Measuring topological charge of partially coherent elegant Laguerre-Gaussian beam

Miao Dong; XingYuan Lu; Chengliang Zhao; Yangjian Cai; Yuanjie Yang

doi:10.1364/OE.26.033035

1. Introduction

Vortex beams have attracted much interest for carrying orbital angular momentum (OAM). The OAM in the propagation direction has the discrete value of lћ per photon, where l can take any integer value, referred to as the topological charge [1]. Since the values of l are theoretically unlimited, the fully coherent beams with OAM exhibit great potential for applications in various fields [2], ranging from optical manipulation [3], quantum information processing [4] to information transfer and communication [5]. Among these studies, measuring the topological charge of vortex beams is necessary and of great significance. Various methods have been proposed to measure the topological charge, such as the intensity distribution [6–8], diffraction pattern [9,10], or interference pattern of a vortex beam [11].

Recently, the study of optical vortices of fully coherent beams has been extended to those of partially coherent beams. Those methods of measuring topological charge for high degree of coherence will be invalid for low degree of coherence. Over the last decade, it has been shown that the coherence functions (or cross-spectral density functions), which are used to mathematically describe a partially coherent field, possess the phase correlation singularities denoting nulls of the cross-spectral density [12–16]. For instance, in 2004, Palacios et al. verified that the cross-correlation function (CCF) maintains a ring dislocation (i.e., correlation singularity) when a vortex is present [17]. In 2007, Rao et al. examined the correlation properties in the focal region of focused partially coherent vortex fields, and it shown that the degree of coherence possesses phase singularities in the focal region [18]. In 2008, Maleev et al. predicted that a low coherence vortex beam can exhibit robust vortex features in the CCF [19]. The partially coherent Laguerre-Gaussian (LG) beam is one typical kind of the partially coherent vortex beams. Much effort has been devoted to studying the correlation singularity of the partially coherent LG beam [20–23]. More recently, the relationship between the number of ring dislocations in the CCF and the mode indices of a partially coherent LG beam has been investigated theoretically and experimentally [24,25]. It was shown that the number of dislocation rings in the far-field CCF is dependent on both the azimuthal mode index (topological charge) and radial mode index, and is equal to 2p + |l| [26]. Recently, Zhao et al. explored the effect of spatial coherence on determining the topological charge of a vortex beam based on the complex degree of coherence (CDOC) [27]. Yang et al. proposed a new kind of correlation function, the double-correlation singularity, which depends on the radial mode index only [28]. Based on the relationship, different techniques for the measurement of the topological charge and OAM of a partially coherent LG beam have been proposed and implemented [26–29].

An elegant Laguerre-Gaussian (ELG) beam, as a typical example of elegant beams, was introduced by Siegman [30–32]. The ELG beam has a more symmetrical form in comparison with the standard Laguerre-Gaussian (SLG) beam since the former has the same complex scaling factor in the argument of both the Laguerre and Gaussian functions, but in the latter the Laguerre part is purely real while the Gaussian part has a complex argument [33–35]. Furthermore, the ELG beam has seen increasing application as an expansion basis for modes of other optical resonators and propagation [35]. In 2009, Wang et al. proposed partially coherent standard and elegant LG beams as a natural extension of corresponding fully coherent beams [36]. It was found that partially coherent ELG beams have many advantages over the corresponding SLG beams under the same circumstance [37–39]. For example, the propagation properties of partially coherent ELG beam are less affected by the turbulence, and the spreading of partially coherent ELG beam is slower through the free space and the turbulent atmosphere [38,39]. Since then, numerous efforts have been made on characterization, propagation, radiation force of partially coherent ELG beams due to their extraordinary characteristics [40–43]. However, to the best of our knowledge, how to measure the topological charge of a partially coherent ELG beam has not been reported yet.

In this study, we theoretically and experimentally explore the topological charge of partially coherent ELG beams. We first derive the analytical expression for the CDOC of a partially coherent ELG beam in the far-field plane and then investigate how the spatial correlation singularity is affected by both mode indices and coherence length. The simulations show that the number of correlation singularities of a partially coherent ELG beam is dependent on the topological charge only. Furthermore, the number of ring dislocations of the far-field CDOC of a partially coherent ELG beam, where the coherence length equals the beam-waist, is just equal to the value of its topological charge. Based on this relationship, we propose a method to determine the topological charge of a partially coherent ELG beam.

