## 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 *W*_{0}, 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

## 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, Laguerre-Gaussian (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 free-space 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 non-diffracted 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 M^{2} 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]

*z*is the propagation distance of light;

*q*

_{0}is the beam parameter at the plane

*z*= 0 with$1/{q}_{0}=1/R+\text{i}2/k{w}_{0}^{2}$;

*R*is the radius of curvature of the initial spherical phase front; ${q}_{0}^{*}$ is the complex conjugate of

*q*

_{0}; $k=2\pi /\lambda $ is the wave number of light;

*λ*is the wavelength;

*w*

_{0}is the beamwidth of the Gaussian envelop;

*l*

_{0}corresponds to the OAM ${l}_{0}\hslash $ carried by the beam, which describes the helical structure of the wave front around a wave front singularity; $r=\left|r\right|,\text{\hspace{0.17em}}r=\left(x,y\right)$ is the two-dimensional position vector in the source plane,

*φ*is the azimuthally angle; ${}_{1}F{}_{1}\left[a,b,x\right]={\displaystyle \sum _{n=0}^{\infty}\frac{{\left(a\right)}_{n}}{n!{\left(b\right)}_{n}}}{x}^{n}$ is the confluent hypergeometric function; $\beta =\left({l}_{0}+1-\text{i}\gamma \right)/2$; ${C}_{{l}_{0},\text{i}\gamma}$ is the normalization constant of WG beams, which is determined by ${C}_{{l}_{0},\text{i}\gamma}=1/\sqrt{{\sigma}_{{l}_{0},\text{i}\gamma}^{0}}$ and ${\sigma}_{{l}_{0},\text{i}\gamma}^{0}$ is given by

*W*

_{0}and

*γ*are real parameters of WG beams; $\xi ={w}_{0}/{W}_{0}$, and

For *l*_{0} = 0 and $\zeta \ll 1$, the beam transverse intensity behaves as a Gaussian beam, and the beam has a doughnut-like structure for ${l}_{0}>0$ [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 $W{{G}^{\prime}}_{l,\text{i}\gamma}\left(r,\phi ,z\right)$ of WG beams as

Based on the discussion in Ref [14, 15], we can write the function $W{{G}^{\prime}}_{l,\text{i}\gamma}\left(r,\phi ,z\right)$ as a superposition of the spiral harmonics $\mathrm{exp}\left(\text{i}l\phi \right)$

Following the same procedure as in Ref [3], the mode probability distribution of the spiral plane mode with phase $\mathrm{exp}\left(\text{i}l\phi \right)$ can be expressed as

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 $\mathrm{exp}\left(\text{i}l\phi \right)$ of WG beams propagating in paraxial turbulence channel, which is named as spiral mode probability distribution and is given by

*ρ*

_{0}is the spatial coherence radius of a spherical wave propagating in slant non-Kolmogorov turbulence [16] and is given by

*α*is the non-Kolmogorov turbulence-parameter, $\Gamma \left(x\right)$ is the Gamma function, ${C}_{n}^{2}(\xi ,\text{\hspace{0.17em}}\theta )$ is the refractive index structure parameter in a slant turbulence channel with units ${\text{m}}^{3-\alpha}$ and

*θ*is the zenith angle [17].

By substituting Eq. (1) into Eq. (8) and utilizing the integral expression [18]

*n*order.

By defining the mode weight (or energy content) for each angular momentum mode *m* with the signal OAM mode *l*_{0}

*z*plane after propagating in non-Kolmogorov turbulence by following relationship

For designated OAM mode *l* = *l*_{0}, ${\wp}_{{l}_{0}}\left(z\right)$ (Eq. (13)) is defined as the detection probability of the signal OAM mode *l*_{0}, which shows the transfer rate of the transmitted OAM state; for $l={l}_{0}\pm \Delta l$, ${\wp}_{l}\left(z\right)$ is defined as the crosstalk probability which denotes the probability of a photon to change its OAM state, and *∆l* denotes the OAM quantum number difference.

