Optics & Laser Technology 50 (2013) 78–86

Contents lists available at SciVerse ScienceDirect

Optics & Laser Technology

0030-39

http://d

n Corr

E-m

huangw

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

Propagation properties of partially coherent higher-order cosh-Gaussianbeam in non-Kolmogorov turbulence

Huafeng Xu a,b, Zhifeng Cui a, Jun Qu a,n, Wei Huang b,n

a Department of Physics, Anhui Normal University, Wuhu 241000, Chinab Laboratory of Atmospheric Physico-Chemistry, Anhui Institute of Optics & Fine Mechanics, Chinese Academy of Sciences, Hefei, Anhui 230031, China

a r t i c l e i n f o

Article history:

Received 17 November 2012

Received in revised form

7 January 2013

Accepted 24 January 2013Available online 22 March 2013

Keywords:

Huygens–Fresnel principle

Partially coherent higher-order cosh-

Gaussian beams

Non-Kolmogorov turbulence

92/$ - see front matter Crown Copyright & 2

x.doi.org/10.1016/j.optlastec.2013.01.025

esponding authors. Tel.: þ86 5533883561.ail addresses: qujun70@mail.ahnu.edu.cn (J. Q

ei@aiofm.ac.cn (W. Huang).

a b s t r a c t

Based on the extended Huygens–Fresnel principle and second-order moments of the Wigner distribu-

tion function (WDF), the analytical formulation for the average intensity, the root-mean-square (rms)

beam width and M2-factor of partially coherent higher-order cosh-Gaussian (ChG) beams propagating

in non-Kolmogorov turbulence are derived. The influences of the beam parameters and the spectrum

parameters associated with the non-Kolmogorov turbulence on the propagation properties of partially

coherent higher-order ChG beams in non-Kolmogorov turbulence are numerically investigated.

Numerical results reveal that partially coherent higher-order ChG beams with higher beam order,

larger decentered parameter and smaller coherence length are less affected by the turbulence.

Furthermore, it is found that the beams will be more affected by the turbulence with smaller inner

scale, larger outer scale or larger structure constant. In addition, numerical results reveal that the

propagation properties of beams in non-Kolmogorov turbulence are largely dependent on the general-

ized exponent a, the structure constant, the inner and outer scale, which is much different from that inthe case of Kolmogorov turbulence. This research may be useful for the practical applications of

partially coherent higher-order ChG beams in connection with optical communications and the remote

sensing.

Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction

For a long time, the studies on beams propagation are mainlybased on Kolmogorov0s power spectrum of refractive indexfluctuations. However, in last several decades, both experimentalresults [1] and theoretical investigations [2] have pointed out thatturbulence in portions of troposphere and stratosphere deviatesfrom predictions of the Kolmogorov model. Since the verticalcomponent is suppressed when the atmosphere is extremelystable, the turbulence is no longer homogeneous in three dimen-sions. Therefore, it is very important and necessary to find newmodels for the solution of this anomalous behavior. Fortunately,Toselli et al. proposed a new theoretical spectrum model, knownas non-Kolmogorov power spectrum [3], by using a power-lawparameter ‘‘a’’ instead of a constant standard exponent value11/3, and a generalized amplitude factor instead of constant value0.033. It is meaningful to analyze the impact of non-Kolmogorovnature of the spectrum on beams propagation when one isdealing with optical communication over slant atmospheric paths

013 Published by Elsevier Ltd. All

u),

or over the horizontal paths in upper layers of the atmosphere.So far, based on this non-Kolmogorov power spectrum, a con-siderable number of interesting work has been presented, suchas second-order statistics of stochastic electromagnetic beams [4],beam propagation factor of partially coherent Laguerre–Gaussianand Hermite–Gassian beams [5,6], propagation of elegantLaguerre–Gaussian beam and cosh-Gaussian beam [7–9], averagespreading of a Gaussian beam array [10,11], angle of arrivalfluctuations for free space laser beam propagation [12], scintilla-tion behavior of cosh Gaussian beams [13] and so on.

In recent years, as a special case of Hermite–sinusoidal-Gaus-sian (HSG) beam, the ChG beam and its propagation characteristicshave attracted extensive attention due to its practical applications[8,9,13–19]. It is well known that a ChG beam can be regarded asthe superposition of four decentered Gaussian beams with thesame waist width. A group of virtual sources that generate a ChGbeam has been proposed [14]. Various intensity profiles can bederived by a suitable choice of beam parameters [8,9] and used inthe space diversity applications in free-space optics (FSO) systems.Considerable theoretical investigations of ChG beams have beenproposed to study the propagation properties in the present ofturbulence atmosphere. For instance, the scintillation [13,16], thepolarization [17], and the reciprocity between the cos- and cosh-Gaussian beams [18] were investigated by Eyyuboğlu. Chu and

rights reserved.

www.elsevier.com/locate/optlastecwww.elsevier.com/locate/optlastechttp://dx.doi.org/10.1016/j.optlastec.2013.01.025http://dx.doi.org/10.1016/j.optlastec.2013.01.025http://dx.doi.org/10.1016/j.optlastec.2013.01.025mailto:qujun70@mail.ahnu.edu.cnmailto:huangwei@aiofm.ac.cnhttp://dx.doi.org/10.1016/j.optlastec.2013.01.025http://dx.doi.org/10.1016/j.optlastec.2013.01.025

