Propagation of Partially Coherent Bessel Gaussian Beams Carrying Optical Vortices in Non Kolmogorov Turbulence 2014 Optics and Laser Technology

7/22/2019 Propagation of Partially Coherent Bessel Gaussian Beams Carrying Optical Vortices in Non Kolmogorov Turbulence

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Propagation of partially coherent BesselGaussian beams carrying

optical vortices in non-Kolmogorov turbulence

Zhiyuan Qin, Rumao Tao, Pu Zhou, Xiaojun Xu, Zejin Liu n

College of Optoelectric Science and Engineering, National University of Defense Technology, Changsha 410073, China

a r t i c l e i n f o

Article history:

Received 2 April 2013

Received in revised form15 July 2013

Accepted 3 August 2013

Keywords:

Propagation

BesselGaussian beams

Non-Kolmogorov turbulence

a b s t r a c t

The analytical formulae for the average intensity of the propagation of partially coherent BesselGaussian

beams (BGBs) with optical vortices in non-Kolmogorov turbulence have been derived based on using a

coherence superposition approximation of decentered Gaussian beams and the extended Huygens

Fresnel principle. The influences of optical vortices, partially coherence and the non-Kolmogorov

turbulence on irradiance distributions are investigated by numerical examples. Numerical results reveal

that the vortex characteristics of the partially coherent BGBs are less affected by the turbulence with

larger topological charge. It is shown that the characteristic of the existence of vortex in the irradiance

distribution has been lost and the doughnut beam spot becomes a circularly Gaussian beam spot during

propagation. The propagation of the beam is different from that in the case of Kolmogorov turbulence

and the propagation properties of partially coherent BGBs in non-Kolmogorov turbulence are closely

related to the beam parameters and the turbulence parameters. The spreading effects due to diffraction

and coherence of initial beams can be neglected after long distance propagation.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the past decades, due to its important applications, such as infree-space optical communication, remote sensing of atmosphere,

and target tracking, the propagation of laser beams through a

turbulent atmosphere has attracted considerable theoretical and

practical interest [1,2]. Various types of laser beams propagating

through turbulent atmosphere have been investigated [311].

However, these studies above cited have been mainly restricted to

the beams without optical vortices and to the case of the ideal

atmospheric turbulence, such as the Kolmogorov model.

Recent experiment results revealed that, in the real atmosphere,

turbulence in some portions of the atmosphere, such as in the

troposphere and the stratosphere, deviates from Kolmogorov's

model, and in the case of laser propagation along the vertical

direction, the turbulence also indicates strongly a non-Kolmogorov

character [1216]. Then a non-Kolmogorov model is presented [17],

which is more general and reduces to the Kolmogorov model only

for the generalized exponent 11/3. It has been reported that,based on this non-Kolmogorov spectrum, laser beams provide a

different property when propagating in non-Kolmogorov turbu-

lence [1216]. On the other hand, laser beams possessing wave-

front singularities known as optical vortices, have become the focus

of many investigations because of their interesting properties as

well as because of their potential applications [1823].

As is well known, Bessel beams are one of the typical examples

of singularity beams and, due to its nondiffracting property in free

propagation and its advantages over other hollow light beams forguiding or trapping atoms, has attracted a lot of attention [2429].

The properties of BGB in turbulent atmosphere have been studied

[3032] and it was both theoretically and experimentally demon-

strated that spatially partially coherent beams are less affected by

turbulence than fully coherent ones [2,3335]. However, the pro-

pagation of partially coherent BGBs in non-Kolmogorov turbulence

has not been examined until now. In this manuscript, our aims are

to investigate the propagation of partially coherent BGBs in non-

Kolmogorov turbulence. Analytical formulas for averaged intensity

distribution are derived and some useful results are found.

2. Analysis of theory

By adopting the Gaussian exponential expansion, A BGB withcharge n can be quite well approximated as a combination of M

decentered Gaussian beams (dGBs), that is [30]

E r!; 0 JnR

w20

!exp

2

w20 in

!exp ik

2

2F

% 1Min

exp R2

4w20

!exp ikx

2 y22F

M1

m 0exp xi=2R cosm

2 yi=2R sin m2w20

im" #

1

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/optlastec

Optics & Laser Technology

0030-3992/$- see front matter & 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.optlastec.2013.08.002

n Corresponding author. Tel.: 86 731 4573763 8111; fax: 86 731 84514127.E-mail addresses: [email protected] , [email protected] (R. Tao),

[email protected] (Z. Liu).

