Animated Solution for Physics - Optics: A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below. Correct shape of the emergent wavefront will be (figures are schematic and not drawn to scale)-

Welcome to a fascinating exploration of wave optics! This problem from JEE Advanced 2020 is a brilliant test of your conceptual understanding of Huygens' principle and how light interacts with varying optical densities. We are not going to crunch heavy numbers here; instead, we will rely purely on physical intuition and geometric visualization. Let's dive in!

Imagine a perfectly straight, parallel beam of light—a plane wavefront—marching steadily through the air. Suddenly, it encounters a peculiar piece of transparent glass. This isn't your standard rectangular slab or a simple spherical lens. It has a flat left surface, but its right surface is wavy, resembling the letter 'B'.

To conquer this problem, we must resist the urge to look at the glass as a single, intimidating object. Instead, we will employ a classic physicist's trick: divide and conquer. We can conceptually slice this glass piece into three distinct horizontal regions: the top section, the middle section, and the bottom section.

Speed and Optical Path

Before we analyze the regions, let's anchor ourselves with a fundamental law of wave optics. When light travels from a rarer medium (like air) into a denser medium (like glass), it slows down. The speed of light in a medium, $v$, is given by:

where $c$ is the speed of light in a vacuum and $\mu$ is the refractive index of the medium.

Because light travels slower in glass, the time it takes for a wavefront to pass through depends entirely on the thickness of the glass it encounters. If a part of the wavefront has to push through a thicker section of glass, it will be delayed more compared to a part passing through a thinner section. This delay causes that specific part of the wavefront to lag behind.

Now, let's apply this logic to our three sliced regions.

1. The Top and Bottom Regions: Look closely at the top and bottom sections of the glass profile. Notice how they bulge outwards to the right? They are thicker in their centers and thinner at their edges. This is the exact geometric profile of a convex (converging) lens.

When our plane wavefront hits these sections, the light rays passing through the very center have to traverse the maximum thickness of glass. Consequently, they suffer the maximum delay. The light passing near the edges of these sections travels through less glass and gets ahead. As a result, the emergent wavefront in these regions will lag in the middle, forming a shape that is concave in the forward direction.

2. The Middle Region: Now, shift your focus to the middle section. It is pinched inwards, making it thinnest in the center and thicker towards its top and bottom edges. This mimics the profile of a concave (diverging) lens.

Here, the story flips. The light rays passing through the exact center of this middle section travel through the least amount of glass. They experience the minimum delay and therefore lead the rest of the wavefront. The emergent wavefront here will bulge forward, forming a convex shape in the direction of propagation.

We have all the pieces of the puzzle; now we just need to snap them together.

- The top part of the wavefront lags (bulges left). - The middle part of the wavefront leads (bulges right). - The bottom part of the wavefront lags (bulges left).

If we draw a continuous, smooth curve connecting these points, we get a wavy line that perfectly matches the shape shown in Option (A).

This problem beautifully illustrates that you don't always need complex ray-tracing equations to predict optical behavior. By simply understanding how optical path length affects the phase of a wavefront, you can intuitively deduce the shape of light as it emerges from even the most complex geometries. Keep visualizing, and the physics will always reveal itself!