9.3.5 Thin Film Interference

• Thin film interference causes the iridescence seen in:
  • Nature on peacock feathers
  • Glossy flower petals
  • Soap bubbles
  • The shiny side of a CD
  • Thin layers - or films - of oil on water
• This phenomenon occurs when light waves reflecting off the top and bottom surfaces of a thin film interfere with one another

The colourful pattern observed on a CD is a result of thin film interference

Conditions for Thin Film Interference

• To see the interference light must be incident on a material which:
  • Is very thin
  • Has a higher refractive index than the medium surrounding it
  • Also transmits light

• The effect is caused by the reflection of waves from the top and bottom surfaces of the thin film

Phase changes at the top and bottom of a thin film

Light incident on the top surface of the thin film (A)

• Part of the light wave reflects at a boundary between a less-dense and a more-dense medium (e.g. Air to Oil where n_air < n_oil ), and a phase change is seen, such that:

A wave reflected at a boundary with a medium of a higher refractive index undergoes a phase change of half a wavelength (), 180° or π rad

A wave incident on the boundary with a more dense material is both reflected and transmitted. The transmitted wave continues without a phase change and the reflected wave has a 180° phase change. What was the peak on the incident wave becomes the trough on the reflected wave.

• You need to be aware of thin-film interference for a wave incident on oil and water
• Light is reflected and transmitted from a boundary with a less dense to a more dense material
  • Air to Water
  • Water to Oil
• Light is transmitted only from a boundary with a more dense to less dense material
  • Oil to Air
• A less dense material has a lower refractive index (n)
  • The refractive index of air is 1
  • The refractive index of water is 1.33
  • The refractive index of oil is 1.4-1.5
  • Normally in thin-film interference situations: n_air < n_water< n_oil

Light is reflected and transmitted at the boundary from a less dense to a more dense material. Light is transmitted only at the boundary from a more dense to a less dense material. Hence, in this diagram P and Q exist but the third unlabelled ray does not.

Light transmitting through the thin film (B)

• Part of the light wave also refracts as it enters the thin film and is transmitted through the material to the bottom surface
• The light wave is now travelling through a denser medium, so it travels at a slower speed and has a shorter wavelength

When the light ray enters a denser medium the wavelength becomes shorter.

Light incident on the bottom surface of the thin film (C)

• If the bottom surface is at a boundary with a more dense material (a higher refractive index e.g. oil-water where  n_oil < n_water) then both of the following will occur:
  • Reflection with a π phase change back into the thin film
  • Refraction of the transmitted light into the next medium
    • This is the same situation as the light wave incident on the top surface of thin film 

Light is transmitted with no phase change between the boundary of a less dense to a more dense material and reflected with a phase change.

• If the bottom surface is at a boundary with a less dense material (a lower refractive index e.g. oil-air where n_air < n_oil ) then the following will occur:
  • Transmission with refraction will occur out of the material
  • No reflection will occur at the boundary

Light transmitting through the thin film and refracting back into the first medium (D)

• Light travelling through the thin film undergoes a wavelength shift due to the density of the plastic
• The wavelength shift is a multiple (normally labelled m) of a half-integer multiple of wavelengths
• The thinner the film, the smaller the value of m 

Light reflecting off the thin-film air boundary undergoes a wavelength shift related to the density of the thin-film. Light reflecting off the air-thin film boundary undergoes a phase change of half a wavelength. This results in constructive interference and a bright light seen by the observer.

Observing Constructive Interference

• Constructive interference will be seen as brighter colours by the observer
  • The light intensity has increased making the colour appear brighter
• The total path difference of the two waves reflected and refracted off the top surface of the thin film must be a multiple of wavelengths for constructive interference to be seen by the observer

The combination of the phase change due to the reflection off the air-film surface and the phase change within the film here, results in constructive interference seen by an observer because the path difference is a multiple number of wavelengths. 

