Abstract

A universal non-line-of-sight (NLOS) ultraviolet single-scatter propagation model in noncoplanar geometry is proposed to generalize an existing restricted analytical model. This generalized model considers that the transmitter and the receiver cone axes lie in the same plane or different planes, where they can be pointed in arbitrary directions. The model is verified by extensive simulations, showing that the proposed model is consistent with the original NLOS single-scatter propagation model and the Monte Carlo model. The path loss performance is further investigated in terms of different noncoplanar geometric settings and path loss dependence is also analyzed for different factors, including scattering volume size, relative position between the scattering volume and the transceiver, and radiation intensity of the transmitter.

©2011 Optical Society of America

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 [46], 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 [1214]. 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 ϕT is located at point T. θT is the Tx apex angle, which equals CTC' between beam axis TC and its projection TC’ onto the ground. A Rx with FOV ϕR is located at point R. θR is the Rx apex angle, which equals DRD' between FOV axis RD and its projection RD’ onto the ground. For more tractable analysis, ϕR/2<θR 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 βT, βT[π,π]. βT is positive if taken counterclockwise from TR and negative otherwise. Similarly, denote the angle between RT and RD’ as Rx off-axis angle βR, βR[π,π]. βR is positive if taken clockwise from RT and negative otherwise.

 figure: Fig. 1

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, tR=DED', as the angle between the plane of RD and RT and the horizontal plane before rotation. Similarly, denote the angle CFC' between the plane of TC and TR and the horizontal plane as Tx axis angle tT. Then, we can get

tT=arctan(tanθT/|sinβT|),tR=arctan(tanθR/|sinβR|).

Thus, the Rx axis angle is π/2 after rotation, and the Tx axis angle tT0 can be given by tT0=f(tT,tR), where f(.) is an explicit algebroidal function with the transceiver geometry parameters tT and tR. f(.) is explained in details in Appendix I. After rotation, denote the Rx apex angle by θR0, the Tx apex angle by θT0, and the Tx off-axis angle by βT0, then use this new geometry scene to build the analytical model. After some algebraic manipulations, these angles are given by Eq. (2)

(θR0,θT0,βT0)=(f1(θR,βR),f2(tT0,θT,βT),f3(tT0,θT,βT)),
where f1(.), f2(.), and f3(.) are the explicit algebroidal functions of θR0, θR0, and βT0 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, RSB between the photon forward direction SB and scattered direction SR (L) is the scattering angle θS. 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 (lmin) and TB (lmax). 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 (θT'=θT0+θ,βT0+φ) uniquely specifies each ray direction, and the maximal possible range of βT0+φ is assumed within [-π, π] [8].

 figure: Fig. 2

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)

δE0=ETkscos(ζ)exp[ke(l+L)]4π(lL)2ΩTP(θS)δV.

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

E0=lminlmaxETkscos(ζ)exp[ke(l+L)]4π(lL)2ΩTIP(θS)δV,
whereΩT=2π[1cos(ϕT/2)] is the Tx solid cone angle, δV=l2cos(θT')δlδφδθ is the differential volume, ET is the beam energy, ks is the atmospheric scattering coefficient, ke is the atmospheric extinction coefficient, and P(θS) 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)

E0=ETkscos(θT')δφδθ8π2[1cos(ϕT/2)]lminlmaxIP(θS)cos(ζ)L2exp[ke(l+L)]δl.

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 SRD. We can obtain Eq. (6)

cosζ=lsinθR0sin(θT0+θ)+|rlcos(θT0+θ)cos(β0+φ)|cosθR0r2+l22rlcos(θT0+θ)cos(β0+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[θmin,θmax] and φ[φmin,φmax], integrating over θ and φ, the received total energy to the detector with area AR is

ER=ETksARθminθmaxcos(θT')φminφmaxlminlmaxIP(θS)cos(ζ)L2exp[ke(l+L)]δlδφδθ8π2[1cos(ϕT/2)].

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

l2+L2+2lLcosθS=r2,(llmin)2+L2+2(llmin)LcosθS=r2+(lmin)22rlmincos(θT0+θ)cos(βT0+φ).

