCHELalso listened to BRZWSTER'Sreport, and his reaction was quite positive. After admitting that the undulatory theory was not able to explain the phenomenon described by BREWSTER, HERSCHEL praised the "indefatigable zeal and industry" of BRZWSTZR in these experiments, and asserted that BR~WSTER'S works "opened an entirely new field of optical discovery.''24 H~RSCHEL'S praise gave BREWSTE~ renewed confidence. When WILLIAMHAMILTON said that he "did not despair" of seeing the phenomenon brought into the undulatory dominion, BR~WSTeR did not hesitate to assert that he saw no way whatsoever of doing so. z5

3. The explanations

After the mid 1830s, when the undulatory theory became dominant in Britain, its proponents were usually not taken aback by new observational and experimental results that, their rivals claimed, could not be explained by the wave system. They simply played down the value or significance of the experi- ments, sometimes just by remaining silent. This occurred, for example, at the 1838 British Association meeting. Besides the problem raised by his new "polarity of light," BREWSTER has also there presented three other papers on several new diffraction phenomena, that, he asserted, FRESNEL'S theory could not explain. Undulationists at the meeting did not offer a word in reply. Their silence embarrassed even HERSCHEL, who was sympathetic to BREWSTER.26

24 The Athenaeum, 1838, 566: 625. 25 The Athenaeum, 1838, 566: 625. 26 The Athenaeum, 1838, 568: 675. "Polarity of Light" during the Optical Revolution 369

BREWSTER'S new polarity, however, was dangerous, primarily because he claimed general significance for it. The urgency stemmed from BREWSTER'S implied taxonomy. If his system of new optical kinds were accepted, then the undulatory's failure to remain consistent with an entire, key category could not be counted as trivial. To make matters worse, BREWSTER had introduced a high- ly contentious notion that implicated the ray-based concept of refrangibility, with its close association to emission ideas. BADEN POWELL supplied the first undulatory account of BREWSTER'S novel discovery. Though a professor of geometry at Oxford, POWELL was the most active advocate of the undulatory theory. From the 1820s to the 1850s, POWELL published more than seventy scientific papers on physical optics and involved himself in almost every debate regarding the new system of optics. 27 He turned his attention to the TALBOT-BREWSTERexperiments during the summer of 1839, and after conducting several experiments of his own, he briefly reported his results at the 1839 British Association meeting. 2s Like both TALBOT and BREWSTER, POWELL believed that the new dark bands in the spectrum were caused by interference between "the two halves of the parallel pencil of each ray, which converges in the eye, and whose breath is equal to the aperture of the pupil. ''29 POWELL however revised TALBOT'S wave analysis by importing into it elements of ray theory. The key to POWELL'S explanation was to consider the retarding effect caused by the prism. POWELL noted that, because of prismatic aberration, all of the light of whatever color emanating from a point source would spread out into a patch on passing through the prism. Dispersion of course in effect creates overlapping patches of different colors, but POWELL'Sinterest at this point in his reasoning involves the light in any one, homogeneously-colored patch. POWELL next argued that the different parts of each such patch had to have been produced by rays that had undergone different degrees of retardation in the prism. The half of the homogeneous patch that lay near the violet end of the spectrum suffered more retardation than the patch's other half simply because it passed through a thicker part of the prism. POWELL now added a bit of wave-front theory. He assumed that every point in each such homogeneous patch itself acted as a new source for waves all of which finally converged onto a single point of the retina after refraction by the eye's lens. These converging waves would then interfere with one another (because they all derived from homogeneous light that had originally emanated from a single point). Using a retarding .plate to cover one half of the beam from a patch will produce differences in retardations between the light that has

a7 Among physicists in nineteenth-century Britain, POWELLwas second in the number of published optical papers. BREWSTERwas first, with more than 100 published papers on optics, and STOKES was third with about 50. No other physicists published more than 30 papers on optics. See Royal Society Catalo#ues of Scientific Papers. 2s POWELL,1839a, 1. 29 POWELL,1839a, 1. 370 X. CHEN passed through the plate and the light that has not passed through it. These differences would however vary according as the plate covered the red or the violet end of the spectrum, because, we will see in a moment, the interference pattern depends on altering the differences in retardation between the violet- and red-tending rays in a given patch, whereas the plate affects only one such group. If the plate lay near the violet end, then it would exacerbate an already- considerable difference in retardation between light from different parts of the patch, thereby making interference bands visible. On the other hand, if the plate covered the red end, then its effect would be to decrease an already small difference in retardations. In which case no visible interference bands would result. 3~ For unknown reasons, POWELLdelayed publishing details of his explanation. However, when he later learned that AIRY had conducted research on the same topic and had produced many new conclusions, POWELL became anxious to put his own work on record, and so he sent in a paper to the Philosophical Magazine. 31 Here POWELL reported a new observation. He found that the retarding plate must have a certain minimum thickness in order to produce the polarity phenomenon. If the plate is thinner than the lower limit, then bands continued to exist even though the plate covered the red end of the spectrum. POWELL believed that his discovery, together with the polar- ity phenomenon found by BREWSTER, could be explained by the account he had presented at the 1839 meeting, namely, by considering both the retarda- tion produced by the plate and that produced by the prism, in the following way. If, POWELL wrote, the plate covered the violet end of the spectrum, then the path differences between the two halves of a circular patch that passed through it would be:

(evlo,pris m -}- Rvio,plate) -- ered, pris m = (Rvio,pris m -- Rred, prlsm ) "t- Rvio,plat e Here, Rvio,pmm, Rred, prism, evio, plate respectively denote the retardations caused by the prism in a given patch near the violet and red ends of the spectrum, and the retardation produced in the plate on the violet-tending ray. No term (Rred, plate) for retardation by the plate on the red-tending ray appears because this ray escapes the plate altogether. In consequence, the path difference became equal to the sum of the original path difference caused by the prism and new one imposed by the plate on the violet-tending light. If, on the other hand, the plate covered the red end of the spectrum, then the path difference between the two involved light pencils would be:

Rvio, prism -- (Rred, prism Jr- ered, plate)= (Rvio, prism -- Rred, prism) -- Rred,plat e

30 According to POWELL, LLOYDalso suggested a similar explanation for the phenom- enon. See POWELL, 1839b, 795. 31 POWELL,1840, 81-85. "Polarity of Light" during the Optical Revolution 371

