Optics Communications 303 (2013) 38–41

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Optics Communications

0030-40http://d

n CorrE-m

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

Spiral spectrum of Laguerre–Gaussian beam propagationin non-Kolmogorov turbulence

YueSong Jiang a, ShuaiHui Wang a,n, JiaHua Zhang b, Jun Ou a, Hua Tang a

a School of Electronic and Information Engineering, Beihang University, Beijing 100191, Chinab Center for Earth Observation and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China

a r t i c l e i n f o

Article history:Received 17 February 2013Received in revised form10 April 2013Accepted 12 April 2013Available online 28 April 2013

Keywords:Spiral spectrumNon-Kolmogorov turbulenceOrbital angular momentumLaguerre–Gaussian beam

18/$ - see front matter & 2013 Elsevier B.V. Ax.doi.org/10.1016/j.optcom.2013.04.013

esponding author. Tel.: +86 1 0823 39 195.ail address: wshcan912@hotmail.com (S. Wan

a b s t r a c t

Analytical formulas for the spiral spectrum of the Laguerre–Gaussian (LG) beam in non-Kolmogorovturbulence have been derived. The influence of the propagation distance, azimuthal index, wavelength,exponent parameter α, inner scale and outer scale on spiral spectrum is investigated. Numerical resultsreveal that spiral spectrum of LG beam in non-Kolmogorov turbulence is more affected by turbulencewith longer distance, larger azimuthal index, smaller inner scale, larger outer scale and larger structureconstant. The spiral spectrum of LG beam in non-Kolmogorov turbulence spreads significantly with theincreasing of exponent parameter α and spreads slightly after reaching a maximum point.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

It has been identified that beams comprising s helical phasefronts, described by a phase term exp ðisφÞ, carry an orbital angularmomentum (OAM) of ℏs per photon in 1992 [1]. Beams which havea expðisφÞ helical phase structure are called optical vortices, havingattracted great attention in optical communication and remotesensing [2–7]. In principle, azimuthal index s can take any integervalue. Compared with the traditional encoding method usingamplitude and polarization, encoding with OAM can significantlyincrease the system capacity.

However, for actual beams, the existence of atmosphericturbulence will induce the spread of spiral spectrum of vortexbeams, leading to the increase of BER and decrease of capacity[4,8–11]. Anguita et al. [8] numerically analyzed the effects ofturbulence on the multichannel free space optical communicationsystem based on OAM-carrying beams. Paterson [6] investigatedthe effects of Kolmogorov turbulence on coherence for commu-nication systems based on OAM. Zhang et al. [12] established thetheoretical models of the OAM crosstalk probabilities for singlephotons propagation in non-Kolmogorov channel with the tilt,coma, astigmatism and defocus aberration. In this paper, the spreadof spiral spectrum of LG beam propagating in non-Kolmogorovturbulence is investigated.

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g).

2. Analytical formulas

LG beams are typical and realizable beams carrying OAM. Thefield distribution of LG at the source plane (z¼0) can be expressedin cylindrical coordinates as [1]

uiðr;φ;0Þ ¼ E0 exp −rw0

� �2" #

Lsp 2rw0

� �2" # ffiffiffi

2p

rw0

!s

expðisφÞ; ð1Þ

where r and φ are the radial and azimuthal coordinates, respectively,w0 is the beam radius of zero-order Gaussian beam at the waist, s isthe azimuthal index, also called topological charge, p is the radialindex, and E0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p!ðpþ jsjÞ!π

pis a constant. When s≠0, the LG

beams contain optical vortices and the intensity cross-sectionsconsist of a radially symmetric shape with no on-axis intensity.

