BACKGROUND AND THEORY

A property of light that is directly related to its monochromaticity is its coherence. The degree of coherence of a source of light is the degree to which that light consists of long, unbroken trains of sinusoidal waves.

Suppose we have a train of pure sine waves having some length in space, and propagating at velocity c.

Figure 1

Mathematically, we might express this train in terms of the field seen by a stationary observer as the waves pass by:

, for 0 < t <

= 0 for t < 0 , or for t > .

Now, we may inquire as to the frequencies present in this wave. Superficially, one might guess that here, we have only one frequency, w_o, since the above equation seems to imply just that. This guess would be wrong, however, and the reason lies in subtle effects connected with the fact that the wave packet turns on and turns off, i.e., that is not continuously oscillating.

The mathematical technique known as Fourier analysis deals directly with this problem. If we are considering any arbitrary function f(t), Fourier analysis shows us that it can be represented as a sum (generally infinite) of simple trigonometric sine or cosine functions of different frequencies and different relative strengths. In particular, the Fourier Integral Transform takes our function f(t) and converts it to a function g(w) that shows directly what the content of various frequencies is in our original f(t) . The simplest example is

,

i.e., an infinitely long train of sine waves. For this case, g(w) is a "delta function", i.e., a single infinitely narrow peak at w = w_o , with no contribution anywhere else, indicating that here, there is indeed only one frequency.

Figure 2

However, when we Fourier transform our short wave train, of length , we discover that a band of frequencies has appeared in g(w) , centered at w_o.

Figure 3

A general result, whose accuracy is sufficient for our needs here, is that a wave train of frequency w_o , truncated to a duration has its frequency spectrum spread over a range of values wide, such that

.

Thus, a short wave "packet" contains a wide spread of frequencies, and a long one has a narrow frequency spectrum.

This relation can also be cast in terms of the length of the packet and the corresponding spread of wavelengths. Since

,

and

,

we have

. ,

or,

,

where again, _o is the length of waves within the packet, is the spread of wavelengths, usually called the bandwidth, and is the packet length, which we usually call the coherence length.

Finally, we introduce the notion of a partially coherent source of light. In its usual meaning, this term refers to a source whose emissions tend to cluster around some particular wavelength _o, with a spread somewhat smaller than _o. Such a source generally appears to the eye to have a distinct color. (However, the eye is a rather poor judge of coherence, as this experiment will demonstrate.) Light from such a source can be thought of as being made up of a stream of wave packets, randomly arriving with respect to one another, and having a mean length

,

which is termed the coherence length of the source.

THE MICHELSON INTERFEROMETER

The interferometer is an optical device which is extremely useful for measurement of the coherence property of light. While there are many interferometer configurations, all bearing the names of their inventors (e.g., Fabry-Perot, Mach-Zender, Twyman-Green, Michelson, etc.),a single underlying principle is involved in all of them: a beam of light is passed through a partially transparent mirror, or "beamsplitter", so that every train of waves in the beam is split into two identical trains, each having (usually) half of the original intensity. Each is sent into a separate path, but subsequently they are again recombined, at which time the two components may interfere destructively, or else interfere constructively, depending upon whether the difference in the path lengths between splitting and recombination is an even or odd number of half-wavelengths.

The Michelson interferometer, which we will use in this experiment, can be schematically described by the following figure:

Figure 4

If we imagine that the rays in the figure represent long wave trains, then the eye receiving the recombined beams will see darkness if

,

and a bright light if

.

A quite special situation arises, however, if we assume finite coherence length, i.e., partial coherence of the light. First, consider the beam again as consisting of a random flood of wave packets where in the first instance, all are much longer than paths l_l and l₂ when each packet recombines with its image at the output, it interferes with itself, so to speak, and if, for example,

,

every packet will be destructively self-canceled and the total output will be zero.

Now, imagine that the coherence length is smaller than ; we get the situation in which one portion of the packet is delayed enough so it fails altogether to overlap its partner at the output, and no interference occurs at all.

Figure 5

Neither of the halves of the split packet will effectively interfere with other packets; since separate packets are in random phase with each other, there will, on the average be as many reinforcing as canceling interference of this sort, and so, no average darkening or lightening of the field results. Thus, we have a convenient means of measuring coherence length: begin with l₁ = l₂, and increase l₂ until the alternating interferences become weaker and just disappear. l₂ - l₁ is then the coherence length.