Solution

Light incident normally on the metallic mirror interferes with the reflected light, generating a standing wave. The electric field has a node at the metal surface $(E = 0)$ and antinodes (intensity maxima) at distances $\lambda/4, 3\lambda/4, 5\lambda/4, \dots$from the surface. The photographic emulsion darkens at those antinodes.

Wavelength

The first darkening (first antinode) occurs at $d_1 = 10^{-5}\ \text{cm}$. Since $d_1 = \lambda/4$:

$\boxed{\lambda = 4 \times 10^{-5}\ \text{cm} = 400\ \text{nm}}$.

Distance between darkened layers

Consecutive antinodes are separated by half a wavelength:

$\boxed{x = \frac{\lambda}{2} = 2 \times 10^{-5}\ \text{cm}}.$