In part II of my series on ‘What is a wave?’, I addressed one of the two most significant behaviors of waves, namely interference, the ability of a wave to ‘interact’ with itself. The second behavior of waves which is extremely significant is diffraction, and we will address it in this post.
Diffraction may be broadly defined as the tendency of a wave traveling in two or more dimensions to spread out as it propagates. The most significant consequence of this spreading is the ability of waves to ‘bend around corners’ when faced with an obstacle. We all have experienced the diffraction of sound waves: if you and a friend stand on opposite sides of a large building (say a farmhouse) in the middle of an open field, you will be able to talk to each other even though there is no direct ‘line of sight’ between you and your friend, and no ability for the sound waves to reflect off of intermediate surfaces. The sound waves wrap around (diffract around) the outside of the farmhouse, allowing communication.
Light waves also diffract, though the effect is much smaller and difficult to detect. This is because the wavelength of visible light (to be discussed in part IV of optics basics) is much smaller than the wavelength of sound. One can demonstrate theoretically that waves only produce appreciable diffraction when interacting with objects of a size comparable to or smaller than the wavelength. Wavelengths of sound can range between millimeters and meters (comparable to a farmhouse) while wavelengths of visible light are on the order of 500 nanometers (0.0000005 meters, or 0.5 micrometers).
To give you an example of how a diffracted field would differ from a field propagated geometrically, we consider the case of light traveling through a narrow slit in a metal plate:
In both pictures, higher light intensity is represented by brighter color. In both pictures, we have a collimated (one-directional) field incident from below upon an aperture in a silver plate (represented in cyan here). The wavelength is taken to be 500 nm, the aperture width is 500 nm (i.e. one wavelength), and overall the images are 4000 nm by 4000 nm.
The first picture indicates the prediction of geometrical optics. Geometric theory models light as traveling along definite paths (rays) in space, indicated on the picture by arrowed lines. Those rays which hit the metal plate are absorbed or reflected, while the rays illuminating the aperture pass directly through unimpeded and form a ‘column’ of light.
The second picture is an exact numerical simulation of Maxwell’s equations, demonstrating what ‘actually’ happens when light illuminates a wavelength-wide slit. Two things are different from the geometric picture: below the aperture, there are horizontal lines of darkness along with the bright lines. These are locations where the incident field from below is interfering with the field which gets reflected from the metal plate. More important for our discussion, however, is what happens when light passes through the aperture: not only are there bright and dark patches indicative of interference, but we can see that the light is spreading in a cone away from the aperture, at odds with the simple geometric prediction. Some of the light has evidently ‘bent around the corner’ on passing through the aperture, and this is what we mean by diffraction.
We’ve mentioned above that diffraction is only appreciable when light interacts with objects of size comparable to the wavelength, and this is true of light passing through the slit, as well. The figure below shows an exact simulation of light passing through an aperture which is 3 wavelengths wide:
This picture can be compared with the geometric one shown previously. Although interference and diffraction effects can still be seen, they are much less prominent in this case, and to a good approximation light is propagating in a narrow ‘column’ of width equal to the aperture. Diffraction effects are still present even at larger aperture sizes, but they become insignificant compared to the overall geometric behavior.
The mathematical theory of diffraction is relatively complicated, and requires a significant amount of vector calculus, so we won’t go into too much detail here. We can understand it a little better, however, by the use of the Huygens construction. Christiaan Huygens (pronounced ‘hoy-gens’ in English, but closer to ‘how-gens’ in Dutch) was a Dutch mathematician/scientist who lived from 1629-1695. He developed the first wave theory of light, and described the propagation of light by means of the Huygens construction: treat every point of a wavefront as the source of a new spherical wave. The position of the new wavefront is constructed by tracing the boundary of all these ‘secondary’ wavelets.
This is illustrated below. The top image shows the construction for an infinite plane wave which is uniformly traveling upward. The construction shows that the wavefronts remain planar. The bottom image illustrates aperture diffraction in Huygens construction; because the wavefront is ‘clipped’ at the edges of the aperture, the wavelets on the edges spread out into the ‘shadow’ region of the aperture:
Borrowing from one of my thesis advisor’s talks, here’s a picture that Huygens himself drew in his manuscript on the subject, showing the secondary wavelets of a candle flame:
Huygens’ work was not enough, however, to conclusively illustrate the wave nature of light. Only when Thomas Young introduced the principle of light interference in 1801 was the theory complete enough to match experiments and make convincing theoretical predictions. That history, however, will have to wait for another post!
Diffraction effects are hugely important in optical science and engineering, and we’ll be addressing the applications and consequences of diffraction as my blog continues. In my last post on optics basics, I’ll introduce some of the basic quantities used to describe waves, such as wavelength and frequency.