Distance Between Mirrors for Image Formation at Source

To determine the distance $d$ between the two concave mirrors such that the final image is formed on the source $S$ itself, we analyze the path of light rays from $S$ through refraction at the slab's surface and reflection from the mirrors.

1. Apparent Position of the Source $S$

The monochromatic point light source $S$ is embedded at the center of the glass slab of refractive index $n_0$ and thickness $t$. Therefore, $S$ is at a real distance of $\frac{t}{2}$ from either face of the slab.

When light rays travel from inside the glass medium ($n_1 = n_0$) into air ($n_2 = 1$), refraction occurs at the plane interface. Using the plane refraction formula: $\frac{n_2}{v} = \frac{n_1}{u}$

Taking the real depth as $u = -\frac{t}{2}$: $\frac{1}{v} = \frac{n_0}{-\frac{t}{2}} \implies v = -\frac{t}{2n_0}$

Thus, to an observer in air (or looking from the concave mirror), the source appears to be at a virtual position $S'$ located at a distance of $\frac{t}{2n_0}$ behind the slab face.

2. Distance of Virtual Source $S'$ from the Mirror

Let $d$ be the total distance between the two concave mirrors. Since the slab is placed symmetrically between them, the distance $x$ from either mirror to the nearest surface of the slab is: $x = \frac{d - t}{2}$

The distance $u$ of the virtual source $S'$ from either mirror is: $u = x + \frac{t}{2n_0} = \frac{d - t}{2} + \frac{t}{2n_0} = \frac{d}{2} - \frac{t}{2}\left(1 - \frac{1}{n_0}\right)$

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3. Conditions for Image Formation at $S$

Case I: Rays retrace their path (Reflection at Center of Curvature)

If the virtual source $S'$ is located at the center of curvature of the concave mirror, light rays strike the mirror normally and reflect back along their original paths, re-focusing at $S$.

Setting $u = 2f$: $\frac{d}{2} - \frac{t}{2}\left(1 - \frac{1}{n_0}\right) = 2f$ $d = 4f + \left(1 - \frac{1}{n_0}\right)t$

This corresponds to Option (A).

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Case II: Rays become parallel after reflection (Reflection at Focus)

If the virtual source $S'$ is located at the principal focus of one mirror, the reflected light rays emerge parallel to the common principal axis.

Setting $u = f$: $\frac{d}{2} - \frac{t}{2}\left(1 - \frac{1}{n_0}\right) = f$ $d = 2f + \left(1 - \frac{1}{n_0}\right)t$

These parallel rays then travel towards the second mirror. Upon reflection from the second mirror, the rays converge towards its focal point. After re-entering the glass slab, refraction causes these converging rays to focus precisely at the point source $S$.

This corresponds to Option (B).

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Conclusion

The correct options are A and B.