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

L=r2+l22rlcos(θT0+θ)cos(βT0+φ),θS=arccos[rcos(θT0+θ)cos(βT0+φ)lr2+l22rlcos(θT0+θ)cos(βT0+φ)].

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.

 figure: Fig. 3

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 βR = 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.

 figure: Fig. 4

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: (ka,ksRay,ksMie)=(9,2.4,2.5)×104m1, 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 (βT,βR,θT,θR,ϕT,ϕR) with r = 50m in Fig. 5 and Fig. 6 .

 figure: Fig. 5

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

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 figure: Fig. 6

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

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Given reference geometry settings (θT,θR,ϕT,ϕR), (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 θT, θR, ϕT, and ϕR in Fig. 6(a)-(d). In Fig. 6(c), specifically, when the Rx off-axis angle βR is 0°, as βT 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 βT 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(βT,βR,θT,θR,ϕT,ϕR) 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.

 figure: Fig. 7

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 tT0 can be calculated according to Eq. (10)

{tT0=tT+(π/2tR)andM=βT/|βT|,ifβRβT>0andtRtTtT0=π[tT+(π/2tR)]andM=βT/|βT|,ifβRβT>0andtR<tTNeedn't to rotate,soθR0=θR,θT0=θTandβT0=βT,ifβR=0or|βR|=πorθR=π/2tT0=tRandM=βR/|βR|,ifβT=0or|βT|=πorθT=π/2tT0=tT(π/2tR)andM=βT/|βT|,ifβRβT<0,

where tT and tR are given by Eq. (1). The Rx apex angle θR0, the Tx apex angle θT0, and the Tx off-axis angle βT0 can be obtained from Eq. (11)

θR0=arccos(cosθR|cosβR|),θT0=arcsin[sintT01cos2θTcos2βT],{βT0=Marctan[cos(tT0)1/(cos2θTcos2βT)1],if(0|βT|π/2)βT0=M{πarctan[cos(tT0)1/(cos2θTcos2βT)1]},if(π/2<|βT|π),

where tT0 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 ≤ |βR| ≤ π/2 and M = −1 when π/2 < |βR| ≤ π. As shown in Fig. 2 (a), when applying laws of trigonometry, we will have

2rcos(ϕR/2)RA=2rsinθR0AA'+McosθR0[(RA')2+r2(TA')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

Ax2+Bx+C=0,

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

A=cos2(ϕR/2)[sinθR0sinθT'McosθR0cosθT'cos(βT0+φ)]2,B=2r[cosθT'cos(βT0+φ)(cos2θR0cos2(ϕR/2))MsinθR0cosθR0sinθT'],C=r2[cos2(ϕR/2)cos2θR0].

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 x1,2=(B±Δ)/(2A), 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

|βT0+φ|A=0=arccos(M(sinθR0sinθT'cos(ϕR/2))cosθR0cosθT'),

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 lmin = lmax holds. If specify Δ = 0, we can find

|βT0+φ|1,2=arccos(±cos2θT'sin2θR0sin2(ϕR/2)[cos2(ϕR/2)cos2θR0]cos2θT'),

where |βT0+φ|1 corresponds to the + term, and |βT0+φ|2 corresponds to −. The third case is that the ray with βT0+φ=±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)

θT1'=arcsin(cos2(ϕR/2)cos2θR0/sinθR0).

The fourth case is that angles θT0' and |βT0+φ|0 make A = 0 and Δ = 0. Then, we have Eq. (15d)

θT0'=arcsin[(cos2(ϕR/2)cos2θR0)/(cos(ϕR/2)sinθR0)]

according to Eq. (15a) and Eq. (15b). After further analysis, we have come to the conclusion that the ray with pointing angle βT0+φ (|βT0+φ||βT0+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 βT0+φ (|βT0+φ|>|βT0+φ|0) will be no longer tangent to the FOV boundary.

1.2 l Limits

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

  • If θR0+ϕR/2π/2 and 0|βR|<π/2

When max(θR0+ϕR/2,θT1')<θT'<π/2, no matter what the value of |βT0+φ| is, the ray always enters and exits the Rx FOV with two intersecting points respectively.