To elucidate the orbital angular momentum content, or spiralspectrum, any optical beam can be decomposed into a superpositionof spiral harmonics expðimφÞ in cylindrical coordinates as [13,14]

uðr;φ; zÞ ¼ 1ffiffiffiffiffiffi2π

p ∑∞

m ¼ −∞amðr; zÞexpðimφÞ; ð2Þ

where amðr; zÞ ¼ ð1=ffiffiffiffiffiffi2π

pÞ R 2π0 uðr;φ; zÞexpð−imφÞdφand the energy of

eachmodem can be given by Cm ¼ R∞0 jamðr; zÞj2r dr. Then the weightof each angular momentum state of any field distribution is deter-mined by

Pm ¼ Cm

∑∞−∞Cq

: ð3Þ

The weight Pm of each angular momentum statem is the detectionprobability of angular momentum statem and the weight Pm in which

Fig. 1. The weight of angular momentum state of LG beam propagating in non-Kolmogorov turbulence for different values of the (a) radial index and the(b) wavelength, where Pms is the weight of angular momentum state while m isequal to transmission OAM state s.

Y. Jiang et al. / Optics Communications 303 (2013) 38–41 39

m¼s is the detection probability of transmission angular momentumstate s. The spread of spiral spectrum of transmission angular momen-tum s reflects the crosstalk of angular momentum s.

With the extended Huygens–Fresnel integral, the complexamplitude of LG beam at propagation distance z from the sourcein free space is represented by

u0ðρ;ϕ; zÞ ¼w0Aw

exp −ρ

w

� �2� � ffiffiffi2

pρ

w

!s

Lsp 2ρ

w

� �2� �exp ð−isϕÞ

�exp ið2pþ sþ 1Þ tan−1 zz0

� �−i

kρ2

2RðzÞ þ ikz� �

; ð4Þ

where ρ and ϕ are the radial and azimuthal coordinates, respec-

tively, k¼ 2π=λ is the wave number,w¼w0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðz=z0Þ2

qis the

beam radius at distance z, z0 ¼ ð1=2Þkw20 is the Rayleigh range,

RðzÞ ¼ z½1þ ðz0=zÞ2�; and A¼ E0½1þ ðz=z0Þ2�−0:5.Using the Rytov approximation, the LG beam which propagates

in weak turbulence at distance z can be represented as [6,12,15]

uðρ;ϕ; zÞ ¼ u0ðρ;ϕ; zÞexp½ψðρ;ϕ; zÞ�; ð5Þwhere ψðρ;ϕ; zÞ is the complex phase perturbation induced byatmospheric turbulence.

amðρ; zÞ2 ¼ 1ffiffiffiffiffiffi2π

pZ 2π

0uðρ;ϕ; zÞexpð−imϕÞdϕ

� 1ffiffiffiffiffiffi2π

pZ 2π

0uðρ;ϕ; zÞexpð−imϕÞdϕ

" #n

¼ 12π

Z 2π

0

Z 2π

0uðρ;ϕ1; zÞexpð−imϕ1Þunðρ;ϕ2; zÞexpðimϕ2Þdϕ1dϕ2;

ð6Þ

where n denotes complex conjugate. Substituting Eqs. (4 and 5)into Eq. (6), we obtain the following expression:

jamðρ; zÞj2 ¼A2

2πw0

2

w2

2ρ2

w2

� �s

Lsp2ρ2

w2

� �� �2exp −

2ρ2

w2

� �

�Z 2π

0

Z 2π

0⟨ψðρ;ϕ1; zÞ þ ψnðρ;ϕ2; zÞ⟩

�exp½iðs−mÞðϕ1−ϕ2Þ�dϕ1dϕ2: ð7Þ

The term ⟨ψðρ; ρ1; zÞ þ ψnðρ; ρ2; zÞ⟩ can be expressed as [16–19]

⟨ψðρ; ρ1; zÞ þ ψnðρ; ρ2; zÞ⟩¼ exp −13π2k2z

ρ1−ρ22Z ∞

0κ3ΦnðκÞdκ

� �:

ð8ÞΦnðκÞ denotes the spatial power spectrum of the refractive-indexfluctuations of the atmospheric turbulence and κ is the magnitude oftwo-dimensional spatial frequency. As a model of the atmosphericturbulence, we adopt the non-Kolmogorov spectrum [20,21]

