1. Introduction

A recent appealing research subject in atmospheric optical communications is non-line-of-sight (NLOS) ultraviolet (UV) communication. Many NOLS UV channel models and communication systems have been developed for civilian and military applications. For the case where the beam of the transmitter (Tx) and the field of view (FOV) of the receiver (Rx) have coplanar axes [1,2], proposed a propagation model based on the prolate-spheroidal coordinate system under the single-scatter assumption that photons emitted by the Tx and received by the Rx are scattered only once in the intersected (common) volume of the Tx beam and the Rx FOV. For tractable analysis [3], proposed an approximate model without integral form by considering the case where the common volume of the Tx beam and the Rx FOV is small. For a large common volume [4], presented a closed-form model based on isotropic scattering and a continuous wave Tx. For noncoplanar Tx and Rx geometries [5], applied trigonometry to develop a model for the case of vertical Rx pointing and arbitrary Tx orientation [6]. relaxed the restriction of vertical Rx pointing and proposed a model for the case in which both the Tx beam and the Rx FOV are above the horizontal plane where the Tx and Rx lie. For simplicity [7], presented a closed-form noncoplanar model for a small common volume. Reference [8] extended the coplanar model of [2] to noncoplanar geometry based on the prolate-spheroidal coordinate system. On the other hand, since multiple scatters may occur when the scattering particle density is high or the propagation distance is long [9–11], developed multiple-scatter models based on the Monte Carlo (MC) method, and some improvements on these models were proposed in [12,13]. In addition, based on extensive field experiments [14], proposed an empirical path loss model for coplanar geometry, and [15] extended it to noncoplanar geometry with vertically pointing Rx.

In this paper, we generalize the models in [5] and [6] to handle the noncoplanar geometry case where the Tx and the Rx can point in arbitrary directions. Our motivation is to complete the work started by [5] and develop an alternative analytical model for NLOS UV channels. The developed model is equivalent to that in [8] in the calculation of the path loss. However, the key ideas of the two models are different. The model in [8] computes the received energy based on the contribution of each spheroidal area within the common volume by taking use of the property of a prolate spheroid, i.e., the sum of the focal radii is constant for a spheroidal surface. Following [5], the idea of our model is to calculate the received energy based on the contribution of each ray emitted by the Tx and intersecting with the Rx FOV.

Moreover, our model is derived based on the spherical coordinate system. In all the existing models [1–15] and field experiments [14,15], the pointing angles of the Tx and the Rx, including their elevation (or inclination) angles and off-axis angles (in noncoplanar geometry), are actually defined according to the definitions of the elevation (or inclination) and azimuth angles in the spherical coordinates. Thus, it is natural and comprehensible to model the UV channel based on the spherical coordinate system. In our model, the Tx is placed at the coordinate origin, then the inclination and azimuth angles specify the emission direction of photons and the radial distance represents the propagation distance of photons before being scattered. The computation of the received energy can be thought of as the simulation of the emission, scattering and detection of photons.

This paper is organized as follows. Section 2 presents the path loss model for NLOS UV single scattering channels based on the spherical coordinate system. In Section 3, numerical results on the path loss for various system geometries are provided, and the validity of this model is verified with the MC model proposed in [12]. Finally, we draw our conclusions in Section 4.

2. NLOS single-scatter propagation model

In NLOS UV communications, the Tx and the Rx are connected together through the atmospheric channel. Figure 1 depicts a sketch of NLOS UV single-scatter propagation in noncoplanar geometry. The Tx is located at point T, i.e., the coordinate origin. The Rx is located at point R on the positive x axis. The baseline distance between the Tx and the Rx is d. Let $C_t$ and $C_r$ denote the Tx beam and the Rx FOV cones, respectively. ${\alpha }_t$ is the Tx half beam angle, and ${\alpha }_r$ is the Rx half FOV angle. ${\theta }_t$ and ${\theta }_r$ are the Tx and Rx elevation angles, respectively, i.e., the angles between the axes of $C_t$ and $C_r$ and their projections onto the horizontal plane (i.e., the xy plane) where the Tx and Rx lie. ${\theta }_t$ (or ${\theta }_r$) is positive when the axis of $C_t$ (or $C_r$) is above the xy plane, and negative otherwise. ${\varphi }_t$ is the Tx off-axis angle, equal to the angle between the projected $C_t$ axis on the xy plane and the positive x axis. ${\varphi }_r$ is the Rx off-axis angle, equal to the angle between the projected $C_r$ axis on the xy plane and the negative x axis. Then, $({\theta }_t,{\varphi }_t)$ and $({\theta }_r,{\varphi }_r)$ determine the pointing directions of the $C_t$ and $C_r$ axes, respectively.

