Autofocusing and spreading of partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence

Accepted Manuscript Title: Autofocusing and spreading of partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence Authors: Ke Cheng, Xinghua Zhu, Yan Zhou, Na Yao, Xianqiong Zhong PII: DOI: Reference:

S0030-4026(18)30979-3 https://doi.org/10.1016/j.ijleo.2018.07.017 IJLEO 61173

To appear in: Received date: Accepted date:

10-5-2018 4-7-2018

Please cite this article as: Cheng K, Zhu X, Zhou Y, Yao N, Zhong X, Autofocusing and spreading of partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence, Optik (2018), https://doi.org/10.1016/j.ijleo.2018.07.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Autofocusing and spreading of partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence

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Ke Cheng1,* , Xinghua Zhu2,1, Yan Zhou1, Na Yao1, Xianqiong Zhong1

1. College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China



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2. College of Intelligent Manufacturing, Sichuan University of Arts and Science, Dazhou 635002, China

Corresponding author E-mail: [email protected] (K. Cheng).

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Abstract

By transferring higher hierarchy Butterfly catastrophe to optical field, the autofocusing

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behavior and beam spreading of partially coherent Butterfly beams modulated by flat-

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topped factor in atmospheric turbulence are investigated, where the input power is fixed. In the process of autofocusing behavior the main lobe remains its location near origin,

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which is different from the movement of main lobe in a lower hierarchy Pearcey beam. The turbulence does not change the autofocusing behavior, but accelerates the decaying in side lobes of butterfly structure. A higher coherence length can lead to a more

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concentrated intensity at autofocusing plane. The analysis of beam spreading also indicates that there exist critical values of propagation distance of zc. A smaller flat-

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topped factor tends to maintain a smaller spreading for zzc the opposite case can be found. Angular spread decreases with an increasing of coherence length, and a smaller beam order shows a larger far field divergence angle. The study of partially coherent Butterfly beams in turbulence provides the possibility of the

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atmospheric laser communication by using higher-order catastrophe optical field. Keywords: Butterfly catastrophe, autofocusing, atmospheric turbulence

1

Introduction Propagation dynamics, spreading and directionality of laser beams in atmospheric 1

turbulence have been widely studied because of these potential applications in space optical communication and remote sensing [1-7]. In general, a partially coherent beam possesses a larger beam spreading in turbulence relative to its fully coherent case, which leads to a smaller energy received by a detector and also hampers long distance propagation [1]. The scintillation index and average bit error rate in partially coherent beams are lower than those in fully coherent beams [2]. On the other hand, the propagation of catastrophe optical field in recent years has

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received extensive attention [8-15]. For example, an optical Airy beam described by the lowest hierarchy fold catastrophe has presents parabolic trajectories and self-healing

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properties [8, 9]. The evolution of the kurtosis parameters of Airy beam modulated by exponential truncation factor through atmospheric turbulence is also analyzed [5]. Subsequently, the cusp catastrophe optical field, i.e., the Pearcey beam exhibits the form-invariance and autofocusing behavior in the free space [10], and the effect of

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atmospheric turbulence on the intensity of Pearcey-Gaussian beam is also studied by

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Belafhal et.al. [11]. A higher hierarchy optical field with Butterfly function is further investigated [14], where the Butterfly catastrophe is given by 

   exp 

i u

6

 a 4u

4

 a 3u

3

 a 2u

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Bu  a 1 , a 2 , a 3 , a 4

2

 .

 a 1 u du

(1)

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In our previous work we found that the autofocusing behavior of the Butterfly beam in free space [15], while for the case of atmospheric turbulence the autofocusing and

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spreading of Butterfly beams have not been dealt with. The main purpose of this paper is to study the autofocusing behavior, meansquared width and angular spread of partially coherent Butterfly beams modulated by

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flat-topped factor through atmospheric turbulence, where the input power is fixed. The results obtained in this paper extend the study of atmospheric optics from the lower

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hierarchy Airy and Pearcey beams to higher-order Butterfly beam.

2 Partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence

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The initial field of a pure Butterfly beam with a1=0, a2=y‫׳‬/y0, a3=x‫׳‬/x0 and a4=0 at

z=0 in the Cartesian coordinate system can be expressed as [14] B u (0, y  y0 , x  x0 , 0 ) 



 

   6  x  3  y  2    exp i u   u    u  d u  x0   y0      

,

(2)

where x0 and y0 are scaling lengths, respectively. It is well known that a pure Butterfly beam with infinite oscillating structures can lead to the infinite energy, so it is difficult 2

to generate a pure Butterfly beam in experiment. To limit its oscillating structure, an exponential truncation factor, a Gaussian function or other factor with decaying form can be used to modulate the pure Butterfly beam. Here we introduce a flat-topped factor [16] to modulate the Butterfly beam, whose electric field at z=0 in a Cartesian coordinate system is written as (  1)

