We model the effects of weak fluctuations on the probability densities and normalized powers of vortex models for the Bessel–Gauss photon beam with fractional topological charge in the paraxial non-Kolmogorov turbulence channel. We find that probability density of signal vortex models is a function of deviation from the center of the photon beam, and the farther away from the beam center it is, the smaller the probability density is. For fractional topological charge, the average probability densities of signal/crosstalk vortex modes oscillate along the beam radius except the half-integer order. As the beam waist of the photon source grows, the average probability density of signal and crosstalk vortex modes grow together. Moreover, the peak of the average probability density of crosstalk vortex modes shifts outward from the beam center as the beam waist gets larger. The results also show that the smaller index of non-Kolmogorov turbulence and the smaller generalized refractive-index structure parameter may lead to the higher average probability densities of signal vortex modes and lower average probability densities of crosstalk vortex modes. Lower-coherence radius or beam waist can give rise to less reduction of the normalized powers of the signal vortex modes, which is opposite to the normalized powers of crosstalk vortex modes.
© 2016 Chinese Laser Press
In recent years, the study of the stable vortex beams whose vortex model per photon can take an arbitrary value within a continuous range, either integer or noninteger, in units of is a newly burgeoning field [1–5]. The propagation property of fractional vortice-imprinted Bessel beams in free space has been given attention both theoretically and experimentally [6–8]. It has been verified that vortices with fractional strength do not propagate unaffected in free space. However, neglecting their evolving intensity and vortex structure, a vortex integrated over the whole beam cross section is invariant under free-space propagation . The fractional Bessel beams (nondiffracting vortex beams) are of special interest due to their properties of divergence-free propagation and self-repair, and have generated widespread interest in the last decade for applications in optical communications [9–13]. Götte proposed a quantum mechanical description of the beams with fractional topological charges , and a high-order Bessel nonvortex beam of fractional type (HOBNVBs-F) was introduced by Mitri [10–12]. Next, we developed the model of average probability densities and the normalized powers of the signal/crosstalk orbital angular momentum modes for the fractional-order Bessel–Gauss (FoBG) photon beams in the turbulent atmosphere of strong irradiance fluctuations . On the other hand, we know that Kolmogorov’s power spectral density model is widely used and accepted to describe wave propagation through atmospheric turbulence, but the experimental data in the last several decades shows that atmospheric turbulence generally possesses a structure different from the Kolmogorov model . The Kolmogorov turbulence is actually an exception to non-Kolmogorov turbulence. When describing the effect of turbulence on fractional Bessel–Gauss photon beams in different conditions, it is better using non-Kolmogorov turbulence than Kolmogorov turbulence, which is especially embodied in its exponent value. As we know, there is almost no discussion with respect to the effects of non-Kolmogorov turbulence on the probability densities and the normalized powers of the vortex modes for FoBG photon beams in the weak and paraxial atmosphere channel.
In this paper, we analyze the influence of non-Kolmogorov turbulence on the vortex mode probability densities and the normalized powers for FoBG beams along the direction of the beam radius. Our aim is to exploit this changing process to present the effects of the weak non-Kolmogorov turbulence on Bessel–Gauss photon beams. We anticipate that this work will be particularly useful when applied to practical application for optical wireless communication. In Section 2, the models of the probability density for vortex modes and the normalized power of vortex per photon per unit length of a transverse slice of the beam for FoBG beams in non-Kolmogorov turbulence are established. The effects of fractional topological charges, turbulence strength (in weak fluctuation region), propagation distance, the non-Kolmogorov turbulence parameter, and wavelength on the probability densities of signal and crosstalk vortex modes of FoBG photon beams along the direction of the receiving plane are researched in Section 3. Conclusions are presented in Section 4.
2. MODE PROBABILITY DENSITY7,15] 7]. To keep it simple, we use to denote the term .
As we know, from the view of quantum theory, the atmospheric turbulence fluctuations disturb the complex amplitude of the propagating photon beam, which is no longer guaranteed to be in the original eigenstate of topological charge. The resulting photon beam now can be regarded as a superposition of the waves with new topological charge. Similar to the discussion of [7,17] for beams with integer topological charge, the FoBG is expanded as an integral with orthogonal basis which carries topological charge ; then the complex amplitude of the FoBG beam can be written as
As we all know, the mode probability density is equal to . In turbulent media, the mode probability density is associated with the ensemble average over turbulent medium, i.e., , where denotes the average over the ensemble of the turbulent atmosphere. By Eqs. (1) and (4) and proceeding with the ensemble average of atmospheric turbulence for , where denotes complex conjugate, the ensemble averaging mode probability density of vortex models of an FoBG beam in paraxial channel is given by
In the far-field region , is given by18] 3, we set without considering the effect of outer scale, and .18,19] and given by 20] 2), (5), and (7) as well as the Bessel function orthogonal completeness 21]. The term indicates the vortex model distribution for different fractional values when changes (see Fig. 1).
The normalized power of vortex per photon per unit length of a transverse slice of the beam is calculated with the expression 7]
Similarly, for , expresses the normalized power of the signal vortex mode for FoBG photon beams and, for , is the normalized power of the crosstalk vortex modes.
3. NUMERICAL DISCUSSION
As we all know, the longer wavelength is beneficial to the propagation of photon beams for less attenuation and lower crosstalk. Besides, it is entirely reasonable that the average probability density of signal vortex modes and signal normalized powers decrease, and the average crosstalk probability of vortex modes and crosstalk normalized powers increase when the propagation distance increases. Then, we do not discuss the influences owed to the wavelength and the propagation distance . To simplify, in the following figures, we now define mode probability as average probability density () of the signal vortex modes, and crosstalk probability as the average probability density () of the crosstalk vortex modes.
