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Metamaterials \(\rightarrow n = – \sqrt{μ_r ε_r}\)

The optical properties of a medium are governed by the relative permitivity \((ε_r)\) and relative permeability (\(μ_r)\). The refractive index is defined as \(\sqrt{μ_rε_r} = n\). For ordinary material \(ε_r > 0\) and \(μ_r > 0\) and the positive sign is taken for the square root.

In 1964, a Russian scientist V. Veselago postulated the existence of material with \(ε_r < 0\) and \(μ_r < 0\). Since then such ‘metamaterials’ have been produced in the laboratories and their optical properties studied.

For such materials \(n = – \sqrt{μ_r ε_r}\) . As light enters a medium of such refractive index, the phases travel away from the direction of propagation.

(i) According to the description above show that if rays of light enter such a medium from air (refractive index =1) at an angle \(θ\) in \(2nd\) quadrant, then the refracted beam is in the \(3rd\) quadrant.

(ii) Prove that Snell’s law holds for such a medium.

\((i)\) Suppose the postulate is true, then two parallel rays would proceed as shown in Figure. Assuming \(ED\) shows a wave front then all points on this must have the same phase. All points with the same optical path length must have the same phase.

Thus \(– \sqrt{ε_rμ_r} AE = BC –\sqrt{ε_rμ_r}CD\) or \( BC = \sqrt{ε_rμ_r}(CD − AE) \)

\(\rightarrow BC > 0, CD > AE\)

As showing that the postulate is reasonable. If however, the light proceeded in the sense it does for ordinary material (viz. in the fourth quadrant, Fig. 2)

Then \(– \sqrt{ε_rμ_r} AE = BC –\sqrt{ε_rμ_r}CD\) or \( BC = \sqrt{ε_rμ_r}(CD − AE) \)

As \(AE > CD, BC < O\), showing that this is not possible. Hence the postulate is correct.

\((ii)\) Prove yourself. \(\ddot \smile\)

Note by Nishant Rai
2 years, 3 months ago

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@Nishant Rai a comprehension based on this was asked in \(JEE\) Tanishq Varshney · 2 years, 3 months ago

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