In this case, the violet-tending light escapes the plate, with the result that his expression has become the difference between the prism-produced difference in retardation and the red-tending one produced by the plate. POWELL thought he had here an explanation for his discovery concerning the effect of plate thickness. If a plate covering the red end of the spectrum has a proper thickness that the retardation produced by it alone is close in magni- tude to the difference produced in the prism between the violet- and red-tending rays in a given patch, then that difference will effectively be annulled and interference will not occur. On the other hand, if the plate is extremely thin, then ered, vlat e will be too small to create any effect, and bands will continue to exist, a2 POWELL'S explanation did not satisfy BREWSTER, because, according to the latter, it rested on an unacceptable approximation. POWELL, BREWSTER noted, had assumed that a whole half of a circular patch suffered the same degree of retardation from the prism. This assumption was clearly grossly inaccurate, for BREWSTER remarked, "every elementary part of the spectrum consists of rays which have passed through all the different thicknesses of that portion of the prism which receives that incident beam of white light. ''33 Therefore, different parts of a patch suffer different degrees of retardation caused by the prism, in which case POWELL'S account could only explain the vanishing of some dark bands but not the disappearance of the whole set. In any case, BREWSTER remarked in his most devastating point, the phenomenon did not require a prism -- it only needed a spectrum. He had observed the same polarity phenomenon by using an interference spectrum, which was not produced by a prism, but rather by a number of parallel grooves cut on a polished steel surface. This experimental fact showed that the "polarity of light" resulted from something other than the retardation of a prism, in which case POWBLL'S account failed. 34 Another account was provided by GEORaE AIRY, the most influential undual- tionist in early nineteenth-century Britain -- his book on optics, written in 1831, was used as reading in preparation for optics questions in the Mathemat- ical Tripos at Cambridge. AIRY was also one of the most dogmatic advocates of the undulatory theory and was known for his fierce counterattacks. In June 1840, he presented a paper to the Royal Society entitled "On the Theoretical Explanation of An Apparent New Polarity of Light." It was latter published in the Philosophical Transactions as the Bakerian lecture for the year. In order to publicize his own explanation of the phenomenon to those familiar with BREWSTER'S work, AIRY also read a simplified version of the paper to the 1840 British Association meeting, although he knew that oral presentation was not an appropriate way to present a complicated mathematical analysis.

32 POWELL'Sexplanation is problematic, because it implies that bands would be visible with no plate present. This problem is caused by his confusingcombination of ray and wave analysis. 33 BREWSTER, 1839, 781. 34 BREWSTER, 1839, 781. Later POWELL reported that he also found the polarity phenomenon in the interference spectrum; see POWELL, 1839b, 795. 372 X. CHEN

In his Bakerian lecture, AIRY provided both a qualitative and a quantitative explanation of BREWSTER'S phenomenon, but these two explanations inconsist- ently categorized and analyzed the phenomenon. In his qualitative account, AIRY, like others before him, did argue that bands would form on account of interference, but only if the spectrum were, as AIRY now supposed, imaged out of focus on the retina. For in that case the homogeneous light from a given point in the spectrum would be imaged to a disk on the retina, and dark bands would form in the disk as a result of interference between light from the point that had passed through the plate and light that had not. These particular interference bands, however, would not be visible precisely because the pris- matic spectrum was out of focus in BREWSTER'S experiment, since the bands from each disk would then confusedly overlap one another. However, AIRY continued, the bands produced by each luminous point in the spectrum were not in fact symmetrically distributed along any line through the center of the image. Because, he argued, the extent of this asymmetry depends on the retardation caused by the plate, bands produced by different luminous points (with different colors) had different degrees of asymmetry. The position of the plate, namely that it covered the red or the violet end, could therefore affect the net effect of overlap among the bands in each image disk. If the thin plate covered the violet end of the spectrum, he wrote, "the bands produced by all these luminous points may be made to coincide, and thereby to produce strong bands, in the confused spectrum formed by the aggregate of all the indistinct images," but if the plate covered the red end, "the bands will be removed further from coincidence than before, and all trace of them Will be lost in the confused spectrum formed by the aggregate. ''35 In this way, AIRY attributed the appearance of the bands to the accumulation of individual luminous images, and explained the polarity phenomenon in terms of either the coincidence or separation of these mutually independent band-images -- a typical form of ray analysis. Nevertheless, AIRY in computing adopted a different approach based on waves instead of rays. He began with equations for each of the unretarded and retarded waves. Since every part of the wave, after refraction through the lens of the eye, originated a wavelet, AIRY integrated these disturbances over the surface of the lens. He thereby determined the intensity of light produced by a single luminous point. After a cumbersome calculation, AIRY found that the intensity of light produced by a luminous point could be represented by the following function:

I = 2 -- G(s)cos (p(s) + 6(s)cos Eqo(s) - R] 36

35 AIRY, 1840, 227. AIRY did not specify how the different positions of the thin plate altered the internal patterns of the bands. He in fact needed to reveal the physical foundation of such an effect, and to determine the exact distribution of these bands. AIRY tried to answer the second question in his 1841 paper, but never gave a hint to the first one. 36 Here G and q~ are integral functions, s is a constant dependent on both the wavelength and the distance from the retina's center, and R is the degree of retardation. "Polarity of Light" during the Optical Revolution 373

According to standard wave practice, the next step would be to determine the total intensity of light at a point on the retina by integrating the effects produced by all of the luminous points in the spectrum. AIRY, however, did not carry through with this step, which would have required some sort of procedure for integrating over both wavelengths and distances. Instead, he used a qualitat- ive method to estimate the aggregate of the function's values with respect to different wavelengths. He concluded that, when the retarding plate covered the red end of the spectrum, the sum of the aggregates of the second and the third elements in the above function became zero. Since the intensity was therefore constant, it followed that no dark bands would be seen in this case. When however the plate covered the violet end the aggregate of the third term became a trigonometric function of the distance to the retina's center, yielding dark bands appeared. 37 AIRY'S computation was clearly built upon the notion of wave-front instead of ray, because he had considered the net effect of an infinite number of wavelets through integration. It is also clear that diffraction had become central to AIRY'S analysis, because the effects of the lens' boundaries (the aperture) were explicitly considered in the integration. It is very important to note that diffrac- tion and interference are two essentially different optical categories, although both of them produce similar patterns with alternate light and dark stripes. From the standpoint of wave analysis, diffraction occurs when the wave front is limited by an obstacle, usually an aperture, and the calculation of diffraction patterns requires an integration over the infinitesimal elements of the wave front. The best examples of interference are, however, produced by a finite (usually small) number of unlimited wave fronts, and the calculation of inter- ference patterns needs only a superposition of these enumerable fronts. It seems however that AIRY did not acknowledge the differences between his qualitative explanation, which was based upon interference and considered only a finite number of independent images, and his quantitative explanation, which relied on diffraction and involved integration. He simply put these two inconsistent versions together in the same article. BREWSTER was present when AIRY read his paper at the 1840 British Associ- ation meeting. He might not have fully understood the details of AIRY'S argu- ment, because it involved a complicated mathematical analysis, and BREWSTZR himself was particularly weak in this respect. 38 BRZWSTZR however seized on a problem in AIRY'S account. To explain the apparent polarity, AIRY has assumed that the prismatic spectrum was out of focus, so that interference bands always existed -- they were simply invisible rather than nonexistent when the thin plate covered the red end of the spectrum. BRZWST~R hoWever insisted that he had always viewed the spectrum in focus in his experiments. He had always seen sharp and distinct Fraunhofer lines, and he had even used the

