# Interference and Colour, Part I - Diffraction Gratings

What gives this beetle its rainbow colours?

- The wavelengths that are reflected with the highest intensity off of an object will determine the colour of the object.
- Interference of the waves reflected off an object is one means by which only certain wavelengths are strongly reflected.
- Constructive interference between the reflected waves for a certain wavelength gives the brightest reflection possible, while destructive interference gives the weakest reflection
- Periodically arranged structures can form 'diffraction gratings' responsible for an observer seeing different colours depending on what angle they are with respect to the grating

**Figure 1**: Examples of colouration of beetles by interference of light 1 Colouration in (a) *Loxandrus rectus *(Carabidae: Harpalinae) and (b) Phalacridae gen. sp. are both due to diffraction gratings while coloration in (c) *Eupholus* sp. (Curculionidae) is due to photonic crystals. Photos courtesy of : J. R. Soc. Interface 2009 **6**, S165-S184

Light reflecting off of an object such as a beetle and traveling to an observer's eye is what allows the observer to see the object. Light has wave-like properties and colour is associated with the wavelength of light. The visible range is from 400 - 700 nm corresponding to the rainbow colours going from blue to red, respectively. An observer will see the object at a colour corresponding to the wavelength that is the most strongly reflected.

Not all the light hitting an object is reflected, some is absorbed or transmitted through the object. We are most familiar with coloration being caused by pigments which are molecules that absorb certain wavelengths better than others. Whatever wavelengths that are poorly absorbed or transmitted will be reflected with relatively high intensity and will determine the colour of the object.

Some beetles have tiny microscopic structures like bumps or pits that reflect light efficiently in all directions. We can consider these as tiny point sources of light. Figure 2 gives a cartoon version of this showing light being scattered in all directions from a bump.

**Figure 2**: Microscopic bumps on an object can scatter light in all directions acting like tiny point sources of light. The wavelength of the light scattered will be the same as that of the incoming light which for visible light is 400-700 nm.

Suppose your eye was at point P in Figure 3. We can see that if we arrange these microscopic structures side by side then waves will travel from each of these point sources towards your eye. If you moved your eye a different set of waves would travel to your eye. The waves that travel to point P overlap on top of one another. What will an observer at point P see? Under certain conditions very intense colours.

**Figure 3**: At any given place in space shown by point P waves from the point sources shown will travel there and interfere.

Consider two waves of equal wavelength and amplitude travelling in the same direction (Figure 4). If these waves also occupy the same region of space they will interfere. According to the 'superposition principle', each point on wave 1 and the corresponding point on wave 2 are added to give a resultant wave whose amplitude can be greater than or less than the amplitude of the two original waves. Figure 4 illustrates this: in each figure two waves are added (red and purple) and the resulting wave is shown in blue. Figure 4a shows two waves perfectly overlapping so that the crest of one wave lies on top of the crest of the other wave. We say that they 'interfere constructively' giving a resultant wave with maximum possible amplitude. In Figure 4b the trough of one wave lies on top of the crest of the other wave. 'Destructive interference' occurs and the amplitude of the resultant wave is zero. Figure 4c and 4d show two cases where the waves add up only partially constructive or partially destructive depending on whether the crest of wave 1 is close to a crest of wave 2 (Fig. 4c) or a trough of wave 2 (Fig 4d).

**Figure 4**: The displacements of the purple and red waves add to give the resultant blue wave. a) shows the purple and red waves interfering constructively. The resultant wave in this case has an amplitude equal to the sum of the amplitudes of the purple and red wave. b) shows the purple and red waves interfering destructively. The resultant wave has zero amplitude because the amplitude of the purple and red waves are the same here. c) shows the crests of the red and purple wave being close so that partial constructive interference results d) shows the crest of one wave being close to the trough of the other so that partial destructive interference results.

A colour can be strongly enhanced when waves reflecting off of an object interfere constructively or be suppressed when waves interfere destructively. The wavelength(s) at which constructive interference occurs will be reflected with the greatest intensity and will therefore determine the colour seen when looking at the object.

Now let us consider the factors involved in determining how two or more waves will interfere (i.e. constructively, destructively or in between) and see how interference leads to the bright coloration of the beetles in Figure 1.

