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Transcript

PHASE DIFFERENCE BETWEEN e-RAY AND o-RAY Natural light incident on the surface of an anisotropic crystal undergoes double refraction and produces two plane polarized waves namely eray and o-ray. The two waves travel along the same direction in the crystal but with different velocities. v0 < ve 1 PHASE DIFFERENCE BETWEEN e-RAY AND o-RAY 2 PHASE DIFFERENCE BETWEEN e-RAY AND o-RAY As a result, when the waves emerge from the rear face of the crystal, an optical path difference would have developed between them. The optical path difference can be calculated as follows Let d be the thickness of the crystal. The optical path for o-ray within the crystal The optical path for e-ray within the crystal The optical path difference between e-ray an o -ray = µod = µed ∆ = = (µe - µo) d 3 PHASE DIFFERENCE BETWEEN e-RAY AND o-RAY Consequently, a phase difference arises between the two waves δ 2π μe μo d λ As the two component waves are derived from the same incident wave, the two waves are in phase at the front face and have emerged from the crystal with a constant phase difference Hence, it may be expected that the waves are in a position to interfere with each other However, as the planes of polarization of o-ray and e-ray are perpendicular to each other, interference cannot take place between e- and o-rays The waves instead combine with each other to give an elliptically polarized wave. 4 Let two light waves traveling in the same direction, x; one wave is polarized in the xy -plane and the other is polarized in xz- plane . We are interested to know the state of polarization of the resultant wave. Let the two orthogonal waves be represented by Ey = E1 cos (kx-ωt) EZ = E2 cos (kx-ωt + δ) (1) (2) The waves are of the same frequency ν= ω/2π. δ is the phase difference between the waves. 5 eq.(2) can be expanded as EZ = E2 cos (kx-ωt) cosδ - E2 sin (kx-ωt) sin δ (4) = E2 cos (kx-ωt) cosδ ± [1- cos2 (kx-ωt) ]1/2 E2 sin δ We find from eqn (1) coskx t Ey = E1 cos (kx-ωt) Ey E1 6 Put this value in Ez EZ = E2 cos (kx-ωt) cosδ ± [1- cos2 (kx-ωt) ]1/2 E2 sin δ EY E Z E2 cos δ E1 E 2y 1 2 E2 sin δ E1 5 2 Ey E2 E y cos δ 1 E2 sin δ E Z E1 E1 On squaring both sides, we obtain E z2 E 2y E22 2 cos δ 2 E1 2E y E z E2 E1 cos δ E22 sin2 δ E 2y E22 2 sin δ 2 E1 7 Rearranging the terms, we get 2 2 E 2E y E z E2 y E2 2 2 2 E z 2 (cos δ sin δ ) cos δ E22 sin2 δ E1 E1 E2 z E 2y E22 2 E1 2E y E z E2 E1 6 cos δ E22 sin2 δ Dividing both the sides E22 by and rearranging the terms , we obtain E2 y E z2 2E y E z 2 cos δ sin δ 2 2 E1 E2 E1 E2 Equation (7) is the general equation of an ellipse. 7 Hence , the tip of the resultant vector traces an ellipse in the yz- plane. Ellipse is constrained within a rectangle having sides 2E1 and 2E2. 8 Special cases :1. When δ= 0, or ±2mπ, the two waves are in phase, cos δ =1 and sin δ = 0, and the eqn.(7) reduces to E2 y 2 E1 2 EZ 2 E2 2E y E z E1 E2 0 2 E y Ez E E 0 2 1 Ey Ez 0 E1 E2 E2 Ez Ey E1 8 9 The above equation represents a straight line, having a slope E2 E 1 Therefore , the equation represents a wave having its plane of polarization making an angle α with respect to the y-axis . It means that the resultant of two plane-polarised waves, which are in phase(i.e., coherent waves), is again a plane-polarised wave. 2.When δ= π, or ± (2m + 1) π ,the two waves are in opposite phase, then cos δ = -1 and sin δ = 0 and the eqn (7) reduces to E y2 Ez2 2 E y Ez 2 0 1 E2 E2 E1 E2 2 E y Ez 0 E1 E2 10 Ey Ez 0 E1 E2 E2 9 Ez Ey E1 This equation represents a straight line of a slope E 2 E 1 Therefore, the equation represents a wave having its plane of polarization making an angle tan1 E2 with respect to the y-axis E 1 It means that the resultant of two plane-polarised waves, which are in opposite phase (i.e, coherent waves), is again a plane-polarised waves. 3. When δ= π/2, or ± (2m + 1) π/2 , then cos δ = 0 and sin δ = 1 and the eqn (7) reduces to 2 E2 E y z 1 10 2 2 E1 E2 11 This is the equation of an ellipse whose major axis and minor axis coincide with y – and z-co-ordinate axes. Therefore, when the two plane polarized waves are out of phase by 90o , their resultant is an elliptically polarized wave. 4. In the particular case, δ= π/2, or E1 = E2 =E0 , eqn. (7) reduces to EY2 E z2 E02 This is the equation of circle. Hence the resultant light is circularly polarized. 