# Thin Film Interference Notes: 9 Facts You Should Know

## Thin film interference definition :

Thin film interference refers to the phenomenon where interference of the light waves that get reflected by the upper and lower sides of a thin film occurs. This interference is capable of either increasing or reducing the light reflected from the film.

## Thin film interference working| Thin film interference explanation

According to optics, a thin film refers to a thin layer of a material having the thickness in the range of sub-nanometer to microns. When light waves fall on the thin film surface then the waves are either reflected back from the upper surface of the material or are transmitted through it. The light waves that manage to get transmitted through the upper surface can suffer from reflection or transmission again from the bottom surface of the thin film.  The amount of light (quantitative description) that can get reflected or transmitted from the surfaces of the material is governed by the Fresnel equations.

At times, the light waves that get reflected by the upper surface interact or interfere with the light waves that get reflected by the bottom surface and form an interference pattern. The level of interference that can be either constructive or destructive between the two reflected light waves is dependent upon the phase difference of the two light waves.

The phase difference between the two waves is again dependent upon the width or thickness of the thin film layer, the index of refraction of the thin film, and the angle at which the initial light wave is incident on the given film layer. Moreover, the refractive index of the medium on the other side of the film boundary also plays a part in shifting the phase by 180° or π radians.

A light wave can suffer from a phase shift of 180° after reflection from the bottom boundary if the index of refraction of the medium that the light is striking is greater than the index of refraction of the medium in which the light was initially traveling. In other words, we can say that if n1 is the refractive index of the first medium and n2 is the refractive index of the film material and it is given that n1 < n2, then the light wave traveling from medium 1 to medium 2, can suffer a phase shift of π radians after reflection.

After the interference from such a medium, the interference pattern of light is observed to form either alternate bright and dark bands or bands of different colors based on the type of the incident light (chromatic or monochromatic or white).

## Condition for destructive interference in thin film

The condition for destructive interference to occur i.e. the condition necessary for the reflected light rays to interfere and cancel each other is that film thickness has to be an odd multiple of 1/4th of the wavelength of the incident light on it. The lightwave belonging to such a wavelength range cannot get reflected and is therefore completely transmitted.

## Condition for constructive interference in thin films

The condition for constructive interference to occur i.e. the condition necessary for the reflected light rays to interfere and reinforce with each other is that film thickness has to be an odd multiple of 1/2 of the wavelength of the incident light on it. In such cases the reflection of the light waves by the thin film boundary increases and the transmission of waves decreases.

## Dependence of thin film interference on the color of light.

Due to the dependence of interference level on wavelength in thin films, it is seen that white light that comprises a number of wavelengths gets reflected and transmitted unevenly. Certain wavelengths or colors of white light get intensified after constructive interference and certain wavelengths or colors suffer from destructive interference and are attenuated. The phenomenon of Thin film interference provides us an explanation about the occurrence of multiple colors of light from soap bubbles and oil films after reflection.

## Anti reflection coatings in terms of thin film interference

The antireflection coatings incorporated in camera lenses and glasses also work on the phenomenon of thin film interference. These are designed so that the relative phase shift in between the beam reflected at the upper and lower boundaries of a thin film is 180°.

## Factors on which the thickness of thin films depend:

The true film thickness or width that is covered by the light waves while passing through it is dependent on two major factors: its index of refraction and the angle of incidence of the incoming light wave. When the refractive index of the medium increases compared to the refractive index of air, the speed of light is decreased. In other words, we can say that the speed of light in a medium is inversely proportional to the refractive index of the medium.

We know that frequency of light remains the same for every medium, therefore the variation in speed occurs due to the change in wavelength of light. For this reason, films are manufactured keeping in mind the wavelength as light travels through the thin film.

When the angle of incidence is zero degrees or the light waves fall normally the thickness of the film is generally 1/4th or 1/2 of the central wavelength of the incident light. When the angle of incidence is oblique, the thickness of the film is given by the product of the cosine of the incident angle with 1/4th or 1/2 of the wavelength. This explains why sometimes we see a variation in color as we change viewing angle. (For a given film width, the color of light is seen to shift from shorter to longer wavelengths when we tilt the angle of incidence from normal position to oblique.)

