Partially coherent vortex beams propagating through slant atmospheric turbulence and coherence vortex evolution

ARTICLE IN PRESS Optics & Laser Technology 42 (2010) 428–433

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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec

Partially coherent vortex beams propagating through slant atmospheric turbulence and coherence vortex evolution Jinhong Li a, Hongrun Zhang b, Baida Lu¨ a, a b

Institute of Laser Physics and Chemistry, Sichuan University, Chendu 610064, China Department of Applied Physics, Sichuan University, Chengdu 610065, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 25 April 2009 Received in revised form 12 August 2009 Accepted 12 August 2009 Available online 15 September 2009

Using the extended Huygens–Fresnel principle and the quadratic approximation of the phase structure function, and taking the Gaussian Schell-model (GSM) vortex beam as a typical example of partially coherent vortex beams, the explicit expressions for the cross-spectral density function and average intensity of GSM vortex beams with topological charge m = + 1 propagating through slant atmospheric turbulence are derived, and used to study the propagation properties of GSM vortex beams in atmospheric turbulence along a slant path and evolution behavior of coherence vortices. It is shown that the spreading of GSM vortex beams along a horizontal path is larger than that along a slant path in the long atmospheric propagation. The propagation through horizontal atmospheric turbulence can be treated as a special case of the altitude-independent structure constant. The position of coherence vortices in slant atmospheric turbulence does not coincide with that in horizontal atmospheric turbulence, and the dependence of position of coherence vortices on the zenith angle, wavelength and reference point is illustrated by numerical examples. A comparison with the previous work is also made. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Atmospheric turbulence Slant path Coherence vortex

1. Introduction The propagation of laser beams through atmospheric turbulence has attracted much attention because of important applications, such as in connection with optical communications, remote sensing and optical radar, etc. [1,2]. Wu et al. first pointed out that partially coherent beams are less sensitive to the effects of turbulence than fully coherent ones [3]. A further study of the meaning of the insensitivity to turbulence of partially coherent beams was made by Wolf et al. [4,5], and the theoretical prediction was confirmed experimentally [6]. The propagation properties of a variety of optical beams propagating through atmospheric turbulence have been extensively studied [7–12]. For some practical applications the propagation of laser beams through slant turbulent atmosphere is required [13,14]. The average intensity of finite laser beams propagating through slant turbulent atmosphere was calculated by Zhang and Wang [13]. Chu et al. studied the propagation of a multi-Gaussian beam through slant turbulent atmosphere by using the average structure constant C¯n2 [14]. This paper is devoted to the study of the propagation properties of partially coherent vortex beams through atmospheric turbulence along a slant path and evolution behavior of their coherence vortices based on the extended Huygens–Fresnel

 Corresponding author.

¨ E-mail addresses: [email protected] (J. Li), [email protected] (B. Lu). 0030-3992/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2009.08.019

principle and the quadratic approximation. In Section 2, taking the Gaussian Schell-model (GSM) vortex beam as a typical example of partially coherent vortex beams, the cross-spectral density function of GSM vortex beams in atmospheric turbulence along a slant path is derived. The evolution of normalized intensity of GSM vortex beams is described in Section 3. The dependence of the position of coherent vortices on the zenith angle, wavelength and reference point is analyzed and illustrated by numerical examples in Section 4. Finally, Section 5 provides the concluding remarks and comparison with the previous work.

2. Cross-spectral density function of a GSM vortex beam through slant atmospheric turbulence Consider a GSM vortex beam whose cross-spectral density function at the plane L=0 is expressed as [15] W ð0Þ ðq1 ; q2 ; 0Þ ¼ ½r1x r2x þ r1y r2y þ i sgnðmÞr1x r2y  i sgnðmÞr2x r1y jmj

exp 

r21 þ r22 w20

!

"

exp 

# ðq1  q2 Þ2 ; 2s20

ð1Þ

where qi (rix, riy) (i= 1, 2) is the two-dimensional position vector at the source plane L= 0, w0 and s0 denote the waist width and spatial correlation length, respectively. sgn(d) specifies the sign function, m is the topological charge, and we take m = + 1.

