A new class of twisted Schell-model array correlated sources are introduced based on Mercer’s expansion. It turns out that such sources can be expressed as superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, on the basis of a stretched coordinate system and a quadratic approximation, analytical expressions for the mutual coherence function of an anisotropic non-Kolmogorov turbulence and the cross-spectral density of a twisted Gaussian Schell-model array beam are rigorously derived. Due to the presence of the twist phase, the beam spot and the degree of coherence rotate as they propagate, but their rotation centers are different. It is shown that the anisotropy of turbulence causes an anisotropic beam spreading in the horizontal and vertical directions. However, impressing a twist phase on source beams can significantly inhibit this effect. For an anticipated atmospheric channel condition, a comprehensive selection of initial optical signal parameters, receiver aperture size and receiver capability, etc., is necessary. Our work is helpful for exploring new forms of twistable sources, and promotes guidance on optimization of partial coherent beam applications.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The manipulation and characterization of partially coherent beams and their novel spatial behaviors under propagation constitute important fundamentals in many areas of optical science and technology [1–6]. In previous decades, most of the studies concerning the spatial coherence properties have been based on Schell-model (SM) sources, in which the spectral degree of coherences obey Gaussian distributions [1–3]. It is found that spatial coherence of a partially coherent beam apparently influences the evolution behaviors of spectral density, degree of coherence, degree of polarization and state of the polarization [7–11]. Therefore, generation of prescribed structure beams with tailored intensity, polarization and phase by means of correlation functions manipulation, and exploration of nontrivial modulation properties have seen a rapid growth of interest. Based on the nonnegative definiteness criteria of the cross-spectral density (CSD), a sufficient condition for devising genuine correlation functions was proposed, and then quickly extended to vector cases [12–14]. In recent years, inspiring significant advances have been achieved in design and experimental generation of nonconventional correlation sources. It has been shown that light beams with peculiar correlation functions exhibit nontrivial propagation properties such as far-field flat-topped, ring-shaped, self-focusing, self-splitting, optical cage and optical lattices [15–27].
Twist phase is an important modulation dimension of partially coherent sources, which was introduced by Simon and later experimentally generated by Friberg [28–31]. Recently, on the basis of the nonnegative constraint of the CSD, Gori and associates introduced a new method to design twisted sources endowed with circular or rectangular symmetry [32–34]. Due to the existence of the twist phase, twisted partially coherent beams not only have the ability to carry orbital angular momentum, but also have better anti-turbulence self-repairing capabilities, which can find applications in optical tweezers, optical imaging and optical communications [35–38]. By using a modal analysis, it is possible to check whether a general correlation source can be twisted by evaluating the eigenvalues. Unfortunately, in addition to limited number of cases, it is usually not an easy work to find out the eigenvalues of a twisted partially coherent source [12, 33].
On the other hand, the study of the interaction of light beams with turbulence has attracted for a long time attention of the scientific community due to many interesting applications such as astronomy, optical communications, remote sensing, and optical imaging [39–41]. Experimental and theoretical results show that light beams with specific structured amplitude, coherence, phase and polarization can help reduce the signal distortion caused by turbulence [42–48]. For many years, atmospheric turbulence described by Kolmogorov’s power spectral density model has been applied extensively in studies of optical wave propagation in the atmosphere. However, it is shown that the atmosphere’s statistical behaviors in the upper troposphere and stratosphere, or along nonhomogeneous path does not always follow the Kolmogorov power spectrum density model, and a general non-Kolmogorov statistics should be taken into account [49–52]. In addition, experimental results also demonstrated that the turbulence in the free atmosphere can be highly anisotropic at large scales [53–56]. Therefore, it is worthwhile to find new spectrum model in order to describe anisotropic non-Kolmogorov turbulence statistical behaviors [49, 51]. In this paper, we introduce a new class of twisted Schell-model array (TSMA) correlated sources by using Mercer’s theory and the nonnegative definiteness criteria of the CSD, and the twistable condition is established as well. In addition, with the help of a stretched coordinate system and a quadratic approximation, a closed form for the mutual correlation function of a spherical wave in anisotropic non-Kolmogorov turbulence is also derived. It is seen that anisotropy of turbulence leads to an anisotropic beam spreading in the horizontal and vertical directions, and optimizing the source beam parameters such as the coherence parameter and the twist phase can significantly inhibit this effect. Our work is helpful for exploring new classes of twisted partially coherent beams, and promotes helpful guidance on optimization of partial coherent beam turbulence applications.
