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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 9 — Apr. 23, 2012
  • pp: 10359–10369
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A single-scatter path loss model for non-line-of-sight ultraviolet channels

Yong Zuo, Houfei Xiao, Jian Wu, Yan Li, and Jintong Lin  »View Author Affiliations


Optics Express, Vol. 20, Issue 9, pp. 10359-10369 (2012)
http://dx.doi.org/10.1364/OE.20.010359


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Abstract

In this paper, a novel single-scatter path loss model is presented for non-line-of-sight (NLOS) ultraviolet (UV) channels. This model is developed based on the spherical coordinate system and extends the previous restricted models to handle the general noncoplanar case of arbitrarily pointing transmitter and receiver. Numerical examples on path loss are illustrated for various system geometries. These results are verified with a Monte Carlo (MC) model, demonstrating the validity of this model.

© 2012 OSA

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

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

,2

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

], 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

3. 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]

], 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

4. 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]

], presented a closed-form model based on isotropic scattering and a continuous wave Tx. For noncoplanar Tx and Rx geometries [5

5. 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]

], applied trigonometry to develop a model for the case of vertical Rx pointing and arbitrary Tx orientation [6

6. H. Xiao, Y. Zuo, J. Wu, H. Guo, and J. Lin, “Non-line-of-sight ultraviolet single-scatter propagation model,” Opt. Express 19(18), 17864–17875 (2011). [CrossRef] [PubMed]

]. 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

7. 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]

], presented a closed-form noncoplanar model for a small common volume. Reference [8

8. M. A. Elshimy and S. Hranilovic, “Non-line-of-sight single-scatter propagation model for noncoplanar geometries,” J. Opt. Soc. Am. A 28(3), 420–428 (2011). [CrossRef] [PubMed]

] extended the coplanar model of [2

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

] 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

9. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009). [CrossRef]

11

11. H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wirel. Commun. Netw. 2010(1), 598572 (2010). [CrossRef]

], developed multiple-scatter models based on the Monte Carlo (MC) method, and some improvements on these models were proposed in [12

12. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef] [PubMed]

,13

13. H. Yin, H. Jia, H. Zhang, X. Wang, S. Chang, and J. Yang, “Vectorized polarization-sensitive model of non-line-of-sight multiple-scatter propagation,” J. Opt. Soc. Am. A 28(10), 2082–2085 (2011). [CrossRef] [PubMed]

]. In addition, based on extensive field experiments [14

14. G. Chen, Z. Xu, H. Ding, and B. M. 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]

], proposed an empirical path loss model for coplanar geometry, and [15

15. L. Wang, Y. Li, Z. Xu, and B. M. Sadler, “Wireless ultraviolet network models and performance in noncoplanar geometry,” in IEEE Globecom 2010 Workshop on Optical Wireless Communications (IEEE, 2010), pp. 1037–1041.

] extended it to noncoplanar geometry with vertically pointing Rx.

In this paper, we generalize the models in [5

5. 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]

] and [6

6. H. Xiao, Y. Zuo, J. Wu, H. Guo, and J. Lin, “Non-line-of-sight ultraviolet single-scatter propagation model,” Opt. Express 19(18), 17864–17875 (2011). [CrossRef] [PubMed]

] 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

5. 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]

] and develop an alternative analytical model for NLOS UV channels. The developed model is equivalent to that in [8

8. M. A. Elshimy and S. Hranilovic, “Non-line-of-sight single-scatter propagation model for noncoplanar geometries,” J. Opt. Soc. Am. A 28(3), 420–428 (2011). [CrossRef] [PubMed]

] in the calculation of the path loss. However, the key ideas of the two models are different. The model in [8

8. M. A. Elshimy and S. Hranilovic, “Non-line-of-sight single-scatter propagation model for noncoplanar geometries,” J. Opt. Soc. Am. A 28(3), 420–428 (2011). [CrossRef] [PubMed]

] 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

5. 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]

], 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

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

15

15. L. Wang, Y. Li, Z. Xu, and B. M. Sadler, “Wireless ultraviolet network models and performance in noncoplanar geometry,” in IEEE Globecom 2010 Workshop on Optical Wireless Communications (IEEE, 2010), pp. 1037–1041.

