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Thin-film interference is the phenomenon that occurs when incident light waves reflected by the upper and lower boundaries of a thin film interfere with one another to form a new wave. Studying this new wave can reveal information about the surfaces from which its components reflected, including the thickness of the film or the effective refractive index of the film medium. Thin films have many commercial applications including anti-reflection coatings, mirrors, and optical filters.

Theory

**Figure 2. White light incident on an oil film.** A colorful interference pattern is observed when light from the sun is reflected from a the top and bottom boundaries of a thin oil film.

A thin film is a layer of material with thickness in the sub-nanometer to micron range. As light strikes the surface of a film it is either transmitted or reflected at the upper surface. Light that is transmitted reaches the bottom surface and may once again be transmitted or reflected. The Fresnel equations provide a quantitative description of how much of the light will be transmitted or reflected at an interface. The light reflected from the upper and lower surfaces will interfere. The degree of constructive or destructive interference between the two light waves is dependent upon the difference in their phase. This difference is dependent upon the thickness of the film layer, the refractive index of the film, and the angle of incidence of the original wave on the film. Additionally, a phase shift of 180° or $\pi$ radians may be introduced upon reflection at a boundary depending on the refractive indices of the materials on either side of the boundary. This phase shift occurs if the refractive index of the medium the light is travelling through is less than the refractive of the material it is striking. In other words, if $n_{1}<n_{2}$ and the light is travelling from material 1 to material 2, then a phase shift will occur upon reflection. The pattern of light that results from this interference can appear either as light and dark bands or as colorful bands depending upon the source of the incident light.

Consider light incident on a thin film and reflected by both the upper and lower boundaries. The optical path difference (OPD) of the reflected light must be calculated in order to determine the condition for interference. Referring to Figure 1, the OPD between the two waves is the following:

OPD=n_{2}({\overline {AB}}+{\overline {BC}})-n_{1}({\overline {AD}})

Where,

{\overline {AB}}={\overline {BC}}={\frac {d}{\cos(\theta _{2})}}

{\overline {AD}}=2d\tan(\theta _{2})\sin(\theta _{1})

Using Snell's Law, $n_{1}sin(\theta _{1})=n_{2}sin(\theta _{2})$

OPD=n_{2}\left({\frac {2d}{\cos(\theta _{2})}}\right)-2d\tan(\theta _{2})n_{2}\sin(\theta _{2})

OPD=2n_{2}d\left({\frac {1-\sin ^{2}(\theta _{2})}{\cos(\theta _{2})}}\right)

OPD=2n_{2}d\cos {\big (}\theta _{2})

Interference will be constructive if the optical path difference is equal to an integer multiple of the wavelength of light, $\lambda$ .

2n_{2}d\cos {\big (}\theta _{2})=m\lambda

This condition may change after considering possible phase shifts that occur upon reflection.

Monochromatic source

In the case where incident light is monochromatic in nature, interference patterns will appear as light and dark bands. Light bands correspond to regions at which constructive interference is occurring between the reflected waves and dark bands correspond to destructive interference regions. As the thickness of the film varies from one location to another, the interference may change from constructive to destructive. A good example of this phenomenon, termed "Newton's rings," demonstrates the interference pattern that results when light is reflected from a spherical surface adjacent to a flat surface. Concentric rings are viewed when the surface is illuminated with monochromatic light.

Broadband source

If the incident light is broadband, or white, such as light from the sun, interference patters will appear as colorful bands. Different wavelengths of light will create constructive interference for different film thicknesses. Different regions of the film will appear to be different colors depending on the local film thickness.

Examples

The type of interference that occurs when light is reflected from a thin film is dependent upon the wavelength and angle of the incident light, the thickness of the film, the refractive indices of the material on either side of the film, and the index of the film medium. Various possible film configurations and the related equations are explained in more detail in the examples below.