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–86 79

Zhou studied the propagation factor [19] and the kurtosis para-meter [20] of ChG beams in a turbulence atmosphere. The proper-ties of cosh-Gaussian and partially coherent cos-Gaussian beamsthrough a paraxial ABCD optical system in turbulent atmospherehave been demonstrated [15,21]. Angular spread of partiallycoherent Hermite–ChG beams through atmospheric turbulencehas been examined [22]. The behaviour of ChG beams and ellipticalChG beams diffracted by a circular aperture in turbulent atmo-sphere were evaluated [23,24]. More recently, the higher-orderChG beams, as the special cases of ChG beams, have also beenextensively investigated [25,26].

Partial coherence, originating from random phase shift and tilt,is a key ingredient of a laser beam. Thus it is of practicalsignificance to incorporate partial coherence into beams anddiscuss the influence of the spatial coherent length on thepropagation properties in turbulence atmosphere. Recently, moreand more attention is being paid to the investigation of partiallycoherent beams in turbulence atmosphere [27–32], which haveshown that partially coherent beams are less affected by turbu-lence than fully coherent beams. But to the best of our knowledge,the propagation of a partially coherent higher-order ChG beam innon-Kolmogorov turbulence has not been reported. In the presentpaper, our main motivations are to understand whether the useof partially coherent higher-order ChG beams as the special casesof ChG beams will improve system performance in turbulentatmosphere, and to study the properties of partially coherenthigher-order ChG beams in non-Kolmogorov turbulence.

2. Theory

2.1. Average intensity and the rms beam width of a partially

coherent higher-order ChG beam in turbulent atmosphere

In the Cartesian coordinate system, taking the z-axis as thepropagation axis, the electric field distribution of higher-orderChG beam in the source plane (z¼0) can be written as [25,26]

Enðx0,y0,0Þ ¼ coshnðO0x0ÞcoshnðO0y0Þexp �

x20þy20w20

!, n¼ 1,2,3,:::

ð1Þ

where n is the beam order, w0 is the initial waist width of theGaussian amplitude distribution, and O0 which has the unit ofm�1, is the parameter associated with the cosh function. Whenn¼0 and n¼1, Eq. (1) reduces to be the well-known Gaussianbeam and ChG beam, respectively. According to [26], the higher-order ChG beam can also be expressed as the form of super-position of nþ1 decentered Gaussian beam with the same waistwidth

Enðx0,y0,0Þ ¼Xnh ¼ 0

Xnl ¼ 0

1

22nn

h

� �n

l

� �exp

p2hþp2l

w20

!exp � ðx0�phÞ

2þðy0�plÞ2

w20

" #

ð2Þ

where ph ¼ ðh�n=2Þdw0 and pl ¼ ðl�n=2Þdw0 with d¼w0O0 beingthe decentered parameter.

The fully coherent beam can be extended to the partiallycoherent one by incorporating the spectral degree of coherenceinto the source beam [27–32], the cross-spectral density functionof a partially coherent higher-order ChG beam is generallycharacterized as

W ð0Þðq01,q02,0,oÞ ¼ Enðq

01,0ÞE

n

nðq02,0Þ

� �¼Xn

h1 ¼ 0

Xnl1 ¼ 0

Xnh1 ¼ 0

Xnl1 ¼ 0

1

24nn

h1

!n

l1

!n

h2

!n

l2

!

expp2h1þp

2l1

w20

" #exp �

ðx01�ph1 Þ2þðy01�pl1 Þ

2

w20

" #

�expp2h2þp

2l2

w20

!exp �

ðx02�ph2 Þ2þðy02�pl2 Þ

2

w20

" #

exp �ðx01�x02Þ

2þðy01�y02Þ2

2s20

" #ð3Þ

where q0 ¼ ðx0 ,y0 Þ is the transverse coordinates in the source plane,

dh i denotes averaging over the field ensemble, s0 is the spatialcorrelation length of partially coherent higher-order ChG beamsat the source plane. Eq. (3) reduces to the cross-spectral density ofa coherent higher-order ChG beams when s0-N.

By using the extended Huygens–Fresnel principle [9,27–32],the cross-spectral density of a partially coherent higher-orderChG beams propagating in turbulence atmosphere can beobtained by

Wðq1,q2,z,oÞ ¼k

2pz

� �2 Z Z Z Zd2q01d

2q02W0ð Þðq01,q

02,0,oÞ

�exp � ik2zðq1�q01Þ

2�ðq2�q02Þ2

h i�Hðq01,q

02,zÞ

� �ð4Þ

where k¼2p/l and l denote the wave number and the wave-length, respectively.

Hðq01,q02,zÞ ¼ exp½cðq1,q

01,zÞþc

nðq2,q02,zÞ�� �

m

¼ expf�Tða,zÞ½ðq1�q2Þ2þðq1�q2Þðq01�q

02Þþðq

01�q

02Þ

2�g ð5Þ

cðq,q0 Þ represents the random part of the complex phase of aspherical wave that propagates from the source point ðq01,0Þ to thereceiver pointðq1,0Þ, and the term of dh im denotes the averageover the ensemble of the turbulence medium statistics, and theasterisk means the complex conjugate. Here, the term Tða,zÞ is setas a quantity to describe the strength of turbulence perturbation,which will be discussed in detail in Section 2.3.