ARTICLE IN PRESS J. Li et al. / Optics & Laser Technology 42 (2010) 428–433

In accordance with the extended Huygens–Fresnel principle [2], the cross-spectral density function of GSM vortex beams propagating through atmospheric turbulence along a slant path is given by 

k

2 ZZ

d2 r1

Wðq0 1 ; q0 2 ; LÞ ¼



/exp½c ðq1 ; q0 1 Þ þ cðq2 ; q0 2 ÞS ¼ exp½12Dc ðq1  q2 ; q0 1  q0 2 Þ;

ð3Þ

0

where Dc(q1  q2, q 1  q 2) represents the phase structure function [2,16–18] Z H Dc ðq1  q2 ; q0 1  q0 2 Þ ¼ 2:914k2 secðxÞ Cn2 ðhÞjð1  ZÞðq1  q2 Þ h0

þ Zðq0 1  q0 2 Þj5=3 dh:

    2 k ik B3 2 2 exp  ðq0 1  q0 2 Þ exp  ðq0 1  q0 2 Þ2 2pL 2L 2    ZZ ZZ v2  d2 u d2 v u2   iðux vy  uy vx Þ 4 !     2 2 ik 0 ik exp  2 u exp ðq 1  q0 2 Þ  u exp  u  v L L w0     ik 0 B2 expðAv2 Þ exp ðq1 þ q20 Þ  v exp  ðq10  q20 Þ  v ; 2 2L 

ð2Þ

where  denotes the complex conjugate, k is the wave number related to the wavelength l by k= 2p/l, q0 i (r0 ix, r0 iy) is the position vector at the L plane, /  S denotes the average over the ensemble, and w(q0 , q) represents the random part of the complex phase of a spherical wave due to the turbulence, and can be written as [16,17]

0

and substituting Eqs. (1) and (3) into Eq. (2), we obtain

ZZ

d2 r2 W ð0Þ ðq1 ; q2 ; 0Þ 2pL   ik exp  ½ðq0 1  q1 Þ2  ðq0 2  q2 Þ2  2L  /exp½c ðq0 1 ; q1 Þ þ cðq0 2 ; q2 ÞS;

Wðq0 1 ; q0 2 ; LÞ ¼

429

ð4Þ

In the literature the quadratic approximation of the phase structure function is extensively used to evaluate the normalized coherence function, spectral degree of coherence and degree of polarization, etc. based on the cross-spectral density function (or mutual coherence function) [17–22], which is shown to be a good approximation for practical situations, and for the slant propagation is given by

ð11Þ with A¼

1 1 B1 ; þ þ 2 2w20 2s20

ð12Þ

Eq. (11) represents the cross-spectral density function of GSM vortex beams with topological charge m= + 1 propagating through slant atmospheric turbulence. If Cn2 is independent of the altitude h, from Eqs. (6)–(8) we obtain B1 = B2 =B3, thus Eq. (11) reduces to the cross-spectral density function of GSM vortex beams through horizontal atmospheric turbulence.

3. Evolution behavior of GSM vortex beams propagating through slant atmospheric turbulence On placing q0 1 = q0 2 = q0 into Eq. (11), the average intensity of GSM vortex beams at the L plane is written as    ZZ v2 d2 u d2 v u2   iðux vy  uy vx Þ 2pL 4 !     2 2 ik ik 0 2 q v : ð13Þ exp  2 u exp  u  v expðAv Þexp L L w0

Iðq0 ; LÞ ¼ Wðq0 ; q0 ; LÞ ¼



k

2 ZZ

Dc ðq1  q2 ; q0 1  q0 2 Þ ¼ B1 ðq1  q2 Þ2 þ B2 ðq1  q2 Þðq0 1  q0 2 Þ

þ B3 ðq0 1  q0 2 Þ2 ;

B1 ¼ 2:914k2 secðxÞ

Z

B2 ¼ 2:914k2 secðxÞ

H

H h0

B3 ¼ 2:914k2 secðxÞ

Z

Making use of the integral formula [23] Z

h0

Z

ð5Þ

H h0

Cn2 ðhÞð1  ZÞ2 dh;

xn expðpx2 þ 2qxÞdx ¼ n!exp

ð6Þ

   2 rffiffiffiffi n E½n=2 1 p k q p q X ; p p p ðn  2kÞ!k! 4q2 k¼0

ð14Þ 2Cn2 ðhÞZð1  ZÞ dh;