## Color of light generated by thin film interference:

After passing through the thin film, constructive or destructive interference occurs and that generates a narrow reflection or transmission bandwidths. Due to the formation of these narrow bandwidths, we cannot distinguish between wavelengths based on color. The light reflected or transmitted comprises a mixture of several wavelengths that are absent from the remaining part of the spectrum.

Such a sighting is also generated by prisms or diffraction gratings. The colors observed in this case rarely belong to the VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange, Red) spectrum and are usually shades of brown, teal, gold, lavender, turquoise, bright blue, and magenta.

We can examine and analyze the reflected or transmitted light wave by a thin film to gather information about the width of the thin-film or the operative index of refraction of the thin-film medium. Thin films are commercially used for a number of purposes such as anti-reflection coatings, anti-reflection camera lenses, mirrors, and optical filters.

## Thin film interference in soap bubble:

The phenomenon of Thin film interference provides us an explanation about the occurrence of multiple colors of light from soap bubbles and oil films after reflection. After passing through the thin film, constructive or destructive interference occurs and that generates a narrow reflection or transmission bandwidths. Therefore, the soap bubble surface acts as a thin film and produces a spectrum of color similar to that of a rainbow.

## Thin film interference derivation | Thin Film interference equation:

Let us consider a scenario in which light waves are incident on thin-film material. These light rays get reflected from both the upper and lower boundaries of the thin film. The optical thickness or the optical path difference (OPD) of the light that gets reflected should be measured for obtaining the conditions for interference.

## Thin film interference diagram

Considering the ray diagram shown below, the optical path difference between the two light waves is given by:

Here,

By using Snell’s law, we can say that

Therefore,

## Constructive interference formula thin film | Destructive interference formula thin film

When the OPD or optical path difference between the two waves is equal to an integral multiple of the given wavelength of light i.e. OPD = mλ, (where m is an integer) then destructive interference can occur. For obtaining constructive interference, the required path length difference (2t) should be equal to an integral multiple of half of the given wavelength.

However, it is observed that this condition of constructive or destructive interference can change depending on the possible phase shifts. However, it is observed that

## Application of thin film interference:

The phenomenon of thin film interference is used for the following applications:

• Anti-reflection coatings: Anti-reflection coatings are used for eliminating or limiting the light reflected by an optical system (mirrors, lenses, etc.) and maximizing or enhancing the light transmitted by such a system. An anti-reflection coating is designed or manufactured in such a way that the light reflected by the optical system generates destructive interference and the light transmitted by the optical system generates constructive interference for a certain color or wavelength of incident light.

Typically, an anti-reflection coating is designed such that it’s optical width or thickness is equal to a quarter-wavelength of the incident light wave and the index of refraction of the medium lies between the refractive index of air and the refractive index of glass. Mathematically, this can be demonstrated by the equations:

nair < ncoatings < nglass

d= λ/(4ncoatings)

• Manufacturing optical instruments: The phenomenon of thin film interference is widely used for manufacturing optical instruments. Optical components such as a lens or a mirror are tested for their accuracy by comparing them with a master while designing and manufacturing them. These optical components are shaped in such a way that they have an accuracy of less than a wavelength over the entire surface of the system.
• Research purposes: Thin film interference can provide information about the index of refraction of a material, its optical thickness, interaction with different wavelengths of light, etc. For this reason, thin film interference is used for analyzing and comparing several different optical mediums.

## Thin film interference questions | Thin film interference example problems | Numerical related to thin film interference:

Complex cameras are designed by using a combination of series of several lenses and mirrors. At times, light rays get reflected from these lens surfaces and reduce the clarity and resolution of the image. These internal stray reflections from lenses are limited by coating the lenses with a thin layer of magnesium fluoride. The anti-reflection coating causes destructive thin film interference and eliminates the stray light.

## What according to you can be the thinnest possible film-width, if the refractive index of the coating is equal to 1.38 and the wavelength it is designed to operate on optimally is 550-nm which is typically the most intense wavelength belonging to the visible spectrum? The refractive index of glass is taken as 1.52.