2. Complex degree of coherence of partially coherent elegant Laguerre- Gaussian beam in far field

The electric field distribution of an ELG beam at the source plane (z = 0) can be expressed as [30,31]:

E (ρ, θ; 0) = {(\frac{ρ}{w_{0}})}^{| l |} L_{p}^{| l |} (\frac{ρ^{2}}{w_{0}^{2}}) \exp (- \frac{ρ^{2}}{w_{0}^{2}}) \exp (- i l θ),

where w₀ is the beam waist width of the fundamental Gaussian mode,

L_{p}^{| l |} (\cdot)

is the Laguerre polynomial with the mode orders p and |l|, and ρ and θ are the radial and angle coordinates, respectively.

The mutual coherence function (MCF) of the electric field for points in the transverse plane, ρ₁ and ρ₂ can be written as:

Γ (ρ_{1}, ρ_{2}) = 〈 E (ρ_{1}) E^{*} (ρ_{2}) 〉,

where

〈 〉

denotes an ensemble average. Here we assume a partially coherent light source in the initial plane with field correlation properties described by a Gaussian-Schell correlator:

C (ρ_{1}, ρ_{2}) = C (| ρ_{1} - ρ_{2} |) = \exp [- \frac{{| ρ_{1} - ρ_{2} |}^{2}}{δ^{2}}],

where δ is the transverse coherence length in the initial plane, and ρ₁, ρ₂ are arbitrary points in the beam.

From Eqs. (1)-(3), the MCF of a partially coherent ELG beam at the source plane z = 0 in a cylindrical coordinate system is expressed as:

\begin{array}{l} Γ (ρ_{1}, ρ_{2}; 0) = {(\frac{ρ_{1} \cdot ρ_{2}}{w_{0}^{2}})}^{| l |} L_{p}^{| l |} (\frac{ρ_{1}^{2}}{w_{0}^{2}}) L_{p}^{| l |} (\frac{ρ_{2}^{2}}{w_{0}^{2}}) \exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w_{0}^{2}}) \\ \times \exp [- \frac{{| ρ_{1} - ρ_{2} |}^{2}}{δ^{2}}] \exp [i l (θ_{2} - θ_{1})] . \end{array}

In the far field, the MCF in the detection plane after the beam has propagated a distance z is given by

Γ^{'} (ρ_{1}^{'}, ρ_{2}^{'};z) = {(\frac{1}{λ z})}^{2} \iint \iint Γ (ρ_{1}, ρ_{2}, 0) \exp [- \frac{i 2 π}{λ z} (ρ_{1}^{'} \cdot ρ_{1} - ρ_{2}^{'} \cdot ρ_{2})] d ρ_{1} d ρ_{2},

where λ is the wavelength.

Substituting Eq. (4) into Eq. (5), converting the cylindrical coordinates into the Cartesian coordinates and making the far field approximation, we obtain:

\begin{array}{l} Γ^{'} (x_{1}, y_{1}, x_{2}, y_{2}) = {(\frac{1}{z w_{0}^{| l |} λ})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} L_{p}^{| l |} [\frac{(x_{10} + i y_{10}) (x_{10} - i y_{10})}{w_{0}^{2}}] L_{p}^{| l |} [\frac{(x_{20} + i y_{20}) (x_{20} - i y_{20})}{w_{0}^{2}}] \\ \times (^{x_{10} + i y_{10}) | l |} (^{x_{20} - i y_{20}) | l |} F x_{0} F y_{0} d x_{10} d x_{20} d y_{10} d y_{20}, \end{array}

where

{Fx}_{0} = \exp [g (x_{10}^{2} + x_{20}^{2}) + \frac{2 x_{10} x_{20}}{δ^{2}} + \frac{i k (x_{1} \cdot x_{10} - x_{2} \cdot x_{20})}{z}],

{Fy}_{0} = \exp [g (y_{10}^{2} + y_{20}^{2}) + \frac{2 y_{10} y_{20}}{δ^{2}} + \frac{i k (y_{1} \cdot y_{10} - y_{2} \cdot y_{20})}{z}],

g = - \frac{1}{δ^{2}} - \frac{1}{w_{0}^{2}} .

The complex degree of coherence (CDOC) of a partially coherent beam between two points (x₁, y₁) and (0, 0) is defined as [27]:

μ (x_{1}, y_{1}, 0, 0, z) = \frac{Γ (x_{1}, y_{1}, 0, 0, z)}{\sqrt{I (x_{1}, y_{1}, z)} \sqrt{I (0, 0, z)}},

where

I (x_{1}, y_{1}, z) = Γ (x_{1}, y_{1}, x_{1}, y_{1}, z) .

Substituting Eq. (6) into Eq. (7), we can obtain the expression for the CDOC of a partially coherent ELG beam.

