Preparing the angles of the collimator

The collimator mirrors are 20 cm long, have a width of 3.8 cm, and are separated 6 cm from each other. Each mirror is placed in the frame of the collimator leaning on 4 points (see figure 2.2). The mirror is pushed against these four supporting points with a screw. The frame of the collimator is designed in such way that each pair of mirrors will be almost perfectly parallel to each other.

Figure 2.2: Attachment of the mirror on the frame of the collimator

To obtain an angle between the two mirrors we have put pieces of silver foil between the supporting point and the coated side of the mirror. The position of the mirrors of the collimator is very critical. The angle $\gamma$ between the two mirrors is approximately 1.4 mrad, which means that the distance between the two mirrors should change 140 $\mu$m over a distance of 10.6 cm. This corresponds to about 10 to 14 foils. Since the foils are not all the same thickness, it is not easy to adjust the angles. Not only should the angle $\gamma$, needed for the curving of the wavefront be correct, but it is also important not to create an angle $\epsilon$, perpendicular to the angle $\gamma$.

Figure 2.3: The angles $\epsilon$ and $\phi _0$, front view of the collimator

This angle is created when the foils in the upper supporting point are not of the same thickness as the foils on the lower supporting point. The angle $\epsilon$ causes the laser light shining in the collimator to slowly drift upwards or downwards away from the middle of the mirrors. Since the atoms of the source are going through the middle of the collimator, the angle $\epsilon$ will cause a smaller overlap between the light and the atoms, making the collimator less efficient. The angle $\epsilon$ can be changed by placing special silver foils of 2 $\mu$m between the upper/lower two supporting points and the mirror when the light was moving respectively downwards/upwards.

Since the angles $\gamma$ and $\epsilon$ are so small, we need a special way to measure them. In Eq. 1.11, $z$ is given as a function of the number of reflections $n$. This can be rewritten to:

$$z = D n (\alpha_0 - \frac{1}{2}\gamma) - \frac{1}{2} n^2\gamma D$$ (2.3)

Now we shine through the two horizontal mirrors of the collimator with a diode laser, and we make sure the light comes out at the other side of the collimator. A diaphragm is inserted directly behind the diode laser to define a smaller spot of light. We now place a diode in the collimator, which can be translated in the z direction over the centerline of the collimator. This diode is sensitive for the laser light used, and can only detect even reflections, since the diode is turned towards one of the mirrors. The diode is placed on an optical rail, and the position of the maximum of the laser light at different reflections can be determined accurately to a few tenths of a millimeter. We now measure the exact position $z$ of the maximum intensity of the laser beam at different values of $n$ (see fig. 2.4). If we make a quadratic fit ( $a n^2 + b n + c$) to these data, the angle $\gamma$ can be evaluated from the quadratic part:

$$\gamma = \frac{2c}{D}$$ (2.4)

Using this fit we find a value for $\gamma$ of $\gamma = 1.39$ mrad. We can evaluate the angle of incidence $\alpha_0$ from Eq. 2.3 by using the linear term $b n$:

$$\alpha_o = \tan^{-1}{(\frac{b}{D})} - \frac {\gamma}{2}$$ (2.5)

Figure 2.4: Correlation between the number of reflections $n$ and the position $z$ in the collimator

We find a value for $\alpha_0$ of $\alpha_0 = 88$ mrad. With use of this angle $\alpha_0$ we can calculate the distance $D^*$ the light has to travel from one mirror to the opposite mirror.

$$D^* = \frac {D}{\cos{\alpha_0}}$$ (2.6)

With the value we found for the angle $\alpha_0$ we find a value for $D^*$ of 6.0233 cm for the pair of mirrors that cools the horizontal transverse velocity. The values for the pair of mirrors that cools the vertical transverse velocity are $\gamma = 1.38$ mrad, $\alpha_0 = 88.7$ mrad and $D^* = 6.0237$ cm.

To measure the angle $\epsilon$ between the two mirrors, we also measure the vertical displacement $y$ of the maximum of the laser light for various values of $n$. The height of the diode can be adjusted by a micrometer. The way to calculate the angle $\epsilon$ is similar to the method of calculating $\gamma$. Equation 2.3 can be adjusted to:

$$y = D^* n (\phi_0 - \frac{1}{2}\epsilon) - \frac{1}{2} n^2\epsilon D^*$$ (2.7)

with $D^* = D\cos{\alpha_0}$ the distance the light has to travel from one mirror to the other, and $\phi _0$ the angle of incidence as shown in fig. 2.3. By searching the maximum intensity of the laser light in the $y$-direction at the maximum intensity in the $z$-direction for different values of $n$, we can fit these data points to equation 2.7. The angle $\epsilon$ can be calculated from:

$$\epsilon = \frac{2c}{D^*}$$ (2.8)

The values found for the two pairs of mirrors are $\epsilon_{horizontal} = 39 \mu$rad and $\epsilon_{vertical} = 13 \mu$rad.

Figure 2.5: Correlation between the number of reflections n and the position y in the collimator

The angle $\epsilon$ causes the light to drift away from the center of the collimator. It is important to know what the maximum value of this drift can be, since the light and the atomic beam should overlap as much as possible. If the laser light is shone in with an angle $\phi_0 = \frac{1}{2} \epsilon n_{total}$, the angle $\epsilon$ will cause the angle $\phi _0$ to decrease, so that it will be zero after half the total number of reflections, and will leave the collimator with an angle -$\phi _0$ at the same height as it is shone in, following a parabolic path through the collimator. The maximum difference in height is the value at $n = \frac{1}{2} n_{total}$ minus the value at $n = 0$. If we use equation 2.7 to calculate the maximum difference in height, assuming the total number of reflections $n_{total}$ to be 40 (which is more than the actual number of reflections), the shift is 0.45 mm. This is small enough compared to the height of the laser light, which is 1 cm.