Diffraction

Light is a transverse electromagnetic wave.  Reflection, refraction, diffraction, and interference are phenomena observed with all waves.   (We first encountered interference when studying mechanical waves (Physics 135).)

A periodic mechanical wave is a periodic disturbance that moves through a medium.  The medium itself goes nowhere.  The individual atoms and molecules in the medium oscillate about their equilibrium position, but their average position does not change.  As they interact with their neighbors, they transfer some of their energy to them.  The neighboring atoms in turn transfer this energy to their neighbors down the line.  In this way the energy is transported throughout the medium, without the transport of any matter.

Each point on a wavefront can therefore be considered a point source for the production of new waves.  In three dimensions, these new waves are spherical waves called wavelets, that propagate outward with the speed characteristic of waves in the medium.  The wavelets emitted by all points on the wavefront interfere with each other to produce the traveling wave.  This is called Huygen's principle.  It also holds for electromagnetic waves.  When studying the propagation of light, we can replace any wavefront by a collection of sources distributed uniformly over the wave front, radiating in phase.

When light passes through a small opening, comparable in size to the wavelength l of the light, in an otherwise opaque obstacle, the wavefront on the other side of the opening resembles the wavefront shown below.

The light spreads around the edges of the obstacle.  This is the phenomenon of diffraction.

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Interference

Two or more waves traveling in the same medium travel independently and can pass through each other.  In regions where they overlap we only observe a single disturbance.  We observe interference.  When two or more waves interfere, the resulting displacement is equal to the vector sum of the individual displacements.  If two waves with equal amplitudes overlap in phase, i.e. if crest meets crest and trough meets trough, then we observe a resultant wave with twice the amplitude.  We have constructive interference.  If the two overlapping waves, however, are completely out of phase, i.e. if crest meets trough, then the two waves cancel each other out completely.  We have destructive interference.

The double slit

If light is incident onto an obstacle which contains two very small slits a distance d apart, then the wavelets emanating from each slit will constructively interfere behind the obstacle.

If we let the light fall onto a screen behind the obstacle, we will observe a pattern of bright and dark stripes on the screen.  This pattern of bright and dark lines is known as a fringe pattern.  The bright lines indicate constructive interference and the dark lines indicate destructive interference.

The bright fringe in the middle of the diagram above is caused by constructive interference of the light from the two slits traveling the same distance to the screen.  It is known as the zero-order fringe.  Crest meets crest and trough meets trough.  The dark fringes on either side of the zero-order fringe are caused by destructive interference.  Light from one slit travels a distance that is 1/2 wavelength longer than the distance traveled by light from the other slit.  Crests meets troughs at these locations.  The dark fringes are followed by the first-order fringes, one on each side of the zero-order fringe.  Light from one slit travels a distance that is one wavelength longer than the distance traveled by light from the other slit to reach these positions.  Crest again meets crest.

The diagram shows the geometry for the fringe pattern.  If light with wavelength l passes through two slits separated by a distance d, we will observe constructively interference at certain angles.  These angles are found by applying the condition for constructive interference, which is

dsinq = ml,  m = 0, 1, 2, ….

The angles at which dark fringes occur can be found be applying the condition for destructive interference, which is

dsinq = (m+1/2)l,  m = 0, 1, 2, ….

If the interference pattern is viewed on a screen a distance L from the slits, then the wavelength can be found from the spacing of the fringes.  We approximately have sinq = z/L and

l = zd/(mL)

where z is the distance from the center of the interference pattern to the mth bright line in the pattern.  This applies as long as the angle q is small, i.e., as long as z is small compared to L

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The single slit

When light passes through a single slit whose width w is on the order of the wavelength of the light, then we observe a single slit diffraction pattern.  Huygen's principle tells us that each part of the slit can be thought of as an emitter of waves.  All these waves interfere to produce the diffraction pattern. Consider a slit of width w as shown in the diagram below.

For light leaving the slit in a particular direction, we may have destructive interference between the ray at the top edge (ray 1)and the middle ray (ray 5).  If these two rays interfere destructively, so do rays 2 and 6, 3 and 7, and 4 and 8.  In effect, light from one half of the opening interferes destructively and cancels out light from the other half.  Ray 1 and ray 5 are half a wavelength out of phase if ray 5 must travel 1/2 wavelength further than ray 1.  We need

(w/2)sinq = l/2 or wsinq = l

for destructive interference to produce the first dark fringe.  Other dark fringes in the diffraction pattern produced by a single slit are found at angles q for which

wsinq = ml.

If the interference pattern is viewed on a screen a distance L from the slits, then the wavelength can be found from the spacing of the fringes.  We approximately have

l = zw/(mL),

where z is the distance from the center of the interference pattern to the mth dark line in the pattern.  That applies as long as the angle q is small, i.e., as long as z is small compared to L

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Diffraction grating

We have seen that diffraction patterns can be produced by a single slit or by two slits.  What happens when light encounters an entire array of identical, equally-spaced slits, called a diffraction grating?

The bright fringes, which come from constructive interference of the light waves from different slits, are found at the same angles they are found if there are only two slits.  But the pattern is much sharper. Why?

For two slits, there is one single position between bright peaks, where the interference is totally destructive.  Between the zero-order and first-order fringes, there is one position which requires that one of the waves travels exactly 1/2 wavelength further than the other to reach it.  For three slits, however, there are two positions where destructive interference takes place.  One is located at the point where the path lengths differ by 1/3 of a wavelength, while the other is located where the path lengths differ by 2/3 of a wavelength.  For 4 slits, there are three positions, for 5 slits there are four positions, etc.  For a diffraction grating with a large number of slits, the pattern is sharp because of the many positions of completely destructive interference between the bright, constructive-interference fringes.

Diffraction gratings, like prisms, disperse white light into its component colors.  The spectral pattern is repeated on either side of the main pattern.  These repetitions are called "higher order spectra".  There are often many of them, each one fainter than the previous one.  If the distance between slits is d, and the angle to a bright fringe of a particular color is q, the wavelength of the light can be calculated.

Exercise  (You can earn up to 5 points extra credit by completing this exercise.)

Problems:

Interference patterns are only observed if the interfering light from the various sources is coherent, i.e. if the phase difference between the sources is constant.  Splitting the light from a single source into various beams guaranties coherence.  Light from two different light bulbs is incoherent and will not produce an interference pattern.  Lasers are sources of monochromatic, (single wavelength), coherent light.  Two lasers can maintain a constant phase difference between each other for relatively long time intervals.