Range

In this section we want to examine the maximum range, at which observers can still determine their location. This maximum range mainly depends on two issues.

Figure 12: The photo detector must be hit by the laser beam at least twice.

The first of these issues is that the photo receiver has to be hit twice by each of the rotating beams in order for the receiver to identify the lighthouse as explained in Section 4.3.2. Figure 12 depicts this situation. It shows a top view of a lighthouse with only one of the two rotating beams at two points in time $t_1$ and $t_2$. At $t_1$, the beam hits the photo detector at distance $d$ from the lighthouse rotation axis the first time. Then, the mirror does one rotation and hits the photo detector a second time at $t_2$. During $t_2 - t_1$, the lighthouse platform has rotated a bit to the left. $l$ denotes the diameter of the photo detector. Assuming a constant diameter $w$ of the laser beam, the distance $d$ at which the photo detector is hit at least twice is given by the following inequality:

$$d < \frac{l+w}{2 \sin (\pi t_\mathrm{mirror} / t_\mathrm{turn})}$$ (14)

With the values of our prototype system $l=5$mm, $w=3$mm, $t_\mathrm{mirror} = 4$ms, $t_\mathrm{turn} = 60$sec we can achieve a theoretical maximum range of about 14m. This value can be improved by increasing $t_\mathrm{turn}$, by decreasing $t_\mathrm{mirror}$, or by defocusing the lasers a bit, such that there is a small angle of beam spread. However, there are certain limits for each of these possibilities. The angle of beam spread is limited by the sensitivity of the photo detector and the output power of the laser. $t_\mathrm{mirror}$ is limited by the possible maximum speed of the mirrors. With MEMS deflectable mirrors such as the one presented in [7], we can achieve $t_\mathrm{mirror} = 1 / 35kHz = 30\mu{}s$. $t_\mathrm{turn}$ is limited by the frequency of location updates needed by the nodes and thus by the degree of node mobility (see Section 4.5.4).

The second issue that limits the maximum range of the system is the speed of the photo detector. Using COTS technology, the beam has to stay on the photo detector for about $t_\mathrm{photo}=10$ns in order to be detected. Depending on the minimum retention period $t_\mathrm{photo}$ of the laser beam on the photo detector, the maximum distance $d$ is limited according to the following inequality:

$$d < \frac{l+w}{2 \sin (\pi t_\mathrm{photo} / t_\mathrm{mirror})}$$ (15)

With the current values of our prototype $t_\mathrm{photo}=200$ns, $t_\mathrm{mirror} = 5$ms, $l=4$mm, $w=3$mm we can achieve a theoretical maximum range of about 27m, giving us an overall range limit of 14m. Again, this value can be improved by reducing $t_\mathrm{mirror}$ and by defocusing the laser with the same limits as above.

The actually measured maximum range, at which the receiver prototype could still detect the base station is about 11 meters. However, the range can be increased by adjusting certain system parameters. A more elaborate system built using fast deflectable MEMS mirrors with values $l=1$mm, $w=20$mm (due to beam spread), $t_\mathrm{mirror}=1$ms, $t_\mathrm{turn} = 60$sec, and $t_\mathrm{photo}=10$ns, for example, could achieve a theoretical maximum range of about 210m (the minimum obtained from Inequalities 14 and 15). Based on our experience, we would expect a practical maximum range of about 120-140m of a system with these parameters, which approximately equals the maximum communication range of 150m during the day for the Berkeley experiments [19].