When θT1'<θT'θR0+ϕR/2, considering the value of |βT0+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 (θT',|βT0+φ|A=0) parallels a certain line of the FOV boundary, so there are two intersecting points when |βT0+φ|<|βT0+φ|A=0, and there is only one intersecting point when |βT0+φ||βT0+φ|A=0.

When θR0+ϕR/2<θT'θT1', 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 |βT0+φ|<|βT0+φ|1 or |βT0+φ|>|βT0+φ|2.

When θT0'<θT'min(θR0+ϕR/2,θT1'), 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 |βT0+φ|<|βT0+φ|1 or |βT0+φ|2<|βT0+φ|<|βT0+φ|A=0, and there is only one intersecting point when |βT0+φ||βT0+φ|A=0.

When θR0ϕR/2θT'θT0', 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 |βT0+φ|<|βT0+φ|1, and there is only one intersecting point when |βT0+φ|>|βT0+φ|A=0.

When θT'<θR0ϕR/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 |βT0+φ|<|βT0+φ|1.

When θR0+ϕR/2<π/2 and θT'=π/2, no matter what the value of |βT0+φ| is, the ray always enters and exits the Rx FOV with two intersecting points respectively.

  • If θR0+ϕR/2>π/2 and 0|βR|<π/2

When θT'>π(θR0+ϕR/2), no matter what the value of |βT0+φ| is, the ray always enters the Rx FOV and remains therein without departure.

When θT1'<θT'π(θR0+ϕR/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 |βT0+φ|<|βT0+φ|A=0, and there is only one intersecting point when |βT0+φ||βT0+φ|A=0.

When θT0'<θT'θT1', 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 |βT0+φ|<|βT0+φ|1 or |βT0+φ|2<|βT0+φ|<|βT0+φ|A=0, and there is only one intersecting point when |βT0+φ||βT0+φ|A=0.

When θR0ϕR/2θT'θT0', 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 |βT0+φ|<|βT0+φ|1, and there is only one intersecting point when |βT0+φ|>|βT0+φ|A=0.

When θT'<θR0ϕR/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 |βT0+φ|<|βT0+φ|1.

When θT'=π/2, the ray becomes vertical, and there is only one intersecting point.

  • If θR0=π/2

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

  • If θR0+ϕR/2>π/2 and π/2<|βR|π

When θT'>π(θR0+ϕR/2), no matter what the value of |βT0+φ| is, the ray always enters the Rx FOV and remains therein without departure.

When θT0'<θT'π(θR0+ϕR/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 |βT0+φ|<|βT0+φ|A=0.

When θR0ϕR/2θT'θT0', 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 |βT0+φ|A=0<|βT0+φ|<|βT0+φ|1, and there is only one intersecting point when |βT0+φ||βT0+φ|A=0.

When θT'<θR0ϕR/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 |βT0+φ|<|βT0+φ|1.

When θT'=π/2, the ray becomes vertical, and there is only one intersecting point.

  • If θR0+ϕR/2π/2 and π/2<|βR|π

When θR0+ϕR/2θT'<π/2, no matter what the value of |βT0+φ| is, there is no intersection between the ray and FOV boundary.

When θT0'<θT'θR0+ϕR/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 |βT0+φ|<|βT0+φ|A=0.

When θR0ϕR/2θT'θT0', 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 |βT0+φ|A=0<|βT0+φ|<|βT0+φ|1, and there is only one intersecting point when |βT0+φ||βT0+φ|A=0.

When θT'<θR0ϕR/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 |βT0+φ|<|βT0+φ|1.

1.3 φ Limits

According to [8], we can find φmax = min = π when π/2θT'θT0+ϕT/2π/2 and Eq. (16)

φmax=φmin=arctan1sin2θT0sin2θT'cos2(ϕT/2)+2sin|θT0|sinθT'cos(ϕT/2)cos(ϕT/2)sin|θT0|sinθT'.

otherwise.

1.4 θ Limits

Since the ray must be within the beam, θ[ϕT/2,ϕT/2]. Also θ should satisfy θT0+θ0 and θT0+θπ/2 according to the Tx pointing geometry. So the integration limits of θ can be obtained from Eq. (17)

θmin=max(ϕT/2,θT0),θmax=min(ϕT/2,π/2θT0).

