The optical properties of a medium are governed by the relative permitivity `(epsi_(r))` and relative permeability `(mu_(r))`. The refractive index is defined as `sqrt(mu_(r)epsi_(r))=n`. For ordinary material `epsi_(r)gt0` and `mu_(r)gt0` and the positive sign is taken for the square root. In 1964, a Russian scientist V. Veselago postulated the existence of material with `epsi_(r)lt0andmu_(r)lt0`. Since then such 'metamaterials' have been produced in the laboratories and their optical properties studied. For such materials `n=-sqrt(mu_(r)epsi_(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 `theta` in `2^(nd)` quadrant, then the refracted beam is in the `3^(rd)` quadrant.
(ii) Prove that Snell's law holds for such a medium.

Explanation

Suppose the given postulate is true.
In that event, two parallel rays entering such a medium from air (refractive index `= 1`) at an angle `theta_(i)` in 2nd quadrant, will be refracted in 3rd quadrant as shwon in Fig.

Let `AB` represent the incident wavefront and `DE` represent the refracted wavefront. All points on a wavefront must be in same phase and in turn, must have the same optical path length.
`:. -sqrt(in_( r) mu_(r )) AE = BC - sqrt(in_( r) mu_(r ))CD or BC = sqrt(in_( r) mu_(r )) (CD - AE)`
If `BC gt 0`, then `CD gt AE`, which is obvious from Fig.
Hence the postulate is resonable.
However, if the light proceeded in the sense it does for ordinary material, (going from 2nd quadrant to 4th quadrant) as shown in Fig. then proceeding as above,
`-sqrt(in_(r ) mu_(r))AE = BC - sqrt(in_(r ) mu_(r )) CD or BC = sqrt(in_(r) mu_(r ))(CD - AE)`
As `AE gt CD`, therfore, `BC lt 0` which is not possible. Hence the given postulate is correct.
(ii) In Fig. `BC = AC sin theta_(i) and CD - AE = AC sin theta_( r)`
As `BC = sqrt(in_(r ) mu_(r)) (CD - AE)`
`:. AC sin theta_(i) = sqrt(in_(r ) mu_(r )) AC sin theta_(r )`
or `(sin theta_(i))/(sin theta_( r)) = sqrt(in_( r)/(mu_(r )) = n`, which proves Snell's law.