In this case, assume that n3 > n2 > n1. For which film-thickness(es) will constructive interference occur for light which is reflected?

Answer 1

To determine the film thicknesses at which constructive interference occurs for light reflected from a thin film with varying refractive indices, such as a film with refractive indices \( n_1 \), \( n_2 \), and \( n_3 \) where \( n_3 > n_2 > n_1 \), we must consider the phase changes that occur upon reflection as well as the path length difference between the rays that reflect off the top and bottom surfaces of the film.

For constructive interference when light is reflected, following considerations apply:

1. Phase Change upon Reflection:

   • When light reflects off a medium with a higher refractive index (from \( n_1 \) to \( n_2 \) or \( n_2 \) to \( n_3 \)), there is a phase shift of \( \pi \) (equivalent to a half-wavelength) upon reflection.
   • When light reflects off from a medium with a lower refractive index (from \( n_2 \) to \( n_1 \)), there is no phase shift.

2. Film Thickness:

   • The path difference for light reflecting from the top and bottom surfaces of the thin film is \( 2t \), where \( t \) is the thickness of the film.

3. Condition for Constructive Interference:

   • To satisfy constructive interference, the overall phase difference must be an integer multiple of wavelengths (\( m\lambda \)) adjusted for any phase shifts:

\[ \text{Total phase difference} = 2t(n_2) / \lambda + \text{phase shift} \]

For the case where light is reflected from:

• Air to film surface (no phase change)
• Film to higher refractive index substrate (phase shift of \( \pi \))

The condition combining these results in:

\[ 2t(n_2) / \lambda + \frac{1}{2} \lambda = m\lambda \]

Where \( m \) is an integer (0, 1, 2, ...). Simplifying this yields:

\[ 2t(n_2) = (m + \frac{1}{2})\lambda \]

This condition can be adjusted to find \( t \):

\[ t = \frac{(m + \frac{1}{2})\lambda}{2n_2} \]

Thus, constructive interference for reflected light occurs at thicknesses given by:

\[ t = \frac{(m + \frac{1}{2})\lambda}{2n_2} \]

Where \( m \) is any non-negative integer (0, 1, 2,...).

This formula provides the film thicknesses where constructive interference of reflected light occurs given the specified conditions.