SYSTEMMODELWe adopt a geometrical model 3 to compare and analyze thetwo different beam modalities. The beam propagation modelFig.

1. Top view of aground-to-train FSO coverage along a straight track DB.of the laser light in this paper follows aGaussian distribution 36, 37.

In our model, a train car has an FSO transceiver installedon the roof, and each BS on the ground has an FSO transceiver. For the sake ofdescription, we focus on ground-to-train communications in this paper. Notethat the establishment of a groundto-train communications link also guaranteesa train-to-ground link because the transmitter and receiver of a transceiverare mutually aligned 13. Therefore, our analysis actually applies to bothlinks.We consider that the transceiver on the train and the BSesalong the track use a wavelength of 850 nm, which is denoted as ?.The 850-nm wavelength is selected because of its availability, reliability,high-performance capabilities, and the lower cost of the transmitter anddetector 12.

We also consider that the transceiver of each BS might beconnected to a fiber-optic backbone where a wavelength between 1530 and 1565 nm(i.e., C-band) is usually employed 38–40. Owing to the differentwavelengthsthattheproposedFSOcommunicationssystemand a fiber-optic backboneoperate, a fiber-to-fiber media converter 41–43 may be needed forwavelength conversion.Fig. 1 shows the geometrical model of the ground-to-trainFSO communications system from a superior view (i.e., as seen from the top).

Inthis figure, we assume that the train travels along line segment DB. d1is the distance between the BS and the track and is set to 1 m 3. d2is the horizontal distance between the BS and the track and it designatesthe location of the shortest coverage point (C) of the beam on the track. ? isthe divergence angle of the laser beam. This angle impacts the beam radius w andthe coverage length L along the track. In Fig.

1, tan? and tan? are tan? = d1/d2, and tan? = d1/(d2 + L). Because ? = ? ? ?, L can be represented as . Denote ?1/2 as half of the divergence angle (i.e., ?1/2 + ? as the tiltangle. The tilt angle is the angle between the optical axis of the beam and thehorizontal axis, which is parallel to the track. This angle affects L because? is afunction of ?and ?.Note that d2affects the tilt angle of the transceiver on the ground.

If d1is kept constant, the tilt angle of the laser beam decreases as d2increases. The height of the BS is the same as the height of the train,which is approximately aboutfour meters above the ground level. AO inFig. 1 is the optical axis of propagation, and z is the distance from the lightsource along the optical axis. The beam radius at distance z is denoted by w(z) and is calculated by 36: (2)where is the beam waist of the laser source at thetransmitter. Here, z = |AH|+ |HO| and |AH| = |AG| + |GH|. Inaddition, the length of the line segment HO can be written as |HO| = (L ? |CH|)cos?.Thus, z canbe given as z= |AG|+ |GH| + (L ? |CH|)cos?.

Therefore, z canbe written as z= Lcos? + xcos?1/2,where r isthe orthogonal offset from the optical axis of propagation of the light beam,which corresponds to the shortest distance between the GO and CB segments at distance z. Forinstance, r isequal to |CG| atthe shortest coverage point C, and it is equal to w(z)when zis equal to |AO|.Considering triangle OHB, we obtain r = (L? |CH|)sin ?. Using a calculation similar to z, r is given as r = Lsin? ?The received power at distance along the track for aGaussian beam is 36 (3)where Ptx is the transmissionpower, and Ac isthe effective light collection area of the receiver.

Ac is given by 44 (4)where n is the refractive index of anoptical concentrator that focuses the incoming light on the photodiode in thereceiver, Ad isthe photosensitive area of the phSYSTEMMODELWe adopt a geometrical model 3 to compare and analyze thetwo different beam modalities. The beam propagation modelFig. 1. Top view of aground-to-train FSO coverage along a straight track DB.of the laser light in this paper follows aGaussian distribution 36, 37.In our model, a train car has an FSO transceiver installedon the roof, and each BS on the ground has an FSO transceiver. For the sake ofdescription, we focus on ground-to-train communications in this paper. Notethat the establishment of a groundto-train communications link also guaranteesa train-to-ground link because the transmitter and receiver of a transceiverare mutually aligned 13.

Therefore, our analysis actually applies to bothlinks.We consider that the transceiver on the train and the BSesalong the track use a wavelength of 850 nm, which is denoted as ?.The 850-nm wavelength is selected because of its availability, reliability,high-performance capabilities, and the lower cost of the transmitter anddetector 12. We also consider that the transceiver of each BS might beconnected to a fiber-optic backbone where a wavelength between 1530 and 1565 nm(i.e., C-band) is usually employed 38–40. Owing to the differentwavelengthsthattheproposedFSOcommunicationssystemand a fiber-optic backboneoperate, a fiber-to-fiber media converter 41–43 may be needed forwavelength conversion.Fig.

1 shows the geometrical model of the ground-to-trainFSO communications system from a superior view (i.e., as seen from the top). Inthis figure, we assume that the train travels along line segment DB. d1is the distance between the BS and the track and is set to 1 m 3. d2is the horizontal distance between the BS and the track and it designatesthe location of the shortest coverage point (C) of the beam on the track. ? isthe divergence angle of the laser beam.

This angle impacts the beam radius w andthe coverage length L along the track. In Fig. 1, tan? and tan? are tan? = d1/d2, and tan? = d1/(d2 + L). Because ? = ? ? ?, L can be represented as . Denote ?1/2 as half of the divergence angle (i.

e., ?1/2 + ? as the tiltangle. The tilt angle is the angle between the optical axis of the beam and thehorizontal axis, which is parallel to the track. This angle affects L because? is afunction of ?and ?.Note that d2affects the tilt angle of the transceiver on the ground. If d1is kept constant, the tilt angle of the laser beam decreases as d2increases. The height of the BS is the same as the height of the train,which is approximately aboutfour meters above the ground level. AO inFig.

1 is the optical axis of propagation, and z is the distance from the lightsource along the optical axis. The beam radius at distance z is denoted by w(z) and is calculated by 36: (2)where is the beam waist of the laser source at thetransmitter. Here, z = |AH|+ |HO| and |AH| = |AG| + |GH|. Inaddition, the length of the line segment HO can be written as |HO| = (L ? |CH|)cos?.

Thus, z canbe given as z= |AG|+ |GH| + (L ? |CH|)cos?.Therefore, z canbe written as z= Lcos? + xcos?1/2,where r isthe orthogonal offset from the optical axis of propagation of the light beam,which corresponds to the shortest distance between the GO and CB segments at distance z. Forinstance, r isequal to |CG| atthe shortest coverage point C, and it is equal to w(z)when zis equal to |AO|.Considering triangle OHB, we obtain r = (L? |CH|)sin ?. Using a calculation similar to z, r is given as r = Lsin? ?The received power at distance along the track for aGaussian beam is 36 (3)where Ptx is the transmissionpower, and Ac isthe effective light collection area of the receiver. Ac is given by 44 (4)where n is the refractive index of anoptical concentrator that focuses the incoming light on the photodiode in thereceiver, Ad isthe photosensitive area of the photodiode in mm2, and ?c isthe half-angle field-of-view (FOV) of the receiver after the lens. For theanalysis in Section V, we use Ad = 7 mm2,?c = 5.

15?,and n = 1.5 3.otodiode in mm2, and ?c isthe half-angle field-of-view (FOV) of the receiver after the lens. For theanalysis in Section V, we use Ad = 7 mm2,?c = 5.15?,and n = 1.5 3.