Single-collision-induced path loss model of reflection-assisted non-line-of-sight ultraviolet communications

Tian Cao; Tianfeng Wu; Changyong Pan; Changyong Pan; Jian Song; Jian Song

doi:10.1364/OE.456619

1. Introduction

Recently, with the emergence of the off-the-shelf ultraviolet light-emitting diodes (LEDs), optical wireless communications exploiting the ultraviolet spectrum, which refers to ultraviolet communications (UVC), has received considerable attention [1]. Commonly, the solar-blind ultraviolet light from 200 to 280 nm is prevalently adopted in UVC because the constituents of solar radiation in that spectral range have been nearly absorbed by ozone [2–4]. Therefore, ultraviolet detectors on the ground can work under the condition of extremely low solar background noise. In addition, thanks to the short wavelength of ultraviolet light, it has to confront strong scattering effects when propagating through the atmosphere, which makes the non-line-of-sight (NLOS) transmission feasible [5]. Therefore, NLOS UVC links can bypass obstacles and relax the requirement of the alignment between the transmitter (Tx) and the receiver (Rx).

In order to predict the path loss of the NLOS UVC channels, the corresponding path loss models have been widely investigated. Generally speaking, ultraviolet photons would suffer more than once scattering events when traversing the atmosphere between the Tx and the Rx. But this process is so complicated that it is hard to solve with analytical methods. Therefore, empirical [6,7] and Monte-Carlo (MC) [8–10] path loss models have been reported. However, empirical path loss models are limited by the applicable conditions because they are formulated by fitting the experimental results. MC path loss models are more general than empirical ones and exhibit excellent accuracy. Nevertheless, MC path loss models are supposed to simulate the traces of a huge number of photons, which is time-consuming. Fortunately, since the NLOS UVC is mainly designed for short-range scenarios, the single-scatter path loss models have been proposed. The single-scatter path loss models assume that a photon is scattered only once before it arrives at the Rx. When the distance between the Tx and the Rx is short and the common volume between the Tx’s beam and the Rx’s field-of-view (FOV) exists, the single-scatter path loss models have comparable accuracy to MC path loss models. Additionally, the single-scatter path loss models have analytical expressions, which would be of help in assessing the path loss of NLOS UVC links and attract a great deal of attention in recent years. In [11], the authors proposed a single-scatter path loss model with the help of the prolate-spheroidal coordinate system for the NLOS UVC in coplanar geometry, where the axes of the Tx and Rx lie on the same plane. In [12], the authors extended the path loss model in [11] to the noncoplanar geometry. Later, Zuo et al. formulated a single-scatter path loss model based on the spherical coordinate system [13]. Recently, Wu et al. proposed a single-scatter path loss model of the diffused-LOS UVC with an obstacle incorporated [14]. Different from the previous works, the authors in [15] proposed a single-scatter path loss model of the LED-based NLOS UVC in the noncoplanar geometry. Under some special conditions, the simplified expressions of the single-scatter path loss model can be further derived [16–18].

However, it is undeniable that the path loss of the NLOS UVC links is very large because the received optical power resulting from the scattering events is weak. In [19], the experiment was conducted when the transceivers were set under canopies. And the corresponding results show that received optical power had been improved due to the reflection of leaves. But currently reported channel models of the NLOS UVC have not considered the effect of reflection except for our recent work [20]. In [20], the channel model of the reflection-assisted NLOS UVC has been proposed in terms of the MC photon-tracing method and verified by experiments. Both scattering and reflection events are called collision-induced events (CIEs). We find that the total received optical power of the reflection-assisted NLOS UVC is dominated by that from the first-order CIEs in short-range scenarios, especially when the path loss result is less than 120dB. Therefore, an analytical single-collision-induced (SCI) path loss model of the reflection-assisted NLOS UVC is proposed in this work, where the path loss caused by the first-order CIE is exclusively considered. More specifically, the expressions of the received optical energy owing to the single-scatter and single-reflection events are derived. Based on them, the expression of the SCI path loss model is further obtained. Finally, MC and experimental results are provided to verify the correctness and effectiveness of the proposed SCI path loss model for the reflection-assisted NLOS UVC.

