Non-line-of-sight ultraviolet single-scatter propagation model

Houfei Xiao; Yong Zuo; Jian Wu; Hongxiang Guo; Jintong Lin

doi:10.1364/OE.19.017864

1. Introduction

The line-of-sight (LOS) wireless optical communication systems, in which the transmitter (Tx) and the receiver (Rx) are in direct view without any object obstructing the path between them, are constrained by difficult alignment of the Tx and the Rx. The non-line-of-sight (NLOS) ultraviolet (UV) communications, on the other hand, is proposed as a new type of atmosphere optics communication technology with the important and potential advantages of short-range, safety, anti-interference and flexible operation, and can be considered as an effective supplement of conventional communication [1].

Radiation in the spectral region between 200nm to 280nm which is called “solar-blind region”, is sorely absorbed by atmospheric gases, so that it is negligible that the solar radiation reaches the ground in this waveband. In addition, the solar-blind UV emitted from the Tx is strongly scattered within the scattering volume [2] and received by the Rx. The two phenomena make the short-range NLOS UV communication is possible.

In recent years, there have been considerate efforts in developing NLOS UV communication link models for many commercial and military applications [1]. NLOS single-scatter propagation model based on the prolate-spheroidal coordinate system was proposed by Reilly et al. in [2,3]. This standard model describes the temporal characteristics of singly scattered radiation by considering the case when the Tx beam and the Rx field-of-view (FOV) have coplanar axes. Thereafter, the simplified single-scatter propagation models were extended by means of a closed-from expression for tractable analysis [4–6], which is applicable to coplanar geometry. In these studies, the common scattering volume was assumed to be small in [4,5], and the performance of NLOS UV links under these approximate models was analyzed in terms of bit error rate and signal to noise ratio of the Rx. Besides, based on isotropic scattering and a continuous wave transmitter, the simulation results of the analytical model in [6] are consistent with that of the available NLOS single-scatter propagation model. Only if the emitting duration of transmitter is longer than the duration of the impulse response, this model remains true. An empirical channel path loss model was presented based on extensive measurements at the 260 nm wavelength for range of up to 100m in [7]. In this work, the author focused on the scenarios where the Tx beam and the Rx FOV axes were coplanar, and only adjusted the Tx and the Rx apex angles. The results showed that the single-scatter assumption leads to a model that is not accurate for small apex angles. In recent works [8,9], the NLOS single-scatter propagation model for noncoplanar geometries was investigated. The author considered the special case of vertical Rx pointing and arbitrary Tx orientation in [8]. The work in [9] proposed an approximate closed-form model to describe link behavior for noncoplanar geometry, and the approximation is best when the Tx apex angle and off-axis angle are relatively small. All the above analytical models base on the single-scatter assumption. Shaw, et al. thought that multiple scattering of photons from atmospheric articles becomes an important consideration when operating at ranges beyond one or two extinction lengths, and the use of a single-scatter approximation is reasonable otherwise [10,11]. To relax this assumption, Monte Carlo (MC) radiative transfer model based on photon migration has been applied to handle the multiple scattering events [12–14]. When the communication link range is shorter, the MC simulation results show a close match with the single-scatter propagation model.

From of the above analysis based on single-scatter assumption, the propagation models already have been developed amply for coplanar geometries but only some special scenarios for arbitrary noncoplanar geometries. The motivation of this work is to generalize the existing single-scatter propagation model in [8] to handle the arbitrary noncoplanar geometry cases, where the Tx beam and the Rx FOV axes can be pointed in arbitrary directions. We first discuss the different intersection cases of the Rx FOV boundary and the infinitesimal solid angle rays within the Tx beam. Then, the boundary of the scattering volume that is formed by the Tx and the Rx cones is defined by the solid geometries.

The organization of this paper is as follows. The universal NLOS single-scatter propagation model is developed in Section 2. Numerical analysis for this model is carried out and link performance is compared with both the standard NLOS single-scatter propagation model and the MC model in Section 3. Some conclusions are drawn in Section 4. Finally two appendixes are given.