Optics & Laser Technology 56 (2014) 182188
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where (j r!j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 y2p

,) is the polar coordinates and r!(x, y).

m nm0, 0 2=M, m m0, while n is a integer termed as aspiral number or topological charge. F is the phase front radius of

curvature. Here, F40 represents a convergent beam and Fo0

represents a divergent beam. Carefully adjusting the parameters n,

M, and R, the results evaluated by Eq. (1) and directly by the BGB

may be quite well consistent. For example, for M12, R4w0 andn1, 2, 3 and 4, over 99% of the optical energy is converted into a

BGB with charge n.By introducing a Gaussian term of the spectral degree of

coherence, the fully coherent BG beam can be extended to the

partially coherent one [3638]. With the help of Eq. (1), cross-

spectral density function of the partially coherent BGBs at the

source plane z0 can be written as

W r!1; r!2; 0

1M2

exp

x

21 x22y21 y22

w20x1x2

2 y1y222s20

ikx21 y21x22 y22

2F

M1

p 0

M1

q 0expinpq0

exp

iRx1 cos p0x2 cos q0 iRy1 sin p0y2 sin q0

w20" #

2In Eq. (2) s0 is the spatial correlation length of the partially

coherent BGB, and k is the wave number related to the wave

length by k 2=.The average intensity distribution on the z plane can be

calculated by using extended HuygensFresnel diffraction integral

incorporating random inhomogeneous media as follows [511]

I r!;z k2z

2 Z11

Z11

Z11

Z11

dr10!dr20

!Wr10

!; r2

0!; 0

exp

ik

2z r!r1

!2 r!r2

!2

exp r!; r

1!n r!; r2! 3

where r1!; r1!

is the complex phase function that depends on theproperties of the turbulence medium. denotes the average over

the ensemble of the turbulent medium, and

exp r!; r1!

n r!; r2!

exp 42k2zZ1

0

Z10

ddn;1J0jr2!

r1!

j( )

4

where is the magnitude of two-dimensional spatial frequency, J0is the Bessel function of the first kind and zero order, and n;denotes the spatial power spectrum of the refractive-index fluc-

tuations of the atmosphere turbulence. Including both the inner-

and outer-scale effects, the non-Kolmogorov spectrum is definedas [12,13]

n; HeC2nexp2=2m2 20=2;0ro1; 3oo4;

5

where H 1d cos =2=42,d denotes the Gammafunction, 0 2=L0 and m c=l0, in which c f5=2dHd2=3g1=5, l0 and L0 are the inner- andouter-scale, respectively. The term eC2n is the generalized structureparameter with units m3

. Applying Eq. (5) into (4), one obtains

[39]

exp r!; r!1n r!; r!2atmosphere exp 1

32k2zT r

!1 r!2j2

o;

6a

where the expression of T can be written as

TZ1

0

3nd

HeC2n

2

2m exp20=2m2=2; 20=2m2402 6b

22022m 2m, and d;d denotes the incomplete Gammafunction.

Upon substituting Eqs. (2), (4) and (5) into Eq. (3), one canobtain that

I r!;z k2zM

2

M1

p 0

M1

q 0expinpq0

Z11

Z11

Z11

Z11

dr10!dr20

!

exp x21 x22 y21 y22

w20x

1x22 y1y222s20

(

ikx21 y21 x22 y22

2F

exp iRx

1 cos p0x2 cos q0 iRy1 sin p0y2 sin q0w20

" #

exp( ik2z r!r1!2 r!r2!2)exp 132k2zTr1!r2!2( )7

which can be rewritten as

I r!;z 1M2

M1

p 0

M1

q 0expinpq0Ixx;zIyy;z 8

with

I;z k

2z

Z11

Z11

d1d

2

exp 21 22

w20

1222s20

ik21 22 2F

( )

expiR

1cos p

0

2cos q

0w20" #exp ik

2z

h1222

i ; x;y

exp 132k2zT122

9

Eq. (9) can be rewritten as

I;z k

2z

Z11

d2exp ik

2z1z

F

1

w20 1

2s201

32k2zT

" #22

(

2 iR cos q0w20

ik2z

!2

)

Z11

d

1exp ik2z 1zF 1w20 12s20 132k2zT" #21(2 1

2s201

32k2zT

!2

iR cos p0w20

ik2z

" #1

)10

Recalling the integral formulaZ11

expC2x2 Dxdx ffiffiffiffi

p

Cexp

D2

4C2

!11

after very tedious but straightforward integral calculations, Eq.

(10) can be written as

Ixx;z 1

exp A1

G21 G22w2

02

z2i

4b1

k2b22b23 !

