Based on the extended HuygensFresnel principle and generalized Stokes theory, the evolution of polarization properties of beams generated by quasihomogenous (QH) sources propagating in clear oceanic water was studied by the use of the oceanic turbulence spatial spectrum function. The results show that the beams have similar polarization selfreconstructed behavior under different turbulence conditions in the far field, but if the propagation distance is not long enough, the degree of polarization (DOP) fluctuates with much more complexity than state of polarization (SOP) of QH beams. The selfreconstructed ability of DOP at the special distance in turbulence would get to the best value if the values of coherence of width were chosen suitably, but for SOP, it has no best value.
I. INTRODUCTION
The QH beams which can be generated by SLMs have been studied and used widely in recent years
[1]
. The coherence and polarization properties of quasihomogenous sources in the far field has been studied by Korotokva
[2]
. The coherence of properties of the field generated by a beam radiated from a quasihomogenous electromagnetic source scattering on QH media has been studied by Xin
[3]
. Chen and Li have been researched the coherence properties and polarization modulation of QH beams scattered from anisotropic media
[4

5]
. The propagation properties of laser beams in random media like turbulence or human tissue are important for applications such as optical imaging, remote sensing and communication systems so more and more researchers are attracted
[6

11]
. The polarization “selfreconstructed” phenomenon in atmosphere turbulence has been known for a long time. The normalized spectrum of the beam generated by QH source after propagating through turbulent media has been proved to be equal to the normalized spectrum of the source
[12]
. The degree of polarization of electromagnetic GaussianSchell beams and partially coherent electromagnetic flattopped beams also have selfreconstructed properties in atmospheric turbulence
[13

14]
. However, recent reports have shown that beams generated from anisotropic sources don’t have polarization selfreconstructed properties in turbulent media
[15

16]
. On the other hand, oceanic turbulence is an important natural random medium but the light propagation though oceanic turbulence is a relatively unexplored topic compared to that in other media. The spatial spectrum model of oceanic turbulence combining temperature and salinity fluctuations is only used by a few articles to study the light propagation in the oceanic waters
[16

18]
.
In this paper, we focused on the polarization features of QH beams propagating through oceanic turbulence. The analytical expressions of onaxis generalized Stokes parameters of QH beams were calculated，then the variations of DOP and SOP of QH beams traveling through oceanic turbulence have been simulated. The selfreconstructed properties of such beams in the far field in turbulent media have been analyzed under different source parameters and turbulence conditions.
II. METHODS
For beams generated by quasihomogenous (QH) sources the spectral density
S
^{(0)}
(
r
) varies much more slowly with position vector than the correlation coefficients
μ_{ij}
^{(0)}
(
r
,
ω
) change with difference of position vector
r_{1}

r_{2}
. The cross spectral density matrix of QH sources can be expressed as:
The superscript (0) denotes quantity pertaining to the incident field. The spectral density can be expressed as the trace of cross spectral density matrix:
(
r_{1}
,
r_{2}
,
ω
),
a_{ij}
(
ω
) is the frequency dependent factor which connects the spectral components:
S_{y}
^{(0)}
(
ω
)=
α
(
ω
)
S_{χ}
^{(0)}
(
ω
).
For
α
=0, the source generated linearly polarized light in the x direction, for
α
→∞, the light is linearly polarized in the y direction. Without loss of generality, the spectral density and correlation coefficients can be described by the GaussianSchell model:
A
is spectral amplitude which we set
A
=1 in this article, and
B_{ij}
is called the coefficient of correlation between the incident field components, for quasihomogenous beams, it can be set as:
B_{xx}
=
B_{yy}
=1,
B_{xy}
=
B^{*}_{yx}
, where the asterisk means complex conjugate.
σ
represents the width of the source,
δ_{ij}
represents the coherent width of the source. Several conditions of the source must be satisfied in order to produce a physically reliable QH beam
[19]
.
The cross spectral density matrix of the beam at the receiving plane can be derived by the extended Huygens Fresnel principle:
Where
r_{1}
,
r_{2}
are two arbitrary position vectors of points on the source plane while
ρ_{1}
,
ρ_{2}
are two position vectors of observation points on the receiving plane.
z
is propagation distance and
is the wave number of the beam. The last term in Eq. (5) represents the correlation function of the complex phase perturbed by random media：
Where
describes the strength of turbulence perturbation.
The model of the spatial power spectrum of refractive index fluctuations of homogenous and isotropic oceanic turbulence can be described as a linearly polynomial of temperature fluctuations and salinity fluctuations:
In Eq. (7),
ξ
is the rate of dissipation of kinetic energy per unit mass of fluid, ranging from 10
^{4}
m
^{2}
s
^{3}
in turbulence surface layer to 10
^{10}
m
^{2}
s
^{3}
in the midwater column.
χ_{T}
is the rate of dissipation of mean squared temperature ranges from 10
^{2}
K
^{2}
s
^{1}
below the oceanic turbulence to 10
^{10}
K
^{2}
s
^{1}
at the midwater column.
η
is the Kolmogorov micro scale. Other parameters in Eq. (7) are:
A_{T}
= 1.863×10
^{2}
,
A_{S}
= 1.9×10
^{4}
,
A_{TS}
=9.41×10
^{3}
,
δ
= 8.284(
κη
)
^{4/3}
+ 12.978(
κη
)
^{2}
; w defines the ratio of temperature to salinity contributions to refractive index spectrum which in oceanic waters varies in the interval [5;0], with 5 and 0 corresponding to dominating temperatureinduced and salinityinduced optical turbulence, respectively.
From Eq. (3)Eq. (6), the cross spectral density matrix of the electromagnetic field on the receiving plane can be expressed as:
As a matter of convenience, we make changes of the spatial arguments:
With the help of Eq. (9), the cross spectral density matrix on the receiving plane is derived:
Where
The polarization properties of an electromagnetic beam at a point in space can be determined by use of Stokes parameters which have recently been generalized from onepoint quantities to twopoint counterparts
[20]
. They contain not only polarization properties but also coherence properties of beams
[21]
and can be measured by Young’s interference experiment
[22

