Thin-film interference

Interference from reflected rays

Thin-film interference occurs when light reflects from the top and bottom surfaces of a thin transparent layer. The two reflected rays overlap and interfere. Depending on wavelength, film thickness, refractive indices, and viewing angle, certain colors are reinforced while others are canceled.

This produces colors in soap bubbles, oil slicks, and coated lenses.

Path difference in a film

For near-normal incidence, light reflecting from the lower surface travels approximately down and back through the film. If the film thickness is $t$ and refractive index is $n$, the extra optical path length is approximately

$2nt$

Optical path length includes refractive index because wavelength is shorter inside the material.

Phase reversal on reflection

Reflection can introduce a phase shift. When light reflects from a boundary leading to a higher refractive index, the reflected wave undergoes a phase shift of

$pi$

equivalent to half a wavelength. Reflection from a boundary leading to lower refractive index does not introduce this phase shift.

This phase reversal is essential in thin-film problems.

One phase reversal case

Suppose a film of refractive index $n_f$ lies between air and glass, with

$n_{air}<n_f<n_{glass}$

Light reflecting from the air-film boundary undergoes a $pi$ phase shift. Light reflecting from the film-glass boundary also reflects from lower to higher index and undergoes a $pi$ phase shift. The two reflection phase shifts cancel relative to each other.

For reflected constructive interference at near-normal incidence,

$2n_ft=mlambda_0$

For reflected destructive interference,

ight)lambda_0$$ where $lambda_0$ is wavelength in vacuum. ## One relative phase reversal For a soap bubble film in air, $$n_{air}<n_{film}$$ at the top surface, but the lower reflection is from film to air, a higher-to-lower boundary. Thus only one reflected ray gains a $pi$ phase shift. In this common case, reflected constructive interference occurs when $$2nt=left(m+rac{1}{2} ight)lambda_0$$ and reflected destructive interference occurs when $$2nt=mlambda_0$$ ## Anti-reflection coatings An anti-reflection coating is designed so reflected rays cancel. For one relative phase reversal, destructive reflection at target wavelength occurs when $$2nt=rac{lambda_0}{2}$$ so $$t=rac{lambda_0}{4n}$$ This is a quarter-wave coating. It reduces reflection and increases transmission for selected wavelengths. ## Colors in thin films White light contains many wavelengths. Different wavelengths satisfy constructive conditions at different thicknesses and angles. Since soap bubble films vary in thickness, different colors appear in different regions. As a film becomes extremely thin, reflected visible light may destructively interfere for much of the visible range, producing dark regions before the film breaks. ## The big idea Thin-film interference comes from superposition of reflected waves with optical path differences and possible reflection phase shifts. Correctly counting phase reversals is the key skill. Thin films explain colorful bubbles, oil slicks, coated optics, and precision optical coatings.