# Scientific Memoirs/1/Experiments on the Circular Polarization of Light

From the Annalen der Physic und Chemie Second Series, vol. v. p. 579
Article III.
Experiments on the Circular Polarization of Light. by H. W. Dove.
From J. C. Poggendorff's Annalen der Physik und Chemie; Berlin, Second Series, vol. v. p. 579.
1. Circular Polarization of Light by Compressed Glasses.
WHEN two systems of waves, of equal intensity, propagated in the same direction, and polarized perpendicularly to each other, differ in their path by an odd number of quarter-undulations, the particles in the resulting system of waves will describe small circles of a similar velocity around their points of equilibrium; that is to say, the light will be circularly polarized. Every means of equally satisfying these two conditions, namely, that of the similar intensity of the system of waves polarized perpendicularly to each other, and that of the determinate difference of path, consisting of an uneven number of quarter-undulations, will therefore furnish a method of circularly polarizing light. Fresnel and Airy have effected this in different ways. The third mode, which I shall here explain, is in practice at least as convenient as those hitherto used, and gives moreover a fuller explanation of the phænomena of compressed and cooled glasses in polarized light.
The condition of the equal intensity of the systems polarized perpendicularly to one another is satisfied by Fresnel by polarizing the incident light in a plane which forms an angle of 45° or 135° with the plane of the total reflexion in a glass parallelopiped. The quantities of light polarized in, and also perpendicularly to the plane of reflexion, are then, according to Fresnel's formula of intensity, equal to each other. He obtains the difference of phases of a quarter-undulation by twice-repeated total reflexion, since after a single one under the given circumstances the periods of vibration of the reflected waves no longer coincide, but exhibit a difference of phases of an $^1$⁄$_8$-undulation.
$a$
$b$
$I_{o}$
$I_{e}$
* ${\begin{aligned}I_{o}&=\cos ^{2}b-\sin 2a\sin 2(a-b)\cos ^{2}\pi \left({\frac {o-e}{\lambda }}\right)\\I_{e}&=\sin ^{2}b+\sin 2a\sin 2(a-b)\sin ^{2}\pi \left({\frac {o-e}{\lambda }}\right),\end{aligned}}$
$\lambda$
$o-e$
$a=45^{\circ }$
* ${\begin{aligned}I_{o}&=\cos ^{2}b-\cos 2b\cos ^{2}\pi \left({\frac {o-e}{\lambda }}\right)\\I_{e}&=\sin ^{2}b+\cos 2b\sin ^{2}\pi \left({\frac {o-e}{\lambda }}\right).\end{aligned}}$
* $o-e=\left[\left({\frac {2n-1}{4}}\right)\lambda \right],$,
* ${\begin{aligned}I_{0}&=\cos ^{2}b-{\frac {1}{2}}\cos 2b={\frac {1}{2}}\\I_{e}&=\sin ^{2}b+{\frac {1}{2}}\cos 2b={\frac {1}{2}}.\end{aligned}}$
The difference of path $o-e$ depends on two quantities; on the thickness of the plate, to which it is in direct proportion, and on the difference of velocity of the two rays which pass through the plate, that is to say, on the constant of double refraction/
Airy's method consists only in varying the thickness of the plate by splitting it, whilst the double refraction remains the same, until the difference between the paths of the rays is equal to an uneven number of quarter-undulations. As biaxal mica under a perpendicular incidence of the light is similar to an uniaxal crystal and best allows splitting into larger plates, its application will therefore be preferable. I, on the contrary, alter the double refraction of the substance, whilst the thickness remains the same, until the required difference of path is obtained.