23]
. The changes of generalized Stokes parameters in optical system or random media are also interesting problems and have received a lot of attention
[24

25]
. The four parameters can be expressed as follows:
where
are Pauli spin matrices:
represents the crossspectral density matrix (CSDM) of the beam:
If we only considered polarization properties of one point, i.e.,
ρ
_{1}
=
ρ
_{2}
=
ρ
, From Eq. (10), each element of the CSDM can be derived as the expression of:
Inserting Eq. (11) into Eq. (12), and using the conditions of
M_{xy}
=
M_{yx}
,
α_{xy}
=
α_{yx}
the analytical expressions of four Stokes parameters can be obtained:
The polarization properties can be expressed as the functions of these four parameters.
We considered the degree of polarization and the state of polarization of QH beams in this paper. DOP is used to describe the polarized portion of a beam. DOP=1 means perfectly polarized beam while DOP=0 means unpolarized beam. The state of polarization (SOP) includes the azimuth angle and ellipticity. The azimuth angle
θ
is defined by the smallest angle formed by the positive xdirection and the direction of major semiaxis of the ellipse.
ε
, which determines the shape of the ellipse, is the ratio of the major semiaxis and the minor semiaxis.
For convenient, we normalized three Stokes parameters by
S
_{0}
as:
The DOP and SOP can be expressed as the function of normalized Stokes parameters:
Because of the uniform polarization properties of quasihomogenous beams, we only need to consider the onaxis point of the beam. And in order to prove the polarization selfreconstructed properties of QH beam, we should study farfield polarization properties of normalized Stokes parameters. First, let’s consider the limit as follow:
Where
s_{i}
(0,
z
,
ω
) are normalized Stokes parameters on the receiving plane while
s_{i}
(0, 0,
ω
) are such parameters on the source plane. From Eq. (13),
can be obtained from Eq. (1)  Eq. (4). Then substituting into the limit of the ratio of two parameters, we get:
In turbulent medium,
so
F
=1, then we proved the selfreconstructed property of
s
_{1}
. Other parameters in Eq. (14) can also be proved similarly, then DOP and SOP selfreconstructed properties can be derived directly. We set follow parameters to judge the ability of the self reconstructed property of QH beams:
Variations of turbulence strength due to temperature and salinity fluctuations. (The parameters in Fig. 1 are : (a) χ_{T}=10^{9} K^{2}(1/s)~10^{10}K^{2}(1/s), ξ =10^{4}m^{2}(1/s)^{3}~,10^{5}m^{2}(1/s)^{3}, w=1.5, 2.5, 3.5, (b) χ_{T}=10^{2}K^{2}(1/s)~10^{10}K^{2}(1/s), ξ =10^{4}m^{2}(1/s)^{3}~,10^{10}m^{2}(1/s)^{3}, w=1.5.)
Variations of onaxis generalized normalized Stokes parameters versus propagation distance z at different α : (a) α =0.5, (b) α =1, (c) α =1.5. Other parameters are λ =0.6328 μm, σ =1 cm, B_{xy}=0.25exp(^{iπ}/6), B_{yx}=0.25exp(^{iπ}/6), δ_{yy}=2δ_{xy}δ_{xx}=1.5δ_{xy}, δ_{xy}=0.2 mm.
These parameters are modulus values of relative difference of the onaxis DOP and SOP between the source plane and the receiving plane at a special distance z in turbulences, the selfreconstructed ability will be better when these values get closer to 0.
III. RESULTS
Figure 1
shows the strength of oceanic turbulence changes with turbulence parameters: w,
χ_{T}
and
ξ
. It can be easily found out that the strength of turbulence T increases with the increase of
χ_{T}
and w, but reduces with the increases of
ξ
.
Figure 2
shows the changes of normalized Stokes parameters with distance at different a. The DOP and SOP we calculated in Eq. (17) and Eq. (19) could be expressed as the function of these parameters, so the behavior of generalized Stokes parameters decides the evolution of DOP and SOP of the beam through propagation.
In
Fig. 3
, we illustrated how the DOP and SOP of the QH beams depend on the parameters of oceanic turbulence. The effect of w on the changes of DOP and SOP has been shown in
