Light on thin films
Equations to relate the thickness of the film, angle of refraction, and wavelength of
light that experiences either constructive or destructive interference.
wavelength change: $$ \lambda' = \lambda / n $$
constructive interference if phase change: $$ 2 n t \cos\theta_2 = (m + 1/2)\, \lambda $$
destructive interference if phase change: $$ 2 n t \cos\theta_2 = (m)\, \lambda $$
where \(\lambda\) is the incident wavelength, \(n\) is the index of refraction,
\(t\) is the thickness of the film, and \(m\) is an integer, for multiples of
the wavelength.
Phase changes depend on the relative indices of refraction, with \(n_1\ \gt n_2\)
resulting in a 180° phase change, and \(n_1 \lt n_2\) resulting in no phase change.
A 180° phase change results in a shifting of the light by half of a wavelength.
Example: Soap bubble
The rays reflecting off the front surface will have a 180° phase shift
because they are reflecting off of soapy water with \(n_r\) ~ 1.34 \( \gt n_i\) = 1 for air.
Ray reflecting off the back surface will not have a phase shift because they are
moving in water and reflecting off of air, so \(n_i \gt n_r\).
Example: Anti-reflective coating on glasses
The rays reflecting off the front surface will have a 180° phase shift
because they are reflecting off of the coating with \(n_r \gt \) 1.
Rays reflecting off the back surface will also have a phase shift because they are
moving in the coated surface and reflecting off of glass, with a higher index
of refraction, so \(n_i \lt n_r\). If the thickness is chosen properly, based on the
equations above, then certain wavelengths will destructively interfere and not produce a reflection
on the glass+coating.
Thin film calculator
Drag the line of interface between the air and the thin film to see the effect
of different thicknesses on the wavelength, as shown in the \(\lambda\) field
above, and in the color of the rays.
Drag the incoming ray to see the effect of incident angle on the reflected color.
Variation in both the angle and thickness can produce a rainbow effect on thin films.
The ray color displayed is an estimate of the constructively
interfering wavelength. Black means the wavelength is outside the visible range of 380-700 nm.
