Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams Yixin Zhang,1, 2, * Mingjian Cheng,1,2 Yun Zhu,2 Jie Gao,1 Weiyi Dan,1 Zhengda Hu1 and Fengsheng Zhao3 1 School of Science, Jiangnan University, Wuxi 214122, China School of Internet of Things, Jiangnan University, Wuxi 214122, China 3 College of Global Change and Earth System Sciences (GCESS), Beijing Normal University, Beijing 100875, China *
[email protected] 2
Abstract: We analyze the effects of turbulence on the detection probability spectrum and the mode weight of the orbital angular momentum (OAM) for Whittaker-Gaussian (WG) laser beams in weak non-Kolmogorov turbulence channels. Our numerical results show that WG beam is a better light source for mitigating the effects of turbulence with several adjustable parameters. The real parameters of WG beams γ and W0, which have significant effects on the mode weight, have no influence on the detection probability spectrum. Larger signal OAM quantum number, shorter wavelength, smaller beamwidth and coherence length will lead to the lower detection probability of the signal OAM mode. ©2014 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (010.1330) Atmospheric turbulence; (010.3310) Laser beam transmission.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
S. Shwartz, M. Golub, and S. Ruschin, “Diffractive optical elements for mode-division multiplexing of temporal signals with the aid of Laguerre-Gaussian modes,” Appl. Opt. 52(12), 2659–2669 (2013). C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). Y. Jiang, S. Wang, J. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre-Gaussian beams propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013). X. Sheng, Y. Zhang, X. Wang, Z. Wang, and Y. Zhu, “The effects of non-Kolmogorov turbulence on the orbital angular momentum of photon-beam propagation in a slant channel,” Opt. Quantum Electron. 43(6–10), 121–127 (2012). J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008). G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009). Y. Zhang, Y. Wang, J. Xu, J. Wang, and J. Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284(5), 1132–1138 (2011). J. Ou, Y. Jiang, J. Zhang, H. Tang, Y. He, S. Wang, and J. Liao, “Spreading of spiral spectrum of Bessel– Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014). B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012). Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014). D. Lopez-Mago, M. A. Bandres, and J. C. Gutiérrez-Vega, “Propagation of Whittaker-Gaussian beams,” In SPIE Optical Engineering Applications. 7430, 743013 (2009). M. A. Bandres and J. C. Gutiérrez-Vega, “Circular beams,” Opt. Lett. 33(2), 177–179 (2008). L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2th ed. (SPIE, 2005). L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13(3), 873–881 (2005). H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence,” J. Mod. Opt. 47(6), 1111–1126 (2000).
#214485 - $15.00 USD Received 30 Jun 2014; revised 14 Aug 2014; accepted 14 Aug 2014; published 4 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022101 | OPTICS EXPRESS 22101
17. H. Chen, X. Sheng, F. Zhao, and Y. Zhang, “Orbital angular momentum entanglement states of Gaussian-Schell beam pumping in low-order non-Kolmogorov turbulent aberration channels,” Opt. Laser Technol. 49, 332–336 (2013). 18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 6th ed. (Academic, 2000).
1. Introduction The mode-division multiplexing is one of recently discussed options for space-division multiplexing [1], the multimode wireless turbulence space optical schemes resorting to orbital angular momentum (OAM) were demonstrated in [2–9]. In particular, LaguerreGaussian (LG) modes with different radial orders p and angular momentum orders l are good candidates for space-division multiplexing [2–9]. One of the biggest challenges that confronts the use of the OAM modes for quantum communication is the effects of the atmospheric turbulence during transmission over long propagation distance. The atmospheric turbulence aberrations can cause the crosstalk among the OAM modes of single photons, reduce information capacity of the communication channel [2, 4, 5], induce the spread of the spiral spectrum of OAM modes [3, 8], and the attenuation or crosstalk among channels in the freespace optical communication systems [6, 7, 9]. The investigation of the probability density of the OAM modes for Hankel-Bessel (HB) non-diffracted beams in paraxial turbulence channel has revealed that the non-diffracted beams can effectively mitigate the effect of the diffraction in free space and the influence of atmospheric turbulence on the