Average intensity and spreading of partially coherent dark hollow beam through the atmospheric turbulence along a slant path

Optik 127 (2016) 7794–7802

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Average intensity and spreading of partially coherent dark hollow beam through the atmospheric turbulence along a slant path Yonggen Xu a,∗ , Ting Yang a , Youquan Dan b , Bin Zhang c , Zairu Ma a , Huanhuan Tian a a b c

School of Science, Xihua University, Chengdu, 610039, China Department of Physics, Civil Aviation Flight University of China, Guanghan 618307, Sichuan, China College of Electronics and Information Engineering, Sichuan University, Chengdu, 610064, China

a r t i c l e

i n f o

Article history: Received 27 March 2016 Accepted 24 May 2016 Keywords: Partially coherent dark hollow beam Inhomogeneous atmospheric turbulence Average intensity Beam spreading

a b s t r a c t The propagations of a partially coherent dark hollow beam (PC-DHB) through the atmospheric turbulence along a slant path (i.e., inhomogeneous atmospheric turbulence) are investigated. Analytical propagation formulae for the average intensity and spreading of PC-DHB in inhomogeneous turbulence are derived based on the extended Huygens-Fresnel integral. It is shown that the dark area of PC-DHB in inhomogeneous turbulence will drop more rapidly with the increase of propagation distance, and the evolution from a DHB to Gaussian beam becomes quicker for a smaller waist width, lower beam order, smaller inner scale of the turbulence and transverse coherence length, and larger zenith angle. Furthermore, PC-DHB spreads in inhomogeneous turbulence more rapidly than the free space, and the saturation propagation distance (SPD) of relative spatial spreading for uplink slant paths with zenith angles of 45◦ or less is about 5 km. The relative spreading becomes quicker for the larger waist width and zenith angle, lower beam order, smaller parameter p and inner scale of the turbulence. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction In past decades, the beams with zero-central intensity called dark hollow beams (DHBs) have been widely studied due to their wide applications in optical communications, atomic optics, medical sciences, modern optics and optical trapping of particles [1–4]. The important theoretical models have been used to describe DHBs and their propagation properties in free space or through paraxial optical system and arbitrary turbulence media [5–14]. Propagation of various types of the beams in atmospheric turbulence has been studied extensively due to their important applications in the free space optical communication, harmonic generation, inertial confinement fusion, optical imaging and optical trapping [15–18]. It is important to find the suitable ways to drop the destructive effect of atmospheric turbulence in the above applications. One of ways for reducing the effect of atmospheric turbulence is using partially coherent beam instead of the fully coherent beam. It has been confirmed successfully through studying the propagation of the various laser beams in turbulence [19–30]. In fact, the propagation of beam in turbulence will be seen as the propagation in the slant path (i.e., the inhomogeneous

∗ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Xu). http://dx.doi.org/10.1016/j.ijleo.2016.05.081 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

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turbulence). The inhomogeneous atmospheric turbulence could influence the propagation properties of laser beams [31,32], such as motion of atmosphere, refractive index fluctuations of the turbulence, propagation distance in the slant path and so on. Up to now, to our knowledge, the propagation properties, such as, average intensity and spreading of PC-DHB in the inhomogeneous atmospheric turbulence have not been studied. Therefore, in this paper, our aim is to study the average intensity and spatial spreading of PC-DHB in the inhomogeneous atmospheric turbulence. Based on the extended HuygensFresnel principle, the analytical formulae for average intensity and spatial spreading of PC-DHB on propagation are derived, and some numerical examples are given in order to illustrate our analytical results. It is shown that the spreading and the average intensity of PC-DHB in inhomogeneous turbulence will depend on the zenith angle of beam propagation in the slant path, the inner scale of the turbulence, the beam order and the waist width. 2. Analytical propagation formulae of PC-DHB in inhomogeneous turbulence The electric field of a DHB of circular symmetry at z = 0 can be expressed as the following finite sum of Gaussian beam [5,6]

  N  (−1)n−1 N



EN0 ( 0 , 0) =

N

−

exp

n

n=1



n 20



 − exp

w02

−

n 20



pw02

,

(1)

