# Bandwidth Density Analysis of Coded Free-Space Optical Interconnects
## 3 System Model
For example, the amount of \(P_{s}\) received by PD9 shown in Figure 1c from VCSEL9 shown in Figure 1b is computed by integrating the two-dimensional normalized optical intensity \((I)\) at the surface of photodetector 9, which is placed at a distance \(D\) from the transmitter. Assuming that the operating mode is the transverse electromagnetic \(mn\) mode (TEM\({}_{mn}\)), where \(m\) is an integer that represents the radial mode order and \(n\) is another integer that represents the angular mode order, then \(P_{s}\) can be formulated as  
\[P_{s,mn}=\iint\limits_{PD9}I_{mn}(r,\theta,D)rdrd\theta=\int_{0}^{2\pi}\int_{ 0}^{\Delta/2}I_{mn}(r,\theta,D)rdrd\theta. \tag{3}\]  
As in [31], the optical intensity \((I)\) of an operating mode TEM\({}_{mn}\) at a plane located at \(Z=D\) from a transmitter can be written as  
\[ \begin{split} I_{mn}(r,\theta,D)&=\frac{4m!}{\pi \times W_{D}^{2}\times(m+n)!\times(\delta_{0n}+1)}\times\left(\frac{\sqrt{2}r} {W_{D}}\right)^{2n}\times\\ &\left(L_{m}^{n}\left(\frac{2r^{2}}{W_{D}^{2}}\right)\right)^{2} \times\exp\left(\frac{-2r^{2}}{W_{D}^{2}}\right)\times\cos^{2}(n\theta),\end{split}  \tag{4}\]  
where \(W_{D}\) is the spot size of the laser beam at the PD array located at a distance \(D\) from the VCSEL array and is given by [31]  
\[W_{D}=W_{i}\sqrt{\left(\frac{2\pi\times\lambda\times D}{A^{2}}\right)^{2}+1},\]where \(W_{i}\) is the beam waist at the transmitting laser, which is equal to \(X/(\sqrt{2}\pi)\), and \(\lambda\) is the wavelength of operation. The  
\[\delta_{0n}= \begin{cases}1,&n=1\\ 0,&\text{otherwise}\end{cases} \]  
and \(L_{m}^{n}(\alpha)\) is the Laguerre polynomial of order \((m,n)\) and is defined as [32]  
\[L_{m}^{n}(\alpha)=\frac{\alpha^{-n}}{m!}\times\exp(\alpha)\times\frac{d^{m}}{ d\alpha^{m}}\big{(}\alpha^{m+n}\times\exp(-\alpha)\big{)},\]  
Herein, we consider a laser beam with four modes of operation, which are the TEM\({}_{00}\), TEM\({}_{01}\), TEM\({}_{10}\), and TEM\({}_{20}\) modes. In addition, we assume that the coupling among the laser modes is negligible. The normalized optical intensity of each mode, respectively, can be written as  
\[I_{00}=\frac{2}{\pi\times W_{D}^{2}}\times\exp\Bigg{(}\frac{-2r^{2}}{W_{D}^{2} }\Bigg{)}\times\Bigg{(}L_{0}^{0}\Bigg{(}\frac{2r^{2}}{W_{D}^{2}}\Bigg{)} \Bigg{)}^{2}, \tag{5}\]  
\[I_{01}=\frac{8r^{2}}{\pi\times W_{D}^{4}}\times\exp\Bigg{(}\frac{-2r^{2}}{W_{ D}^{2}}\Bigg{)}\times\cos(\theta)\times\Bigg{(}L_{0}^{1}\Bigg{(}\frac{2r^{2}}{W_{ D}^{2}}\Bigg{)}\Bigg{)}^{2}, \tag{6}\]  
\[I_{10}=\frac{2}{\pi\times W_{D}^{2}}\times\exp\Bigg{(}\frac{-2r^{2}}{W_{D}^{2} }\Bigg{)}\times\Bigg{(}L_{1}^{0}\Bigg{(}\frac{2r^{2}}{W_{D}^{2}}\Bigg{)} \Bigg{)}^{2}, \tag{7}\]  
and  
\[I_{20}=\frac{2}{\pi\times W_{D}^{2}}\times\exp\Bigg{(}\frac{-2r^{2}}{W_{D}^{2} }\Bigg{)}\times\Bigg{(}L_{2}^{0}\Bigg{(}\frac{2r^{2}}{W_{D}^{2}}\Bigg{)} \Bigg{)}^{2}, \tag{8}\]  
where \(L_{0}^{0}(\alpha)=1\), \(L_{0}^{1}(\alpha)=1\), \(L_{1}^{0}(\alpha)=1-\alpha\), and \(L_{2}^{0}(\alpha)=1-2\alpha+(1/2)\alpha^{2}\). Upon substituting the optical intensities found in (5)-(8) into (3), we can find the \(P_{s}\) received by the intended photodetector, which is in this case PD9, due to each mode separately. The crosstalk noise (\(N\)) shown in Figure 1c can be evaluated by finding the amount of incident power on the photodetectors that are not intended to receive the power from VCSEL9. The noise of the fundamental mode TEM\({}_{00}\) is defined as  
\[ \begin{split} N_{00}&=\frac{2}{\pi\times W_{D}^{2 }}\sum\limits_{j=1}^{8}\iint\limits_{PD_{j}}\exp\Bigg{(}\frac{-2r^{2}}{W_{D}^{ 2}}\Bigg{)}rdrd\theta\\ &=\frac{4\times 2}{\pi\times W_{D}^{2}}\int_{0}^{2\pi}\int_{0}^{ \Lambda/2}\exp\Bigg{(}\frac{-2r_{1}^{2}}{W_{D}^{2}}\Bigg{)}rdrd\theta\\ &+\frac{4\times 2}{\pi\times W_{D}^{2}}\int_{0}^{2\pi}\int_{0}^{ \Lambda/2}\exp\Bigg{(}\frac{-2r_{2}^{2}}{W_{D}^{2}}\Bigg{)}rdrd\theta.\end{split}  \tag{9}\]  
where \(r_{1}^{2}=(r\cos(\theta)-\Delta)^{2}+(r\sin(\theta))^{2}\) and \(r_{2}^{2}=(r\cos(\theta)-\Delta)^{2}+(r\sin(\theta)+\Delta)^{2}\).