## 3. Numerical discussion

In this section, we discuss the numerical results of the detection probability spectrum, the detection probability, crosstalk probability and the mode weight of the signal OAM mode for WG beams in non-Kolmogorov turbulence by using the analytical equations in the previous section with the radius of curvature of the initial spherical phase front *R* = $5\times {10}^{4}\text{m}$and the zenith angle *θ* = ${0}^{\circ}$.

In Figs. 1(a)-1(b), we investigate the impact of source model for WG beams (a) and LG beams (b) (based on Eq. (14) in [3]) on the detection probability of the signal OAM mode. To this end, we fix the parameters of both WG and LG beams as wavelength *λ* = 1550nm, the beamwidth *w*_{0} = 0.02m, the refractive index structure parameter ${C}_{n}^{2}={10}^{-15}{\text{m}}^{3-\alpha}$, *α* = 3.97, the real parameters of WG beams *W*_{0} = $1\times {10}^{-3}$, *γ* = 1, and vary the propagation distance *z* from 0 to 1000m and the signal OAM mode quantum number *l*_{0} from 1 to 4. Comparing the subplots from Figs. 1(a)-1(b), it can be said that WG beams provide excellent performance improvement on the detection probability of the signal OAM mode for LG beams in the weak turbulent regimes. However, as propagation distance $z\ge 500\text{m}$, the decrease of the detection probability of the signal OAM mode for WG beams is more noticeable with the increase of *z*. Figures 1(a)-1(b) also show that increasing quantum number *l*_{0} will lead to a significant effect on the detection probability of the signal OAM mode for both WG and LG beam, and the detection probability decreases with the increase of quantum number *l*_{0} and the propagation distance z. It is because the OAM modes with larger quantum number *l*_{0} will have a larger radius after propagation and will result in an increase in the negative effects of turbulence, therefore it can lead to the decrease of the detection probability of the signal OAM mode.

With the same parameters setting used in Figs. 1(a)-1(b), we evaluate the crosstalk probability with the OAM quantum number difference *∆l* = |*l*-*l*_{0}| = 1 and the detection probability spectrum of the signal OAM mode *l*_{0} = 1, 2, 3 and 4 for WG beams in Fig. 2 and Fig. 3. Figure 2 shows that the crosstalk probability increases with the increasing of the signal OAM mode quantum number *l*_{0} and propagation distance *z*. As shown in Fig. 3, the majority of the crosstalk for the WG modes happens with the immediate neighboring modes (*∆l* = 1), and the other crosstalk effect on the detection of the signal OAM mode for WG beams can be ignored, from this point, we can said that the resistant ability to turbulence interference of WG beams is much better than LG beams [5, 7]. It is also observed that the detection probability of the signal OAM mode decreases with the increasing of the quantum number *l*_{0}.

Figure 4 explores the impact of the beamwidth *w*_{0} on the detection probability of the signal OAM mode *l*_{0} = 1 for WG beams by changing *w*_{0} = 0.01m, 0.02m and 0.03m in the different turbulence ${C}_{n}^{2}$ from ${10}^{-16}$ to ${10}^{-15}$, under the conditions with *λ* = 1550nm, *α* = 3.97, *W*_{0} = $1\times {10}^{-3}$, *γ* = 1 and *z* = 1000m. We can see that increasing beamwidth *w*_{0} will lead to a significant improvement on the detection of the signal OAM mode for WG beams. This is reasonable, since the decrease of *w*_{0} will result in stronger diffraction and larger mode crosstalk at the receiver. These results show that increasing the beamwidth of the laser beams is a good choice to improve the SNR of the communication system in turbulent atmosphere.