2. NLOS single-scatter propagation model

Within different ray-pointing scenes, it is possible that the ray intersects the Rx FOV boundary twice by entrance and departure, or only once and remains in the FOV after entering, or there is no intersection when the ray doesn’t enter the Rx FOV. The relative position between the ray and the Rx FOV can be distinguished in accordance with varying geometry. The first step is to calculate the contribution of a single ray to the received energy and next step is to evaluate the total beam contribution by integration.

A transmit/receive solid geometry sketch of NLOS single-scatter propagation is shown in Fig. 1(a) . A Tx with beam angle $ϕ_{T}$ is located at point T. θ_T is the Tx apex angle, which equals $∠ C T C'$ between beam axis TC and its projection TC’ onto the ground. A Rx with FOV $ϕ_{R}$ is located at point R. θ_R is the Rx apex angle, which equals $∠ D R D'$ between FOV axis RD and its projection RD’ onto the ground. For more tractable analysis, $ϕ_{R} / 2 < θ_{R}$ is restricted, this scene means Rx FOV cone doesn’t intersect the ground and meets the need of realistic communication application. The angle between transceiver baseline TR (r) and TC’ is defined as Tx off-axis angle β_T, $β_{T} \in [- π, π]$ . β_T is positive if taken counterclockwise from TR and negative otherwise. Similarly, denote the angle between RT and RD’ as Rx off-axis angle β_R, $β_{R} \in [- π, π]$ . β_R is positive if taken clockwise from RT and negative otherwise.

Fig. 1 Single-scatter propagation solid geometry and axis angle sketch.

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It’s difficult to directly analyze the geometry relation. In order to facilitate the calculation, rotate the transceiver around the TR axis simultaneously, and make the point D projected on the TR axis, as shown in Fig. 1(b). Define the Rx axis angle, $t_{R} = ∠ D E D'$ , as the angle between the plane of RD and RT and the horizontal plane before rotation. Similarly, denote the angle $∠ C F C'$ between the plane of TC and TR and the horizontal plane as Tx axis angle t_T. Then, we can get

t_{T} = \arctan (\tan θ_{T} / | \sin β_{T} |), t_{R} = \arctan (\tan θ_{R} / | \sin β_{R} |) .

Thus, the Rx axis angle is π/2 after rotation, and the Tx axis angle $t_{T}^{0}$ can be given by $t_{T}^{0} = f (t_{T}, t_{R})$ , where f(.) is an explicit algebroidal function with the transceiver geometry parameters t_T and t_R. f(.) is explained in details in Appendix I. After rotation, denote the Rx apex angle by $θ_{R}^{0}$ , the Tx apex angle by $θ_{T}^{0}$ , and the Tx off-axis angle by $β_{T}^{0}$ , then use this new geometry scene to build the analytical model. After some algebraic manipulations, these angles are given by Eq. (2)

(θ_{R}^{0}, θ_{T}^{0}, β_{T}^{0}) = (f_{1} (θ_{R}, β_{R}), f_{2} (t_{T}^{0}, θ_{T}, β_{T}), f_{3} (t_{T}^{0}, θ_{T}, β_{T})),

where f₁(.), f₂(.), and f₃(.) are the explicit algebroidal functions of

θ_{R}^{0}

,

θ_{R}^{0}

, and

β_{T}^{0}

with corresponding geometry parameters. f₁(.), f₂(.), and f₃(.) are explained in details in Appendix I, respectively.

Figure 2(a) depicts the solid geometry after rotation. A ray emitted from T intersects the FOV boundary twice by entrance point A and departure point B. For a scattering point S on AB, $∠ R S B$ between the photon forward direction SB and scattered direction SR (L) is the scattering angle θ_S. Points A', S', B' within the horizontal plane are the projections of points A, S, B, respectively. The lower and upper limits of TS (l) are specified by TA (l_min) and TB (l_max). φ is the angle between TC' and TA'. Moving TA' from TC' (counterclockwise) yields a positive φ, and negative otherwise. A cone (called the equal-θ cone) centered on the ground with the normal direction originated at T, and the cone intersects with the plane CTC' at TB”, as shown in Fig. 2(b). Assume the ray is on the surface of the cone, then the angle θ between TB” and TC uniquely specifies the cone where the ray lies. θ is positive if taken clockwise from the beam axis and negative otherwise. So $(θ_{T}^{'} = θ_{T}^{0} + θ, β_{T}^{0} + φ)$ uniquely specifies each ray direction, and the maximal possible range of $β_{T}^{0} + φ$ is assumed within [-π, π] [8].

