Intensity and Polarization Properties of the Partially Coherent Laguerre Gaussian Vector Beams With Vortices Propagating Through Turbulent Atmosphere 2014 Optics and Laser Technology

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Intensity and polarization properties of the partially coherentLaguerreGaussian vector beams with vortices propagating throughturbulent atmosphere

Haiyan Wang a,b,n, Hailin Wang a, Yongxiang Xu a, Xianmei Qian b

a Department of Physics, Nanjing University of Science and Technology, Nanjing 210094, PR Chinab Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Science, Hefei 230031, PR China

a r t i c l e i n f o

Article history:

Received 5 January 2013

Received in revised form

28 April 2013

Accepted 26 June 2013

Keywords:

Atmospheric turbulence

Optical vortices

Polarization

a b s t r a c t

By using the extended HuygensFresnel principle, the analytical expressions for the cross-spectral

density matrix of the partially coherent LaguerreGaussian vector beams with vortices propagating

through atmospheric turbulence are derived theoretically in detail, and used to study the intensity

and polarization properties of partially coherent LaguerreGaussian beams in atmospheric turbu-

lence. It is found that the variations of the intensity of the completely polarized part and the

completely unpolarized part are closely related with the strength of atmospheric turbulence, the

topological charge and the beam width in the source plane. The spectral degree of polarization of the

partially coherent LaguerreGaussian vector beams with vortices tends to a certain value that is

different from the source plane after a sufficiently long propagation distance in turbulent atmo-

sphere. Furthermore, this value is dependent of the strength of turbulent atmosphere, the topological

charge and the beam width in the source plane. The polarization property of the partially coherent

LaguerreGaussian vector beams with vortices can be modulated by modulating the topological

charge and the beam width in the source plane.

& 2013 Published by Elsevier Ltd.

1. Introduction

In the past decades, characterization, generation and propa-

gation of vector beam have been studied extensively due to its

important applications in free-space optical communications,

optical imaging, active laser radar systems and remote sensing

[17]. Recently, more and more attention is being paid to the

vortex beams. Such vortex beams (so called because the phase

circulates about the central null, much like a fluid circulating a

drain) have been investigated for various applications ranging

from being used as information carriers in laser communications

[8] to being employed as optical tweezers and spanners [9].Much works have been done to study the propagation of vortex

beams through nonlinear media, as well as free space propaga-

tion [1013]. From a practical point of view, the Laguerre

Gaussian (LG) beam is the most interesting. It should be noted

that certain LG beams, whose transverse field distribution fea-

tures an azimuthal angular dependence of the form expil,

where l is the topological charge (also azimuthal index), carry

orbital angular momentum (OAM) and are examples of the so-

called optical vortices [14]. Apart from their fundamental impor-

tance, LG beams have found applications ranging from optical

manipulation of BoseEinstein condensates to optical imaging

[1519]. Propagation dynamics of optical vortices in LG beams had

been investigated [20]. Greg and Dipankar [21,22] studied the

principle of topological charge conservation and the trajectory of

an optical vortex propagating in atmospheric turbulence. Chen

explored the intensity distribution and the degree of polarization

of the stochastic electromagnetic LG vortex beam propagating in

free space [23]. Zhong et al. [24] investigated the polarization

properties of partially coherent LG (PCLG) beams in turbulent

atmosphere. Chen and Li [25] have examined the scintillation index

of LG beams propagating in simulated atmospheric turbulence. A

partially polarized partially coherent vector beam can be locally

represented as a sum of completely polarized beam and a com-

pletely unpolarized beam [26]. In this paper, our aim is to explore

the intensity (including the completely polarized and unpolarized

parts) and the polarization properties of a PCLG vector beam with

optical vortices in turbulent atmosphere. Firstly, applying the

extended HuygensFresnel principle, the analytical expressions for

the cross-spectral density matrix of the PCLG vector beams with

vortices have been derived in turbulent atmosphere at an arbitrary

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Optics & Laser Technology

0030-3992/$- see front matter & 2013 Published by Elsevier Ltd.

http://dx.doi.org/10.1016/j.optlastec.2013.06.026

n Corresponding author at: Department of Physics, Nanjing University of Science

and Technology, Nanjing 210094, PR China. Tel.: +86 25 84303071.