2. Generation of twisted Schell-model array (TSMA) coherent sources
For a generic partially coherent beam in space-frequency domain, its complete CSD with respect to arbitrary position vectors and can be expressed as 11, 22]Eq. (2) reduces to a Gaussian function, and the corresponding CSD reduces to a standard twisted Gaussian Schell-model (TGSM) source.
It is well known that light beams generated from SM sources have the ability to possess a twist phase 
Recently, based on the nonnegative definiteness criteria of the CSD, Borghi and associates demonstrated a method to determine whether a correlation structure is capable of carrying a twist phase by means of Mercer’s expansion. For any twisted partially coherent source, a Mercer’s expansion of the CSD reads as [29, 33]33]Eq. (8), we have used the following identity57], Eq. (8) turns out to beEq. (10) that the eigenvalues of a twisted SMA source has the same form of a twisted -correlated Gaussian Schell-model source except for the sum item . Under the condition of. Equation (10) reduces to Eq. (6) associated with a twisted SM source. Since the Laguerre polynomials are nonnegative for negative values of their argument, it is easily to find that the nonnegative condition of Eq. (10) is the same as a twisted SM source. Consequently, we have proven that SMA correlated sources have the ability to be twisted and the twistable condition is.
3. Propagation of a twisted Gaussian Schell-model array (TGSMA) beam through anisotropic turbulence
In this section, we further extend the work to study the propagation of a TGSMA beam in anisotropic turbulence (Fig. 1). For the sake of convenience, let us consider the source beam has an elliptical Gaussian spectral density and the CSD is expressed as
Under the framework of the narrow-angle propagation assumption, the propagation the CSD of a TGSMA beam in weak turbulence can be evaluated by means of the generalized Huygens-Fresnel integral 41], the last term is the mutual correlation function of turbulence having the form
Here, let us consider a generalized non-Kolmogorov anisotropic power spectrum introduced by Andrews .
For further analysis of anisotropic atmospheric turbulence, we change the stretched coordinate system for the anisotropic spectrum back to an isotropic one by means of the following substitutionsEqs. (14)-(16) into Eq. (13), and integrate over, we haveEq. (17) has the same form as the mutual correlation function in an isotropic turbulence. Then, by making use of the identityEquation (17) turns out to beEq. (19) coincides with Eq. (24) in . Although a complicated numerical calculation is possible for further study, this seems not very fruitful in understanding the propagation characteristics. We shall, therefore, consider a quadratic approximation of Eq. (19), which then can be solved in a closed form. This has been shown to be a good approximation for practical situations . Thus, the mutual correlation function of a spherical wave in anisotropic turbulence can be expressed asEquation (23) provides a convenient analytical expression to study the statistical properties of the TAGSM beam on propagation through anisotropic turbulence such as the spectral density and the DOC
4. Numerical analysis
In this section, we numerically analyze the propagation properties of a TGSMA beam propagating through anisotropic non-Kolmogorov turbulence. The source beam parameters the turbulence parameters are set as unless different values are specified. Figure 2 depicts the contour plots of the normalized spectral density distribution and the corresponding cross line of a TGSMA beam at different propagation distances. Due to the reciprocal relationship between source coherence and the far-field spectral density distribution, the initial elliptical GSMA beam spot gradually evolves into an array structure in isotropic turbulence [see Figs. 2(a1)-2(a3)]. Meanwhile, Fig. 2(a4) illustrates that an array spectral density will further evolve into a flat-top distribution as the propagation distance further increases due to the turbulence statistics. A comparison of Figs. 2(a1)-2(a4) and Figs. 2(b1)-2(b4) clearly shows that the spectral density distribution within a few kilometers is more sensitive to the source beam parameters than turbulence statistics. While for sufficient distance transmission, the spectral density distribution is determined at most by turbulence statistics, and the anisotropy of turbulence results in an anisotropic beam spreading. Figures 2(c) and 2(d) illustrate the effects of the twist phase and the initial coherence parameter on the spectral density. It is clearly seen that the twist phase not only gives rise to a rotation of beam spot, but also accelerates the beam diffraction, which is similar to that of conventional TGSM beams . As can been seen from Fig. 2(d4) that each far-field elliptical beamlet gradually turns into a circular spot as the initial coherence decreases. This implies that reducing the initial coherence can also help resist the turbulence-induced anisotropy [55,56].