 

Fig. 1 NLOS UV single-scatter propagation in noncoplanar geometry.

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According to the single-scatter assumption, photons emitted by the Tx are scattered only once somewhere in the common volume of $C_t$ and $C_r$ before being received by the Rx. In Fig. 1, V denotes the common volume of $C_t$ and $C_r$. A ray emitted from the Tx is scattered once by the elemental volume $\delta V$. We define the parameters about this ray based on the spherical coordinate system as follows: its forward direction is specified by the inclination (zenith) angle $\theta$and the azimuth angle $\varphi$, and the distance from the scattering volume $\delta V$ to the Tx is specified by the radial distance r. The scattering angle ${\theta }_s$ defines the angle between the ray’s forward direction and its scattered direction toward the Rx. $\zeta$ defines the angle between the scattered direction and the $C_r$ axis. Let $r^{\prime }$ denote the distance from the scattering volume $\delta V$ to the Rx.

Following the propagation theory in [2], the energy scattered from the elemental volume $\delta V$ and received by the Rx is expressed by

In the spherical coordinate system, the elemental volume$\delta V$ equals $r^2\sin \theta \delta \theta \delta \varphi \delta r$. Thus, the total energy scattered from the common volume V and received by the Rx can be calculated by

From Fig. 1, we can see that the model expressed by Eq. (2) actually simulates the propagation of photons: photons are emitted by the Tx with pointing angles $(\theta ,\varphi )$, then scattered after traveling distance r, and finally detected by the Rx after traveling distance $r^{\prime }$. The integral limits on $\theta$ and $\varphi$ are to restrict the emitted ray within the Tx beam, and the limits on r correspond to the intersections between the ray and the Rx FOV.

Through some algebraic operations, we can obtain

In the following, we discuss how to compute the integral limits on $\theta$, $\varphi$ and r.

2.1 θ Limits

In the spherical coordinate system, the inclination angle $\theta$ is restricted to the interval $[0,\pi ]$. Since an emitted ray should be inside the Tx beam, $\theta \in \left[\pi /2-({\theta }_t+{\alpha }_t),\pi /2-({\theta }_t-{\alpha }_t)\right]$. Combining these two conditions, the integral limits on $\theta$ can be calculated by

2.2 $\varphi$ Limits

The analysis of the integral limits on the azimuth angle $\varphi$ is illustrated in Fig. 2 . Note that the $\varphi$ limits are calculated to keep an emitted ray with angle $\theta$ inside the Tx beam.

 

Fig. 2 Example of limits on the azimuth angle $\varphi$.

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In Fig. 2(a), the Tx is located at the coordinate origin, and $C_{\theta }$ is the cone where an emitted ray with the inclination angle $\theta$ lies. $C_{\theta }$ can be given by the implicit equation:

$S_{\theta }$ is the conic section where $C_{\theta }$ intersects the plane $z=z_0$. The equation of $S_{\theta }$ can be obtained by substituting $z=z_0$ into Eq. (7), so $S_{\theta }$ is a circle with radius $R_{\theta }=z_0\tan \theta$.

In Fig. 2(a), $C_t$ has the elevation angle ${\theta }_t$ and the off-axis angle ${\varphi }_t=0$. Then, $C_t$ can be expressed by the implicit equation

$S_t$ is the conic section formed by intersecting $C_t$ with the plane $z=z_0$. The equation of $S_t$ can be obtained by substituting $z=z_0$ into Eq. (8). $S_t$ is either: an ellipse when ${\theta }_t-{\alpha }_t>0$ or ${\theta }_t+{\alpha }_t<0$; a parabola when ${\theta }_t-{\alpha }_t=0$ or ${\theta }_t+{\alpha }_t=0$; or half of a hyperbola when ${\theta }_t-{\alpha }_t<0$ and ${\theta }_t+{\alpha }_t>0$. A special case is that when ${\alpha }_t=\pi /2$, $C_t$ degenerates to a plane and $S_t$ is a straight line. Figure 2(a) shows the case in which $S_t$ is half of a hyperbola.

