Theory of the collimator

The collimator consists of four rectangular mirrors, positioned in two pairs of two mirrors opposite to each other (see fig. 1.3).

Figure 1.3: Collimator with light coupled in at the back

In figure 1.4 a one-dimensional collimator is shown. The two laser beams enter the pair of mirrors under an angle of $\alpha_0$ with the plane perpendicular to the mirrors at the back side of the collimator from above and below. Both lasers have the same angle of incidence $\alpha_0$ and the same frequency, which makes the one-dimensional collimator symmetric in the plane between the two mirrors.

Figure 1.4: One-dimensional collimator with light coupled in at the back

If an atom moves through the collimator, it will interact with the light coming both from above and from below. An atom with an upward transverse velocity component will, due to the Doppler effect, be more resonant with the laser beam coming from the upper mirror, and will absorb more photons from this laser beam than from the beam coming from the lower mirror. The transverse velocity will decrease due to this net momentum transfer. When the atom is moving parallel to the centerline of the collimator, it absorbs the same number of photons from the upper and the lower mirror because the Doppler shift with both lasers is the same (the collimator is symmetric around the centerline), and both lasers have the same detuning to the red. The atom will not receive a net momentum transfer in the direction perpendicular to the center plane of the collimator. Since the transverse velocity of the atoms in the collimator will become smaller as the atoms go through the collimator, the atoms will shift out of resonance due to change in Doppler-shift (Eq. 1.5). To keep the atoms in resonance over the complete length of the collimator, a small angle between the two mirrors is introduced, which curves the wave front of the laser light. If the angle $\gamma$ between the two mirrors is chosen well, the change of the direction of the velocity vector $\vec v$ is in phase with the change of the wave vector $\vec k$, so that $\vec k \cdot \vec v$ is constant. If an atom leaves the source under an angle bigger than $\beta_{\rm max}$, the capture angle, the decrease of the transverse velocity is not fast enough to keep up with the change of angle of the laser light, and the atoms will have a nonzero transverse velocity when they leave the collimator. These atoms will not reach the channeltron at the end of the setup, and are lost for our experiments.