In order to examine the effects of turbulence parameters on the far-field spectral density, Fig. 3 illustrates the spectral density distribution of a TGSMA beam at z = 5km for different turbulence parameters. In the absence of the twist phase, the anisotropy of turbulence leads to an anisotropic beam spreading in the horizontal and vertical directions [see Figs. 3(a1)-3(a4)]. For the case of, the long axis of elliptical beamlets is along the y (vertical) direction, while horizontal (long axis is along the x direction) elliptical beamlets correspond to. Moreover, due to the inverse square relationship between spectral density and the anisotropy parameter, a smaller anisotropic parameter means a stronger turbulence effect . It is of interest to find from Figs. 3(c1)-3(c4) that the twist phase-induced diffraction increases dramatically with the growth of the twist phase, and plays a major role in determining the spectral density profile. Therefore, a reasonable optimization of the twist phase is beneficial to suppress the effects of turbulence. One possible reason is that the twist phase leads to a more rapid beam diffraction. As a result, a relatively larger beam spot helps to enhance the ability to reduce the turbulence effect [40, 46].
Figure 4 illustrates the evolution of the on-axis spectral density as a function of power law and anisotropic factor for different values of the twist phase and coherence parameter at z = 5km. It is clearly seen from Fig. 4(a1) that a GSMA beam without the twist phaseis more sensitive to the turbulence statistics than a TGSMA beam. Meanwhile, as the increase of the value of the twist phase, the effect of turbulence decreases rapidly. Besides, a significant impact of the anisotropic factor on the on-axis spectral density mainly appears nearfor a GSMA beam. Similar theoretical simulations for beam width can be found in . The physical interpretation of the maximum value of spectral density appearing nearis that turbulence tends to vanish. On the other extreme of, the physical explanation is that phase effects dominate in the form of random tilts, which cause beam wander. Compared with Figs. 4(a1)-4(a3) and Figs. 4(b1)-4(b3), one finds that the dependence of the beam on turbulence statistics decreases as the coherence parameter decreases. Therefore, in other words, an appropriate reduction in initial coherence and use of the twist phase can effectively reduce the effects of turbulence on signal beams. However, it is worthwhile to note that it is very difficult to achieve very long-distance transmission of optical signal applications by manipulating the initial beam parameters [42, 43].
Now, let’s turn our attention to evaluate the evolution properties of the DOC of a TGSMA beam in anisotropic turbulence. Figure 5 shows the DOC and the corresponding cross line of a TGSMA beam at several different propagation distances. It is clearly seen that for relatively short distance turbulence transmission, the initial coherence and the twist phase play a significant role in determining the DOC, while the anisotropy parameter has little effects on the DOC. However, it is worth noting that for sufficiently long distance transmission, the spatial coherence property are mainly determined by turbulence statistics [40, 42]. In addition, it is of interest to find that although the spectral density and the DOC rotate during transmission, their rotation center are different. The rotation center of the spectral density splits as the beam splits during transmission, while the rotation center of the DOC remains invariant.
Figure 6 illustrates the DOC and the corresponding cross line of a TGSMA beam at several different propagation distances. It is interesting to find that the DOC rotates clockwise in the case of, while it rotates counterclockwise for . In addition, as the increase of the twist phase, the array structure of the DOC gradually appears, which is completely opposite to the evolution of spectral density. In other words, the ability of a TGSMA beam to resist anisotropic turbulence is enhanced as the twist phase increases. In Fig. 7, we display the behavior of the DOC at z = 5km as a function of vertical anisotropic factor or power law for different twist phase. In the absence of the twist phase [see Fig. 7(a1)] for a conventional Kolmogorov turbulence, the DOC is insensitive to the vertical anisotropic parameter and maintains Gaussian distribution. While for a TGSMA beam, Figs. 7(a2) and 7(a3) show that the DOC has been a significant change around. Meanwhile, it is found that the DOC array structure alongdirection significantly enhanced with the increase of anisotropic parameter, which means that the influence of turbulence is weakened. In addition, a remarkable change in the DOC near the boundary value of power lawindicates that the turbulence strength changes sharply. Furthermore, a comparison of Figs. 7(b2) and 7(b3) shows the array structure of the DOC grows apparently with the increase of the twist phase. This implies that the use of a twisted partially coherent beam is beneficial for suppressing the effect of turbulence.