ð7Þ

Cn2 ðhÞZ2 dh;

ð8Þ

Z = 1 (h  h0)/(H h0) for the uplink propagation, H denotes the altitude between the ground level and the receiver, h0 the height above ground level of the transmitter, L=(H h0)sec(x) is propagation distance in a slant path, and x the zenith angle. Cn2(h) describes the variation of the structure constant versus the altitude h. If Cn2(h) is given, Eqs. (6)–(8) can be integrated analytically, one of the most widely used models is the Hufnagel– Valley (H–V) model which is given by [2]

the straightforward integral calculations yield   2   k2 ðD2 r0 2 þ CÞ D Iðr0 ; LÞ ¼  D exp  r0 2 C 4L2 AC 3 ! k2 p2 w20 ðG2 r0 2 þ FÞ G2 r0 2  ; exp F 32L2 F 3 where C¼

ð9Þ

2 k2 þ ; w20 4AL2

ð16Þ

k2 w20 ; 8L2

ð17Þ

F ¼ Aþ

Cn2 ðhÞ ¼ 0:00594ðu=27Þ2 ð105 hÞ10 expðh=1000Þ þ2:7  1016 expðh=1500Þ þCn2 ð0Þexpðh=100Þ;

ð15Þ

D¼

k2 ; 4AL2

ð18Þ

2

where u is the wind speed, Cn (0) a nominal value at ground level. In our calculations, we use the H–V5/7 model with u= 21 m/s and Cn2(0) = 1.7  10 14 m  2/3. Introducing two new variables of integration u¼

q1 þ q2 2

; v ¼ q1  q2 ;

ð10Þ

ik ð19Þ 2L Eq. (15) with Eqs. (6), (12) and (16)–(19) provides the analytical propagation expression for the average intensity of GSM vortex beams through slant atmospheric turbulence, which depends on the altitude between the ground level and the receiver H, the G¼

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J. Li et al. / Optics & Laser Technology 42 (2010) 428–433

height above ground level of the transmitter h0, the zenith angle x, wavelength l and correlation length s0. Fig. 1 gives the normalized average intensity distributions I(r0 x, r0 y, L)/I(rx, ry, 0)max of a GSM vortex beam through atmospheric turbulence along a slant path with x =601 (solid curves) and a horizontal path (dotted curves) versus the slanted axis r0 r, where I(rx, ry, 0)max denotes the maximum intensity at the source plane L=0, the calculation parameters are l =1.06 mm, w0 =3 cm, s0 =2.7 cm and h0 =0. As can be seen, an intensity profile with central dip whose minimum is equal to a half of the maximum intensity appears at L=1.035 km, and at L=2.18 and 2.9 km a flat-topped profile and a Gaussian profile appear, while the GSM vortex beam propagates through horizontal atmospheric turbulence (dotted curves). However, the central half-maximum intensity deep, flat-topped and Gaussian profiles appear at L=1.27, 4.85, and 10 km, respectively, for the GSM vortex beam propagating through slant atmospheric turbulence (solid curves). Consequently, the evolution of GSM vortex beams along a horizontal path is faster than that along a slant path. Additionally, Fig. 1 indicates that the spreading of the GSM vortex beams along a horizontal path is larger than that along a slant path in the long atmospheric propagation (say, LZ5 km). Therefore, the slant path is more beneficial to the beam propagation through atmospheric turbulence in comparison with the horizontal propagation. The result is physically reasonable, because the structure constant Cn2(h) in Eq. (9) decreases with increasing altitude h, so that the effect of turbulence along the slant path is less than that along the equal horizontal path.