The 'path length difference' between two waves will affect the relative positions of the overlapping waves. The path length difference between two waves can be defined as how much further one wave travels from its starting point than another wave before the two waves overlap. In Figure 3 for example, the wave from the bottom bump has a longer distance to travel to point P than the other two waves. In Figure 5a we can see that if the dashed lines denote where the waves originated then wave A and B have a half wavelength path length difference. The two waves are shown separated to make the figure visually simple but if they were to overlap then there would be destructive interference. In Figure 5b wave A and B have a path length difference equal to one wavelength and constructive interference results. Figure 5c and 5d show that if you add in whole wavelengths to the path length difference the interference properties do not change.

**Figure 5**: Wave A and B at their starting points are at the same part of their oscillaton cycle. The two waves are considered to overlap but are shown separately for visual simplicity. The extra distance travelled by wave B is called the path length difference. a) shows how destructive interference results for a path length difference of half a wavelength. b) constructive interference results when the path length difference is a whole wavelength. c) and d) show that adding in full wavelengths to the cases shown in a) and b) does not affect the resulting interference.

We can see that, as drawn in Figure 5, constructive and destructive interference holds for path length differences, , which obey the following equations:

, (constructive) Equation 1

, (destructive) Equation 2

Here is the wavelength and is a non-negative integer.

So far, we have seen that we can get bright colours by constructive interference, and that constructive interference can result from differences in the path length. In other words, if two waves are reflected from slightly different points of a surface they can interfere constructively at the location of our eyes because of tiny differences in their path length. Remember, a visible wavelength is only ~ half a micrometer, much less than the width of a human hair ( ~ 80 micrometers).

** Figure 6**: The two waves from the point sources A and B meet at a distant point (similar to point P in figure 3) and interfere. Because the distance to the point of overlap is large compared to the distance d between the sources we make the approximation that the waves are parallel. Using this approximation geometrical arguments give the path length difference between the waves as .

Figure 6 shows how to determine the path length difference between the waves travelling from two adjacent point sources. We are considering that the waves are almost parallel because the observation point where they overlap and interfere (point P as shown in figure 3) is far away from the point sources compared to the distance between the point sources. The condition for constructive interference is given in Equation 1. The path length difference, , is given geometrically by

Equation 3

where is the angle point P makes with respect to the normal of the surface on which the point sources are located. Finally from equation 1 and 3 we have

, Equation 4

for constructive interference, where the subscript on is to remind us that for every there is a different . Inserting numbers into equation 4, you will find that is typically a few micrometers, much smaller than any typical observation distance justifying our assumption of parallel waves.

You are probably wondering why we considering only two waves from two points. Shouldn't there be lots of waves from many points contributing to the brightness at the location of our eyes? That is correct and now that we know how two waves interfere constructively, we will extend our result to many waves

**Figure 7**: Three point sources are uniformly arranged along a line. Waves that meet and interfere at some distant observation point P (not shown) are considered. The path length difference between any adjacent pair is the same and this means when one adjacent pair interferes constructively, all three waves interfere constructively. This result can be extended for more than three point sources so long as they are uniformly arranged along a line.

Figure 7 shows an example of three waves interfering constructively. The waves start from three point sources and are uniformly arranged a distance from each other so that the path length differnece between any two adjacent point soures is identical. This means that if the wave from A and B interfere constructively then so will waves B and C. We see that when the condition for constructive interference is achieved at point P for any adjacent pair of waves then all the waves will interfere constructively at point P and the amplitude at point P would be the greatest possible. In other words, when any one of the waves has a crest at point P all the other waves from all the other points would also have a crest at P for constructive interference. It is important to realize that for a given wavelength constructive interference is only satisfied for a set of specific angles for which equation 4 is satisfied (see Figure 8). For any other angles for this wavelength partial constructive or destructive interference will result. The net result is that one wavelength (colour) is strongly enhanced over all other wavelengths at a certain observation angle. The more waves interfere (more point sources), the greater is the enhancement.

**Figure 8**: For a given wavlength, constructive intefernce only occurs at certain angles as described in equation 4. We are considering for the moment only one wavelength (corresponding to red colour) to be emitted from the uniformly arranged point sources shown in the inset circle. The locations at which constructive interference can be observed at a certain distance away from the sources are marked by red squares. On a screen this 'interference pattern' would show up as bright red bands separated by dark bands. If your eye was at one of the angles indicated in the figure you would see bright red. Moving your eye away from that angle you would not see the colour red as brightly and at some angles (which we have not discussed) you would see zero intensity of red light corresponding to destructive interference.

