How Do You See Interference Between Independent Lasers?

This is adapted from an answer to a question at the Physics Stack Exchange site. The questioner asked:

It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true? How fast should photo-detectors be for observing of the interference of beams from two of the "best available" lasers?

This is a question about the itnerference of light waves, which is traditionally demonstrated via the famous "double slit" experiment, where a single laser is sent through a barrier with two narrow slits cut in it. The light waves passing through the two slits overlap with each other, and produce a characteristic pattern of bright and dark spots, like this image from MIT showing three different lasers going through a double-slit, producing patterns with different spacing:

As I said, this is traditionally done with a single laser and two slits. In principle, you ought to be able to do it with two completely independent lasers of the same wavelength, and see an interference pattern that way. In practice, though, this is very hard to do, as the phase difference between two independent lasers will jump around randomly at a very high rate. At any given instant, the two lasers will interfere with each other to make a nice double-slit pattern, but the positions of the bright and dark spots will shift around randomly, so fast that your eye can't follow them.

In principle, though, you ought to be able to "see" this sort of pattern using an electronic detector system. The question from Stack Exchange is, essentially, what would you need to see this sort of thing? (At least, that's how I interpreted the question...)

The key number for this is time scale for the fluctuations of the phase. The pattern between the lasers would be steady for a time on the order of the "coherence time" of the lasers. A good rule of thumb for finding the coherence time for a fairly ordinary laser is that the coherence length of the laser is roughly equal to the length of the cavity, and the coherence time is the coherence length divided by the speed of light. A typical gas laser of the sort you would readily find on the shelves of your local physics department stockroom has a cavity that is roughly one foot long, which means a coherence time of the time required for light to travel one foot, which is one nanosecond (one of the rare cases where American units are more convenient than SI units).

So, you would want to be able to pick up the pattern in less than a nanosecond, which means your detector needs to be able to handle count rates of at least one gigahertz. That's not too difficult to manage-- you can get photodiodes with a bandwidth of 50-60GHz pretty easily. In order to see a spatial pattern, though, you really want a linear array of these at the very least, and a CCD camera would be even better. You also need to be able to pick up multiple photons in that span, in order to be able to see the pattern, so you can clearly resolve the difference between bright and dark fringes-- I would say you probably want at least 100 photons/pixel in the bright fringes to be able to get decent fringe contrast. This is doable with fairly basic lasers-- a few milliwatts in the red range of wavelengths is about 10¹⁵ photons per second, or about 10⁶ photons per nanosecond, so you can spread that out over a few thousand pixels and still be safe. And, of course, you need a low level of "dark counts" in your detector, so it can readily tell the difference between a bright spot with 100 photons and a dark spot with none.

That's a pretty challenging set of detector requirements. You want, essentially, a CCD detector with single-photon sensitivity and a bandwidth of 100 GHz. I don't think you're going to find that just lying around. Even the fast CCD detectors people use for real-time monitoring of BEC experiments and the like have a frame readout time of a millisecond or so (at least, that's the time between frames in the movies they usually show, and I assume that if they could go faster, they would), and those aren't single-photon sensitive. Single-photon detectors tend to be things like single avalanche photodiodes, which are easily damaged by photon count rates of a few tens of kilohertz. You might be able to construct a detector with these specs, but it would be an extremely difficult problem, and not something that would be worth doing just to see a spatial interference pattern between two independent lasers.

If you can bump the coherence time of the lasers up by a factor of 1000 (which is not a trivial undertaking) then it becomes a little easier to do, but it's still very difficult-- you're looking for a CCD with a frame readout time of a microsecond or so, able to handle photon counts in the megahertz range. Which is still a very hard problem. Another factor of 100-1000 would put you solidly in the region where you could reasonably expect to find a detector that would do the job, but then you're talking coherence times of a tenth of a second or so, and that requires some really exceptional laser design-- we're talking the sort of lasers that they use for optical clock experiments.

This is an interesting problem, though. You will often hear the statement "A photon interferes only with itself" thrown around (I've seen it attributed to both Wigner and Dirac, but I'm not sure which said it first), which leaves open the possibility that photons from two completely independent sources might not be able to interfere with each other. Because of that, people have looked closely at this process, and in fact, the existence of a spatial interference pattern between two independent lasers was verified way back in 1967 by Pfleegor and Mandel. They even did it with single photons.

The Pfleegor and Mandel experiment is pretty clever, as you would expect of an experiment doing this with 1960's technology. It's kind of involved, though, so explaining their trick will wait for another post.