On substituting from Eqs. (3) and (5) into Eq. (4), we obtain theaverage intensity of partially coherent ChG beams propagating innon-Kolmogorov turbulence

Iðq,zÞ� �

¼ k2

4a1a2z2Xn

h1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

1

24nn

h1

!n

l1

!n

h2

!n

l2

!

expb21xþb

21y

a1þb22xþb

22y

a2

!ð6Þ

where

E¼ 12s20þTða,zÞ, a1 ¼

1

w20þ 1

2s20þ ik

2zþTða,zÞ, a2 ¼ an1�

E2

a1

b1x ¼ph1w20þ ik

2zx, b2x ¼

ph2w20� ik

2zxþ E

a1b1x, b1y ¼

pl1w20þ ik

2zy,

b2y ¼pl2w20� ik

2zyþ Ea1

b1y ð7Þ

We have further studied the rms beam width to examine thespreading properties of partially coherent higher-order ChGbeams in non-Kolmogorov turbulence, and the rms beam widthis defined as [10,11,31]

WrðzÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4

Z 1�1

Z 1�1

r2/Iðx,y,zÞSdxdy=Z 1�1

Z 1�1

/Iðx,y,zÞSdxdy

s, ðr¼ x,yÞ

ð8Þ

On substituting from Eq. (6) into Eq. (8), the analyticalformulae for the rms beam width of partially coherent higher-order ChG beams in non-Kolmogorov turbulence is derived.Owing to the symmetry of the rms beam widths in x-direction

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–8680

and in y-direction, we obtain

WxðzÞ ¼WyðzÞ ¼

ffiffiffiffiffiffiffiffiffi4G1G2

sð9Þ

with

G1 ¼Xn

h1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!1

a3þ2 x

21

a23

!

expp2h1þp

2l1

a1w40þt21þt22a2

þ x21þx

22

a3

!ð10aÞ

G2 ¼Xn

h1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!

expp2h1þp

2l1

a1w40þt21þt22a2

þ x21þx

22

a3

!ð10bÞ

a3 ¼k2

4z21

a1þ 1a2

E

a1�1

� �2" #, t1 ¼

1

w20

E

a1ph1þph2

� �, t2 ¼

1

w20

E

a1pl1þpl2

� �

ð10cÞ

x1 ¼ik

2z

ph1a1w20

þ t1a2E

a1�1

� �" #, x2 ¼

ik

2z

pl1a1w20

þ t2a2E

a1�1

� �" #

ð10dÞ

Eqs. (6)–(10) can be used conveniently to study the averageintensity and the rms beam width of partially coherent higher-order ChG beams in non-Kolmogorov turbulence.

2.2. WDF and M2-factor of partially coherent higher-order ChG

beams in turbulent atmosphere

By using the paraxial form of the extended Huygens–Fresnelprinciple, the cross-spectral density of partially coherent higher-order ChG beams propagating in turbulence atmosphere can beobtained by [5,6,27–29]

Wðq,qd,zÞ ¼k

2pz

� �2ZZWðq0 ,q0d,0Þexp

ik

zðq�q0 ÞUðqd�q0dÞ �

�Hðqd,q0d,zÞ�

d2q0d2q0d ð11Þ

To evaluate the above equation, it is convenient to introducenew variables of integration

q0 ¼

q01þq022

, q0d ¼ q01�q

02, q¼

q1þq22

, qd ¼ q1�q2 ð12Þ

and the cross-spectral density in the source plane can beexpressed as

Wðq0 ,q0d,0Þ ¼Gðq01,q02,0Þ ¼G q

0 þq0d2

,q0�

q0d2

,0

� �ð13Þ

Eq. (11) can be expressed in the following alternative form[5,6,27–29]

Wðq,qd,zÞ ¼1

2p

� �2ZZ ZZW q00,qdþ

z

kjd,0

� �exp �iqUjdþ iq00Ujd�Hðqd,qdþ

z

kjd,zÞ

h id2q00d2jd ð14Þ

In Eq. (14), the term exp �H qd,qdþ zk jd,z� �

represents theeffect of the turbulence, and H can be written as

Hðqd,q0d,zÞ ¼z2

k2jd

2þ3 zkjdqdþ3qd2

� �Tða,zÞ ð15Þ

where kd ¼ 9jd9¼ ðk2dxþk2dyÞ

1=2, and rd ¼ 9qd9¼ ðr2dxþr2dyÞ

1=2, and

W q00,qdþz

kjd,0

� ¼W x00,xdþ

z

kkdx,y00,ydþ

z

kkdy,0

�

¼Xn

h1 ¼ 0

Xnl1 ¼ 0

Xnh1 ¼ 0

Xnl1 ¼ 0

1

24nn

h1

!n

l1

!n

h2

!n

l2

!