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

1. Z. Xu and B. M. Sadler, “Ultraviolet communication: potential and state-of-art,” IEEE Commun. Mag. 46(5), 67–73 (2008). [CrossRef]  

2. M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A 8(12), 1964–1972 (1991). [CrossRef]  

3. D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. A 69(3), 464–470 (1979). [CrossRef]  

4. Z. Xu, “Approximate performance analysis of wireless ultraviolet links,” in Proc. IEEE Intl. Conf. on Acoustics, Speech, and Signal Proc. (IEEE, Honolulu, 2007).

5. Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008). [CrossRef]   [PubMed]  

6. H. Yin, S. Chang, X. Wang, J. Yang, J. Yang, and J. Tan, “Analytical model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 27(7), 1505–1509 (2010). [CrossRef]   [PubMed]  

7. G. Chen, Z. Xu, H. Ding, and B. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express 17(5), 3929–3940 (2009). [CrossRef]   [PubMed]  

8. L. Wang, Z. Xu, and B. M. Sadler, “Non-line-of-sight ultraviolet link loss in noncoplanar geometry,” Opt. Lett. 35(8), 1263–1265 (2010). [CrossRef]   [PubMed]  

9. L. Wang, Z. Xu, and B. M. Sadler, “An approximate closed-form link loss model for non-line-of-sight ultraviolet communication in noncoplanar geometry,” Opt. Lett. 36(7), 1224–1226 (2011). [CrossRef]   [PubMed]  

10. G. A. Shaw, M. Nischan, and M. Iyengar et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000). [CrossRef]  

11. G. A. Shaw, A. M. Siegel, and J. Model et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004). [CrossRef]  

12. A. N. Witt, “Multiple scattering in reflection nebulae I: A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977). [CrossRef]  

13. H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007). [CrossRef]  

14. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Non-line-of-sight ultraviolet communication channel characterization: modeling and validation,” Proc. Of SPIE Photonics and Optics – Free Space Laser Communications IX, San Diego, CA, August 2–3, 2009.

References

  • View by:

  1. Z. Xu and B. M. Sadler, “Ultraviolet communication: potential and state-of-art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
    [Crossref]
  2. M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A  8(12), 1964–1972 (1991).
    [Crossref]
  3. D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. A 69(3), 464–470 (1979).
    [Crossref]
  4. Z. Xu, “Approximate performance analysis of wireless ultraviolet links,” in Proc. IEEE Intl. Conf. on Acoustics, Speech, and Signal Proc. (IEEE, Honolulu, 2007).
  5. Z. Xu, H. Ding, B. M. Sadler, and G. Chen, “Analytical performance study of solar blind non-line-of-sight ultraviolet short-range communication links,” Opt. Lett. 33(16), 1860–1862 (2008).
    [Crossref] [PubMed]
  6. H. Yin, S. Chang, X. Wang, J. Yang, J. Yang, and J. Tan, “Analytical model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 27(7), 1505–1509 (2010).
    [Crossref] [PubMed]
  7. G. Chen, Z. Xu, H. Ding, and B. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express 17(5), 3929–3940 (2009).
    [Crossref] [PubMed]
  8. L. Wang, Z. Xu, and B. M. Sadler, “Non-line-of-sight ultraviolet link loss in noncoplanar geometry,” Opt. Lett. 35(8), 1263–1265 (2010).
    [Crossref] [PubMed]
  9. L. Wang, Z. Xu, and B. M. Sadler, “An approximate closed-form link loss model for non-line-of-sight ultraviolet communication in noncoplanar geometry,” Opt. Lett. 36(7), 1224–1226 (2011).
    [Crossref] [PubMed]
  10. G. A. Shaw, M. Nischan, M. Iyengar, and et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000).
    [Crossref]
  11. G. A. Shaw, A. M. Siegel, J. Model, and et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004).
    [Crossref]
  12. A. N. Witt, “Multiple scattering in reflection nebulae I: A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
    [Crossref]
  13. H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
    [Crossref]
  14. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Non-line-of-sight ultraviolet communication channel characterization: modeling and validation,” Proc. Of SPIE Photonics and Optics – Free Space Laser Communications IX, San Diego, CA, August 2–3, 2009.