37 AIRY, 1840, 225-239. 3s For BREWSTER'S own confession, see BREWSTERto BROUGHAM, 8/25/1849, Univer- Sity College London, Brougham Collection, 26.643. 374 X. CHZN lines as a reference to count the number of dark bands. AIRY'S assumption was accordingly unfounded, and his analysis fell with it. 39 AIRY in response provided a new mathematical analysis for wave fronts in a paper published in 1841. 4o When the spectrum was in focus, AIRY found that he needed to change only one parameter in his equations representing the disturbances produced by the unretarded and the retarded waves, thereby reducing the distance between the retina and the focal point of the eye's lens to zero. This change actually simplified the computation and allowed him directly to solve the integral equations that described the disturbances produced by a single luminous point. AIRY thereby obtained a simple trigonometric function for the intensity of light produced by a luminous point, which has the following form:

I = (sin c0/co)2 cos2(co - R/2). ,1 With this function in hand, AIRY might have followed standard wave prac- tice to determine the intensity of light produced by the spectrum. But he did not do so; instead, he presented a different analysis that was probably in- fluenced by his earlier, qualitative account based on rays. AIRY now tried to illustrate quantitatively how the thin plate altered the distribution of the bands, which he could not do in his 1840 paper. On the basis of the new trigonometric function, AIRY calculated the values of the intensities produced by twelve rays (with different wavelengths) that suffer different degrees of retardation, and he represented these results by a group of curves (Fig. '5). Assuming that the sum of the intensities could be determined by adding up the effects produced by the twelve rays, AIRY explained the polarity phenomenon by showing graphically how the retarding plate affected the aggregate of the curves. If the plate covered the red end, according to AIRY, the aggregate could be represented by supposing the second curve in Fig. 5 to be moved toward the right hand by 30 degrees, the third to be moved even further to the right by 60 degrees, and so on. The result was that the large ordinates of one curve would be added to the small ones of another, which would produce an approximately constant overall value. Thus, no interference bands would be visible. On the otherhand, if the plate covered the violet end, the aggregate could be represented by superposing the curves after they were moved toward the left (the second curve by 30 degrees, the third by 60 degrees, and so on). In this way, the interference effect would be enhanced, and the bands became visible. r With these graphical representations, AIRY was confident that the investigation of the apparent polarity of light "may now be considered as sufficiently complete, and (I conceive) as perfectly satisfactory.''43

39 The Athenaeum 1840, 679: 870. 40 AIRY, 1841, 1-10. 41 Here, ~o is a constant dependent on both the wavelength and the distance fi'om the center of the retina; R is the degree of retardation. 42 For more detail of AIRY's mathematical analysis, see Appendix l. 43 AIRY, 1841, 9. For more about AIRY'S graphical analysis, see Appendix 1. "Polarity of Light" during the Optical Revolution 375

Curves representing by their ordinates the values of (sin0~)2cos2 (a)_R)

Values of w

-180 -90 0 +90 +180

R = 2 n:rt"

R=2na:+60 ~

R =2nat+ 120 ~

R=2n~+ 180 ~

R=2nar+240 ~ _ --

R=2ner+300 ~ _ .-, . ~ .

R = 2 net+ 360 ~ ~ -,~-~-- ~ - 180 -90 0 +90 +180

R= 2 nJr+420 ~ .~

R = 2 nzc+ 480 ~

R=2nzr+540 o _ ~ ~r'T'~PF3~

R = 2 nzc+ 600 ~ _ ....

R=2n~+660 o ~ _ _.~-rf iI f Jl i fr ~ t - 180 -90 0 +90 +180 Values of o)

Fig. 5. AIRY'S graphical account of the polarity phenomenon. Source: AIRY, "Supplement to A paper", p. 10.

In addition to explaining the polarity phenomenon, AIRY found that his 1841 analysis had two novel results. First, he derived a reciprocal relation between the intervals of the bands and the size of the pupil, or the radius of the object glass if a telescope was used -- the larger the size of the pupil (or the object lens), the smaller the intervals between bands. Second, he found that the polarity phenomenon was sensitive to the diameter of the pupil -- if the size of 376 X. CHEN the pupil was beyond a certain limit, interference bands appeared no matter which end the plate covered, and so no polarity phenomenon could be seen. These novel predictions, however, caused AIRY trouble. Once again, BREW- STER might not have followed AIRY'S mathematical analysis, but he was able to show that AIRY'S predictions conflicted with observation. BREWSTER conducted several experiments, in which he varied "the diameter of the pupil from its greatest expansion to its greatest contraction, and the diameter of the object- glass from four inches to a quarter of an inch," but nevertheless found that the appearance and magnitude of the bands remained the same. 44 After carefully repeating the experiments, BREWSTER confidently reported that there was no direct relation between the appearance and magnitude of the bands and the diameter of the pupil or of the object-glass. 45 BREWSTER also questioned AIRY'S qualitative account, which, on the basis of the out-of-focus assumption, argued that the bands might be invisible due to the diffused images formed on the retina. Subsequently when AIRY considered the case of a spectrum in focus, he continued nevertheless to assume that a luminous point formed a diffused image in the eye. From the standpoint of geometrical optics, AIRY was obviously mistaken, because a point source should form a point image under these circumstances. 46 Thus BREWSTER asserted that AIRY'S assumption was "quite untenable," and "cannot for a moment be admit- ted." Because of these problems, both in AIRY'S qualitative and in his mathemat- ical accounts, BREWSTER concluded that "the phenomena which I discovered are still unexplained by the undulatory theory, and may still be regarded as indica- tive of a new species of polarity, till they are brought under the dominion of some general principle. ''47

4. The debate

BREWSTER'S successes in exposing the difficulties and problems in both POWELL'S and AIRY'S accounts encouraged him to launch a full-scale attack against the undulatory theory, and this attack stirred up heated debates at the British Association. Although in these debates BREWSTER alone faced almost all