 

N is a binomial coefficient,  0 = (x0 , y0 ) denotes the position vector in the n source plane, w0 is the beam waist width of DHB, and 0 < p < 1. A partially coherent beam which has a DHB intensity distribution and a Gaussian spatial correlation can be given by the following cross-spectral density function (CDF) [22,26–28] where N is the order of the circular DHB,

WN0 ( 1 ,  2 ; 0) =

  N ×

N N    (−1)n+n



exp n

N

N2

n=1 n =1

n 2 − 21 w0

 



n

 − exp

−

n 21 pw02

 



× exp

−

n  22 w02



 − exp

−

n  22 pw02



 × exp −

( 1 −  2 )

2

,

(2)

2g2

where g denotes the transverse coherence length of source,  j = (x j , y j )(j = 1, 2) are the position vectors of two arbitrary points in the source plane. According to the extended Huygens-Fresnel principle, the CDF of PC-DHB through the inhomogeneous atmospheric turbulence can be expressed as [26–28] WN (, d ; z) =

k 2  2z

W ( ,  d ; 0) × exp

ik −

z

[( −  ) · (d −  d )] − 0.5Dw (d ,  d ; z)

d2   d2   d .

(3)

where k = 2/ is the optical wave number,  is the wavelength, z is the propagation distance. In Eq. (3), we have used the central abscissa coordinate systems, that is,  = ( 1 +  2 )/2,

(4)

 d =  1 −  2 ,

(5)

 = (1 + 2 )/2,

(6)

d = 1 − 2 .

(7)

where j = (xj , yj )(j = 1, 2) are the position vectors of two arbitrary points in the observation plane. In Eq. (3). the two-point spherical wave structure function Dw (d ,  d ; z) [26–28] is given by

 

2 2

Dw (d ,  d ; z) = 8 k z ×



1

d 0

∞

[1 − J0 (| d + (1 − )d |)]˚n (, z)d.

(8)

0

where  is the magnitude of spatial wavenumber, J0 (·) denotes the Bessel function of the first kind and zero order. ˚n (, z) = Cn2 (h)˚ n () [31,32] is the spatial power spectrum of the inhomogeneous turbulence. Cn2 (h), which is related to propagation distance of beam, is the structure constant of the refractive index fluctuations of the turbulence.

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Substituting Eqs. (4) and (5) into Eq. (2), the initial CDF of PC-DHB in the incident plane can be expressed as WN0 ( ,  d ; 0) =

N N    (−1)n+n

   N

N

N2



n=1 n =1  n n    × exp −r1  2 − r2  ·  d − r3  2d − exp −s1  2 − s2  ·  d − s3  2d − exp −t1  2 − t2  ·  d − t3  2d ,   2   2



+ exp −u1 

(9)

− u2  ·  d − u3  d

where n

r1 =

+

w02 n

s1 =

n

t1 =

pw02

w0

w02

n

n

, s2 = 2

n

n

w0

pw02

, t2 = 2

n

, u2 = 2

pw0

n

n + n

w0

4w02

, r3 = 2

−

w02

+

pw02

−

pw0

+

n

u1 =

n

, r2 = 2

+

w02

n

n

pw02

n

, s3 = 2

pw0

4w02

n

n

w0

4pw02

−

n

+

, t3 = 2

−

n

, u3 = 2

pw0

1 2g2

.

(10)

n

+

4pw02

+

n 4w02

n + n 4pw02

+

+ +

1 2g2 1 2g2

1 2g2

.

(11)

.

(12)

.

