Reflection assisted beam propagation model for obstructed line-of-sight FSO links

Abstract

A non-line-of-sight (NLOS) infra-red reflection based beam propagation model is proposed as a supplement to conventional terrestrial free space optical (FSO) communication system. This ray propagation model lets tactically positioned optical reflectors to smartly exploit the aggregated advantages of mirror characteristics to bridge the existent communication gap between two FSO nodes due to inclined or obstructed line-of-sight view. Additionally, a numerical framework of the proposed system is presented that analytically explores the optical losses induced by harmonic distortions and the resultant beam divergence at the receiver. The impact of the different reflectors on the traversing beam is then investigated through an experimental FSO test-bed set in an outdoor environment in terms of phase shifts, divergence loss, noise margin and maximum achievable link length. Matlab based simulations, based on the experimental outcomes, envisages that concave reflectors can effectively compensate the turbulence induced signal fading and restrict the beam divergence loss; thereby, improving the maximum achievable NLOS FSO link length.

Appendices

Appendix A: Estimation of zenith angle of the optical reflector

Here, 3 FSO nodes with 1 reflector (n = 1, i = {0, 1, 2}, n ∈ i) are considered for collecting the laser beam, relayed from Tx (i = 0) node, at the mirror installed at ith node i.e. (i = 1), and then precisely reflect it within FOV of Rx (i = 2) node. Figure 14 illustrates an alternate obstruction-less NLOS beam propagation path in the y–z plane when the LOS view between Tx and Rx is blocked.

Fig. 14

Side view of the ray propagation path from Tx to Rx node via reflector node

Within the ray-pointing loci, the arms of the triangle, with vertices as the FSO nodes, represent the propagation path of the beam from i = 0 to i = 2, via the mirror MM′ installed at reflector node i = 1. Consequently, the interior angle of the triangle formed at the node i = 1, is bisected by a line N so that the beam incident angle θ _I equals the reflected angle θ _R (to satisfy the laws of mirror reflection (Halliday 2013)). This implies that the mirror plane MM′, traced as a perpendicular to mirror’s normal N, may bear an angular tilt of $\phi_{i = 1}^{ze}$ from z-axis at node i = 1, and is obtained as,

$$\varphi_{i = 1}^{ze} = \frac{1}{2}\arcsin \left[ {\left( {\frac{{\left( {h_{i = 2} - h_{i = 0} } \right)\cos ec\varphi_{(0,2)} }}{{\left( {h_{i = 1} - h_{i = 0} } \right)\cos ec\varphi_{(0,1)} }}} \right)\sin \left( {\varphi_{(0,2)} - \varphi_{(2,1)} } \right)} \right]\,\,\, - \varphi_{(2,1)},$$

(A.1)

after applying the trigonometric manipulations. Here, the angular tilt ϕ_(i,j) of the beam from node i to j is given by

$$\,\varphi_{(i,j)} = \arctan \left( {{{\left| {h_{i} - h_{j} } \right|} \mathord{\left/ {\vphantom {{\left| {h_{i} - h_{j} } \right|} {l_{i,j} }}} \right. \kern-0pt} {l_{i,j} }}} \right),$$

(A.2)

and is uniquely specified for node installation heights: h_i=2 > h_i=1 > h _obstacle  > h_i=0. Note that l_i=0,i=2 denotes the horizontal distance between nodes: 0 and 2, and the zenith angle is positive when it is measured in clockwise direction along z-axis. By following a similar approach, the zenith angles of n reflectors, installed at their respective nodes, in a NLOS FSO link with i (= 0, 1, …, n + 1) nodes can be easily gauged in x–z and y–z planes.

Appendix B: Estimation of azimuth angle of the optical reflector

Figure 15 illustrates an alternate obstruction-less NLOS beam propagation path in the x–y plane when the LOS view between Tx and Rx nodes is blocked. Here, the NLOS FSO link comprises of 3 static link nodes with 1 reflector (n = 1, i = {0, 1, 2}, n ∈ i) for collecting the laser beam relayed from Tx (i = 0) node. Subsequently, the mirror installed at ith node i.e. (i = 1) precisely reflect it within FOV of Rx (i = 2) node. Note that the beam traversal paths (L_i=1,i=2, L_i=1,i=0) between the FSO nodes forms a triangle, with blocked LOS axis, with interior angles (θ_i=0, θ_i=1, θ_i=2). Thereafter, trace a reflector plane MM′ through the reflector node (i = 1) perpendicular to the angle bisector of θ_i=1, such that θ_i=1 = θ _I +θ _R and the beam incident angle θ _I equals the beam angular tilt of $\phi_{i = 1}^{a}$ from x-axis and is given by reflected angle θ _R . During such ray drafting procedure, the mirror plane MM′ will bear an

$$\varphi_{i = 1}^{a} = \varphi_{(0,1)} - \theta_{I,i = 1},$$

(B.1)

where the beam incidence angle at node i = 1, with further geometric manipulations, is given as,

$$\begin{aligned} \theta_{I,i = 1} = \,90^{ \circ } - \frac{1}{2}\left[ {\arcsin \,\,\,\left( {\frac{{L_{0,1} }}{{L_{0,2} }} \cdot \sin \,\,\left( {\varphi_{(0,1)} + \varphi_{(0,2)} } \right)} \right)} \right. \hfill \\ \left. {\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \arcsin \,\,\,\left( {\frac{{L_{1,2} }}{{L_{0,2} }} \cdot \sin \,\,\left( {\varphi_{(0,1)} + \varphi_{(0,2)} } \right)} \right)} \right]. \hfill \\ \end{aligned}$$

(B.2)

and the angular tilt ϕ_(i,j) of the beam from node i to j is given by: ϕ_(i,j) = arcsin(L _i,j /l _i,j ). By following a similar approach, the azimuth angles of n reflectors, installed at their respective nodes, in a NLOS FSO link with i (= 0, 1, …, n + 1) nodes can be easily gauged.

Fig. 15

Top-view of the ray propagation path from Tx to Rx node via reflector node