12 when e-ray and o- ray overlap on each other after emerging from an anisotropic crystal plate, they cannot produce interference fringes as in a double slit experiment On the other hand, they combine to produce different states of polarisation depending upon their optical path difference When the optical path difference is 0 or an even or odd multiple of λ/2, the resultant light wave is linearly polarized. When the optical path difference is λ/4, the resultant light wave is elliptically polarised In the particular instance when the wave amplitudes are equal and the optical path difference is λ/4, the resultant light wave is circularly polarised. 13 QUARTER WAVE PLATE It is a thin plate of birefringent crystal having the optic axis parallel to its refracting faces and its thickness adjusted in such a way that : when a plane polarized light propagates through it , a path difference of quarter-wave (λ/4) is introduced between the e-ray and o-ray. Thus, for a quarter wave plate λ μe μo d 4 λ d 4μ e μ o 14 So a phase difference δ between e-ray and o-ray will be 2π π δ 90 λ 2 A quarter-wave plate is used in producing elliptically or circularly polarized light. It converts plane polarized light into elliptically or circularly polarized light depending upon the angle that the incident light makes with the optic axis of the quarter wave plate. 15 Action of quarter wave plate on elliptically and circularly polarised light consider elliptically polarised light being incident on a quarter wave plate Elliptically polarised light may be viewed as made up of two coherent plane polarised waves of different amplitudes, and differing in phase by 90°. the quarter wave plate introduces an additional phase difference of 90° leading to a total phase difference of 180° between the two component waves 16 When they emerge out of the plate, they combine to form linearly polarised wave, the action of the quarter wave plate on circularly polarised light wave is similar. Circularly polarised light incident on a quarter wave plate is converted into linearly polarised light. HALF WAVE PLATE A half wave plate is a thin plate of birefringent crystal having the optic axis parallel to its refracting faces and its thickness chosen such that it introduces a half-wave (λ/2) path difference (or a phase difference of 180°) between e-ray and o-ray. 17 As a result, when they emerge from the rear face of the crystal, an optical path difference developed between them: λ μe μo d 2 λ d 2μ e μ o A half wave plate introduces between e-ray and o-ray a phase difference δ given by 2π δ π 180 λ 18 The half wave plate rotates the plane of the incident light through an angle 2θ. This inverts the handedness of elliptical or circular polarized light changing right to left and vice versa. 19 PRODUCTION AND ANALYSIS OF PLANE, CIRCULAR AND ELLIPTICALLY POLARIZED LIGHT PLANE POLARIZED LIGHT When a beam of monochromatic light is allowed to pass through a Nicol prism, it splits up into two parts: ordinary and extraordinary rays. During the transmission of these rays through the prism, the ordinary ray is totally reflected back by a Canada balsam layer, whereas the extraordinary ray is transmitted through the Canada balsam. Hence, the emergent light is only the extraordinary ray, which is 20 plane-polarized. ANALYSIS OF POLARIZED LIGHT The unknown polarized light is allowed to fall normally on a polarizer. The polarizer is slowly rotated through a full circle and observe the intensity of the transmitted light. If the intensity of the transmitted light is extinguished twice in one full rotation of the polarizer, then the incident light is plane polarized. 21 Circularly Polarized Light When two mutual perpendicular coherent linear vibrations of same amplitude and period are superimposed with phase difference π/2, the resultant motion will be circular motion. These can easily be obtained by allowing a beam of monochromatic plane-polarized light to be normally incident on a quarter wave plate. 