A51 How Do Interferometers Work

Light is a wave, and two waves with the same frequency add together by summing amplitudes A and a. If the waves are moving in opposite directions, the result is a standing wave, like a guitar string. If the waves are moving in the same direction, the result is a wave that can have an amplitude as much as A + a or as little as A — a, depending on the respective phases. (See Fig. A-11.) This wave is the result of interference.

One would seldom expect two waves to line up in direction well enough to see visible, reasonably stationary, interference effects in nature. Indeed, this effect is almost unknown using diffuse white-light sources. The exception can be found in the case of thin films.

With thin films, an incident light beam is reflected twice in quick succession by two very nearly parallel surfaces. The most common way of achieving parallel surfaces in nature is with liquids. Thus, our first experience with thin-film interference is usually found in the colors of a soap bubble. Thin-film interference also occurs in some insect wings, bird feathers, and mollusk shells. It even has a name that was coined before physical understanding was achieved, iridescence, after Iris, ancient goddess of the rainbow.

The color in a soap bubble is caused by the reflection from the outside of the bubble interfering with the reflection from the inside. (If you want to try an experiment, the colors are easier to see on a soap film still held in the loop.) The strength of the reflection from the air-to-soapy-water interface is about the same as the strength from the soapy-water-to-air interface, so the soap bubble interference should have high contrast. A color is favored at those frequencies for which one reflection is out of phase with the other by an even multiple of the color's wavelength.

What happens to those colors for which the conditions aren't favorable? A wave contains energy, and that energy doesn't go away when the wave is canceled. If one path has been blocked, it goes in another direction. The light passes straight through.

The reason we see colors at all is because soap bubbles are so thin. If the surfaces had two reflections with a phase delay amounting to 1000 wavelengths of blue light, it would be 999 wavelengths of not-so-blue light, 998 even-less-blue, and so forth. So many frequencies would be favored that the reflected light would be colorless again. The spectral response would resemble a pocket comb, with many selected frequencies jutting up. However, our eyes are not sensitive to such fine structure and we would see only white light.

We also don't see bubble colors in ordinary windows for the same reason. In fact, this color effect is only seen at a certain stage of the ephemeral life of a bubble. A fresh soap film is thick, so one sees many frequencies adding together to a rough approximation of white. As it evaporates, it passes through an iridescent stage where the colors depend on the local thickness. Stripes of red, yellow, green, blue, and purple appear briefly. Then comes a stage when all colors of reflection are equally discouraged. The bubble is colored a pale white or a white tinged with yellow once again.

The bubble is still behaving as a barrier when it appears pale yellow, but soon it will get so thin that the wavelength of light is incapable of sensing the interface at all. If the bubble is long-lived, careful inspection will reveal an area of the film where it is actually invisible. The layer is so thin that it cannot materially impede the forward progress of the wave.

If, instead of using white light, we illuminate a fresh film with one color only, what would we see? As the film dries up, it goes from invisible to visible at any given location, depending on whether that wavelength is favored to reflect. The bubble looks tattered and frayed depending on how fast it is drying at a certain location, and onlookers are able to tell it is still whole only by looking at the edges.

The interference principles exemplified in a soap bubble can help us to understand how interferometers work. If we can generate one perfect wavefront, or at least a known wavefront, we can use it to interfere with the unknown wavefront of our telescopes. If the wavefronts are identical, the interference will be the same over the whole aperture.

A simple example is given to illustrate the ideas behind this principle. Let's say we wish to test a flat used for a diagonal mirror. We set it up in the configuration of Fig. A-12 with a known optical flat, illuminate it from above with fairly pure yellow-green light, and get far enough away in a vertical direction so that perspective does not cause any effects of its own.4 Here the thin film is not glass or soapy water, but the wedge of trapped air.

If the diagonal is flat and the blocks (actually, thin strips of paper) are of the same thickness, one would detect a uniform brightness depending on the relative phases of the two reflections. Such precision is almost impossible to achieve by luck alone. More likely one will see, as in Fig. A-12a, a thin film of air amounting to N wavelengths on one side to N + n wavelengths on the other. If we follow the course of light as it passes through the glass and reflects off the two layers, we see that the two reflections have a phase difference of 2N wavelengths on the short side of the wedge and 2N + 2n wavelengths on the other side (to within a constant). One can subtract the 2N phase difference since it is the same over the whole piece and concentrate on the 2n. From the discussion with soap bubbles above, the expectation is that light stripes should appear at the locations where the interfering phases appear separated by 0, 1 wave, 2 waves, etc. and dark bars should appear where the phases are separated by :/2 wave, 3/2 wave, 5/2 wave, etc. Because the phase difference results from reflection, which doubles the distances light propagates, a bar occurs for every increase of a half wavelength in the separation of the plates. Furthermore, the bars should be straight, as in Fig. A-12b, because the separation varies only in the direction of offset.

If the diagonal is not flat, however, the bars curve as in Fig. A-12c. We may easily view the dark bars (called "fringes") as contours of the tested piece separated by a half wavelength of light. By stretching a dark thread across the test setup, we can even estimate the amount of curvature. This case is out of line by not quite half a fringe so the tested piece is smoothly curved by a little less than :/4 wavelength.

This test can easily be generalized to other shapes. If the known surface is a convex paraboloid, a Newtonian telescope maker just needs to test and polish until the bars are straight. The truth is, no one works this way because all of the pieces made would need to have precisely the same focal

4 In practice, such a test is conducted with a mild condensing lens just above the two pieces

(Fizeau's interferometer). Such an arrangement concentrates the returning light and rectifies the incident diffuse beam so that the optician can draw close.

unknown surface unknown surface

Fig. A-12. Interference from a thin Rim of air is used to test the quality of a Bat surface.

length (for no good reason). An optician needs a powerful incentive before incurring the expense of a curved testing piece. A camera manufacturer who may have a tight tolerance on thousands of spherical parts has a valid justification to make one. A telescope tester has easier roads to follow.

Most easily interpreted interferometer tests have one feature in common. They always start with a single beam divided into two parts. One becomes the "reference" beam, and the other beam is transmitted through the optics. The reference wavefront must be conditioned into the same converging shape as it would attain had it passed through perfect optics. Finally, the beams must be brought together with only a few wavelengths of tilt. After all of these diversions, the two beams should have roughly the same amplitude in order to provide good contrast between the bright fringes and the dark ones. For example, if the lower surface were aluminized in the example above, the fringes would be very weak. One interfering beam would originate from the transparent glass (maybe 4% - 5% reflectivity) and the other would bounce strongly off the aluminum coating (92% reflectivity). The coating should be removed to use interferometry in this case.

Another condition that must be maintained is approximately equal path length. The light source ultimately consists of a molecular or atomic transition, with a finite line width, implying through the Heisenberg indeterminacy principle, a localization of the light beam. A typical stream electromagnetic energy originating from a single transition is a wave packet about

0.5 m long. If the beam is split, half to enter a compact "perfect" conditioner (where it is immediately returned) and the other half continuing on to the optics, then these two path lengths must not differ by more than 0.5 m if the two beams are to successfully interfere. This situation is eased considerably by using a laser, which has a longer coherence length.

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