To alter the refraction of rays in a crystallized lamina by pressure or change of temperature, so that it may exhibit the desired effect in a given thickness. would afford no convenient practical arrangement. It is, however, very easy by means of pressure or cooling to change the uncrystallized into a double-refracting body, which gives precisely the required effect. In the apparatus proposed by Fresnel, consisting of ​four prisms, by whih the double refraction of the glass is directly indicated, one of the two images which arise is polarized parallel to the axis of compression and the other perpendicular to it; whence it follows that the axis of the double refraction coincides with the axis of compression. If a square or circular plate of glass therefore is compressed so that the axis of compression forms an angle of 45° or 135° with the plane of primitive polarization, the light passing through the centre of the glass at a certain degree of the pressure will be circularly polarized. Let us now suppose a division of a circle so placed upon the incident ray that the plane of polarization passes through the points 90° and 270°; then, if the axis of compression passes through 45° and 225°, a plate of Iceland spar cut perpendicularly to the axis exhibits in the light passing through the centre of the compressed glass, instead of the black cross, rings in the second and fourth quadrants (on the right side above and on the left side below) advanced forwards by a quarter-interval from the centre, and on the contrary in the same proportion approaching nearer to the centre when in the first and third quadrant (on the left above and on the right below). Exactly the reverse takes place when the axis of compression passes through the points of division 135° and 315°. Hence we see that the angles which in the parallelopiped of Fresnel are formed by the plane of the twice-repeated total internal reflexion with the plane of primitive polarization, must be equal to the angles under which the plane perpendicular to the axis of compression is inclined towards the plane of primitive polarization, when the same phænomena are to be produced by both those arrangements.
No further particular explanation is now required to show that during a complete revolution of the plate in its plane round the perpendicular incident ray as an axis of revolution, the light is polarized four times rectilinearly and four times circularly; rectilinearly when the compressing screw acts on the points 0°, 90°, 180°, 270°, that is to say, when the axis of compression is perpendicular to the plane of primitive polarization or lies within it; and on the contrary, it is polarized circularly when that point of action corresponds to the points of division 45°, 135°, 225°, 315°, whilst 45° and 225°, as also 135° and 315°, exhibit a similar effect.
By a combination of two compressed plates and two tourmaline plates, so that the mutually perpendicular axes of compression of the glass plates, which are between the crossed tourmaline plates, form with their axes an angle of 45°, a lamina of Iceland spar laid between the glass plates exhibits the rings without a cross with the black spot in the centre, and complementary ones on the contrary when we make the axes of the tourmalines or the axes of compression of the glass plates parallel to each other. If we make an axis of compression parallel to a tourmaline plate we obtain displacement of the rings in the four quadrants by a ​quarter-interval; but the phænomenon is in that case not reciprocal, as a revolution here takes place similar to that which occurs when we look from the opposite side at an electric current in which the circuit is complete, and which is made to proceed in a circular form ; the first and third quadrant then become the second and fourth, and vice versâ. By placing the tourmaline axes and the axes of compression parallel severally to each other, we obtain the phænomena of rectilinearly polarized light.
If between the crossed mirrors we insert a round or square plate compressed to a certain degree, so that the axis of compression coincides with one of the planes of reflexion of the mirror, we see upon it a black cross with white vacant spaces at the corners. If by means of the plate of Iceland spar these four white vacant spaces be examined, we find that those which belong to the same diagonal are similar to each other, but in opposition to the two white vacant spaces of the other diagonal; and it will be found that the light proceeding from them is circularly polarized, in the one diagonal to the right and in the other to the left. Hence it directly follows, that when the pllate is turned in its plane 90°, all the white vacant spaces have exactly exchanged their effect in the diagonals. The plates I made use of in these experiments were 11½ lines in diameter, and 3¾ lines in thickness.
2. Circular Polarization by Cooled Glasses.
I carefully cooled a glass cube of 17 lines each side, so that when the diagonals of the surface of the cube turned towards the eye form with the plane of polarization an angle of 45°, it exhibited between the crossed mirrors in the centre a dark cross, and in the four corners only the white surrounding it. The light of the four white vacant spaces was exactly similar to the light of the four white vacant spaces of the compressed plate, when their axis of compression lay perpendicularly to, or within, the plane of polarization. By turning the cube excentrically round the ray perpendicularly escaping through one of the white vacant spaces, as round an axis of revolution, similar variations are produced, whilst at 90° revolution the diagonals interchange their effect. Instead of turning the cube round, it may, in order to obtain the same variations, be so moved that two of the parallel sides of the surface turned towards the eye are carried forwards perpendicularly to their direction, whilst the other two advance in their own path. We pass from the white vacant space of the one diagonal into that of the other. The combinations of the cooled glasses, for the purpose of analysing circularly a circularly polarized light, explain themselves. In order to obtain the system of rings without the cross with the black spot in the centre, they must be combined as in Plate II. fig. 5.
