228s12-l08

# 228s12-l08 - Physics 228 Today Polarization Scattering...

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Unformatted text preview: Physics 228 Today: Polarization, Scattering Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Summer research internships may still be possible - see: http:/ /www.youtube.com/watch?v=5lcSGNDRonw&feature=youtu.be or http:/ /see.orau.org/ First exam (Thursday, Feb 23, 9:40 PM) room assigments: Physics Lecture Hall: sections 03, 16, 20, 22 - Profs. Salur and Basirnia ARC 103: sections 02, 04, 05, 06, 14, 15, 21 - Profs Cizewski, Zamolodchikov You are allowed 1 formula sheet (both sides may be used, any size font you want), pencils, and a calculator calculator. Monday, February 13, 2012 Linear Polarization Polarization refers to the orientation of the electric (and magnetic) ﬁelds of a light wave. For a wave moving to +z, we have written: yB ￿ (z, t) = E0 cos(kz − ω t)ˆ E x E ￿ B (z, t) = B0 cos(kz − ω t)ˆ y x But the orientation of the ﬁelds could be different. For example: ￿ E (z, t) = E0 cos(kz − ω t)ˆ y or ￿ B (z, t) = −B0 cos(kz − ω t)ˆ x (ˆ + y ) xˆ ￿ E (z, t) = E0 cos(kz − ω t) √ 2 (ˆ − x) yˆ ￿ B (z, t) = B0 cos(kz − ω t) √ 2 y E B x By E and B must be perpendicular, but can be in any direction. Monday, February 13, 2012 E x Circular Polarization The polarization direction does not have to be ﬁxed - it can rotate. Consider ``right handed’’ circularly polarized light: B ￿ E (z, t) = E0 cos(kz − ω t)(cos(ω t)ˆ + sin(ω t)ˆ) y x y ￿ B (z, t) = B0 cos(kz − ω t)(cos(ω t)ˆ − sin(ω t)ˆ) y x As the light heads towards us, we see the ﬁelds rotating CCW. There is also ``left handed’’ circularly polarized light. B y ￿ E (z, t) = E0 cos(kz − ω t)(cos(ω t)ˆ − sin(ω t)ˆ) x y ￿ B (z, t) = B0 cos(kz − ω t)(cos(ω t)ˆ + sin(ω t)ˆ) y x As the light heads towards us, we see the ﬁelds rotating CW. You can also see that if you add the RH to the LH light, the ``sin’’ terms have opposite signs and cancel, we we get linearly polarized light with E in the x direction, B in the y direction. Any direction of linearly polarized light can be represented as a sum of RH + LH circularly polarized light, and vice versa. Monday, February 13, 2012 E x E x Circular Polarization The picture from the text... Monday, February 13, 2012 Elliptical Polarization If the two linearly-polarized waves we add have different amplitudes, the sum is an elliptically polarized wave. The amplitude will, for example, be bigger when it is in the ±x direction than in the ±y direction. Monday, February 13, 2012 Pick a Direction? With a rope, you can make a wave traveling to the right with the displacement in the vertical or in the horizontal direction. You can also ``polarize’’ the wave build a ﬁlter that only allows waves with displacement in a particular direction to pass. By waving the end of rope around in a circle we generate a ``corkscrew’’ wave, which is polarized by a vertical slit. Monday, February 13, 2012 Polarization of Light Sources As we discussed earlier, if, as in the top of the broadcast antenna shown to the left, you make the electrons oscillate vertically, then you get light with a vertically polarized E ﬁeld. But if, as in the bulb, there is no preferred direction, the light is unpolarized - the E ﬁeld of the light at any point varies randomly with time. PhET simulation Monday, February 13, 2012 Polarizing Light iClicker 1 of 3 We can polarize light, similar to how we polarize the transverse wave on a rope. We use materials that have slots smaller than the wavelength of light. Consider the slotted metal grid. Can we polarize microwaves with a few cm wavelength with it? a) No. b) Yes. E is parallel to the slot. c) Yes. E ⊥ slot. d) Yes. But E is not oriented any particular way. Recall that at the surface of a metal, E|| vanishes, but E⊥ does not. e) Yes, but you really need small holes rather than slots. Demo Monday, February 13, 2012 Polarizing Visible Light Although the wavelength of visible light is < 1 μm, we can linearly polarize it using arrays of molecules, as in a polaroid ﬁlter. The idea is the same as with microwaves and the metal plate: if the electric ﬁeld orientation can accelerate electrons in the material - E is parallel to the long molecules - the electrons will accelerate, absorbing the energy from the ﬁeld and screening out the ﬁeld components in that direction. If the electrons cannot be accelerated, no energy is absorbed and