3. Numerical simulations

To illustrate the effect of coherence length on determining the topological charge based on the CDOC, numerical calculations were performed by using Eqs. (6) and (7). Figure 1 gives some simulation results of CDOC of a partially coherent ELG beam with a radial mode index p = 1 at the plane z = 20z₀ for various coherence lengths (Rayleigh length z₀ = kw₀²/2, λ = 532nm, w₀ = 100λ). It is seen from Fig. 1 that the number of ring dislocations in the contour graph is equal to |l| when δ is small, and the ring dislocations between outermost and inner rings get weaker when δ increases. Thus the topological charge of a partially coherent ELG beam can be measured by its CDOC in the far field.

Fig. 1 Distribution (contour graph) of CDOC of a partially coherent ELG beam for different values of l and δ.

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Figure 2 shows the effect of both radial mode indices p and topological charge l = 1, 2, 3 on the CDOC of partially coherent ELG beams. From Fig. 2 we can see that the radial mode index doesn’t affect the number of the ring dislocations in the far-field CDOC, in other words, the number of ring dislocations is dependent on the topological charge l only, which is much different from the case of partially coherent standard Laguerre-Gaussian beams [26].

Fig. 2 Distribution (contour graph) of CDOC of partially coherent ELG beams with δ = w₀ for different values of l and p.

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4. Experimental results

Now we carry out some experiments to verify the aforementioned theoretical analysis. Figure 3 shows the experimental device for generating partially coherent ELG beam and measuring its focused mutual correlation function (MCF). Based on the conclusion in [44], an anomalous vortex beam will evolve into an ELG beam in the far field (or in the focal plane) in free space. A coherent light with wavelength λ = 532nm emitted from solid-state laser passes through an expanding system and then is focused on the rotating ground glass disk (RGGD) by L1 (f = 80mm), producing a partially coherent beam with Gaussian statistics. The influence of the rotating speed of RGGD on degree of coherence can be negligible in this experiment since it only influences the degree of coherence in the time domain. After passing L2 (f = 150mm) and Gaussian amplitude filter (GAF), the transmitted beam becomes a Gaussian Schell model beam, whose intensity and degree of coherence satisfy Gaussian distribution. The generated partially coherent source in turn goes through the spatial light modulator (SLM1, loading hologram of anomalous hollow beam) and spiral phase plate (SPP) to evolve into an anomalous vortex beam. After a long distance transmission (z = 2.6m) and passing through the focal lens (L3), we get a focused ELG beam. In order to measure the MCF $Γ (ρ, 0)$ of the focused partially coherent ELG beam, we put the SLM2 on the measured plane. Based on the self-referencing-holography method in [45], we load a stop function $T (ρ_{1})$ and then we in turn load three different phase perturbations $γ δ (ρ - ρ_{0})$ (0, 2π/3, −2π/3) using SLM2. Here, we consider only the conditions of zero reference point, so the phase perturbation is located at the center of the focal ELG beam, i.e. $γ δ (ρ - 0)$ . The perturbing ELG beam arrives at the Fourier plane and the intensity information is recorded by the charge-coupled device (CCD) which can be expressed as:

\begin{matrix} I (k) = \iint \iint [T (ρ_{1}) + γ δ (ρ_{1} - 0)] {[T (ρ_{2}) + γ δ (ρ_{2} - 0)]}^{*} \\ \times Γ (ρ_{1}, ρ_{2}) \exp [- i 2 π k (ρ_{1} - ρ_{2})] d^{2} ρ_{1} d^{2} ρ_{2} \\ = I_{0} (k) + γ γ^{*} Γ (ρ_{0}, ρ_{0}) \\ + γ \int {[T (- (ρ - ρ_{0})) Γ (- (ρ - ρ_{0}), ρ_{0})]}^{*} \exp [- i 2 π k \cdot ρ] d ρ \\ + γ^{*} \int T (ρ + ρ_{0}) Γ (ρ + ρ_{0}, ρ_{0}) \exp [- i 2 π k \cdot ρ] d ρ . \end{matrix}

Corresponding to three different phase perturbations γ, we obtain three intensity patterns. Then we take an inverse Fourier transformation:

Fig. 3 Experimental set up for measuring the mutual correlation function of the focused partially coherent elegant Laguerre-Gaussian beam. BE, beam expander; L1, L2, L3 and L4, lenses; RGGD, rotating ground glass disk; GAF, Gaussian amplitude filter; SLM1 and SLM2, spatial light modulators; SPP, spiral phase plate; BS, beam splitter; CCD, charge coupled device.

