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Questiоn 1: Opticаl Prоperties оf Metаls Mg hаs an electron density of 8.61 x 1028 m-3 and a DC conductivity of σ0=2.26×107Ω-1m-1{"version":"1.1","math":"σ0=2.26×107Ω-1m-1"}. The optical properties of Mg follow the Drude model well and there are no bound electron transitions that impact the plasma frequency. (a) Find the plasma frequency and the relaxation time. (b) Find the real and imaginary parts of the dielectric constant at a wavelength of 500 nm. (c) Find the reflection coefficient of the Mg at 500 nm for normal incidence. The reflection coefficient for normal incidence, R, given by: in which ni’ and ni” are the real and imaginary parts of the refractive index of medium i. (d) Make a rough sketch of nmetal’ and the reflection coefficient at normal incidence as a function of frequency from ω = 0 s-1 to ω = 1017 s-1. Clearly indicate the plasma frequency in this plot. Question 2: Negative Refraction Consider the negative refraction phenomena as shown in Figure (1). The photonic crystal has a constant frequency contour as shown in Figure (2). Assuming that at ω{"version":"1.1","math":"ω"} = 0.226 *2π{"version":"1.1","math":"π"}c/a, (a is the lattice constant), the constant frequency contour in the wavevector space is a circle, centered at M, with a radius of approximately 0.1*2π{"version":"1.1","math":"π"}/a. (Figure 2, the right panel). The crystal is truncated with interfaces that are perpendicular to the wavevector at M, and outside the crystal is air. Assume that the incident beam has a direction that is sufficiently close to the normal direction of the crystal: (a) What is the relation between θ1{"version":"1.1","math":"θ1"} and θ2{"version":"1.1","math":"θ2"}? (b) What is the direction of the beam as it emerges from the other side of the crystal (i.e. the dashed arrow in Figure (1))? Why? (c) Suppose the source is at a distance d1 from one surface of the crystal, and that the thickness of the crystal is d, what is the distance d2 of the image from the other interface of the crystal? Congratulations, you are almost done with this exam. DO NOT end the Honorlock session until you have submitted your work to Brightspace When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to submit your work: Exam 1 Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.