Fig. 2 System geometry and ray pointing.

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The relative position of Tx and Rx has not been changed after rotation, so there is no influence on scattering volume. Following the propagation theory in [2], the energy per unit area scattered from the differential volume δV and received by the detector can be calculated according to Eq. (3)

δ E_{0} = \frac{E_{T} k_{s} \cos (ζ) \exp [- k_{e} (l + L)]}{4 π {(l L)}^{2} Ω_{T}} P (θ_{S}) δ V .

Then the received energy contributed by a single ray to the unit area detector is

E_{0} = \int_{l_{\min}}^{l_{\max}} \frac{E_{T} k_{s} \cos (ζ) \exp [- k_{e} (l + L)]}{4 π {(l L)}^{2} Ω_{T}} I P (θ_{S}) δ V,

where

Ω_{T} = 2 π [1 - \cos (ϕ_{T} / 2)]

is the Tx solid cone angle,

δ V = l^{2} \cos (θ_{T}^{'}) δ l δ φ δ θ

is the differential volume, E_T is the beam energy, k_s is the atmospheric scattering coefficient, k_e is the atmospheric extinction coefficient, and P(θ_S) is the single-scatter phase function. The dimensionless indicator function I = 1 when the ray intersects with the Rx FOV and I = 0 otherwise. After some algebraic manipulations, Eq. (4) can be written in the form as Eq. (5)

E_{0} = \frac{E_{T} k_{s} \cos (θ_{T}^{'}) δ φ δ θ}{8 π^{2} [1 - \cos (ϕ_{T} / 2)]} \int_{l_{\min}}^{l_{\max}} \frac{I P (θ_{S}) \cos (ζ)}{L^{2} \exp [k_{e} (l + L)]} δ l .

As in Fig. 2(a), ζ is the angle between the Rx FOV axis RD and a vector pointing from R to S, which equals $∠ S R D$ . We can obtain Eq. (6)

\cos ζ = \frac{l \sin θ_{R}^{0} \sin (θ_{T}^{0} + θ) + | r - l \cos (θ_{T}^{0} + θ) \cos (β^{0} + φ) | \cos θ_{R}^{0}}{\sqrt{r^{2} + l^{2} - 2 r l \cos (θ_{T}^{0} + θ) \cos (β^{0} + φ)}} .

As mentioned earlier, the direction of a ray is specified by two angles θ and φ, so the received energy contributed by the entire beam can be obtained from aggregating the energy contributed by a single ray. Assume $θ \in [θ_{\min}, θ_{\max}]$ and $φ \in [φ_{\min}, φ_{\max}]$ , integrating over θ and φ, the received total energy to the detector with area A_R is

E_{R} = \frac{E_{T} k_{s} A_{R} \int_{θ_{\min}}^{θ_{\max}} \cos (θ_{T}^{'}) \int_{φ_{\min}}^{φ_{\max}} \int_{l_{\min}}^{l_{\max}} \frac{I P (θ_{S}) \cos (ζ)}{L^{2} \exp [k_{e} (l + L)]} δ l δ φ δ θ}{8 π^{2} [1 - \cos (ϕ_{T} / 2)]} .

Referring to Fig. 2 (a), applying the cosine rule to triangles ΔTSR and ΔASR, we have

\begin{array}{l} l^{2} + L^{2} + 2 l L \cos θ_{S} = r^{2}, \\ {(l - l_{\min})}^{2} + L^{2} + 2 (l - l_{\min}) L \cos θ_{S} = r^{2} + {(l_{\min})}^{2} - 2 r l_{\min} \cos (θ_{T}^{0} + θ) \cos (β_{T}^{0} + φ) . \end{array}

Solving the above Eq. (8) for L and θ_S, we can obtain Eq. (9)

\begin{array}{l} L = \sqrt{r^{2} + l^{2} - 2 r l \cos (θ_{T}^{0} + θ) \cos (β_{T}^{0} + φ)}, \\ θ_{S} = \arccos [\frac{r \cos (θ_{T}^{0} + θ) \cos (β_{T}^{0} + φ) - l}{\sqrt{r^{2} + l^{2} - 2 r l \cos (θ_{T}^{0} + θ) \cos (β_{T}^{0} + φ)}}] . \end{array}

The integration limits for l, φ, and θ have been given in Appendix II.