the wave passes through. Monday, February 13, 2012 Single Polarizer Algebra: light has components relative to polarizer direction of: E|| = (E.n)n cos(kz-ωt) = Ecos(φ)cos(kz-ωt) which passes through E⊥ = [E-(E.n)n] cos(kz-ωt) = Esin(φ)cos(kz-ωt) which is blocked Note: from here on, by || we mean light polarized Demo in the direction the polarizer passes, ... Monday, February 13, 2012 What intensity of light makes it though a polarizer? Since I ∝ E , and E 2 || / Eincident = cos(φ), the intensity after the polarizer is Iout = Iincident cos2(φ). If the incident light is unpolarized, the average cos2(φ) is 1/2, so Iout = Iincident/2. Demo Monday, February 13, 2012 Polarizer + Analyzer -or- Polarizers Crossed at Arbitrary Angles ``Malus’s Law’’ Monday, February 13, 2012 Crossed Polarizers Demo Monday, February 13, 2012 iClicker 2 of 3 I have 3 polarizers in a row, with the 1st in the x direction, the 2nd rotated 45o, and the 3rd in the y direction. What fraction of the light that makes it through the 1st polarizer also makes it through the 3rd polarizer? a) 0 = cos(φ1-φ3) = cos(90). b) 1/4 = cos2(45) x cos2(45). c) 1/2 = cos(45) x cos(45). d) It depends on whether the light is in the +x or -x direction initially. e) It depends on whether the light is linearly polarized or circularly polarized? Monday, February 13, 2012 We have cos2(45) two times. It does not matter what the polarization was before the previous polarizer, only what it was before the current one. Polarization in Reﬂection When light is incident upon a surface, it generally partially refracts and partially reﬂects. The ``plane of incidence’’ is the plane that contains the incident and reﬂected light rays. The electric ﬁeld can be split into a component in the plane of incidence, and a component perpendicular to the plane of incidence (also parallel to the surface). Normally each component is partially reﬂected and partially transmitted, but not to the same degree, so the reﬂected / transmitted light is partially polarized. Monday, February 13, 2012 Brewster’s angle Brewster’s angle is the angle for which θp + θb = 90o: the reﬂected and refracted rays are 90o apart. From Snell’s Law, na sin(θp) = nb sin(θb) na sin(θp) = nb sin(90-θp) na sin(θp) = nb cos(θp) tan(θp) = nb/na For this angle, the reﬂected ray is polarized completely perpendicular to the plane of incidence, and the component of the E ﬁeld in the plane of incidence is completely refracted. Monday, February 13, 2012 Brewster’s angle For this angle, the reﬂected ray is polarized completely perpendicular to the plane of incidence, and the component of the E ﬁeld in the plane of incidence is completely refracted. Practical application: reﬂected sunlight has large horizontal polarization, the glare of which can be largely eliminated with vertical polarizing sunglasses. Monday, February 13, 2012 Scattering of Light by Air Why are clouds white? Why is the sky blue? Thin clouds are white because they scatter all wavelengths of light. Thick clouds turn dark / gray because too much of the light is absorbed. The sky blue because the scattering of light is proportional to f4 or 1/ λ4. Similarly, in the evening the sun appears more red because the blue light scatters out. Blue scatters more by a factor of ≈ (750/450)4 ≈ 8. The scattered blue light is polarized. You can see that the sunlight can accelerate atoms in the yz plane, but the sunbather sees acceleration in the xz plane. The sunbather sees linearly (z) polarized light. Monday, February 13, 2012 Calcite crystal from Furrfu, Wikimedia Commons Calcite is a material where the index of refraction depends on the polarization of the light (birefringence). Thus the light is refracted into two images, each polarized differently. We can see this with a polarizer. Monday, February 13, 2012 iClicker 3 of 3 Light is normally incident on two polarizers that are crossed at an angle of 60 degrees. (It might help to know that cos(60) = 1/2.) What fraction of incident unpolarized light intensity is transmitted through both polarizers? a) 1/2. b) 1/4. c) 1/8. d) 1/16. e) It depends on whether the incident light is unpolarized linearly or unpolarized circularly. Monday, February 13, 2012 There is a factor of 1/2 going from upolarized light to polarized light by the ﬁrst polarizer, then a factor of cos2(60) = 1/4 for the second polarizer. 1/2 x 1/4 = 1/8. ...
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