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doubly refracting force is at a maximum, and when a ray is incident in this plane, the resulting extraordinary and ordinary rays are both in the same plane.
If a plate of Iceland spar, cut perpendicularly to the principal or shortest axis, be placed in the polariscope, the polarizing and analyzing plates being crossed, we observe coloured curves or concentric rings intersected by a rectangular black cross, the arms of which meet at the centre of the rings (fig. 32).
The coloured curves or rings are called the lines of equal tint, or isochromatic lines (from Io-os equal and xpanariKos coloured). In this and other uniaxial crystals, they are disposed in concentric circles, and are similar to Newton's rings seen by reflection.
If we revolve the plate of Iceland spar on its axis, the rings and cross preserve the same position; but if either the polarizing or analyzing plate be rotated, some remarkable changes occur.
Suppose the analyzing plate to be turned 45° round the incident ray in a left-handed direction, we observe that the original or primary coloured rings grow fainter or more dilute, and the cross seems to shift its position to the left, while its blackness lessens and is replaced by another set of rings, which alternate with, and are complementary to, the original curves (fig. 33).
If the analyzing plate be rotated 45° further in the same direction, that is 90° to the first or original position, the black cross is replaced by a white one, and the original set of coloured rings is succeeded by a second or complementary set, the rings of which are intermediate to the original ones, and are similar to Newton's rings seen by transmission (fig. 34).
If the system of rings with a black cross (fig. 32) were superposed in the system with the white cross (fig. 34) white light would be reproduced.
If the incident polarized light be white, the rings consist of compound tints produced by the superposition on each other of rings formed by each of the homogeneous rays composing white light.
Of course, if the rings of all the colours were of the same size, the resulting system would consist of black and white rings; but being of different dimensions, we obtain a system of different colours. In this case, the cross is either black or white, not coloured.
If the incident polarized light be homogeneous the rings consist of rings of the colour of the light employed separated by black rings. Thus, suppose red light to be used, the rings will be alternately red and black; whereas if blue light be employed, they will be alternately blue and black. Their size varies with the colour of the light: red produces the largest, violet the smallest system of rings. In all cases in which homogeneous light is employed the cross is either a black or a coloured one.
The radii of the bright rings are as the square roots of the odd numbers, 1, 3, 5, 7, &c.; while those of the dark rings are as the square roots of the even numbers, 2, 4, 6, 8, &c. In other words, the squares of the diameters of the bright rings are as the odd numbers, 1, 3, 5, 7,&c; while the squares of the diameters of the dark rings are as the even numbers, 2, 4, 6, 8, &c
Squares of the dia- ( Bright rings meters of the ... J Dark rings
The actual diameter and breadth of the rings are increased by diminishing the thickness of the crystalline plate. To speak more precisely, the radii of the rings are inversely as the square root of the thickness of the plate; and, therefore, the rings are smaller with a thick plate than with a thin one. Thus while a plate of a given thickness will produce a system of rings, the whole of which can be seen at once, a plate considerably thinner will give rings of so much larger diameter and greater breadth, that the whole system cannot be taken in at once by the eye. It is obvious, therefore, that the comparative doubly refracting power of two uniaxial crystals may be ascertained by observing the size of the rings produced by plates of equal thickness: with a powerful doubly refracting crystal the rings are less than with a crystal possessing this property in a weaker degree. In fact, the radii of the rings are inversely as the doubly refracting power of the crystal.
Let us now endeavour to explain generally the origin of the coloured rings and of the cross, according to the undulatory hypothesis; and, for precision and brevity of description, I shall suppose that tourmaline plates are used in the polariscope both for polarizing and analyzing.
The first tourmaline plate polarizes the light which is then incident on the Iceland spar. In their progress through the latter, some of the polarized rays suffer double refraction, others are transmitted without undergoing this change. For there are Fig. 35.
two rectangular planes of polarization of the luminous rays in Iceland spar, one of the plane of polarization of the ordinary rays, the other of the extraordinary rays; and in those parts of the crystal in which the plane of polarization of the incident light coincides with either of the planes of polarization of the rays in the crystal, no double refraction occurs. On the other hand, in those parts of the crystal in which neither of its planes of polarization coincide with the plane of the incident polarized light, double refraction ensues.
All the ordinary rays which emerge from the crystal, are polarized in planes which pass through the principal axis of the crystal: while the extraordinary rays will be polarized in planes perpendicular to these. Let fig. 35 represent the crystalline plate cut peri pendicularly to the axis e. The radiating white lines represent the planes of polarization for the ordinary rays, and the circular white lines the planes of polarization for the extraordinary rays.
The two sets of rays (that is, the ordinary and the extraordinary) form two cones of refracted rays, having a common axis coincident with the axis of the crystal. The summit, or apex of each cone, will be at the eye of the observer; and the diameter of the base of the cone will of course vary according to its distance from the eye. The different rays, of which each cone is made up, undergo different changes. Those which form the axis of the cone, traverse the plate at a perpendicular incidence, and, therefore, are not refracted; while those which pass through the plate obliquely,undergo double refraction. The ordinary or the extraordinary rays forming the same cone have not all an equal intensity at different parts of its circumference. For if the plane of polarization of the incident light be identical with or parallel to a b, fig. 35, it is evident, that while the intensity of the ordinary rays will be at a maximum in the plane a b, and at a minimum or nil in a direction perpendicular to this c d, the intensity of the extraordinary rays will be at a maximum in the plane c d, and at a minimum or nil in a direction perpendicular to this, a b.