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E ( r , 0 )  E 0 

N   n

N

 n ( x  y )    exp    Bu  0 , y  y 0 , x  x 0 , 0  , 2  w0    2

2

(3)

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n 1

N       n 

where

n 1

represents the binomial coefficients, w0 is the waist width, E0 is constant

P 



n 0 0 c 

2

E ( r ,0 ) d r 

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factor for a fixed input power and (x‫׳‬, y‫ )׳‬is the component of r‫ ׳‬in transverse plane. Assume that the input power P of the Butterfly beam at z=0 is fixed and it can be expressed by ,

(4)

2

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where n0, ε0 and c denote refractive index, dielectric constant and vacuum velocity,

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respectively. Fig. 1 gives the intensity profile of Butterfly beams modulated by flat-

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topped factor, where the input power P=1W is fixed. From the Figs. 1(a)-(d), one can find that the increasing of flat-topped factor leads to the expansion of the main lobe.

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However this expansion results in the reduction in intensity of main lobe under the fixed power condition.

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For a partially coherent Butterfly beam, the cross-spectral density function at z=0 reads as [17] 0 

( r1, r 2 , 0 )  E  r1, 0  E



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W

 r 2 , 0 

 E  r1, 0  E



 r 2 , 0  exp

 ( r 1  r 2 )    2    2

,

(5)

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where σ is the correlation length. Based on the extended Huygens-Fresnel integral, the propagation of a partially

coherent Butterfly beam in atmospheric turbulence is characterized by [18]

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W ( r1 , r 2 , z )  (



k

2 z

 exp   r1 , r1   

)



2

  d r 

2

1

d r 2 W 2

0 

 ik 2 2  ( r1, r 2 , 0 ) exp  ( r1  r1 )  ( r 2  r 2 )  2z 



 r 2 , r 2  ,



(6)

m

where k denotes the wave number related to the wavelength λ by k=2π/λ, r=(x, y) is the two-dimensional position vector at the plane z, the asterisk is complex conjugation and 

m

represents the ensemble average of the turbulent medium statistics given by 3



exp   r1, r1   



m

 0  0 . 545 C n k z  2

Here

  r   r     r   r   r  r   ( r  r ) 2  2 1 2 1 2 1 2  exp   1  2  0   2

 r 2 , r 2  2

3 / 5

is the spatial coherence radius with

.

(7) 2

being the

Cn

refraction index structure constant. By letting r1=r2=r in Eqs. (6) and (7), the average intensity of partially coherent

2

4 B1 B 2 z

 ds



exp

 i  1



k T



1

  k exp   

6 2

4 B2 y0 s  s 2

6

4

2

x

 y

2

4z

4 B 2 x0  s 2

3

2



2  B0   1       1    B2  B1     B 1 

2

 kx

1

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I ( x, y, z) 

2

E0

2 B 2 x 0 z  kxB

2

 ky

2 B 2 y 0 z  kyB



2 B 1 B 2 y 0 z  Bu  0 ,  ,  , 

0

,

(8)

with  B0s  y    2 iB 1 B 2 x 0  2

1

B0 



2



1



,

2

i

 1

2

4 B1 x 0

 j

2





kjB



2 iB 1 zj 0

2



1 2

 B0s  x   2 iB 1 B 2 x 0 



2

3

1



N

0 2

,

T 

4 B1 B 2 x 0

kj



iB

2

w0

1



2

0

kjB



2 iB 1 B 2 zj 0

ik



,

2z

(  1)

n 1

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

n

B1 



  ,    

  ,    

N

3

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1

M



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 

n 1

N

N   n



U



2 B1 B 2 x 0 z 

0



 exp s

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Butterfly beams in turbulence at the z plane turns out to be

B2 

n 2



w0

 N (  1)   N  n  1

n 1



2 3

1



2

N      n 

 iB 0 i   2 2 2  4 B1 B 2 y 0  4 B1 y 0



2

1



2



ik 2z

,

   

, (9)

2



B0

,

(10)

B1

(11)

2

0

, (j=x or y).

2

(12)

2 iB 1 B 2 zj 0

The mean-squared beam width and angular spread are defined by [19] I  x , y , z  dxdy

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2 j

z  

4







j



2

I  x , y , z  dxdy

and



j

z  

w lim

z 

j

z 

,

(j=x or y).

(13)

z

From Eqs. (8)-(12), one can find that the average intensity is associated with the

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coherence length σ, flat-topped factor N and the refraction index structure constant

2

Cn

.

Further, the special spatial structures of partially coherent Butterfly beams will lead to the difference of mean-squared beam width and angular spread in x- and y- directions, respectively.