So far as I am aware, we possess as yet no direct experiments upon the double refraction of the cooled glass; and as in the theory of the ​so-called moveable polarization the double refraction was not considered as a necessary consequence of the appearance of its colour in the rectilinearly polarized light, it is desirable to confirm by new experiments the proofs that these colours originate in the difference of path of the rays passing through the glass. The following therefore, for the explanation of the colours upon the principle of interference, seems to me not unimportant.
When a ray polarized rectilinearly in the azimuth of 45°, after two total reflexions in the interior of a Fresnel's parallelopiped, exhibits a difference of phase of a quarter-undulation, between the quantities of light polarized perpendicularly to each other, of uniform intensity, this difference will in this case, after four reflexions, become a half-undulation; the ray consequently will be again polarized rectilinearly, but perpendicularly to the plane of primitive polarization. After six reflexions it is again circular, but left-handed, if after the two reflexions it was right-handed, since the azimuth of the rectilinearly polarized incident light is now – 45° instead of + 45°. Finally, after eight reflexions the plane of the restored polarization coincides with that of the primitive one. The explanation of the observed phænomena of circular polarization in the above-mentioned experiments, depended upon making the difference of path of the two rays exactly equal to the quarter- undulation, by means of a determinate change of heat in the interior of the brody made use of, its thickness remaining unaltered. If this explanation is correct, precisely the same phænomena would be obtained by gradual heating as by successive reflexions in the interior of the Fresnel's rhomboid, but with this difference, that instead of the direction of the polarization varying by successive steps we should expect a continual transition through all degrees of elliptic polarization. The experiments confirm this perfectly. They must of course be made in homogeneous light.
3. Phænomena during the Heating and Cooling of the Glasses.
The apparatus (Plate II. fig. 1.) more particularly described in the succeeding paper was adjusted before a monochromatic lamp giving yellow light, so that the plate of Iceland spar in the ring $l$, cut perpendicularly to the axis, exhibited distinctly the black rings with the dark cross, when the glass cube reduced by a new heating and cooling to perfect loss of action upon polarized light, was thus intei-posed between $k$ and $o$, before the Nicol's polarizing prism. In order to heat it conveniently over a lamp, the three-sided prism or rod be, carrying all the polarizing arrangements, was placed in such a manner in its case as to bring those arrangements from their vertical situation over the rod to a position in which they projected on one side of it; their position as represented in the figure must therefore be imagined as altered 120°. In
(Improve this image)
Dove's Apparatus for the Polarization of Light.
​the ring $m$ the screw was withdrawn a turn, in order that the motion of the rings, either away from the central point or towards it, might be the more easily observed.
The lamp having been lighted, the black cross began directly to open in the centre; the circular arcs in the second and fourth quadrant receded from the central point, whilst the first and third approached it. After some time the dark, arcs of the odd quadrants exactly corresponded with the bright vacant spaces of the even ones; the light was circularly polarized,and the difference of path was a quarter-undulation. Whilst this was going on, with the exception of the points proceeding from the centre which remained black, the dark cross had become brighter and brighter. When it had entirely disappeared, the arcs, growing shorter at their ends, had gradually advanced, so that the two black spots proceeding from the centre formed with the parts approaching each other from the two other quadrants, the inner ring, separated by four bright intervening vacant spaces. All the other rings were in the same state. The figure given by the Iceland spar had thus changed, precisely as if the polarizing prism had been revolved 90°; the light was therefore polarized linearly and perpendicularly to tlie plane of primitive polarization: the difference of path of both rays was a half-undulation. On a further heating, as the difference of path became three quarters of an undulation, the light was again circularly polarized, with the difference, however, that now the rings in the first and third quadrant were the nearest, those in the second and fourth the more distant; in which case the direction of the motion of the arcs in the single quadrants naturally remained the same. Finally, when the difference of path amounted to an entire undulation, the white cross became darkened into a perfect black; the arcs previously separated closed in whole circles; the light was polarized rectilinearly in the same direction as at the beginning of the experiment. The lamp was now removed and the opposite phænomena were observed in regular succession during the cooling of the apparatus; consequently the action of the glass, becoming gradually heated from below upwards, upon the incident light, is as follows. The particles of æther, which at first vibrate rectilinearly, begin to open into ellipses, the excentricity of which diminishes continually, until they become circles. The axis which at first was the ​larger now becomes the smaller one, and vice versâ. With increasing excentricity the elliptic vibrations, which are perpendicular to the initial ones, pass directly over them. During all this process, the direction of the vibrations did not change; supposing it to have been from left to right, it remained so. When however the second rectilineal vibration opens into an elliptic one, and the direction of the motion has become inverted, the vibration now takes place from left to right, supposing it to have been before from riglit to left. The vibrations then return through circular again into the initial vibrations.