�exp � 2w20

x002� 1e2

xdþz

kkdx

� 2þ2Ahx00 þBh xdþ

z

kkdx

� " #

�exp � 2w20

y002� 1e2

ydþz

kkdy

� 2þ2Aly00 þBl ydþ

z

kkdy

� " #

ð16Þ

with 1=e2 ¼ ð1=2w20Þþð1=2s20Þ, Ah ¼ ph1þph2� �

=w20, Bh ¼ph1�ph2� �

=w20, Al ¼ pl1þpl2� �

=w20, and Bl ¼ pl1�pl2� �

=w20It is well known that the WDF can characterize partially

coherent beams in space and in spatial frequency domain simul-taneously and can be expressed in terms of the cross-spectraldensity [5,6,27–29]

hðq,h,zÞ ¼ k2p

� �2 Z 1�1

Wðq,qd,zÞexpð�ikhUqdÞd2qd ð17Þ

where vector h¼ ðyx,yyÞ denotes an angle of propagation, kyx andkyy is the wave vector component along the x-axis and y-axis,respectively.

Based on the second-order moments of WDF, the M2-factor ofbeams can be defined as [27–29]

M2ðzÞ ¼ k q2� �

h2D E

� qUh� �2� 1=2

¼ k x2� �þ y2� �Þð y2xD E

þ y2yD E

Þ�ð xyx� �

þ yyy� �� 2� �1=2

ð18Þ

According to definition, moments of the order n1þn2þm1þm2of WDF, it can be expressed as

xn1 yn2ym1x ym2y

D E¼ 1

P

ZZxn1 yn2ym1x y

m2y hðq,h,zÞd

2qd2h ð19Þ

where

P¼ZZ

hðq,h,zÞd2qd2h

¼pw20

2

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!

�expw202ðAh2þAl2Þ

� �ð20Þ

Substituting Eqs. (14)–(17) into Eq. (19), after some tediousintegration, we obtain

q2� �

¼ x2� �þ y2� �

¼ 1P

pw202

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!

expw202ðAh2þAl2Þ

� �

� 4e2

z2

k2þ

w202þ 4z

2

k2Tða,zÞ

� �� z

2

k2ðBh2þBl2Þþ

w404ðAh2þAl2Þ

� �ð21Þ

h2D E

¼ y2xD E

þ y2yD E

¼ 1P

pw202

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!

�expw202ðAh2þAl2Þ

� �4

k21

e2þ3Tða,zÞ

� �� 1

k2ðBh2þBl2Þ

� �ð22Þ

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–86 81

qUh� �

¼ xyx� �

þ yyy� �

¼ 1P

pw202

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!exp

w202ðAh2þAl2Þ

� �

� zk2

4

e2þ6Tða,zÞ

� �� z

k2ðBh2þBL2Þ

� �ð23Þ

Substituting Eqs. (21)–(23) into Eq. (18), the expression ofM2-factor for partially coherent higher-order ChG beams inturbulent atmosphere can be expressed as

M2ðzÞ ¼ k x2� �þ y2� �Þð y2xD E

þ y2yD E�

� xyx� �

þ yyy� �� �2h i1=2

¼ k 1P

pw202

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!8<:

8<:�exp

w202ðAh2þAl2Þ

� �� 4

e2z2

k2þ

w202þ 4z

2

k2Tða,zÞ

� ��

� z2

k2ðBh2þBl2Þþ

w404ðAh2þAl2Þ

�� 1P

pw202

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

8<:

�n

h1

!n

l1

!n

h2

!n

l2

!exp

w202ðAh2þAl2Þ

� �

� 4k2

1

e2þ3Tða,zÞ

� �� 1

k2ðBh2þBl2Þ

� ��

� 1P

pw202

1

24n

Xnh1 ¼ 0

Xnl1 ¼ 0

Xnh2 ¼ 0

Xnl2 ¼ 0

n

h1

!n

l1

!n

h2

!n

l2

!8<:�exp

w202ðAh2þAl2Þ

� �� z

k24

e2þ6Tða,zÞ

� �� z

k2ðBh2þBL2Þ

� ��2)1=2

ð24Þ

2.3. Turbulence model

In the above sections, the term T(a,z) is set as a quantity todescribe the strength of turbulence perturbation, and can beexpressed as [3–13]

Tða,zÞ ¼ 13p2k2z

Z 1�1

k3Fnðk,aÞdk ð25Þ

here k is the magnitude of two-dimensional spatial frequency andFnðk,aÞ denotes the spatial power spectrum of the refractive

Fig. 1. The evolution of normalized average intensity distributions of coherent and pturbulence.

index fluctuations of the atmosphere turbulence. Including boththe inner- and outer-scale effects, the non-Kolmogorov spectrumis defined as [3–13]

Fnðk,aÞ ¼ AðaÞ ~C2

n

exp½�ðk2=k2mÞ�ðk2þk20Þ

, 0rko1 , 3oao4 ð26Þ

where k0 ¼ 2p=L0 with L0 being the outer scale parameter andkm ¼ cðaÞ=l0 with l0 being the inner scale parameter,cðaÞ ¼ ½Gð5�a=2ÞUAðaÞU2p=3�½1=ða�5Þ�, the parameter a is the power-law exponent or the spectral index, AðaÞ ¼ Gða�1Þcosðap=2Þ=4p2with GðdÞ being the gamma function, the term ~C

2

n is the generalizedstructure parameter with unit of m3�a. When the power lawapproaches the limiting value a¼3, the function AðaÞ approacheszero and consequently the refractive-index power spectral densityvanishes.

