The turbulent effects of strong irradiance fluctuations on the probability densities and the normalized powers of the orbital angular momentum (OAM) modes are modeled for fractional Bessel Gauss beams in paraxial turbulence channel. We find that the probability density of signal OAM modes is a function of position deviation from the beam center, and the farther away from the beam center the detection position is, the smaller the probability density is. For fractional OAM quantum numbers, the average probability densities of signal/crosstalk modes oscillate along the beam radius except the half-integer. When the beam waist of source decreases or the irradiance fluctuation increases, the average probability density of the signal OAM mode drops. The peak of the average probability density of crosstalk modes shifts to outward of the beam center as beam waist gets larger. In the nearby region of beam center, the larger the quantum number deviation of OAM, the smaller the beam waist and the turbulence fluctuations are, the lower average probability densities of crosstalk OAM modes are. Especially, the increase of turbulence fluctuations can make the crosstalk stronger and more concentrated. Lower irradiance fluctuation can give rise to higher the normalized powers of the signal OAM modes, which is opposite to the crosstalk normalized powers.
© 2015 Optical Society of America
The progress in optical quantum communication over the last few years have enabled the development of the orbital angular momentum (OAM) of stable vortex beams whose OAM per photon can take an arbitrary value within a continuous range, either integer or non-integer in units of [1–7]. It was demonstrated theoretically that the fractional topological charges associated with center vortices and instantons are necessarily grouped into integral charges on a global scale, but are uncorrelated with each other on any shorter scale . Several experiments have verified that light beam may carry fractional OAM when the beam propagates in free-space [8–13]. Such as, S. H. Tao et al. emphasize that a higher-order fractional Bessel beam will be ideal for investigating the behavior of orbital angular momentum at the opening of the fractional Bessel beam . W. M. Lee et al. examine the evolution of optical beams with a fractional phase step hosted within a Gaussian beam by experimental analysis of both the phase and intensity distribution . S. Vyas et al. experimentally demonstrate the generation of a fractional topological charge beam on the basics of vortex lens . This type of light beam has an additional line singularity from beam center to its outward, where the intensity also vanishes.
The fractional Bessel beams (nondiffracting vortex beams) are of special interest due to their properties of divergence-less propagation and self-repair, and have generated widespread interest in the last decade for optical communication [14–17]. J. B. GÖTTE proposed a quantum mechanical description of the beams with fractional topological charges  and a high-order Bessel non-vortex beam of fractional type (HOBNVBs-Fα), was introduced by F. G. Mitri [16,17]. It was found the light modes with fractional orbital angular momentum can be applied to the field of where range from quantum entanglement  to optical manipulation . However few researchers have addressed the turbulence effects of strong irradiance fluctuations on OAM modes of fractional Bessel Gauss (FBG) beams.
To understand the turbulence effects of strong irradiance fluctuations on the receiving signals at each point in the receiving plane and the receiving value of acceptor, for FBG beams with orbital angular momentum . In this paper, we outline the influences of atmospheric turbulence on the orbital angular momentum mode probability densities and the normalized powers for FBG beams along the direction of the beam radius. In particular, we draw attention to the models of mode probability density for signal or the crosstalk OAM modes and the normalized power of OAM per photon per unit length of a transverse slice of the beam of a FBG beams in strong turbulence are established. The effects of the fractional order of OAM modes, turbulence strength (in strong fluctuation region), propagation distance and wavelength on the mode probability densities of the signal and crosstalk OAM modes of FBG beams along the direction of in receiving plane are researched in section 3. Conclusions are presented in Section 4. Our results provide us with new insight into the behavior of the OAM of fractional Bessel Gauss beams in the context of fractional beams and elucidate the connection between traditional optical communication and quantum theory to represent beams with fractional OAM.
2. Ensemble averaging mode probability density
In the strong irradiance fluctuation region  and in the half-space, the normalized complex amplitude of FBG beams in a cylindrical coordinate system can be expressed as9,19]9,19]. The diffractive spread of beams propagation in turbulence-free channel from = 0 to 1000m is given by Fig. 1.
In a quantum description beams with fractional OAM are superpositions of states of different OAM , the FBG is expanded as an summation with basis which carries OAM of , then the complex amplitude for FBG can be written as [6,20]
In quantum description, is the mode probability density of the mode at the position . In turbulent media, the mode probability density is associated with the ensemble average over turbulence medium, i.e., where represents the ensemble average of the atmospheric turbulence. By Eqs. (1) and (4) and proceeding the ensemble average of the atmospheric turbulence for , and the denotes complex conjugate. In paraxial channel(), the ensemble averaging mode probability density of OAM models of FBG beam, which is marked with, is given by21].
Based on the modified Rytov method [18,21], the effective atmospheric spectral model is given by21], the filter function are given by22].
On spherical wave approximation and by the approximation of generalized hypergeometric functions in , we have the large and small scale portions of the wave structure function for a spherical wave
For strong irradiance fluctuations and by approximation , we have
Based on the integral expression Eqs. (2), (5) and (6), we have the mode probability density of the OAM for FBG beams in paraxial turbulence channel24].