The light proceeding from the cube was now circularly analysed, by means of the interposition of a lamina of mica $f$ of a proper thickness between the plate of Iceland spar and the analysing prism. The axis of this lamina lay so that the segments of the arcs were removed from the central point to the first and third quadrants. When the cube was yet unheated, its action was thus in direct opposition to its action in the first degree of its heating. When, proceeding from this point, the rings without the cross and with the black spot in the centre were formed, this spot, on the heat being increased, divided itself into two, which removed themselves from the centre into the second and fourth quadrants, and after having passed through the figure in the circular light, closed into a circle with the arcs proceeding from the first and third quadrants, so as to produce the system of rings with a bright centre, which would have been obtained at the very beginning by turning the polarizing prism 90°. The arcs, approaching nearer to the central point from the first and third quadrants, formed then the opposite circular figure, and united themselves at last in the centre into a black spot, whilst all the arcs closed themselves into circles. In this process, the phænomena before described of the linear analyses will again be easily recognised as a conditional element, without the necessity of particularly describing the alteration in form of the rings before they disunite into separate arcs.
To make circular light incident, is simply to add to the difference of phases produced by the heated cube a constant quantity, viz. ${\frac {2n-1}{4}}$ or ${\frac {2n+1}{4}}$ undulations ; that is to say, to alter the starting-point of the experiment. Having therefore inserted the lamina of mica $g$ between the polarizing prism and the heated cube, I obtained by linear analysis the phænomena first described, and by circular analysis those last described, beginning at another starting-point.
4. Phænomena in the different Colours of the Spectrum.
$\lambda$
$\lambda$
$\lambda _{\prime }$
* $o-e=m\lambda ,\qquad o-e=m_{\prime }\lambda _{\prime },$
​then will
* $m-m_{\prime }=(o-e)\left({\frac {1}{\lambda }}-{\frac {1}{\lambda _{\prime }}}\right).$
As ${\frac {1}{\lambda }}-{\frac {1}{\lambda _{\prime }}}$ is a constant quantity for a definite substance, the difference $m-m_{\prime }$ will be proportional to the quantity $o-e$. Hence it follows, That when for one definite colour the light is circularly polarized by an interposed crystallized lamina, it may for the other colours be linearly and oppositely circularly polarized, and that the difference between the single colours increases with the thickness of the lamina and with the intensity of the double refraction.
If the incident light is circular for the centre of the spectrum, when the diiference of path is ¼ for ths centre, the light is not yet linear for the extreme limits of the spectrum. If it is here linear in the red, with a ½ undulation difference of path, in the blue it is circular. With ¾ difference of path in the red, it will, if it is circular to the right, be linear in the blue, and circular to the left in the extreme violet. Linear light in the red, with difference of path 1 , gives on the left in the green a circular light, in the indigo a linear light perpendicular thereto and approaching the circular on the right in the extreme violet; finally, on the left, circular in the red, with difference of path $^5$⁄$_4$, will give linear in the yellow, circular on the right where the blue passes into the indigo, and perpendicular to it linear at the commencement of the violet, and so forth. In order to prove this by experiment an equilateral prism of Guinand's flint glass was placed upright, so that after the removal of the condensing-lens $p$ the red end of the spectrum fell exactly upon the aperture e of the Nicol's polarizing prism. The cube had by gradual heating exhibited the phænomena which corresponded to a difference of path of ¼,½ ¾, undulation, and the other coloured rays were brought into the axis of the polarizing-apparatus, and the alteration of the Iceland spar figure examined. This might easily be accomplished without revolving the prism, as the height of the instrument may be altered at pleasure by means of the sliding-tube, as may its inclination by means of the motion of the prismatic rod. Mica plates of various thicknesses were examined in the same manner as the heated cube. The changes may be seen most beautifully when, beginning with the violet, the instrument is slowly lowered in the sliding-tube through the single colours of the spectrum. The gradual transitions are, in respect to the difference of colours from one end of it to the other, exactly the same as those which are obtained by the heating and cooling of the cube.
