A thin film of soap solution (n=1.33)^(n=1.33) has air on...

A thin film of soap solution $(n=1.33)^{(n=1.33)}$ has air on either side and is illuminated normally with white light. Interference minima are visible in the reflected light only at wavelengths of 400,480 , and $600 \mathrm{~nm}$ in air. What is the minimum thickness of the film? Example $23-6$

Key Concepts

Thin Film Interference

Thin film interference is a phenomenon where light waves reflected from the top and bottom surfaces of a thin film combine and interfere. The interference—constructive or destructive—depends on the phase difference between the two reflected rays, which is influenced by both the optical path difference inside the film and any phase changes that occur upon reflection, making the film appear colorful or dull at different thicknesses.

Optical Path Difference

The optical path difference is the extra distance that light travels within the film compared to the directly reflected light. This additional path, given by twice the product of the film’s refractive index and its thickness for normal incidence, determines the phase difference between the interfering beams. It is a key parameter in setting up the condition for constructive or destructive interference at particular wavelengths.

Phase Change on Reflection

When light reflects off a boundary where the refractive index increases, it undergoes a phase change of ? (equivalent to a half-wavelength shift). In a thin film with air on both sides, one of the reflections typically involves such a phase shift while the other does not. Accounting for this phase change is crucial when deriving the conditions for interference minima or maxima in the reflected light.

Interference Minima Conditions

For the reflected light to exhibit a minimum intensity (destructive interference), the optical path difference plus any phase shifts must equal an odd multiple of half wavelengths. This condition allows one to relate the film thickness, refractive index, and wavelength of light to determine at which thicknesses minimal reflection occurs. The lowest-order minimum gives the minimum film thickness that supports this phenomenon.

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