22 ANALYSIS OF CIRCULARLY POLARIZED LIGHT If the intensity of the transmitted light remains constant on rotation of the polarizer, then the incident light is either circularly polarized or unpolarized. To distinguish between circularly polarized and unpolarized light, we take the help of a quarter wave plate. 23 The light is first made to be incident on the quarter wave plate and then it passes through the polarizer. If the incident light is circularly polarized, the quarter wave plate converts it into plane polarized light. When this linearly plane polarized light passes through the polarizer, it would be completely extinguished twice in one full rotation of the polarizer. 24 If the intensity of the transmitted light stays constant, then the incident light is unpolarized. 25 ELLIPTICALLY POLARIZED LIGHT (Production) Unpolarized light is first converted to plane polarized ( Nicol prism). The plane polarized light is then made incident on a quarter wave plate. When this emerges out of the crystal produces elliptically polarized light. ANALYSIS If the intensity of the transmitted light varies between a maximum and a minimum value but does not become extinguished in any position of the polarizer, then the incident light is either elliptically polarized or partially polarized 26 To distinguish between elliptically polarized and unpolarized light, we take the help of a quarter wave plate If the incident light is elliptically polarized, the quarter wave plate converts it into a plane polarized beam. When this linearly polarized light passes through the polarizer, it would be extinguished twice in one full rotation of the polarizer. if the transmitted light intensity varies between a maximum and a minimum without becoming zero, then the incident light is partially polarized 27 OPTICAL ACTIVITY For a crossed Nicol position in which the polarizer and analyzer are perpendicular no light comes out of the analyzer. If we place quartz plate cut with faces perpendicular to the optic axis; inbetween polarizer and analyzer, light comes out of the analyzer. Quartz has the property to turn the plane of vibration. When plane polarized light enters the quartz, its plane of vibration is rotated gradually. The amount of rotation by which the plane of vibration is turned depends upon the thickness of the quartz plate and the wavelength of incident light. 28 polarizer Quartz plate Two Crossed Nicol analyser I Optic Axis Rotation of vibration 29 The property of turning the plane of vibration is known as optical activity and substances that show optical activity are known as optically active substances. . Calcite does not have the property to rotate the plane of vibration, therefore, it is not an optically active substance. 30 Substances such as quartz, sugar solution, turpentine and cinnabar, etc. are optically active substances. Substances which rotate the plane of vibration in clockwise direction or to the right are called dextrorotatory or right-handed rotation. Righthanded rotation means that it is right-handed for an observer. 31 A substance that rotates the plane of vibration in the anticlockwise direction or to the left is called levorotatory substance. It is observed that some quartz crystals are dextrorotatory while others are levorotatory. The amount of rotation in a solution depends upon the concentration of the solution. 32 SPECIFIC ROTATION The angle by which the plane of vibration rotates depends on the thickness of the substance density of the material or concentration of the solution wavelength of light the temperature 33 The specific rotation of a solution for a given temperature and wavelength of light is the rotation of plane of vibration produced by one decimeter length of solution when the concentration of solution is one gram per c.c. The specific rotation S is When length is taken in cm, then 34 FRESNEL’S THEORY OF OPTICAL ROTATION Fresnel’s theory of optical rotation is based on the fact that any plane polarized vibration may be regarded as the resultant of two circularly polarized vibrations rotating in opposite direction with the same velocity or frequency. 35 He assumed that a plane polarized light on entering a crystal along optic axis is resolved into two circularly polarized vibrations rotating in opposite directions with the same angular velocity. 