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other way. The reflected light goes vertically up the page, reflects from end mirror 1, which is at a small angle so the reflected beam does not quite go right back along the same path. The reflected beam goes down the page and part of it goes right through the beam-splitting mirror. (Part of this beam reflects from the beam splitter, but we are not concerned with this portion.) This path is leg 1 of the interferometer. The 50% of the original beam that goes through the beam splitter hits end mirror 2, which is also at a small angle. This reflected beam travels back to the left, almost retracing its original path. It reflects from the beam splitter. (The portion that goes through the beam splitter is unimportant for our considerations.) The reflected portion heads down the page. This path is leg 2 of the interferometer. The result is that the two beams, one that traveled leg 1 and one that traveled leg 2, come together after traveling the same distance and cross at a small angle in the “overlap region” shown by the circle in Figure 3.4. This crossing of the light waves is like the crossing of the sound waves in Davies Symphony Hall that caused the interference problems.

In Figure 3.4 the light beams are drawn as lines, but in any real experiment the beams have a width. The x direction shown in the figure is perpendicular to the bisector of the angle (the line that splits the angle) made by the crossing beams. Since the angle is small, the x direction is basically perpendicular to the propagation direction of the beams, and in the figure it is the horizontal direction. A blowup of what is seen along the x direction in the overlap region is shown in the lower right portion of the figure. In the graph the vertical axis is the intensity of the light, I, and the horizontal axis is the position along x. Because the beams cross at a small angle, the phase relationship between them varies along the x direction, and there are alternating regions of constructive and destructive interference. The intensity of the light varies from a maximum value to zero back to the maximum, and again to zero, and so on. The crossed light waves form regions of constructive and destructive interference. At the intensity maxima, the light waves are in phase (0°—see Figure 3.2), and they add constructively to give increased amplitude. At the zeros of intensity, the light waves are 180° out of phase (see Figure 3.3), and they add destructively, to exactly cancel. This pattern can be observed by placing a piece of photographic film or a digital camera in the overlap region to measure the intensity at the different points along the x direction.

For a small angle, the fringe spacing, that is, the spacing, d, between a pair of intensity peaks or nulls is given by d = λ/θ, where λ is the wavelength of light, and θ is the angle between the beams in radians (1 radian = 57.3 degrees). If 700 nm red light is used, and the angle between the beams is 1°, the fringe spacing is 40 μm or 1.6 thousandths of an inch. These fringes can be seen with film or a good digital camera. If the angle is 0.1°, the fringe spacing is 0.4 mm, which you can see by eye. If the angle is 0.01° (an exceedingly small angle), the fringe spacing is 4 mm (about a sixth of an inch), which you can easily see by eye. To have 4 mm fringes, the beams that cross must be much larger in diameter than 4 mm.

As discussed, in the classical description, light is an electromagnetic wave, and the intensity is proportional to the square of the electric field amplitude (size of the wave in Figure 3.1). In the following, we are not going to worry about units. By including a lot of constants, the units in the following all work out, but they are unimportant for our purposes here. Take the electric field in one of the beams in one leg of the interferometer to have an amplitude of 10. Then the intensity is 100 (102 = 100 = 10×10). The other beam also has I = 100. These are the intensities when we are not observing in the beam overlap region. When the beams are separated, the sum of their intensities is 200.