] and field experiments [14

14. G. Chen, Z. Xu, H. Ding, and B. M. 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]

,15

15. L. Wang, Y. Li, Z. Xu, and B. M. Sadler, “Wireless ultraviolet network models and performance in noncoplanar geometry,” in IEEE Globecom 2010 Workshop on Optical Wireless Communications (IEEE, 2010), pp. 1037–1041.

], 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

12. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef] [PubMed]

]. 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
Fig. 1 NLOS UV single-scatter propagation in noncoplanar geometry.
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 Ct and Cr denote the Tx beam and the Rx FOV cones, respectively. αt is the Tx half beam angle, and αr is the Rx half FOV angle. θt and θr are the Tx and Rx elevation angles, respectively, i.e., the angles between the axes of Ct and Cr and their projections onto the horizontal plane (i.e., the xy plane) where the Tx and Rx lie. θt (or θr) is positive when the axis of Ct (or Cr) is above the xy plane, and negative otherwise. ϕt is the Tx off-axis angle, equal to the angle between the projected Ct axis on the xy plane and the positive x axis. ϕr is the Rx off-axis angle, equal to the angle between the projected Cr axis on the xy plane and the negative x axis. Then, (θt,ϕt) and (θr,ϕr) determine the pointing directions of the Ct and Cr axes, respectively.

According to the single-scatter assumption, photons emitted by the Tx are scattered only once somewhere in the common volume of Ct and Cr before being received by the Rx. In Fig. 1, V denotes the common volume of Ct and Cr. A ray emitted from the Tx is scattered once by the elemental volume δV. 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 θand the azimuth angle ϕ, and the distance from the scattering volume δV to the Tx is specified by the radial distance r. The scattering angle θs defines the angle between the ray’s forward direction and its scattered direction toward the Rx. ζ defines the angle between the scattered direction and the Cr axis. Let r denote the distance from the scattering volume δV to the Rx.

Following the propagation theory in [2

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

], the energy scattered from the elemental volume δV and received by the Rx is expressed by
δEr=EtArksP(cosθs)cosζ4πΩtr2r2eke(r+r)δV,
(1)
whereΩt=2π(1cosαt) is the Tx solid cone angle, Et is the Tx beam energy, Ar is the Rx detection area, P(cosθS) is the scattering phase function, ks and ke are the atmospheric scattering and extinction coefficients, respectively. Let ka denote the atmospheric absorption coefficient such that ke=ka+ks.

In the spherical coordinate system, the elemental volumeδV equals r2sinθδθδϕδr. Thus, the total energy scattered from the common volume V and received by the Rx can be calculated by
Er=EtArks4πΩtθminθmaxϕminϕmaxrminrmaxP(cosθs)cosζsinθr2eke(r+r)δθδϕδr,
(2)
where [θmin,θmax], [ϕmin,ϕmax] and [rmin,rmax]are the integral limits on θ, ϕ and r, respectively.

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 (θ,ϕ), then scattered after traveling distance r, and finally detected by the Rx after traveling distance r. The integral limits on θ and ϕ 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

r=d2+r22drsinθcosϕ,
(3)
cosθs=dsinθcosϕrr,
(4)
cosζ=r[sinθrcosθcosθrsinθcos(ϕ+ϕr)]+dcosθrcosϕrr.
(5)

In the following, we discuss how to compute the integral limits on θ, ϕ and r.

2.1 θ Limits

In the spherical coordinate system, the inclination angle θ is restricted to the interval [0,π]. Since an emitted ray should be inside the Tx beam, θ[π/2(θt+αt),π/2(θtαt)]. Combining these two conditions, the integral limits on θ can be calculated by

θmin=max[0,π/2(θt+αt)],
(6a)
θmax=min[π,π/2(θtαt)].
(6b)

2.2 ϕ Limits

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

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

z=cosθx2+y2+z2.
(7)

Sθ is the conic section where Cθ intersects the plane z=z0. The equation of Sθ can be obtained by substituting z=z0 into Eq. (7), so Sθ is a circle with radius Rθ=z0tanθ.

