# Bandwidth Density Analysis of Coded Free-Space Optical Interconnects
## 3 System Model  
In this section, we provide a design description of the FSOI system and a definition of the bandwidth density (BD) in terms of data rate and channel spacing. The data rate depends mainly on both the optical power received by the intended receiver and the optical crosstalk noise; hence, clear mathematical definitions of both of them are provided. The FSOI system is composed of two arrays--one as a transmitter and the other as a receiver--with a given interconnect distance of D between the arrays; see Figure 1a. The transmitter array contains nine identical transmitters of vertical-cavity-surface-emitting lasers (VCSELs) arranged as demonstrated in Figure 1b. Likewise, the receiver array contains nine identical photodetectors (PDs) arranged as shown in Figure 1c. The design parameters X, Y, and \(\Delta\) marked on the figures represent the VCSEL's and photodetector's diameters and the channel spacing, respectively. In fact, the exact number of channels, the distance between the source and detector arrays, and the other parameters of the FSOI system model depend on the application for which the system would be deployed. For example, in this study, we particularly used nine channels within a rectangular array shape to demonstrate the ability of using the rotational symmetry of the photodetector distribution to simplify the mathematical calculations of the crosstalk noise. From Figure 1c, it is clear that the crosstalk noise received by photodetectors 1, 3, 5, and 7 is the same, and the crosstalk noise received by photodetectors 2, 4, 6, and 8 is also the same. In addition, it should be mentioned that we used rectangular source and detector arrays--not other possibly used shapes such as hexagonal or circular shapes--for many reasons. These include that rectangular arrays are simple and easy to fabricate, align, and scale to any size. Moreover, when it comes to process  
Figure 1: (**a**) Cross-sectional view of the FSOI system model and two-dimensional schematic representations of (**b**) nine-VCSEL array and (**c**) nine-photodetector array. each photodetector in the array, many signal processing and optimization techniques are naturally aligned with rectangular arrays [27; 28; 29]. The proposed FSOI system does not utilize lenses in its design. Actually, using or not using lenses completely depends on the application and the FSOI system design parameters. For example, if we try to employ the system on small printed circuit boards, such as the ones used in smart phones, where the distance between the source and the detector arrays is in the range of millimeters, the laser beam diffraction could be negligible, and using lenses in such cases would be just a waste of resources. However, if the distance between the source and the detector arrays is large, like the one mentioned in [30] where the distance between the arrays is in the range of 3 m, lenses might be used to mitigate the effect of optical interference. The BD of an FSOI channel is defined as the ratio of the data rate (\(R\)) and channel spacing squared, which is written as [19]  
\[\text{BD}=\frac{R}{\Delta^{2}}. \tag{1}\]  
The relationship between the BD and the used SNR as well as the FSOI model parameters can be seen from the definition of \(R\), which is given by [19]  
\[R=\frac{\left(P_{t}\times P_{s}-\text{SNR}\times P_{t}\times N\right)^{2}}{ \left(\text{SNR}\times\text{NEP}\right)^{2}}, \tag{2}\]  
where \(P_{t}\) represents the channel's transmitted power, \(P_{s}\) is the optical power received by the intended receiver, \(N\) is the crosstalk noise, and NEP is the noise equivalent power that characterizes the amplifier thermal noise. From (1) and (2), the BD of an FSOI channel can be determined by calculating \(P_{s}\) and \(N\) for designated \(P_{t}\), NEP, SNR, and \(\Delta\) values.