4. Evolution of coherence vortices After tedious but straightforward integration of Eq. (11), the closed-form expression for the cross-spectral density function of GSM vortex beams with m= + 1 propagating through slant atmo-

spheric turbulence is expressed as   ik B3 Wðq0 1 ; q0 2 ; LÞ ¼ exp  ðr0 21  r0 22 Þ  ðq0 1  q0 2 Þ2 2 2L    M2  M1   iðM3  M4 Þ ; 4

ð20Þ

where M1 ¼

M2 ¼

k2 Bx By ðD2x þ D2y þ CÞ 4L2 AC 3 k2 w20 ðG2x þG2y þ FÞ 8L2 F 3

exp

D2x þ D2y C

! ;

ð21Þ

!   G2x þ G2y k2 w20 0 0 2 exp ; exp  ð q  q Þ 1 2 F 8L2 ð22Þ !

M3 ¼

  k2 w20 0 k2 w0 Bx Dx Gy D2x G2y 2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  exp þ ; ð r  r Þ 1y 2y C F 8L2 4L2 2AC 3 F 3

ð23Þ

M4 ¼

!   D2y G2x k2 w20 0 k2 w0 By Dy Gx 2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  exp þ ; ð r  r Þ 1x 2x C F 8L2 4L2 2AC 3 F 3

ð24Þ

   2  B2 k2 Bx ¼ exp  ðr0 1x  r0 2x Þ2 ðr0 1x þ r0 2x Þ2 exp 2 16A 16AL   ikB2 0 2 02 exp  ðr 1x  r 2x Þ ; 8AL    ik ikB2 k2 0 0 ðr0 1x  r0 2x Þ þ ð r þ r Þ þ 1x 2x ; 4AL L 4AL2

Dx ¼

1 2

Gx ¼

 2 2    k w0 B2 1 ik 0 0 0 ð  r  r Þ ; ðr 1x þ r0 2x Þ þ 1x 2x 2 2 2L 4L2

ð25Þ

ð26Þ

ð27Þ

Fig. 1. The normalized average intensity distributions I(r0 x, r0 y, L)/I(rx, ry, 0)max of a GSM vortex beam through atmospheric turbulence along a slant path (solid curves) and along a horizontal path (dotted curves) versus the slanted axis r0 r.

ARTICLE IN PRESS J. Li et al. / Optics & Laser Technology 42 (2010) 428–433

due to the symmetry, By, Dy, and Gy can be obtained by the replacement of r0 1x, r0 2x in Bx, Dx and Gx with r0 1y, r0 2y, respectively. The spectral degree of coherence is defined as [24]

mðq0 1 ; q0 2 ; LÞ ¼

Wðq0 1 ; q0 2 ; LÞ ½Iðq0 1 ; LÞIðq0 2 ; LÞ1=2

;

(3.50, 1.29 cm), (1.60, 10.80 cm) in Fig. 2(b), respectively. Fig. 2(c) and (d) indicates that the position of coherence vortex varies with the variation of the zenith angle x. The positions of two coherence vortices approach to those along a horizontal path when the zenith angle x-901. For example, for x = 89.991 two coherence vortices with m0 = 71 are located at (3.50, 1.29 cm), (1.60, 10.80 cm), respectively, in Fig. 2(c)–(d) which are consistent with those in Fig. 2(b). The position of coherence vortices of GSM vortex beams through slant atmospheric turbulence versus the wavelength l and reference point q0 1 are depicted in Fig. 3(a)–(d), for x = 601 and the other calculation parameters are the same as those in Fig. 2. From Fig. 3(a)–(d) we see that the position of coherence vortices depends on the wavelength l and the choice of the reference point, but the number and topological charge of coherence vortices remain unchanged.

ð28Þ

where I(q0 i, L)= W(q0 i, q0 i, L) is given by Eq. (15). The position of coherence vortices is determined by [25] Re½mðq0 1 ; q0 2 ; LÞ ¼ 0;

ð29aÞ

Im½mðq0 1 ; q0 2 ; LÞ ¼ 0;

