## Abstract

We develop a novel model of the probability density of the orbital angular momentum (OAM) modes for Hankel-Bessel beams in paraxial turbulence channel based on the Rytov approximation. The results show that there are multi-peaks of the mode probability density along the radial direction. The peak position of the mode probability density moves to beam center with the increasing of non-Kolmogorov turbulence-parameters and the generalized refractive-index structure parameters and with the decreasing of OAM quantum number, propagation distance and wavelength of the beams. Additionally, larger OAM quantum number and smaller non-Kolmogorov turbulence-parameter can be selected in order to obtain larger mode probability density. The probability density of the OAM mode crosstalk is increasing with the decreasing of the quantum number deviation and the wavelength. Because of the focusing properties of Hankel-Bessel beams in turbulence channel, compared with the Laguerre-Gaussian beams, Hankel-Bessel beams are a good light source for weakening turbulence spreading of the beams and mitigating the effects of turbulence on the probability density of the OAM mode.

© 2014 Optical Society of America

## 1. Introduction

In recent years, a number of articles address the use of the OAM states of light as a basis set for impressing coding information onto light field propagation in turbulence atmosphere [1–10]. The Kolmogorov atmospheric turbulence aberrations cause the crosstalk among the OAM states of single photons, reduce information capacity of the communication channel [1,2] and induce the spread of the spiral spectrum of OAM modes [3]. In the case of non-Kolmogorov turbulence, the crosstalk among orbits increases as the non-Kolmogorov parameter increases [4] and the turbulence also induces attenuation and crosstalk among multichannel free-space optical communication channels [5,6]. The degradation in mode quality results in crosstalk between OAM modes [7]. The effects of atmospheric turbulence tilt, defocus, astigmatism, coma and Z-tilt corrected residual aberrations on the orbital angular momentum measurement probability of photons propagating in Kolmogorov/non-Kolmogorov turbulence channel are different [8–10]. As we know, there is almost no discussion with respect to the effects of turbulence on the probability distribution of the OAM modes of Hankel-Bessel(HB) beams in paraxial optical system.

In this paper we model the effects of turbulence on the probability density of the OAM modes or mode crosstalk for HB beams in the paraxial atmosphere channel.

## 2. Mode probability density of OAM

In the weak atmospheric turbulence region and at any point in the half-space $z>0$, the complex amplitude of HB beams can be expressed as

*z*

^{2}+

*r*

^{2})

^{1/2}≈

*z*+

*r*

^{2}/2

*z*, traveling scalar wave $H{B}_{{l}_{0}}^{}\left(r,\phi ,z\right)$ has the form [11]

Using the superposition of the plane waves with phase $\mathrm{exp}\left(\text{i}l\phi \right)$ [3], the function $H{B}_{para}\left(r,\phi ,z\right)$ can be written as

*β*(

_{l}*r*,

*z*) is given by the integral

By substituting Eq. (2) into Eq. (4) and based on the integral expression [12]

*I*

_{n}(

*η*) is the Bessel function of second kind with $n$ order. We have the mode probability densities of HB beams in paraxial turbulence channel at

*r*

*ρ*

_{0}is the spatial coherence radius of a spherical wave propagating in the non-Kolmogorov turbulence [13] and is given by

*α*is the non-Kolmogorov turbulence-parameter, Г(

*α*) denotes the Gamma function and ${C}_{n}^{2}$ is the generalized refractive-index structure parameter with units m

^{3-α}.

## 3. Numerical results

Numerical calculations of the probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ and the crosstalk probability densities ${\left|{\beta}_{l}(r,z)\right|}^{2}$ of OAM mode of HB beams along the direction of $r$ in receiving plane are in Figs. 1, 2, 3, 4, and 5. We additionally give a comparative analysis with Laguerre-Gaussian(LG) beams [14], which have been studied intensively at present.