E-mail address: [email protected] (H. Wang).

Optics & Laser Technology 56 (2014) 16
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point in the receiver plane. Then, the formulas for the intensity of

the completely polarized and unpolarized parts as well as the

spectral degree of polarization are obtained. Finally, some numerical

results are illustrated.

2. Theoretical analysis

Based on the unified theory of coherence and polarization, the

second-order coherence and polarization properties of the beam

can be characterized by a cross-spectral density matrix of the

electric field, defined by the formula [27]:

W2

s1; s2; 0; Wxxs1; s2; 0; Wxys1; s2; 0;

Wyxs1; s2; 0; Wyys1; s2; 0;

" #1

where

Wijs1; s2; 0; En

i s1; 0;Ejs2; 0; i;j x;y

Exand Ey denote the components of the random electric vector,

along two mutually orthogonal x and y directions perpendicular to

the z-axis. s1 and s2 are the coordinates of two arbitrary points at

the source plane. The asterisk stands for the complex conjugate.

Angle bracket represents the average, taken over an ensemble of

realizations of the electric field in the sense of the coherence

theory in the space-frequency domain. is the frequency, which

can be omitted later for brevity.

The element of the cross-spectral density matrix of the PCLG

vector beams with vortices at the source plane can be expressed in

the form

W0ij r1; r2; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

W0i r1;W0

j r2;q

0ij r1r2; 2

where W0i represents the spectral density of the component Ei of

the electric field and 0ij is the spectral degree of correlation

between the components Ei and Ej in the source plane. These

quantities can be determined experimentally. The spectral degree

of correlation satisfies the inequality j0ij j1. Assume 0ij r1r2;

has a Gaussian profile [28,29], i.e.

0ij r1r2 Bij expr1r2

2=2s2gij 3

where sgij is a positive constant characterizing the correlation

length in the source plane.

The elements of the cross-spectral density matrix of the PCLG

beam with vortices can be expressed as [23,24]

W0ij r1; r2; 1; 2; 0 AiAjBij

2r1r2s

2I

lexp

r21 r22

s2I

expil12expr1r22=2s2gij 4

where sI is the beam width, l is the topological charge, and r;

represent the modulus and the azimuth of the position vector in

the source plane.Using the paraxial form of the generalized HuygensFuneral

principle, the elements of the cross-spectral density matrix of the

PCLG vector beams with vortices propagating in a turbulent

atmosphere are given by [21,24,26]

Wij1;2;z k

2

4z2

Z20

Z20

Z10

Z10

W0ij r1; r2; 1; 2; 0

exp ik

2zr11

2 ik

2zr22

2

!expr1;1;z r2;2;zr1r2dr1 dr2 d1 d2

5

where ; represent the modulus and the azimuth of the

position vector in the output plane. k is the wave number related

to the wave length by k 2=, and r; stands for the random

part of the complex phase of a spherical wave due to the

turbulence. The angular bracket denotes averaging over the

ensemble of turbulent media, which can be expressed as [30]

expr1;1;z r2;2;z

exp0:5Dr1r2;12

expfMr1r22 r1r212 12

2g 6

M 2k2

z3

Z1

0

k3nd 7

where the quantityR1

0 k3nkdk describes the effect of turbu-

lence, nk being the spectrum of the refractive index fluctuations

that can be characterized by the Tatarskii model or the Kolmo-

gorov model, where is 2=l, where l is the size of eddies. The

quantity Mcan be represented as 0:5465C2nl1=30 k

2zfor the Tatarskii

spectrum and as 0:49C2n6=5k

12=5z6=5 for the Kolmogorov spectrum,

with C2n being the refraction index structure constant which

describes how strong the turbulence is and l0 being the inner

scale of turbulence.