ð29bÞ

where Re and Im denote the real and imaginary parts of m(q 1, q0 2, L), respectively. Eqs. (20), (28), (29a) and (29b) imply that the position of coherence vortices depends on the zenith angle, wavelength, correlation length, propagation distance, and reference point q0 1 in general. Fig. 2(a)–(d) represents curves of Re m =0 and Im m = 0 of a GSM vortex beam through slant atmospheric turbulence for x =601 (Fig. 2(a)) and along a horizontal path (Fig. 2(b)) at the propagation distance L=7 km and the position of coherence vortices versus the zenith angle x (Fig. 2(c)–(d)), the calculation parameters are l = 1.06 mm, w0 = 3 cm, h0 = 0, q01 =(5, 5 cm), s0 =4 cm, and m= + 1. From Fig. 2(a) and (b) we see two coherence vortices with m0 = 71, when the GSM vortex beam propagates through atmospheric turbulence along a slant path and along a horizontal path, but there are different positions. For example, the coherence vortices with m0 = 71 are located at (  0.21,  0.99 cm), (13.50, 31.31 cm) in Fig. 2(a) and at

5. Concluding remarks In this paper, based on the extended Huygens–Fresnel principle and the quadratic approximation of the phase structure function, the explicit expressions for the cross-spectral density function and average intensity of GSM vortex beams with m = + 1 propagating through slant atmospheric turbulence have been derived and used to study their propagation properties and evolution behavior of coherent vortices. The spreading of GSM vortex beams along a slant path is smaller than that along a horizontal path in the long atmospheric propagation distance, thus the slant path is more

15

ρ′ 2x (cm)

ρ′2y (cm)

0

34 33 32 31 30 29

431

6 4 2 0 -2 -4 -6

10

6 4 2

5

0 80

0 -5

0

5 10 ρ′2x (cm)

0

15

82

84

86

88

20 40 60 zenith angle ξ (degree)

90 80

15 30 25 ρ′ 2y (cm)

ρ′2y (cm)

10

5

0

30 25 20 15 10 5 0 80 82 84 86 88 90

20 15 10 5 0

-5 -5

0

5 ρ′2x (cm)

10

15

0

20 40 60 zenith angle ξ (degree)

80

Fig. 2. The curves of Re m = 0 and Im m =0 of a GSM vortex beam through atmospheric turbulence along a slant path (a) and along a horizontal path (b) at L= 7 km and the position of coherent vortices versus the zenith angle x (c), (d), ‘‘K’’ m0 = + 1, ‘‘J’’ m0 =  1.

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30

40 30

20 ρ′2x (cm)

ρ′2x (cm)

20 10 0

0

-10 -20 1

2 3 4 wavelength λ (um)

-10

5

50

60

40

50

30

40

20 10 0

2

3

4 5 6 7 ρ′ = ρ′1y (cm)

8

9 10

4 5 6 7 ρ′ = ρ′1y (cm)

8

9 10

30 20 10

-10 -20

1

1x

ρ′2y (cm)

ρ′2y (cm)

10

0 1

2 3 4 wavelength λ (um)

5

1

2

3

1x

Fig. 3. The position of coherence vortices at L= 7 km of a GSM vortex beam through atmospheric turbulence along a slant path versus the wavelength l (a), (b) and the reference point q0 1 (c), (d). ‘‘K’’ m0 = + 1, ‘‘J’’ m0 =  1.

beneficial to the beam propagation through atmospheric turbulence in comparison with the horizontal path. The propagation of partially coherent vortex beams through horizontal atmospheric turbulence can be regarded as a special case through slant one, where the structure constant is independent of the altitude. The position of coherence vortices in slant atmospheric turbulence does not coincide with that in horizontal atmospheric turbulence, and the dependence of position of coherence vortices on the zenith angle, wavelength and reference point have been illustrated numerically. In comparison with the previous work [13,14], the expression (5) for Dc(q1  q2, q01  q0 2) in our paper instead of Dc(q1  q2) in Ref. [13] has been used to derive the closed-form formulas of both the average intensity and crossspectral density, thus the evolution behavior of coherence vortices can be studied by using Eqs. (20)–(29a, b). The average structure constant C¯n2 in Ref. [14] was taken. It means that the slant atmosphere propagation with altitude-dependent Cn2(h) is equivalently treated as the horizontal atmospheric propagation with average C¯n2. On the contrary, throughout our mathematical manipulations the altitude-dependent Cn2(h) has been taken into consideration, so that the results for the average intensity obtained in this paper are more accurate. We believe that a deep understanding of the propagation properties of partially coherent vortex beams through slant turbulent atmosphere would be useful for their potential applications e.g., in optical communications [26] and their parameter optimization.

