Thin-film interference occurs when light reflects from the top and bottom boundaries of a thin transparent layer. The two reflected rays can reinforce or cancel each other, depending on their optical path difference and whether either reflection introduces a phase reversal.
For a film of refractive index \(n\), thickness \(t\), and internal propagation angle \(\theta\), the round-trip optical path difference is
\[
\delta = 2nt\cos\theta.
\]
This is the extra optical distance traveled by the ray that enters the film, reflects from the lower surface, and emerges again.
The interference condition depends on reflection phase shifts. If there is no reflection phase reversal, then constructive interference occurs when
\[
\delta = m\lambda,
\]
and destructive interference occurs when
\[
\delta = \left(m+\frac12\right)\lambda.
\]
If there is one reflection phase reversal, the conditions are swapped:
\[
\text{constructive: }\delta=\left(m+\frac12\right)\lambda,
\qquad
\text{destructive: }\delta=m\lambda.
\]
This is the common textbook case for a film where one reflection occurs from lower to higher refractive index and the other does not.
Consider the sample values
\[
t=100\text{ nm},\qquad n=1.5,\qquad \theta=0,\qquad \lambda=550\text{ nm}.
\]
Since \(\cos 0 = 1\),
\[
\delta = 2nt = 2(1.5)(100\text{ nm}) = 300\text{ nm}.
\]
So the optical path difference is
\[
\delta = 300\text{ nm}.
\]
Comparing this with the wavelength,
\[
\frac{\delta}{\lambda}=\frac{300}{550}\approx 0.545.
\]
This is close to one-half. Therefore, with one reflection phase reversal, the interference is close to constructive for this wavelength, while in the no-shift case it would be close to destructive. This shows why tracking reflection phase changes is essential.
Thin-film interference explains the colors seen in soap bubbles, oil films, anti-reflection coatings, and many optical devices. Different wavelengths satisfy the interference conditions differently, so white light can produce iridescent colors. The observed color changes when the film thickness changes or when the angle of viewing changes.
At more advanced level, thin-film optics extends to multiple layers, complex amplitude reflection coefficients, and full coating design. But the basic formula used here,
\[
\delta = 2nt\cos\theta,
\]
together with the phase-shift rule, is the correct starting point for introductory interference problems.
This calculator uses that standard model. It computes the path difference, compares it with the selected wavelength, classifies the interference condition, and helps visualize the phenomenon with both a graph and a contained thin-film animation.