2. System model and single-collision-induced path loss analysis

A reflection-assisted NLOS UVC system is demonstrated in Fig. 1, where the Rx and the Tx locate at the origin and the point of $(0,r,0)$, respectively. Therefore, the vector ${\mathbf {r}}$ from the Rx to the Tx is ${\left [ {0,r,0} \right ]^{\rm {T}}}$, where the superscript “T” is the transpose operator. There is an infinite plane P in the considered reflection-assisted NLOS UVC system. The z-axis is placed to be perpendicular to P. The corresponding intersection is point $p$, whose coordinates are $\left ( {0,0,h} \right )$, where $h$ is positive. The shape of Rx’s FOV is conical, the full angle of which is denoted by ${\beta _R}$. An LED is adopted by the Tx. The light emission pattern of the LED can be modeled by the Lambertian distribution, which can be expressed as [21]

(1)$${\cal L}\left( {{\gamma _T}} \right) = \left\{ {\begin{array}{cc} {\frac{{{m_L} + 1}}{{2\pi }}{{\cos }^{{m_L}}}\left( {{\gamma _T}} \right),} & {0 \le {\gamma _T} \le \frac{\pi }{2}}\\ {0,} & {\frac{\pi }{2} < {\gamma _T} \le \pi } \end{array}} \right.,$$

where ${\gamma _T}$ is the angle of the emitted light deviated from the Tx’s axis. ${m_L}$ is the Lambertian order of the LED and can be obtained as $\frac {{ - \ln 2}}{{\ln \cos \left ( {{\beta _T}/2} \right )}}$. ${\beta _T}$ denotes the full angle at the half illuminance of the LED. In addition, the direction vectors of the Rx’s and the Tx’s axes are represented by ${{{\boldsymbol{\mathrm{\mu}}}}_{R}} = {\left [ {\sin {\theta _R}\cos {\phi _R},\sin {\theta _R}\sin {\phi _R},\cos {\theta _R}} \right ]^{\rm {T}}}$ and ${{{\boldsymbol{\mathrm{\mu}}}}_{T}} = {\left [ {\sin {\theta _T}\cos {\phi _T},\sin {\theta _T}\sin {\phi _T},\cos {\theta _T}} \right ]^{\rm {T}}}$, respectively. ${\theta _R}$ and ${\theta _T}$ are the inclination angles of the Rx’s and the Tx’s axes, respectively, measured from the positive z-axis. ${\phi _R}$ and ${\phi _T}$ are the azimuth angles of the Rx’s and the Tx’s axes, respectively, measured from the positive x-axis.

Fig. 1. The diagram of the reflection-assisted NLOS UVC system.

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In this work, the single-CIE condition is considered. That is, the emitted photons experience the CIE only once before arriving at the Rx. Therefore, when an optical impulse with the energy of ${Q_T}$ is transmitted by the Tx, the received optical energy ${Q_R}$ can be written as

(2)$${Q_R} = Q_R^{sca} + Q_R^{ref},$$

where $Q_R^{sca}$ and $Q_R^{ref}$ are the received optical energy resulting from the single-scatter event and single-reflection event, respectively. Both of them are computed in the following subsections.

2.1 Derivation of $Q_R^{sca}$

The single-scatter condition in an NLOS UVC system is illustrated in Fig. 2. When the first-order scattering event occurs within the FOV, the single-scatter path loss model is available. The received optical energy for the LED-based NLOS UVC system under the single-scatter condition can be written as [15]

(3)$$\begin{aligned} Q_R^{sca} = & \int_{{\theta _{\min }}}^{{\theta _{\max }}} \int_{{\phi _{\min }}}^{{\phi _{\max }}} \int_{{r_{{2_{\min }}}}}^{{r_{{2_{\max }}}}} \\ & {\frac{{{\cal L}\left( {{\gamma _T}} \right){Q_T}{k_s}{A_r}\exp \left[ { - {k_e}\left( {{r_1} + {r_2}} \right)} \right]\cos \zeta {\mathop{\rm P}\nolimits} \left( {\cos {\theta _s};\gamma ,g,f} \right)}}{{r_1^{2}}}\sin \theta {\mathop{\rm d}\nolimits} \theta {\mathop{\rm d}\nolimits} \phi {\mathop{\rm d}\nolimits} {r_2}} \end{aligned},$$