Hence those rays which are incident on the crystal in the plane a b, traverse the plate without having their plane of polarization charged, emerge as ordinary rays, and, by the subsequent action of the analyzing plate, form two arms of the rectangular cross, a b (fig. 36). Those rays which are incident on the crystal at any point of the line c d also traverse the crystal without having their plane of Fig. 36. polarization changed, but they emerge as ex
a traordinary rays, and, by the subsequent ac
g^ tion of the analyzer, form the remaining two
E arms of the rectangular cross c d (fig. 36).
c Extraordinary rays, a The two sets of polarized incident rays which I thus traverse the crystal,without having their
41 plane of polarization changed, and emerge,
O the one as the ordinary, the other as the
* extraordinary rays, form either a black or a
white cross, according as they are either suppressed or transmitted by the second or analyzing tourmaline. If the two tourmalines be crossed the rays are suppressed—if they coincide the rays are transmitted. In the first case we perceive a black cross, in the second a white one.
Fio. 37. Thus, then, all the rays which emerge
from the second surface of the crystal, at any point of the two lines a b, c d (fig. 37), will not be divided into two, nor have
their planes of polarization altered. But
all the polarized rays which are incident on the crystal in any direction intermediate between the positions a b and c d suffer 6 double refraction, since their planes of pola
rization coincides neither with the plane of polarization of the ordinary, nor with that of the extraordinary rays; that is, the vibrations of the incident rays are resolved into two sets, one which forms the ordinary rays, and the other perpendicular to it, which forms the extraordinary rays. The two systems of waves, produced by these two sets of vibrations, proceed through the crystal with unequal velocities and describe different paths; consequently they emerge in different phases, that is, in a condition for suffering interference by the action of the analyzer. In my last lecture, however, I so fully explained the agency of the analyzer in giving rise to the phenomena of colour, that I need not now enter further into it. I shall, therefore, only add, that the coloured rings owe their origin to interference.
A circumstance which affects the formation of the rings, is the inclination of the polarized rays to the optic axis of the crystal. In the axis itself, where the arms of the cross pass athwart each other, no colour is produced, consequently there can be, in this position, no double refraction. But those rays which suffer double refraction and produce colour, traverse the crystal obliquely, and at an inclination to the optic axis, and the obliquity or inclination augments in proportion as we recede from the centre or axis. Now the effect of an increase in the inclination of the rays to the optic axis is equivalent to an increase of thickness in the crystal. Hence it is obvious why we have rin"-s,
and not an uniform tint, as in the case of the thin films of selenite described in our last lecture. Moreover, it is obvious that at equal distances around the axis the inclinations will be the same, and consequently the similar tints will be found at equal distances from the axis; in other words, the lines of equal tint or isochromatic lines will be disposed in concentric circles.
That the tints of the system of rings accompanying the black cross should be complementary to those which accompany the white cross, will be readily understood from what was stated in the last lecture respecting the office of the analyzing plate.
The rings of the two systems do not occupy the same position, but are transposed; that is, the bright rings of the one system occupy the position of the dark rings of the other system. The cause of this is obvious—the rings of the two systems are produced by different rays. The two sets of rays which successively pass through the tourmaline analyzing plate in its two positions, would, if this plate were not interposed, pass simultaneously and produce an uniform tint of the same colour as that of the incident light. In other words, without the analyzer neither cross nor rings would be perceived.
But why, it may be asked, is the maximum brilliancy of the rings at the middle of the four quadrants; that is, in lines or directions which are equidistant from the two nearest arms of the cross? Because it may be replied, it is at these spots that the ordinary and extraordinary rays (produced by double refraction) are equal. On either side of these directions, the ordinary ray has either a greater or less intensity than the extraordinary one.
Iceland spar has, as I have already stated, a negative or repulsive axis; and I shall take this opportunity of explaining the method used by Dr. Brewster for distinguishing whether the axis of a crystal be positive or negative. Take a film of selenite (sulphate of lime), and mark on it the neutral axes; then, by a little wax, attach it to a plate of Iceland spar (cut so as to show the rings), and place them in the polariscope. If the film by itself produces the red of the second order, it will now, when combined with the Iceland spar, obliterate part of the red ring of the second order in two alternate and opposite quadrants (either a c and b d, or ad and b c, figs. 32 and 34). The line of the film which crosses these two quadrants at right angles to the rings is the principal axis of selenite, and should be marked as such. Then if we wish to examine whether any other system of rings is positive or negative, we have only to cross the rings with the principal axis " by interposing the film: and if it obliterates the red ring of the second order in the quadrant which it crosses, the system will be negative; but if it obliterates the same ring in the other two quadrants which it does not cross, then the system will be positive. It is of no consequence what colour the film polarizes, as it will always