In Fig. 2(a), Ct has the elevation angle θt and the off-axis angle ϕt=0. Then, Ct can be expressed by the implicit equation

xcosθt+zsinθt=cosαtx2+y2+z2.
(8)

St is the conic section formed by intersecting Ct with the plane z=z0. The equation of St can be obtained by substituting z=z0 into Eq. (8). St is either: an ellipse when θtαt>0 or θt+αt<0; a parabola when θtαt=0 or θt+αt=0; or half of a hyperbola when θtαt<0 and θt+αt>0. A special case is that when αt=π/2, Ct degenerates to a plane and St is a straight line. Figure 2(a) shows the case in which St is half of a hyperbola.

The ϕ limits can be determined by the intersection of Ct and Cθ, i.e., the intersection of St and Sθ. Figure 2(b) shows the top view of St and Sθ on the plane z=z0, where E and F are the intersection points of St and Sθ. Then, E and F determine the upper and lower limits on ϕ, respectively. Since ϕt=0, E and F are symmetric about the xz plane. Let ϕmax0 and ϕmin0 denote the upper and lower limits on ϕ for the case of ϕt=0, respectively. Thus, ϕmax0=ϕmin0.

Let (xE,yE) denote the coordinates of E. By substituting z=z0 and Eq. (7) into Eq. (8) and solving for x, xE can be given as

xE=z0cosθt(cosαtcosθsinθt).
(9)

When the intersection points exist, Rθ|xE| and RθyE0. Thus, when Rθ|xE|, yE=Rθ2xE2, and ϕmax0=arctan(yE/xE). When Rθ<|xE| or θt=π/2, St contains Sθ, so ϕmax0=π. In summary, when ϕt=0, the ϕ limits can be calculated by

ϕmin0=ϕmax0={arctan(yE/xE),Rθ|xE|π,Rθ<|xE|orθt=π/2.
(10)

When ϕt0, St is rotated by angle ϕt around the z axis to the position of St, as illustrated in Fig. 2(b). E and F are rotated to E and F, respectively. Thus, the integral limits on ϕ can be given by

[ϕmin,ϕmax]=[ϕmin0+ϕt,ϕmax0+ϕt].
(11)

2.3 r Limits

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

In Fig. 3, the Rx is located at point R(d,0,0), and Cr has the elevation angle θr and the off-axis angle ϕr. Then, Cr can be represented with the explicit equation

(cos2αrcos2ϕrcos2θr)(xd)2+sin(2ϕr)cos2θr(xd)y+(cos2αrsin2ϕrcos2θr)y2+sin(2θr)cosϕr(xd)zsin(2θr)sinϕryz+(cos2αrsin2θr)z2=0.
(12)

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

xcosθrcosϕrycosθrsinϕrzsinθrdcosθrcosϕr.
(13)

In Fig. 3, a ray L emitted by the Tx is specified with (θ,ϕ). The coordinates of a point on L can be expressed by (rsinθcosϕ,rsinθsinϕ,rcosθ), where r>0. Substituting it into Eq. (12), a quadratic equation for the radial distance r can be expressed as
ar2+br+c=0,
(14)
where the coefficients a, b and c are computed by

a=cos2αr[sinθrcosθcosθrsinθcos(ϕ+ϕr)]2,
(15a)
b=2dsinθ[cos2θrcosϕrcos(ϕ+ϕr)cos2αrcosϕ]dcosϕrsin(2θr)cosθ,
(15b)
c=d2(cos2αrcos2ϕrcos2θr).
(15c)

Solving Eq. (14) yields: a real solution r0=c/b when a=0 and b0; or two real solutions r1,2=(b±Δ)/(2a) when a0 and Δ=b24ac0,where r1 corresponds to the smaller one and r2 is the greater one, i.e., r2r1.