where $k_s$ and $k_e$ denote the scattering and extinction coefficients, respectively. ${k_e} = {k_s} + {k_a}$, where ${k_a}$ is the absorption coefficient. ${k_s} = {k_{s,r}} + {k_{s,m}}$, where ${k_{s,r}}$ and ${k_{s,m}}$ are Rayleigh and Mie scattering coefficients, respectively. ${A_r}$ is the detection area of the Rx. ${{\mathbf {r}}_{\mathbf {1}}}$ and ${{\mathbf {r}}_{\mathbf {2}}}$ are the vectors from the Tx and Rx, respectively, to the place that the scattering event happened. The Euclidean norm of ${{\mathbf {r}}_{\mathbf {1}}}$ and ${{\mathbf {r}}_{\mathbf {2}}}$ are denoted by ${r_1}$ and ${r_2}$, respectively. That is, ${r_1} = \left \| {{{\mathbf {r}}_{\mathbf {1}}}} \right \|$ and ${r_2} = \left \| {{{\mathbf {r}}_2}} \right \|$. $\zeta$ is the angle between ${{\mathbf {r}}_2}$ and ${{{\boldsymbol{\mathrm{\mu}}} }_{{R}}}$. ${\mathop {\rm P}\nolimits } \left ( {\cos {\theta _s};\gamma,g,f} \right )$ is the scattering phase function and given in Eq. (4) of [15], where ${\theta _s}$ represents the scattering angle; $\gamma$, $g$, and $f$ are atmospheric model parameters. $\theta$ and $\phi$ denote the inclination and azimuth angles of ${{\mathbf {r}}_2}$, respectively. When the plane P does not exist, the intervals of integration for $\theta$, $\phi$, and $r_2$ are constrained by the whole FOV and have been given in [15]. Since the plane P exists in the investigated NLOS UVC system, the interval of integration for $r_2$ should be updated. Let ${r_p}$ be the distance from the Rx to the intersection between the plane P and the direction vector ${{\boldsymbol{\mathrm{\mu}}}'} = {\left [ {\sin \theta \cos \phi,\sin \theta \sin \phi,\cos \theta } \right ]^{\rm {T}}}$. We have ${r_p} = \frac {h}{{\cos \theta }}$. Given that the interval of integration for $r_2$ is obtained as $\left [ {{{\hat r}_{2_{\min } }},{{\hat r}_{2_{\max } }}} \right ]$ according to the left column in page 3 of [15], the limits of integration for ${r_2}$ can be further updated as follows:

1) If ${r_p} > {\hat r_{2_{\max } }}$, then $\left [ {{r_{2_{\min } }},{r_{2_{\max } }}} \right ] = \left [ {{{\hat r}_{2_{\min } }},{{\hat r}_{2_{\max } }}} \right ]$;
2) If ${\hat r_{2_{\max } }} \ge {r_p} \ge {\hat r_{2_{\min } }}$, then $\left [ {{r_{2_{\min } }},{r_{2_{\max } }}} \right ] = \left [ {{{\hat r}_{2_{\min } }},{r_p}} \right ]$;
3) If ${r_p} < {\hat r_{2_{\min } }}$, then $\left [ {{r_{2_{\min } }},{r_{2_{\max } }}} \right ] = \emptyset$.

2.2 Derivation of $Q_R^{ref}$

The single-reflection condition in an NLOS UVC system is demonstrated in Fig. 3. The normal vector of the plane P is denoted by ${{\mathbf n}_{\rm P}} = {[0,0, - 1]^{\rm {T}}}$. Let $S$ be the common area that is determined by the intersection between the plane P and the cone of the FOV. When the first-order reflection event happened in the common area, the optical signal could arrive at the Rx via the single-reflection event. Let ${\rm {d}}s$ represent the differential area in the common area. The optical energy collected by ${\rm {d}}s$ can be obtained as

(4)$${\rm{d}}Q_R^{s} = \frac{{{Q_T}{\cal L}\left( {{\gamma _T}} \right)\exp \left( { - {k_e}{r_1}} \right)\cos {\theta _i}{\rm{d}}s}}{{r_1^{2}}},$$

where ${r_1} = \left \| {{{\mathbf {r}}_{\mathbf {1}}}} \right \|$ and ${{\mathbf {r}}_{\mathbf {1}}}$ is the vector from the Tx to ${\rm {d}}s$ under the single-reflection condition. ${\theta _i}$ is the angle of incidence. In addition, ${{\mathbf {v}}_{\mathbf {s}}}$ denotes the direction vector of the specular reflection of ${{\mathbf {r}}_{\mathbf {1}}}$. We model the reflection pattern of the plane P as the Phong model, which can be expressed as [22,23]

