Q.Some materials fabricated in the laboratory (metamaterials) have a negative refractive index. A ray of light travels in air (medium 1) and is incident on the flat surface of such a negative-index material (medium 2), which occupies the region below the horizontal interface. The dotted line drawn perpendicular to the interface is the normal, and the incident ray arrives from the upper left, making an angle of incidence $i$ with the normal. Four possible outcomes are offered for the transmitted/refracted ray. Which one correctly describes what happens in a negative-refractive-index medium?

Concept

Snell's law at the interface: $n_1 \sin i = n_2 \sin r$, here $n_1 = +1$ (air) and $n_2 < 0$. Then

A negative $\sin r$ means the refraction angle is measured to the opposite rotational sense — geometrically the refracted ray falls on the same side of the normal as the incident ray, rather than crossing to the other side as in a normal medium.

Why this is (A)

For ordinary (positive-index) refraction, an incident ray from the upper-left continues into the lower medium on the right of the normal. Reversing the side (the negative-index signature) sends it into the lower medium on the left of the normal — still travelling forward (downward) into medium 2, but bent the "wrong" way. That is exactly option (A). …

Method: Handling the Sign Reversal in Negative-Index (Metamaterial) Refraction

Use this whenever a question involves a medium with $n < 0$ — Snell's law still applies formally, but the geometric meaning of a negative angle needs care.

Steps

Step 1: Write Snell's law with the actual signed index

Do not assume $n_2 > 0$ by default — substitute the given (possibly negative) value directly.

Step 2: Interpret the sign of $\sin r$ geometrically

For an ordinary medium ($n_2 > 0$), $\sin r$ comes out positive and the refracted ray crosses to the opposite side of the normal from the incident ray — this is the everyday refraction picture. When $n_2 < 0$, $\sin r$ comes out negative, which geometrically means the refracted ray stays on the same side of the normal as the incident ray instead of crossing over. …