Note that in the spherical coordinate system, (θ,ϕ) 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 (θ,ϕ).

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

r[cosθrsinθcos(ϕ+ϕr)sinθrcosθ]dcosθrcosϕr.
(16)

Comparing it with Eq. (5), we can see that Eq. (16) means cosζ0, which holds true since ζ=αrπ/2 for a point on the surface of Cr. Therefore, the real solutions of Eq. (14) represent the intersection points of L and Cr only when they satisfy the condition

{r>0,cosζ0.
(17)

With different θ and ϕ, L may intersect Cr 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 Cr (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 Cr.

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 βr, which is the angle between the Cr axis and the negative x axis, as shown in Fig. 3. According to the solid geometry, we can obtain βr=arccos(cosθrcosϕr).

2.3.1 βrαr>0 and βr+αr<π

In this case, the x axis is outside Cr, as shown in Fig. 4
Fig. 4 Cases of intersection between L and Cr when the x axis is outside Cr.
. In the figure, Cr (shown in dashed line) is the lower part of the double cone defined by Eq. (12) relative to R, and L(in dashed line) is the negative part of the line (θ,ϕ) relative to T.

Figure 4 covers all the possible cases in which L intersects Cr, including transition to single intersection, single intersection, and double intersection [5

5. 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]

]. In Fig. 4(a), L intersects Cr once and remains therein after entering, and L does not intersect Cr or Cr, so Eq. (14) has r0>0. In Fig. 4(b), L intersects Cr once by entrance and never exit, and L intersects Cr once, then Eq. (14) has r2>0and r1<0. In Fig. 4(c), L intersects Cr twice by entrance and departure, so Eq. (14) has r2r1>0.

In other cases where L intersects Cr, or L intersects Cr or Cr, 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

[rmin,rmax]={[r0,+),r0>0[r2,+),r1<0andr2>0[r1,r2],r1>0,otherwise.
(18)

2.3.2 βrαr0

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

In Fig. 5(a), L is inside Cr since being emitted and then exit through the intersection point, and L does not intersect Cr, so Eq. (14) has r0>0. In Fig. 5(b), L intersects Cr once by departure and then intersects Cr once by entrance, then Eq. (14) has r2r1>0. In Fig. 5(c), L and L intersects Cr once respectively, so Eq. (14) has r2>0and r10. It can be seen that in the three cases, L always intersects Cr once by departure.

In Fig. 5(d), L only intersects Cr once and L remains inside Cr, so Eq. (14) has r00. In Fig. 5(e), L intersects Cr and Cr once respectively, then Eq. (14) has 0r2r1. It can be seen that since the Tx is inside Cr, L always remains in Cr in the two cases.

Therefore, the integral limits on r can be calculated by

[rmin,rmax]={[0,r0],r0>0[0,r1],r1>0[0,r2],r10andr2>0[0,+),otherwise.
(19)

2.3.3 βr+αrπ

In this case, the x axis is inside Cr, and Cr 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 Cr, i.e., single intersection [5

5. 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]

]. In the case, L intersects Cr once by departure and then intersects Cr once by entrance, then Eq. (14) has r2r10. Therefore, the integral limits on r is given by

[rmin,rmax]={[r2,+),r10,otherwise.
(20)

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

12. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef] [PubMed]

] 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 Et/Er. 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

16. A. Bucholtz, “Rayleigh-scattering calculations for the terrestrial atmosphere,” Appl. Opt. 34(15), 2765–2773 (1995). [CrossRef] [PubMed]

] and a generalized Henyey-Greenstein function [17

17. A. S. Zachor, “Aureole radiance field about a source in a scattering-absorbing medium,” Appl. Opt. 17(12), 1911–1922 (1978). [CrossRef] [PubMed]

], respectively,
PR(cosθs)=3[1+3γ+(1γ)cos2θs]4(1+2γ),
(21)
PM(cosθs)=(1g2)[1(1+g22gcosθs)3/2+f0.5(3cos2θs1)(1+g2)3/2],
(22)
where γ, g and f are model parameters.

