Animated Solution for Physics - Optics: A thin slice is cut out of a glass cylinder along a plane parallel to its axis. The slice is placed on a flat plate as shown. The observed interference fringes from this combination shall be

The phenomenon of interference is one of the most beautiful demonstrations of the wave nature of light. When we think of interference patterns formed by thin films, the classic example that comes to mind is Newton's Rings. In that setup, a plano-convex spherical lens is placed on a flat glass plate, creating a thin film of air whose thickness varies radially. Because of the spherical symmetry, the locus of points with equal thickness forms concentric circles, resulting in a circular fringe pattern.

But what happens if we change the geometry of the lens?

In this problem, instead of a spherical lens, we are given a thin slice of a glass cylinder cut parallel to its axis. This slice is placed on a flat glass plate.

Imagine looking at the cross-section of this setup. It looks exactly like the plano-convex lens. However, the crucial difference lies in its 3D geometry. A cylinder possesses translational symmetry along its central axis.

When light shines down from above, it reflects off the bottom curved surface of the cylinder and the top flat surface of the glass plate. These two reflected rays interfere with each other. The condition for constructive or destructive interference depends entirely on the path difference between these rays, which in turn depends on the thickness of the air film, $t$, at any given point.

To determine the shape of the interference fringes, we must find the locus of points where the air film thickness $t$ is constant.

Let's define a coordinate system where the central line of contact between the cylinder and the plate is the y-axis, and the x-axis is perpendicular to it along the flat plate. The thickness of the air film $t$ at a horizontal distance $x$ from the central axis can be approximated using the geometry of a circle (for small $x$):

where $R$ is the radius of the cylinder.

Notice that the thickness $t$ depends only on the distance $x$. It does not depend on the position along the y-axis (parallel to the cylinder's axis). Therefore, if you move along a line parallel to the cylinder's axis, the distance $x$ remains constant, and consequently, the thickness $t$ remains perfectly constant.

Since the locus of constant thickness is a straight line parallel to the axis of the cylinder, the locus of constant path difference is also a straight line. As a result, the bright and dark fringes we observe will be perfectly straight lines.

Now, let's address the spacing between these straight fringes. Are they equally spaced?

For a dark fringe to form, the path difference must satisfy the condition for destructive interference. Considering the phase change of $\pi$ upon reflection from the denser glass plate, the condition for dark fringes is:

Substituting our expression for $t$:

This equation tells us the position $x_n$ of the $n$-th dark fringe. Because $x_n$ is proportional to the square root of $n$ ($\sqrt{n}$), the distance between consecutive fringes ($x_{n+1}-x_n$) is not constant.

As $n$ increases (moving further outwards from the central axis), the difference $\sqrt{n+1}-\sqrt{n}$ decreases. Physically, this means that the curved surface of the cylinder pulls away from the flat plate more rapidly as you move outwards. To accumulate the next $\lambda /2$ path difference requires a smaller horizontal step.

Therefore, the fringes are straight, but their spacing decreases as we go outwards.

By carefully analyzing the geometry of the air film, we can confidently conclude that the interference pattern consists of straight lines parallel to the axis of the cylinder. This elegant result highlights how the spatial symmetry of the physical setup directly dictates the symmetry of the resulting optical phenomena.