Polarization and Birefringence

Transcription

1 Unit 3 Polarization and Birefringence Introduction Phenomena in which polarized light plays a part are frequently surprising, often beautiful, and of considerable practical use. The series of experiments suggested below permits observation of three schemes for producing polarized light -- by use of selective absorption (Polaroids), by reflection, and by use of doubly refracting (birefringent) materials. (For birefringent materials, the index of refraction has different values for the two senses of polarization of light.) Most of your measurements will be qualitative or semi-quantitative, but do make careful notes of what you observe. Given that these topics are often not covered in lecture, we have provided an appendix to discuss the relevant physics concepts of polarization and birefringence. Most texts do a credible job on polarization, so our discussion of that will be brief in that appendix. Equipment Description You will need two white light sources, a simple desk lamp or bulb for most of the studies, and a penlight when you need a more directed beam. You should have a protractor for measuring angles and a ruler for measuring lengths. In addition there is a kit that contains, among other things: several pieces of Polaroid film, at least two of which are rectangular, having their pass axis along one of the edges two simple pieces of birefringent material that will look identical, except one is thicker than the other a piece of Polaroid that has a birefringent sheet laminated to it a small mirror a small sheet of flat glass a Lucite shape for investigating stress lines some stretched cellophane tape Experiments (A) Look at a light source through two superposed sheets of Polaroid, and note how the intensity varies as they are rotated relative to each other. Sketch how the transmitted intensity varies as one Polaroid is rotated relative to the other through 360. Does your observation agree with the theory of "Malus' Law", which is that I = I 0 cos 2 θ? (B) Between two crossed Polaroids (i.e., θ = 90 so that the throughput is minimal) insert a third Polaroid sheet and rotate this third one without disturbing the other two. Sketch the transmitted intensity as the middle Polaroid is turned through 360.

2 Give an explanation of how the insertion of the middle Polaroid can permit light to pass through the system. (C) Originally unpolarized light which has been reflected from an interface is found to be partially or completely polarized. For dielectric materials, there is an angle of incidence for which the reflected light is expected to be completely polarized. This angle is called Brewster's angle, θ B, and it is found that tan θ B = n 2 /n 1 where n 1 and n 2 are the indices of refraction of the medium from which the light comes and the medium into which it is directed. [Your textbook has a nice description of this phenomenon!] Devise a simple experimental setup, using the pen-light as a source and one of the Polaroids, by means of which you can measure the value of θ B for the piece of flat glass and, hence, get a value for the index of refraction of the glass. Here assume that the index of refraction of air is unity. Suitable measurements with the meter stick will give sufficient accuracy on this index (try for 5% or 10%) for our purposes. Besides giving a value for the glass index, your observation should also tell you which is the pass direction of your Polaroid, i.e., the direction in which the E-vector of the transmitted light points. (D) We now turn to some observations of the behavior of birefringent materials (refer to notes given in the Appendix). Properties of this birefringent materials in your kit do not change greatly as the wavelength ranges over the visible spectrum, so you can make your observations with white light. (D.1) You have two such rectangular samples: one is a quarter wave plate ( 1/4- wv ) and the other a half-wave plate ( 1/2-wv ). The challenge is to determine which plate is which! [Do not assume that the thicker one is the 1/2-wv -- these may not even be of the same material!!] Devise some definitive test and carry it out. As you begin this test, you should determine how the fast and slow polarization directions are lined up relative to the edges of the sheets. However, you will not be able to tell which is the fast and which the slow direction without additional apparatus. (D.2) To make some systematic observations, place the quarter-wave plate with its slow (or fast) polarization direction at 0, 30, 45, and 90 to the plane of polarization of light coming from a Polaroid. Use another Polaroid to determine the polarization state of the light emerging from the quarter-wave plate. Repeat the same steps with the halfwave plate. Make sure you can interpret everything you see in terms of properties of polarized light. (E) One of the sheets is a laminate of a 1/4-wv plate with a Polaroid. You should be able to identify this laminate by a simple test. The slow and fast directions of the 1/4- wv layer are arranged to be at 45 with the pass direction of the Polaroid. (E.1) What will be the polarization state of light entering the Polaroid side and coming out the 1/4-wv side? Check this experimentally. (E.2) If you turn the plate over, what is the state of polarization of the light entering the 1/4-wv side and coming out from the Polaroid side? Check this experimentally too.