The $\varphi$ limits can be determined by the intersection of $C_t$ and $C_{\theta }$, i.e., the intersection of $S_t$ and $S_{\theta }$. Figure 2(b) shows the top view of $S_t$ and $S_{\theta }$ on the plane $z=z_0$, where E and F are the intersection points of $S_t$ and $S_{\theta }$. Then, E and F determine the upper and lower limits on $\varphi$, respectively. Since ${\varphi }_t=0$, E and F are symmetric about the xz plane. Let ${\varphi }_{\max }^0$ and ${\varphi }_{\min }^0$ denote the upper and lower limits on $\varphi$ for the case of ${\varphi }_t=0$, respectively. Thus, ${\varphi }_{\max }^0=-{\varphi }_{\min }^0$.

Let $(x_E,y_E)$ denote the coordinates of E. By substituting $z=z_0$ and Eq. (7) into Eq. (8) and solving for x, $x_E$ can be given as

When the intersection points exist, $R_{\theta }\ge \left|x_E\right|$ and $R_{\theta }\ge y_E\ge 0$. Thus, when $R_{\theta }\ge \left|x_E\right|$, $y_E=\sqrt{R_{\theta }^2-x_E^2}$, and ${\varphi }_{\max }^0=\mathrm{arc}\tan (y_E/x_E)$. When $R_{\theta }<\left|x_E\right|$ or ${\theta }_t=\pi /2$, $S_t$ contains $S_{\theta }$, so ${\varphi }_{\max }^0=\pi$. In summary, when ${\varphi }_t=0$, the $\varphi$ limits can be calculated by

When ${\varphi }_t\ne 0$, $S_t$ is rotated by angle ${\varphi }_t$ around the z axis to the position of ${S^{\prime }}_t$, as illustrated in Fig. 2(b). E and F are rotated to $E^{\prime }$ and $F^{\prime }$, respectively. Thus, the integral limits on $\varphi$ can be given by

2.3 r Limits

Figure 3 demonstrates the analysis of the integral limits on the radial distance r.

 

Fig. 3 Example of limits on the radial distance r.

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In Fig. 3, the Rx is located at point $R(d,0,0)$, and $C_r$ has the elevation angle ${\theta }_r$ and the off-axis angle ${\varphi }_r$. Then, $C_r$ can be represented with the explicit equation

Note that Eq. (12) actually defines a double cone and $C_r$ is its positive part relative to vertex R, which can be chosen by the restriction

In Fig. 3, a ray L emitted by the Tx is specified with $(\theta ,\varphi )$. The coordinates of a point on L can be expressed by $(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )$, where $r>0$. Substituting it into Eq. (12), a quadratic equation for the radial distance r can be expressed as

Solving Eq. (14) yields: a real solution $r_0=-c/b$ when $a=0$ and $b\ne 0$; or two real solutions $r_{1,2}=(-b\pm \sqrt{\Delta })/(2a)$ when $a\ne 0$ and $\Delta =b^2-4ac\ge 0$,where $r_1$ corresponds to the smaller one and $r_2$ is the greater one, i.e., $r_2\ge r_1$.

Note that in the spherical coordinate system, $(\theta ,\varphi )$ specifies a straight line passing through the coordinate origin. L is only the positive part of this line relative to T. Therefore, the real solutions of Eq. (14) generally correspond to the intersection points between the double cone defined by Eq. (12) and the line $(\theta ,\varphi )$.

In the spherical coordinate system, Eq. (13) can be expressed as

Comparing it with Eq. (5), we can see that Eq. (16) means $\cos \zeta \ge 0$, which holds true since $\zeta ={\alpha }_r\le \pi /2$ for a point on the surface of $C_r$. Therefore, the real solutions of Eq. (14) represent the intersection points of $L$ and $C_r$ only when they satisfy the condition

With different $\theta$ and $\varphi$, L may intersect $C_r$ twice by entrance and departure, or only once by entrance or departure; or there is no intersection at all. A special case is that L is tangent to $C_r$ (i.e., touching but not intersecting). In order to determine the integral limits on r, we need to distinguish the cases of intersection between L and $C_r$.

