Interference in Thin Films

In everyday life, the interference of light most commonly gives rise to easily observable effects when light impinges on a thin film of some transparent material. For instance, the brilliant colours seen in soap bubbles, in oil films floating on puddles of water, and in the feathers of a peacock's tail, are due to interference of this type.
Suppose that a very thin film of air is trapped between two pieces of glass, as shown in Figure. If monochromatic light (e.g., the yellow light from a sodium lamp) is incident almost normally to the film then some of the light is reflected from the interface between the bottom of the upper plate and the air, and some is reflected from the interface between the air and the top of the lower plate. The eye focuses these two parallel light beams at one spot on the retina. The two beams produce either destructive or constructive interference, depending on whether their path difference is equal to an odd or an even number of half-wavelengths, respectively.

Interference of light due to a thin film of air trapped between two pieces of glass.
Let

be the thickness of the air film. The difference in path-lengths between the two light rays shown in the figure is clearly ${\mit\Delta}=2\,t$ . Naively, we might expect that constructive interference, and, hence, brightness, would occur if ${\mit \Delta}=m\,\lambda$ , where

is an integer, and destructive interference, and, hence, darkness, would occur if ${\mit\Delta}=(m+1/2)\,\lambda$ . However, this is not the entire picture, since an additional phase difference is introduced between the two rays on reflection. The first ray is reflected at an interface between an optically dense medium (glass), through which the ray travels, and a less dense medium (air). There is no phase change on reflection from such an interface, just as there is no phase change when a wave on a string is reflected from a free end of the string. (Both waves on strings and electromagnetic waves are transverse waves, and, therefore, have analogous properties.) The second ray is reflected at an interface between an optically less dense medium (air), through which the ray travels, and a dense medium (glass). There is a $180^\circ$ phase change on reflection from such an interface, just as there is a $180^\circ$ phase change when a wave on a string is reflected from a fixed end. Thus, an additional $180^\circ$ phase change is introduced between the two rays, which is equivalent to an additional path difference of $\lambda/2$ . When this additional phase change is taken into account, the condition for constructive interference becomes

$\begin{displaymath} 2\,t=(m+1/2)\,\lambda, \end{displaymath}$

where is an integer. Similarly, the condition for destructive interference becomes

$\begin{displaymath} 2\,t = m\,\lambda. \end{displaymath}$

For white light, the above criteria yield constructive interference for some wavelengths, and destructive interference for others. Thus, the light reflected back from the film exhibits those colours for which the constructive interference occurs.
If the thin film consists of water, oil, or some other transparent material of refractive index

then the results are basically the same as those for an air film, except that the wavelength of the light in the film is reduced from $\lambda$ (the vacuum wavelength) to $\lambda/n$ . It follows that the modified criteria for constructive and destructive interference are

$\begin{displaymath} 2\,n\,t=(m+1/2)\,\lambda, \end{displaymath}$

and

$\begin{displaymath} 2\,n\,t = m\,\lambda, \end{displaymath}$

Thin-film interference

Constructive and destructive interference of light waves is also the reason why thin films, such as soap bubbles, show colorful patterns. This is known as thin-film interference, because it is the interference of light waves reflecting off the top surface of a film with the waves reflecting from the bottom surface. To obtain a nice colored pattern, the thickness of the film has to be on the order of the wavelength of light.
Consider the case of a thin film of oil floating on water. Thin-film interference can take place if these two light waves interfere constructively:

the light from the air reflecting off the top surface
the light traveling from the air, through the oil, reflecting off the bottom surface, traveling back through the oil and out into the air again.

An important consideration in determining whether these waves interfere constructively or destructively is the fact that whenever light reflects off a surface of higher index of refraction, a 180° phase shift in the wave is introduced.

Light in air, reflecting off just about anything (glass, water, oil, etc.) will undergo a 180° shift. On the other hand, light in oil, which has a higher n than water does, will have no phase shift if it reflects off an oil-water interface. Note that a shift by 180° is equivalent to the wave traveling a distance of half a wavelength.
To get constructive interference, the two reflected waves have to be shifted by an integer multiple of wavelengths. This must account for any phase shift introduced by a reflection off a higher-n material, as well as for the extra distance traveled by the wave traveling down and back through the film. With the oil film example, constructive interference will occur if the film thickness is 1/4 wavelength, 3/4 wavelength, 5/4, etc. Destructive interference occurs when the thickness of the oil film is 1/2 wavelength, 1 wavelength, 3/2 wavelength, etc.

In the case of 1/4 wavelength, the wave reflected off the top surface is shifted by 1/2 a wavelength by the reflection. The wave traveling through the film has no phase shift, but travels a total down-and-back distance of 1/2 wavelength, meaning that it will be in phase with the wave reflected from the top. On the other hand, if the film thickness is 1/2 wavelength, the first wave gets a 1/2 wavelength shift and the other gets a wavelength shift; these waves would cancel each other out.
Note that one has to be very careful in dealing with the wavelength, because the wavelength depends on the index of refraction. The film thickness, for constructive interference in the example above, has to be 1/4 (or 3/4 or 5/4 or ...) of the wavelength of the light in the oil. This wavelength is related to the wavelength in vacuum (which differs negligibly from the wavelength in air) by:

The cancellation (destructive interference) of reflected light waves is utilized to make non-reflective coatings. Such coatings are commonly found on some camera lenses or binocular lenses, and often have a bluish tint. The coating is put over glass, and the coating material generally has an index of refraction less than that of glass. In that case, then, both reflected waves have a 180° phase shift, and a film thickness of 1/4 wavelength (in the film) would produce a net shift of 1/2 wavelength, resulting in cancellation.

For non-reflective coatings, then, the minimum film thickness required is:

where n is the index of refraction of the coating material.
Note that you have to be very careful to account for whether a phase shift occurs at an interface where reflection is taking place. In some cases, the minimum film thickness required for constructive interference is a quarter of the wavelength; in other cases, the minimum film thickness must be half a wavelength for constructive interference to take place. It all depends on whether or not a phase shift occurs for reflections at both interfaces, one interface, or neither interface.

Interference in Wedge Shaped Film (Reflected Rays)

Thin Film Interference A film of thickness from 0.5 to 10 m is a transparent medium of glass, mica, air enclosed between glass, soap film, etc. When the light is made incident on this thin film partial reflection and partial refraction occur from the top surface of the film. The refracted beam travels in the medium and again suffers partial reflection and partial refraction at the bottom surface of the film. In this way several reflected and refracted rays are produces by a single incident ray. As they moves are superimposed on each other and produces interference pattern. Interference in Parallel Film ( Reflected Rays) Consider a thin film of uniform thickness ‘t’ and refractive index bounded between air. Let us consider monochromatic ray AB is made incident on the film, at B part of ray is reflected (R 1 ) and a part is refracted along BC.At C The beam BC again suffer partial reflection and partial refraction, the reflected beam CD mov...

Optics for Optometry

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