44 BREWSTER,1842, 12. 45 See BREWSTER, 1842, 8. Later POWELL confirmed one of BREWSTER's experimental findings that the contraction of the pupil did not enlarge the intervals between the bands; see POWELL, 1846, 4. 46 From the perspective of wave optics, however, a luminous point, due to aberration, always forms a slightly diffused image after passing through a lens or a prism, even though it is viewed in focus. AIRY would not have needed the out-of-focus assumption in his qualitative account, if he had appealed the aberration effect. But it is not clear whether AIRY had thought closely about his issue, because he did not openly respond to BREWSTER's criticism. 47 BREWSTER, 1845, 7--8. "Polarity of Light" during the Optical Revolution 377 of the first-rank undulationists in Britain, including AIRY, POWELL, HERSCHEL, HAMILTON, MACCULLAGH, CHALLIS, and LLOYD, he made his point so success- fully that, in the end, he actually convinced some of his rivals that the polarity problem remained unsolved. BREWSTER did not concentrate his rhetoric entirely on the undulatory theory's specific failures here or elsewhere. Instead, he proposed a discussion of the general merit of the theory, though it had dominated the field for more than a decade. BREWSTERinsisted that the theory still failed to explain entire classes of well-observed and distinctly marked phenomena. One of these was his own recently discovered "polarity of light," which remained unexplained despite the best efforts of POWELL and AIRY. The second class of unexplained phenomena was the one he had discovered more than a decade ago: the phenomena of selective reflection by grooved surfaces. Here, a polished metallic surface with equal and equidistant grooves was incapable of reflecting a single ray of homogeneous light at several angles of incidence, whereas it reflected that ray freely at intermediate angles. *s Explanations of these "extraordinry facts," BREW- STER asserted, were beyond the power of the undulatory theory. Its failures here gave BREWSTER SUfficient reason to reject the theory altogether: "Notwithstand- ing the great power of the undulatory theory in explaining phenomena, and its occasional success in predicting them, I have never been able to consider it as a representation of that interesting assemblage of facts which constitute Physical Optics. ''49 BREWSTER should have expected nothing but fierce counterattacks from the undulatory camp, because he was openly challenging the status of the already- influential undulatory theory. Nevertheless, he received a qualifiedly sympath- etic response from an undulationist, namely JAMES MAcCULLAGH, professor of mathematics at Trinity College, Dublin. Educated at Trinity College, Dublin, MAcCuLLAGH completely endorsed FRESNEL'S theory in his early optical re- searches. His faith in undulatory principles did not, however, also extend to the theory's physical foundation, which he began to explore in the late 1830s. MAcCULLAGH decided that he could not develop a workable account for crystal- line reflection using the kinds of assumptions that had been put to use (with only partial success in any case) by FRESNEL himself for ordinary reflection. He instead sought to develop a general account, or what he termed a "dynamical theory," for reflection and refraction, by working with a potential function for an elastic solid. 5~ MAcCULLAGtt found that he had to dispense with terms in the general expression for the potential that had been developed by GEORGEGREEN, the implication being that the appropriate medium could not be considered elastic in the usual sense. In a letter to HERSCHELin 1846, MAcCuLLA~H wrote

48 BREWSTER, 1829, 301-16. BREWSTERinsisted that these selective reflections had nothing to do with interference and diffraction, and remained problematic for both optical theories. 49 BREWSTER,1845, 7. 5o MACCULLAGH,1839 (reprinted 1880), 187-217. 378 X. CHEN that "I have thought a good deal (as you may suppose) on the subject -- but have not succeeded in acquiring any definite mechanical conception ... One thing only I am persuaded of, that the constitution of the ether if it ever should be discovered, will be found to be quite different from any thing that we are in the habit of conceiving, though at the same time very simple and very beauti- ful.TM Because he failed to establish a mechanical basis within the generally- accepted undulatory framework, MACCuLLAGH'S view of the theory's integral structure became more skeptical, as he insisted -- based on his own experience

-- that much remained to be worked out. After listening to BREWST~R'S complaints at the 1842 British Association meeting, MAcCULLAGH expressed his sympathy, and admitted that undulation- ists still "knew so little of the undulatory theory." The major problem, as MAcCtJLLAGH admitted, consisted in the theory's obscure physical foundation, for without a firm ground here the theory relied entirely on the application of the principle of interference. Bereft of any "physical foundation," MAcCULLAGH argued, undulationists "knew nothing absolutely of the undulatory theory," although they were able to use it to explain many things in very beautiful way. He went on to suggest that perhaps the research style of the theory, namely, its employment of purely mathematical investigation, was responsible for the ne- glect of physical inquiry, s2 Agreeing with MAcCULLAGH on this point at any rate, BREWSTER expressed his strong discontent with the current tendency in the field to overlook the importance and value of experimental inquiry. The undulatory theory had explained several "grosset phenomena," but had been supported in a way that held back optical science, by discouraging all experimental research. People who knew very little of the subject had praised the theory as perfect, and had even ventured to place it on the same level as the theory of universal gravi- tation. 53 These people, BREWSTER continued, held up those facts explained by the theory as great discoveries, while they ignored other far more interesting and valuable facts simply because they were either hostile to or unexplained by the theory, s4 To support his criticism that undulationists discouraged experimental re- searches, BREWSTER complained to the audience about one of his recent experi- ences. In 1841 he had, BREWSTER remarked, submitted a paper containing mainly experimental results on polarization to the Royal Society, but the

5t MACCULLAGHto HERSCHEL, 10/2/1846, Royal Society Library, Herschel Papers, HS. 12.11 52 The Athenaeum, 1842, 769: 662; Literary Gazette, 1842, 1332: 554. Because of their conflicts in several priority issues, MACCULLAGH saw HAMILTON as a competitor. His comment on purely mathematical investigation might have been a criticism of Hamilton's research style. For more about the personal relationship between MACCULLAGH and HAMILTON, see HANKINS, 1980, 93-94, 167-168. 53 BREWSTER here referred to AIRY, and WHEWELL. ~* Literary Gazette, 1842, 1332: 534. "Polarity of Light" during the Optical Revolution 379

