Coherence

Interference is possible only because of coherence. To cohere means to stick together, usually two or more similar objects. Two waves are said to be coherent, for example, if there is a definite and fixed phase relation between them. In a sense, the waves stick together. Equation (3.42) is a coherent superposition of two waves if the phase difference A^> is fixed. Now suppose that this phase difference varies in time much more slowly than the (instantaneous) amplitude of either wave but much more rapidly than the time response of a detector. If A^> ranges at least over all values from —n to n, the time average of cos(A^) is zero. The two waves are said to be incoherent, and the total power transmitted by them is the sum of the individual transmitted powers.

 — r P

Figure 3.3: An opaque screen with two holes in it separated by a distance 2d illuminated by an extended source (disc) of radius a. Each point on the disc, which radiates independently of every other point, is imagined to excite two waves, one from each hole (in reality, it is the screen that radiates). These two waves combine at P, the phase difference between them determined by the path difference s2 — si.

Figure 3.3: An opaque screen with two holes in it separated by a distance 2d illuminated by an extended source (disc) of radius a. Each point on the disc, which radiates independently of every other point, is imagined to excite two waves, one from each hole (in reality, it is the screen that radiates). These two waves combine at P, the phase difference between them determined by the path difference s2 — si.

Coherent waves must arise from coherent sources. An array of sources is coherent if all its elements bear a fixed spatial relationship to each other. The elements may move and still be coherent, as long as they move in unison. Only a coherent array of sources can give rise to an interference pattern, by which is meant a distribution of radiant power that is not the sum of the radiant powers from each source. Two identical flashlights, for example, do not yield an interference pattern. Shine them onto the same spot on a wall and this illuminated spot is twice as bright as it would be with only one flashlight; the flashlights are mutually incoherent.

Suppose that we illuminate a coherent array of scatterers. Is this sufficient to give an interference pattern? Not unless the array is illuminated by a source of light that itself is coherent to some degree.

Sunlight is sometimes said to be incoherent. Laser light is often said to be coherent. But complete incoherence and coherence are extremes never realized in nature. Every light source is coherent to a degree, and even lasers are incoherent to a degree. One way of specifying the degree of coherence of a light source is by its lateral coherence length. To determine this length as simply as possible, consider the two-slit interference problem found in elementary physics textbooks (Fig. 3.3). An opaque screen with two slits (holes) in it is illuminated on one side of the screen, and an observing screen is placed on the other side. Some textbooks show another screen with a single slit or hole between the source (e.g., a light bulb) and the opaque screen with the slits, although the purpose of this mysterious single hole may not be explained. The screen with the two holes in it is a coherent array because the holes are fixed and do not move.

Consider an extended source of light, an incandescent disc, for example, of radius a centered on the axis of the system. The distance between the screen and this source is r. Every point on the source radiates independently of every other point. Waves originating from different points bear no fixed and definite phase relationship. And yet an interference pattern still is possible if the source is sufficiently small. The holes are separated by a distance 2d and the center of the coordinate system lies equidistant between them; the coordinates of the two holes are (0, d, 0) and (0, -d, 0). The source (disc) is centered on the x-axis.

Consider first the source point (-r, 0,0). In the usual textbook treatment of two-slit interference (criticized in Sec. 3.1), a wave emitted by a source point illuminates the two holes, which magically become sources of outgoing waves (despite the embarrassing fact that this is physically impossible). These two waves combine (interfere) at observation points on the observing screen. The radiant power at these points depends on the phase shift between the two waves, which in turn depends on the different paths they traversed. For simplicity take the observation point P to lie on the axis. With this assumption the difference in paths from the holes to P is zero. The distance from the source point (-r, 0,0) at the center of the disc to either hole in the screen is the same. Thus the two waves that interfere at P are exactly in phase (no phase shift).

But now consider a source point at the edge of the disc (-r, a, 0). The distance from this point to the nearest hole is si = y'r2 + (d- a)2 (3.46)

and the distance to the farthest hole is s2 = v/r2 + (d+a)2. (3.47)

If d and a are much less than r, the path difference is approximately

r which corresponds to a phase difference [see Eq. (3.33)] /As\ 2n 2ad

This phase difference, the greatest possible, is for points lying on the rim of the incandescent disc. The phase difference is zero for the source point at the center of the disc. Hence all phase differences lie between 0 and Ay.

If the maximum phase difference is much less than 2n, all points on the source will give essentially the same interference pattern at the observation point. This condition will certainly be satisfied if

The solid angle Q (see Sec. 4.1.1) subtended by the source at the holes is na2/r2, and hence Eq. (3.50) can be written

The quantity A/ VQ, called the lateral coherence length, is a property of the source. To determine if illumination of a coherent array can yield an interference pattern we need to know the linear dimensions of the array relative to the coherence length. The condition in Eq. (3.51) is a bit strong. We still can get an interference pattern even if d is only less than the coherence length. The mystery of the screen with a single hole placed between an extended source and a screen with two slits now disappears. The function of this first hole is to reduce the angular spread of the light illuminating the slits. Nowadays the two slits are likely to be illuminated by lasers, but before they existed interference patterns were possible using ordinary incandescent light bulbs illuminating a small hole to provide a source with a lateral coherence length larger than the slit spacing.

Note that all points of the source are mutually incoherent: the wave from one point is completely uncorrelated with waves from all other points. Yet this source can still yield interference patterns. Waves from different points do not interfere with each other but rather with themselves (by way of the secondary waves they excite in the slit screen), and if the coherence length is large compared with the slit spacing, each of these separate interference patterns is approximately the same.

At visible wavelengths the coherence length of sunlight is about 50 pm, and hence sunlight can give rise to interference patterns. One example is the corona (Sec. 8.4.1). The coherent array that produces this pattern is a single cloud droplet, with dimensions of order 10 pm, less than the coherence length of sunlight.

Although a single cloud droplet can produce an interference pattern, an array of droplets cannot. That is, waves from different droplets do not interfere. A typical number density of droplets in a cloud is 200 cm-3, which corresponds to an average distance between droplets of about 2000 pm, much greater than the coherence length of sunlight. Even if cloud droplets were fixed in space (or at least fixed relative to each other), the waves from different droplets would not interfere, and the power scattered by N droplets still would be N times the power scattered by one.

A striking demonstration of how the coherence properties of an array of scatterers and that of the light illuminating them combine to yield different patterns can be done using a laser with a beam spreader.

An ordinary sheet of white paper in sunlight appears uniformly bright, but that same paper illuminated by a laser exhibits a speckle pattern (laser speckle), a mottled pattern of dark and bright regions. The paper is a coherent array: all the fibers in it are fixed in place. And the coherence length of the laser is huge compared with the separation between fibers. Now illuminate a glass of milk with this same laser. The speckle pattern disappears: the milk is uniformly bright. Milk is a suspension of fat globules, the average separation between which is much smaller than the lateral coherence length of the laser. But the globules are in constant motion, and hence do not form a coherent array. Trying to obtain an interference pattern with a suspension of jiggling fat globules would be like trying to obtain a two-slit interference pattern with slits moving rapidly and randomly toward and away from each other.