For non-Kolmogorov model, the integration over the spatialpower spectrum in Eq. (25), one obtains

Tða,zÞ ¼ 13p2k2z

Z 10

k3Fnðk,aÞdk

¼ p2k2zAðaÞ ~C

2

n

6

k2�am bexpðk20=k2mÞGð2�a=2,k20=k2mÞ�2k4�a0a�2

ð27Þ

with b¼ 2k20�2k2mþak2m.However, when a¼11/3, A(a)¼0.033, L0¼N, l0¼0 and

~C2

n ¼ C2n, the spectrum expressed in Eq. (26) reduces to conven-

tional Kolmogorov spectrum [3,10,11], and the spatial powerspectrum of the refractive index fluctuations can be written as[31,32]

FnðkÞ ¼ 0:033C2nk�11=3 ð28Þ

for Kolmogorov model [31,32]

Tð11=3,zÞ ¼ 1=r20 ð29Þ

where r0 ¼ ð0:545C2nk

2zÞ-3=5 is the coherence length (induced bythe atmospheric turbulence) of a spherical wave propagating inthe turbulent medium with C2n being the structure constant.

3. Numerical results and analysis

In this section, the evolution behavior of the normalizedaverage intensity, the spreading of the rms beam width and thenormalized M2-factor of partially coherent higher-order ChGbeams in non-Kolmogorov turbulence are discussed in detail. In

artially coherent higher-order ChG beams propagating through non-Kolmogorov

Fig. 2. Normalized average intensity distribution at z¼1.5 km, the rms beamwidth and the normalized M2-factor of partially coherent higher-order ChG beams

for different values of beam order on propagation in non-Kolmogorov turbulence.

The calculation parameters are n¼3, w0¼0.02 m O0¼80 m�1, and s0¼0.02 m,

Fig. 3. Normalized average intensity distribution at z¼2 km, the rms beam widthand the normalized M2-factor of partially coherent higher-order ChG beams for

different values of the decentered parameter on propagation in non-Kolmogorov

turbulence. The calculation parameters are n¼3, and s0¼0.02 m. The othercalculation parameters are same as Fig. 1.

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–8682

the source plane, the intensity profile of a partially coherenthigher-order ChG beam can be obtained as a Gaussian-like beam,a flat-topped beam and a dark-hollow beam by a suitable choiceof beam parameters; however, it is independent of the initialcoherence. Without loss of generality, we take the dark-hollowbeam as an example. The evolution behaviors of normalizedaverage intensity distributions of coherent and partially coherenthigher-order ChG beams propagating through non-Kolmogorovturbulence are depicted in Fig. 1. It is necessary to note that thenon-Kolmogorov spectrum parameters are set as a¼3.8, L0¼1 m, l0¼0.01 m and ~C

2

n ¼ 1� 10�14 m3�a, and the other beam

parameters are adopted as n¼3, l¼0.85 mm, w0¼0.02 m andO0¼60 m�1. From Fig. 1 we can see that, with increase inpropagation distance through non-Kolmogorov turbulence, the

central dip of the hollow shape gradually disappears, the darkhollow beam tends to a flattened distribution and finally evolvesinto a Gaussian-like profile, which agrees well with the resultsreported in [8,9,26]. Moreover, under the same conditions, thecentral dip of the fully coherent higher-order ChG beam or thebeam with higher spatial coherence can be sustained for longerpropagation distance. This may be caused by the fact that thefluctuation of the source field increases as the coherencedecreases. Figs. 2–4 show the normalized average intensitydistribution, the rms beam width and the normalized M2-factorof partially coherent higher-order ChG beams for different valuesof beam order, decentered parameter and initial coherence on

Fig. 4. Normalized average intensity distribution at z¼2 km, the rms beam widthand the normalized M2-factor of partially coherent higher-order ChG beams for

different values of the initial coherence on propagation in non-Kolmogorov

turbulence. The other calculation parameters are n¼3, w0¼0.02 m, andO0¼60 m�1.

Fig. 5. Normalized average intensity distribution at z¼2 km, the rms beam widthand the normalized M2-factor of partially coherent higher-order ChG beams on

propagation both in Kolmogorov turbulence and in non-Kolmogorov turbulence

for different outer scale and inner scale, respectively. The beam calculation

parameters are n¼3, w0¼0.02 m, O0¼60 m�1 and s0¼0.02 m, and the turbulenceparameters are set as a¼3.667, and ~C

2

n ¼ 1� 10�14 m3�a for non-Kolmogorov

turbulence, and C2n ¼ 1� 10�14 m-2=3 for Kolmogorov turbulence.

