# Interference Fringes 101

What are interference fringes?  What shape are they?  Why do they occur in the wild?  If these are questions that you have pondered/are pondering, then you have come to the right page!  To understand more about interference and the fringes that result, one must understand wave motion.  In our case, the waves are light, but the following arguments work for any kind of wave whether it be a wave on a string, a sound wave, or a wave in a pool.  Each wave originates from a source, whether they be the same source our two different ones.  Two independent waves from different sources, when the interfere with each other will interfere only with intensity, NOT amplitude.  Due to the differences in phase of the independent sources, when the waves reach a point of intersection it will not be possible to combine them into stationary waves.  This type of wave is described as incoherent or as you may have heard the term, incoherent light.  Now, if we have light emitted from a single source, the phase of the light will be the same at all times because as the source changes phase the light emitted instantaneously changes phase with the source.   Therefore, the waves from the same source have the ability to superpose because their amplitudes combine, NOT their intensity.  This kind of wave or light is called coherent.  Thus, it is only possible to produce stationary waves/effects by using two or more waves ( in our case light beams) that originate from the same point on the same source.  This superposition of waves results in what we know as fringes.

As simple as that sounds, the mathematics behind fringes takes some thought.  To understand the fringes on the Michelson Interferometer, we know we have the following situation: In the above picture, M1 and M2' are the plane images produced by the separate paths of the Michelson Interferometer inclined at an angle alpha.  The eye of the observer is positioned at O, and R is the foot of the perpendicular from O on to the plane with M1.   The ray OR exists along the z-axis, thus P is a point (x,y).  We will assume the light the observer sees at O originates at P, so the angle between them is theta.  The path difference of the rays from the planes M1 and M2' is 2dCos(theta) where d is the separation of the planes at P. From geometry, we arrive at the following relationships: Thus, if the film thickness at R is e, the thickness at P is .  Knowing a little about constructive interference, specifically that it occurs at (where n is the order of interference, d is the thickness at P, and lambda is the wavelength) we can substitute in the above relationships.  The substitutions result in the following equation responsible for the shapes of the fringes: Above, we merely substituted in our equations for cosine of theta and the thickness at P.

Now that we have a relationship that governs the behavior of the fringes, we can look at the special cases!  The first case involves the two planes M1 and M2' intersecting at the perpendicular R.  If this were to occur, the thickness at point P e would be zero.   Our equation therefore reduces to: This relation further reduces as follows: .  The result is an equation that is linear, or a line when the angle alpha is small.  The fringes are equidistant from each other and appear localized on the plane M1.  This condition only holds for z>>x+y.  As it turns out, these fringes tend to be fairly weak and hard to photograph.  Here is a picture of fairly linear fringes from a HeNe laser: The above fringes are almost linear and very difficult to distinguish.   This brings us to our second special case, when the two planes are parallel.  When parallel, the angle alpha between the two planes is zero.     Our fringe governing equation then reduces to: If we do a little manipulation and squaring of terms, we get a nice result! If the observer is at a fixed z, the right hand side of the above equation is a constant which yields the equation of a circle!  Hence, our fringes are spherical when the surfaces are parallel.  Here are some pictures of circular fringes from a HeNe laser (left) and sulfur light (two on right):   Clearly, the fringes above are spherical, althought much more clearly shown by the HeNe laser light.

Another characteristic of amplitude combining waves are beats.  Beats result from the superposition of two or more waves with slightly different frequencies travelling in the same direction.  Hence, the waves are in and out of phase periodically which results in an alternation between constructive and destructive interference, or temporal interference.  Now, it becomes clear why one cannot observe beat patterns using a HeNe laser in the Michelson Interferometer, because the light is only of one wavelength (632.8 nm)!  However, when one uses sodium or the "ever so elusive" white light, one can observe these beat patterns.  They are characterized by the fringes going from really distinct to blurry.  To view an .AVI file that shows one of these coveted sodium beats, click here (....if you have a lot of time that is....the file is approximately 1 meg and takes a while to download if you have a modem!  Otherwise, if you have a T1 connection you might as well splurge!!.

### References:

1. Tolansky, S.  An Introduction to Interferometry.   William Clowes and Sons, Ltd: London.  1966.

2. Serway, Raymond A.  Physics For Scientists and Engineers with Modern Physics.  Saunders College Publishing: Chicago.  1992.