It seems that we need some very special conditions to get this interference effect such as a regular pattern of point sources. Regular patterns are not rare in nature. Take a look at the hair on your head, for example. Since each root needs a certain space, your hair is roughly uniformly spaced. So if there are certain discrete spots that reflect light in all directions and are uniformly spaced, then chances are good that we get interference which results in coloration phenomena. We can say that a periodic pattern or texture can lead to light interference if the spacing is right. A man-made example of this is a CD or DVD 2. You may have noticed when you look at a CD you see a rainbow of colours as you tilt the CD at different angles. On a microscopic level the CD has many tiny metallic pits evenly spaced. As light hits a pit it scatters in all directions acting like a point source.

Figure 8 shows how for a given wavelength, in the direction of your eyes, the waves from the point sources will add up and interfere. With many waves adding up at the same point of observation (P in Figure 2), only one wavelength will satisfy equation 4 for all waves. For visible light being emitted from the point sources, if you move your eyes to a different location, the angle changes and equation 4 is satisfied for a different wavelength (colour). If you move your observation point far enough, the rainbow patterns repeats because the path length difference has now changed by another wavelength. The number counts how often this happens. This is illustrated in Figure 9 below. An alternative to changing your observation point is changing the angle of the CD. In both cases, the angle between your observation point and the surface normal is changed.

** Figure 9**: Similar to figure 8 except now we consider the point sources emitting all visible wavelengths. Different wavelengths will interfere constructively at different angles and so a screen placed behind the point sources would have rainbow patterns on it which would repeat for different orders. The dashed lines indicate angles from the horizontal. The order for all wavelengths occurs at so that a bright white fringe occurs there.

In nature one can find examples of colours formed by periodic structures. We have discussed 1 dimensional structures where the point sources form a line (called a 1 D diffraction grating). In nature this can be achieved such as in Figure 9 where lines are placed side by side.

**Figure 10**: a) Scanning electron microscope image of the shell of Sphaeridiinae gen. sp. (Hydrophilidae) 1. The image shows lines all oriented along the same direction. The blue line indicates the line along which the one dimensional point sources line up so that these structures can be treated as a one dimensional grating. b) Picture of Sphaeridiinae gen. sp. with the 0,1,2,3 indicating the from equation 4 for bright fringes of the visible wavelengths. Photos courtesy of : J. R. Soc. Interface 2009 **6**, S165-S184

2 and 3 dimensional arrays of microscopic structures are also used in nature ( Figures 11 and 12). For two dimensions, the explanation and phenomena of different colours being seen at different angles is very similar as we have seen for one dimension. There are also three dimensional arrays, called 'photonic crystals', that have more complicated interference and colour effects.

**Figure 11**: a) Scanning electron microscope image of the shell of *Pallodes* sp. (Nitidulidae)1. The microscopic structure forms points sources in a line along two directions so that this forms a 2D diffraction grating. b) Picture of *Pallodes* sp. (Nitidulidae). Photos courtesy of : J. R. Soc. Interface 2009 **6**, S165-S168

**Figure 12**: a) Scanning electron microscope image of the shell of *Pachyrrhynchus congestus pavonius *(Curculionidae: Entiminae)1 which shows complicated arrays of periodic structure in 3D forming a photonic crystal. These crystals give rise to amazing coloration effects such as in the photograph shown in b) of *Pachrrhynchus gemmatus*. Photos courtesy of: J. R. Soc. Interface 2009 **6**, S165-S184

Another means of causing only certain wavelengths of light to be intensely reflected off a beetle is to have the shell composed of thin layers of alternating material. We will discuss this in an extra article on thin-film interference. The layers form a periodic pattern across the thickness of the shell. The physics is similar to our one-dimensional grating case discussed in this article but has additional complications.

- 1. a. b. c. d. J. R. Soc. Interface 2009
**6**, S165-S184 - 2. Wikipedia.
*Diffraction Grating*(online). http://en.wikipedia.org/wiki/Diffraction_grating [31 July 2011].

© Physics and Astronomy Outreach Program at the University of British Columbia (Janelle Van Dongen 2011-07-31)

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