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\begin{matrix} F T^{- 1} [I (k)] (k) = F T^{- 1} [I_{0} (k)] (k) + γ γ^{*} Γ (ρ_{0}, ρ_{0}) δ (ρ) \\ + γ {[Γ (- (ρ - ρ_{0}), ρ_{0}) T (- (ρ - ρ_{0}))]}^{*} . \\ + γ^{*} [Γ (ρ + ρ_{0}, ρ_{0}) T (ρ + ρ_{0})] \end{matrix}

Neglecting the $γ γ^{*} Γ (ρ_{0}, ρ_{0}) δ (ρ)$ , three functions can help resolve the MCF of the focused ELG. The complex degree of coherence (CDOC) of partially coherent ELG is defined as $μ (ρ, 0) = Γ (ρ, 0) / \sqrt{I (ρ) I (0)}$ . Therefore, we need to measure the intensity on the focal plane and the zero point.

Figure 4 shows the experimental and theoretical results of CDOC with w₀ = 0.62mm, δ = 0.35mm, where p = 1 and l = 1, 2, 3. Here, we can see that the number of ring dislocations in the focal-field CDOC is equal to the topological charge. The results are consistent with the theoretical prediction, and it means that one can measure the topological charge of the partially coherent ELG by observing its CDOC in the far field.

Fig. 4 Experimental results of complex degree of coherence for the partially coherent elegant Laguerre-Gaussian beam with w₀ = 0.62mm, δ = 0.35mm, p = 1 and: (a) l = 1; (b) l = 2; (c) l = 3, and the corresponding theoretical results (d),(e),(f).

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5. Conclusions

In conclusion, we have studied the relationship between CDOC of the partially coherent ELG beam and the topological charge theoretically and experimentally. We also have investigated the effect of coherence length on CDOC of partially coherent ELG beams. Our results show that the radial mode order doesn’t affect the number of the ring dislocations in the CDOC. Furthermore, the number of ring dislocations in the CDOC of a partially coherent ELG beam is equal to the value of topological charge |l| for partial coherence cases. Based on this relationship, we may measure the topological charge of a partially coherent ELG beam by observing its CDOC. Our results will be useful for optical communication, optical trapping and singular optics.

Funding

National Natural Science Foundation of China (11874102, 11474048, 11774250 and 91750201); National Natural Science Fund for Distinguished Young Scholars (11525418); Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

References

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

2. Y. Yang, G. Thirunavukkarasu, M. Babiker, and J. Yuan, “Orbital-Angular-Momentum Mode Selection by Rotationally Symmetric Superposition of Chiral States with Application to Electron Vortex Beams,” Phys. Rev. Lett. 119(9), 094802 (2017). [CrossRef] [PubMed]

3. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

4. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum,” Phys. Rev. Lett. 91(22), 227902 (2003). [CrossRef] [PubMed]

5. I. B. Djordjevic, “Deep-space and near-Earth optical communications by coded orbital angular momentum (OAM) modulation,” Opt. Express 19(15), 14277–14289 (2011). [CrossRef] [PubMed]

6. J. Long, R. Liu, F. Wang, Y. Wang, P. Zhang, H. Gao, and F. Li, “Evaluating Laguerre-Gaussian beams with an invariant parameter,” Opt. Lett. 38(16), 3047–3049 (2013). [CrossRef] [PubMed]

7. Z. Xu, T. Zhu, D. Cheng, J. Long, Z. Huang, R. Liu, P. Zhang, H. Gao, and F. Li, “Accurate and practical method for characterizing Laguerre-Gaussian modes,” Appl. Opt. 53(8), 1644–1647 (2014). [CrossRef] [PubMed]

8. Y. Han and G. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. 36(11), 2017–2019 (2011). [CrossRef] [PubMed]

9. G. C. G. Berkhout and M. W. Beijersbergen, “Method for Probing the Orbital Angular Momentum of Optical Vortices in Electromagnetic Waves from Astronomical Objects,” Phys. Rev. Lett. 101(10), 100801 (2008). [CrossRef] [PubMed]

10. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Light’s Orbital Angular Momentum,” Phys. Rev. Lett. 105(5), 053904 (2010). [CrossRef] [PubMed]

11. G. Anzolin, F. Tamburini, A. Bianchini, G. Umbriaco, and C. Barbieri, “Optical vortices with starlight,” Astron. Astrophys. 488(3), 1159–1165 (2008). [CrossRef]

12. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003). [CrossRef] [PubMed]

13. G. V. Bogatyryova, C. V. Fel’de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28(11), 878–880 (2003). [CrossRef] [PubMed]

14. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45(3), 539–554 (1998). [CrossRef]

15. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1), 117–125 (2003). [CrossRef]