Φnðκ; αÞ ¼ AðαÞ ~C2nexp½−ðκ2=κm2Þ�ðκ2 þ κ02Þα=2

;0≤κo∞;3oαo4; ð9Þ

where ~C2n is the structure constant of the refractive-index fluctua-

tions of the turbulence, AðαÞ ¼ Γðα−1Þcosðaπ=2Þ=4π2; κ0 ¼2π=L0; κm ¼ cðαÞ=l0; in which cðαÞ ¼ ½Γð5−ðα=2ÞÞAðαÞð2π=3Þ�½1=ðα−5Þ�;L0 is the outer scale and l0 is the inner scale. Let T ¼ R∞

0 κ3ΦnðκÞdκ,the term T is expressed as [19]

T ¼Z ∞

0κ3ΦnðκÞdκ

¼ AðαÞ ~C2n

2κ2−αm β expðκ20=κ2mÞΓð2−α=2; ðκ20=κ2mÞÞ−2κ4−α0

α−2; ð10Þ

where β¼ 2κ02−2κm2 þ ακm2 and Γ denotes the gamma function.

Substituting Eqs. (8) and (10) into Eq. (7), jamðρ; zÞj2becomes

amðρ; zÞj2 ¼ exp −23π2k2zρ2T

� �Z 2π

0

Z 2π

0exp

2π2k2zρ2cosðϕ1−ϕ2Þ3

T

" #�exp½iðs−mÞðϕ1−ϕ2Þ�dϕ1dϕ2: ð11Þ

By use of the formulas [22]

expðcos ϕÞ ¼ ∑∞

l ¼ −∞ilIlðxÞexpðilϕÞ; ð12Þ

Z 2π

0expðimϕÞ ¼

2π ðm¼ 0Þ0 ðm≠0Þ ;

(ð13Þ

where Ilð⋅Þ is the lthorder modified Bessel function of the first kindthe function jamðρ; zÞj2 can be expressed as

jamðρ; zÞj2 ¼2πA2w0

2

w2

2ρ2

w2

� �s

Lsp2ρ2

w2

� �� �2

�exp −2ρ2

w2 −23π2k2zρ2T

� �Im−s

2π2k2zρ2

3T

!: ð14Þ

Therefore, the weight of each angular momentum state of LGbeam at the z plane after propagating in non-Kolmogorov

Y. Jiang et al. / Optics Communications 303 (2013) 38–4140

turbulence can be expressed as

Pm ¼ Cm

∑∞−∞Cq

; ð15Þ

where

Cm ¼ 2π2A2w02

w2

Z R

0

2ρ2

w2

� �s

Lsp2ρ2

w2

� �� �2

�exp −2ρ2

w2 −23π2k2zρ2T

� �Im−s

2π2k2zρ2

3T

!ρdρ: ð16Þ

3. Numerical results and analysis

Here we illustrate numerical results of the spiral spectrum ofLG in non-Kolmogorov turbulence by using the analytical formulasin the previous section.

Fig. 1 plots the weight of angular momentum state of LG beampropagating in non-Kolmogorov turbulence (λ¼ 632:8 nm inFig. 1(a) and s¼1 in Fig. 1(b)) for different values of the azimuthalindex and the wavelength, respectively. Other parameters are

Fig. 2. The spiral spectrum of LG beams propagating in non-Kolmogorov turbu-lence at a fixed distance z¼ 1 km for different α: (a) α¼3.3 and (b) α¼ 3:8. Otherparameters are s¼ 1; p¼ 0; w0 ¼ 0:01 m; λ¼ 632:8 nm; ~C

2n ¼ 1� 10�14 m3−α ; l0 ¼

0:01m; and L0 ¼ 1 m:

p¼ 0; w0 ¼ 0:01 m; α¼ 3:8; ~C2n ¼ 1� 10−14 m3−α; l0 ¼ 0:01 m; and

L0 ¼ 1 m. It is shown that the spiral spectrum of LG beams spreadssignificantly with the increase of propagation distance z, and thespiral spectrum spreads lesser with smaller azimuthal index s andlarger wavelength λ in turbulence.