(13)

Substituting Eqs. (6)–(13). into Eq. (3), the CDF of PC-DHB through the inhomogeneous atmospheric turbulence can be expressed as 1  (−1)n+n WN (, d ; z) = 4 N2 N



×

1 r1

exp



∞

exp



−∞

−˛2 d2

N



n=1 n =1



−˛1 d2

   N

N

n

n

− ˇ1 d · d − c1 d2

− ˇ2 d · d − c2 d2



1 + i · d − 0.5Dw (d ,  k ; z) d d − s1 

+ i · d − 0.5Dw (d



, 



1 k ; z) d d − t1

 



2

exp −˛3 d2 − ˇ3 d · d − c3 d2 + i · d − 0.5Dw (d ,  k ; z) d2 d +



2



1 u1

∞



∞

−∞

(14)

−∞∞

exp −˛4 d2 − ˇ4 d · d − c4 d2 + i · d − 0.5Dw (d ,  k ; z) d2 d ,

−∞

where ˛1 = r3 − ˛2 = s3 − ˛3 = t3 −

r22 4r1 s22 4s1 t22 4t1

˛4 = u3 −

, ˇ1 =

k2 − r22 z 2 − 2ikr2 z r 2 z + ikr2 2r3 z r3 z 2 . − 2 , c1 = 2 + k 2r1 k k 4r1 k2

(15)

, ˇ2 =

s2 z + iks2 k2 − s22 z 2 − 2iks2 z 2s3 z s3 z 2 − 2 . , c2 = 2 + k 2s1 k k 4s1 k2

(16)

, ˇ3 =

t 2 z + ikt2 k2 − t22 z 2 − 2ikt2 z 2t3 z t3 z 2 − 2 . , c3 = 2 + k 2t1 k k 4t1 k2

(17)

u22 4u1

 k = d +

, ˇ4 =

u2 z + iku2 k2 − u22 z 2 − 2iku2 z 2u3 z u3 z 2 − 2 + . , c4 = 2 k 2u1 k k 4u1 k2

(18)

z  . k d

(19)

where d = (dx , dy ) is the position vector in the spatial-frequency domain [22]. For the case of g → ∞, Eq. (9) is reduced to a fully coherent DHB, the initial CDF can be given by WN0

( , 





×

d ; 0) =

N N    (−1)n+n

   N

N2 n n=1 n =1 exp −r1  2 − r2  ·  d − r  3  2d − exp + exp −u1  2 − u2  ·  d − u 3  2d ,







N



n





−s1  2 − s2  ·  d − s 3  2d − exp −t1  2 − t2  ·  d − t  3  2d



(20)

where r  3 = r1 /4, s 3 = s1 /4, t  3 = t1 /4, u 3 = u1 /4.

(21)

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Substituting Eqs. (20) and (21) into Eq. (3), then the CDF of the fully coherent DHB in inhomogeneous atmospheric turbulence can be easily obtained (i.e., using r  3 , s 3 , t  3 , u 3 instead of r3 , s3 , t3 , u3 in Eqs. (10)–(18), respectively). Letting d = 0 in Eq. (14), the average intensity of PC-DHB through the inhomogeneous atmospheric turbulence can be given by

  

1  (−1)n+n IN (, z) = WN (, 0; z) = 4 N2 N

N



×

1 r1



∞

exp



n

n=1 n =1



−c1 d2

−∞

N



N



n

1 + i · d − 0.5D w (0, d ; z) d d − s1 



1 t1

exp −c2 d2 + i · d − 0.5D w (0, d ; z) d2 d −









2



∞

−∞

exp −c4 d2 + i · d − 0.5D w (0, d ; z) d2 d .



∞

−∞



exp −c3 d2 + i · d − 0.5D w (0, d ; z) d2 d +

1 u1



(22) ∞

−∞

where





1

D w (0, d ; z) = 82 k2 z ×

∞

d 0

0

z [1 − J0 ( d )]˚n (, z)d. k

(23)

For /k < < 1, using the approximation of the Bessel function of zero order [19] z 1 J0 ( d ) ≈ 1 − 4 k

z k

2 d

.

(24)

Then Eq. (22) is rewritten as 1  (−1)n+n IN (, z) = 4 N2 N

N



×

1 r1

exp





  

n=1 n =1



∞

−c1 d2

exp

 −∞ −c2 d2



+ i

N

n

n



· d − 2 z 3 d2



· d − 2 z 3 d2

+ i

N



2

 0∞ 

0

1

∞

 2 d

0 1

0 ∞  2 d

exp −c4 d2 + i · d − 2 z 3 d2



1 ˚n (, z) d d d − s1

0

0

2



1 ˚n (, z) d d d − t1



∞

2

3

 d



∞

3

 d

2

0

exp −c3 d2 + i · d − 2 z 3 d2

1



1



˚n (, z)3 d d2 d +





2

˚n (, z)3 d d d

1 u1



∞

−∞

(25)



−∞ ∞ −∞

.