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–86 83

propagation in non-Kolmogorov turbulence, respectively. For thecase of beam order n¼1, the partially coherent higher order ChGbeam reduces to the partially coherent ChG beam. As can beexpected by Ref. [26], the conversion of the average intensitydistribution of partially coherent higher-order ChG beams withsmaller beam order or decentered parameter is quicker than thatwith larger one. One finds from Figs. 2 and 3(b) that the partiallycoherent higher-order ChG beam with the large beam order ordecentered parameter has the large effective beam size in thesource plane; however, the beam with smaller beam parametersspreads more rapidly with increasing the propagation distance,and the apparent distinction will disappear after propagating over

a sufficiently long distance. As shown in Fig. 4 (a) and (b), thebehavior of the normalized average intensity distribution and therms beam width are also closely determined by the initialcoherence of the beam. As seen from Figs. 2–4(c), the normalizedM2-factor increase more rapidly with smaller beam order, smallerdecentered parameter or larger initial coherence. One comes tothe conclusion that we can use partially coherent higher orderChG beams instead of ChG beams to improve system performancein turbulence atmosphere.

Figs. 5–7 show the normalized average intensity distribution,the rms beam width and the normalized M2-factor of partially

Fig. 6. Normalized average intensity distribution at z¼2 km, the rms beam widthand the normalized M2-factor of partially coherent higher-order ChG beams in

Kolmogorov turbulence and in non-Kolmogorov turbulence for different para-

meter a, respectively. The beam calculation parameters are the same as Fig.5, theturbulence parameters are set as L0¼1 m, l0¼0.01 m, and ~C

2

n ¼ 1� 10�14 m3�a for

non-Kolmogorov turbulence, and C2n ¼ 1� 10�14 m-2=3 for Kolmogorov turbulence.

Fig. 7. Normalized average intensity distribution at z¼2 km, the rms beam width andthe normalized M2-factor of partially coherent higher-order ChG beams for different

values of the decentered parameter on propagation in non-Kolmogorov turbulence. The

other calculation parameters are n¼3, w0¼0.02 m, O0¼60 m�1,s0¼0.02 m, a¼3.8,L0¼1 m, and l0¼0.01 m.

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–8684

coherent higher-order ChG beams on propagation in turbulentatmosphere for different outer scale, inner scale, parameter a andstructure constant, respectively. From Figs. 5–7(a) it is seen thatthe hollow shape evolves into a Gaussian profile more rapidly inthe turbulence with smaller inner scale, larger outer scale, smallerparameter a or larger structure constant. In Fig. 5(b) and (c), inorder to compare the results in non-Kolmogorov turbulence withthat in Kolmogorov turbulence, the turbulence condition is set tobe a¼3.667 and ~C

2

n ¼ 1� 10�14 m3�a for non-Kolmogorov turbu-

lence, and C2n ¼ 1� 10�14 m-2=3 for Kolmogorov turbulence. It is

clear from Fig. 5(b) and (c) that the propagation properties of

partially coherent higher-order ChG beams in non-Kolmogorovturbulence are much different from that in the case of conven-tional Kolmogorov turbulence. It is largely dependent on theouter scale and inner scale. The rms beam width and the normal-ized M2-factor of the beams spread more rapidly with decreasinginner scale l0 or increasing outer scale L0 in non-Kolmogorovturbulence. It is seen from Fig. 6(b) and (c) that the propagationproperty of partially coherent higher-order ChG beams are alsoclosely dependent on the exponent a in non-Kolmogorov turbu-lence. In addition, as indicated by Fig. 7, the rms beam width andthe normalized M2-factor of the beams spread more rapidly in theturbulence for a larger structure constant. The results can beinterpreted physically as follows. The inner scale l0 forms thelower limit of the inertial range and the outer scale L0 forms the

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–86 85

upper limit of the inertial range. It is to be noted that decrease ininner-scale parameter and increase in outer-scale parametercorresponds to atmosphere turbulence with more intensity.The beam will be affected more in a stronger turbulence becauseof the larger wave front randomness. As a result, the partiallycoherent higher-order ChG beam will be more affected in non-Kolmogorov turbulence with smaller inner scale, larger outerscale or larger structure constant.

Fig. 8 shows the rms beam width of partially coherent higher-order ChG beams versus the propagation distance z in non-Kolmogorov turbulence with ~C

2

n ¼ 10�14 m3�a and in free space

for different values of initial coherence, other parameters are thesame as Fig. 1. It is clear from Fig. 8 that when the initialcoherence is very small, there is no distinct difference betweenthe spreading properties of partially coherent higher-order ChGbeams in turbulence and in free space. As the initial coherenceincreases, the difference becomes apparent, and the difference isquite evident for fully coherent beams. One comes to the conclu-sion that partially coherent higher-order ChG beams with smallerinitial coherence length are less affected by the turbulencealthough the beams with smaller initial coherence have greaterspreading both in turbulent atmosphere and in free space.The physical interpretation is that the beams with lower initialcoherence will suffer larger wave front randomness on theirpropagation, and the existence of substantial original randomnesson wave front reduces the effect of turbulent atmosphere.The results are in good agreement with previous results of[30,31].

