Q.An infinitely long solid cylinder of radius $R$ is made of an unusual exotic material whose refractive index is exactly $-1$. The cylinder is placed between two parallel horizontal planes (a lower plane and an upper plane) whose normals point in the $y$ direction; the axis and centre O of the cylinder lie on the $y$-axis, midway, with the cylinder's circular cross-section in the $xy$-plane. A narrow laser beam is fired straight upward (in the $+y$ direction) from the lower plane, at a horizontal distance $x$ from the vertical diameter of the cylinder (so $x$ is the perpendicular distance of the beam from the axis, i.e. its impact parameter). Find the range of $x$ for which the light emitted from the lower plane fails to reach the upper plane.

Incidence angle

The beam travels along $+y$ and strikes the lower surface of the cylinder. With impact parameter $x$, the radius to the strike point makes the normal, and the angle of incidence $i$ satisfies

(For $x>R$ the beam misses the cylinder entirely and goes straight to the upper plane.)

Negative refraction: deviation at each surface

Snell's law with $n_2=-1$ gives $\sin i=|n_2|\sin r=\sin r$, so $|r|=i$, but the refracted ray lies on the same side of the normal as the incident ray (the negative-index signature). Resolving the ray into components along and across the normal, the tangential component reverses while the normal component keeps going forward. Starting from the upward direction $(0,1)$, the ray inside the cylinder becomes

i.e. it is rotated by $2i$. Tracing the chord to the far surface and refracting again (by symmetry the second incidence angle is also $i$), the emergent ray direction works out to

So the total deviation is $4i$.

Condition for not reaching the upper plane

The emergent ray has vertical (upward) component $\cos 4i$. It can climb to the upper plane only if this is positive. Hence the light fails to reach the top when

Convert to $x$

Using $x=R\sin i$: …

Method: Ray-tracing through a negative-index element by tracking total deviation

This method applies whenever a ray must be traced through an optical element made of a negative-refractive-index material, and you need the condition (on an angle, impact parameter, or position) under which the emergent ray does or doesn't reach a target.

Steps

Step 1: Convert the given geometric parameter into an incidence angle.

Whenever a ray strikes a curved (circular/spherical) surface at some offset from the axis — an impact parameter, a height, a lateral distance — that offset fixes the angle between the ray and the local normal (which, for a circle, is along the radius to the strike point). For an impact parameter $x$ on a cylinder of radius $R$,

Always state the geometric relation first; every later step is expressed in terms of $i$, and you convert back to the original variable only at the end.

Step 2: Apply Snell's law recognising the sign behaviour of negative refraction, not just its magnitude.

For an ordinary medium ($n>0$), a refracted ray crosses to the opposite side of the normal and deviates by $|i-r|$. For a negative-index medium, $|r|=i$ in magnitude (Snell's law $n_1\sin i = n_2 \sin r$ still holds numerically) but the refracted ray stays on the same side of the normal as the incident ray — so the ray direction is rotated, at that single surface, by a full $2i$ rather than the small ordinary deviation. This doubling is the key general fact to reuse for any negative-index surface, not a coincidence of this problem.

Step 3: Sum the deviation over every surface the ray crosses, using the geometry's symmetry.

For a body with two refracting surfaces related by symmetry (e.g. entry and exit points on a circular cross-section symmetric about the chord), the same incidence angle $i$ recurs at the second surface, so the total deviation is simply the deviation-per-surface added up ($2i+2i=4i$ for two surfaces). Track the ray as a direction vector (or angle from a reference axis) so that "deviation" always means "how much the direction vector has rotated," which composes additively across surfaces. …