# Homework1 - Homework 1 Answers 95.658 Spring 2011...

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Homework 1 Answers, 95.658 Spring 2011, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell Problem 1 Following the derivation in class, keep the negative-valued dispersion relation instead of the positive one (this corresponds to meta-materials instead of standard materials). Derive the equivalent to Snell's law that meta-materials obey and sketch out what this leads to. SOLUTION: Plugging a plane wave solution into the wave equation led to the dispersion relation: k 2 − 2 = 0 which when solved for the wave number k gives: k  For metamaterials we keep the negative-valued solution: k =−  We previously plugged the plane wave solution into Faraday's law to find that the fields are perpendicular: k × E 0 = B 0 If we use the dispersion relation, we can simplify this: k × E 0 =− 1  B 0 k × E 0 =− c n B 0 In metamaterials, the electric field E , the magnetic field B , and the propagation vector k are still all perpendicular to each other, but they now form a left-handed triple instead of a right-handed triple as shown in the diagram. That is why meta-materials are also known as “left-handed” materials. k E B Standard Materials: Right-Handed System k E B Metamaterials: Left-Handed System

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We can calculate the Poynting vector to get a sense of where the energy is going. We must remember to use the real parts of the fields: S = E × H S = E 0 cos k x − t × H 0 cos k x − t  Use H = 1 B S = 1 E 0 × B 0 cos 2 k x − t Use the relation above: S =− E 0 × k × E 0 cos 2 k x − t S =− E 0 2 cos 2 k x − t k To find the time-averaged energy flow, we integrate over some time T in the usual way and let T become large, leading to: < S > =− 1 2 E 0 2 k In metamaterials, the energy flows at the same rate as in standard materials, but in the opposite direction as the wave vector. This seems counter-intuitive that the energy would flow in the opposite direction as the wave-fronts are traveling, but this has been verified experimentally. In deriving the law of refraction, we matched up the incident wave k and transmitted wave k ' at the boundary to find: k x z = 0 = k ' x z = 0 Carrying through the dot-products with the understanding that the x vector at the boundary is always in the z plane gives us: k sin i = k 'sin t Suppose the region of the incident wave is filled with standard material and the region of the transmitted wave is filled with meta-material. Using the respective dispersion relations gives us:  sin i =− '  '  sin t n sin i = n 'sin t where n' < 0
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