0

where ˚n (, z) = Cn2 (h)˚ n (), h = z cos  is the propagation height of PC-DHB in turbulence of the slant path,  denotes the zenith angle of beam propagation in the slant path in turbulence. In Eq. (25), letting the turbulence terms cz satisfies the following expression:

 cz = 2 z 3



1

 2 Cn2 (h)d 0

∞

 ˚ n ()3 d

,

(26).

0

where Cn2 (h) is the well-known Hufnagel-Valley (H-V5/7 ) model [31]:

v

Cn2 (h) = 0.00594

pw

27

2 

10

10−5 h

exp

−h

1000

+ 2.7 × 10−16 exp

−h

1500

+ 1.7 × 10−14 exp

−h

100

.

(27)

where vpw is root-mean-square (rms) wind velocity. It should be noted  that 2the  Tatarskii spectrum for the spectral density  , where m = 5.92/l0 with l0 being the inner of the index of refraction fluctuations, then ˚n () = 0.033−11/3 exp −2 /m scale of the turbulence.

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Y. Xu et al. / Optik 127 (2016) 7794–7802

Fig. 1. Normalized average intensity for the different parameters, (where I-turbulence denotes inhomogeneous atmospheric turbulence, H-turbulence

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7799

Substituting Eq. (27) into Eq. (26), we can obtain −1/3 14 z

−55 2 cz = ⎧1.3358 × 10 vpw l0

⎨

QB =12

×

⎩

qB =1

−1/3 4 z

+4.426 × 10−16 l0





(−1)qB −1 QB ! (QB − qB + 1)!

z cos 

× exp −

1500

z cos 

exp −

100

−

(cos )11

1000 z cos 



cos 



qB

z cos 

× exp −

1500 z cos 

1000

exp − −1/3 4 z



3  100

z cos 

1500



−2 −



cos 

z cos 

exp −



+ (−1)QB QB ! −

z cos 

+ 2.786 × 10−14 l0

−1

+2 −

−



100

−



−1

1000 z cos 

1500 2 z cos 

100 z cos 

QB +1 

z cos 

exp −

z cos 

exp −

1500

z cos 

× exp −

100

1000



+2 −



−2 −



−1

1500 z cos 

3 (28)

2 100 z cos 

.

Substituting Eq. (28) into Eq. (25), and using the following integral formula [33,34].





∞

√



 exp s

exp −s2 x2 ± qx dx = −∞



q2 4s2



.(s > 0)

(29)

we obtain (after some vector tedious integrations) the following expression: 1  (−1)n+n IN (, z) = 4 N2 N

N



n=1 n =1





   N

N

n

n



2

1



2

1

 1

  exp −   −   exp −   −   r1 c1 + cz 4 c1 + cz s1 c2 + cz 4 c +c t1 c3 + cz     2 z

×

exp −



2

4 c3 + cz

 +



1

u1 c4 + cz

 exp − 

2

4 c4 + cz



(30)

.

Eq. (30). is the analytical formula that used for describing the average intensity of propagation of PC-DHB through the inhomogeneous atmospheric turbulence. It can provide the transformation rule and be a powerful tool to study propagation properties of the PC-DHB in the inhomogeneous turbulence. Especially, Eq. (30). will reduce to the average intensity PC-DHB in free space when parameter cz = 0. The rms spatial width of PC-DHB in the inhomogeneous atmospheric turbulence can be defined as [26–28]

  2 1  IN (, z)d2  2  wNz = .

(31)

IN (, z)d2 

Substituting Eq. (22) into Eq. (31), and using the following formula [26–28].



∞

exp(−i · d )d2  = (2)2 ı(d ).

