Page 157 - ITU Journal Future and evolving technologies – Volume 2 (2021), Issue 2

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 2

Laser beam at given x value
r

Electric field intensity
of laser I(r, x)

x

x R
θ w b (x)
w 0
x
θ ψ=2θ

Fig. 4 – A Gaussian pro ile of a single laser beam propagating in the ‑direction.

the time to steer between consecutive steering positions. is the radius of curvature of the wavefront of the beam
If is the maximum speed of the incoming target, then it at an axial distance , given by ( ) = [1 + ( 2
) ],
is required that ≫ in order to accurately detect and
s
localize the moving object. and ( ) is the Gouy phase. The divergence of the laser
beam in Fig. 4 is represented by angle for ≫ as

( )
3. DETECTION OF A TARGET = lim arctan ( b ). Similarly, the apex angle of the
→∞
2
In this section, a Gaussian laser beam is considered and cone is given as = 2 , and solid angle Ω = sin .
inc
the blockage to the path of a Gaussian beam is mathemat‑ Moreover, the incident magnetic ield H ( , ) polarized
ically modeled. The model is then used to detect the pres‑ in the ‑direction is given as H (inc) ( , ) = ̂ y (inc) ( , ),
ence of a target. where is the impedance of the free space. 0
0
The incident intensity distribution is given by
3.1 Mathematical modeling of blockage of a
laser beam Re( (inc) × ∗(inc) )
(inc) ( , ) = ,
Laser beams considered in this work are modeled as 2
Gaussian beams [19]. Consider a single Gaussian beam il‑ | | 2 2 −2 2
0
lustrated in Fig. 4. The direction of the propagation of the = 2 0 ( ( ) ) exp ( ( ) ), (3)
2
beam is along the ‑direction and the beam is polarized in 0 b b
the ‑direction. The incident electric ield E (inc) ( , ) for where | 0 | 2 is the intensity at the beam’s waist. In the case
2 0
the beam at a distance from the source, using paraxial of blockage, the incident electric ield is divided into re‑
approximation, given as [20] (rfl)
lected and transmitted electric ields given as E ( , ),
(tx)
E (inc) ( , ) = and E ( , ), respectively. Therefore, the incident elec‑
(rfl)
(inc)
tric ield can be written as E ( , ) = −E ( , ) +
0 − 2 2 (tx) (rfl)
̂ z ( ) exp ( ( ) ) exp ( − ( + 2 ( ) − ( ))), E ( , ). The re lected component, E ( , ) is mainly
0
2
b
b
(1) specular as the wavelength is signi icantly small com‑
pared to the size of the target. The transmitted compo‑
nent is signi icantly small compared to the re lected com‑
where represents the radial distance from the beam’s ponent for solid surface targets. The re lected and trans‑
central axis, is the electric ield amplitude at origin and mitted intensity distributions represented, respectively,
0
at time instance = 0. ( ) is the width of the beam as (rfl) ( , ) and (tx) ( , ) are
0
b
along the direction of propagation, which is given as [20]
2 (inc)
(rfl) ( , ) = Γ ( , ),
1

( ) = √1 + , (2) (tx) ( , ) = Γ ( , ), (4)
2 (inc)
0
b
2
∣ (rfl) ( , )∣
2 where Γ = is the re lection coef icient, and
where is the Rayleigh range given by = 0 0 , 1 ∣ (inc) ( , )∣

is the wavelength, and is the index of refraction of the ∣E (tx) ( , )∣
0
free space. In (1), = 2 0 is the wavenumber, ( ) Γ = ∣E (inc) ( , )∣ is the transmission coef icient.
2

© International Telecommunication Union, 2021 143