The atmospheric phase function is a weighted average [17

17. A. S. Zachor, “Aureole radiance field about a source in a scattering-absorbing medium,” Appl. Opt. 17(12), 1911–1922 (1978). [CrossRef] [PubMed]

] given as
P(cosθs)=ksRksPR(cosθs)+ksMksPM(cosθs),
(23)
where ksR and ksM are the Rayleigh and Mie scattering coefficients, respectively, such that ks=ksR+ksM. According to [9

9. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009). [CrossRef]

,12

12. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef] [PubMed]

], the model parameters (γ,g,f) are chosen as (0.017, 0.72, 0.5), and the atmospheric coefficients (ka,ksR,ksM) are chosen as (8.02,2.66,2.84)×104m−1 at wavelength λ=260nm.

In the simulations, the Rx detection area Ar is set to 104m2, and the Rx half FOV angle and the Tx half beam angle (αr,αt) are set to (20°,15°). Figure 6
Fig. 6 Simulation results on the path loss of NLOS UV channels.
illustrates the changes of the path loss in decibels for different pointing angles (θr,θt,ϕr,ϕt) and baseline distance d. In the figure, “SS” denotes our single-scatter model, and “MC” denotes our implementation of the MC model in [12

12. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef] [PubMed]

]. Figure 6(a) shows the path loss values as d varies from 10m to 100m for two cases: (θr,θt)=(5°,10°) and (θr,θt)=(30°,20°)with (ϕr,ϕt)=(10°,30°). For θr=5° and ϕr=10°, βr11.2°<αr so that the Tx is within the Rx FOV. For θr=30° and ϕr=10°, βr31.5°>αr, 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 (θr,θt)=(5°,10°) is always smaller than that for (θr,θt)=(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 ϕt for two cases: (θr,θt)=(5°,10°) and (θr,θt)=(30°,20°) when ϕr=10° and d=50m. We can also see that our model is in accordance with the MC model. In the case of (θr,θt)=(5°,10°), since the Rx FOV contains the Tx, the path loss is always finite for any ϕt in the range [-180°,180°]. In the case of (θr,θt)=(30°,20°), when −150°<ϕt<-30°, the Rx FOV does not intersect the Tx beam, so the path loss becomes infinite.

4. Conclusions

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.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities 2012.

References and links

1.

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

2.

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

3.

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]

4.

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]

5.

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]

6.

H. Xiao, Y. Zuo, J. Wu, H. Guo, and J. Lin, “Non-line-of-sight ultraviolet single-scatter propagation model,” Opt. Express 19(18), 17864–17875 (2011). [CrossRef] [PubMed]

7.

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]

8.

M. A. Elshimy and S. Hranilovic, “Non-line-of-sight single-scatter propagation model for noncoplanar geometries,” J. Opt. Soc. Am. A 28(3), 420–428 (2011). [CrossRef] [PubMed]

9.

H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm. 27(9), 1535–1544 (2009). [CrossRef]

10.

H. Yin, S. Chang, H. Jia, J. Yang, and J. Yang, “Non-line-of-sight multiscatter propagation model,” J. Opt. Soc. Am. A 26(11), 2466–2469 (2009). [CrossRef] [PubMed]

11.

H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wirel. Commun. Netw. 2010(1), 598572 (2010). [CrossRef]

12.

R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A 28(4), 686–695 (2011). [CrossRef] [PubMed]

13.

H. Yin, H. Jia, H. Zhang, X. Wang, S. Chang, and J. Yang, “Vectorized polarization-sensitive model of non-line-of-sight multiple-scatter propagation,” J. Opt. Soc. Am. A 28(10), 2082–2085 (2011). [CrossRef] [PubMed]

14.

G. Chen, Z. Xu, H. Ding, and B. M. 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]

15.

L. Wang, Y. Li, Z. Xu, and B. M. Sadler, “Wireless ultraviolet network models and performance in noncoplanar geometry,” in IEEE Globecom 2010 Workshop on Optical Wireless Communications (IEEE, 2010), pp. 1037–1041.

16.

