# Making Fast Light

As described in the Fast-Light Tutorial, fast light is generated by creating a fast group velocity vg. The group velocity is given by vg = c / (n + ω dn/dω) where n is the index of refraction, c is the speed of light in vacuum, and ω is frequency. This should be evaluated at the central pulse frequency ω0.

From this equation, we see that we can make vg very large by making the denominator (n + ω dn/dω) very small. We can do this by making the derivative dn/dω large and negative. So, all we have to do to make fast light is find a material for which the index n(ω) is a decreasing function of ω.

## An Absorption Line

 A fast-light pulse traveling through an absorber, from the Budker group tutorial. Notice that the output pulse is smaller than the input pulse even though the output pulse has been blown up 10,000,000,000,000,000,000,000,000,000,000,000 times! That's only because this example assumes lots of absorption, though. The dashed line represents what the pulse would do if it traveled through vacuum. Note that inside the medium, the pulse actually travels backward.

It turns out that it's not hard to find such materials. All materials have some frequency ranges where dn/dω is negative. For the most part, materials have anomalous dispersion (that's what it's called when dn/dω < 0) in frequency regions where they absorb light. This is true because of the relationship between the index of refraction n and the absorption coefficient α. This relationship is described by the Kramers-Kronig relations.

It has been known for more than a century that the group velocity becomes fast in absorption lines (regions of absorption). In fact, it was this knowledge that led to the controversy when Einstein developed his theory of relativity about a century ago. Until then, it was believed that information propagated at the group velocity. Sommerfeld and Brillouin quickly showed theoretically that pulses become very distorted (and no longer move at the group velocity) under such conditions. However, they were using square pulse, not smooth pulses!

In the early 1970's, Garrett and McCumber showed (again, theoretically) that smooth pulses can propagate undistorted and with fast group velocities faster than c. Their predictions were verified in the early 80's by Chu and Wong, and again more clearly by Ségard and Macke.

The problem with using an absorption line is, of course, the absorption. The faster you make the pulse go, the more it is absorbed! Fortunately, another technique was discovered that does not require absorption.

## A Gain Doublet

 A Raman gain level diagram. Blue represents the probe and red represents the pump. A photon of each type comes in and two probe photons leave. The atom also changes from one of the lower states to the other.

The important part of the absorption line is not really the absorption itself, but the fact that the absorption decreases on both sides. This requirement is also satisfied by the region between two gain lines. Gain and absorption are really opposite effects, so in some sense, more gain is equivalent to less absorption.

This idea was first realized by Steinberg and Chiao in 1994 and was implemented in 2000 by Wang, Kuzmich, and Dogariu. Using this technique, it is possible to create fast group velocities without absorption. In fact, it usually results in a little gain!

 A photograph of a potassium vapor cell. You can see some solid potassium coating the cell wall near the center. Click for a high-resolution version.

We use (as did Wang et al.) a process called Raman gain to create our gain lines. This process involves the combination of a raw medium (in our case, a potassium vapor in a cell like the one shown to the left) and a strong laser beam (called the pump) to amplify a weak beam (called the probe). The process involves the creation of a new probe photon by both a change in atomic state and the annihilation of a pump photon. This process is represented graphically in the figure to the right.

The Raman gain process has several features that make it appealing for this application. We can create two gain lines by simply injecting two pump beams. For each of these, the frequency of the pump beam determines the frequency of the gain. The intensity (or brightness) of the pump beam determines the amount of gain. So, by adjusting the frequencies and intensities of the two pump beams, we can create a gain doublet with nearly arbitrary separation and strength.

## The two-zone technique

 The experimental two-zone setup. Each pump interacts with the probe only in one region.

Unfortunately, we discovered that when two strong pumps are used, they interact with each other via the atoms in a very complex way. This behavior interferes dramatically with the fast light process.

When we were discussing this problem with Lijun Wang, he suggested that we use two pump beams, but not in the same region. The Raman process that we are interested in is linear in the probe, which means that having the probe interact with the two pumps separately is the same as having it interact with them simultaneously. Based on his suggestion, we came up with the following scheme.

## Generating pulses

 An acousto-optic modulator (AOM). Incoming light interacts with the sound waves in the crystal to change the light's frequency and direction.

With the medium prepared, there still remains the task of generating optical pulses. To do this, we use a device called an acousto-optic modulator, or AOM. An AOM is a very versatile device that can be used for many different purposes; changing the intensity of a light beam, changing the frequency of a light beam. or changing the direction of a light beam. We use ours for the first two.

 A photograph of the AOM that we use. The scale is set by the hand holding the base. Click for a close-up.

AOMs work by creating sound waves in a crystal. When light propagates through the crystal, it can interact with the sound waves. Some of the light is changed in frequency and direction of propagation. The amounts by which the frequency and direction change are related to the frequency of the sound waves. Obviously, when the sound wave is turned off, this effect stops. So, by varying the volume of the sound field, we can vary the intensity of the deflected beam. By changing the frequency of the sound field, we can vary the frequency of the deflected beam. This is important to us because the frequency of the probe pulses must be between the gain frequencies.

In our experiment, we generate our pulses by varying the power on the sound waves. We can control the precise shape of the pulses by using an arbitrary waveform generator. This allows us to program into the computer any shape we want and have an optical pulse with that shape emerge from the AOM!