To further learn about the influence of other parameters on thespreading properties of partially coherent higher-order ChGbeams in non-Kolmogorov turbulence, numerical results arecompiled in Fig. 9. The plot, shown in Fig. 9(a), illustrates theimpact of a variation on the spreading performance of partiallycoherent higher-order ChG beams in non-Kolmogorov turbulencefor different initial coherence width. As indicated by Fig. 9(a),there is a remarkable increase of the rms beam width with theincreasing of a and consequently a major penalty on systemperformance. However, after it reaches its maximum value(approximately a¼3.078), the curve changes its slope and gradu-ally decreases because the term A(a) begins to decrease to zero.The obvious physical interpretation of a approaching 3 is that theturbulence power spectrum tends to vanish. On the other hand, inthe limiting case of a being close to 4, the power spectrumcontains fewer eddies of high wave numbers, i.e. the wave fronttilt is the primary aberration, leading to an improvement on thebeam spreading. It is indicated, in non-Kolmogorov turbulence,

Fig. 8. The rms beam width of partially coherent higher-order ChG beams in non-Kolmogorov turbulence and in free space with different values of initial coherence.

The other parameters are the same as Fig. 1.

Fig. 9. The rms beam width of partially coherent higher-order ChG beams as afunction of a, s0, w0 and O0 in the reference plane of z¼2 km in non-Kolmogorovturbulence.

that the beam spreading is largely dependent on the generalizedexponent a, which leads to the results much different from those inthe case of Kolmogorov turbulence with a¼11/3. Fig. 9(b) is a plotof the rms beam width of partially coherent higher-order ChGbeams versus the initial Gaussian waist width w0 for differentvalues of the cosh-part parameter O0. It is shown that the rmsbeam width decreases sharply with the increasing of w0 when w0is extremely small; however, with further increases of w0, the rmsbeam width gradually increases, and the beams with larger O0spread more quickly. The plot shown in Fig. 9(c) illustrates theimpact of the cosh-part parameter O0 variation on the beamsspreading for different values of beam order, and the Gaussianwidth is set to w0¼0.02 m. Notice that, when the parameter O0 isvery small (O0r30 m�1), the rms beam width does not seem to

H. Xu et al. / Optics & Laser Technology 50 (2013) 78–8686

change appreciably with O0, and there is no distinct differencebetween the spreading properties of partially coherent higher-order ChG beams with different beam orders. The results can bephysically interpreted as follows. For a small value of the para-meter w0 or O0, the intensity profile of the partially coherenthigher-order ChG beam is similar to a Gaussian distribution, andthe corresponding effective beam width is equal to the Gaussianwidth w0 in the source plane, so it almost shows the samespreading properties. The larger w0, O0 or n means the largereffective beam width. This is the physical reason why the rmsbeam width of partially coherent higher-order ChG beamsincreases with the further increase of w0, O0 and n.

4. Conclusions

In the present paper, the analytical formulas for the averageintensity, the rms beam width and M2-factor of partially coherenthigher-order cosh-Gaussian beams in non-Kolmogorov turbu-lence have been derived by use of the extended Huygens–Fresnel principle and definition of the WDF. We mainly concen-trate on the influences of different beam parameters and turbu-lence parameters on the propagation characteristics of partiallycoherent higher-order ChG beams in non-Kolmogorov turbulence.It has been found that partially coherent higher-order ChG beamswith higher beam order, larger decentered parameter and smallercoherence length are less affected by the turbulence. It meansthat one may use partially coherent higher-order ChG beams asspecial cases of ChG beams to improve the system performanceon propagation in turbulent atmosphere. The decreasing innerscale l0, or increasing outer scale L0 or the structure constant isequivalent to the increasing strength of the turbulence, and as aresult, the partially coherent higher-order ChG beam will be moreaffected. Results show that the inner and outer scale, the expo-nent value a and the structure constant have serious influenceson the beam propagation properties through non-Kolmogorovturbulence and is different from that in the case of Kolmogorovturbulence; that is, a¼11/3. It is to be noted that our results willbe useful for the practical applications of the partially coherenthigher-order ChG beams, such as remote sensing and opticalcommunications.

Acknowledgements

Jun Qu acknowledges the support by the AnHui Provincial NaturalScience Foundation of China under Grant no.11040606M154.Wei Huang acknowledges the support by the National NaturalScience Foundation of China under Grant no. 21033008.

References

[1] Belenkii MS, Barchers JD, Karis SJ, Osmon CL, Brown JM, Fugate RQ.Preliminary experimental evidence of anisotropy of turbulence and the effectof non-Kolmogorov turbulence on wave front tilt statistics. Proceedings ofSPIE 1999;3762:396–406.

[2] Gurvich A, Belen0kii M. Influence of stratospheric turbulence on infraredimaging. Journal of the Optical Society of America A 1995;12:2517–22.

[3] Toselli I, Andrews LC, Phillips RL, Ferrero V. Free-space optical systemperformance for laser beam propagation through non-Kolmogorov turbu-lence. Optical Engineering 2008;47:026003.

[4] Shchepakina E, Korotkova O. Second-order statistics of stochastic electro-magnetic beams propagating through non-Kolmogorov turbulence. OpticsExpress 2010;18:10650–8.

[5] Luo H, Xu H, Cui Z, Qu J. Beam propagation factor of partially coherentLaguerre–Gaussian beams in non-Kolmogorov turbulence. Progress In Elec-tromagnetics Research M 2012;22:205–18.

[6] Chu X. Evolution of beam quality and shape of Hermite–Gaussian beam innon-kolmogorov turbulence. Progress In Electromagnetics Research2011;120:339–53.