3. Numerical examples

The universal noncoplanar propagation model we proposed in this paper is applied to analyze the NLOS UV single-scatter propagation characteristics, and it is verified by extensive simulations. The simulation results of the universal propagation model in Eq. (7) are shown in Fig. 3 , indicating that this proposed model is fairly consistent with the standard NLOS single-scatter propagation model [2] in coplanar geometry.

Fig. 3 Simulation results of the proposed model and the standard model with path loss (per cm²).

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We further compare this universal propagation model with MC model in noncoplanar geometry, as shown in Fig. 4 . In order to analyze expediently, all the normalized power values of these two models are obtained by unitary processing according to the power values for corresponding Rx off-axis angle β_R = 5°. The simulation results show that, for a set of given geometry parameters, the normalized power value computed by the MC model is larger than the value computed by the generalized model because the MC model takes multiple scattering interactions effect into account, which accords with the conclusion of [14]. Due to reduced scattering volume, the received power values for both models decrease as the degree of separation of the Tx beam and the Rx FOV axes increases. Separately, in the scenario of oversized Rx off-axis angle, the Tx beam cone no longer intersects the Rx FOV, and the received power will be zero, which is different from the MC model because there are some photons will reach the FOV of the Rx after multiple scattering interactions with the atmospheric constituents.

Fig. 4 Simulation results of the proposed model and Monte Carlo model with normalized power.

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In order to further evaluate the proposed universal propagation model, typical numerical examples are presented with different geometric links. The parameters in this model are chosen as follows: $(k_{a}, k_{s}^{R a y}, k_{s}^{M i e}) = (9, 2.4, 2.5) \times 10^{- 4} m^{- 1}$ , γ = 0.017, g = 0.72 and f = 0.5 [5]. The link path loss per unit area cm² is considered under different geometric settings $(β_{T}, β_{R}, θ_{T}, θ_{R}, ϕ_{T}, ϕ_{R})$ with r = 50m in Fig. 5 and Fig. 6 .

Fig. 5 Path loss per cm² with reference geometry settings.

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Fig. 6 Path loss per cm² versus some geometry angles.

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Given reference geometry settings $(θ_{T}, θ_{R}, ϕ_{T}, ϕ_{R})$ , (30°, 60°, 15°, 30°), the effects of the Tx and the Rx off-axis angles on path loss are demonstrated in Fig. 5. In order to compare the performances with Fig. 5 clearly, we change the values of θ_T, θ_R, $ϕ_{T}$ , and $ϕ_{R}$ in Fig. 6(a)-(d). In Fig. 6(c), specifically, when the Rx off-axis angle β_R is 0°, as β_T increases within (40°, 60°) or (120°, 140°), the scattering volume changes significantly, which causes the path loss changing dramatically. The path loss is unacceptably large and hasn’t been depicted in Fig. 6(c), as the received energy can be ignored with β_T in (60°, 120°). The path loss also hasn’t been depicted in Fig. 6(d) when it is oversize.

Although the effects of the transceiver geometry parameters on path loss in noncoplanar geometry aren’t plain as them in coplanar geometry, Fig. 5 and Fig. 6 show that the closer the relative position between the Tx beam and the Rx FOV cones is, or the closer the relative position between the scattering volume and the transceiver baseline is, or the more concentrative the radiation of the Tx beam becomes, the less path loss will be, and vice versa. Specifically, when the Tx beam cone doesn’t intersect the Rx FOV or the scattering volume decreases to zero, path loss increases to infinite. While the geometry scene is in the transition between no intersection and the nonzero scattering volume, path loss changes dramatically.

