# Interference

In photography, a two-dimensional real image (that is, an image existing in the plane of converging light originating from an object) is recorded by means of a focusing lens and photographic emulsion (or image sensor). By contrast, the holographic process records not the image of the object, but the interference between the light from the object and the light from a reference beam. The interference pattern preserves all of the information regarding the light originating from the object; thus permitting the eventual production of a three-dimensional image with parallax intact. Evidently, the theory of holography starts with the rudiments of interference.

The light we experience around us is composed of a great spectrum of frequencies, and each ray of light is travelling in a random direction; consequently, we seldom notice interference in light on a daily basis. If two lamps of equal intensity are turned on in a room, where light from both lamps hits, the intensity from each lamp at that point is summed together and the spot brightens. However, when light of one frequency intersects with more light of the same frequency (like when a laser beam is split and each beam crosses paths), interference occurs.

Light has electrical and magnetic field components that sinusiodally oscillate in planes perpendicular to each other and the direction of travel (See Figure 1). The wavelength of the light is the spatial distance over which the wave repeats. Wavelength (&lamda;) and frequency (ν) are related by the speed (c) of the light . All three quantities are constant within a uniform medium. When a light source is 'coherent' it produces light rays whose electrical and magnetic components are aligned and of the same wavelength and speed. Often, instead of drawing individual rays, a diagram will mark where each ray has its crest or valley, ie. it draws a wavefront. For the point source in Figure 1c, the solid-line wavefronts are circles and mark the wave crests of the infinitely many rays.

Figure 1

a. b. c.

a. A light ray with electrical and magnetic components shown; b. Wavefronts produced by a point source that emits rays; c. Wave amplitude and wavelength.

Like any water or radio waves, when light waves are superimposed with their electrical and magnetic components aligned, their functions are summed. Thus, if two waves are in phase (ie. their aplitudes line up), their amplitudes add together--constructive interference-- and when they are out of phase, their amplitudes are substracted--destructive interference (See Figure 2).

Figure 2 The top line is constructive interference and the bottom line is destructive interference

Consider the two point sources of light of a certain frequency in Figure 3a. Each source emits a spherical wave, but for now, we will consider only the 2-dimensional cross-section. The wavefronts from each point source mark the crests of the waves emitted at a certain time. Where wavefronts intersect, there will be total constructive interference. Constructive interference occurs all along the dashed lines at all times. Because the valley of a wave occurs exactly halfway between each crest, exactly halfway between the dashed lines there is total destructive interference at all times.

Figure 3

a.

b.

Figure 3b illustrates the appearance of the interference pattern with each curve mapping the regions of maximal amplitude of the waves. Since each point source emits a spherical wave, one need only rotate this image along the axis between the point sources to see that the interference pattern is a set of hyperboloids. These hyperboloids are stationary in space so long as the two point sources emit mutually coherent spherical wavefronts.

Of course, in actuality, one could not see this pattern floating in space; a viewing screen must be placed. The fringes (Figure 4b) observed on the screen correspond to the intersection of the interference pattern with the screen on the side closest to the sources (Figure 4a).

Figure 4

a.     b.

a. A viewing screen placed in the interference field.   b. Pattern observed on a viewing screen as placed

Along from each source, the separation of adjacent concentric circles is half a wavelength (0.5λ). For a very small λ such as with visible light, the separation between wavefronts is infinitesimal and the intersections of the two sets of wavefronts will form a set of hyperbolas determined by RA-RB=nλ (See Figure 5).

Consider now a quasi-rhombus formed by four wavefronts as in Figure 5.

Figure 5

For a very small wavelength (λ) such as is found with visible light, we can consider the geometry of this quasi-rhombus to be that of an actual rhombus. Thus, the separation of hyperbolas is the length of diagonal DF, 0.5λ/sinα.

The situation changes (hyperboloids become elipsoids and sines become cosines) as the direction of light from the sources changes, but fringe separation always satisfies the equation:

For my purposes, both point sources are exactly as I described previously.

The situation where each light source produces a beam instead of a spherical wave is in some sense the same as a ray from each wavefront intersecting (See Figure 6). Instead of quasi-rhombi forming, exact rhombi form and the equations are exactly the same.

Figure 6

Left: Two points emitting spherical waves. Right: Two coherent collimated beams intersecting; exact rhombi are observable. Notice that if one were to zoom in on the two rays in the spherical diagram, the case of two beams would be apparent.

 Thomas Mittiga Summer 2012 Laser Teaching Center Stony Brook University