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## Thin Films

Thin films, such as the oil slick in the photo to the right, can often produce a rainbow of colors. To get this effect, the material thickness must be on the order of the wavelength of visible light, around 380 - 750 nanometers. The effect is produced through a combination of refraction and light wave interference. It can also be seen on common items such as compact disks, soap bubbles and the mother-of-pearl lining of shells. Some animals have microscopic structures in their feathers, scales, or hair that use a similar effect to produce vibrant colors with limited pigmentation.

### Interference

The path difference between parallel rays of light hitting a thin film can be used to calculate the wavelength of maximum or minimum reflection for a given incident angle and index of refraction. If the difference in the distance travelled by rays that reflect off the bottom of the film to the distance travelled by rays reflecting off the top surface is a multiple of the wavelength, and there is not phase change, there will be constructive interference and it will be visible.

## 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.

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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.

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