transmission [10]. However, the number of adjustable parameters of HB beams is limited, which is the disadvantage for the optimization design of the system for resistance against turbulence interference. Compared with HB beams, Whittaker-Gaussian (WG) laser beams are the nondiffracted beams with several adjustable parameters. So it is more significant to study the transmission rules of WG beams in turbulent atmosphere. The propagation through complex ABCD optical systems, normalization factor, beamwidth, the quality M2 factor and the kurtosis parameter of the WG beams had been studied in [11]. However, to the best of our knowledge, there are almost no discussion with respect to the effects of turbulence on the transmission of the OAM modes for WG beams in slant non-Kolmogorov turbulence channels. In this paper we model the effects of turbulence on the detection probability spectrum and the mode weight of the OAM mode for WG beams through in a slant non-Kolmogorov turbulence atmosphere. 2. The probability distribution of spiral plane modes In cylindrical coordinates, the electric field distribution of WG modes at the z plane in a paraxial channel of free atmospheric turbulence can be described as [11, 12] WGl0 ,iγ ( r , ϕ , z ) = Cl0 ,iγ × ( Pr
(1 − z / q )
* iγ /2 −1/2 0
ikr 2 exp iγ /2 +1/2 (1 + z / q0 ) 2 ( z + q0 )
)
2 l0 /2
(1)
F ( β , l0 + 1, Pr 2 ) exp ( il0ϕ )
1 1
where z is the propagation distance of light; q0 is the beam parameter at the plane z = 0 with 1 / q0 = 1 / R + i2 / kw02 ; R is the radius of curvature of the initial spherical phase front; q0* is the complex conjugate of q0; k = 2π / λ is the wave number of light; λ is the wavelength; w0 is the beamwidth of the Gaussian envelop; l0 corresponds to the OAM l0 carried by the beam, which describes the helical structure of the wave front around a wave front singularity; r = r , r = ( x, y ) is the two-dimensional position vector in the source plane, φ is the azimuthally angle;
( a )n n x n = 0 n ! ( b )n ∞
1
F1 [ a, b, x ] =
is the confluent hypergeometric function;
#214485 - $15.00 USD Received 30 Jun 2014; revised 14 Aug 2014; accepted 14 Aug 2014; published 4 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022101 | OPTICS EXPRESS 22102
β = ( l0 + 1 − iγ ) / 2 ; Cl ,iγ is the normalization constant of WG beams, which is determined 0
by Cl0 ,iγ = 1 / σ
and σ l00 ,iγ is given by
0 l0 ,iγ
σ l0 ,iγ =
π l0 ! w02 2
0
2 l0 +1
ζ 2l
0
(1 − iζ
2
/ 4)
1+ l0 − iγ
1 + l0 − iγ 1 + l0 + iγ ζ4 × F , ; l0 + 1; 2 2 16 + ζ 4
(2)
where W0 and γ are real parameters of WG beams; ξ = w0 / W0 , and 2 (3) −2 2 w ( z + 1) (1 / R ) + 4 ( kw02 ) + 2 z / R For l0 = 0 and ζ 1 , the beam transverse intensity behaves as a Gaussian beam, and the beam has a doughnut-like structure for l0 > 0 [12]. In the weak atmospheric turbulence region [13] and at any point in the half-space z>0, as an approximate, we can express the complex amplitude WGl′,iγ ( r, ϕ , z ) of WG beams as P=
2 0
{
}
2
WGl′,iγ ( r, ϕ , z ) = WGl0 ,iγ ( r , ϕ , z ) exp ψ 1 ( r , ϕ , z )
(4)
where ψ 1 ( r, ϕ , z ) is complex phase perturbation of spherical waves propagating through turbulence. Based on the discussion in [14, 15], we can write the function WGl′,iγ ( r, ϕ , z ) as a superposition of the spiral harmonics exp ( ilϕ ) WGl′,iγ ( r, ϕ , z ) =
1 2π
∞
β ( r, z ) exp ( −ilϕ )
l =−∞
l
(5)
where β l ( r, z ) is given by the integral
1 2π WGl′,iγ ( r, ϕ , z ) exp [ ilϕ ] dϕ (6) 2π 0 Following the same procedure as in [3], the mode probability distribution of the spiral plane mode with phase exp ( ilϕ ) can be expressed as
β l ( r, z ) =
1 2π 2π WGl′,iγ ( r, ϕ , z ) WGl′,i∗γ ( r, ϕ , z ) exp −il (ϕ − ϕ ′ ) dϕ ′dϕ (7) 2π 0 0 By substituting Eq. (4) into Eq. (7) and averaging over turbulence ensembles, we have the ensembles averaging mode probability distribution of the spiral plane waves with phase exp ( ilϕ ) of WG beams propagating in paraxial turbulence channel, which is named as spiral mode probability distribution and is given by 2
β l ( r, z ) =
βl ( r, z )
2
1 2π
2π
2π
0
0
WG γ ( r,ϕ , z )WG γ ( r,ϕ , z ) exp −il (ϕ − ϕ ′) dϕ ′ dϕ (8) × exp − ( 2r − 2r cos (ϕ ′ − ϕ ) ) / ρ
=
∗ l0 ,i
l0 ,i
2
2
2 0
#214485 - $15.00 USD Received 30 Jun 2014; revised 14 Aug 2014; accepted 14 Aug 2014; published 4 Sep 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.022101 | OPTICS EXPRESS 22103
where represent an ensemble average over the turbulence statistics, ρ0 is the spatial coherence radius of a spherical wave propagating in slant non-Kolmogorov turbulence [16] and is given by 1/(α − 2)
(α − 2 ) /2 3−α 8 2 2 Γ Γ 2 α − 2 α − 2 ρ0 = z π 1/2 k 2 Γ 2 − α C 2 (ξ , θ )(1 − ξ / z )α − 2 dξ n 2 0
, 3