Assuming parameters as *w*_{0} = 0.02m, ${C}_{n}^{2}={10}^{-15}{\text{m}}^{3-\alpha}$, *γ* = 1, *W*_{0} = $1\times {10}^{-3}$ and *z* = 1000m, Fig. 5 depicts the detection probability of the signal OAM mode *l*_{0} = 1 for WG beams versus non-Kolmogorov parameter α for different wavelength *λ* = 1550nm, 850nm and 632.8nm. It is shown that the detection probability increases with the increase of wavelength *λ.* From Figs. 4 and 5,we also can observe that the increases of the values of ${C}_{n}^{2}$ and *α* result in the decrease of the detection probability, and increasing *α* from 3.07 to 3.97, the detection probability of the signal OAM mode (*l*_{0} = 1) will give rise to decrease only about 10%, and the influence is obvious when *α* from 3.77 to 3.97. Although the influence of the zenith angle is not displayed here explicitly, the change tendency of the detection probability versus the zenith angle of the slant non-Kolmogorov turbulence channel is analogous to that versus associated ${C}_{n}^{2}$ in Fig. 4. It is clearly that the smaller ${C}_{n}^{2}$ and *α* will cause the larger coherence length ${\rho}_{0}$, and the larger ${\rho}_{0}$ will attribute the improvement in the detection probability to a smaller ratio of the mode crosstalk.

Figures 6(a)-6(b) exhibit the effect of the real parameter of WG beams *γ* on the detection probability and the mode weight of the signal OAM mode *l*_{0} = 1 for WG beams by changing *γ* from 0 to 2 in the different turbulence ${C}_{n}^{2}$ from $5\times {10}^{-16}$, $7.5\times {10}^{-16}$ to ${10}^{-15}$. Here, we set parameters as *λ* = 1550nm, *w*_{0} = 0.02m, *α* = 3.97, *W*_{0} = $1\times {10}^{-3}$ and *z* = 1000m. Figure 6(a) shows that the decrease of the value of *γ* has no significant improvement on the detection probability of the signal OAM mode for WG beams. However, in Fig. 6(b), it is observed that *γ* has a significant effect on the mode weight of each OAM mode and the mode weight decrease with the increase of *γ*.

Figures 7(a)-7(b) show the detection probability and the mode weight of the signal OAM mode *l*_{0} = 1 for WG beams versus the real parameter of WG beams *W*_{0} from 0 to $1\times {10}^{-3}$ for propagation distance *z* = 1000m, 1500m and 2000m, and parameters *λ* = 1550nm, *w*_{0} = 0.02m, ${C}_{n}^{2}={10}^{-15}{\text{m}}^{3-\alpha}$, *α* = 3.97 and *γ* = 1. Analogous to *γ*, the effect of *W*_{0} on the detection probability of the signal OAM mode for WG beams is negligible, but the mode weight decreases with the increase of the value of *W*_{0}. These results show that, although *γ* and *W*_{0} have no effect on the detection probability spectrum, decreasing the values of *γ* and *W*_{0} is also a good choice to improve the mode weight (energy content) of the signal OAM mode.

## 4. Conclusions

We have quantitatively described the effects of turbulence on the detection probability spectrum and the mode weight of the OAM for WG beams in non-Kolmogorov turbulence channels. The influence of the OAM quantum number, propagation distance, beamwidth, the refractive-index structure parameter, non-Kolmogorov parameter, wavelength and the real parameters of WG beams *γ* and *W*_{0} are discussed. Our results show that the detection probability of the signal OAM mode decreases with the increasing of the signal OAM mode quantum number *l*_{0}, propagation distance, non-Kolmogorov turbulence parameters and the generalized refractive index structure parameters, and with the decreasing of the wavelength and beamwidth of the beams. The crosstalk probability of OAM mode is symmetrical and has an opposite trend of the detection probability. Although *γ* and *W*_{0} has no effect on the detection probability of the signal OAM mode, the mode weight decreases with the increasing of the real parameters of WG beams *γ* and *W*_{0}. Compared with LG beam, WG beam is a better light source for mitigating the effects of turbulence on the detection probability of the signal OAM mode in the weak turbulence regime due to its nondiffraction property.

## Acknowledgments

This work is supported by the graduate student research innovation project of Jiangsu province general university and Fundamental Research Funds for the Central Universities of China (Grant no. 1142050205135370).

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