Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, increasing reflection at some wavelengths and decreasing it at others. When white light is incident on a thin film, this effect produces colorful reflections.

Thin-film interference explains the multiple colors seen in light reflected from soap bubbles and oil films on water. It is also the mechanism behind the action of antireflection coatings used on glasses and camera lenses. If the thickness of the film is much larger than the coherence length of the incident light, then the interference pattern will be washed out due to the linewidth of the light source.

The reflection from a thin film is typically not individual wavelengths as produced by a diffraction grating or prism, but rather are a mixture of various wavelengths. Therefore, the colors observed are rarely those of the rainbow, but rather browns, golds, turquoises, teals, bright blues, purples, and magentas. Studying the light reflected or transmitted by a thin film can reveal information about 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

In optics, 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 depends on the difference in their phase. This difference in turn depends on 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 ${\displaystyle \pi }$ 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 index of the material it is striking. In other words, if ${\displaystyle n_{1}<n_{2}}$ and the light is travelling from material 1 to material 2, then a phase shift occurs 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 the ray diagram above, the OPD between the two waves is the following:

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

Where,

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

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

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

${\displaystyle {\begin{aligned}OPD&=n_{2}\left({\frac {2d}{\cos(\theta _{2})}}\right)-2d\tan(\theta _{2})n_{2}\sin(\theta _{2})\\&=2n_{2}d\left({\frac {1-\sin ^{2}(\theta _{2})}{\cos(\theta _{2})}}\right)\\&=2n_{2}d\cos {\big (}\theta _{2})\\\end{aligned}}}$

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

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

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

Monochromatic source

Where incident light is monochromatic in nature, interference patterns 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 observed when the surface is illuminated with monochromatic light. This phenomenon is used with optical flats to measure the shape and flatness of surfaces.

Broadband source

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

Phase interaction

The figures show two incident light beams (A and B). Each beam produces a reflected beam (dashed). The reflections of interest are beam A’s reflection off of the lower surface and beam B’s reflection off of the upper surface. These reflected beams combine to produce a resultant beam (C). If the reflected beams are in phase (as in the first figure) the resultant beam is relatively strong. If, on the other hand, the reflected beams have opposite phase, the resulting beam is attenuated (as in the second figure).

The phase relationship of the two reflected beams depends on the relationship between the wavelength of beam A in the film, and the film's thickness. If the total distance beam A travels in the film is an integer multiple of the wavelength of the beam in the film, then the two reflected beams are in phase and constructively interfere (as depicted in the first figure). If the distance traveled by beam A is an odd integer multiple of the half wavelength of light in the film, the beams destructively interfere (as in the second figure). Thus, the film shown in these figures reflects more strongly at the wavelength of the light beam in the first figure, and less strongly at that of the beam in the second figure.

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 (${\displaystyle n_{\rm {air}}=1}$) and the film has an index that is larger than 1 (${\displaystyle n_{\rm {film}}>1}$). 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 (${\displaystyle n_{\rm {air}}<n_{\rm {film}}}$). 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 ${\displaystyle n_{\rm {film}}>n_{\rm {air}}}$. The condition for interference for a soap bubble is the following:

${\displaystyle 2n_{\rm {film}}d\cos(\theta _{2})=\left(m-{\frac {1}{2}}\right)\lambda }$  for constructive interference of reflected light

${\displaystyle 2n_{\rm {film}}d\cos {\big (}\theta _{2})=m\lambda }$  for destructive interference of reflected light

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

Oil film

In the case of a thin oil film, a layer of oil sits on top of 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. Zdroj:https://en.wikipedia.org?pojem=Thin-film_interference
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