(5)$$I\left( {{\theta _1},{\theta _2}} \right) = \underbrace {\xi \frac{{\cos {\theta _1}}}{\pi }}_{diffuse} + \underbrace {\left( {1 - \xi } \right)\frac{{{m_s} + 1}}{{2\pi }}{{\cos }^{{m_s}}}{\theta _2}}_{specular},$$

where ${\theta _1}$ is the angle between ${{\mathbf n}_{\rm P}}$ and $- {{\mathbf {r}}_{\mathbf {1}}}$, where ${{\mathbf {r}}_{\mathbf {1}}}$ is the vector from the Rx to ${\rm {d}}s$. ${\theta _2}$ is the angle between ${{\mathbf {v}}_{\mathbf {s}}}$ and $- {{\mathbf {r}}_{\mathbf {1}}}$. As shown in Eq. (5), the Phong model consists of a diffuse component and a specular component. $\xi \in \left [ {0,1} \right ]$ is the percentage of diffusely reflected light in whole reflected light. ${m_s}$ describes the directivity of the specular components. It should be noted that $\xi$ and ${m_s}$ describe the reflective characteristics of the plane and are functions of the position on the plane. They can be directly substituted into the proposed SCI path loss model after the reflective characteristics are measured. Therefore, we consider them constant hereinafter to make our description easier to understand. By considering the light reflected by ${\rm {d}}s$ as a secondary light source, the received optical energy by the Rx from ${\rm {d}}s$ can be further obtained as

(6)$${\rm{d}}Q_R^{ref} = {\rho _P}I\left( {{\theta _1},{\theta _2}} \right)\frac{{{A_r}\cos \zeta }}{{r_2^{2}}}\exp \left( { - {k_e}{r_2}} \right){\rm{d}}Q_R^{s},$$

where ${\rho _P}$ is the reflection coefficient of the plane P. ${r_2} = \left \| {{{\mathbf {r}}_2}} \right \|$ and ${{\mathbf {r}}_2}$ is the vector from the Rx to ${\rm {d}}s$ under the single-reflection condition. $\zeta$ is the angle between ${{\mathbf {r}}_2}$ and ${{{\boldsymbol{\mathrm{\mu}}}}_{R}}$. By integrating Eq. (6) over the common area, the received optical energy under the single-reflection condition can be written as

(7)$$\begin{aligned} Q_R^{ref} & = \iint_S {{\rm{d}}Q_R^{ref}}\\ & = {Q_T}{\rho _P}{A_r} \iint_S {\frac{{ {\cal L} \left( {{\gamma _T}} \right)\exp \left[ { - {k_e}\left( {{r_1} + {r_2}} \right)} \right]I\left( {{\theta _1},{\theta _2}} \right)\cos \zeta \cos {\theta _i}}}{{r_1^{2}r_2^{2}}}} {\rm{d}}s \end{aligned}.$$

Mathematically, by dividing the plane P into many small grids and considering the ${\rm {d}}s$ as the area of a grid, Eq. (7) can be approximately solved through accumulating the reflected optical energy by each grid. However, since the infinite plane is considered here, the computational complexity and accuracy of the above-mentioned method are poor. Here, we will derive the intervals of integration for Eq. (7) in a polar coordinate system. First, the cone of Rx’s FOV is rotated around the z-axis until that the projection of the Rx’s axis onto the x-y plane lies on the x-axis, as is shown in Fig. 4. The pole of the polar coordinate system on the plane P is placed at the point $p$. The polar axis of the polar coordinate system is the projection of the x-axis onto the plane P. Let $l$ and $\alpha$ denote the radius and the azimuth angle of the polar coordinate system on the plane P. Thus, Eq. (7) can be rewritten as

(8)$$Q_R^{ref} = {Q_T}{\rho _P}{A_r}\int_{{l_{\min }}}^{{l_{\max }}} {\int_{{\alpha _{\min }}}^{{\alpha _{\max }}} {\frac{{ {\cal L} \left( {{\gamma _T}} \right)\exp \left[ { - {k_e}\left( {{r_1} + {r_2}} \right)} \right]I\left( {{\theta _1},{\theta _2}} \right)\cos \zeta \cos {\theta _i}}}{{r_1^{2}r_2^{2}}}l{\rm{d}}l{\rm{d}}\alpha } },$$

where

(9a)$${r_2} = \left\| {{{\mathbf{r}}_2}} \right\| = \left\| {{{\left[ {l\cos \alpha ,l\sin \alpha ,h} \right]}^{\rm{T}}}} \right\| = \sqrt {{l^{2}} + {h^{2}}},$$

