Circularly polarized Light.

142. We have hitherto supposed that the incident light is plane polarized, and that it is analysed by an instrument which could plane polarize common light. If however the light which has passed through the polarizer, or the light which emerges from the crystal, is passed through an apparatus which could circularly polarize plane polarized light, the rings and brushes undergo certain modifications, which we shall proceed to consider.

143. Circularly polarized light may be either produced by passing plane polarized light through a Fresnel's rhomb, which is an instrument which will be explained in a subsequent chapter; or by passing it through a quarter undulation plate, which consists of a thin plate of uniaxal crystal cut parallel to the axis, of such a thickness, that it produces a difference between the retardations of the ordinary and extraordinary waves, which is equal to a quarter of a wave-length.

Let y be the angle which the principal section of the quarter undulation plate makes with the plane of polarization of the incident light. Then since 0 - E = 4, the vibrations on emergence parallel and perpendicular to the principal section of the quarter undulation plate, are

sin y cos 2π (t/T - Ox), and cos y sin 2π (t/T – O/X). We therefore see that the effect of the plate is to convert plane polarized light into elliptically polarized light; if however y = 1, the emergent light is circularly polarized.

144. We shall first suppose, that the quarter undulation plate (or the Fresnel's rhomb) is placed between the polarizer and the crystal, so that the light incident upon the latter is circularly polarized. The vibrations incident upon the crystal may be taken to be cos 2πt/T in the principal plane, and sin 2πt/τ perpendicularly to the principal plane; hence if þ = 2π (t/τ — O/X), the vibration on emerging from the analyser is

cos (+2π (0 – E)/λ} cos a + sin & sin z,

where a is the angle which the principal section of the analyser makes with the principal plane of the incident ray. Whence the intensity of the emergent light, is proportional to

I2 = 1-sin 2a sin 2π (0 - E)λ(19).

CIRCULARLY POLARIZED LIGHT.

153

From this expression we see, that I can never vanish unless sin 2a sin 2π (0 - E)λ = 1; hence there are no brushes.

When the crystal is a plate of Iceland spar, cut perpendicularly to the axis, - E varies as r2, whence (19) may be written

=

.(20),

(2n + 1)π/k, the

π, or π, and a nπ/k, I is constant,

If we assign any constant value to r, say r intensity along this circle is zero at the points a maximum when a 3, or 3. When and equal to 1; and when 2= (2n+) π/k, the intensity is a maximum when aπ or, and a minimum when a =

=

=

. Hence the general appearance of the pattern is, that the brushes are absent, whilst the rings in the first and third quadrants are pulled out, whilst those in the second and fourth are pushed in.

Any other case can be discussed in a similar manner; and the appearance is of an analogous character, when the incident light is plane polarized and circularly analysed.

145. We shall lastly consider the case in which the light is circularly polarized and circularly analysed. In order to accomplish this, we must place another quarter undulation plate between the crystal and the analyser, with its principal plane inclined at an angle of 45° to the principal plane of the latter.

Let y be the angle between the principal section of the quarter undulation plate, and that of the crystal. Then on emerging from the plate, the vibrations parallel and perpendicular to the principal section are of the form

whence on emerging from the analyser, the vibration is

accordingly

sin (x + y) - sin {x + y + 2π (0 − E)/^} ;

I2 = 4 sin2 π (0 – E)/λx.

It therefore follows that the rings are of the same form as when the light is plane polarized and analysed, but that there are no brushes.

EXAMPLES.

1. If a horizontal ray is first polarized in a vertical plane, then passed through a plate of crystal with its axis inclined at an angle to the vertical, then through a film which retards by a quarter undulation, light polarized in a vertical plane; show that the emergent light is polarized in a plane inclined to the vertical, at an angle equal to half the retardation of phase due to the plate of crystal.

2. Plane polarized light is incident normally on a plate of uniaxal crystal cut parallel to its axis, and is then passed through a parallel plate of crystal, which could circularly polarize plane polarized light. Prove that the emergent light will be plane polarized if

tan a tan ẞ sin 2πk/λ- tan y cos 2πk/λ;

where a is the angle between the principal plane of the first plate and the plane of polarization of the incident light, ẞ is the angle between the principal plane of the second plate and the plane of polarization of the emergent light, y is the angle between the principal planes of the first and second plates, and k is the equivalent in air to the relative retardation of the ordinary and extraordinary rays caused by the first plate.

