# Active Control Loop of the BOROWIEC SLR Space Debris Tracking System
## 2 Satellite Laser Ranging Process
### Laser Beam Propagation
#### 2.3.1 Beam Divergence  
Each pulsed laser source is defined by the physical parameters, among which we distinguish: pulse energy (uJ, mJ), pulse width (ps, ns), repetition (Hz, kHz), beam divergence (mrad). With reference to [PERSON]'s radar link equation (Equation (1)) [19], the probability of registering a photon scattered from an orbital object strongly depends on the energy of the laser pulse and laser beam divergence. This dependence is as follows:  
\[n_{pe}=\eta_{q}\bigg{(}E_{T}\frac{\lambda}{hc}\bigg{)}\eta_{t}G_{t}\sigma \bigg{(}\frac{1}{4\pi R^{2}}\bigg{)}^{2}A_{r}\eta_{T}T_{a}^{2}T_{c}^{2}, \tag{1}\]  
where \(\eta_{q}\) is the detector quantum efficiency, \(E_{T}\) is the laser pulse energy, \(\lambda\) is the laser wavelength, \(h\) is [PERSON]'s constant, \(c\) is the velocity of light in a vacuum, \(\eta_{t}\) is the transmit optics' efficiency, \(G_{t}\) is the transmit gain, \(\sigma\) is the orbital object cross-section, \(R\) is the slant range to the object, \(A_{r}\) is the effective area of the telescope receive aperture, \(\eta_{r}\) is the efficiency of the receive optics, \(T_{a}\) is the one-way atmospheric transmission, and \(T_{c}\) is the one-way transmissivity of cirrus clouds (when present). The transmitter gain \(G_{t}\) of the system with the use of a Gaussian laser beam is defined as:  
\[G_{t}=\frac{8}{\theta_{t}^{2}}exp\bigg{[}-2\bigg{(}\frac{\theta}{\theta_{t}} \bigg{)}^{2}\bigg{]}, \tag{2}\]  
where \(\theta_{t}\) is half of the divergence angle between the axis of the laser beam and the area of intensity \(1/e^{2}\) and \(\theta\) is the error of the positioning uncertainty of the emitted beam. The geometrical divergence defines the degree of enlargement of the cross-sectional area of the laser beam with the distance and is expressed in mrad. Laser measurements are made for orbital objects from the regions from LEO to GEO, so the distance between the SLR station and the orbital object ranges from about 350 km to 42,000 km. The aim of the measurement process is to propagate the laser beam between the ground station and the orbital object. The laser source emits a laser beam with given physical parameters and a known starting diameter, which is usually a few millimeters (<1 cm) in size. Taking into account the large distance separating the ground station and the orbital object (>350 km), the geometric divergence parameter of the beam determines its cross-section size on the  
Figure 5: Block diagram of the laser ranging process. orbit. Standard laser sources for SLR needs are characterized by a geometric divergence of the beam at the level of 0.5 mrad-1.0 mrad (1.0 mrad = 206.26 arcsec). Assuming a 1 cm diameter of the laser beam with a divergence of 1 mrad, for an orbital object distance of 350 km, the orbital beam's cross-sectional area is 700 m (for a 42,000 km distance, it is 84 km). This fact is of no great importance when tracking of cooperative objects with low-energy laser emission sources. The energy density that is deposited at a given orbital altitude in this case is not critical. However, in the case of tracking of uncooperative objects in order to obtain a photon event in the form of scattering a portion of light from the object and its subsequent detection, a high energy density deposited on the object's surface is required. Therefore, it forces the use of systems correcting the divergence of the laser beam. Such a system is classically composed of a set of lenses for which the distance relation between them affects the final beam divergence. In this way, it is possible to correct the divergence parameter to a minimum value of \(\theta\in(1^{\prime\prime},5^{\prime\prime})\). For example, when using the divergence correction system, assuming a 1 cm diameter of the output laser beam with a divergence of 0.02 mrad, for a distance of the orbital object of 350 km, the cross-sectional size of the beam on the orbit is 14 m (for 42,000 km, it is 1.6 km, respectively).