1.derive the expression for path difference in the transmitted media for thin film interference.

To derive the expression for path difference in transmitted media for thin film interference, we need to consider how light behaves when it encounters a thin film, such as a soap bubble or an oil slick on water. The phenomenon of thin film interference arises from the constructive and destructive interference of light waves reflected from the top and bottom surfaces of the film.

Understanding Thin Film Interference

When light strikes a thin film, part of it is reflected off the top surface, while the rest enters the film, reflects off the bottom surface, and then exits back into the air. The key to understanding the path difference lies in the phase changes that occur during reflection and transmission.

Key Concepts

• Phase Change: When light reflects off a medium with a higher refractive index, it undergoes a phase change of π (or 180 degrees). No phase change occurs when light reflects off a medium with a lower refractive index.
• Path Length: The distance traveled by light in the film contributes to the overall path difference between the two waves.

Deriving the Path Difference

Let's denote the following variables:

• n: Refractive index of the film
• d: Thickness of the film
• λ: Wavelength of light in a vacuum

When light enters the film, it travels a distance of d and then reflects off the bottom surface. The light wave that reflects off the bottom surface travels an additional distance of d before it exits the film. Therefore, the total distance traveled by the light wave in the film is:

2d (since it goes down and comes back up).

Adjusting for Wavelength in the Film

However, the wavelength of light changes when it enters a medium with a different refractive index. The wavelength in the film is given by:

λ' = λ/n

where λ' is the wavelength in the film, and n is the refractive index of the film. Thus, the effective path length in terms of the wavelength in the film becomes:

Path Length = 2d/n

Considering Phase Changes

Now, we must account for the phase change that occurs when light reflects off the top surface of the film. Since the top surface reflects light from a lower to a higher refractive index, there is a phase shift of π. This means that for constructive interference, we need to consider this phase shift when calculating the total path difference.

Final Expression for Path Difference

The total path difference Δ between the two waves can be expressed as:

Δ = 2d/n + λ/2 (for constructive interference, accounting for the phase shift).

For destructive interference, the expression would be:

Δ = 2d/n - λ/2.

Summary

In summary, the path difference for thin film interference in transmitted media is influenced by the thickness of the film, the refractive index, and the wavelength of light. The derived expressions help us understand the conditions for constructive and destructive interference, which are crucial in applications ranging from anti-reflective coatings to the colorful patterns seen in soap bubbles.