The mode probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ of HB beams versus *r* for the wavelength *λ* = 1550nm, propagation distance *z* = 1km, *α* = 11/3 and the refractive index structure parameter ${C}_{n}^{2}={10}^{-14}$m^{3-}* ^{α}* is shown in Fig. 1(a), where the effects of different initial mode indices

*l*

_{0}= 1,2,3 are considered. It is clear that the maximum of the ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ moves away from

*r*= 0 as the value

*l*

_{0}increases, and the ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ increases with the increasing of

*l*

_{0}. The increasing of the OAM quantum number

*l*

_{0}gives rise to the broadening of the first ring of the OAM mode probability. Larger

*l*

_{0}can be the better choice for HB beams to propagate in the turbulent channel. The probability density ${\left|{\beta}_{l}(r,z)\right|}^{2}$of the mode crosstalk between the OAM modes

*l*

_{0}and

*l*is shown in Fig. 1(b). Clearly, the ${\left|{\beta}_{l}(r,z)\right|}^{2}$ between the OAM modes

*l*

_{0}and

*l*

_{0}± 1 are larger than that of between the OAM states

*l*

_{0}and

*l*

_{0}+ 2, and the maximum crosstalk of the mode probability density moves away from

*r*= 0 while the values Δ

*l*=

*l*±

*l*

_{0}increase. Compared with LG beams in the same parameters in Figs. 1(c) and 1(d), the spreading of LG beams is more seriously for different initial mode indices and LG beams have only one peak. Additionally, the mode probability density of the HB beams is much larger than that of LG beams, but the crosstalk probability density of the HB beams is smaller.

Also the mode probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ and crosstalk probability density ${\left|{\beta}_{l}(r,z)\right|}^{2}$ of HB beams versus different non-Kolmogorov turbulence-parameters *α* = 3.07, 3.37, 3.67 and 3.97 with the same wavelength *λ* = 1550nm, *z* = 1km, mode indice *l*_{0} = 1 and the refractive index structure parameter ${C}_{n}^{2}={10}^{-14}$m^{3-}* ^{α}* is presented in Figs. 2(a)-2(c). Clearly, the peak position of the mode probability densitiy moves to the beam center as the non-Kolmogorov turbulence-parameter

*α*increases. Note, the mode probability densities are the same at the values

*α*= 3.37 and

*α*= 3.67. It can be seen in Figs. 2(a)-2(c) that the ring breadth of the probability density of the mode or mode crosstalk is independent on the non-Kolmogorov property of the turbulence, while the mode probability density at the non-Kolmogorov turbulence-parameter

*α*= 3.07 is larger than that of

*α*= 3.97, that is to say the probability density of the crosstalk mode at

*α*= 3.97 is larger than that of

*α*= 3.07. We should choose smaller

*α*to hold more energy in the turbulent atmosphere. Moreover, we draw comparisons with the situation in LG beams (Figs. 2(d) and 2(e)). Their mode probability density and crosstalk probability density are almost the same.

It is depicted in Fig. 3(a) that the mode probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ versus *r* with the mode indice *l*_{0} = 1, *λ* = 1550nm, propagation distance *z* = 1km and *α* = 11/3 for the structure parameters ${C}_{n}^{2}={10}^{-16},{10}^{-15}$ and ${10}^{-14}$m^{3-}* ^{α}*. From Fig. 3(a), it is observed that increasing ${C}_{n}^{2}$ results in a lower mode probability density, and increasing ${C}_{n}^{2}$, the peak position

*r*of the mode probability density moves to beam center. The effects of the refractive index structure parameter ${C}_{n}^{2}$ on the probability density of the mode crosstalk in the case of

*l*= 2, and

*l*

_{0}= 1 are depicted in Fig. 3(b).Note the ring breadth of the probability density of the mode or mode crosstalk is independent on the ${C}_{n}^{2}$. Clearly, the smaller the refractive index structure parameter, the smaller probability density of mode crosstalk is achieved. Figures 3(c) and 3(d) are the situation of LG beams. The mode probability density and crosstalk probability density of HB beams and LG beams are in the same level.

We plot the mode probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ versus *r* in Fig. 4(a) for propagation distance *z* = 0.5 and 1km, assuming the mode indice *l*_{0} = 1, wavelength *λ* = 1550nm, *α* = 11/3 and the refractive index structure parameter ${C}_{n}^{2}={10}^{-14}$m^{3-}* ^{α}*. As is indicated in Fig. 4(a), the mode probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ decreases with the increasing of the propagation distance

*z*, and increasing

*z*, the broader breadth of the first ring of the mode probability density is brought about. Moreover, the maximum value position of the mode probability density moves away from

*r*= 0 as the propagation distance

*z*increases. As is shown in Fig. 4(b), the longer propagation distance, the larger crosstalk of the mode probability densities of the OAM modes. Clearly, increasing the propagation distance, also results in broader breadth of the first ring of the probability density of the mode crosstalk. By comparison to LG beams in Figs. 4(c) and 4(d), the mode probability density of LG beams is larger, but the spreading of the main peak of HB beams is much smaller.