Solution:

For obtaining destructive interference here,

2t= λn2/2

Let the wavelength in the film be λn2 and is given by

λn2= λ/n2

Therefore, the thickness t can be given by

t = (λ/n2)/4 = (550 nm/1.38) /4 =99.6 nm

Note: The anti-reflective coating films like the one mentioned in this question are considered to be one of the most efficient ways of generating destructive interference with the use of the thinnest possible layer. This also provides a reduced stray intensity of light belonging to a broader spectrum and over a broader range of incident angles.

Anti-reflecting coating is named after its function of reducing the reflection of a particular wavelength. However, wavelengths other than the mentioned one can partially pass through the filter i.e. they are not canceled entirely. These anti-reflective coatings are also used for making automobile windows and sunglasses.

## Find the three smallest possible optical width of a soap bubble that can generate constructive interference for light belonging to the red spectrum having a wavelength of 650 nm? The refractive index of the soap bubble is considered to be equal to the refractive of water in this case.

Solution: Here, n1 = n= 1.00 for air

n2 = 1.333 for soap (equivalent to water).

A shift of λ/2 occurs for the ray that gets reflected from the top surface of the soap bubble. The ray that suffers from reflection from the bottom surface does not experience any shift.

For obtaining constructive interference, the required path length difference (2t) should be equal to an integral multiple of half of the given wavelength.

Therefore, the first three possible length difference values are λn/2, 3λn/2, and 5λn/2.

For obtaining destructive interference, the required path length difference should be equal to the integral multiple of the given wavelength.

Therefore, the first three possible length difference values are 0, λn, and 2λn.

So,

Constructive interference can take place when

2tc= λn/2, 3λn/2, and 5λn/2, and so on

Therefore, the smallest possible constructive width or thickness tc is equal to:

tc= λn/4 = (λ/n)/4 = (650 nm/1.333)/4 =122 nm

The second possible value of thickness that can provide constructive interference is tc = 3λn/4, Therefore, tc = 366 nm.

Similarly, the third possible value of thickness that can provide constructive interference is t′′c = 5λn/4, therefore, t′′c = 610 nm.

Note: From the above question we can observe that if the incident light was purely red, then we could observe bright and dark bands that increase uniformly in terms of thickness.

The position of the first possible dark band would be at 0 thickness, then the first possible bright band could be positioned at 122 nm thickness, then second dark band at 244 nm, bright band at 366 nm, dark band at 488 nm, and bright band at 610 nm. If the soap bubble had a uniform variation of thickness throughout, such as a smooth wedge, then the band pattern obtained would be distributed evenly in space.

## Why don’t we see interference in thick films?

Light sources are generally not found to be infinitely small in the practical world. Light waves travel as a beam having a certain width. This means that light waves are incident on the surface of a material over a range of angles. For thin films, the angles cover approximately the same amount of optical path difference and generate an interference pattern.

However, for thick layered materials, the optical path difference at different angles is not the same. At certain angles, the light waves show constructive interference, while some angles show destructive interference. The resultant pattern, therefore, gets canceled out and we are unable to see any interference.

## Why is a broad source of light needed to observe the interference pattern by a thin film?

If we consider a narrow source or a point source of light for observing interference, then it will be able to illuminate only a small selective portion of the thin film. In other words, the human eye will be able to see only a certain section of the thin film. Due to this, it will be rather impossible to observe the entire interference pattern.

In contrast to this, when we use a broader source of light, the light waves illuminate the entire surface at considerably different incident angles and reflect back a parallel beam to the human eye. This helps in viewing the entire interference pattern formed by the thin film.

## How do you find the minimum thickness of a thin film?

The minimum required thickness t of the thin film is given by the equation t = (λ/n2)/4. Where n2 is the refractive index of the thin film.

Conclusion : In this thin film interference notes tutorial we are done with the discussion on thin film interference, Equation,Working,Dependence, Applications,Problems and few Frequently Asked Questions . To know more about light energy click here.