A. Bucholtz, “Rayleigh-scattering calculations for the terrestrial atmosphere,” Appl. Opt. 34(15), 2765–2773 (1995). [CrossRef] [PubMed]

17.

A. S. Zachor, “Aureole radiance field about a source in a scattering-absorbing medium,” Appl. Opt. 17(12), 1911–1922 (1978). [CrossRef] [PubMed]

OCIS Codes
(060.4510) Fiber optics and optical communications : Optical communications
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 29, 2012
Revised Manuscript: April 9, 2012
Manuscript Accepted: April 12, 2012
Published: April 20, 2012

Citation
Yong Zuo, Houfei Xiao, Jian Wu, Yan Li, and Jintong Lin, "A single-scatter path loss model for non-line-of-sight ultraviolet channels," Opt. Express 20, 10359-10369 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-9-10359


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References

  1. D. M. Reilly and C. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. Am.69(3), 464–470 (1979). [CrossRef]
  2. M. R. Luettgen, J. H. Shapiro, and D. M. Reilly, “Non-line-of-sight single-scatter propagation model,” J. Opt. Soc. Am. A8(12), 1964–1972 (1991). [CrossRef]
  3. 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]
  4. 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. A27(7), 1505–1509 (2010). [CrossRef] [PubMed]
  5. 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]
  6. H. Xiao, Y. Zuo, J. Wu, H. Guo, and J. Lin, “Non-line-of-sight ultraviolet single-scatter propagation model,” Opt. Express19(18), 17864–17875 (2011). [CrossRef] [PubMed]
  7. 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]
  8. M. A. Elshimy and S. Hranilovic, “Non-line-of-sight single-scatter propagation model for noncoplanar geometries,” J. Opt. Soc. Am. A28(3), 420–428 (2011). [CrossRef] [PubMed]
  9. H. Ding, G. Chen, A. K. Majumdar, B. M. Sadler, and Z. Xu, “Modeling of non-line-of-sight ultraviolet scattering channels for communication,” IEEE J. Sel. Areas Comm.27(9), 1535–1544 (2009). [CrossRef]
  10. H. Yin, S. Chang, H. Jia, J. Yang, and J. Yang, “Non-line-of-sight multiscatter propagation model,” J. Opt. Soc. Am. A26(11), 2466–2469 (2009). [CrossRef] [PubMed]
  11. H. Ding, Z. Xu, and B. M. Sadler, “A path loss model for non-line-of-sight ultraviolet multiple scattering channels,” EURASIP J. Wirel. Commun. Netw.2010(1), 598572 (2010). [CrossRef]
  12. R. J. Drost, T. J. Moore, and B. M. Sadler, “UV communications channel modeling incorporating multiple scattering interactions,” J. Opt. Soc. Am. A28(4), 686–695 (2011). [CrossRef] [PubMed]
  13. H. Yin, H. Jia, H. Zhang, X. Wang, S. Chang, and J. Yang, “Vectorized polarization-sensitive model of non-line-of-sight multiple-scatter propagation,” J. Opt. Soc. Am. A28(10), 2082–2085 (2011). [CrossRef] [PubMed]
  14. G. Chen, Z. Xu, H. Ding, and B. M. Sadler, “Path loss modeling and performance trade-off study for short-range non-line-of-sight ultraviolet communications,” Opt. Express17(5), 3929–3940 (2009). [CrossRef] [PubMed]
  15. L. Wang, Y. Li, Z. Xu, and B. M. Sadler, “Wireless ultraviolet network models and performance in noncoplanar geometry,” in IEEE Globecom 2010 Workshop on Optical Wireless Communications (IEEE, 2010), pp. 1037–1041.
  16. A. Bucholtz, “Rayleigh-scattering calculations for the terrestrial atmosphere,” Appl. Opt.34(15), 2765–2773 (1995). [CrossRef] [PubMed]
  17. A. S. Zachor, “Aureole radiance field about a source in a scattering-absorbing medium,” Appl. Opt.17(12), 1911–1922 (1978). [CrossRef] [PubMed]

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