In this paper, the Kolmogorov spectrum and a quadratic

approximation of the 5=3 power law for Rytov's phase structure

function are employed, which is accepted to be valid not only for

weak fluctuations but also for strong ones. Eq. (6) can be expressedas

exp r1;1;z r2;2;z

exp 1

20r1r2

2 r1r212 122

h i( )8

where 0 0:545C2nk

2z3=5 denotes the coherence length of a

spherical wave propagating through turbulence.

Substituting (8) and (4) into (5), we can express the elements of

cross-spectral density matrix for a PCELG beams with vortices in

the receiver plane as follows:

Wij1;2;1;2;z AiAjBijk

2

4z2

exp 1

2

0

21 22( )exp

ik

2z

2122& '

exp21220

cos 12

Z20

Z20

Z10

Z10

2r1r2s

2I

l

exp 1

s2I

1

20

1

2s2gij

r21 r

22

( )exp

ik

2zr21r

22

& '

expikr11 cos 11

z

ikr22 cos 22

z

& '

exp 1

20r11 cos 11

( )exp

1

20r12 cos 21

( )

exp1

2

0

r21 cos 12( )exp 1

2

0

r22 cos 22( )exp

r1r2s

2gij

2r1r220

cos 12

( )

expfil12gr1r2dr1 dr2 d1 d2 9

Using following equations [31]:

expikr

zcos

!

n 1

n 1inJn

kr

z

exp in 10

Z20

exp il1 r1r2s

2gij

2r1r220

cos 12

" #d1

2expil2Ilr1r2s

2

gij

2r1r22

0 11

H. Wang et al. / Optics & Laser Technology 56 (2014) 162


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Z20

expimd2 m 0

0 m0

(12

Eq. (9) can be simplified as

Wij1;2;z AiAjBijk

2

z2exp

1

2021

22

( )exp

ik

2z21

22

& '

exp

21220 cos 12

n 1

n 1Z1

0Z1

0

2r1r2s2I

l

exp 1

s2I

1

20

1

2s2gij

r21 r

22

( )exp

ik

2zr21r

22

& '

Jnkr11

z

Jn

kr22z

Jn

r1120

Jn

r1220

Jn

r2120

Jn

r2220

Inlr1r2s

2gij

2r1r220

expin12r1r2dr1 dr2 d1 d2

13

where Jn denotes the Bessel function of order n and Inl denotes

the modified Bessel function of order n l.

In Eq. (5), if we set 1 2 , the elements of the cross-spectral density matrix in the output plane can be expressed as

Wij;z AiAjBijk

2

z2

n 1

n 1

Z10

Z10

2r1r2s

2I

lexp

1

s2I

1

20

1

2s2gij

r21 r

22

( )

exp ik

2zr21r

22

& 'Jn

kr1

z

Jn

kr2

z

Inl

r1r2s

2gij

2r1r220

r1r2dr1 dr2

14

The average intensity of the PCLG vortex beam at the output

plane is expressed as

I;z Tr2W;z Wxx;z Wyy;z 15

where Tr denotes the trace of the matrix.

Assume that the x- and y-components of the electric field are

uncorrelated at each source point (i.e., jBxyj jByxj 0). According

to Eq. (4), we can obtain the off-diagonal elements of the cross-

spectral density matrix in the source plane that tend to zero. That

is to say, W0xy r1; r2; 0 W0

yx r1; r2; 0 0ij.