Soap bubble

In the case of a soap bubble light travels through air and strikes a soap film. The air has a refractive index of 1 ( $n_{air}=1$ ) and the film has an index that is larger than 1 ( $n_{film}>1$ ). The reflection that occurs at the upper boundary of the film (the air-film boundary) will introduce a 180° phase shift in the reflected wave because the refractive index of the air is less than the index of the film ( $n_{air}<n_{film}$ ). Light that is transmitted at the upper air-film interface will continue to the lower film-air interface where it can be reflected or transmitted. The reflection that occurs at this boundary will not change the phase of the reflected wave because $n_{film}>n_{air}$ . The condition for interference for a soap bubble is the following:

2n_{film}d\cos(\theta _{2})=\left(m-{\frac {1}{2}}\right)\lambda

for constructive interference of reflected light

2n_{film}d\cos {\big (}\theta _{2})=m\lambda

for destructive interference of reflected light

Where $d$ is the film thickness, $n_{film}$ if the refractive index of the film, $\theta _{2}$ is the angle of incidence of the wave on the lower boundary, $m$ is an integer, and $\lambda$ is the wavelength of light.

Oil film

In the case of a thin oil film, a layer of oil sits atop a layer of water. The oil may have an index of refraction near 1.5 and the water has an index of 1.33. As in the case of the soap bubble, the materials on either side of the oil film (air and water) both have refractive indices that are less than the index of the film. $n_{air}<n_{water}<n_{oil}$ . There will be a phase shift upon reflection from the upper boundary because $n_{air}<n_{oil}$ but no shift upon reflection from the lower boundary because $n_{oil}>n_{water}$ . The equations for interference will be the same.

2n_{oil}d\cos(\theta _{2})=\left(m-{\frac {1}{2}}\right)\lambda

for constructive interference of reflected light

2n_{oil}d\cos {\big (}\theta _{2})=m\lambda

for destructive interference of reflected light

Anti-reflection coatings

An anti-reflection coating eliminates reflected light and maximizes transmitted light in an optical system. A film is designed such that reflected light produces destructive interference and transmitted light produces constructive interference for a given wavelength of light. In the simplest implementation of such a coating, the film is created so that its optical thickness $dn_{coating}$ is a quarter-wavelength of the incident light and its refractive index is greater than the index of air and less than the index of glass.

n_{air}<n_{coating}<n_{glass}

d=\lambda /(4n_{coating})

A 180° phase shift will be induced upon reflection at both the top and bottom interfaces of the film because $n_{air}<n_{coating}$ and $n_{coating}<n_{glass}$ . The equations for interference of the reflected light are:

2n_{coating}d\cos {\big (}\theta _{2})=m\lambda

for constructive interference

2n_{coating}d\cos(\theta _{2})=\left(m-{\frac {1}{2}}\right)\lambda

for destructive interference

If the optical thickness $dn_{coating}$ is equal to a quarter-wavelength of the incident light and if the light strikes the film at normal incidence $(\theta _{2}=0)$ , the reflected waves will be completely out of phase and will destructively interfere. Further reduction in reflection is possible by adding more layers, each designed to match a specific wavelength of light.

It should also be noted that interference of transmitted light will be completely constructive for these films.

Applications

Thin films are used commercially in anti-reflection coatings, mirrors, and optical filters. They can be engineered to control the amount of light reflected or transmitted at a surface for a given wavelength. A Fabry-Pérot etalon takes advantage of thin film interference to selectively choose which wavelengths of light are allowed to transmit through the device. These films are created through deposition processes in which material is added to a substrate in a controlled manner. Methods include chemical vapor deposition and various physical vapor deposition techniques.

Thin films are also found in nature. Many animals have a layer of tissue behind the retina, the Tapetum lucidum, that aids in light collecting. The effects of thin-film interference can also be seen in oil slicks and soap bubbles.

Ellipsometry is a technique that is often used to measure properties of thin films. In a typical ellipsometry experiment polarized light is reflected off a film surface and is measured by a detector. The complex reflectance ratio, $\rho$ , of the system is measured. A model analysis in then conducted in which this information is used to determine film layer thicknesses and refractive indices.

References

Fowles, Grant R. (1989), "Multiple-Beam Interference", Introduction to Modern Optics, Dover
Greivenkamp, John (1995), "Interference", Handbook of Optics, McGraw-Hill
Hecht, Eugene (2002), "Interference", Optics, Addison Wesley
Knittl, Zdeněk (1976), Optics of Thin Films; An Optical Multilayer Theory, Wiley {{citation}}: Missing or empty |title= (help)