Acknowledgment This work was supported by the National Natural Science Foundation of China under the Grant no. 10874125.

References [1] Strohbehn JW. Laser beam propagation in the atmosphere. New York: Springer-Verlag Press; 1978. [2] Andrews LC, Phillips RL. Laser beam propagation through random media. Bellingham, Washington: SPIE Press; 1998. [3] Wu J, Boardman AD. Coherence length of a Gaussian-Schell beam and atmospheric turbulence. J Mod Opt 1991;38:1355–63. [4] Gbur G, Wolf E. Spreading of partially coherent beams in random media. J Opt Soc Am A 2002;19:1592–8. [5] Shirai T, Dogariu A, Wolf E. Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence. J Opt Soc Am A 2003;20: 1094–102. [6] Dogariu A, Amarande S. Propagation of partially coherent beams: turbulenceinduced degradation. Opt Lett 2003;28:10–2. [7] Shirai T, Dogariu A, Wolf E. Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence. Opt Lett 2003;28:610–2. [8] Eyyuboglu HT, Baykal Y. Transmittance of partially coherent cosh-Gaussian, ˘ cos-Gaussian and annular beams in turbulence. Opt Commun 2007;278: 17–22. [9] Cai Y, Lin Q, Eyyuboglu HT, Baykal Y. Off-axis Gaussian Schell-model beam ˘ and partially coherent laser array beam in a turbulent atmosphere. Opt Commun 2007;278:157–67. [10] Li J, Yang A, Lu¨ B. Comparative study of the beam-width spreading of partially coherent Hermite-sinh-Gaussian beams in atmospheric turbulence. J Opt Soc Am A 2008;25:2670–9. [11] Young CY, Gilchrest YV, Macon BR. Turbulence induced beam spreading of higher order mode optical waves. Opt Eng 2002;41:1097–103. [12] Konyaev PA, Lukin VP, Sennikov VA. Effect of phase fluctuations on propagation of the vortex beams. Atmos Oceanic Opt 2006;19:924–30. [13] Zhang Y, Wang G. Slant path average intensity of finite optical beam propagating in turbulent atmosphere. Chin Opt Lett 2006;4:559–62. [14] Chu X, Liu Z, Wu Y. Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path. J Opt Soc Am A 2008;25:74–9. [15] Maleev ID, Palacios DM, Marathay AS, Swartzlander GA. Spatial correlation vortices in partially coherent light: theory. J Opt Soc Am B 2004;21: 1895–900. [16] Leader JC. Atmospheric propagation of partially coherent radiation. J Opt Soc Am 1978;68:175–85. [17] Wang SCH, Plonus MA. Optical beam propagation for a partially coherent source in the turbulent atmosphere. J Opt Soc Am 1979;69:1297–304.

ARTICLE IN PRESS J. Li et al. / Optics & Laser Technology 42 (2010) 428–433

[18] Yura HT. Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium. Appl Opt 1972;11:1399–406. [19] Eyyuboglu HT, Baykal Y, Cai Y. Complex degree of coherence for partially ˘ coherent general beams in atmospheric turbulence. J Opt Soc Am A 2007;24: 2891–901. [20] Ji X, Ji G. Spatial correlation properties of apertured partially coherent beams propagating through atmospheric turbulence. Appl Phys B 2008;92:111–18. [21] Salem M, Korotkova O, Dogariu A, Wolf E. Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere. Waves Random Media 2004;14:513–23.

433

[22] Zhu Y, Zhao D, Du X. Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere. Opt Express 2008;16:18437–42. [23] Gradshteyn IS, Ryzhik IM. Table of integrals, series and products. New York: Academic Press; 2007. [24] Mandel L, Wolf E. Optical coherence and quantum optics. Cambridge: Cambridge University Press; 1995. [25] Gbur G, Visser TD. Coherence vortices in partially coherent beams. Opt Commun 2003;222:117–25. [26] Gbur G, Tyson RK. Vortex beam propagation through atmospheric turbulence and topological charge conservation. J Opt Soc Am A 2008;25:225–30.