It was shown in [32] that a completely polarized partially

coherent beam is depolarized (i.e., becomes partially polarized)

after propagation, thus it is important to analyze its state of

polarization. The cross-spectral density matrix of a partially

polarized partially coherent beam at point can be locally

represented as a sum of completely polarized beam and a

completely unpolarized beam [26]

2W;;z 2Wu;;z 2Wp;;z 16

where

W2 u

;;z A;;z 0

0 A;;z

17

W2 p

;;z B;;z D;;z

Dn;;z C;;z

18

with

A;;z 12

"Wxx;;z Wyy;;z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWxx;;zWyy;;z

2 4jWxy;;zj2

q #

19 1

B;;z 12

"Wxx;;zWyy;;z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Wxx;;zWyy;;z2

4jWxy;;zj2q #

19 2

C;;z 12

"Wyy;;zWxx;;z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWxx;;zWyy;;z

2 4jWxy;;zj2

q #19 3

D;;z Wxy;;z 19 4

The degree of polarization of the PCLG vortex beams with

vortices at point is defined by the expression [28]

P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14Det2W;;z

Tr2W;;z2s 20where Det stands for the determinant of the matrix.

Substituting (15) into (20) with 1 2 and considering

jBxyj jByxj 0, we can obtain the polarization degree across the

output-plane which is given by

P;z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4Wxx;;zWyy;;z

Wxx;;z Wyy;;z2

s21

Applying the above derived formulas, the polarization as well

as the intensity of the completely polarized part and the

Fig.1. The influence of the strength of atmospheric turbulence on the variation of the on-axis intensity of the completely polarized part and the completely unpolarized part

of the PCLG vector beams with vortices propagating through turbulent atmosphere. (a) C2n 0 and (b) C

2n 10

13

m2=3 .

H. Wang et al. / Optics & Laser Technology 56 (2014) 16 3


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completely unpolarized part of the PCLG vector beams with

vortices propagating through turbulent atmosphere can be

directly investigated.

3. Numerical calculation and analysis

Now we study the numerical results of the intensity and the

spectral degree of polarization for the PCLG vector beams with

vortices propagating through turbulent atmosphere by using the

formulas derived in the above section. The parameters of numer-

ical simulation about the PCLG vector beam with vortices in thesource plane are: Ax Ay 2; Bxx Byy 1; 632:8 nm;sI 3 mm; l 2; sgxx 3 mm, and sgyy 6 mm. The refraction

index structure constant is C2n 1013 m2=3.

Figs. 13 are intended to show the on-axis intensity distribu-

tion of the PCLG vector beams with vortices propagating through

turbulent atmosphere, its completely polarized and unpolarized

parts. One finds from Figs. 13 that the on-axis intensity of the

completely polarized and unpolarized parts depends on the

propagation condition and the parameters of the source beam.

Both the completely polarized part and the completely unpolar-

ized part have oscillation in the near field of the source plane.

With the increasing of the propagation distance, the on-axis

intensity of the completely polarized and unpolarized parts

increases first. Due to the scattering and beam wandering in the

atmospheric turbulence, the on-axis intensity of the completely

polarized and unpolarized parts decreases with the increasing of

the propagation distance. Fig. 1 indicates that the oscillation in

near field is stronger and the peak value is higher when the PCLG

vector beams are propagating in turbulent atmosphere than in free

space. Generally speaking, the intensity of the completely unpo-

larized part is larger than that of the completely polarized part.

Comparing with Fig. 2(a) and (b), one can find that the on-axis

intensity of the PCLG vector beam with l 0 is higher than that of

the PCLG vector beam with l 4, because there is no dark hollow

in the center of the source beam for l 0. Fig. 3 shows that

variation in the beam width of the source beam can change theintensity distribution of the completely polarized and unpolarized

parts obviously. Although in the near field, the completely unpo-

larized part is in dominated position, the intensity of the com-

pletely polarized part can exceed that of the completely

unpolarized part with the increasing of the propagation distance

for the PCLG vector beams with higher beam width in the

source plane.