(9b)$${r_1} = \left\| {{{\mathbf{r}}_1}} \right\| = \left\| {{{\mathbf{r}}_2} - {\mathbf{r}}} \right\| = \sqrt {{l^{2}} + {h^{2}} + {r^{2}} - 2lr\sin \alpha },$$

(9c)$$\cos {\gamma _T} = \frac{{{\mathbf{r}}_1^{\rm{T}}{{{{\boldsymbol{\mathrm \mu}}} }_T}}}{{{r_1}}} = \frac{{\sin {\theta _T}\left[ {l\cos \left( {\alpha - {\phi _T}} \right) - r\sin {\phi _T}} \right] + h\cos {\theta _T}}}{{\sqrt {{l^{2}} + {h^{2}} + {r^{2}} - 2lr\sin \alpha } }},$$

(9d)$$\cos \zeta = \frac{{{\mathbf{r}}_2^{\rm{T}}{{{{\boldsymbol{\mathrm \mu}}} }_R}}}{{{r_2}}} = \frac{{l\sin {\theta _R}\cos \left( {\alpha - {\phi _R}} \right) + h\cos {\theta _R}}}{{\sqrt {{l^{2}} + {h^{2}}} }},$$

(9e)$$\cos {\theta _i} ={-} \frac{{{\mathbf{r}}_1^{\rm{T}}{{\mathbf{n}}_{\rm P}}}}{{{r_1}}} = \frac{h}{{\sqrt {{l^{2}} + {h^{2}} + {r^{2}} - 2lr\sin \alpha } }},$$

(9f)$$\cos {\theta _1} ={-} \frac{{{\mathbf{r}}_2^{\rm{T}}{{\mathbf{n}}_{\rm P}}}}{{{r_2}}} = \frac{h}{{\sqrt {{l^{2}} + {h^{2}}} }},$$

(9g)$$\cos {\theta _2} ={-} \frac{{{\mathbf{r}}_2^{\rm{T}}{{\mathbf{v}}_s}}}{{\left\| {{{\mathbf{r}}_2}} \right\| \cdot \left\| {{{\mathbf{v}}_s}} \right\|}} = \frac{{{h^{2}} + lr\sin \alpha - {l^{2}}}}{{\sqrt {{{\left( {{l^{2}} + {h^{2}}} \right)}^{2}} + \left( {{l^{2}} + {h^{2}}} \right)\left( {{r^{2}} - 2lr\sin \alpha } \right)} }}.$$

$\left [ {{l_{\min }},{l_{\max }}} \right ]$ and $\left [ {{\alpha _{\min }},{\alpha _{\max }}} \right ]$ are the intervals of integration in terms of $l$ and $\alpha$, respectively, and further derived as follows.

1) $\left [ {{l_{\min }},{l_{\max }}} \right ]$: The limits of integration for $l$ locate at the polar axis because the cone of the FOV has been rotated. If ${\theta _R} - \frac {{{\beta _R}}}{2} \ge \frac {\pi }{2}$, FOV cannot intersect with the plane P. Hence $Q_R^{ref}$ equals 0. If ${\theta _R} \le \frac {{{\beta _R}}}{2}$, FOV contains the pole and ${l_{\min }} = 0$. Otherwise, we have ${l_{\min }} = h\tan \left ( {{\theta _R} - \frac {{{\beta _R}}}{2}} \right )$. If ${\theta _R} + \frac {{{\beta _R}}}{2} \ge \frac {\pi }{2}$, the intersection of FOV and plane P is not closed. At this point, ${l_{\max }}$ will be infinite. Otherwise, ${l_{\max }}$ can be obtained as $h\tan \left ( {{\theta _R} + \frac {{{\beta _R}}}{2}} \right )$.
2) $\left [ {{\alpha _{\min }},{\alpha _{\max }}} \right ]$: For the given $l$, we can calculate ${\alpha _0}$ in the FOV-rotated scenario. Under this condition, the equation of the cone of FOV can be written as $(10)$$\frac{{x\sin {\theta _R} + z\cos {\theta _R}}}{{\sqrt {{x^{2}} + {y^{2}} + {z^{2}}} }} = \cos \frac{{{\beta _R}}}{2}.$$$
Substituting $x = l\cos {\alpha _0}$, $y = l\sin {\alpha _0}$, and $z = h$ into Eq. (10), we have
$(11)$$\cos {\alpha _0} = \frac{{\sqrt {{l^{2}} + {h^{2}}} \cos \frac{{{\beta _R}}}{2} - h\cos {\theta _R}}}{{l\sin {\theta _R}}}.$$$
If $\frac {{\sqrt {{l^{2}} + {h^{2}}} \cos \frac {{{\beta _R}}}{2} - h\cos {\theta _R}}}{{l\sin {\theta _R}}} \le - 1$, the circle defined by $l$ and ${\alpha _0}$ is wrapped by the common area. Hence ${\alpha _0}$ is equal to $\pi$. Otherwise, ${\alpha _0} = \arccos \left ( {\frac {{\sqrt {{l^{2}} + {h^{2}}} \cos \frac {{{\beta _R}}}{2} - h\cos {\theta _R}}}{{l\sin {\theta _R}}}} \right )$. Finally, we need to rotate the FOV back to the original position. Therefore, we have $\left [ {{\alpha _{\min }},{\alpha _{\max }}} \right ] = \left [ {{\phi _R} - {\alpha _0},{\phi _R} + {\alpha _0}} \right ]$.

