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Paper-II Chapter-Polarization of Light Nicol Prism Paper-II Chapter-Polarization of light Nicol Prism: Construction: A Nicol prism is an optical device, made by a calcite crystal having a length to breath ratio as 3 : 1. The end faces AB and CD a of a calcite crystal ABCD are properly ground to make the angles in the principal section 710 and 1090 . The crystal is then cut into two haves along the plane A0B0, perpendicular to both the principal section and end faces. The two faces are polished optically flat and joined together by a transparent liquid (canada balsam) of refracted index µ = 1:55. The crystal is finally enclosed in a tube blackened inside and formation of Nicol prism is complete. Use of Nicol prism as polarizer: Inside the crystal the unpolarise light PM splits into O-ray and E-ray. Again, we have µe < µc < µo, where µc is the r.i. of canada balsam. So when the O-ray reaches the balsam layer, it passes from a denser to a rarer medium. Since the length of the crystal is large, the O-ray is usually incident at the at the calcite balsam surface at an angle greater than its critical angle (690) and gets totally reflected to be finally absorbed by the tube enclosing the crystal. The E-ray is transmitted by the calcite-balsam surface to emerge from the Nicol as plane polarized light with vibrations parallel to the principal section. Use of Nicol prism as analyzer: 1 Step I: When an unpolarise ray is incident on a Nicol prism the emergent ray is plane polarised with vibration in the principal section of P. If this emergent ray falls on a second Nicol A whose principal section is parallel to that of P, the vibrations will be in the principal section of A and the ray behaves as E-ray in it and will be transmitted completely. The Intensity will be maximum. Step II: Now if A is rotated such that its principal section perpendicular to that of P, the ray will behaves as O-ray inside A. So it gets lost by total reflection. No light will emerge from A and the Nicols P and A are said to crossed. Step III: If A is further rotated till its principal section is again parallel to that of P, the intensity of the emergent light will again be a maximum. So, a rorating Nicol shows intensity variation with zero minimum. Limitations: A Nicol prism cannot be used in highly convergent or divergent beams. Malus law: It states that if a completely plane polarized light is incident on an analyser, the intensity of the emergent light varies as the square of the cosine of the angle between the planes of transmission of the polariser and the analyser. i.e. 2 Iθ = Icos θ 2 Proof: Let a be the amplitude of light transmitted by polariser and θ the angle between the planes of transmission of the polariser and the analyser. This plane polarised light is incident on the analyger. The amplitude a be resolved into two components: (i) asinθ perpendicular to the plane of transmission of the analyser and is elimi- nated. (i) acosθ along to the plane of transmission of the analyser and only is freely transmitted by the analyser. Intensity of light emerging from the analyser is 2 2 2 Iθ = a cos θ = Icos θ Where I = a2 is the intensity of polarised light incident on analyser. Case I: If the polariser and analyser are parallel to each other, then θ = 0 or 1800, Iθ = I Case I: If the polariser and analyser are perpendicular to each other, then θ = 900, Iθ = 0 Effect of polariser on transmission of polarised light: (i) If unpolarised light is incident on a polariser, the transmitted light will be linearly polarised light. The intensity of the transmitted polarised light will be half the intensity of unpolarised light incident on the polarizer. (ii) The intensity of the transmitted light does not change on rotation of the polariser. (iii) If partially polarised light is incident on a polarizer, the intensity of the transmitted light will vary from Imax to Imin in one full rotation of the polarizer. 