Fig. 4 indicates that the spectral degree of polarization of the

PCLG vector beams with vortices is closely related with the

strength of atmospheric turbulence, the topological charge as well

as the correlation property in the source beam. As shown as in

Fig. 4, the on-axis spectral degree of polarization has oscillation

property in the near field that is similar to its intensity distribu-

tion. With the increase of the propagation distance, the spectral

Fig. 2. The influence of the strength of the topological charge on the variation of the on-axis intensity of the completely polarized part and the completely unpolarized part

of the PCLG vector beams with vortices propagating through turbulent atmosphere. (a) l 0 and (b) l 4.

Fig. 3. The infl

uence of the strength of the coherence of the source beam on the variation of the on-axis intensity of the completely polarized part and the completelyunpolarized part of the PCLG vector beams with vortices propagating through turbulent atmosphere. (a) sI 2 mm and (b) sI 4 mm.



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degree of polarization increases firstly and then decreases exceptfor the PCLG vector beams with l 0. When the propagation

distance is sufficiently long, the spectral degree of polarization

tends to a certain constant value. From Fig. 4(a), one can see that

the spectral degree of polarization of PCLG vector beams propa-

gating in free space tends to a higher constant value than that of

PCLG vector beams propagating in turbulent atmosphere. The

stronger the strength of atmospheric turbulence, the lower the

constant value due to the depolarization property of atmospheric

turbulence. Fig. 4(b) indicates that the spectral degree of polariza-

tion of PCLG vector beams with l 0 increases with the increasing

of propagation distance, and then tends to a constant value that is

higher than that of PCLG vector beams with higher topological

charge l. Fig. 4(c) shows that the spectral degree of polarization of

the PCLG vector beams with wider beam width is lower than thatof the vector beam with narrower beam width property. From

Fig. 4(b) and (c), one can conclude that the distribution of the

spectral degree of polarization of the PCLG vector beams during

the propagation can be controlled by modulating the topological

charge and the beam width in the source beam. It is obviously

shown in Fig. 4 that PCLG vector beams with lower topological

charge and poor coherence can reduce the effect of the atmo-

spheric turbulence on the polarization properties.

4. Conclusion

In this paper, based on the extended HuygensFresnel princi-

ple, the explicit expressions for the cross-spectral density matrix

function of PCLG vector beams with vortices propagating throughatmospheric turbulence have been derived and used to study their

propagation properties and evolution behavior of completely

polarized and unpolarized parts in the PCLG vector beams. It is

found from numerical results that the on-axis intensity of the

completely polarized and unpolarized parts of the PCLG vector

beams with vortices has oscillatory behavior in the near field of

the source plane. When the PCLG vector beams with vortices are

propagating in turbulent atmosphere, the on-axis intensity of the

completely polarized and unpolarized parts increases firstly and

then decreases with the increasing of the propagation distance,

which is closely related with the strength of atmospheric turbu-

lence, the topological charge and the beam width of the source

beam. The investigation also shows that the spectral degree of

polarization of the PCLG vector beams acquires a particular valueat a certain distance in turbulent atmosphere, which is different

from the value in the source plane. Furthermore, this value is

dependent on the strength of atmospheric turbulence, the topo-

logical charge and the beam width in the source plane. The results

obtained in this paper would be useful for studying the propaga-

tion dynamics of stochastic electromagnetic vortex beams in

atmospheric turbulence

Acknowledgments

This project was supported by the National Natural Science

Foundation of China (11204135), Research Fund for the Doctoral

Program of Higher Education of China (20113219120039), and

Fig. 4. The spectral degree of polarization of the PCLG vector beams with vortices propagating through turbulent atmosphere (a) with different C2n; (b) with different

topological charge l; and (c) with different sI.

H. Wang et al. / Optics & Laser Technology 56 (2014) 16 5


6/6

Open Research Fund of Key Laboratory of Atmospheric Composi-

tion and Optical Radiation, Chinese Academy of Sciences, China.

The author is indebted to the reviewers for valuable comments.

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