February 10, 2008 / Vol. 6, No. 2 / CHINESE OPTICS LETTERS 79
Propagation of Helmholtz-Gauss beams in
weak turbulent atmosphere
Yixin Zhang (
ÜÜÜ
ººº
###
) and Tuo Zhu (
ÁÁÁ
ÿÿÿ
)
School of Science, Southern Yangtze University, Wuxi 214122
Received June 13, 2007
Based on the Ryt ov approximation of light propagation in weak turbulent atmosphere, the closed-form
expressions of field and average irradiance of each one of the four fundamental families of Helmholtz-Gauss
(HzG) beams: cosine-Gauss beams, stationary Mathieu-Gauss beams, stationary parabolic-Gauss beams,
and Bessel-Gauss beams, which are propagating in weak turbu lent atmosphere, are obtained. The results
show that the field and average irradiance can be written as the product of four factors: complex amplitude
depending on the z-coordinate only, a Gaussian beam, a factor of complex phase perturbation induced by
atmospheric turbulence, and a complex scaled version of the transverse shape of the non-diffracting beam.
The effect of weak atmospheric turbulence on irradiance distribution of th e HzG beam can be ignored.
OCIS codes: 010.1300, 010.1290, 010.1330.
Propagation of light through atmospher ic turbulence has
recently attracted renewed attention due to the emer-
gence of high-capacity free-space optical communication
systems. Propagatio n in the atmosphere is significantly
influenced by turbulence. Therefore, it is usually desired
to find ways to reduce the turbulent effects on the prop-
agating optical beam
[1−5]
. A great number of studies
about using partially coherent light sources as a method
for reducing the turbulent effects and improving the sys-
tem performance have been reported. Non-diffracting
beams as new light sources have attracted attention since
Durnin firs t r eported the generation of Bessel beams in
1987
[6]
. Thereafter, several exact non-diffracting solu-
tions of the wave equation have also been re ported, for
instance Mathieu beams in elliptic c oordinates
[7,8]
and
parabolic beams in par abolic coordinates
[9]
. It was ob-
served that the disturbance of turbulent atmosphere on
non-diffracting light beam is less than that of conven-
tional light beams
[10]
. In this paper, we study the propa-
gation of a non-diffracting Helmholtz-Gauss (HzG) beam
in weak turbulent atmosphere.
Let us suppose that a monochromatic wave E (~r
ρ
, z)
with time dependence exp (−jωt) propa gates in z direc-
tion which ha s a disturbance a cross the plane z = 0 given
by
E
0
(~r
ρ
) = exp
−r
2
ρ
/w
2
0
W (~r
ρ
, k
ρ
) , (1)
where ~r
ρ
= (x, y) = (ρ, ϕ) denotes the transverse coor-
dinates, W (~r
ρ
, k
ρ
) is the transverse pattern of an ideal
non-diffracting beam W (~r
ρ
, k
ρ
) exp (jk
z
z), and w
0
is the
waist size of a Gaus sian envelope. The transverse (k
ρ
)
and longitudinal (k
z
) components of the wave vector
~
k
satisfy the relation
~
k
2
=
~
k
2
ρ
+
~
k
2
z
.
The transverse distribution W (~r
ρ
, k
ρ
) of the ideal
non-diffracting beam fulfills the two-dimensional (2D)
Helmholtz equation and can be expr e ssed as a superpo-
sition of plane waves whose transverse wave numb e rs k
ρ
are restricted to a single value, that is
[11]
W (~r
ρ
, k
ρ
) =
π
Z
−π
˜
E (ϕ) exp [jk
ρ
(x cos ϕ + y sin ϕ)] dϕ, (2)
where
˜
E (ϕ) is the angular spectrum of the ideal non-
diffracting beam. Because non-diffracting beams can be
expanded in terms of plane waves, E (~r) is given by
E (~r) =
w
0
w (z)
exp
"
−
r
2
ρ
w
2
(z)
+ j
kz +
kr
2
ρ
2R(z)
− Θ (z)
!#
×exp
r
2
0
w
2
(z)
−
r
2
0
w
2
0
exp
−j
kr
2
0
2R (z)
W
~r
ρ
µ
, k
ρ
, (3)
with r
2
= r
2
ρ
+ z
2
, w (z) = w
0
1 + z
2
/z
2
R
1/2
, µ
−1
=
[w
0
/w (z)] exp [−jΘ (z)], R (z) = z + z
2
R
/z, Θ (z) =
arctan (z/z
R
), r
0
= k
ρ
w
2
0
/2. Here z
R
= kw
2
0
/2 is the
usual Rayleigh range of a Gaussian beam
[11]
. E quation
(??) is a solution o f the homogeneous Helmholtz equation
under the paraxial r e gime throughout the whole space.
Based on the Rytov approximation
[2,3,12]
, we give the
expression for the field E (~r) at any point in the atmo-
spheric turbulence half-s pace z > 0 as
E (~r) =
w
0
w (z)
exp
"
−
r
2
ρ
w
2
(z)
+ j
kz +
kr
2
ρ
2R(z)
− Θ (z)
!#
×exp
r
2
0
w
2
(z)
−
r
2
0
w
2
0
exp
−j
kr
2
0
2R (z)
×W
~r
ρ
µ
, k
ρ
exp [ψ
1
(~r)] , (4)
where ψ
1
is the complex phase pe rturbation due to weak
turbulence and can be express e d as
ψ
1
= χ + jS, (5)
in which χ and S are the log amplitude and the phase,
respectively, at a point in the output plane.
1671-7694/2008/020079-04
c
2008 Chinese Optics Letters