A beam of parallel light, originating from a distant source, is first incident on a convex lens with focal length $f_2$.
Subsequently, the light passes through the lens and then reflects from a concave mirror having a focal length $f_1$.
The concave mirror is placed at a distance $d$ from the convex lens.
For the light rays to retrace their original path and ultimately emerge from the lens as a parallel beam heading back towards the distant source, the separation distance $d$ between the lens and the mirror must be:

Question

Prepp Team · Accepted Answer

Analyzing Parallel Light Ray Path with Lens and Mirror

This problem involves understanding how parallel light rays interact with a convex lens and a concave mirror. We are looking for the specific distance d between these optical components that ensures the light rays, after reflection, return along their original path and emerge from the lens as a parallel beam. Key concepts include focal length, image formation, and the conditions for light ray reversal.

Optical System Overview

The setup consists of the following elements:

A beam of parallel light from a distant source.
A convex lens with focal length $f_2$.
A concave mirror with focal length $f_1$.
The mirror is placed at a distance $d$ from the lens.

We need to determine the value of $d$ for the rays to retrace their path.

Step-by-Step Derivation for Light Ray Retracing

Let's break down the process step-by-step:

Lens Action: The initial beam of parallel light rays is incident on the convex lens ($f_2$). According to the properties of lenses, parallel rays converge at the focal point on the opposite side of the lens. This convergence point is located at a distance $f_2$ from the lens.
Object for the Mirror: These converging rays then travel towards the concave mirror, which is positioned at a distance $d$ from the lens. The point where the rays converge acts as the object for the concave mirror. The distance of this object point from the mirror is calculated as $d - f_2$. For this analysis, we assume $f_1$ and $f_2$ represent the magnitudes (positive values) of the focal lengths.
Condition for Path Retracing: For the light rays to retrace their original path after reflection from the concave mirror, they must strike the mirror surface perpendicularly. This only happens if the rays are directed precisely towards the center of curvature of the mirror. Therefore, the object point (the convergence point of the rays from the lens) must coincide with the center of curvature of the concave mirror.
Mirror's Center of Curvature: The center of curvature (C) of a spherical mirror lies at twice its focal length from the mirror's pole. For the concave mirror with focal length $f_1$ (magnitude), the distance of the center of curvature from the mirror is $2f_1$.
Calculating the Distance $d$: We equate the distance of the object point from the mirror with the distance of the center of curvature from the mirror: Object distance from mirror = $d - f_2$ Center of curvature distance from mirror = $2f_1$ Setting these equal, we get: $d - f_2 = 2f_1$ Solving this equation for the separation distance $d$: $d = 2f_1 + f_2$
Emergence as Parallel Beam: When rays strike the mirror at its center of curvature, they reflect back along the same path. These rays then travel back towards the lens. For these rays to emerge from the convex lens as a parallel beam (heading back towards the original source), they must be incident on the lens as if they are coming from its focal point on the object side. The condition $d = 2f_1 + f_2$ ensures this entire process occurs correctly, satisfying both the retracing of the path upon reflection and the parallel emergence upon refraction.

Resulting Separation Distance

The condition required for the light rays to retrace their path and emerge as a parallel beam is determined by the calculated relationship between the distances and focal lengths.

The separation distance $d$ between the convex lens and the concave mirror must be:

$d = 2f_1 + f_2$