1 6007 spring 2011 problem set 8 electromagnetic waves

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Unformatted text preview: have a lot of incentive for writing a great problem! 1 6.007 Spring 2011 Problem Set 8: Electromagnetic Waves at Boundaries Problem 8.1 – Snell’s Law In ray optics, it is useful to use Snell’s Law at an interface between two materials: n1 sin(θ1 ) = n2 sin(θ2 ). (a) Imagine a (collimated) beam of light being shone down to an air-water interface from 5 cm above the water at 45◦ from the normal. The index of air is taken to be 1 and the index of water is taken to be 1.33. A fish is swimming at 15 cm horizontal distance away from the light as shown. At what depth will the fish see the beam of light? 45± 5 cm n1=1 n2=1.33 d 15 cm (b) Now the fish is 15 cm deep, looking up at the water at 30◦ as shown and there is a fly skimming the water surface 1 cm above the water and changing his position. Can the fish see the fly? Please explain your answer. 1 cm cm n1=1 n2=1.33 15 cm 30± 2 6.007 Spring 2011 Problem Set 8: Electromagnetic Waves at Boundaries Problem 8.2 – Frustrated Total Internal Reflection This problem explores the phenomenon of frustrated total internal reflection and the more general math that goes with it. In lecture, we discussed what happens when total internal reflection is frustrated by bringing a second medium (e.g., glass) into the evanescent field of the reflected wave. Figure 1 shows a schematic of the physical setup of frustrated internal reflection, where light which would be reflected internally inside a glass waveguide is able to transmit across an air gap into another piece of glass. Figure 1: Schematic of frustrated internal reflection. To simplify our modeling of the above system, we’ll look only in the direction across the air gap, assuming that the incoming angle is such that total internal reflection occurs inside the first piece of glass, and therefore, the field in the air gap is evanescent. Figure 2 shows a schematic in 1D of the glass-air-glass transition. I II III ~ ~ ~ n1 n1 n2 Incident Wave z 0 d x Figure 2: 1D schematic of coupling via evanescent field. We set up the following equations for the electric field in the various regions in Figure 2, setting the incident wave’s magnitude to 1 so that the reflection and transmission coefficients (rn and tn ) can be solved for directly: Ey,I (x, t) = ej (ωt−k1x x) +...
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