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i . eucematmestch rh. = a, |, 262 Physical Opties, bf, : +, aS SS Ke, - lou OPTICAXIS «, ze ss, o> SF, er -—----* =, Oo, E————Jop ric AXis| t E, 1 (by, @m, (Fig. 1), , On entering the crystal, the amplitude of the incident light, wave- splits up into two components, A cos 8 along PE and, Asin§ along PO. The component Acosé@ having vibrations +, parallel to the optic axis forms the E-wave and the component, A sin @ having vibrations perpendicular to the optic axis forms the, O-wave. According to the Huygens’ construction for double, refraction, the two waves travel in the plate along the same path, but with different velocities (Fig. 1 b), the E-wave being faster., Hence the two waves emerge from the plate with a phase, difference 8 (say) between them. Thus if A sin wt be the incident, wave, the two emergent plane-polarised E- and O-waves would be, represented respectively by, , x = Acos 6 sin (wf+é8), and y = Asin 6 sin wt., Let us put A cos @=a and A sin 6=b. Then we have, x = asin (wi+), and y = bsin ot. ‘, The nature of the resultant vibration can be obtained by climinating ¢ from eq. (i) and (ii). From eq. (i), we have, x, , — sin wt cos 8+-cos wt sin §, , = sin wt cos $-+4/(1 —sin* wf) sin 3., But from eq. (ii), sin wt=y/b., , oe =- $ cos3+/(1- 5) sind, , , , +8, , or (z- $ cord )' = ('-#) sin? 3, or 4 — B cossasint. «a (iil), , Circularly and Elliptically Polarised Light 263, , This, in general, represents an ellipse. Hence the light emerging, from the crystal plate is, in general, elliptically polarised., , Special Cases—(¥) If the thickness of the plate be such that., vn, then cos8=1 and sin8=0. Then eq. (iii), , , , , , , , 8=0, 2n, 4m, .., , gives, , n (E-3)9, , or Ea (2 = $) =0, , or y= ce x. . (Fig. 2) ...(iv), , This represents a pair of coincident straight lines through the origin, having a positive slope 5/a (Fig. 2). This means that the emergent, light is linearly-polarised with the same direction of vibration as, the incident light., , , , van ¢ ay 5oe See ae ; then cos 8=—1 and sin $8=0., =e Seo, , “(Eras, , or a (2 + +) =O), , o ty Fox, Fi), , This is again a pair of coincident straight lines but with a slo}, (—b/a). Hence the emergent light is linearly-polarised with the, vibration direction making an angle 2 tan-* ja =28 with that, oe the incident light (Fig. 3) This case is the bests of a half-wave, plate., (iii) If 87/2, 3n/2, 5x/2, 7x/2, -., sin? 3=1. The eq. (iii) reduces to, Boe, Haha. (vi), This represents an ellipse with its axes along x and y directions., Hence the light emerging from the plate is eliptically polarised,, the axes of the ellipse being along and perpendi to the optic, axis,, If 87/2, 5x2, 9n/2, ...-...- .then x- and y-components of the, , .. then cos 8=0 and

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264 Physical Optics, elliptic vibration, by eq. (i) and (ii), are y, given by, , eo, ‘=, 2, , }, y=b sin ot. —, These equations show that the ellipse is, described by the tip of light vector rotating counter-clockwise (Fig. 4). The emer- —, gent light is said to be ‘left-handed’, elliptically polarised light. (Fig. 4), To determine the direction of rotation of the emergent elliptically, polarised wave, let us locate the tip of the rotating vector at ¢=0 and t=Ar., , At t=0, x=a, y=0., At f=At, xoa, yob wAr,, , Thus,.as seen from Fig. 4, the tip of the vector rotates counter-clock wise., But, if 8=3x/2, 7x/2, 1177/2, elliptic vibration are, , }, y=b sin wt., , The ellipse is now described clockwise i, (Fig. 5), and the light is said to be ‘right- 1 fa,, , , , , , , , , , , , -.., the components of the, , , , , , , , handed’ elliptically polarised light., At 1=0, x=—a, y=0, At rAt, xer—a, yoeb wf., Thus, as scen from Fig. 5, the tip of the vector ke -a —}, Totates clockwise, (Fig. 5), If, in the present case, 6=45°, then we have a=A cos 45°,, 6=A sin 45°. The eq. (vi) now reduces to, , xt y2=ai_, Hence in this case the emergent light is ‘circularly’ polarised light,, (This case is the basis of a quarter-wave plate), , Ifd=x/2, 5n/2, 9n/2,...... the circle is described counterclockwise (left-handed circularly polarised light), but if 8=3n/2,, Trj2, Vix]2,...<0: the circle is described clockwise (right-handed, circularly polarised light)., , ___ Thus we sec that the plane-polarised and circularly-polarised, lights are,the special cases of elliptically-polarised light., , \Q. 2. What are quarter-wave and half-wave plates. Explain, their use im the study of different types of polarised light., (Meerut 86, 80, Lucknow 85, Agra 85, Kanpur 88, Rohilkhand 79,, Gorakhpur 80, Rajasthan 84), _ Ans, Quarter-wave Plate—A doubly-refracting crystal plate, having a thickness such as to produce a path difference of 2/4,, , , , , , , , , , , , an, , Circularly and Elliptically Polarised Light 265, , or a phase difference of x/2, between the ordinary and extraordinary wave is called a ‘quarter-wave plate’ or \/4-plate,, , Let us consider a plane, parallel plate cut from a doublyrefracting crystal so as to have, its faces parallel to the optic |, axis, Let a beam of mono, , , , , , , , , , , , , , , , , , , , , chromatic light of wavelength A ; Fear, be incident normally on the plate, Sars iL =, It is broken up into O and E WAI, waves. By Huygens’ construc- x Fal t, tion for double refraction, both —$—<OPric AXIS, , =, the waves travel along the same esa, , , , , , path perpendicular to faces but, with diferent velocities (Fig. 6). =o, In the case of a negative aio, crystal, such as calcite, the, E-waye travels faster than the :, O-wave, so that 4. > ., where (Fig. 6), #» and y, are the principal refractive indices of the crystal for the, O and E-wayes,, , paths, und E waves in the plate are jf and pt. There: ‘ore, the path, difference between the two waves on emerging is, , (He — He) t, If the plate is to act as a quarter-wave plate, this path difference, should be equal to 4/4, i.e.,, , , , , , , , , , , , (i — a) t= >, , é A, or t= iG=ae, For a positive crystal, such as quartz, p, > pw, , so that, , a, ‘= Tl =a)’, , Hence a plate of thickness given by this equation will serve as, quarter-wave plate for the particular wavelength A,, , The quarter-wave plate is used for producing circularly and, elliptically polarised light. In conjunction of a Nicol prism, it is, used for analysing all kirfds of polarised light., , Half-waye Plate—A doubly-refracting crystal plate having a, thickness such as to produce a path difference of4/2, or a phase, difference of x, between the ordinary and extraordinary waves is, called a ‘half-wave plate’ or 4/2-plate., , ’, ], :

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266 Physical Optics, , If ¢ be the thickness of such a plate, then for a negative, crystal, such as calcite, we have, , a, (te — He) f= >, , t= bs, , 2 (ts — fe)”, If linearly-polarised light is passed through a A/2-plate, the, emergent light is also linearly-polarised but its direction of, vibration is inclined at 26 to that in the incident light, where 6 is, the angle between the incident vibration and the principal section, of the plate. Hence such a plate is used in polarimeters as, , half-shade devices to divide the field of view into two halves, presented side by side., , or, , 2/4 and 2/2-plates are often made either of quartz by