In the following, we discuss how to determine the integral limits on r based on the real solutions of Eq. (14). To do this, we define an angle ${\beta }_r$, which is the angle between the $C_r$ axis and the negative x axis, as shown in Fig. 3. According to the solid geometry, we can obtain ${\beta }_r=\mathrm{arc}\cos (\cos {\theta }_r\cos {\varphi }_r)$.

2.3.1 ${\beta }_r-{\alpha }_r>0$ and ${\beta }_r+{\alpha }_r<\pi$

In this case, the x axis is outside $C_r$, as shown in Fig. 4 . In the figure, ${C^{\prime }}_r$ (shown in dashed line) is the lower part of the double cone defined by Eq. (12) relative to R, and $L^{\prime }$(in dashed line) is the negative part of the line $(\theta ,\varphi )$ relative to T.

 

Fig. 4 Cases of intersection between L and $C_r$ when the x axis is outside $C_r$.

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Figure 4 covers all the possible cases in which L intersects $C_r$, including transition to single intersection, single intersection, and double intersection [5]. In Fig. 4(a), $L$ intersects $C_r$ once and remains therein after entering, and $L^{\prime }$ does not intersect $C_r$ or ${C^{\prime }}_r$, so Eq. (14) has $r_0>0$. In Fig. 4(b), $L$ intersects $C_r$ once by entrance and never exit, and $L^{\prime }$ intersects ${C^{\prime }}_r$ once, then Eq. (14) has $r_2>0$and $r_1<0$. In Fig. 4(c), $L$ intersects $C_r$ twice by entrance and departure, so Eq. (14) has $r_2\ge r_1>0$.

In other cases where $L$ intersects ${C^{\prime }}_r$, or $L^{\prime }$ intersects $C_r$ or ${C^{\prime }}_r$, none of real solutions of Eq. (14) satisfies the condition (17). Therefore, by analyzing the solutions of Eq. (14), the integral limits on r can be calculated from

2.3.2 ${\beta }_r-{\alpha }_r\le 0$

In this case, the x axis is inside $C_r$, and $C_r$ points towards the Tx, i.e., towards the negative direction of the x axis. Figures 5(a) to 5(e) give all the possible cases in which L intersects or remains in $C_r$.

 

Fig. 5 Cases of intersection between L and $C_r$ when the x axis is inside $C_r$.

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In Fig. 5(a), $L$ is inside $C_r$ since being emitted and then exit through the intersection point, and $L^{\prime }$ does not intersect $C_r$, so Eq. (14) has $r_0>0$. In Fig. 5(b), $L$ intersects $C_r$ once by departure and then intersects ${C^{\prime }}_r$ once by entrance, then Eq. (14) has $r_2\ge r_1>0$. In Fig. 5(c), $L$ and $L^{\prime }$ intersects $C_r$ once respectively, so Eq. (14) has $r_2>0$and $r_1\le 0$. It can be seen that in the three cases, L always intersects $C_r$ once by departure.

In Fig. 5(d), $L^{\prime }$ only intersects $C_r$ once and L remains inside $C_r$, so Eq. (14) has $r_0\le 0$. In Fig. 5(e), $L^{\prime }$ intersects $C_r$ and ${C^{\prime }}_r$ once respectively, then Eq. (14) has $0\ge r_2\ge r_1$. It can be seen that since the Tx is inside $C_r$, L always remains in $C_r$ in the two cases.

Therefore, the integral limits on r can be calculated by

2.3.3 ${\beta }_r+{\alpha }_r\ge \pi$

In this case, the x axis is inside $C_r$, and $C_r$ points away from the Tx, i.e., towards the positive direction of the x axis. Figure 5(f) gives the only case where L intersects $C_r$, i.e., single intersection [5]. In the case, $L$ intersects ${C^{\prime }}_r$ once by departure and then intersects $C_r$ once by entrance, then Eq. (14) has $r_2\ge r_1\ge 0$. Therefore, the integral limits on r is given by

3. Numerical examples

In this section, the proposed model is applied to compute the path loss of NLOS UV channels. Numerical results for different Tx and Rx pointing geometries are presented.