Z. Qin et al. / Optics & Laser Technology 56 (2014) 182188 183


3/7

exp 1w20

22x2 2G1 G2

z2w20i

kb2b3

x

( )12a

Iyy;z 1

exp A2

G23 G24w20

2 z

2i

4b1k2b24b25

!

exp

1

w2022y2

2G3

G4

z2w20i

k b4

b5

y ( )12b

with

A1 z2R2 cos p0 cos q02

k22w20

z2

2

3T;zz

T;zF

1zs20

!224 z

w20

3T;zz

T;zF

1zs20

! 1

w20T;z 1

2s20

!#13a

A2 z2R2 sin p0 sin q02

k22w2

0

z2

2

3T;zz

T;zF

1zs2

0 !2

24

zw20

3T;zz

T;zF

1zs20

! 1

w20T;z 1

2s20

!#13b

G1 z2R

2k

2cos p0zw20

3T;zz

T;zF

1zs20

! cos p0 cos q0

" #13c

G2 z2R

2k

2cos q0zw20

3T;zz

T;zF

1zs20

! cos p0 cos q0

" #13d

G3 z2R

2k

2sin p0zw20

3T;zz

T;zF

1zs20

! sin p0 sin q0

" #13e

G4 z2R

2k

2sin q0zw20

3T;zz

T;zF

1zs20

! sin p0 sin q0

" #13f

b1 2T;z

w20zk

212z3

13g

b2 kR

w20z2

1 cos p0 244

cos p0 cos q0

13h

b3 kR

w20z2

1 cos q0 244

cos p0 cos q0

13i

b4 kR

w20z2

1 sin p0 244

sin p0 sin q0

13j

b5 kR

w20z2

1 sin q0 244

sin p0 sin q0

13k

1 1z

F; 2

2z

kw20; 3

2z

kw0s0; 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8z2T;z

k2

w20

s13l

Fig. 1. Normalized intensity profiles of BGBs for different n with 3.33. (a) n1, (b) n2, (c) n3 and (d) n4.

Z. Qin et al. / Optics & Laser Technology 56 (2014) 182188184


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2 21 22 23 24 13m

T;z 132k2Tz 13n

with 22022m 2m and d;d denoting the incompleteGamma function. The parameters 1, 2, 3, and 4 describe theinfluence of the geometrical magnification, diffraction, coherence

of initial beams and turbulence, respectively. Here we employed a

quadratic approximation of the 5/3 power law for Rytov's phase

structure function, which has been shown to be a good approximation

for the second-order moments for practical situations [4042], and has

been used widely [513]. Because this paper only concerns the average

intensity, the quadratic approximation is suitable.

Eq. (8) with Eqs. (12) and (13) is the main results of the

manuscript, which provide a useful and reliable tool to study the

propagation properties of BGBs with optical vortices in the non-

Kolmogorov turbulence.

Fig. 2. Normalized 3D intensity profiles of BGB at several propagation distance.



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3. Numerical calculation and analysis

For the sake of simplicity, we assume that the BGB emitting

from the transmitter is collimated, namely, F 1. The normalizedaverage intensity is defined as

INormalized r!;z I r!;z

P014

where P0 is the total power at initial plane and is given as

P0 Z1

1I r!; 0d2 r!

w20

2M2

M1

p 0

M1

q 0expinpq0exp

R2

4w201 cos pq0

( )15

The normalized transversal average intensity distributions at

several propagation distances for four different values of topolo-

gical charge n are shown in Fig.1 with 1 m, w0 0.2 m, M12,R4w0 and s0 0.5 m. The parameters for the non-Kolmogorovturbulence is l0 0.001 m, L0 5 m, 3.33 and eC2n 1 1014m3To learn about the structure of the intensity distribution of a partially

coherent BGBs, we also calculated in Fig. 2 the normalized 3D-irradiance distribution and the corresponding contour graph at

several propagation distances in non-Kolmogorov turbulence with

3.33. One sees from Figs. 1 and 2 that the doughnut beam profiledisappears gradually and the on-axis irradiance increases during

propagation within certain propagation distance, the beam spot with

flat-topped profiler can be formed at certain propagation distance and

the doughnut beam spot becomes a circularly solid beam spot in the

far field, which means the characteristic of the existence of vortex in

the irradiance distribution has been lost during propagation, and can

be explained by the fact that the atmospheric turbulence degrades

the coherence of the beam, which results in the interesting irradiance

distribution during propagation. One also finds from Fig. 1 that, for

smaller values of the topological charge, such annihilation occurs

earlier (at around 2.5 km for n1), as opposed to n4 case whichoccurs at longer propagation distance (at around 4.5 km). This means

that vortex characteristic of the partially coherent BGBs with largertopological charge is less affected by the turbulence.