Fig. 2. The diagram of the single-scatter condition.

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Fig. 3. The diagram of the single-reflection condition.

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Fig. 4. Illustration of the polar coordinate on the plane P.

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2.3 SCI path loss model

Substituting Eqs. (3) and (8) into Eq. (2), the SCI path loss model of reflection-assisted NLOS UVC systems can be obtained as

(12)$$L ={-} 10{\log _{10}}\left( {\frac{{{Q_R}}}{{{Q_T}}}} \right)\quad {\rm{(dB)}}.$$

In order to observe the effects of the single-scatter and single-reflection events on the SCI path loss, the path loss caused by the single-scatter and single-reflection events in reflection-assisted NLOS UVC systems can be further expressed as

(13)$${L_{sca}} ={-} 10{\log _{10}}\left( {\frac{{Q_R^{sca}}}{{{Q_T}}}} \right)\quad {\rm{(dB)}}$$

and

(14)$${L_{ref}} ={-} 10{\log _{10}}\left( {\frac{{Q_R^{ref}}}{{{Q_T}}}} \right)\quad {\rm{(dB)}},$$

respectively. Next, we will compare the computational complexity between the proposed SCI path loss model and the MC photon-tracing path loss model in [20] in terms of the required number of loops during calculation. To get stable path loss results via MC in the next section, $10^{7}$ photons are needed in simulation. Therefore, the number of loops for MC photon-tracing path loss model is at least $10^{7}$. The proposed SCI path loss model includes a triple integral in Eq. (3) and a double integral in Eq. (8). To solve those two integrals numerically, Gauss-Legendre quadrature rule can be adopted. It is found that by setting the order of Legendre polynomial to 30 can obtain accurate results. Hence, the required number of loops for the proposed SCI path loss model is $30^{3}+30^{2}=27900$, which is far less than that for the MC photon-tracing path loss model.

3. Results and discussion

In this section, the results of the proposed SCI path loss model of reflection-assisted NLOS UVC systems are computed via Eq. (12) and demonstrated in detail. Additionally, simulation results based on the MC photon-tracing method [20] are provided to verify the correctness and effectiveness of the proposed SCI path loss model. In the legends of figures, MC results are labeled “MCI”, which is the abbreviation of “multiple-collision-induced”. The parameters adopted in the calculation for the ultraviolet light with the wavelength of 266nm are extracted from [15] and [20]. Those parameters are given as follows: ${\beta _T} = {60^ \circ }$, ${\beta _R} = {30^ \circ }$, ${k_{s,r}} = 0.24$ km$^{-1}$, ${k_{s,m}} = 0.25$ km$^{-1}$, ${k_a} = 0.9$ km$^{-1}$, $\gamma = 0.017$, $g = 0.72$, $f = 0.5$, ${A_r} = 1.94$ cm$^{2}$, and ${\rho _P} = 0.1$.