2011 (1)

2010 (2)

2009 (1)

2008 (2)

2007 (1)

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

2004 (1)

G. A. Shaw, A. M. Siegel, J. Model, and et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004).
[Crossref]

2000 (1)

G. A. Shaw, M. Nischan, M. Iyengar, and et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000).
[Crossref]

1991 (1)

M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A  8(12), 1964–1972 (1991).
[Crossref]

1979 (1)

D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. A 69(3), 464–470 (1979).
[Crossref]

1977 (1)

A. N. Witt, “Multiple scattering in reflection nebulae I: A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
[Crossref]

Chang, S.

H. Yin, S. Chang, X. Wang, J. Yang, J. Yang, and J. Tan, “Analytical model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 27(7), 1505–1509 (2010).
[Crossref] [PubMed]

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

Chen, G.

Ding, H.

Iyengar, M.

G. A. Shaw, M. Nischan, M. Iyengar, and et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000).
[Crossref]

Jia, H.

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

Luettgen, M. R.

M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A  8(12), 1964–1972 (1991).
[Crossref]

Model, J.

G. A. Shaw, A. M. Siegel, J. Model, and et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004).
[Crossref]

Nischan, M.

G. A. Shaw, M. Nischan, M. Iyengar, and et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000).
[Crossref]

Reilly, D. M.

M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A  8(12), 1964–1972 (1991).
[Crossref]

D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. A 69(3), 464–470 (1979).
[Crossref]

Sadler, B.

Sadler, B. M.

Shao, Z.

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

Shapiro, J. H.

M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A  8(12), 1964–1972 (1991).
[Crossref]

Shaw, G. A.

G. A. Shaw, A. M. Siegel, J. Model, and et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004).
[Crossref]

G. A. Shaw, M. Nischan, M. Iyengar, and et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000).
[Crossref]

Siegel, A. M.

G. A. Shaw, A. M. Siegel, J. Model, and et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004).
[Crossref]

Tan, J.

Wang, L.

Wang, X.

Warde, C.

D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. A 69(3), 464–470 (1979).
[Crossref]

Witt, A. N.

A. N. Witt, “Multiple scattering in reflection nebulae I: A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
[Crossref]

Xu, Z.

Yang, J.

H. Yin, S. Chang, X. Wang, J. Yang, J. Yang, and J. Tan, “Analytical model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 27(7), 1505–1509 (2010).
[Crossref] [PubMed]

H. Yin, S. Chang, X. Wang, J. Yang, J. Yang, and J. Tan, “Analytical model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 27(7), 1505–1509 (2010).
[Crossref] [PubMed]

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

Yin, H.

H. Yin, S. Chang, X. Wang, J. Yang, J. Yang, and J. Tan, “Analytical model of non-line-of-sight single-scatter propagation,” J. Opt. Soc. Am. A 27(7), 1505–1509 (2010).
[Crossref] [PubMed]

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

Astrophys. J. Suppl. Ser. (1)

A. N. Witt, “Multiple scattering in reflection nebulae I: A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
[Crossref]

IEEE Commun. Mag. (1)

Z. Xu and B. M. Sadler, “Ultraviolet communication: potential and state-of-art,” IEEE Commun. Mag. 46(5), 67–73 (2008).
[Crossref]

J. Opt. Soc. A (2)

M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. A  8(12), 1964–1972 (1991).
[Crossref]

D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. A 69(3), 464–470 (1979).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (3)

H. Yin, J. Yang, S. Chang, H. Jia, Z. Shao, and J. Yang, “Analysis of several factors influencing range of non-line-of-sight UV transmission,” Proc. SPIE 6783, 67833E, 67833E-6 (2007).
[Crossref]

G. A. Shaw, M. Nischan, M. Iyengar, and et al.., “NLOS UV Communication for Distributed Sensor Systems,” Proc. SPIE 4126, 83–96 (2000).
[Crossref]

G. A. Shaw, A. M. Siegel, J. Model, and et al.., “Field Testing and Evaluation of a Solar-Blind UV Communication Link for Unattended Ground Sensor,” Proc. SPIE 5417, 250–261 (2004).
[Crossref]

Other (2)

Z. Xu, “Approximate performance analysis of wireless ultraviolet links,” in Proc. IEEE Intl. Conf. on Acoustics, Speech, and Signal Proc. (IEEE, Honolulu, 2007).