[7] Xu H, Cui Z, Qu J. Propagation of elegant Laguerre–Gaussian beam in non-Kolmogorov turbulence. Optics Express 2011;19:21163–73.

[8] Chu X, Qiao C, Feng X. The effect of non-Kolmogorov turbulence on thepropagation of cosh-Gaussian beam. Optics Communications 2010;283:3398–403.

[9] Tao R, Si L, Ma Y, Zhou P, Liu Z. Relay propagation of partially coherent cosh-Gaussian beams in non-Kolmogorov turbulence. Progress In ElectromagneticsResearch 2012;131:495–515.

[10] Zhou P, Ma Y, Wang X, Zhao H, Liu Z. Average spreading of a Gaussian beamarray in non-Kolmogorov turbulence. Optics Letters 2010;35:1043–5.

[11] Wu G, Guo H, Yu S, Luo B. Spreading and direction of Gaussian–Schell modelbeam through a non-Kolmogorov turbulence. Optics Letters 2010;35:715–7.

[12] Toselli I, Andrews LC, Phillips RL, Ferrero V. Angle of arrival fluctuations forfree space laser beam propagation through non kolmogorov turbulence.Proceedings OF SPIE 2007;6551:65510E-1.

[13] Eyyuboğlu H. Scintillation behavior of cos, cosh and annular Gaussian beamsin non-Kolmogorov turbulence. Applied Physics B 2012;108:335–43.

[14] Zhang Y, Song Y, Chen Z, Ji J, Shi Z. Virtual sources for a cosh-Gaussian beam.Optics Letters 2007;32:292–4.

[15] Chu X. Propagation of a cosh-Gaussian beam through an optical system inturbulent atmosphere. Optics Express 2007;15:17613–8.

[16] Eyyuboğlu H. Annular, cosh and cos Gaussian beams in strong turbulence.Applied Physics B 2011;103:763–9.

[17] Eyyuboğlu H, Baykal Y, Cai Y. Degree of polarization for partially coherentgeneral beams in turbulent atmosphere. Applied Physics B 2007;89:91–7.

[18] Eyyuboğlu H, Baykal Y. Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere. Optics Express2004;12:4659–74.

[19] Zhou G. Generalized beam propagation factors of truncated partially coher-ent cosine-Gaussian and cosh-Gaussian beams. Optics and Laser Technology2010;42:489–96.

[20] Chu X. Moment and kurtosis parameter of partially coherent cosh-Gaussianbeam in turbulent atmosphere. Applied Physics B 2011;103:1013–9.

[21] Zhou G, Chu X. Propagation of a partially coherent cosine-Gaussian beamthrough an ABCD optical system in turbulent atmosphere. Optics Express2009;17:10529–34.

[22] Yang A, Zhang E, Ji X, Lü B. Angular spread of partially coherent Hermite–cosh-Gaussian beams propagating through atmospheric turbulence. OpticsExpress 2008;16:8366–80.

[23] Chu X, Ni Y, Zhou G. Propagation of cosh-Gaussian beams diffracted by acircular aperture in turbulent atmosphere. Applied Physics B 2007;87:547–52.

[24] Du X, Zhou G. Propagation of elliptical cosh-Gaussian beams and off-axialelliptical cosh-Gaussian beams in apertured optical systems. Applied PhysicsB 2007;88:267–72.

[25] Zhou G, Zheng J. Beam propagation of a higher-order cosh-Gaussian beam.Optics and Laser Technology 2009;41:202–8.

[26] Zhou G. Propagation of a higher-order cosh-Gaussian beam in turbulentatmosphere. Optics Express 2011;19:3945–51.

[27] Zhu S, Cai Y, Korotkova O. Propagation factor of a stochastic electromagneticGaussian Schell-model beam. Optics Express 2010;18:12587–98.

[28] Dan Y, Zhang B. Beam propagation factor of partially coherent flat-toppedbeams in a turbulent atmosphere. Optics Express 2008;16:15563–75.

[29] Yuan Y, Cai Y, Qu J, Eyyuboglu H, Baykal Y. Propagation factors of Hermite–Gaussian beams in turbulent atmosphere. Optics and Laser Technology2010;42:1344–8.

[30] Wang F, Cai Y, Eyyuboğlu H, Baykal Y, C- il C. Partially coherent elegantHermite–Gaussian beams. Applied Physics B 2010;100:617–26.

[31] Wang F, Cai Y, Eyyuboğlu H, Baykal Y. Average intensity and spreading ofpartially coherent standard and Elegant Laguerre–Gaussian beam in turbu-lent atmosphere. Progress In Electromagnetics Research 2010;103:33–56.

[32] Zhong Y, Cui Z, Shi J, Qu J. Polarization properties of partially coherentelectromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere.Applied Physics B 2011;102:937–44.

Propagation properties of partially coherent higher-order cosh-Gaussian beam in non-Kolmogorov turbulenceIntroductionTheoryAverage intensity and the rms beam width of a partially coherent higher-order ChG beam in turbulent atmosphereWDF and M2-factor of partially coherent higher-order ChG beams in turbulent atmosphereTurbulence model

Numerical results and analysisConclusionsAcknowledgementsReferences