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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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a comprehension based on this was asked in $$JEE$$ · 2 years, 3 months ago