#### The optical properties of a medium are governed by the relative permittivity (εr) and relative permeability (μr ). The refractive index is defined as  $\sqrt{\mu, \epsilon }=n$     For ordinary material εr > 0 and μr > 0, and the positive sign is taken for the square root. In 1964, a Russian scientist V. Veselago postulated the existence of material with εr < 0 and μr < 0. Since then, such ‘metamaterials’ have been produced in the laboratories and their optical properties studied. For such materials $n=\sqrt{\mu, \epsilon }$ As light enters a medium of such refractive index, the phases travel away from the direction of propagation. (i) According to the description above shows that if rays of light enter such a medium from the air (refractive index = 1) at an angle in 2nd quadrant, them the refracted beam is in the 3rd quadrant.(ii) Prove that Snell’s law holds for such a medium.

Again consider figure (i), let AB represent the incident wavefront and DE represent the refracted wavefront. All point on a wavefront must be in same phase and in turn, must have the same optical path length.

$\\Thus\; \; \; \; \; -\sqrt{\epsilon_r \mu_r }AE=BC -\sqrt{\epsilon_r \mu_r }CD\\\\Or\; \; \; \; \; BC=\sqrt{\epsilon_r \mu_r }\left ( CD-AE \right )\\\\BC>0,CD>AE$

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)

$\\Then\; \; \; \; \; -\sqrt{\epsilon_r \mu_r }AE=BC -\sqrt{\epsilon_r \mu_r }CD\\\\Or\; \; \; \; \; BC=\sqrt{\epsilon_r \mu_r }\left ( CD-AE \right )\\\\i\! f \; \; \; \; BC>0,then \;CD>AE$

Which is obvious from Fig. (i). Hence, the postulate is reasonable.

However, if the light proceeds in the sense it does for ordinary material, (going from the second quadrant to 4th quadrant)as shown in Fig. (i). then proceeding as above,

$-\sqrt{\epsilon_r \mu_r }AE=BC -\sqrt{\epsilon_r \mu_r }CD\\\\Or\; \; \; \; \; BC=\sqrt{\epsilon_r \mu_r }\left ( CD-AE \right )\\\\As \; \; \; \; AE>CD \;Ther\! f\! ore \;BC<0$

Which is not possible. Hence, the postulate is correct

(ii) From Fig (i),

$\; \; \; \; \; \; BC=AC \sin \theta,\\ and\: \; \; \; \;CD-AE=AC \sin \theta \\As\; \; \; \; \; \; BC=-\sqrt{\mu_r \epsilon_r }(AE-CD)\\\\ \therefore \; \; \; \; \; \; AC \sin \theta_i =-\sqrt{\epsilon_r\mu_r }AC \sin \theta\\\\or\; \; \; \; \; \; \; \; \frac{\sin \theta_i}{\sin \theta_r}=\sqrt{\epsilon_r\mu_r }\;n$

Which proves Snell's law