The normalized power of OAM per photon per unit length of a transverse slice of the beam is calculated with the following expression 9]. Similarly, for , expresses the normalized power of the signal OAM mode for FBG beams, and for , is the normalized powers of the crosstalk OAM modes.
3. Numerical discussion
To investigate the influences of the strong irradiance fluctuations fractional order of OAM modes , the deviation of quantum number, the Rytov variance and beam waists on the average probability density of signal OAM modes and signal normalized powers, in this section, we employ the method of numerical calculation to analyze the Eqs. (16) and (17). Obviously, the average probability density of signal OAM modes and signal normalized powers decrease, and the average crosstalk probability of OAM modes and crosstalk normalized powers increase when the propagation distance increases. Then we do not discuss the influences owed to the propagation distance . To simplify, in figures, we write ensemble average probability density () of the signal OAM modes to mode probability, and write ensemble average probability density () of the crosstalk OAM modes to crosstalk probability.
In this section, we first study the effects of , , and on the mode probability densities of the signal () and crosstalk () OAM modes for FBG beams along the direction of beam radius in receiving plane. The simulation results are shown in Figs. 2-5. The parameters are taken as = 1550nm, = 1km, = 0.1m, = 5.5, = 0.001, = 11/3, = 0.001m, = 10, = 0, 1, 2 and 3, unless otherwise indicated. Finally, we evaluate the effects of the irradiance fluctuations on the normalized powers in parameter or . The simulation results are shown in the final figure.
Figure 2 plots the performance of of FBG beam versus from 0 to 0.2m for the different . Figure 2 demonstrates that the decays rapidly and tends to a stable value when r increases. As can be seen from Fig. 2, the peak of moves outward and downward clearly as gets larger. When is large enough (in Fig. 2, >0.05m), and tend to be same. The results show that we can obtain the higher signal-noise ratio (SNR) only in the circle region of radius <0.05m, and the center of the circle region is beam center. So in the next context, we adopt the circle region of radius <0.05m.
In Fig. 3, is plotted as a function of for mode crosstalk = 0 and 1. Figures 3 (a) and 3(b) is for = 5.1, 5.3, 5.5, 5.7 and 5.9, respectively. As is seen from Fig. 3, are the vibration curves for = 5.1, 5.3, 5.7 and 5.9, and these vibration curves are around the curve of = 5.5. Figure 3(a) indicates that decrease with the increasing of , while decay after the increase in the region close to circle area to circle area in Fig. 3(b). It is interesting to note that when the value of is half integer, its curve is without vibration in Figs. 3(c) and 3(d). Obviously, the closer the value of to half integer, the closer its curve tends to the image of = 5.5. In order to facilitate the beam properties, we set = 5.5.
Figure 4 indicates the variation of against for selected beam waist = 0.02m, 0.05m and 0.10m. Figure 4 shows that increasing can rise . From Fig. 4(a), it can be seen that the lower is, the faster decays. And the signal difference of adjacent channel increases as the detection position away from the beam center. As is shown by Fig. 4(b), all the curves pick up and then fall, and the value of keeps growing as gets larger.
Figure 5 illustrates that the variation of along the radial direction from 0 to 0.05m. For this example we have chosen = 5, 10 and 30. According to Fig. 5(a), as increases, decreases slightly but not obvious. It’s clear that the influence of turbulence on is more apparent as increases in the strong fluctuation region. Near the central area of the beam, increases with the increasing of and the main energy still remains at the transmit signal channel. However, as the detection position away from the beam center, the energy of crosstalk channel increases immediately and is almost as strong as the energy that remains in the signal channel. When detection position is far away from the beam center in receiving plane, and tend to be same. For the case of = 5, the energy of crosstalk channel at = 0.05m is almost the same as the energy of signal channel. The peak of the shifts to the beam center, and half width of the peak narrows down as increases, i.e. the increase of can make the crosstalk noise stronger and more concentrated.
Finally, in Fig. 6, we describe the effects of Rytov variance on the normalized powers in the case that the inner scales tend to 0. Comparing the curves in Fig. 6, as increases, the normalized power of the signal OAM mode reduces more and the normalized powers of the crosstalk OAM modes increase more. As is seen from this figure, the normalized powers of the signal or crosstalk OAM modes are the complicated functions of beam waist.
In conclusion, the model of average probability densities and the normalized powers of the signal/crosstalk OAM modes for FBG beams in the turbulent atmosphere of strong irradiance fluctuations have been developed. This study has revealed that average mode probability and average crosstalk probability are the same when beam radius is large enough. For FBG beams, when varies, and oscillate except half-integer, and the larger the parameter is, the greater the amplitude of oscillation curves will be. For beams of = 5.5, the average mode probability decays with the decreasing of the beam waist or the increasing of irradiance fluctuations along beam radius, and average crosstalk probability drops rapidly after the slow growth as increases, namely average crosstalk probability curves have peaks. Near the central region of beam, enlarging or minifying and can make drop down. The normalized powers of signal or crosstalk OAM modes are the complicated functions of beam waist. Furthermore, lower can more efficient reduce the turbulence effect on the normalized power of signal OAM modes, which is opposite to crosstalk normalized powers .
This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128) and the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174).
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