Interference and Colour, Part II - Thin Film Interference

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What does this beetle and a soap bubble have in common?

Big Ideas: 
  • When light travels to the boundary of two different media, part of the light is reflected and part is transmitted.
  • When two boundaries are very close we call this a thin-film. The two reflected beams off of the two boundaries can interfere.
  • It is the reflections off of objects that allow us to see objects. If the reflections interfere constructively only at a specific wavelength then the object will appear as the colour associated with that wavelength.
  • By having multiple layers of thin-films so that a large percentage of the light is reflected, the colour that interferes constructively for all those reflections will appear very bright.
  • Path length difference between two waves does not alone fully determine the conditions for constructive and destructive interference.

Figure 1: Examples of coloration of beetles by thin-film interference1.  (a) Cicindela scutellaris scutellaris (Carabidae: Cicindelinae), (b) Amargyminae gen. sp.  (Tenebrionidae) (c) Phanaeus vindex (Scarabaeidae: Phanainae).  Photos courtesy of J. R. Soc. Interface 2009 6, S165-S184

When sunlight travelling through the air hits a soap bubble you can see bright colours on the soap skin.   The reason for this is interference.  In this case, the interference is between the waves reflected from the outer surface and the inner surface of the soap bubble.  Why isn't the light just passing through the transparent soap bubble?  Generally, a small portion of the light will be reflected whenever it encounters a boundary between two media.  In our case, the boundaries are air/(soap-)water on the outside and (soap-)water/air at the inner surface of the bubble.  The optical properties of the media are characterized by their refractive indices.  The general principle is illustrated in Figure 2. 

Figure 2:  As light wave A travels from a medium of index of refraction n1 to a medium of index of refraction n2, part of the light will be transmitted through the boundary (wave B) and part will be reflected off of the boundary (wave C).  The arrows show the direction which the waves are travelling.

 

Figure 3:  A thin film is formed by sandwiching a thin layer of medium (here with index of refraction n2) between two other media (here with n1 and n3).  Medium 1 and 3 could be the same type of media.  Because of the reflections at the two boundaries wave C and F are formed and can interfere.  Constructive interference will result in a bright reflection being seen.  Perfect destructive interference will result in no reflection being seen.

Let's consider a layer of material sandwiched in between two other media (Figure 3).  In case of soap bubbles we have a thin layer of water between thick layers of air and n3 = n1.  Any time a light wave encounters a boundary where the index of refraction changes, a transmitted and reflected wave are created.  For example, in Figure 3 we can see wave B gives rise to wave D and E, also wave E gives rise to wave F and G and so on.

Th bright colours of soap bubbles are due to constructive interference between wave C and F shown in Figure 3.  In our article on diffraction gratings, we showed that the interference between two waves perceived by an observer is related to a difference in path length between the two waves as they travel towards the observer.  Once we can express the path length difference, one might expect to use the same equation as in our previous diffraction grating article, $ \Delta x = m \lambda, m = 0,1,2,... $ for constructive interference.  It turns out however that with thin-film interference there are two complications that we need to take into accout: the wavelength of light changes as it travels in a different medium and 'phase changes' introduced by reflection.  We will discuss these two complications later but let us first see what the path length difference, $ \Delta x  $, is by looking at Figure 4.

 

Figure 4:  A replica figure of figure 3 except this figure more clearly shows that the light forming wave F passes through medium 2 twice before interfering with wave C.

 Figure 4 is the same as figure 3 but slightly simplified to show that the light forming wave F travels through medium 2, bounces off the layer between medium 2 and 3 and then travels back through medium 2 again before overlapping with wave C.  This extra distance travelled by the wave forming wave F is the path length difference.  The lateral difference between the two waves is usually negligible for angles close to normal incidence, which we will consider here.   When the incident wave A is normal to the boundary between medium 1 and 2 ( i.e. coming in straight down in Figure 4) the path lenght difference between wave C and F is simply twice the thickness of medium 2, or $ \Delta x = 2 t_{2} $.

In our article on diffraction gratings, we mentioned that constructive interference is greatly enhanced (gives very bright colours) when many waves interfere.  We have a similar effect in thin film when we have many thin layers.  This brings us back to our beetles.

Figure 5:  a)  Alternating layers of refractive index n1 and n2 can be used in nature to provide bright reflections at a specific wavelength.  b) TEM cross section of a Cincindela Scutellaris shell showing these multilayer reflectors are used in nature for coloration1,2. c) Picture of Cincindela scutellaris scutellaris1.  Photos courtesy of:  J. R. Soc. Interface 2009 6, S165-S184.

The bright coloration in some beetles can occur by their shells having multiple alternating layers of organic material as shown in figure 4.  The layers alternate such that one layer has refractive index n1 with thickness t1 and the next has index of refraction n2 with thickness t2 and so on.  In Figure 5b) though the exact composition of the layers is not known, the index of refraction n1 of the lighter material is n1 = 1.5 and that of the darker n2 = 2 2.  The lower index material is thought to be a protein and that of the darker melanin2.  Many reflections (A, B, C, D... shown in figure 5) will be generated and interfere.  If wave B interferes constructively with both wave A and C then A and C also interfere constructively and all the waves will add up constructively giving the brightest possible reflection.  Having many layers gives many interfering reflections and a large percentage of the intensity of the incoming light can be reflected.

How do we determine what the thickness t1 and t2 should be in order to have constructive interference at a certain wavelength $ \lambda $?    We could try using our knowledge of path length difference for A and B being $ \Delta x = 2 t_{1} $ and the condition for constructive interference given in part I  as $ \Delta x = m \lambda $ , $  m = 1,2,3... $ to find the values for t1 and t2.  Note m = 0 was not included because the film can not have zero thickness.  This approach gives $  \Delta x = 2t_{1} = m \lambda $ so that $ t_{1} = m \frac{\lambda}{2} $, $ m = 1,2,3, ... $.   Similarly the path length difference for wave B and C is $ \Delta x = 2 t_{2} $ which would also give $ t_{2} = m \frac{\lambda}{2} $.  This approach then predicts that $ t_{1} = t_{2} = m \frac{\lambda}{2} $ for constructive interference between waves A, B and C.  The thickness of the layers of medium 1 and 2, however, are not found to be the same by observation in real life with microscopes.  What is the reason and how do we define the conditions of  constructive interference?  We will see in the next article that the wavelength of light changes depending on the medium it is travelling in.    Taking wave A and B for example in Figure 5, wave A reflects off of a boundary travelling from medium 1 to medium 2,  whereas wave B reflects off of a boundary travelling from medium 2 to medium 1.  We will see one of the reflections is considered a 'hard' reflection where the wave encounters a medium with higher index n.  For the case of a soap bubble this would be when a wave travelling in air reflects off of water.  The other reflection is considered a 'soft' reflection where the wave encounters a medium with lower index n.  For the case of a soap bubble this would be when a wave travelling in water reflects off of air.  How these considerations changes the condition for constructive interference and destructive interference is explained in the next article.

  • 1. a. b. c. J. R. Soc. Interface 2009 6, S165-S184
  • 2. a. b. c. J. exp. Biol 117, 87-110 (1985)

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