Accuracy

In this section we want to examine which factors influence the accuracy of the system. For this, we have to examine errors that can occur during the measurement of $t_\mathrm{beam}$ and $t_\mathrm{turn}$. From a measurement point of view the two are identical, since they are both an amount of time elapsed between two beam sightings. Therefore we will use $t$ as a genus for the two and $\Delta t$ as the absolute error of $t$. The following list contains possible causes for measurement errors:

• Vibrations: Due to their fast rotation, the mirrors and thus the reflected beams suffer from small vibrations, resulting in a small angle $\epsilon$ of beam spread, which is about $0.05^\circ$ in our prototype. Assuming $\sin \epsilon = \epsilon$ (since $\epsilon \approx 0$), the resulting error is $\Delta t \le t_\mathrm{turn} \frac{\epsilon \sqrt{d^2+h^2}}{2 \pi d}$ for an observer located at distance $d$ from the lighthouse rotation axis and at height $h$ over the lighthouse center.
• Lower bound on time $t_\mathrm{mirror}$ for one mirror rotation: Since we can measure elapsed time only when the rotating laser beam hits the photo detector, the accuracy of $t_\mathrm{beam}$ and $t_\mathrm{turn}$ is limited by the speed of the rotating mirrors (i.e., $t_\mathrm{mirror}$). The resulting error is $\Delta t \le t_\mathrm{mirror}$.
• Flutter of platform rotation: The relative error in lighthouse rotation speed $\rho_\mathrm{lh}$ causes an error in $t$. $\rho_\mathrm{lh}$ is mainly caused by the flutter of the motor driving the lighthouse platform. The motor used in our prototype has a flutter of 0.1%. The resulting error is $\Delta t \le t \rho_\mathrm{lh}$.
• Variable delays: There is a variable time offset between the laser beam hitting the photo detector and the interrupt handler reading the clock. On the path from the photo detector to the interrupt handler are many sources of variable delay, such as hardware and interrupt latency. The actual value of this error pretty much depends on what is currently happening on the computer, but is typically small compared to the other sources of errors.
• Clock resolution: The minimum time unit $t_\mathrm{min}$ that can be measured by the clock limits the time resolution for measurement of $t$. The Linux laptop we used has $t_\mathrm{clockres}=1\mu$s. On the ATMEL we used a 16-bit counter to implement a clock with $t_\mathrm{clockres}=50\mu$s. The resulting error is $\Delta t \le t_\mathrm{clockres}$.
• Clock drift: The maximum relative error $\rho_\mathrm{clock}$ in the clock rate also causes an error in $t$. A typical value is $\rho_\mathrm{clock}=10^{-5}$ both on Linux and the ATMEL. The resulting error is $\Delta t \le t \rho_\mathrm{clock}$.

In our prototype systems, the clearly dominating errors are caused by vibrations, limited $t_\mathrm{mirror}$, and flutter of platform rotation. The use of deflectable MEMS mirrors can both drastically reduce vibrations and $t_\mathrm{mirror}$. The flutter of platform rotation can be reduced to about 0.01% by using electronically stabilized motors as used, for example, in turntable drives. By this, we expect a possible reduction of $\Delta t$ by a factor of about 10.

Note, however, that the errors resulting from these three main sources can be modeled by a Gaussian noise source. This means that averaging over a large number of measurements helps to reduce the error.