In this paper, we also implement the MC model developed in [12] as an alternative method to compute the path loss. It is a physics-based channel model, which probabilistically models each event occurring during the propagation of photons, involving the emission by the Tx, the interaction with the atmosphere (including absorption and scattering), and the detection by the Rx. A combination of MC and analytical methods are used to simulate the propagation of a large number of independent photons and estimate the overall detection probability for all orders of scattering of photons. The path loss can be given by the reciprocal of the detection probability. Since the MC model is used to verify the results computed by our model, we consider only first-order scattering in this model.

The path loss is defined as the ratio of transmitted and received energy, given by $E_t/E_r$. It is a function of the system geometry, including the Tx beam angle, Rx FOV angle, Tx and Rx pointing angles, communication range, as well as the optical properties of the atmosphere. In the simulations, the Tx and Rx baseline distance is up to 100 meters. For this relatively short range, atmospheric turbulence effects are neglected. We consider two types of scattering: Rayleigh and Mie. The former refers to atmospheric molecules much smaller than the UV wavelength, and the latter refers to atmospheric aerosols approaching the UV wavelength. The Rayleigh and Mie scattering phase functions follow a generalized Rayleigh model [16] and a generalized Henyey-Greenstein function [17], respectively,

The atmospheric phase function is a weighted average [17] given as

In the simulations, the Rx detection area $A_r$ is set to ${10}^{-4}$m², and the Rx half FOV angle and the Tx half beam angle $({\alpha }_r,{\alpha }_t)$ are set to $(20°,15°)$. Figure 6 illustrates the changes of the path loss in decibels for different pointing angles $({\theta }_r,{\theta }_t,{\varphi }_r,{\varphi }_t)$ and baseline distance d. In the figure, “SS” denotes our single-scatter model, and “MC” denotes our implementation of the MC model in [12]. Figure 6(a) shows the path loss values as d varies from 10m to 100m for two cases: $({\theta }_r,{\theta }_t)=(5°,10°)$ and $({\theta }_r,{\theta }_t)=(30°,20°)$with $({\varphi }_r,{\varphi }_t)=(10°,30°)$. For ${\theta }_r=5°$ and ${\varphi }_r=10°$, ${\beta }_r\approx \text{11}\text{.2}°<{\alpha }_r$ so that the Tx is within the Rx FOV. For ${\theta }_r=30°$ and ${\varphi }_r=10°$, ${\beta }_r\approx \text{31}\text{.5}°>{\alpha }_r$, then the Rx FOV is above the xy plane. From this figure, we can see that our model and the MC model are consistent. The path loss for $({\theta }_r,{\theta }_t)=(5°,10°)$ is always smaller than that for $({\theta }_r,{\theta }_t)=(30°,20°)$ at the same baseline distance. In both cases, the path loss increases as the baseline distance increases. Figure 6(b) shows the dependence of the path loss on ${\varphi }_t$ for two cases: $({\theta }_r,{\theta }_t)=(5°,10°)$ and $({\theta }_r,{\theta }_t)=(30°,20°)$ when ${\varphi }_r=10°$ and $d=50$m. We can also see that our model is in accordance with the MC model. In the case of $({\theta }_r,{\theta }_t)=(5°,10°)$, since the Rx FOV contains the Tx, the path loss is always finite for any ${\varphi }_t$ in the range [-180°,180°]. In the case of $({\theta }_r,{\theta }_t)=(30°,20°)$, when −150°$<{\varphi }_t<$-30°, the Rx FOV does not intersect the Tx beam, so the path loss becomes infinite.

 

Fig. 6 Simulation results on the path loss of NLOS UV channels.

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4. Conclusions

In this paper, we have developed a path loss model for NLOS UV single scattering channels by generalizing the previous restricted models to handle the noncoplanar case of arbitrarily pointing Tx and Rx. This model is derived clearly based on the spherical coordinate system, in which the inclination and azimuth angles specify the emission direction of photons and the radial distance represents the propagation distance of photons before being scattered. The computation of the received energy can be thought of as the simulation of the emission, scattering and detection of photons.

Finally, numerical examples on path loss of short-range NLOS channels are provided for various Tx and Rx pointing geometries. The validity of this model is verified with MC simulations. In future work, we will apply this model to estimate the received power and achievable data rates for field experiments.