A typical example in Fig. 7 presents that the geometry settings $(β_{T}, β_{R}, θ_{T}, θ_{R}, ϕ_{T}, ϕ_{R})$ have great impact on path loss. With reference geometry settings of (15°, 15°, 30°, 60°, 15°, 30°) in relatively small baseline distance, a larger Tx apex angle yields larger scattering volume and more received photons and corresponding signal strength, which means smaller path loss. A smaller Rx apex angle yields larger scattering volume and shorter distance between the scattering volume and the transceiver baseline and more increase in signal strength, so that there is more decrease in path loss. The path loss decreases rapidly as the Rx FOV increases, because the scattering volume increases rapidly while there is no visible change of the distance between the scattering volume and the transceiver baseline. The path loss increases as the Tx beam angle increases, because the larger Tx solid cone angle leads to more decentralized radiation regardless of the increased scattering volume.

Fig. 7 Path loss per cm² versus range.

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4. Conclusion

The author has developed a universal propagation model based on NLOS UV single-scatter propagation theory, where the Tx beam and the Rx FOV axes can be pointed in arbitrary directions. Numerical evaluations show that our universal single-scatter propagation model agrees with the standard model [2] in coplanar geometry and MC model in noncoplanar geometry, concluding that the proposed model is accurate and can be used to analyze the NLOS UV communication links. Path loss dependence was studied for different parameters, indicating that the closer the relative position between the Tx beam and the Rx FOV cones is, or the closer the relative position between the scattering volume and the transceiver baseline is, or the more concentrative the radiation of the Tx beam becomes, the less path loss will be, and vice versa. In practice, we should pay attention to two things when we adjust the various geometric parameters. Firstly, adjust one of the Tx beam and the Rx FOV axes to another as close as possible, so that the scattering volume will be larger. Secondly, put the scattering volume as close to the transceiver baseline as possible, so as to reduce the path length of photons. Meeting these two requirements will be more conducive to the reception of scattering irradiance.

Appendix I

After rotation the Tx axis angle $t_{T}^{0}$ can be calculated according to Eq. (10)

{\begin{cases} t_{T}^{0} = t_{T} + (π / 2 - t_{R}) and M = β_{T} / | β_{T} |, if β_{R} \cdot β_{T} > 0 and t_{R} \geq t_{T} \\ t_{T}^{0} = π - [t_{T} + (π / 2 - t_{R})] and M = - β_{T} / | β_{T} |, if β_{R} \cdot β_{T} > 0 and t_{R} < t_{T} \\ Needn't to rotate, so θ_{R}^{0} = θ_{R}, θ_{T}^{0} = θ_{T} and β_{T}^{0} = β_{T}, if β_{R} = 0 or | β_{R} | = π or θ_{R} = π / 2 \\ t_{T}^{0} = t_{R} and M = - β_{R} / | β_{R} |, if β_{T} = 0 or | β_{T} | = π or θ_{T} = π / 2 \\ t_{T}^{0} = t_{T} - (π / 2 - t_{R}) and M = β_{T} / | β_{T} |, if β_{R} \cdot β_{T} < 0 \end{cases},

where t_T and t_R are given by Eq. (1). The Rx apex angle $θ_{R}^{0}$ , the Tx apex angle $θ_{T}^{0}$ , and the Tx off-axis angle $β_{T}^{0}$ can be obtained from Eq. (11)

\begin{array}{l} θ_{R}^{0} = \arccos (\cos θ_{R} | \cos β_{R} |), \\ θ_{T}^{0} = \arcsin [\sin t_{T}^{0} \cdot \sqrt{1 - \cos^{2} θ_{T} \cos^{2} β_{T}}], \\ {\begin{cases} β_{T}^{0} = M \cdot \arctan [\cos (t_{T}^{0}) \cdot \sqrt{1 / (\cos^{2} θ_{T} \cos^{2} β_{T}) - 1}], i f (0 \leq | β_{T} | \leq π / 2) \\ β_{T}^{0} = M \cdot {π - \arctan [\cos (t_{T}^{0}) \cdot \sqrt{1 / (\cos^{2} θ_{T} \cos^{2} β_{T}) - 1}]}, i f (π / 2 < | β_{T} | \leq π) \end{cases}, \end{array}

where $t_{T}^{0}$ and M are given by Eq. (10).