In this work, we have introduced a new class of TGSMA correlated sources based on the nonnegative definiteness criteria of the CSD and a Mercer’s expansion. It turns out that such sources can be expressed as the superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, we extend our work to evaluate the statistical properties of a TGSMA beam in an anisotropic non-Kolmogorov turbulence. It has been seen that the twist phase not only gives rise to a rotation of beam spot, but also accelerates the beam diffraction. One also finds that the rotation center of the spectral density splits as the beam splits during transmission, while the rotation center of the DOC remains invariant. Moreover, the DOC distribution rotates clockwise in the case of, while a counterclockwise rotation corresponds to. The anisotropy of turbulence causes an anisotropic beam diffraction in the horizontal and vertical directions. However, using a twisted partially coherent beam can significantly inhibit this effect. It is worth noting that optimizing the initial beam parameters such as the coherence and the twist phase can help enhance turbulent resistance. However, this results in a larger beam spot and a reduced power incident on the receiver. Therefore, a comprehensive selection of initial beam parameters, receiver aperture size and receiver capability, etc., for the anticipated atmospheric channel conditions is necessary. Our work is helpful for exploring new classes of twisted partially coherent beams, and promotes helpful guidance on optimization of partial coherent beam turbulence applications.
National Natural Science Foundation of China (NSFC) (11504172, 91750201); National Natural Science Fund for Distinguished Young Scholar (11525418); Natural Science Foundation of Jiangsu Province (BK20150763); China Postdoctoral Science Foundation (7131701018); Postdoctoral Science Foundation of Jiangsu Province (7131707317); Fundamental Research Funds for the Central Universities (30917011336, 30916014112-001); Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents.
1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
2. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
3. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of partially polarized light fields (Springer Science & Business Media, 2009).
4. J. C. Dainty, Laser speckle and related phenomena (Springer science & business Media, 2013).
5. G. Gbur and T. D. Visser, “The structure of partially coherent fields,” Prog. Opt. 55, 285–341 (2010). [CrossRef]
6. Y. Cai, Y. Chen, J. Yu, X. Liu, and L. Liu, “Generation of partially coherent beams,” Prog. Opt. 62, 157–223 (2017). [CrossRef]
7. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8(2), 282–289 (1991). [CrossRef]
8. E. Wolf and D. F. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996). [CrossRef]
11. S. Zhu, J. Wang, X. Liu, Y. Cai, and Z. Li, “Generation of arbitrary radially polarized array beams by manipulating correlation structure,” Appl. Phys. Lett. 109(16), 161904 (2016). [CrossRef]
13. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross–spectral density matrices,” J. Opt. A 11(8), 085706 (2009). [CrossRef]
18. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91, 013832 (2014).
19. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014). [CrossRef] [PubMed]
24. S. Zhu, Y. Chen, J. Wang, H. Wang, Z. Li, and Y. Cai, “Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization,” Opt. Express 23(26), 33099–33115 (2015). [CrossRef] [PubMed]
25. M. W. Hyde, S. Bose-Pillai, D. G. Voelz, and X. Xiao, “Generation of vector partially coherent optical sources using phase-only spatial light modulators,” Phys. Rev. Appl. 6(6), 064030 (2016). [CrossRef]
26. M. Santarsiero, R. Martínez-Herrero, D. Maluenda, J. C. G. de Sande, G. Piquero, and F. Gori, “Partially coherent sources with circular coherence,” Opt. Lett. 42(8), 1512–1515 (2017). [CrossRef] [PubMed]
28. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). [CrossRef]
29. R. Simon, N. Mukunda, and K. Sundar, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10(9), 2008 (1993). [CrossRef]
31. A. T. Friberg, B. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994). [CrossRef]
35. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]
37. Y. Cai and S. Zhu, “Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain,” Opt. Lett. 39(7), 1968–1971 (2014). [CrossRef] [PubMed]
39. A. Ishimaru, Wave propagation and scattering in random media (Academic, 1978).
40. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, 2005).
45. G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107(2), 021104 (2016). [CrossRef]
47. J. Wang, S. Zhu, H. Wang, Y. Cai, and Z. Li, “Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence,” Opt. Express 24(11), 11626–11639 (2016). [CrossRef] [PubMed]
49. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008). [CrossRef]
50. I. Toselli, O. Korotkova, X. Xiao, and D. G. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov anisotropic turbulence,” Appl. Opt. 54(15), 4740–4744 (2015). [CrossRef] [PubMed]
51. L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014). [CrossRef]
52. J. Wang, S. Zhu, and Z. Li, “Vector properties of a tunable random electromagnetic beam in non-Kolmogrov turbulence,” Chin. Opt. Lett. 14(8), 80101–80105 (2016). [CrossRef]
53. A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Random Complex Media 4(3), 297–306 (1994). [CrossRef]
55. F. Wang, I. Toselli, J. Li, and O. Korotkova, “Measuring anisotropy ellipse of atmospheric turbulence by intensity correlations of laser light,” Opt. Lett. 42(6), 1129–1132 (2017). [CrossRef] [PubMed]
57. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).