In the same manner the phænomena, when the incident light is circularly analysed by a mica plate inserted before the Iceland spar, are throughout similar to those before described. Instead of the homogeneous rays of the spectrum, we can of course also employ in these experiments a monochromatic lamp or absorption by coloured glasses. It is only when the light has been circularly polarized in one colour ​through a plate of definite thickness that we can determine, whether the difference of path of the two rays be or ${\frac {2n-1}{4}}$ or ${\frac {2n+1}{4}}$ undulation. If, however, the same plate is examined in the different parts of the spectrum, we obtain by the experiments just mentioned $n$ itself. It is manifest that if we wish to obtain by refraction phænomena of circular polarization in white light, it is advisable so to determine the thickness of the plate or the temperature of the glass that the difference of path for the central rays will become ¼ undulation. For this purpose I use the flame of alcohol coloured yellow by common salt or nitrate of soda.
5. Phænomena of Colours of combined Crystals in White Light.
It now becomes easy to account for the complicated phænomena of colours obtained by the insertion of a crystallized plate parallel to the axis and of any given thickness behind a crystallized lamina cut perpendicularly to the axis. For as the light is circularly polarized for one colour on the right, for the other on the left, and rectilinearly for an intermediate one, the black tufts on their two sides assume different colours: the phænomena in the even quadrants differ essentially from those in the odd ones, but the rings of colours in both are essentially different from the succession of colours in Newton's rings. The phænomenon may be previously determined from the known values of the indices of refraction, the length of waves for the homogeneous rays of the spectrum, and the thickness of the plate ; but it may also be experimentally exhibited by adjusting the condensing-lens $p$ of the apparatus so that the spectrum in the aperture of the Nicol's polarizing prism e be concentrated to white; a confirmation, the frequent repetition of which, however, is not advisable, on account of the intensity of the light of the apparatus.
6. Phænomena of Colours in Twin-Crystals.
In passing from the artificial combinations of two crystals to natural twin-crystals we have to distinguish them into three classes: namely, the axes of the united individual crystals are either perpendicular or parallel to each other, or they are inclined at some angle with one another. The section is always to be made perpendicular to the axis of one of the individual crystals. Though the first case may immediately give the phænomena just mentioned, yet, as far as I am aware, it does not occur with transparent crystals, whilst the second case may occasion the phænomena of colours with biaxal crystals only. Thus, if (as for instance in arragonite,) a very thin crystal is so united with another that its crystallographic axis lies parallel to the axis of the crystal which is divided by it into two parts, these two parts (since the optical axes of this lamella render perceptible, however small, angles with the bounding planes,) will operate as double-refracting prisms upon the light passing through these axes, because their optical axes do not lie in the plane of ​the axes of the lamella. The particular construction of this natural polarizing apparatus described by Erman, which from the thinness of the lamella exhibits the systems of rings of an unusual size, and considerably removed toward their optical axes on account of the obliquity of the surface of emergence, is obtained optically by comparing these systems of rings seen without previous polarization, in size and position, with those which evolve light previously polarized rectilinearly and afterwards analysed also around the optical axes of the including individuals, of which the one serves for the polarizing, the other for the analysing arrangement. That this last is the case, is moreover apparent from the following observation, that when a tourmaline is revolved before the crystal viewed in ordinary light, one of the systems of rings disappears alternately without changing its form. As however the phænomenon remains the same when the crystal is revolved, the same holds good for the polarizing prism, with which also the alterations of intensity of the rings agree when the crystal is viewed with the naked eye in rectilinearly polarized light. A decisive proof, however, that the individual behind polarizes rectilinearly, lies, as it seems to me, in the following fact, that the rings seen with the naked eye do not take the form which corresponds with the light when this is circularly incident.