3. A small beam of circularly polarized light is incident on one of the parallel faces of a plate of uniaxal crystal, which is cut parallel to its axis, the angle of incidence being small; and the crystal is then made to revolve round a common normal to its plane faces, whilst the direction of the incident pencil remains unchanged. It is found, that when the axis of the crystal lies in the plane of incidence, the emergent light is circularly polarized in the opposite direction to the incident light; and when the axis of the crystal is at right angles to the plane of incidence, the emergent light is circularly polarized in the same direction as the incident light. Prove that if the axis of the crystal were inclined at an angle to the plane of incidence, the emergent light would be polarized either in or perpendicularly to that plane.

4. If n equal and similar plates of a crystal be laid upon each other, with their principal directions arranged like steps of a uni

form spiral staircase, and a polarized ray pass normally through them; prove that the component vibrations of the emergent ordinary and extraordinary rays are of the form

X cos 2πt/T + Y sin 2πt/T,

where X and Y are of the form A cos ny + B sin ny, where cos y = cos & cosa; a being the angle between the principal directions of two consecutive plates, and 28 the difference of phase between the ordinary and extraordinary rays in passing through one plate.

Determine also the condition, that a ray originally plane polarized may emerge plane polarized.

5. The extraordinary wave normal OQ in a uniaxal crystal, whose optic axis is OA, makes a constant direction with a given direction OP. Show that the mean of the displacements irrespective of sign, which are parallel to the plane POA as the position of OQ varies, will be a minimum, when OA and OP are at right angles to one another.

6. If a biaxal crystal is bounded by two parallel planes perpendicular to the axis of greatest elasticity, and if 0, be the angles of inclination to this axis of the two emergent rays, situated in the plane containing the optic axes at the point of emergence, prove that

b-c2 cosec2 0 = a2 (a2 cot2 + c2).

7. A pencil passing through a feebly doubly refracting plate, is defined by two small holes through which it has to pass, the holes being situated in a line perpendicular to the plate, and on opposite sides of it; show that whatever be the law of double refraction, when the thickness of the plate and the distances of the holes vary, the angle in air between the two pencils which can pass, varies as

h

where h is the thickness of the plate, and k, k' the distances of the holes from the surfaces respectively next them.

8. The surfaces of a plate of uniaxal crystal are nearly perdicular to the axis of the crystal; show that if polarized light be incident nearly perpendicularly to the faces, and afterwards analysed and received on a screen, the rings will be sensibly the same as

would have been formed if the surfaces had been perpendicular to the axis, but shifted in the direction of the projection of the axis through a distance proportional to pa, where μ is the refractive index for the ordinary ray, and a the angle between the axis of the crystal and a normal to the surfaces of the plate.

9. Plane polarized light of amplitude c passes in succession through two plates of crystal cut parallel to the axis, and is then analysed by a Nicol's prism. The inclinations of the principal planes of the two crystals, and of the Nicol's prism to the plane of polarization of the incident light, are a, a + ẞ, and a +B+y; p and q are the retardations of phase due to the two crystals respectively. Prove that if AO be drawn equal to c cos a cos B cos y,

and

AB = c sin a sin ẞ cos y,

BC= c cos a sin ẞ sin 7,

CD = c sin a cos ẞ sin y;

and if AB, BC, CD make with AO angles respectively equal to p, q and p+q; then OD will be the amplitude, and the supplement of CDO will be the retardation of phase of the emergent ray.

10. The end of a Nicol's prism, in which air is substituted for balsam, is a rhombic face inclined at an angle to the axis of the crystal, and the prism is sawn so that the layer of air contains that axis. If the axes of the ellipsoid in the wave-surface corresponding to a sphere of unit radius in air be (2:3) -1, (2·6) −1, the cosines of the angles of incidence for the extinction of the ordinary and extraordinary rays are respectively equal to