3 (iv) If plane polarised light is incident on a polarizer, the intensity of the transmit- ted light will vary from zero to a maximum value in one full rotation of the polarizer. (v) If circularly polarised light is incident on a polarizer, the intensity of the transmitted light stays constant in any position of the polariser. hence the intensity is same for all position of the polariser. (vi) In case of elliptically polarised light, the intensity of the light transmitted through the polariser varies with the rotation of the polariser from Imax to Imin. Imax is found when the polariser axis coincides with the semi-major axis of the ellipse. Imin is found when the polariser axis coincides with the semi-minor axis of the ellipse. Huygens' theory of double refraction: Huygens' explained the phenomenon of double refraction by his theory of sec- ondary wavelets. Here we give below his extended theory. (a) When a wavefront is incident on a doubly refracting crystal, every point on the crystal becomes the origin of two wavefronts,one the ordinary (O) and the other extraordinary (E). (b) Ordinary wavefront corresponds to ordinary rays which obey the laws of re- fraction and have the same velocity in all direction. The ordinary wavefront is thus spherical. (c) Extraordinary wavefront corresponds to extraordinary rays which do not obey the laws of refraction and have different velocities in different directions. The ex- 4 traordinary wavefront is thus an ellipsoid of revolution, with the optical axis as the axis of revolution. (d) The velocity of the O-ray and E-ray is the same along the optic axis. For this the sphere and the ellipsoid touch each other at points that lie on the optic axis of the crystal. (e) In negative crystal the ellipsoid lies outside the sphere, S being the source implying that in such crystal the E-ray wavefront travels faster than the ordinary wavefront, except along the optic axis. In positive crystal the sphere is outside the ellipsoid implying that in such crystal the O-ray wavefront travels faster than the extraordinary wavefront, except along optic axis. Refractive indices of a crystal: Refractive index of a substance is defined as c n = v 5 where c is the velocity of light in vacuum v is the velocity of light in the substance Refractive index of the crystal for O-ray is c nO = = constant vo because, the wave velocity of O-ray is a constant being the same in all directions and for all orientation of the optic axis. Refractive index of the crystal for E-ray is c ne = = not constant ve because, the wave velocity of E-ray is not a constant but varies with directions. So r.i. of the crystal for the E-ray will be different for different directions. The principal r.i. for E-ray in positive crystal is c ne = (ve)min The principal r.i. for E-ray in negative crystal is c ne = (ve)max In positive crystal, ne > n0 In negative crystal ne < no Birefringence or amount of double refraction of a crystal is defined as 4n = ne − no For +ve crystal, ne > n0, hence 4n = +ve For -ve crystal, ne < n0, hence 4n = −ve Huygens' construction of wave surfaces on the plane of incidence in Uniaxial crys- tal: 6 Case I: Optic axis inclined to upper face but lying in the plane of inci- dence: Let a plane wavefront AB be incident obliquely on the upper face of a uniaxial -Ve crystal (e.g. Calcite) cut such that the optic axis lies in the plane of incidence along AX, but inclined to the upper face at some angle. On the crystal surface, the point A of the wavefront AB is the centre of both O-ray and E-ray. At the same time, the disturbance from B reaches C, O-ray travels a distance AD and e-ray travels AF, so BC AD AF t = = = c vo ve Hence, v BC AD = BC o = c no With A as centre and BC as radius, we draw a circle cutting the optic axis at X. The no circle gives the position of O-wave surface. v BC AF = BC e = c no With A as centre and BC as radius, we draw a circle cutting the optic axis at X. In (ne)min -Ve crysta ne < no, and (ne)min = no along the optic axis. So AX is the semi-minor axis of ellipse. The semi-major axes AF. If an ellipse is drawn with given minor and major axes touching the circle at X, it gives the position of E-wave surface. Tangents CD and CF drawn from C on O- and E-wave surfaces gives the O- and E-wave refracted wavefronts. 7 Case II: Optic axis parallel to upper face but lying on the plane of inci- dence: (a) Oblique incidence: As the optic axis is parallel to the crystal face and lies in the plane of incidence, we get the O-wave and E-wave surface as shown in fig. The circle and ellipse will touch at X along the optic axis AX, CD and CF are O-ray and E-ray refracted wavefronts corresponding to the incident wavefront AB. (b) (i)Normal incidence: This is illustrated in fig. Here CD and FG are the O- and E-refracted wavefronts corresponding to the incident wavefront AB.The refracted wavefronts (CD for O-rays and FG for E-ray) are parallel to each other and also to the crystal surface.
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