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Non-line-of-sight ultraviolet communication channel characterization: modeling and validation,” Proc. Of SPIE Photonics and Optics – Free Space Laser Communications IX, San Diego, CA, August 2–3, 2009.

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Figures (7)

Fig. 1
Fig. 1 Single-scatter propagation solid geometry and axis angle sketch.
Fig. 2
Fig. 2 System geometry and ray pointing.
Fig. 3
Fig. 3 Simulation results of the proposed model and the standard model with path loss (per cm2).
Fig. 4
Fig. 4 Simulation results of the proposed model and Monte Carlo model with normalized power.
Fig. 5
Fig. 5 Path loss per cm2 with reference geometry settings.
Fig. 6
Fig. 6 Path loss per cm2 versus some geometry angles.
Fig. 7
Fig. 7 Path loss per cm2 versus range.

Equations (20)

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t T = arctan ( tan θ T / | sin β T | ) , t R = arctan ( tan θ R / | sin β R | ) .
( 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 ) ) ,
δ E 0 = E T k s cos ( ζ ) exp [ k e ( l + L ) ] 4 π ( l L ) 2 Ω T P ( θ S ) δ V .
E 0 = l min l max E T k s cos ( ζ ) exp [ k e ( l + L ) ] 4 π ( l L ) 2 Ω T I P ( θ S ) δ V ,
E 0 = E T k s cos ( θ T ' ) δ φ δ θ 8 π 2 [ 1 cos ( ϕ T / 2 ) ] l min l max I P ( θ S ) cos ( ζ ) L 2 exp [ k e ( l + L ) ] δ l .
cos ζ = 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 r 2 + l 2 2 r l cos ( θ T 0 + θ ) cos ( β 0 + φ ) .
E R = E T k s A R θ min θ max cos ( θ T ' ) φ min φ max l min l max 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 ) ] .
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 ) .
L = r 2 + l 2 2 r l cos ( θ T 0 + θ ) cos ( β T 0 + φ ) , θ S = arccos [ r cos ( θ T 0 + θ ) cos ( β T 0 + φ ) l r 2 + l 2 2 r l cos ( θ T 0 + θ ) cos ( β T 0 + φ ) ] .
{ t T 0 = t T + ( π / 2 t R ) and M = β T / | β T | , if β R β T > 0 and t R t T t T 0 = π [ t T + ( π / 2 t R ) ] and M = β T / | β T | , if β R 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 β T < 0 ,
θ R 0 = arccos ( cos θ R | cos β R | ) , θ T 0 = arcsin [ sin t T 0 1 cos 2 θ T cos 2 β T ] , { β T 0 = M arctan [ cos ( t T 0 ) 1 / ( cos 2 θ T cos 2 β T ) 1 ] , i f ( 0 | β T | π / 2 ) β T 0 = M { π arctan [ cos ( t T 0 ) 1 / ( cos 2 θ T cos 2 β T ) 1 ] } , i f ( π / 2 < | β T | π ) ,
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 ] .
A x 2 + B x + C = 0 ,
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 ) ( 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 ] .
| β T 0 + φ | A = 0 = arccos ( M ( sin θ R 0 sin θ T ' cos ( ϕ R / 2 ) ) cos θ R 0 cos θ T ' ) ,
| β T 0 + φ | 1 , 2 = arccos ( ± 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 ' ) ,
θ T 1 ' = arcsin ( cos 2 ( ϕ R / 2 ) cos 2 θ R 0 / sin θ R 0 ) .
θ T 0 ' = arcsin [ ( cos 2 ( ϕ R / 2 ) cos 2 θ R 0 ) / ( cos ( ϕ R / 2 ) sin θ R 0 ) ]
φ max = φ min = arctan 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 ' .
θ min = max ( ϕ T / 2 , θ T 0 ) , θ max = min ( ϕ T / 2 , π / 2 θ T 0 ) .

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