Council of the Society rejected its publication. As one of the oldest members of the society, and author of more than thirty papers in the Philosophical Transactions, BREWSTER felt humiliated. He believed that the Council rejected the paper solely because it was experimental and contained results and views hostile to the undula- tory theory. This rejection was a clear indicator that the process of discouraging experiment researches had spread to such an extent that "even learned societies were so completely under the incubus of the undulatory theory.''Ss Among the undulationists who listened to BREWSTER'S attack, MAcCULLAG~ was the only one who was sympathetic. Most could accept neither BREWSTER'S critiques nor the doubt cast on the theory. Nevertheless, they admitted, as BREWSTER had claimed, that neither AIRY nor POWELL could explain the "polar- ity of light," and that the phenomenon was still a problem for the undulatory theory. What they did was to reduce the damage, arguing that this was a purely local, minor failure. For example, HERSCHEL asked the audience to suspend their judgment, not to put the theory on trial for life or death based just upon BREWSTER'S discovery. Similarly, HAMILTON reminded the audience that, al- though undulationists considered BREWSTER'S discovery to be (at present) in- explicable, "it would not be supposed that the wave men were wavering, or that the undulatory theory was at all undulatory in their minds." HAMILTON said that at least the Dublin wave men, retained as strong a conviction as ever of the substantial truth of the undulatory theory. 56 Even the most stubborn undulationists such as AIRY and POWELL recognized the compelling power of BREWSTER'S critique. Although they did not accept BREWSTER'S charges, and felt that they had been able to explain the phenomenon, they knew that their arguments were weak. Thus, after BREWSTER presented his critique at the 1845 British Association meeting, AIRY was clearly not willing to carry on the debate. He complained that he was not aware of BREWSTER'S plan to discuss the subject until he saw the announcement about half an hour before the meeting began, and he said that his memory on the subject was so imperfect that he did not even remember the details of his own account. AIRY accordingly declared that under these circumstances he was totally unprepared to debate the matter, and refused to have any substantial discussion with BREWSTER.57

5. The solution

BREWSTER'S victory at the 1842 and the 1845 British Association meetings made the problem of the "polarity of light" a significant, recognized issue and

55 BREWSTERbelieved that AIRY, who acting as referee of the Philosophical Transac- tions, was responsible for the rejection and had done this entirely from his personal feelings. See BREWSTERto BROVaHAM, 12/4/1841, University College, London, Brougham Collec- tion, 26.624. 56 The Athenaeum, 1842, 769: 662; Literary Gazette, 1842, 1332: 534. 57 The Athenaeum, 1845, 924: 699. 380 X. CHZN

Refractive liquid Thin plate The eye

/ / Hollow prism The lens

Fig. 6. POWELL'Sexperimental design. forced undulationists to continue their researches on the matter. POWELL in this period continued to work on the "polarity" problem. Around 1847, he de- veloped a new experimental apparatus that, he felt, could still produce the polarity. The device consisted of a hollow glass prism containing highly refrac- tive liquid and a plate of glass. The plate of glass was inserted vertically, covering the upper half of the liquid (Fig. 6). POWZLL believed that his appar- atus was similar to the one used by BREWSTER: he changed only the position of the thin plate, moving it from in front of the eye to within the prism. Although he did not use a telescope to view the spectrum as BREWSTER had, he apparently believed that these alterations did not result in important differences. The apparatus, however, revealed something new. Surprisingly, with some combina- tions of retarding medium and liquid, such as glass with water, or glass with oil of turpentine, the dark bands crossing the spectrum disappeared entirely, just as B~EWSTE~ had discovered when the plate covered the red end of the spectrum. This new disappearance of the dark bands could hardly be attributed to a "polarity" of light. According to POWELL, interference between the retarded and unretarded light accounted for both his new discovery and BREWSTER'S polarity phenomenon. POWELL sent a paper to the Royal Society reporting his discovery. 5s Since he classified his finding as a rather simple case of the well-known phenomenon of interference, he had to justify the significance of his work. At the beginning of his paper, POWELLadmitted that, given the advanced state of the theory of light, his topic -- a case of interference of unpolarized light -- could hardly be deemed of sufficient importance to form the subject of a paper for the Society. However, POWELL claimed that the matter he was going to discuss was a case "which by no means stands isolated, but offers analogies with other classes of phenomena which have excited considerable interest and discussion, especially with regard to what has been termed, perhaps improperly, a 'polarity' in the prismatic rays.''59

58 POWELL, 1848, 213--26. 59 POWELL, 1848, 213. "Polarity of Light" during the Optical Revolution 381

POWELL'S research, which was mainly experimental, drew the attention of GEogaz STOKES, one of the most skillful undulationists of the day. STOKES had graduated from Cambridge in 1842, and he was one of a new generation of proponents who had received their education after the undulatory theory was already embedded in university curricula in Britain. He had not himself experi- enced the conversion from the old theoretical framework to the new one. Unlike those from the older generation who had very diverse interests and seldom specialized in a narrow field, STOKES' intellectual concerns were highly specialized, focusing mainly on those physical subjects that required mathemat- ical analysis. With this superb analytical skills, STOKES was able to handle very complicated phenomena that had escaped the powers of his predecessors. In correspondence with POWELL between 1847 and 1848, STOI(ES gave a detailed mathematical treatment of the "polarity" phenomenon. His analysis was so fertile that POWEJ~L later complained of an "embarras de richesses" and sugges- ted that STOKES write a separate, summary paper. 6~ Accepting POWELI?S sugges- tion, STOKES Wrote a paper summarizing his analysis and sent it to the Royal Society. 61 In the first section of his paper, STOKES reviewed several existing explana- tions of the polarity phenomenon, primarily POWELL's and AIRY'S . According to STOKES, both POWELL and AIRy in their explanations had analyzed the pheno- menon in terms of rays, adding only the principle of interference. They had also assumed that the only modification occurred when rays passed the prism and the retarding plate, and that the form and magnitude of the aperture (that is, the object lens of the telescope or the pupil of the eye) need not be taken into consideration. They accordingly attributed the phenomenon to the interference between retarded and unretarded ray. 62 STOKES labeled both POWELL'S and AIRY'S accounts "imperfect theory of interference." Their problems, as STOKES remarked, lay in two unacceptable assumptions. First, the AI~v-PowELL accounts supposed an annihilation of light when the phases of two interfering rays were in opposition. According to STOKES, "light is never lost by interference... The effect of interference [is], not to annihilate any light, but only to alter the 'distribution of the illumination'.''63 Thus, bright bands would also appear in the spectrum if interference were really the cause, but that had not been seen in the experiment. Second, the Amy- POWELL accounts supposed that the wave of light, after passing through the lens of either the eye or the telescope, still had an unbroken front. They had accordingly treated waves in effect as though they were rays following the courses given them by geometrical optics, and they had attributed the pheno- menon to the interference between two or a finite number of rays. However, it was incorrect to ignore the role of the aperture in these experiments, because

