Find the variation of refractive index as function of y such that a ray entering at origin at grazing incidence follows a parabolic path y=x^2?

To find the variation of the refractive index as a function of \( y \) such that a ray entering at the origin with grazing incidence follows a parabolic path described by \( y = x^2 \), we can utilize Snell's law and the principles of geometric optics. The key here is to understand how the refractive index affects the path of light as it travels through different media.

Understanding the Problem

A ray of light entering at grazing incidence means it strikes the boundary of the medium at a very shallow angle. The path described by \( y = x^2 \) suggests that the light is bending in a parabolic trajectory. To achieve this, the refractive index must vary with \( y \). We can denote the refractive index as \( n(y) \).

Applying Snell's Law

Snell's law states that:

• \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \)

In our case, since the ray enters at grazing incidence, \( \theta_1 \) is very small, and we can approximate \( \sin(\theta_1) \approx \theta_1 \). If we assume the ray is moving from air (where \( n_1 \approx 1 \)) into a medium with a variable refractive index \( n(y) \), we can express the relationship as:

• \( \theta_1 \approx n(y) \sin(\theta_2) \)

Finding the Relationship

To derive the refractive index as a function of \( y \), we need to relate the angle of incidence to the path of the ray. The path \( y = x^2 \) implies that the slope of the path at any point \( x \) is given by the derivative:

• \( \frac{dy}{dx} = 2x \)

This slope represents the tangent of the angle \( \theta \) that the ray makes with the horizontal axis. Thus, we can express the angle as:

• \( \tan(\theta) = 2x \)

Relating Refractive Index to Path Curvature

Using the small angle approximation, we can relate the refractive index to the curvature of the path. The curvature \( K \) of the path can be defined as:

• \( K = \frac{d^2y}{dx^2} = 2 \)

In optics, the refractive index can be related to the curvature of the path of light. For a ray following a parabolic trajectory, the refractive index must change in such a way that it compensates for the curvature. A common relationship in optics is:

• \( n(y) = n_0 + k \cdot y \)

where \( n_0 \) is the refractive index at \( y = 0 \) and \( k \) is a constant that determines how quickly the refractive index changes with \( y \).

Final Expression for Refractive Index

To ensure that the ray follows the parabolic path, we can set \( k \) based on the curvature we derived. Thus, we can express the refractive index as:

• \( n(y) = 1 + 2y \)

This means that the refractive index increases linearly with \( y \), which allows the light ray to bend appropriately to follow the parabolic trajectory \( y = x^2 \).

Summary

In summary, the variation of the refractive index as a function of \( y \) that allows a ray entering at the origin at grazing incidence to follow the path \( y = x^2 \) is given by:

• \( n(y) = 1 + 2y \)

This relationship captures how the refractive index must change to accommodate the bending of light along the specified parabolic path.