36 2. In an optically inactive substance these two circular components travel with the same speed along the optic axis. Hence at emergence they give rise to a plane polarized light without any rotation of the plane of polarization. 37 3. In an optically active crystal, like quartz , two circular components travel with different speeds so that relative phase difference is developed between them. In dextro-rotatory substance vr>vl and in leavo rotatory substance vl>vr 38 4. On emergence from an optically active substance the two circular vibrations recombine to give plane polarized light whose plane of vibration has been rotated w.r.t that of incident light through a certain angle. 39 So This explanation was based on the following assumptions: 1. A plane polarized light falling on an optically active medium along its optic axis splits up into two circularly polarized vibrations of equal amplitudes and rotating in opposite directions –one clockwise and other anticlockwise. 2. In an optically inactive substance these two circular components travel with the same speed along the optic axis. Hence at emergence they give rise to a plane polarized light without any rotation of the plane of polarization. 3. In an optically active crystal, like quartz , two circular components travel with different speeds so that relative phase difference is developed between them. 4. In dextro-rotatory substance vR>vL and in leavo rotatory substance vL>vR.. 5. On emergence from an optically active substance the two circular vibrations recombine to give plane polarized light whose plane of vibration has been rotated w.r.t that of incident light through a certain angle depends on the phase diff between the two vibrations. Plane polarized means resultant of R and L. MATHEMATICAL TREATMENT Consider a plane polarized light beam of amplitude a, incident on a material. According to Fresnel, it splits up in to two components ‘R’ and ‘L’. Their components in the direction of X and Y are •For clockwise circular vibrations ‘R’: If angle between A and R is ωt; a x1 sin(t ), 2 a y1 cos(t ) 2 A R L •For anti clockwise circular vibrations ‘L’: If angle between A and L is also ωt (same angular speedO for optically inactive material) a x 2 sin(t ), 2 a y 2 cos(t ) 2 y B x 42 Therefore the resultant vibration be along the x axis : x x1 x 2 0 along the y axis : y y1 y 2 a cos(t ) Plane of vibration is along original direction. The result shows that two oppositely directed circular motions of equal velocity combine to give linear motion along the direction of motion (optically inactive material) 43 FOR OPTICALLY ACTIVE SUBSTANCES According to Fresnel the two circular components are propagated through the plate with different angular speeds. So when they emerges out of the crystal there is a phase difference between them. Suppose clockwise component moves faster compared to anticlockwise component. 44 •FOR CLOCKWISE VIBRATIONS: a x1 sin(t ) 2 a y1 cos(t ) ........(1 ) 2 A For anti clockwise vibration: 2 A’ L angle is (ωt) R O a x 2 sin t 2 a y 2 cos t ........(2 ) B 2 The resultant displacement along the two axes are, y y1 y 2 x x1 x 2 a a a a cost cos t sin(t ) sin t 2 2 2 2 a sin cos t ....(i) a cos cos t .......(ii ) 2 2 2 2 45 •So the vibrations represented by equation (i) and (ii) are perpendicular to each other. They have the same period and phase. •Therefore the resultant vibration is plane polarized and it makes an angle /2 with the original direction. •So plane of vibration has rotated through an angle /2 on passing through the crystal. •Dividing (i) by (ii) we get sin x 2 tan y cos 2 2 This is equation of straight line inclined at /2 with y-axis. That is with the vibrations of incident light. 46 Let refractive indices of the clockwise and anticlockwise vibrations be R and L , ‘t’ be the thickness of the quartz plate, then the optical path difference between the two components will be L R t •Corresponding phase difference will be 2 t L R •Angle of rotation of plane of vibration will be c c t L R t or 2 vL vR If velocities vR vL Right handed optical active material. •In case of left handed optically active crystals R L t or c c t and vL vR 2 vR vL 47