3 (E.3) An amusing trick to play with this laminate is to place it directly on top of a mirror. One way up, reflections in the mirror can be seen; the other way up there is almost complete blackout. Explain this phenomenon. (Hint: consider that, with the laminate one way up, the light passes effectively through a Polaroid, two 1/4-wv plates in sequence, which are equivalent to one 1/2-wv plate, and then through Polaroid once more, with the pass axis parallel to that of the first Polaroid. When the laminate is turned over, this sequence becomes 1/4-wave plate, two parallel Polaroids, 1/4-wv plate, and you can show that this arrangement permits some light to pass.) This idea is used in some liquid crystal displays in which little segments of the crystals can be made birefringent by applying voltage to them, allowing the image behind the crystal to appear or vanish at will! (F) Stress in structural members can be evaluated by making models of the system under study in a material that becomes birefringent when stressed. Lucite is such a material. Arrange the Lucite L test piece between two crossed sheets of Polaroid and hold this collection up to the light without stressing the test piece. Describe your observations as the piece is put under stress (try not to break the test pieces, pleases!) and explain how this is consistent with your notions of birefringence. Appendix on Polarization Most textbooks give some discussion of linear and circular polarization of light, so this appendix will be a brief summary. One chooses the direction of propagation to be x, and looks into the beam of light (i.e., looking along x). The polarization of the light is defined for the motion of the tip of the electric field vector as time progresses. [In this discussion, x, y, z are unit vectors!] For example, if the electric field is given by E LP = E 0 (cos α y + sin α z) cosωt the tip of the field vector traces out a line in the y-z plane at angle α with respect to the y axis as time evolves, as shown in the accompanying sketch. This is the general case of a linearly polarized wave. Note that the two components in z and in y are in phase with each other; as the z component of E decreases, so does the y component. The time averaged value of E*E, which is proportional to the intensity, is just E 0 2 /2. This angle α is a physical angle, one that can actually be measured with a protractor in the lab. If one changes this phase relationship so that the y and z components peak a quarter of a period apart in time the field vector would be as E EP = (E 0 cos α) cosωt y ± (E 0 sin α) sinωt z with the + or - determined by whether the z component leads of lags the y component. Now as the y component decreases, the z component increases and vice versa. To simplify the picture take α = 45 so that we have E CP = E 1 cosωt y ± E 1 sinωt z With E 1 = E 0 / 2. The time dependence of the tip of E CP is shown in the figure below and traces out a circle; hence this situation, equal components in the two transverse directions with a phase difference of π/2, is called circular polarization. [Recall that phase angles cannot be measured with protractors or rulers and that only phase differences matter to

4 the physics!] Now the magnitude of the electric filed vector E CP remains constant in time as E*E = E 12. Many natural phenomena emit circularly polarized light. As extension we see that E EP above, with unequal components along y and z, will have its field vector trace out an ellipse and be called elliptical polarization. One can also see from this construction that taking the linearly polarized form and introducing a half-period lag of one component with respect to the other again gives linearly polarized light, but at an angle of α with respect to y. Two empirical aspects: (1) If elliptically polarized light is viewed through a Polaroid, the transmitted intensity varies as the Polaroid is rotated. The ratio of the maximum to the minimum intensity will be the square of the ratio of the major to minor axes of the ellipse traced out by the tip of E EP ; the ratio is just tan α. (2) If you view circularly polarized light through a Polaroid, the intensity remains unchanged as you rotate the Polaroid. In this respect the result is just the same as if you had viewed unpolarized light. However, circularly polarized light has a rigid phase discipline which unpolarized light lacks, and there are ways of distinguishing between the two! Appendix on Birefringence Certain anisotropic materials present different refractive indexes to light whose electric field vector points in different directions with respect to the material. I.e, the index of refraction is a function of the angle of E with repsect to some internal axis of the material. Several common crystals, such as calcite and mica, display this property. Cheap cellophane tape also is birefringent; in this case, the birefringence results from a unidirectional stretching which takes place during manufacture and which tends to align the long molecules parallel to the "run" of the tape. Describing the propagation of light waves in such a birefringent medium is surprisingly complicated, in the general case. However, in our experiments it will suffice to define two perpendicular axis, s and f, on the surface of the sheet of material, for slow and fast. Light with E along f travels faster in the material than light with E along s; since v=c/n, ts means n s > n f. If E entering a birefringent material has components along both of these axes, the slow component will leave the material lagging the fast component. A birefringent plate which introduces just π/2 radians of phase difference between E s and E f is called a quarter-wave plate. If it introduces π radians, it is a half-wave plate. In the experiment, you will be playing tricks with polarized light and quarterwave or half-wave plates. To check your understanding of what goes on, make certain that you understand and agree with the following empirical statements: (3) When plane-polarized light passes through a quarter-wave plate whose slow and fast axes are at 45 to the plane of polarization, circularly polarized light emerges from the plate. (4) When circularly polarized light strikes a quarter-wave plate (in any orientation), plane-polarized light emerges. The plane of polarization lies at 45 between the slow and fast axes of the plate.

5 α α Figure 1: The electric field vector, E LP, of the linearly polarized wave described in the appendix at three times: t = 0, t = π/4ω, and t = π/ω. As time evolves the tip of the E vector traces out a line. Figure 2: The electric field vector, E CP, of the circularly polarized wave described in the appendix at three times: t = π/4ω, t = 3π/4ω, and t = 3π/2ω. As time evolves the tip of the E vector traces out a circle.

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