3 Numerical result and analysis Numerical calculations based on the Eqs. (8)-(13) in this section are performed to 4

explore the propagation of partially coherent Butterfly beams in atmospheric turbulence. The calculation parameters λ=633nm, w0=1mm, x0=y0=100μm, Cn2=10−13m−2/3 and P=1W are fixed unless otherwise stated. The propagation of the fully coherent Butterfly beam in free space has been studied in our previous work, where the autofocusing behavior and a distinct butterfly structure are found. The intensity evolution of partially coherent Butterfly beams in atmospheric turbulence is further given in Fig. 2, and the evolution in free space is also compared.

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As shown in Figs. 2 (b1), (b2), (d1) and (d2), the turbulence does not change the

autofocusing behavior, but it can accelerate the decaying of the side lobes comparing

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with the free space.

Fig. 3 gives the effect of coherence length and flat-topped factor on the autofocusing behavior of Butterfly beams in x-direction. Comparing with partially coherent field, fully coherent field has a clearer side lobe and it also presents a larger

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intensity at autofocusing plane. For example, for N=1 the peak intensities are 2.84×103 and 3.21×103 at focal plane for σ=w0 and σ→∞, respectively. One can see that the

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autofocusing lengths in Fig. 3 (a4) and (b4) are 0.71ze and 0.73ze for σ=w0 and σ→∞,

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respectively. As the order of flat-topped factor gets bigger, the peak intensity at focal

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plane becomes smaller. The phenomenon is consistent with the distribution at source plane. Furthermore, a decrease in coherence length and the order of flat-topped factor

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can lead to a slightly reduction in autofocusing length. Fig. 4 describes the effect of coherence length on maximal intensity at autofocusing plane for different order of flat-topped factor. The maximal intensity tends

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to increase with coherence length, which indicates that the intensity of fully coherent Butterfly beam at autofocusing plane is more concentrated than that of partially

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coherent Butterfly beam. The maximal intensity for a lower order is larger than that for a higher order. For example, the intensity at autofocusing plane are 3.12×103, 1.91×103 and 1.33×103 V2/m2 for N=1, N=3 and N=6, respectively. Fig. 5 shows the evolution of the mean-squared width of partially coherent

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Butterfly beams for different order of flat-topped factor, where the Gaussian Schellmodel beam is also compared. For the mean-squared width there exist critical values of propagation distance. From Fig. 5, one can see that the critical values are zc=29.7ze and 16.5ze in x- and y- direction, respectively. For z<29.7ze, the larger order leads to a smaller beam width, but for z>29.7ze the beam width of a smaller order increase sharply during propagation in atmospheric turbulence, which leads to the beam width of a 5

smaller order corresponding to a larger spread. In addition, the partially coherent Butterfly beams also present a larger beam width than Gaussian Schell-model beam. The effect of coherence length on the angular spread for different order of flattopped factor is depicted in Fig. 6. As can be seen, the x- and y- directional angular spread decreases with an increasing of coherence length. A smaller order shows a larger far field divergence angle, but for a larger order of flat-topped factor, i.e., N=3 or N=5, the angular spread is less sensitive to the beam order. The comparison shows that the

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directivities of the Gaussian-Schell beams are better than those of partially coherent

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Butterfly beams.

Conclusion

The propagation of partially coherent Butterfly beams modulated by flat-topped factor through atmospheric turbulence is studied, where the input power of the incident

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beam for different order of flat-topped factor is fixed. An autofocusing behavior in

N

turbulence can be found and a distinct butterfly structure near autofocusing plane is also

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present. The effect of coherence length and flat-topped factor on autofocusing behavior, maximal intensity, Mean-squared width and angular spread is analyzed. The results

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show that the turbulence does not change the autofocusing behavior, but it leads to a further decaying of the side lobes comparing with the free space. The decaying of side

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lobes in partially coherent field is faster than that in fully coherent field. The increasing of intensity at autofocusing plane is accompanied as the coherence length increases.

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The analysis of spreading indicates that there exist critical values of propagation distance of zc. For zzc. The partially coherent

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Butterfly beams present a larger beam width than Gaussian Schell-model beam. Angular spread decreases with an increasing of coherence length. A smaller beam order shows a larger far field divergence angle.

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The results obtained in this paper may be useful for analyzing the evolution of

autofocusing behavior for partially coherent Butterfly beam in turbulence, and extend the propagation dynamics in turbulence from a lower catastrophe optical field, i.e., Airy or Pearcey beam, to a higher Butterfly catastrophe optical field.

Acknowledgements This work was supported by the Natural Science Foundation of Sichuan Education 6

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Committee under Grant no. 17ZA0072.