(32)

−∞

we obtain



wNz =

 ×

1  (−1)n+n M0 N2 N

N

n=1 n =1 4r3 z 2 + − 2 r1 k2 r1

1

1 2 . +T (z, )



r22 z 2 r12 k2

   N n

  −

N n

1 s12

+

s2 z 2 4s3 z 2 − 22 s1 k2 s1 k2

  −

1 t12

+

t 2 z2 4t3 z 2 − 22 t1 k2 t1 k2

  +

1 u21

+

u2 z 2 4u3 z 2 − 22 u1 k2 u1 k2

 (33)

denotes homogeneous atmospheric turbulence and F-space denotes free space)  = 632.8nm, vpw = 21m/s, (a) w0 = 50mm, p = 0.8, N = 5, l0 = 10mm,  = /4, g = 20mm, (b) z = 1km, p = 0.8, N = 5, l0 = 10 mm,  = /4, g = 20mm, (c) z = 1km, w0 = 50mm, N = 5, l0 = 10mm,  = /4, g = 20mm, (d) z = 1 km, w0 = 50mm, p = 0.8, l0 = 10mm,  = /4, g = 20mm, (e) z = 1 km, w0 = 50mm, p = 0.8, N = 5, l0 = 10mm, g = 20mm, (f) z = 1 km, w0 = 50mm, p = 0.8, N = 5, l0 = 10mm, g = 20mm, (g) z = 1 km, w0 = 50mm, p = 0.8, N = 5, l0 = 10mm,  = /4, (h) z = 2 km, w0 = 50mm, p = 0.8, N = 5, l0 = 10mm,  = /4, g = 20mm.

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where T (z, ) is related to the inhomogeneous atmospheric turbulence, it is given by T (z, ) =

42 z 2 cos 



∞



z cos 

˚ n ()3 d



1−

0

0

h z cos 

2

Cn2 (h)dh,

(34)

and M0 =

  N N     N N (−1)n+n 1 N2

n=1 n =1

n

n

×

r1

−

1 1 1 − + s1 t1 u1

.

(35)

It should be noted that the analytical formula for Eq. (34). can be easily obtained through substituting Eq. (27) into Eq. (34), and then the analytical formula for rms spatial width of PC-DHB can be obtained by using the method of deriving Eq. (28). Letting cz = 0 in Eq. (33), the rms spatial width of PC-DHB in free space is given by

 w

Nz

=

1  (−1)n+n M0 N2 N

N



     N N n

n

×

r22 z 2 4r3 z 2 + − r1 k2 r12 r12 k2 1

⎫ 1 ⎪     ⎬ t22 z 2 u22 z 2 2 4t3 z 2 4u3 z 2 1 1 . − + − + + − t1 k2 u1 k2 t12 t12 k2 u21 u21 k2 ⎪ ⎭ n=1 n =1

  −

s22 z 2 4s3 z 2 + − s1 k2 s12 s12 k2 1

 (36)

In the derivations of Eqs. (33) and (36), we have used the following formula of the Dirac delta function:









x2 + y2 exp i xdx + ydy







dxdy = −(2)2 ı(2) (dx )ı(dy ) + ı(2) (dy )ı(dx ) .

(37)

where ı(·) denotes the Dirac delta function and ı(2) (·) is its second derivative. For the sake of comparison, the normalized rms spatial width of PC-DHB in inhomogeneous atmospheric turbulence are defined as wr =

wNz . w Nz

(38)