60 LARMOR,1907, 115. 61 STOKES, 1848, 227-42. 62 STOKES, 1848, 228-30. 63 STOKES, 1848, 234--35. 382 X. C~e>~

Thin plate / Refractive liquid n / ~erture The eye r

Hollow prism, t Telescope The lens

Fig. 7. STOKES' experimental design. both the object lens of the telescope and the eye's pupil could alter the shape of the front. With broken fi'onts, waves could not be treated as rays, and the cause of the polarity phenomenon was no longer interference. To unpack the "polarity" on correct grounds, STOKES first replicated the experiment. He improved POWELL'S design by using a telescope to view the spectrum, which was p!aced behind the retarding plate but in front of the eye (Fig. 7). This new experimental design did not yield any new result, but it demonstrated that the object lens of the telescope played a very important role, for it functioned like a diffracting aperture. An aperture also existed in POWELL'S experiment, but that was the puPil of the eye. BREWSTER'S experiment had employed a telescope, but it had been put in front of the retarding plate. According to STOKES, only the aperture behind the retarding plate was critical in altering light distribution. Thus, it was again the eye's pupil that functioned as an aperture in BREWSTER'S experiment. The crucial role of the aperture in BREWSTER'S and POWELL'S experiments, as STOKES saw it, had been ignored because the effect of the eye's pupil was usually overlooked. But when a tele- scope was used and was put behind the retarding plate, the function of the object lens as an aperture was clearly brought out. STOKES argued that both the formation of the dark bands and their disappearance were the results of the redistribution of illumination triggered by the retarding plate and the object lens: "the explanation of the polarity of the bands depends on diffrac- tion. ',64 In the second section of his paper STOKES calculated the distribution of light in the experiment on the basis of the integral formulation originally developed by FRESNEL. Starting with a point source emitting homogeneous light, STOKES first determined the disturbances produced by the unretarded and retarded fronts after passing through the aperture, and the intensity of light caused by these disturbances when the spectrum was viewed in focus. Up to this point, his

64 STOKES, 1848, 229. "Polarity of Light" during the Optical Revolution 383 results were essentially identical to AIRY'S presented in the 1841 paper. 65 Differ- ences emerged after the next steps. Unlike AIRY, who worked with a finite set of rays, STOKES worked with continuous fronts. He took a next step to calculate the intensity of light with the source being a line of homogeneous light, by integrating the intensities caused by all luminous points on the line. Finally STOKES determined the intensity of light with the source being a spectrum, by integrating the intensities caused by all homogeneous lines in the spectrum. 66 The results of these calculations were very impressive. STOKES was able to derive a formula which showed that, when the retarding plate covered the violet end of the spectrum, the intensity of light at the retina was a trigonometric function of the distance to the retina center, in such a manner that dark bands appeared alternately. The formula also showed that, when the retarding plate covered the red end of the spectrum, the intensity was constant and no dark bands could be seen, whatever the size of the aperture (the pupil or the object lens) might be. This result exactly accommodated BREWSTER'S experiments, and corrected one of the mistakes made by AIRY, whose mathematical analysis implied that, under certain conditions, dark bands could appear when the plate covered the red end of the spectrum. STOKES' calculation also clarified a con- fusion caused by AIRY. AIRY'S analysis predicted that the interval between bands could become broader by contracting the aperture of the pupil, but neither BREWSTER nor POWELLwas able to observe such an effect. According to STOKES' analysis, AIRY had incorrectly assumed that his computation could be applied under all conditions. On the contrary, the reciprocal relation between the size of the aperture and the width of the interval held only when the thickness of the retarding plate was at a certain optimal value. This explained why neither BREWSTER nor POWELL had been able to confirm AIRY'S prediction: they prob- ably used plates with other thicknesses. STOKES had thus demonstrated that the polarity phenomenon was completely explicable by the undulatory theory. After his temporary victory at the 1845 British Association meeting, BREWSTER had continued to keep his eye on the issue of polarity, and he did not miss POWELL'S and STOKES' findings. BREWSTER might not have fully understood STOKES' mathematical analysis, but he definitely grasped the implications of POWELL'S experimental discoveries, which contradicted BREWSTER'S understand- ing of the phenomenon. The fact that the dark bands disappeared at certain combinations of the retarding medium and the liquid conflicted with BREWSTER'S notion of polarity, which referred to an optical asymmetry sensitive only to spatial variation. BREWSTER'S taxonomy of optical asymmetry could not incor- porate this new experimental discovery, and so POWEEL'S experimental appar- atus stimulated BREWSTER to develop a new conception. In 1847, BREWSTER

65 STOKES'result was also a trigonometric function, but more complicated than AIRY'S because STOKES considered the situation when the retarding plate did not cover exactly one half of the spectrum. If the plate covered exactly one half of the spectrum, STOKES' result was identical to Airy's except a differencein the constant; see Appendix 2. 66 For the detail of STOKES' integration and the results, see Appendix 2. 384 X. CHEN found that the edges of thin plates could produce diffraction bands similar to those in the polarity experiment. 67 POWELL'S experimental apparatus, in which the thin plate was inserted in a hollow prism filled with liquid, now suggested to BREWSTER that the dark bands in the polarity experiment might actually be caused by the edge of the thin plate. After carefully studying the internal diffraction fringes produced by fine objects such as a needle, BREWSTER finally convinced himself that the phenomenon he had considered as indicating a new species of polarity of light was merely the internal diffraction fringes produced by the edge of the thin plate, and rendered visible by the action of the prism. BREWSTER openly announced this new view at the 1852 British Association meeting, and in his 1853 edition of A Treatise on Optics. 6s After a more than ten-year heated dispute that had involved almost every major actor in the field of physical optics, the debate on the polarity of light finally ended in 1852 when BREWSTER openly admitted that the phenomenon was not a problem for the undulatory theory, although he still did not accept the current undulatory explanation, which attributed the phenomenon to the diffraction produced by the aperture.

Conclusion

The extended debate concerning the "polarity of light" was hardly a mar- ginal one since so many important practitioners in the field were actively involved. On the one side there was DAVID BREWSTER, the most prestigious opponent of the undulatory theory. On the other side, much of the undulatory camp turned out in full force. Almost every first-rank undulationist in Britain, including GEORGE AIRY, BADEN POWELL, WILLIAMWHEWELL, JOHN HERSCHEL, HUMPHREY LLOYD, WILLIAMHAMILTON, and JAMES MACCULLAGH, in one way or another took part in the fight. BREWSTER'S success in this debate, though temporary, was very unusual, considering that the undulatory theory of light had established its superiority in mathematical explanation and novel prediction more than a decade ago. The polarity debate suggests that the result of a scientific debate does not always coincide with the judgment of those factors articulated by scientists of the day themselves, in particular their evaluation of explanatory power. Scientific prac- tice involves other factors as well, factors that are seldom articulated in scient- ific discourse. The focus of the debate on the "polarity of light" was not justification or comparison at the level of explanation. As we saw, individual anomalies hardly troubled the undulatory theory after it became dominant in the mid 1830s. BREWSTER evidently understood this, and he saw that, to have an impact, he had to turn his singular anomaly into an entirely new class of phenomena. The