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References 1. G. Gbur, E. Wolf, Spreading of partially coherent beams in random media, J. Opt. Soc. Am. A, 19 (2002) 1592-1598. 2. J. C. Ricklin, F. M. Davidson, Atmospheric optical communication with a Gaussian Schell beam, J. Opt. Soc. Am. A, 20 (2003) 856-866. 3. X. Ji, Z. Pu, Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence, Appl. Phys. B, 93 (2008) 915-923.

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4. Y. Gu, G. Gbur, Scintillation of Airy beam arrays in atmospheric turbulence, Opt. Lett. 35 (2010) 3456-3458.

5. X. Chu, Evolution of an Airy beam in turbulence, Opt. Lett. 36 (2011) 2701-2703.

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6. H. T. Eyyuboğlu, Scintillation behavior of Airy beam, Opt. Laser Technol. 47 (2013) 232-236.

7. J. Zhu, X. Li, H. Tang, K. Zhu, Propagation of multi-cosine-Laguerre-Gaussian

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correlated Schell-model beams in free space and atmospheric turbulence, Opt.

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Express, 25 (2017) 20071-20086.

8. G. A. Siviloglou, J. Broky, A. Dogariu, D. N. Christodoulides, Ballistic dynamics

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of Airy beams, Opt. Lett. 33 (2008) 207-209.

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9. H. I. Sztul, R. R. Alfano, The Poynting vector and angular momentum of Airy beams, Opt. Express 16 (2008) 9411-9416.

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10. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, M.R. Dennis, Autofocusing and self-healing of Pearcey beams, Opt. Express 20 (2012) 18955-18966. 11. F. Boufalah, L. Dalil-Essakali, H. Nebdi, A. Belafhal, Effect of turbulent

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atmosphere on the on-axis average intensity of Pearcey–Gaussian beam, Chin. Phys. B 25 (2016) 264-269.

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12. K. Cheng, G. Lu, X. Zhong, Energy flux density and angular momentum density of Pearcey-Gauss vortex beams in the far field, Appl. Phys. B 123 (2017) 60-70.

13. A. Zannotti, M. Rüschenbaum, C. Denz, Pearcey solitons in curved nonlinear photonic caustic lattices, J. Opt. 19 (2017) 094001.

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14. A. Zannotti, F. Diebel, M. Boguslawski, C. Denz, Optical catastrophes of the swallowtail and butterfly beams, New J. Phys 19 (2017) 053004.

15. K. Cheng, G. Lu, Y. Zhou, X. Zhong, The Poynting vector and angular momentum density of the autofocusing Butterfly-Gauss beams, Opt. Laser Technol. 105 (2018) 23-34. 16. Y. Li, Light beams with flat-top profiles, Opt. Lett. 27 (2002) 1007-1009. 8

17. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. 18. R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition, SPIE Press, Bellingham, 2005. 19. A. E. Siegman, New developments in laser resonators, Proc. SPIE: Optical

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Resonators, 1224 (1990) 2-14.

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Figure captions Fig. 1 (Color online) The intensity profile of Butterfly beams modulated by flat-topped factor at z=0, where the input powers for different order of flat-topped factor are fixed to 1W. Fig. 2 (Color online) The intensity evolution of partially coherent Butterfly beams in free space and atmospheric turbulence. Top row: Cn2=0; Bottom row: Cn2=10−13m−2/3. N=3, σ=w0

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Fig. 3 (Color online) The effect of coherence length and flat-topped factor on the autofocusing behavior in atmospheric turbulence. length for different order of flat-topped factor.

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Fig. 4 (Color online) The maximal intensity at autofocusing plane versus coherence Fig. 5 (Color online) Mean-squared width in x- and y-direction versus propagation distance z/ze for different order of flat-topped factor. σ=w0.

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Fig. 6 (Color online) The effect of coherence length on angular spreads in the x- and y-

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A

N

direction for different order of flat-topped factor.

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(a) N=1

(c) N=6

M

y/mm

A

(b) N=3

ED

x/mm

A

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Fig. 1

11

(c1)

(b2)

(c2)

(d1)

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N

A M ED

A

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Fig. 2

12

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(b1)

x/mm

(a2)

z=1.5ze

z=ze

y/mm

(a1)

z=0.85ze

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z=0.5ze

(d2)

σ=w0

σ→∞

(a1)

(b1)

N=1

(a2)

(b2)

(a3)

(b3)

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N=3

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X/mm Z/Ze

CC E

PT

ED

M

A

N

U

N=6

A

Fig. 3

13

A ED

PT

CC E

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SC R

U

N

A

M

Fig. 4

14

A ED

PT

CC E

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SC R

U

N

A

M

Fig. 5

15

A ED

PT

CC E

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SC R

U

N

A

M

Fig. 6

16