Eqs. (33) and (36) are the main analytical formulae of the present paper, which provide with a convenient way for studying the propagation properties of PC-DHB through the inhomogeneous atmospheric turbulence and in free space, respectively. 3. Numerical examples Now we study the propagation properties of PC-DHB in the inhomogeneous atmospheric turbulence by using the analytical formulae derived in the previous section. Fig. 1 shows the normalized average light intensity I(x, 0, z)/I(x, 0, z)max through using Eq. (30). The calculation parameters are  = 632.8nm, g = 20mm, vpw = 21m/s. It can be seen from Fig. 1(a) that the dark area of PC-DHB in turbulence will disappear gradually with the increase of propagation distance, and it is quicker than that of the free space. Fig. 1(b)–(e) and (g) indicate that the dark area of PC-DHB in inhomogeneous atmospheric turbulence will drop more rapidly, and the evolution from a DHB to Gaussian beam becomes quicker for a smaller waist width, smaller parameter p, lower beam order, smaller inner scale of the turbulence and transverse coherence length. Fig. 1(f) indicates that the dark area of PC-DHB in turbulence drops rapidly with the increase of the zenith angle. Fig. 1(h) indicates that the evolution from a DHB to Gaussian beam becomes quicker in turbulence than in free space. The evolution from a DHB to a Gaussian beam in turbulence become quicker when zenith angle increases from  = 0 (beam propagation is in the vertical direction) to  = /2. These results are different from the case of the homogeneous atmospheric turbulence ( = /2) [5,6]. Using Eqs. (33). and (36), (37), and the calculation parameters are  = 632.8nm, g = 20mm, vpw = 21m/s. The normalized rms spatial widths for the different waist width, beam order, parameter p, inner scale of the turbulence and the zenith angle  versus the propagation distance z in inhomogeneous atmospheric turbulence or in free space are shown in Fig. 2. It can be found from Fig. 2(a) that the PC-DHB spreads in inhomogeneous turbulence more rapidly than the free space, but it spreads more slowly than the homogeneous turbulence. The reason is that the turbulence will become gradually weak with the increase of the vertical height, and the effect of turbulence can be neglected when the vertical height is large. Fig. 2(b–f) shows that as the propagation distance of PC-DHBs in the inhomogeneous atmospheric turbulence, the relative spatial spreading at first increases rapidly, then increases slowly, and finally approaches to a constant. The saturation propagation distance (SPD) of the relative spatial spreading can be defined as the distance zs from the initial plane to where the relative beam width is ten times less than its maximum. As shown in Fig. 2(b–f), SPD of the relative spatial spreading is about 5 km for zenith angles of 45◦ or less. And the relative spatial spreading becomes quicker for the larger waist width and zenith angle, lower beam order, smaller parameter p and inner scale of the turbulence.

Y. Xu et al. / Optik 127 (2016) 7794–7802

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Fig. 2. Rms spatial width (Relative rms spatial width) versus propagation distance z,  = 632.8nm, g = 20mm, vpw = 21m/s, (a) w0 = 15mm, p = 0.8, N = 10, l0 = 1.6cm,  = /4, (b) For the inhomogeneous turbulence p = 0.8, N = 10, l0 = 1.6cm,  = /4, (c) For the inhomogeneous turbulence w0 = 15mm, N = 10, l0 = 1.6cm,  = /4, (d) For the inhomogeneous turbulence w0 = 15mm, p = 0.8, l0 = 1.6cm,  = /4, (e) For the inhomogeneous turbulence w0 = 15mm, p = 0.8, N = 10,  = /4, (f) For the inhomogeneous turbulence w0 = 15mm, p = 0.8, N = 10, l0 = 1.6cm.

4. Conclusions In conclusion, we have derived some analytical formulae for the average intensity and rms spatial width of PC-DHB in inhomogeneous atmospheric turbulence based on the extended Huygens-Fresnel integral. The average intensity and beam spatial spreading of PC-DHB in turbulence have been investigated comparatively and numerically. It is shown that the dark

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area of PC-DHB in inhomogeneous atmospheric turbulence will drop more rapidly with the increase of propagation distance, and the evolution from a DHB to Gaussian beam becomes quicker for a smaller waist width, smaller parameter p, lower beam order, smaller inner scale of the turbulence and transverse coherence length, and larger zenith angle. PC-DHBs spread in inhomogeneous turbulence rapidly with the increases of propagation distance and the zenith angle of uplink slant path. SPDs of spatial spreading for uplink slant paths with zenith angles of 45◦ or less is about 5 km. The relative spatial spreading becomes quicker for the larger waist width and zenith angle, lower beam order, smaller parameter p and inner scale of the turbulence. The results are different from the case of the homogeneous atmospheric turbulence ( = /2). 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