Fig. 3
(a1)(a3) and the effect of
χ_{T}
and
ξ
has been shown in
Fig. 3
(b1)(b3).
In
Fig. 4
, the influence of coherent width and the coefficients
α_{ij}
(
ω
) on the changes of selfreconstructed properties of DOP of QH beams were revealed. While
δ_{yy}
and
δ_{xx}
are kept fixed,
c
_{0}
is the coefficient to decide the value of
δ_{xx}
as the form
δ_{xx}
=
c
_{0}
δ_{xy}
.
In
Fig. 5
, the selfreconstructed behavior of azimuth angle
θ
and the degree of ellipticity
ε
were judged as the same method of DOP we plotted their variation versus
c
_{0}
.
Variations of the onaxis DOP and SOP versus propagation distance z under different turbulence conditions. a1, b1 are the changes of DOP, b1, b2 are the changes of θ, c1,c2 are the changes of ε. We set α =1, and other parameters are same with Fig. 2. The parameters of oceanic turbulence are set as (a1, b1, c1) χ_{t} =10^{6}K^{2}(1/s), ξ =10^{8} m^{2}(1/s)^{3}. solid curve: w=1.5, dashed curve: w=2.5, dotdashed curve: w=3.5. (a2, b2, c2) w=1.5. solid curve: χ_{t} =10^{6}K^{2}(1/s), ξ =10^{8} m^{2}(1/s)^{3}, dashed curve: χ_{t} =10^{7}K^{2}(1/s), ξ =10^{8} m^{2}(1/s)^{3}, dotdashed curve: χ_{t} =10^{7}K^{2}(1/s), ξ =10^{9} m^{2}(1/s)^{3}.
Variations of P_{d} with c_{0} on the receiving plane at the propagation distance z=10 km under different α. c_{0} is the ratio of δ_{xx} to δ_{xy}. Other parameters are same with Fig. 2. δ_{xy}=0.2 mm, δ_{xx}=c_{0}δ_{xy}, (a) σ =1 cm, δ_{yy}=0.5δ_{xy}, (b) σ =1 cm, δ_{yy}=δ_{xy}.
Variations of ε_{d} and θ_{d} with c_{0} on the receiving plane at the propagation distance z=10 km under different α (a) σ =1 cm, δ_{yy}=0.5δ_{xy}, (b) σ =1 cm, δ_{yy}=0.5δ_{xy}. Other parameters are same with Fig. 2.
IV. DISCUSSION
(1) Both DOP and SOP fluctuate at the range of 0.01 km1 km. DOP and SOP get more fluctuations in weak turbulence and get larger selfreconstructed distance than that in strong turbulence.
(2)
S
_{2}
and
S
_{3}
in ocean waters fluctuate less than
S
_{1}
during the propagation but they will all reach the initial value if the propagation distance is long enough.
(3) DOP varies with more complexity with distance in ocean waters than SOP. The selfreconstructed distance of both decreases with the increase of
χ_{t}
and w, but increases with the increases of
ξ
.
(4) When
δ_{yy}
=
δ_{xy}
,
P_{d}
reaches the minimum value, and it’s almost equivalent for different
α
，but if
δ_{xx}
≠
δ_{xy}
, as what have shown in
Fig. 4
(b),
P_{d}
will get to minimum value at different
c
_{0}
for different
α
.
(5)
ε
_{d}
and
θ
_{d}
increase as increases but will change slowly when
c
_{0}
>1. It means when other parameters are kept fixed, the coherence width
δ_{xx}
should be controlled to be smaller than
δ_{xy}
in order to obtain better SOP selfreconstructed properties, but these two variables of SOP have no minimum values with the variation of
c
_{0}
.
V. CONCLUSION
According to our analysis, the normalized Stokes parameters will be affected by temperature and salinity fluctuations so the DOP and SOP will also be affected. And
s
_{1}
fluctuates much more than
s
_{3}
and
s
_{3}
, DOP varies with more complexity than SOP. Furthermore, both of normalized generalized Stokes parameters of QH beams have proved to be equal to the initial value if the propagation distance is long enough, so the selfreconstructed properties of DOP and SOP of such beams can also be proved. The choice of coherence width of the QH beams can help us to get the best selfreconstructed ability of DOP, but for two parameters of SOP, there are no best choices of these parameters. Our work may be helpful to develop underwater optical imaging or communication system.
Acknowledgements
The work is supported by National Natural Science Foundations of China (Grant No.61077012, 61107011).
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