Appendix II

1. Common single-scattering volume

1.1 Cone equation

Different Rx pointing inevitably leads to different solid geometry scenes, and there are instantaneous influences on the received energy. Make M = 1 when 0 ≤ |β_R| ≤ π/2 and M = −1 when π/2 < |β_R| ≤ π. As shown in Fig. 2 (a), when applying laws of trigonometry, we will have

2 r \cos (ϕ_{R} / 2) R A = 2 r \sin θ_{R}^{0} A A' + M \cos θ_{R}^{0} [{(R A')}^{2} + r^{2} - {(T A')}^{2}] .

In order to analyze the path of a photon, firstly discuss the running distance of the photon before being scattered, and then obtain the distance traveled after being scattered by some geometrical manipulations. Denote the length of TA as x, applying the cosine rule and laws of trigonometry to some triangles, Eq. (12) becomes the following equation

A x^{2} + B x + C = 0,

which is in the form of quadratic equation where the parameters A, B and C are computed by

\begin{array}{l} A = \cos^{2} (ϕ_{R} / 2) - {[\sin θ_{R}^{0} \sin θ_{T}^{'} - M \cos θ_{R}^{0} \cos θ_{T}^{'} \cos (β_{T}^{0} + φ)]}^{2}, \\ B = 2 r [\cos θ_{T}^{'} \cos (β_{T}^{0} + φ) \cdot (\cos^{2} θ_{R}^{0} - \cos^{2} (ϕ_{R} / 2)) - M \sin θ_{R}^{0} \cos θ_{R}^{0} \sin θ_{T}^{'}], \\ C = r^{2} [\cos^{2} (ϕ_{R} / 2) - \cos^{2} θ_{R}^{0}] . \end{array}

In Eq. (14), if A = 0, the only one root of Eq. (13) is x₁ = -C/B. Otherwise, there will be two roots, which are $x_{1, 2} = (- B \pm \sqrt{Δ}) / (2 A)$ , where Δ = B² - 4AC, x₁ corresponds to the greater term, x₂ corresponds to another. For the sake of latter analysis, some special cases should to be considered. The first case is A = 0, which means the ray emitted from T parallels a certain line of the Rx FOV boundary. After some algebraic manipulations, we can obtain

| β_{T}^{0} + φ |_{A = 0} = \arccos (\frac{M (\sin θ_{R}^{0} \sin θ_{T}^{'} - \cos (ϕ_{R} / 2))}{\cos θ_{R}^{0} \cos θ_{T}^{'}}),

when A = 0. The second case is Δ = 0, which means the ray is a tangent to the FOV boundary, that is to say, points A and B coincide and l_min = l_max holds. If specify Δ = 0, we can find

| β_{T}^{0} + φ |_{1, 2} = \arccos (\pm \sqrt{\frac{\cos^{2} θ_{T}^{'} \sin^{2} θ_{R}^{0} - \sin^{2} (ϕ_{R} / 2)}{[\cos^{2} (ϕ_{R} / 2) - \cos^{2} θ_{R}^{0}] \cos^{2} θ_{T}^{'}}}),

where $| β_{T}^{0} + φ |_{1}$ corresponds to the + term, and $| β_{T}^{0} + φ |_{2}$ corresponds to −. The third case is that the ray with $β_{T}^{0} + φ = \pm π / 2$ is a tangent to the FOV boundary, and this ray has the greatest apex angle within all the rays which are tangent to the FOV boundary. Referring to Eq. (15b), we can get the apex angle by Eq. (15c)

θ_{T 1}^{'} = \arcsin (\sqrt{\cos^{2} (ϕ_{R} / 2) - \cos^{2} θ_{R}^{0}} / \sin θ_{R}^{0}) .

The fourth case is that angles $θ_{T 0}^{'}$ and ${| β_{T}^{0} + φ |}_{0}$ make A = 0 and Δ = 0. Then, we have Eq. (15d)

θ_{T 0}^{'} = \arcsin [(\cos^{2} (ϕ_{R} / 2) - \cos^{2} θ_{R}^{0}) / (\cos (ϕ_{R} / 2) \sin θ_{R}^{0})]

according to Eq. (15a) and Eq. (15b). After further analysis, we have come to the conclusion that the ray with pointing angle $β_{T}^{0} + φ$ ( $| β_{T}^{0} + φ | \leq | β_{T}^{0} + φ |_{0}$ ) will be tangent to the FOV boundary only once, as the apex angle of this ray is increased form 0 to π/2. Otherwise, the ray with pointing angle $β_{T}^{0} + φ$ ( $| β_{T}^{0} + φ | > | β_{T}^{0} + φ |_{0}$ ) will be no longer tangent to the FOV boundary.

