## 3 OLF Transmission System Case Study  
In this paper, we devised a system that uses OLF as an array of transmitting and receiving slots for free space and underwater communication systems. This technique will reduce the cost of implementing multiple transmitters at each location. The OLF transmission system includes a light transmitter/receiver, OLF, and a termination, as shown in Figure 6.  
The transmitter has \(\mathbf{N}\) slots and the light power at the input of the \(\mathbf{n}\)-th slot is given by:  
\[P_{in}^{n}=P_{t}-L_{0}-\Big{[}\sum_{i=1}^{n-1}\Big{(}\gamma z+L_{b}^{i}+P_{out }^{i}\Big{)}\Big{]}, \tag{30}\]  
where \(\gamma\) is the fiber attenuation in dB for a straight fiber per unit length, Pt is the transmitted light power, L0 is the loss in the first section of the OLF, z is the distance between slots, \(L_{b}^{i}\) is the core loss in the ith slot given in Equation (26) and \(P_{out}^{i}\) is the leaked light power from the ith slot given in Equation (23). The radiated light power from the nth slot can be written as:  
\[P_{out}^{n}=\eta_{n}(P_{in}^{n}-L_{b}^{n}). \tag{31}\]  
Here, we add subscript for \(\eta_{n}\) to allow for different efficiencies to be used for each slot. Figure 6: OLF as MIMO Transmission System. In order to control the transmitted power at a specific slot, we use a different bending radius, such that the closer the slot to the source, the larger the bend radius is chosen. The receiver has \(M\) slots that combine the received light into the detector. This structure forms a MIMO system with received light signal written as:  
\[\mathbf{y}=H\mathbf{x}+\mathbf{z}, \tag{32}\]  
where  
\[y= \begin{bmatrix}y_{1}\\ \vdots\\ y_{M}\end{bmatrix} ,\;x= \begin{bmatrix}x_{1}\\ \vdots\\ x_{N}\end{bmatrix} ,\;z= \begin{bmatrix}z_{1}\\ \vdots\\ z_{M}\end{bmatrix} \text{and}\;H= \begin{bmatrix}h_{11}&\ldots&h_{1N}\\ \vdots&\ddots&\vdots\\ h_{M1}&\ldots&h_{MN}\end{bmatrix} , \tag{33}\]  
\(\mathbf{z}\) is an AWGN and the transmitted signal \(\mathbf{x}_{i}\) is given by:  
\[\mathbf{x}_{i}=\sqrt{P_{out}^{u}}\mathbf{s}(\mathbf{t}), \tag{34}\]  
where s(t) is the message signal and the received signal \(\mathbf{y}_{\text{j}}\) is given by:  
\[\mathbf{y}_{j}=\sum_{i=1}^{N}\mathbf{h}_{ji}\sqrt{P_{out}^{u}}\mathbf{s}(\mathbf{t}-\mathbf{\tau} _{i})+z_{j}, \tag{35}\]  
where \(\mathbf{\tau}_{i}\) is the propagation delay in transmitting OLF. The received signal can be rewritten as:  
\[\mathbf{y}_{j}=\sum_{i=1}^{N}\mathbf{h}_{ji}\sqrt{\eta_{n}\big{(}P_{in}^{n}-L_{bj}^{n} \big{)}}\mathbf{s}(\mathbf{t}-\mathbf{\tau}_{i})+z_{j}. \tag{36}\]  
The combined observed signal at the detector is given by:  
\[\mathbf{y}_{r}=\sum_{j=1}^{M}\bigg{(}\sum_{i=1}^{N}\mathbf{h}_{ji}\sqrt{\eta_{n}\big{(} P_{in}^{n}-L_{bj}^{n}\big{)}}\mathbf{s}\big{(}\mathbf{t}-\mathbf{\tau}_{i}-\mathbf{\delta}_{j} \big{)}+z_{j}\bigg{)}, \tag{37}\]  
where \(\mathbf{\delta}_{j}\) is the propagation delay in receiving OLF. We can design the receiving slots to be separated to compensate for the transmission delay in fixed scenarios, such as:  
\[\mathbf{t}_{0}=\mathbf{\tau}_{i}-\mathbf{\delta}_{j}\forall\mathbf{j},\mathbf{i}. \tag{38}\]  
This led to:  
\[\mathbf{y}_{r}=\mathbf{s}(\mathbf{t}-\mathbf{t}_{0})\!\!\sum_{j=1}^{M}\!\bigg{(}\sum_{i=1}^{N }\mathbf{h}_{ji}\sqrt{\eta_{n}\big{(}P_{in}^{n}-L_{bj}^{n}\big{)}}+z_{j}\bigg{)}. \tag{39}\]  
Since a different coupling angle may be obtained at each receiving slot, the received signal at the receiver detector may be averaged. This effect can be modeled as a variation in the channel coefficient \(\mathbf{h}_{ji}\), such that the slots outside the acceptance cone (depending on the distance between transmitter and receiver) will have \(\mathbf{h}_{ji}=0\). In this case, the OLF transmission system forms a MIMO system with diversity combiner at the receiver side [21].