Figure 5 gives the path loss versus the range of the reflection-assisted NLOS UVC systems in coplanar geometry with different parameters of the Phong model considered. Here, $\left ( {\xi,{m_s}} \right )$ are set to (0.5,10), (0.1,10), and (0.5,2) in Fig. 5 (a), (b), and (c), respectively. In addition, we set ${\theta _T} = {\theta _R} = {30^ \circ }$, ${\phi _T} = - {90^ \circ }$, ${\phi _R} = {90^ \circ }$, and $h = 50$ m. It can be seen from Fig. 5 (a), (b), and (c) that the results of the proposed SCI path loss model have good agreement with those of the first-order CIE-induced path loss obtained from MC simulations, which verifies the correctness of the proposed SCI path loss model. In addition, the results of the proposed SCI path loss model are almost identical to those of overall path loss, i.e., multiple CIE-induced path loss, obtained from MC simulations. This is because the total received optical power of the reflection-assisted NLOS UVC is dominated by that from the first-order CIEs, which shows the effectiveness of the proposed SCI path loss model. Besides, the path loss of the reflection-assisted NLOS UVC system is lower than that of the NLOS UVC system without the reflective plane, which is labeled “Only Sca” in the legend and obtained through MC simulations. This is because the received optical energy in the NLOS UVC system becomes more concentrated via the reflective plane, and hence the path loss of the NLOS UVC system reduces. More specifically, it can be found that as the range increases, the curves of the proposed SCI path loss model are closer to, even coincide with, those of corresponding single-reflection events-induced path loss obtained from Eq. (14) and labeled “Ref” in the legend. However, the received optical energy due to single-scatter events becomes increasingly less in the reflection-assisted NLOS UVC system as the range increases. Therefore, the results of the single-scatter events-induced path loss in the reflection-assisted NLOS UVC system, which are labeled “Sca” and obtained from Eq. (13), become large as the range increases. Furthermore, the received optical energy from the tiny grids on the plane P are provided in Fig. 5 (d)-(h), which are obtained by setting $Q_T=1$J and considering the ${\rm {d}}s$ in Eq. (7) as the area of a tiny grid on the plane P. Here, the steps of a tiny grid on the x-axis and y-axis are both set to 0.5m. It can be seen that given the geometrical parameters here, the shape of the common area is an ellipse. From Fig. 5 (d)-(f), the concentration of the reflected optical energy moves from the left margin of the common area to the right margin as the range increases. And the reflected optical energy from the plane P in Fig. 5 (e) is much higher than that in Fig. 5 (d) and (f). This is because, with the given geometric parameters, such as the inclination angles and the azimuth angles, of the considered NLOS UVC system, more optical energy can arrive at the Rx via reflection in the middle of the given range interval than that close to either end. Therefore, the path loss of the reflection-assisted NLOS UVC system at $r = 50$ m is lower than that at $r = 10$ m and $r = 100$ m. Figure 5 (e), (g), and (h) give the distribution of the received optical energy reflected by the plane P with different parameters of the Phong model at $r = 50$ m. It can be seen from Fig. 5 (e) and (g) that when ${m_s}$ is fixed, the smaller $\xi$ is, the higher reflected optical energy is. This is because the specular reflection of the plane P becomes stronger as $\xi$ decreases. It can be found from Fig. 5 (e) and (h) that when $\xi$ is fixed, the smaller ${m_s}$ is, the lower reflected optical energy is. This is because the directivity of the specular reflection becomes worse as ${m_s}$ decreases. Additionally, we have measured the execution time of the SCI and the MCI path loss models to obtain Fig. 5 (a) at one personal computer with a processor of 3.7GHz and a memory of 16GB. MCI path loss model costs about 536.88s whereas the proposed SCI path loss model only needs 0.28s, which exhibits a significant improvement, i.e., over three magnitudes, in computational complexity.

Fig. 5. (a), (b), and (c) plot the path loss versus the range of the reflection-assisted NLOS UVC systems in coplanar geometry with $\left ( {\xi,{m_s}} \right )$ equaling (0.5,10), (0.1,10), and (0.5,2), respectively. (d)-(h) show the distributions of the received optical energy by the Rx due to the reflection on the plane P under the conditions that are directly labeled in this figure.