• According to the data booklet, the formula for the path difference for constructive interference is: 

2dn = 

• Where: 
  • d = thickness of the film
  • n = refractive index of medium
  • m = an integer related to the refractive index and thickness of the medium
    • The thinnest film exists when m = 0
  •  λ = wavelength of the light in air

• Constructive interference occurs when two waves are out of phase by an integer number of wavelengths The wave reflected from A has undergone a phase change so now has a path difference of
• The wave coming from C inside the film must have a phase difference of a half-integer multiple of wavelengths So, the total phase difference would be + multiple of
• In the formula for the path difference for constructive interference: 2dn = 
• So, the thinnest film exists when m = 0
  • So the multiple of  would just be one
  • When m = 1 the multiple would be  
  • A common question is to be asked to calculate the thickness of the thinnest film

Observing Destructive Interference

• Darker colours are observed through destructive interference
  • The light intensity has decreased making the colour appear darker
• The total path difference of the two waves reflected and refracted off the top surface of the thin film must be a half-integer multiple of wavelengths for destructive interference to be seen by the observer

The combination of the phase change due to the reflection off the air-film surface and the phase change within the film here, results in destructive interference seen by an observer because the path difference is a half integer multiple number of wavelengths. 

• According to the data booklet, the formula for the path difference for destructive interference is: 

2dn = mλ 

• Where: 
  • d = thickness of the film
  • n = refractive index of medium
  • m = an integer related to the refractive index and thickness of the medium
  • λ = wavelength of the light in air

• Destructive interference occurs when two waves are out of phase by a half-integer number of wavelengths The wave reflected from A has undergone a phase change so now has a path difference of
• The wave coming from C inside the film must have a phase difference of a half-integer multiple of wavelengths  So, the total phase difference would be + multiple of = 
• In the formula for the path difference for destructive interference: dn = 
• So, the thinnest film exists when m = 1
• Other films exist with destructive interference for only odd values of m
  • When m = 2 for example constructive interference and not destructive takes place as = λ 

Distance Travelled inside the film

• Light travels the same distance within a thin film whether it undergoes constructive or destructive interference
• Light enters from the top surface of the thin film, passes through the thin film of thickness d 
• Light is then reflected off the bottom surface travelling a further distance of d to the original surface
• So, the total distance travelled whether the light interferes constructively or destructively is 2d

The reflected wave travels a total distance of twice the thickness of the film

 

Model the thin film as a parallel-sided, rectangular ‘slice’ with a thickness of ‘d’

Uses of Thin Film Interference

• Thin films can prevent light from being reflected
• Due to the conservation of energy, this increases the light which is transmitted through a medium
• This effect is utilised in:
  • Camera lenses - these often have a coating to prevent reflection
  • Solar cells - this is to ensure a high proportion of incident light can be captured

The Mathematics of Constructive and Destructive Interference

• Consider a thin film, such as a soap bubble, modelled as a very thin, parallel-sided rectangle, with thickness d and refractive index higher than that of the surrounding air, i.e. n > 1
  • If the thin film is sitting on top of a different denser medium e.g. water then the following situations will not apply because of the phase change obtained from the reflection at the oil-water boundary

A thin film can be modelled as a parallel-sided, rectangular ‘slice’ with a thickness of ‘d’

• Constructive interference occurs when:

The thickness of the coating is

• This is because the light from the bottom of the film has travelled an extra distance (there and back)
• Since the first ray has undergone a phase change of π, or half a wavelength, , the condition for constructive interference is:

Path difference, 

• Where:
  • d = thickness of the film (m)
  • m = an integer (generally taken as m = 0)
  • λ₀ = wavelength of light in soap (m)
• The value of λ₀ can be obtained using the relationship:

• Where:
  • λ = wavelength of light in a vacuum (m)
  • n = refractive index of the film
• The condition for constructive interference can, therefore, be rewritten in terms of λ as:

• Rearranging this gives:

• Destructive interference occurs when:

The thickness of the coating is

• This is because the light from the bottom of the film has travelled an overall distance of λ
• This time, the condition for destructive interference is:

Path difference, 

• The value of λ₀ can be obtained using the relationship:

• The condition for destructive interference can, therefore, be rewritten in terms of λ as:

• Rearranging this gives:

Worked Example

Step 1: List the known quantities

• • Refractive index of coating, n = 1.31
  • Wavelength, λ = 540 nm

Step 2: Use the data booklet to find the conditions for thin-film interference

Constructive Interference	Destructive Interference

Step 3: Consider the phase changes at the boundaries

• • Firstly, at the boundary with the coating, which has higher optical density, phase change = π
  • Secondly, at the boundary with the lens, which has higher optical density, phase change = π

Step 4: Determine the correct equation to use 

• • To prevent reflection, this requires destructive interference, so the equation to use is:

• • For a minimum thickness, m = 0

Step 5: Rearrange to make thickness, d, the subject and calculate

= 100 nm