1.2 l Limits

We find l_min and l_max firstly. When the ray both enters and exits the Rx FOV, we have l_max = x₁ and l_min = x₂. Single intersection occurs as the ray enters the Rx FOV and remains therein without departure, we have l_max = ∞ and l_min = x₁. The photons between l_min and l_max within this ray have opportunities to be received by the detector.

● If $θ_{R}^{0} + ϕ_{R} / 2 \leq π / 2$ and $0 \leq | β_{R} | < π / 2$

When $\max (θ_{R}^{0} + ϕ_{R} / 2, θ_{T 1}^{'}) < θ_{T}^{'} < π / 2$ , no matter what the value of $| β_{T}^{0} + φ |$ is, the ray always enters and exits the Rx FOV with two intersecting points respectively.

When $θ_{T 1}^{'} < θ_{T}^{'} \leq θ_{R}^{0} + ϕ_{R} / 2$ , considering the value of $| β_{T}^{0} + φ |$ from 0 to π, the situation of intersection between the ray and FOV boundary changes from two intersecting points to only one intersecting point. The ray with pointing angle $(θ_{T}^{'}, | β_{T}^{0} + φ |_{A = 0})$ parallels a certain line of the FOV boundary, so there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{A = 0}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | \geq | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{R}^{0} + ϕ_{R} / 2 < θ_{T}^{'} \leq θ_{T 1}^{'}$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection, and then changes to two intersecting points. This means that points A and B coincide twice, so there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ or $| β_{T}^{0} + φ | > | β_{T}^{0} + φ |_{2}$ .

When $θ_{T 0}^{'} < θ_{T}^{'} \leq \min (θ_{R}^{0} + ϕ_{R} / 2, θ_{T 1}^{'})$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection, then to two intersecting points, and finally to only one intersecting point. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ or $| β_{T}^{0} + φ |_{2} < | β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{A = 0}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | \geq | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{R}^{0} - ϕ_{R} / 2 \leq θ_{T}^{'} \leq θ_{T 0}^{'}$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection, and then changes to only one intersecting point. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | > | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{T}^{'} < θ_{R}^{0} - ϕ_{R} / 2$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ .

When $θ_{R}^{0} + ϕ_{R} / 2 < π / 2$ and $θ_{T}^{'} = π / 2$ , no matter what the value of $| β_{T}^{0} + φ |$ is, the ray always enters and exits the Rx FOV with two intersecting points respectively.

● If $θ_{R}^{0} + ϕ_{R} / 2 > π / 2$ and $0 \leq | β_{R} | < π / 2$

When $θ_{T}^{'} > π - (θ_{R}^{0} + ϕ_{R} / 2)$ , no matter what the value of $| β_{T}^{0} + φ |$ is, the ray always enters the Rx FOV and remains therein without departure.

When $θ_{T 1}^{'} < θ_{T}^{'} \leq π - (θ_{R}^{0} + ϕ_{R} / 2)$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to only one intersecting point. There are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{A = 0}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | \geq | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{T 0}^{'} < θ_{T}^{'} \leq θ_{T 1}^{'}$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection, then to two intersecting points, and at last to only one intersecting point. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ or $| β_{T}^{0} + φ |_{2} < | β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{A = 0}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | \geq | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{R}^{0} - ϕ_{R} / 2 \leq θ_{T}^{'} \leq θ_{T 0}^{'}$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection, and then changes to only one intersecting point. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | > | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{T}^{'} < θ_{R}^{0} - ϕ_{R} / 2$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ .

When $θ_{T}^{'} = π / 2$ , the ray becomes vertical, and there is only one intersecting point.

● If $θ_{R}^{0} = π / 2$

This case means the Rx pointing is vertical, and it has been analyzed detailedly in [8].

● If $θ_{R}^{0} + ϕ_{R} / 2 > π / 2$ and $π / 2 < | β_{R} | \leq π$

When $θ_{T}^{'} > π - (θ_{R}^{0} + ϕ_{R} / 2)$ , no matter what the value of $| β_{T}^{0} + φ |$ is, the ray always enters the Rx FOV and remains therein without departure.