First, we study the effects of the fractional order of topological charge , the fractional parameter , the non-Kolmogorov turbulence coherence radius (in weak fluctuation region), and beam waist on the average probability densities along the direction of for FoBG beams. The simulation results are shown in Figs. 2–5. In Fig. 2, we evaluate performance of average probability densities along the direction of the beam radius for FoBG beam with , , and , 1, 2, 3, and 4. The parameters for the simulation of atmospheric turbulence channel are as follows: , , and . From Fig. 2, we can see that crosstalk probability decreases with the increase of topological charge deviation and it drops rapidly after the slow growth with the increase of . When is large enough (in Fig. 2, ), mode probability and crosstalk probability tend to be the same. The results show that we can obtain the higher signal crosstalk probability ratio only in the circle region of radius , and the center of the circle region is the beam center. So in the next context, we adopt the circle region of radius .
In Fig. 3, the average probability density is plotted as a function of the beam radius for topological charge deviation and 1, with , , , , and . Figures 3(a) and 3(b) are for , 3.3, 3.5, 3.7, and 3.9, respectively. As is seen from Fig. 3, the for , 3.3, 3.7, and 3.9 are the vibration curves which are around the curve of . Figure 3(a) indicates that mode probability decreases with the increase of , while the crosstalk probability decays after the increase in the area close to the circle area in Fig. 3(b). Figures 3(a) and 3(b) show that the curve is without vibration when the value of is half-integer. And the more close the value of is to half-integer, the closer its curve tends to the curve with . In order to facilitate the beam properties, we set .
Figures 4 and 5 present how the various parameters of atmospheric turbulence affect the average probability densities . Figure 4(a) is a plot of the average mode probability versus the varied beam radius for the different non-Kolmogorov turbulence parameters . The curve associated with corresponds to Kolmogorov turbulence. The curves in Fig. 4(a) are consistent with  in that the smaller comes with the larger in the receiving plane for any given propagation distance; the reason is that the spatial coherence radius tends to infinity when approaches 3. That is, the interruption of the turbulence on the photon beams is very weak, and a high is acquired. However, when is close to 4, the wave aberration caused by turbulence is a pure wavefront tilt, which shifts the beam off-axis and results in the beam center being away from the receiving plane . Thus, decreases as approaches 4. The relationship between the average crosstalk probability and the of non-Kolmogorov turbulence is revealed by Fig. 4(b). When the propagation distance is given, the of FoBG photon beams increases rapidly when changes from 3.07 to 3.97 near the beam center regions (about ). When detection position is far away from the beam center in the receiving plane, all the curves in Fig. 4(b) tend to be same.
Figure 5 is constructed to assess how the non-Kolmogorov turbulence coherence radius affects the average probability densities with varied beam radius . The curves in Fig. 5(a) show that increasing makes mode probability decrease sooner, while increasing doesn’t make the signal probability density change apparent. It is important to note that crosstalk probability increases with the lower near the beam center (about ), and the main energy still remains at the transmit signal channel. However, crosstalk probability decays and the energy of the adjacent channel decreases immediately as the detection position moves away from the beam center. For instance, all the curves slowly tend to overlap; namely, the effects of different value on crosstalk probability densities become less distinct when .
Figure 6 investigates the average probability as a function of beam radius for a FoBG beam with topological charge deviation and 1, , and for , 0.05 m, 0.08 m, and 0.10 m, respectively. The parameters of the atmospheric turbulence channel for the simulation are as follows: , , and . Figure 6(a) shows that increasing makes mode probability decay slowly. The results of Fig. 6(b) reveal that higher corresponds to the higher crosstalk probability densities and makes the curve peak away from the beam center.
For the sake of simplicity, we will discuss the impact of on the mode probability densities and normalized powers instead of the basic parameters and . The normalized powers of signal and crosstalk vortex modes for the parameters , , , and , 1, 2, 3, and 4 are shown in Fig. 7. Here we see that the crosstalk-normalized powers between vortex modes decrease with the increasing of .
Finally, the effects of and on the normalized powers of signal or crosstalk vortex modes per photon per unit length of a transverse slice for FoBG beams are shown in Fig. 8. To do that, we fix the parameters and . Lower can give rise to increase of the normalized power of the signal vortex mode and decrease of the normalized powers of the crosstalk vortex modes . Obviously, greater can enhance the normalized power of the signal vortex mode and reduce the normalized powers of crosstalk vortex modes.
In conclusion, the model of probability densities and the normalized powers of signal or crosstalk vortex modes for FoBG photon beams in weak fluctuation turbulent atmosphere have been developed. Our results show that average mode probability and average crosstalk probability are the same when beam radius is large enough. For fractional order Bessel–Gauss beams, when varies, oscillates except is half-integer. For the half-integer beams, the increases of and result in the degradation of , and average crosstalk probability drops rapidly after the slow growth as increases, namely average crosstalk probability curves have peaks. Near the central region of the beam, enlarging or minimizing , , and can make the decrease. Furthermore, lower and can give rise to less reduction of the turbulence effect on the normalized power of signal vortex mode. Compared with signal normalized powers, the opposite changes in tendency of crosstalk-normalized powers are observed.
This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128), the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174), and the Fundamental Research Funds for the Central Universities (JUSRP51517).
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