16. D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21(11), 2097–2102 (2004). [CrossRef] [PubMed]

17. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander Jr., “Spatial Correlation Singularity of a Vortex Field,” Phys. Rev. Lett. 92(14), 143905 (2004). [CrossRef] [PubMed]

18. L. Rao and J. Pu, “Spatial correlation properties of focused partially coherent vortex beams,” J. Opt. Soc. Am. A 24(8), 2242–2247 (2007). [CrossRef] [PubMed]

19. I. D. Maleev and G. A. Swartzlander Jr., “Propagation of spatial correlation vortices,” J. Opt. Soc. Am. B 25(6), 915–922 (2008). [CrossRef]

20. T. van Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009). [CrossRef]

21. G. Gbur, T. D. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A, Pure Appl. Opt. 6(5), S239–S242 (2004). [CrossRef]

22. Y. L. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282(5), 709–716 (2009). [CrossRef]

23. T. van Dijk and T. D. Visser, “Evolution of singularities in a partially coherent vortex beam,” J. Opt. Soc. Am. A 26(4), 741–744 (2009). [CrossRef] [PubMed]

24. Y. Yang, M. Mazilu, and K. Dholakia, “Measuring the orbital angular momentum of partially coherent optical vortices through singularities in their cross-spectral density functions,” Opt. Lett. 37(23), 4949–4951 (2012). [CrossRef] [PubMed]

25. A. Y. Escalante, B. Perez-Garcia, R. I. Hernandez-Aranda, and G. A. Swartzlander Jr., “Determination of angular momentum content in partially coherent beams through cross correlation measurements,” Proc. SPIE 8843, 884302 (2013). [CrossRef]

26. Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, and K. Dholakia, “Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity,” New J. Phys. 15(11), 113053 (2013). [CrossRef]

27. C. Zhao, F. Wang, Y. Dong, Y. Han, and Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012). [CrossRef]

28. Y. Yang and Y. D. Liu, “Measuring azimuthal and radial mode indices of a partially coherent vortex field,” J. Opt. 18(1), 015604 (2016). [CrossRef]

29. H. D. L. Pires, J. Woudenberg, and M. P. van Exter, “Measurement of the orbital angular momentum spectrum of partially coherent beams,” Opt. Lett. 35(6), 889–891 (2010). [CrossRef] [PubMed]

30. A. E. Siegman, “Hermite-gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]

31. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18(1), 177–184 (2001). [CrossRef] [PubMed]

32. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

33. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]

34. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]

35. W. Nasalski, “Exact elegant Laguerre-Gaussian vector wave packets,” Opt. Lett. 38(6), 809–811 (2013). [CrossRef] [PubMed]

36. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009). [CrossRef] [PubMed]

37. T. Yang, Y. Xu, H. Tiana, Q. Du, D. Die, and Y. Dan, “Comparative study of propagation properties of partially coherent standard and elegant Hermite-Gaussian beams in inhomogeneous atmospheric turbulence,” Optik (Stuttg.) 127(22), 10772–10779 (2016). [CrossRef]

38. F. Wang and Y. Cai, “Average intensity and spreading of partially coherent standard and elegant Laguerre- Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res 103, 33–56 (2010).

39. Y. Xu, Y. Li, and X. Zhao, “Intensity and effective beam width of partially coherent Laguerre-Gaussian beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 32(9), 1623–1630 (2015). [CrossRef] [PubMed]

40. T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017). [CrossRef] [PubMed]

41. H. Xu, H. Luo, Z. Cui, and J. Qu, “Polarization characteristics of partially coherent elegant Laguerre-Gaussian beams in non-Kolmogorov turbulence,” Opt. Lasers Eng. 50(5), 760–766 (2012). [CrossRef]

42. Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013). [CrossRef]

43. C. Zhao and Y. Cai, “Trapping two types of particles using a focused partially coherent elegant Laguerre-Gaussian beam,” Opt. Lett. 36(12), 2251–2253 (2011). [CrossRef] [PubMed]

44. Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013). [CrossRef] [PubMed]

45. Y. Shao, X. Lu, S. Konijnenberg, C. Zhao, Y. Cai, and H. P. Urbach, “Spatial coherence measurement and partially coherent diffractive imaging using self-referencing holography,” Opt. Express 26(4), 4479–4490 (2018). [CrossRef] [PubMed]

Measuring topological charge of partially coherent elegant Laguerre-Gaussian beam

Abstract

1. Introduction

2. Complex degree of coherence of partially coherent elegant Laguerre- Gaussian beam in far field

3. Numerical simulations

4. Experimental results

5. Conclusions

Funding

References

Cited By

Figures (4)

Equations (13)

Optics Express