Fig. 3 shows the normalized intensity profiles for different

beam parameters and turbulence parameters at z6 km. One canfind from Figs. 1 and 3 that the beam spot spreads more rapidly for

a larger n, eC2n and smaller s0, . One can also find that thepropagation of the partially coherent BGBs in non-Kolmogrov

turbulence is different from that in the case of Kolmogorov

turbulence, that is, 11/3E3.67. As shown in Fig. 3(c),Fig. 4 shows the normalized on-axis irradiance distribution of a

partially coherent BGB along z in a turbulent atmosphere for

different beam parameters and turbulence parameters. It is

revealed that the on-axis irradiance of a partially coherent BGB

increases gradually during propagation until its value reaches its

maximal at certain propagation distance. As z increases further,

the value of on-axis irradiance decreases due to the spreading of

the beam spot. The increase of the on-axis irradiance means that

the light energy concentrating into the central part of the beam

from the outer ring during propagation. On the other hand, due to

the spreading of the beam spot, the on-axis energy decreases. It

also shows that the conversion from a doughnut beam spot to a

circularly solid beam spot becomes quicker for a smaller n, s0,

Fig. 3. Normalized intensity profiles of BGBs for different beam parameters and turbulence parameters with z6 km. (a) s0 0.05 m, 3.33, eC2

n 1 1014m3

(b) n3, 3.33, eC2n 1 1014m3 (c) n3, s0 0.05 m, eC2n 1 1014m3 and (d) n3, s0 0.05 m, 3.33.



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and larger eC2n. From aforementioned discussions, one comes to theconclusion that the propagation properties of partially coherent

BGBs in non-Kolmogorov turbulence are closely related to the

beam parameters and the turbulence parameters.

Because a BGB can be regarded as a superposition of dGB, the

spreading of a BGB can be reflected by the change of the waist

width (see Eq. (13m)). The variation of , 2, 3 and 4 with

propagation distance is shown in Fig. 5. From Fig. 5 and Eq. (13m)

we can see that the variation ofis much small when 2, 3 and 4are much less than unity, and is mainly determined by 4(turbulence) when the propagation distance is larger than 15 km.

This can be understood easily from Eqs. (13l)(13n), because 2and 3 are proportional to z, and 4 is proportional to z

3/2, the

effects due to diffraction and coherence of initial beams can be

neglected if z is larger enough.

4. Conclusions

BGBs are one of the typical examples of singularity beams and,

by representing the BGB as the combination of the dGBs, its

propagation in non-Kolmogorov turbulence is studied analytically

based on the extended HuygensFresnel diffraction integral. In

comparison with [30], where the propagation of BGBs throughKolmogorov atmospheric turbulence, and the spreading and direc-

tion of BGBs through non-Kolmogorov atmospheric turbulence

were investigated, in our paper, the effect of non-Kolmogorov

power spectrum on the propagation and the average intensity

and vortex evolution has been studied in detail. In patircular,

our results have been interpreted physically by examining the

nonmonotonic variation of on-axis irradiance distribution of a

partially coherent BGB for different beam parameters and turbu-

lence parameters. We have derived the analytical propagation

formulae for the average irradiance of a partially coherent BGB,

and investigated the irradiance and spreading of a partially

coherent BGB with optical vortices in non-Kolmogorov turbulence.

Our results show that the characteristic of the existence of vortex

in the irradiance distribution has been lost and the doughnut

Fig. 4. Normalized on-axis irradiance of a partially coherent BGB along zin non-Kolmogorov turbulence for different values of beam parameters and turbulence parameters.

(a) s0 0.05 m, 3.33, eC2n 1 1014m3 (b) n3, 3.33, eC2n 1 1014m3 (c) n3, s0 0.05 m, eC2n 1 1014m3 and (d) n3, s0 0.05 m, 3.33.

Fig. 5. The variation of, 2, 3 and 4 with propagation distance where 11, n3,3.67, eC2n 1 1014m3.



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beam spot becomes a circularly Gaussian beam spot during

propagation. The vortex characteristics of the partially coherent

BGBs are less affected by the turbulence with larger topological

charge. The propagation properties are different from that in the

case of Kolmogorov turbulence; that is, 3.67. The propagationproperties of partially coherent BGBs in non-Kolmogorov turbu-

lence are closely related to the beam parameters and the turbu-

lence parameters. The effects due to diffraction and coherence of

initial beams can be neglected after long distance propagation.These should be taken into consideration for designing and

evaluating the performance of vortex beam propagation in a real

environment.

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