AcademyInterference
Academy
Thin-Film Interference
Level 1 - Physics topic page in Interference.
Principle
Thin-film interference comes from superposing light reflected from the top and bottom surfaces of a film. Reflection phase shifts and optical path difference decide whether a color is bright or dark.
Notation
\(t\)
film thickness
\(\mathrm{m}\)
\(n\)
film refractive index
1
\(\lambda_0\)
wavelength in vacuum or air
\(\mathrm{m}\)
\(\lambda_f\)
wavelength inside the film
\(\mathrm{m}\)
\(m\)
interference order
1
Method
Derivation 1: Film wavelength and path
At near-normal incidence, the extra distance inside the film is approximately down and back through the film.
Wavelength in film
\[\lambda_f=\frac{\lambda_0}{n}\]
Optical path difference
\[\Delta=2nt\]
Derivation 2: Reflection phase shifts
Reflection from a boundary to higher refractive index adds a half-cycle phase shift. Reflection from higher to lower index does not.
Higher-index reflection
\[\Delta\phi_{\mathrm{refl}}=\pi\]
Lower-index reflection
\[\Delta\phi_{\mathrm{refl}}=0\]
Derivation 3: One phase reversal
The most common coating case has exactly one reflection phase reversal.
Reflected bright
\[2nt=\left(m+\frac12\right)\lambda_0\]
Reflected dark
\[2nt=m\lambda_0\]
Rules
For reflected light with exactly one phase reversal:
Constructive reflection
\[2nt=\left(m+\frac12\right)\lambda_0\]
Destructive reflection
\[2nt=m\lambda_0\]
For reflected light with zero or two phase reversals, these conditions swap.
Quarter-wave coating
\[t=\frac{\lambda_0}{4n}\]
Examples
Question
Find the minimum nonzero thickness of an anti-reflection coating with
\[n=1.25\]
for \[\lambda_0=500\,\mathrm{nm}\]
assuming one phase reversal.Answer
\[t=\frac{\lambda_0}{4n}=\frac{500\,\mathrm{nm}}{4(1.25)}=100\,\mathrm{nm}\]
Checks
- Always count reflection phase reversals before choosing bright or dark conditions.
- Use wavelength in air or vacuum in \(2nt=m\\lambda_0\) forms.
- At oblique incidence, path geometry changes; these rules assume near-normal incidence.
- A film can be bright for one wavelength and dark for another.