A plane wave is incident on a layered interface as shown in the figure. The indices of refraction of the three nonpermeable media are n₁, n₂, n₃. The thickness of the intermediate layer is d. Each of the other media is semi-infinite.

(a) Calculate the transmission and reflection coefficients (ratios of transmitted and reflected Poynting's flux to the incident flux), and sketch their behavior as a function of frequency for n₁ = 1, n₂ = 2, n₃ = 3; n₁ = 3, n₂ = 2, n₃ = 1; and n₁ = 2, n₂ = 4, n₃ = 1.

(b) The medium n₁ is part of an optical system (e.g., a lens); medium n₃ is air (n₃ = 1). It is desired to put an optical coating (medium n₂) on the surface so that there is no reflected wave for a frequency w₀. What thickness d and index of refraction n₂ are necessary?

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