cutting, it parallel to the optic axis or by splitting thin sheets of mica, along cleavage planes. Mica is a negative biaxial crystal but the, angle between the two axes is very small. Quartz is a positive, crystal and has got no cleavage planes. It has, therefore, to be, cut and its faces have to be polished to make them optically, plane., , Q.3. (a) Plane-polarised light is normally incident on, quarter-waye plate. Discuss the conditions under which ellipticallypolarised light, plane-polarised light and circularly-polarised light, can be obtained. (Agra 69), , (b) What will be the state of polarisation of the light, emerging from a quarter-wave plate when the incident light is, circularly-polarised, unpolarised ? (Rajasthan 83), , (c) If a quarter-waye plate and a half-wave plate be given, to you, how would you proceed to distinguish them from each other ?, (Kanpur 88, 84, Bundelkhand 86, Awadh 85, Rohilkhand 82, Vikram 84), (@ How would you change a left-handed circularly-polarised, light beam into a right-handed circularly-polarised beam ?, (Gorakhpur 81, Agra 80, Meerut 76), , Ans. (a) Action of Quarter-waye Plate on Plane-polarised, Light—When plane-polarised light falls normally on a quarter, wave plate, the emerging light is, in general, elliptically pola- *, , rised, Let us prove it., , Let A be the amplitude of vibration of the incident light, wave and @ the angle between the direction of the incident, vibration and the optic axis of the A/4-plate. On entering the, plate, the incident wave is splitted into two (plane-polarised), components, One component has vibrations parallel to the optic, axis (E-wave) and the other has vibrations perpendicular to the, optic axis (O-waye) ; the amplitudes of vibration being Acosé, , ieee de, , , , Circularly and Elliptically Polarised Light 267, , and A sin 6 respectively. On emergence from the plate, the two, components have a path difference of 4/4 or a phase diflerence of, x/2 (introduced by the plate),, , Let A cos @=aand Asin@=d. Ifthe axes of coordinates, X and Y be taken along and perpendicular to the optic axis respectively,-the equations of the two component vibrations can be, written as, x = asin (w+ 5) =a COS wt (i), and y = b sin of. + (ii), Eliminating ¢ between eq. (i) and (ii), the equation of the resultant emergent vibrations is, eT ye *, ota =l, «-(iii), which represents an ellipse whose major and minor axes are along, , and perpendicular to the optic axis respectively. Hence the emergent light is, in general, elliptically polarised., , There are three special cases :, (i) If the incident light has vibrations parallel to the optic, , axis (0=0), then a=A cos (=A and 6=A sin 0=0, The equations, , (@) and (ii) became, x = Acos wt, and y=0., This shows that the emergent light is plane-polarised having, vibrations parallel to the optic axis, just as the incident light., , _ i) If the incident light has vibrations perpendicular to the, optic axis (8=90"), then a=0 and b=d. Again, the emergent, light is plane-polarised having vibrations -perpendicular to the, optic axis, just as the incident light., , (iii) If the incident light has vibrations inclined at 45° to, the optic axis (0=45°), then a= and the equation of the emergent, , vibrations is, x+y=a=} At Ba., This shows that the emergent light is circularly-polarised., , z For values of @ other than 0, 45° and 90°, the emergent light, is elliptically polarised., , If the light inoident on the 4/4-plate is circularly-pola~, rised, Re cieseree can be resolved ane plane-polarised O, and E components having a phase difference of x/2. The plate, removes this phase difference so that plane-polarised, emerges.