67 BREWSTER, 1847, 33. 68 BREWSTER, 1852, 25; 1853, 170. "Polarity of Light" during the Optical Revolution 385 critical step in BREWSTER'S attack was therefore his introduction of a novel taxonomy for optical asymmetry with "polar refrangibility" as one of the major categories, though he never explicitly discussed the point. Similarly, the temporary setback BREWSTER gave the undulationists had nothing to do with the intrinsic power of the theory. FRESNEL had developed effective physical models and mathematical procedures for solving problems related to interference and diffraction as early as the mid 1820s. Through HERSCl~EL'S introduction, most undulationists in Britain in the mid 1830s knew how to apply FRESNEL'S analysis to common problems related to interference and diffraction. The direct cause of AIRY'S and POWELL'S failures in explaining the polarity phenomenon was not that they did not understand FRESNEL'S models, but that they incorrectly categorized the phenomenon as interference due to refraction rather than diffraction by aperture. The debate on the "polar- ity of light" thus shows the important role of classification in the development of science. Classification is often the precondition for the evaluation of an explanation. Whether a theory can explain a particular phenomenon, or whether it can be justified by certain kind of empirical evidence, frequently depends on the underlying taxonomic system. The debate on the "polarity of light" also shows the important role of analytic skills. For example, BREWSTER'S new taxonomy of optical asymmetry that classified the polarity phenomenon as a special kind was fully consistent with his own skills, which were grounded on ray analysis. From a ray-based perspective, new properties could be added without undermining the identity of these fundamental objects of theory, because rays were not assumed to be individually visible, but rather to have observable effects only when grouped together in populous collectivities. Thus, it was legitimate to assign rays such properties as side and degree of refrangibility, and to classify the polarity phenomenon as a special kind of asymmetry similar to polarization. POWELL'S and AIRY'S own mistakes in classification, that is, their assumption that the polarity phenomenon belonged to interference, were also related to their skills. Both POWELL and AIRY had used outdated ray analysis in their undulatory explanations by assuming that waves followed exactly the courses assigned to them by geometrical optics. Because of this assumption, they had ignored the critical role of the aperture, and had focused entirely on the modifications caused by the prism and the retarding plate, and so they had naturally classified the phenomenon as one of simple interference. Similarly, the key to STOKES' final success in ending the debate was not that he had developed any new explanatory model or procedure, but rather that he was expert in analysing fronts. From the perspective of fronts, the aperture (either the object lens of the telescope or the pupil of the eye) was critical, because it could alter the shape of the wave. The polarity phenomenon became, in STOKES' hands, the result of the redistribution of illumination caused by the boundaries of the prism, the retarding plate, and the aperture, and was therefore to be classified as diffraction. Its explanation thereby became simply conventional. The specific experimental designs were also critical in the debate. AIRY'S analysis was based upon BREWSTER'Sexperimental design, in which the telescope 386 X. CHEN was put in front of the retarding thin plate. However, STOKES' analysis was built on a new experimental arrangement, in which the thin plate was put in front of the telescope. These two designs produced the same observational result, and STOKES showed that his mathematical solution could be applied, with a very trivial modification, to BREWSTER'S experiment. 69 Yet the arrangements were significantly different. In BREWSTER'S experiment, the eye rather than the tele- scope's object lens functioned as an aperture, but the role of the eye was usually ignored in physical optics. BREWSTER'S experiment accordingly exhibited many similarities to an exemplary case of interference -- YOUNG'S double slit experi- ment: in these two experiments the light from a single source was split into two parts and then recombined. Here apertures played no role. In STOKES' experi- ment, by contrast, the object lens of the telescope functioned as the aperture, the impact of which was much more easily recognized. This arrangement exhibited strong similarity to an exemplary case of diffraction -- FRESNEL'S experiment with a circular aperture. Thus, it was natural for STOKES to classify the phenomenon as diffraction, but difficult for AIRY to do so. The debate on the "polarity of light" clearly indicates the critical role of classification, analytic apparatus, and experimental design in scientific debates. One thing common to these factors is that they were not fully articulated in the debate. Excepting STOKES, no one in the debate gave any argument to justify their classification of the subject matter. The reason for adopting a specific analytic apparatus also remained tacit in the whole debate. The issue of analytic apparatus was so deeply buried that even AIRY did not realized the inconsist- ency between his qualitative and qunatitative accounts. Similarly, except for brief descriptions, no one in the debate gave any justification for their experi- mental designs, probably because designing experiments involved activities of knowing how, or applications of skills, which were difficult to be precise about. Thus, to have a full historical understanding of the long-term debates during the revolutionary change of optics, it is necessary to go beyond the limits of explicit arguments, wherein discussion hovers about physical models or ex- planatory 15ower. We instead need a historiographical perspective that fully appreciates the importance of the many unarticulated factors involved in de- bate, an incomplete list of which includes the practice of classification, the application of analytic skills, and the process of experimentation of the sort that the polarity debate so well illustrates.

Appendices

1. Airy's mathematical analysis (1841)

1) A~RY first calculated the intensity of light produced by a point source. In Fig. 8, the wave of light, when it reaches its focus after passing the lens of the

69 STOKES, 1848, 240. "Polarity of Light" during the Optical Revolution 387

N (x,y) Y

M (a,b)

/ Z Thin plate / / X The lens The retina

Fig. 8. AIRY'S mathematical analysis.

eye, has a spherical surface with radius c. With the assumption that the spectrum is viewed in focus, the distance of the retina from the lens is also c. Consider a fixed point M on the retina with the ordinates (a, b), whose distance from the center of the lens is e. Consider a point N on the spherical surface, whose coordinates are (x, y). The distance between N and M is approximately e - (by~e). Note that every part of the spherical surface is simultaneously the origin of a small wavelet. The disturbance at M produced by a small portion of the spherical surface (dx dy) at N can be represented by:

sin2-ff(vt-~)dxdy=sin2--ff(vt-e+!y)dxdy. (1)

Suppose that the retarding plate covers one half of the lens with the radius of h. The disturbance at M produced by the unretarded front is:

0 h f fsin(v,-e+!y)dx+. (2) -h -h The disturbance at M produced by the retarded front is:

h h

f f -T e +!y)-R]dxdy. (3) o -h Here R is the retardation measured by phase difference. 388 X. CHEY

By solving these two integral equations, AIRY determined the intensity of light at M produced by the whole front, which is:

rcbh I = 4hZ(~)2cos2(co _R) where co- 2e" (4)

