Deflection of Monochromatic Light Wave Through Gradient Slab

To determine the deflection of the monochromatic light wave emerging from the glass slab, we analyze the optical path length and wavefront orientation as the wave traverses the medium.

1. Refractive Index Profile

Let $y$ be the vertical coordinate measured from the bottom of the slab ($y = 0$) to the top ($y = h$). The refractive index $n(y)$ increases linearly from $n_1$ to $n_2$: $n(y) = n_1 + \left( \frac{n_2 - n_1}{h} \right) y$

2. Optical Path Length (OPL)

A plane wave is incident normally on the left face of the slab. The optical path length traversed by the light wave at height $y$ through the slab of thickness $d$ is given by: $\text{OPL}(y) = n(y) \cdot d = \left[ n_1 + \left( \frac{n_2 - n_1}{h} \right) y \right] d$

3. Wavefront Orientation and Deflection Angle

The phase $\phi(y)$ of the wave emerging at the right face of the slab is: $\phi(y) = \phi_0 - \frac{2\pi}{\lambda_0} \text{OPL}(y)$

Since $n_2 > n_1$, light at the top of the slab ($y = h$) travels through a region of higher refractive index, meaning its phase velocity is slower compared to the light at the bottom ($y = 0$). Consequently, the wave at the top experiences a larger phase delay, causing the emerging wavefront to tilt backward at the top.

The direction of wave propagation (the ray direction) is always perpendicular to the wavefront. The tangent of the deflection angle $\theta$ relative to the horizontal ($x$-axis) is given by the gradient of the optical path length along the $y$-direction: $\tan \theta = \frac{d(\text{OPL})}{dy} = d \frac{dn}{dy} = \frac{(n_2 - n_1)d}{h}$

Since $\frac{dn}{dy} > 0$, the wave vector tilts upward. Therefore, the deflection angle is: $\theta = \tan^{-1} \left[ \frac{(n_2 - n_1)d}{h} \right] \quad \text{(deflected upwards)}$

4. Conclusion and Option Analysis

• Option A: Incorrect.
• Option B: Correct. The light wave deflects up by an angle $\tan^{-1} \left[ \frac{(n_2 - n_1)d}{h} \right]$.
• Option C: Incorrect.
• Option D: Correct. The angle $\theta$ depends only on the difference $(n_2 - n_1)$ and the geometric parameters $d$ and $h$, and not on the individual values of $n_1$ and $n_2$.

Correct Answer: (B) and (D)