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The path loss of the reflection-assisted NLOS UVC systems in noncoplanar geometry with regard to ${\phi _T}$ is provided in Fig. 6. Here, we set $r = 100$ m, $h = 50$ m, ${\theta _T} = {\theta _R} = {30^ \circ }$, ${\phi _R} = {90^ \circ }$, $\xi = 0.5$, and ${m_s} = 10$. It can be found in Fig. 6 (a) that the results of the proposed SCI path loss model match well with those of MC simulations, which manifests that the proposed SCI path loss model can also work well in noncoplanar geometry. In addition, the reflective plane P can also reduce the path loss of the NLOS UVC systems in noncoplanar geometry. The distributions of the optical energy received by the Rx due to reflection on the plane P are illustrated in Fig. 6 (c)-(d) when ${\phi _T}$ equals ${180^ \circ }$, ${270^ \circ }$, and ${360^ \circ }$, respectively. It can be seen that when ${\phi _T}$ equals ${270^ \circ }$, the received optical energy is the highest among Fig. 6 (c)-(d). This is because the axes of the Tx’s beam and the Rx’s FOV are in the same plane and towards each other when ${\phi _T}$ equals ${270^ \circ }$. Under this condition, more optical energy with ${\phi _T} = {270^ \circ }$ can be received via reflection than that with other ${\phi _T}$s. Therefore, the path loss of the reflection-assisted NLOS UVC system with ${\phi _T} = {270^ \circ }$ is lower than that with ${\phi _T} = {180^ \circ }$ and ${360^ \circ }$.

In Fig. 7, the path loss results of the reflection-assisted NLOS UVC systems with different inclination angles of the Tx and the Rx obtained from experiments [20], the proposed SCI path loss model, and MC simulations are exhibited. The experiment was conducted under canopies of roadside trees. The canopies act as the reflective plane. The ranges are set to 5 m and 10 m in Fig. 7 (a) and (b), respectively. The used parameters are extracted from [20] in accordance with the experimental settings. Specifically, we set $h = 5$ m, ${\phi _T} = - {90^ \circ }$, ${\phi _R} = {90^ \circ }$, $\xi = 1$, and ${\rho _P} = 0.014$. Since $\xi$ equals 1, the parameter ${m_s}$ does not take effect. The parameters of the Phong model are obtained by fitting the experimental path loss results when $r$ equals 10 m. Hence, the results of the proposed SCI path loss match well with the corresponding experimental results in Fig. 7 (b). In addition, in Fig. 7 (a), it can be found that the proposed SCI path loss model can also work well when $r$ equals 5 m using the same parameters of the Phong model obtained in the scenario with $r = 10$ m. The gap between the curves of the proposed SCI path loss model and experiments is about 3dB, which is a reasonably small error because the reflective characteristics of the canopies are not necessarily identical everywhere. Therefore, the proposed SCI model is effective in assessing the path loss of the reflection-assisted NLOS UVC systems in real environments.

Fig. 6. (a) gives the path loss of the reflection-assisted NLOS UVC system versus ${\phi _T}$. (b), (c), and (d) provide the distributions of the received optical energy by the Rx due to the reflection on the plane P with ${\phi _T} = {180^ \circ }$, ${270^ \circ }$, and ${360^ \circ }$, respectively.

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Fig. 7. Experimental and analytical path loss results of the reflection-assisted NLOS UVC systems versus ${\theta _T}$ (${\theta _R}$) with (a) $r = 5$ m and (b) $r = 10$ m.

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4. Conclusion

In this work, the SCI path loss model of the reflection-assisted NLOS UVC systems was proposed. Specifically, the analytical expressions of the received optical energy caused by the single-scatter and single-reflection events were derived. Based on them, the analytical expression of the proposed SCI path loss model was further obtained. MC simulations and experimental results were provided to verify the correctness and effectiveness of the proposed SCI path loss model. It was shown that the results of the proposed SCI path loss model match well with those of MC simulations in both coplanar and noncoplanar geometry. In addition, a reflective plane can improve the received optical energy and hence decrease the path loss of the NLOS UVC systems. Moreover, the reflective plane with parameters of smaller $\xi$ or larger ${m_s}$ can reduce the path loss of the reflection-assisted NLOS UVC systems. This work could be of great help for theoretically assessing the path loss of the reflection-assisted NLOS UVC systems. In the future, we will attempt to derive analytical expressions of path loss for reflection-assisted NLOS UVC systems with multiple-order CIEs, such as two-order and three-order CIEs, when long-distance scenarios are under investigation. In addition, SCI path loss of NLOS UVC systems assisted by reflective objects with representative shapes will be also studied.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Single-collision-induced path loss model of reflection-assisted non-line-of-sight ultraviolet communications

Abstract

1. Introduction

2. System model and single-collision-induced path loss analysis

2.1 Derivation of $Q_R^{sca}$

2.2 Derivation of $Q_R^{ref}$

2.3 SCI path loss model

3. Results and discussion

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (20)

Optics Express