The third case, in which the axis of the lamina growing into the other is inclined at an angle toward the axis of the including crystal, is also of importance for uniaxal crystals. The modification of the system of rings round the axis of the including crystal thus produced must coincide with that in two exactly central plates when a crystallized lamina of definite thickness is inserted between them. As that lamina may here be replaced by another similarly acting crystal, this case may be treated in the same way without difficulty. Among seven plates of Iceland spar exhibiting a deviation from the usual system of rings, I found two which produced a very regular figure, namely, a black cross with curves alternately osculating, which appeared to me to be circles and lemniscates; the interior curve was completely entwined into a figure of 8. If the plate is turned in its own plane, the interior part of the system of rings consists of four triangular vacant spaces. I obtained precisely the same phaenomena by inserting a lamina of mica of definite thickness between two plates exactly centred and producing the regular system of rings, and by turning that lamina in its own plane.
7. Experiments on Circular Polarization by other Modifications.
Fluor spar is the only crystallized substance of the regular system which I have examined with respect to the effect of an unequal distribution of temperature within the body. The fragment I used in this instance was quite colourless and transparent, 1½ inch long, and was lent to me for these experiments by Professor Weiss. At a heat in which the difference of path had become $^5$⁄$_4$ undulation in the glass cube, it ​exhibited throughout no effect upon the reclilinearly polarized light, although, in order to increase the difference of heat, I was continually cooling its upper end with sulphuric æther, whilst the lower end stood upon the hot steel plate. Sonorous plates vibrating transversely acted neither upon the linear nor the circular incident light. But it is well known that Biot obtained a flash of light between the cross mirrors by the longitudinal vibrations of long strips of glass. Although in the experiments made with reference to this, the cross of the Iceland spar figure appeared to me to open, yet those experiments stand in need of being repeated with a better acoustic apparatus.
8. Difference between the Action of Glass when it is Heating and when it is Cooling.
Two square plates 3 lines thick, the side of one 11½ lines, and that of the other 13¼ lines, produced on being heated at first a circular light on the right, and then a rectilinearly polarized one; on their cooling, however, after they had returned to the rectilinear through the circular one on the right, they produced circular light on the left. The reason of this phænomenon is as follows: The lower end of the glass plate heated upon the hot steel plate cools when the lamp is taken away quicker than the upper one, to which heat is also communicated by con- duction. After some time therefore the centre of the plate becomes its warmest part. As the lower end, standing upon the rapidly cooled conductor of heat, becomes still cooler, the warmer spot moves upwards until finally the upper angle becomes the warmest. That this is truly the reason of the phænomenon may be seen by examining the cooling plate between the crossed mirrors. The four white vacant spaces of the diagonals do not disappear on the spot where they had been formed; the lower ones rather move upwards, so that the dark cross becomes changed into two parallels, which are intersected by a perpendicular line. Finally, the central white vacant spaces dislodge the upper ones, whilst those newly arrived from below occupy the lower spot. By heating the plate so that its lower part constantly preserves the strongest heat, the progress of the phænomena must of course be more simple.
The action of a determined point of a cooled or compressed glass as a circularly polarizing apparatus, in the homogeneous rays of the spectrum, gives immediately the elements of determination for the colour which the glass presents in rectilinearly polarized light.
* ↑ Precisely the same succession of phænomena may naturally be produced by the gradual increase of pressure or its relaxation. With the plates, however, which I had employed I was able to carry it only as far as a difference of path of three quarters of an undulation in the proximity of the points of action of the screw. On applying a stronger pressure the plates broke. Now it is evident that when a cooled glass plate, which in white light exhibits a regular series of colours proceeding from black, is interposed, in homogeneous light the same phænomena will be observed in the plate of Iceland spar, if it be slowly moved along before the aperture of the polarizing prism. The thicker the plate the nearer to each other are the differently-acting vacant spaces.
* ↑ Brewster says in reference to the colours which fluor spar acquires by rapid cooling, "Fluor spar was very slightly affected."