Fig. 2 plots the spiral spectrum of LG beam propagating in non-Kolmogorov turbulence for different α (α¼ 3:3 in Fig. 2(a) andα¼ 3:8 in Fig. 2(b)). The parameters are s¼1, z¼ 1 km,andλ¼ 632:8 nm; other parameters are the same as in Fig. 1. We candeduce that the spread of spiral spectrum varies with exponentparameter α. Fig. 3 plots the dependence of the weight of angularmomentum state m¼ 1 on α. Other parameters are the same asFig. 2. It is shown that the spiral spectrum spreads considerablywith the increase of α. But after reaching the maximum point(approximately 3.1), the spiral spectrum spreads slightly. Thisphenomenon is closely related with the term T in Eq. (16). It isseen from Eq. (10) that T describes the strength of turbulence, thatis, the the larger T, the stronger the turbulence. Therefore, thespiral spectrum spreads significantly when T is large (Fig. 4).

Fig. 3. The weight P1 of angular momentum state m¼ 1 of LG beam through non-Kolmogorov turbulence as a function of α for z¼ 1 km. Other parametersare s¼ 1; p¼ 0; w0 ¼ 0:01m; λ¼ 632:8 nm; ~C

2n ¼ 1� 10�14 m3−α ; l0 ¼ 0:01m; and

L0 ¼ 1 m:

Fig. 4. T as a function of α. The parameters are ~C2n ¼ 1� 10�14 m3−α ; l0 ¼

0:01m; and L0 ¼ 1 m:

Fig. 5. The weight P1 of angular momentum state m¼1 of LG beam through non-Kolmogorov turbulence as a function of z for different values of l0, L0 and ~C

2n . Other

parameters are s¼1, p¼0, w0¼0.01 m, and λ¼632.8 nm.

Y. Jiang et al. / Optics Communications 303 (2013) 38–41 41

Fig. 5 plots the weight P1 of angular momentum state m¼1 ofLG beam propagating in non-Kolmogorov turbulence as a functionof z for different values of inner scale l0, outer scale L0 and

structure constant ~C2n. The parameters are L0 ¼ 1m; ~C

2n ¼ 1�

10�14 m3−α in Fig. 5(a), l0 ¼ 0:01 m; ~C2n ¼ 1� 10�14 m3−αin Fig. 5

(b) and l0 ¼ 0:01 m; L0 ¼ 1 m in Fig. 5(c), α¼ 3:8 and otherparameters are the same as in Fig. 2. As indicated by Fig. 5, spiralspectrum of LG beam spreads more significantly in non-Kolmogorov turbulence for smaller l0, larger L0 and larger struc-ture constant ~C

2n. The inner scale l0 forms the lower limit of the

inertial range and the outer scale L0 forms the upper limitof the inertial range. Smaller inner scale l0 or larger outer scaleL0 means there are more turbulence cells in turbulence. Therefore,LG beams are more affected and the spiral spectrum spreads moresignificantly.

Conclusions

In this paper, analytical formulas for spiral spectrum of LGbeam in non-Kolmogorov turbulence have been derived. Resultsindicate that spiral spectrum of LG beam in non-Kolmogorovturbulence is less affected by turbulence with shorter distance z,smaller azimuthal indexm, larger inner scale l0, smaller outer scaleL0 and smaller structure constant ~C

2n. The spiral spectrum of LG

beam in non-Kolmogorov turbulence spreads significantly withthe increasing α and spreads slightly after reaching a maximumpoint. These results may be useful in practical communication.

Acknowledgments

This work was supported by the Project of National NaturalScience Foundation of China under Grant 61101005, and by the100 Talents Program of the Chinese Academy of Sciences (Remotesensing and assessment of drought and derivative disasters underglobal change). The authors are thankful for these supports.

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