When $θ_{T 0}^{'} < θ_{T}^{'} \leq π - (θ_{R}^{0} + ϕ_{R} / 2)$ , the situation of intersection between the ray and FOV boundary changes from only one intersecting point to no intersection. So there is only one intersecting point when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{R}^{0} - ϕ_{R} / 2 \leq θ_{T}^{'} \leq θ_{T 0}^{'}$ , the situation of intersection between the ray and FOV boundary changes from only one intersecting point to two intersecting points, and then changes to no intersection. So there are two intersecting points when $| β_{T}^{0} + φ |_{A = 0} < | β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | \leq | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{T}^{'} < θ_{R}^{0} - ϕ_{R} / 2$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ .

When $θ_{T}^{'} = π / 2$ , the ray becomes vertical, and there is only one intersecting point.

● If $θ_{R}^{0} + ϕ_{R} / 2 \leq π / 2$ and $π / 2 < | β_{R} | \leq π$

When $θ_{R}^{0} + ϕ_{R} / 2 \leq θ_{T}^{'} < π / 2$ , no matter what the value of $| β_{T}^{0} + φ |$ is, there is no intersection between the ray and FOV boundary.

When $θ_{T 0}^{'} < θ_{T}^{'} \leq θ_{R}^{0} + ϕ_{R} / 2$ , the situation of intersection between the ray and FOV boundary changes from only one intersecting point to no intersection. So there is only one intersecting point when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{R}^{0} - ϕ_{R} / 2 \leq θ_{T}^{'} \leq θ_{T 0}^{'}$ , the situation of intersection between the ray and FOV boundary changes from only one intersecting point to two intersecting points, and then changes to no intersection. So there are two intersecting points when $| β_{T}^{0} + φ |_{A = 0} < | β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ , and there is only one intersecting point when $| β_{T}^{0} + φ | \leq | β_{T}^{0} + φ |_{A = 0}$ .

When $θ_{T}^{'} < θ_{R}^{0} - ϕ_{R} / 2$ , the situation of intersection between the ray and FOV boundary changes from two intersecting points to no intersection. So there are two intersecting points when $| β_{T}^{0} + φ | < | β_{T}^{0} + φ |_{1}$ .

1.3 φ Limits

According to [8], we can find φ_max = -φ_min = π when $π / 2 - θ_{T}^{'} \leq θ_{T}^{0} + ϕ_{T} / 2 - π / 2$ and Eq. (16)

φ_{\max} = - φ_{\min} = \arctan \frac{\sqrt{1 - \sin^{2} θ_{T}^{0} - \sin^{2} θ_{T}^{'} - \cos^{2} (ϕ_{T} / 2) + 2 \sin | θ_{T}^{0} | \sin θ_{T}^{'} \cos (ϕ_{T} / 2)}}{\cos (ϕ_{T} / 2) - \sin | θ_{T}^{0} | \sin θ_{T}^{'}} .

otherwise.

1.4 θ Limits

Since the ray must be within the beam, $θ \in [- ϕ_{T} / 2, ϕ_{T} / 2]$ . Also θ should satisfy $θ_{T}^{0} + θ \geq 0$ and $θ_{T}^{0} + θ \leq π / 2$ according to the Tx pointing geometry. So the integration limits of θ can be obtained from Eq. (17)

θ_{\min} = \max (- ϕ_{T} / 2, - θ_{T}^{0}), θ_{\max} = \min (ϕ_{T} / 2, π / 2 - θ_{T}^{0}) .

Heretofore, all the derivations of variables required for evaluation of the received energy in Eq. (7) have been completed.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (2009RC0404).

References and links

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Non-line-of-sight ultraviolet single-scatter propagation model

Abstract

1. Introduction

2. NLOS single-scatter propagation model

3. Numerical examples

4. Conclusion

Appendix I

Appendix II

1. Common single-scattering volume

1.1 Cone equation

1.2 l Limits

1.3 φ Limits

1.4 θ Limits

Acknowledgments

References and links

Cited By

Figures (7)

Equations (20)

Optics Express