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268 Physical Optics, , Ifthe incident light is unpolarised, the emergent light is, also unpolarised,, , (c) Distinction between Quarter-waye and Half-wave Plates—, To distinguish between a quarter-wave and a half-wave plate, planepolarised light is made to fall normally on each of them. The, emergent light is then examined with a Nicol prism rotating about, the direction of light as axis for different oricntations of the plates,, , In case of A/4-plate, the light emerging from the plate may, be tically-polarised, circularly-polarised, or plane-polarised, depending upon the orientation of the plate. Hence on examining, through the rotating Nicol we shall observe (i) variation in intensity, with non-zero minimum, when the emergent light is elliptically, polarised, (ii) no variation in intensity when the emergent light is, cireularty-polarised, ) variation tensity with zero minimum, when the emergent light is plane-polarised,, , In case of half-wave plate which creates a phase difference of, x between the two components, the state of polarisation of the, emergent light is the same as that of the incident light. Therefore,, the light emerging from the half-wave plate will be planepolarised for al! orientations of the plate. Hence rotation of, the Nicol will always give variation in intensity with zero, minimum,, , {d) Conversion of Left-handed Circularly-Polarised Light, into Right-handed Circularly-Polarised Light—To convert lefthanded circularly-polarised light into right-handed circularlypolarised light (or vice-versa), a phase difference of x must be, introduced in the given light. Hence the given light should be, passed normally through a /2-plate (which introduces a phase, difference’ of x)., , \Q. 4. Describe how, with the help of a Nicol prism and a, quarter-wave plate, plane-pelarised light, circularly-polarised light, and ciliptically-polarised light are produced and detected., , (Meerut 895, 89, 88, 86, 85, Purvanchal 89, Allahabad 86,, Agra 83, Kanpur 84, 83, Rohilkhand 86, Ranchi 86,, Riwa 85, Indore 84, Rajasthan 87, 84, 82), , Ans. Production of Plane-Polarised Light—To produce, lane-polarised light, a beam of ordinary light is sent through a, icol prism in a direction almost parallel to the long edge of, , the prism. Inside the tism, the beam is broken up into two, components, O and E. The O-component is totally refiected at, the Canada Balsam layer and is absorbed. The E-component, emerges out. It is plane-polarised with its vibration parallel to, the shorter diagonal of the end face of the Nicol., , Detection—To detect plane-polarised light, it is examined, through Another Nicol prism rotating about the dircction of, Propagation of light. If the intensity of the emerging light, Yarics with zero minimum, the light is plane-polarised,, , , , , , , , , , , Circularly and Elliptically Polarised Light 269, , SreU aE, polarised light can be produced by allowing plane-polarised light, obtained from a Nicol prism to fall normally on a quarter-wave, Plate such that the direction of vibration in the incident plancpolarised light makes an angle of 45° with the optic axis of the, plate,, , Inside the plate, the incident wave of amplitude A (say) is, divided into an E component A cos 45° parallel to the optic axis,, and an O component 4 sin 45° perpendicular to the optic axis., These components emerge from the plate with a phase difference, of x/2., , Let 4 cos 45°=A sin 45°=a. If the axes of x and y be, taken along and perpendicular to the optic axis, then the, emerging components can be written as, , , , x=asin (wr + 3) = acos wt wei), and y = asin wt, 7) ++e(ii), Eliminating ¢ between cq. (j) and (ii), the resultant vibration is, x*-|-ytea?,, , , , w Hence the fight emerging from the, A/4-plate ts circularly-polarised,*, , Detection —The <ircularly-polarised light, when secn through, a rotting Nicol, shows no variation in intensity, It thus, + Hence to confirm that the given, ht is circularly-polarised it is first passed through a A/4-platet, (which converts it into planc-polarised light) and the through, , ag N The light now shows a variation in intensity, rouminimum,, , If the given light were unpolarised, it would have remained unchanged, by the a/4-plite. Henee on passing through the Nicol the light would have, shown no v: ion in intensity., , Production of Elliptically-Polarised Light ~The cllipticallypolarised light can be produced by allowing plane-polarised light, obtained from a Nicol prism to fall normally on a quarter-wave, that the direction of vibration in the incident planekes an angle other than 0, 45° and 90” with the, of the plate. An appropriate angle is 30°., , In this case, the incident wave is divided inside the plate, into Eand O components of waegual amplitudes A cos 30’ and, A sin 30° respectively which emerge from the plate with a phase, difference of x'2., , , , , , , , , , , , , , , *Q. How can you convert plane-pélarised light ints cieculadys, , polarised light ? (Allahabad 87, Agra 88), this A/4-plate is other than that which produced the circularly, Polurised ligh