This expression also represents, according to AIRY, the intensity produced by a line source parallel to the edge of the retarding plate. 2) Next AIRY determined the intensity of light produced by a spectrum. Instead of using integration, AIRY computed the values of intensity for every 10 degrees of co and every 60 degrees of R. He summed up the results in a table (Fig. 9). Because co reflects the distance of M to the center of the retina, every row in the table shows the intensities at a certain point of the retina. Because R is related to wavelength, every column in the table displays the intensities produced by light with a particular wavelength. AIRY further presented these results in a group of curves, which we have seen in Fig. 5. Because the numbers in the table recur after every six columns, AIRY expanded his numerical results into twelve curves, representing the inten- sities produced by twelve light beams. When the retarding plate covered the red end of the spectrum, AIRY reason- ed, the aggregate could be represented graphically by superposing twelve curves in Fig. 5 together after moving the second curve toward the right by 30 degrees, the third by 60 degrees, and so on. The result was a flat line, because the large ordinates of one curve had been added to the small ones of another. The same aggregation could also be represented numerically, by combining the last num- ber of the first column in the table (Fig. 9) with the last but three in the second column, the last but six in the third column, and so on. The result also proved that the intensity of light distributed evenly in the retina, and no interference bands would be seen. When the retarding plate covered the violet end, the aggregation could be represented by similar ways. Graphically, the aggregation was equal to super- posing all curves after moving the second curve toward the left by 30 degrees, the third curve by 60 degrees, and so on. Numerically, the aggregation was equal to combining the first number of the first column with the fourth in the second column, the seventh in the third column, and so on. In both cases, all minimum ordinates corresponded to each other, and so did the maximum ordinates. Thus, the interference bands could be vividly seen.

2. Stokes' mathematical analysis (1848)

1) Like AIRY, STOKES also began with a calculation of the intensity of light at M produced by a luminous point. In this part, there were no substantial differences between STOKES' and AIRY'S analyses, except that the former con- sidered several more complicated situations, which included, for example, the retarding plate did not divide the spectrum evenly. STOKES also obtained "Polarity of Light" during the Optical Revolution 389

Table of (-~)2 cos2 (o9 -R )

Values of R Values of (o 0 ~ 60 ~ 120 ~ 180 ~ 240 ~ 300 ~

- 17g ~5 20 8 0 4 ]7 - 165 2~9 230 is 17 17 123 - 155 610 738 499 133 6 245

- 145 lO5O 1553 1~86 51s be 280 -- 135 1372 2560 2560 1372 184 184 -- 125 1413 3527 4s 2as2 767 zz -- 115 1109 4168 6164 5101 2043 47 - ]05 567 4s 7896 7896 4231 I 567 -- 95 84 3618 9032 10912 7378 1964 -- 85 104 2453 9216 13631 I]28~ 4519 -- 75 1111 1111 8~94 15476 15475 8294 -- 65 3473 148 6396 15970 19294 13046

-- 55 7~98 169 3962 14884 22013 18221

-- 45 12346 1654 1654 12346 23038 23038

-- 35 180fO 4796 204 8835 ~2060 ~6652 -- 95 23474 9402 s 5104 19175 28360

-- 15 27778 14887 1994 1994 14886 s -- 5 30154 20389 5427 23] 9996 24959 + 5 30154 24959 9996 231 5427 20388 + 15 27778 s 14887 1994 1994 14887 + 25 s s 19175 5104 2]7 940s + 35 18020 s s 8835 204 4794 + 45 12346 23038 ~3038 12346 1654 ]654 + 55 7~98 18s 22013 I4884 3962 169 + 65 3473 13045 19s 15970 6396 148 + 75 1111 8294 15475 15475 8294 1111 + 85 104 4519 11s 13631 9216 2453 + 95 84 1964 7378 10912 903s 3618 + 105 567 567 4231 7896 7896 4231 + 115 1109 47 9043 5101 6164 4168 + 125 1413 33 767 s 4261 3527 + 135 1372 184 184 1372 2560 2560 + 145 1050 ~79 1s 515 1~86 1553 + 155 610 s 6 133 499 738 +165 229 123 16 17 1~3 230 + 175 25 17 ~ 0 8 ~0

Fig. 9. AIRY's numerical table. Source: AIRY, "Supplement to A paper", p. 4.

a trigonometric function representing the intensity of light at M produced by the whole front. If we replace StoKEs' initial conditions with AIRY'S, STOKES' expression can be rewritten as the following: 4h2 . 27rbh dTzbh R) I = 4Q~ sin ~c c~ \~cc (5) where / 2c . 2ztah'~2 390 X. CHEN

2) STOKES next determined the intensities produced by a line of homogene- ous light parallel to the edge of the retarding plate, or the distribution of intensity along the x axis. Because each element of the line must be regarded as an independent source, an aggregation along the x axis is needed to determine the intensities due to the whole line. This requires integrating (5) with respect to x from -oo to + oo (note that the constant a in (5) now becomes the variable x). The result is:

+co l' 4h2 2 rcbh 2[~bh 2rchx~2 = sm G cos R) f I2@hxSin-~-c j dx

--oo

- ~ sm ~cos 2\2c " (6)

3) Finally, STOKES determined the intensities produced by a spectrum, or the distribution of intensity along the y axis. This requires integrating (6) with respect to y from -oo to + oo (note that the constant b in (6) now becomes the variable y). Because the retardation also changes with respect to y, it can be represented by the following approximation: R = R' + wy.

Here, R' is the retardation at the center of the spectrum, and w is a constant that is positive when the retarding plate covers the violet side but negative otherwise. The intensity function produced by a spectrum then becomes;

+oo f82chZ. 27chy2 (Tchy2 R'+wy)2~h 2 R' I" = T sin ~ c~ \ 2c 2 dy = --)~c + 27~ cos ~-. (7)

-oo

Here,

~ch 2~ch T= fsin cosgx~Xz; 9=w-- 2-~-" By differentiation with respect to g, STOKES proved that 7~ in (7) was zero when w was smaller than zero and larger than a certain value (4rch/2c); 7j was a positive constant when w was between zero and that particular value. The integrated intensity is thus:

2Tc2h 4rch I"- , when -oo

2rc2h 4~h I"- +2OcosR, when 0

Hence, when the plate covers the red end of the spectrum, that is, when w is negative, the intensity is constant and no bands will be seen. When the plate covers the violet end, that is, when w is positive (given that its thickness is not over a certain limit), the intensity changes with respect to y and bands appear.

Acknowledgment. I would like to thank JED BUCHWALD for his critical comments on earlier drafts of this article, which helped me strengthen the arguments and avoid a number of mistakes.

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California Lutheran University Thousand Oakes, California

(Received April 3, 1996)