It is shown in Figs. 5(a) and 5(b) that the mode probability density ${\left|{\beta}_{{l}_{0}}(r,z)\right|}^{2}$ and the crosstalk probability density ${\left|{\beta}_{l}(r,z)\right|}^{2}$ versus *r* for wavelengths *λ* = 690, 785, 850 and 1550nm, assuming the mode indice *l*_{0} = 1, propagation distance *z* = 0.5km, *α* = 11/3 and the refractive index structure parameter ${C}_{n}^{2}={10}^{-14}$m^{3-}* ^{α}*. It can be seen from Fig. 5(a) that the mode probability density decreases as the wavelength decreases, while in Fig. 5(b) the probability density of the mode crosstalk increases as the wavelength decreases. When the wavelength increases, the width of the first ring of the mode probability and the probability density of the mode crosstalk all become broader. The energy which the larger

*λ*holds is larger, but its noise is larger too. It is the opposite way round-the mode probability density of LG beams raises as

*λ*decreases(Figs. 5(c) and 5(d)). The probability density of LG beams is much larger than HB beams, and in similar matters, the spreading of the main light spot is smaller.

## 4. Conclusions

In summary, in this paper, a novel statistical model for the probability density of the OAM modes or mode crosstalk of HB beams in the atmospheric turbulence is developed. It is shown that, there are multi-peaks of the probability density of the modes or mode crosstalk along the *r* axis. The peak position of the mode probability density moves away from *r* = 0 with the increasing of the OAM quantum number *l*_{0}, the propagation distance *z* and the wavelength *λ* of HB beams and the decreasing of the non-Kolmogorov turbulence-parameters *α* and the generalized refractive-index structure parameters ${C}_{n}^{2}$. Decreasing *l*_{0} or increasing ${C}_{n}^{2}$ and *z* result in decreasing of the mode probability density, but the mode probability densities are the same at the values of *α* = 3.37 and *α* = 3.67. We should choose larger *l*_{0}, smaller ${C}_{n}^{2}$ to gain more energy. The probability density of the OAM mode crosstalk is increasing with the decreasing of *λ* and the deviation Δ*l* of OAM quantum number. In addition, the ring breadth of the probability density of the mode or mode crosstalk is independent with *α* and${C}_{n}^{2}$, however it depends on *z*, *λ* and *l*_{0}. Smaller *λ* and *l*_{0} are the better choice to mitigate beam spreading in the turbulence atmospheric. Compared with LG beams in the same parameters, we can conclude that although HB beams have multi-peak mode probability densities, most of the energy is located between *r* = 0 and *r* = 5cm, which is really a small region. The spreading of LG beams is more seriously than HB beams. Similarly to the case of no turbulence [11], HB beams still have the characteristic of focusing in turbulence channel.

## References and links

**1. **C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. **94**(15), 153901 (2005). [CrossRef] [PubMed]

**2. **F. E. S. Vetelino and R. J. Morgana, “Model validation of turbulence effects on orbital angular momentum of single photons for optical communication,” Proc. SPIE **7685**, 76850R (2010). [CrossRef]

**3. **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). [CrossRef]

**4. **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). [CrossRef]

**5. **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). [CrossRef] [PubMed]

**6. **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). [CrossRef] [PubMed]

**7. **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). [CrossRef] [PubMed]

**8. **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). [CrossRef]

**9. **J. Wang, J. Jia, J. Xu, Y. Wang, and Y. Zhang, “The probability of orbital angular momentum states of single photons with Z-tilt corrected residual aberration in a slant path turbulent atmosphere,” Optik **122**(11), 996–999 (2011). [CrossRef]

**10. **S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express **20**(1), 452–461 (2012). [CrossRef] [PubMed]

**11. **V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Hankel-Bessel laser beams,” J. Opt. Soc. Am. A **29**(5), 741–747 (2012). [CrossRef] [PubMed]

**12. **I. S. Gradshteyn and I. M. Ryzhik, *Table of Integrals, Series and Products*, 6th ed. (Academic, 2000).

**13. **X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. **37**(13), 2607–2609 (2012). [CrossRef] [PubMed]

**14. **F. Li, H. Tang, Y. Jiang, and J. Ou, “Spiral spectrum of Laguerre-Gaussian beams propagating in turbulent atmosphere,” Acta Phys. Sin. **60**(1), 014204 (2011).