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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 2 — Jan. 27, 2014
  • pp: 1350–1358
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The cross correlation function of partially coherent vortex beam

Pan Feng Ding and Jixiong Pu  »View Author Affiliations


Optics Express, Vol. 22, Issue 2, pp. 1350-1358 (2014)
http://dx.doi.org/10.1364/OE.22.001350


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Abstract

This work presents theoretical analysis on the cross correlation function (CCF) of partially coherent vortex beam (PCVB), where the relation of the number of the rings of CCF dislocations and orbital angular momentum (OAM) of PCVB is analyzed in detail. It is shown that rings of CCF dislocations do not always exist, and depend on the coherence length, the order of PCVB and location of observation plane, although the CCF indicates topological charge to some degree. Comprehensive analysis of the CCF of PCVB and numerical simulations all validate such phenomenon.

© 2014 Optical Society of America

1. Introduction

The concept of vortex beam was first introduced by Nye and Berry in 1974 [1

1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]

]. In recent years, a lot of literature focuses the property and application of this special beam carrying orbital angular momentum [2

2. E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]

7

7. E. Mari, F. Tamburini, G. A. Swartzlander Jr, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012). [CrossRef] [PubMed]

]. Among the investigations, measurement of the topological charge (related with the OAM) of vortex beam becomes an important embranchment and most of them are focused on the coherent case [8

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

13

13. M. E. Anderson, H. Bigman, L. E. E. De Araujo, and J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B 29(8), 1968–1976 (2012).

]. An illuminative research on the CCF of PCVB was presented in [14

14. 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]

]. More recently, the relation between the topological charge of a partially coherent beam and its CCF has been studied theoretically and experimentally [15

15. 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]

, 16

16. Y. J. 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). [PubMed]

]. Based on the analysis of CCF of PCVB, it is revealed by numerical simulation that the number of ring dislocations of the far-field CCF of PCVB is just equal to the value of its topological charge [16

16. Y. J. 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). [PubMed]

]. Moreover, the detail of the spatial correlation singularity affected by both the radial and azimuthal mode indices is investigated in a partially coherent light field [17

17. 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]

]. However, it is noteworthy that the studies were focused on the case of far field. It is interesting to find that for very low coherence, the near-field CCF has clear ring dislocations, namely, the topological charge may be detected by the CCF in the near field rather than in the far field.

2. The CCF of PCVB during propagation

Generally, we assume a partially coherent beam carrying a Gaussian amplitude and mth-order vortex, of which the mutual correlation function (MCF) in the original plane can be described as
Γ(r1,r2,0)=E02(r1r2w2)mexp[|r1r2|2Lc2r12+r22w2im(ϕ2ϕ1)],
(1)
where E0 and w are the characteristic amplitude and beam size, LC is the transverse coherence length, r1,r2 and ϕ1,ϕ2 are the transverse position vector and azimuthal coordinates in the source plane (z = 0), respectively. Based on paraxial approximation, the MCF of PCVB after propagating a distance z can be written as [18

18. Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008). [CrossRef]

20

20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

]
Γ(ρ1,ρ2,z)=(k/2πz)2Γ(r1,r2,0)exp(ik2z[(ρ1r1)2(ρ2r2)2])dr1dr2,
(2)
where k=2π/λ is the wave number and ρ1,ρ2 is the transverse position vector coordinates in the observation plane. By substituting Eq. (1) into Eq. (2) and by settingρ1=ρ2=ρ, the CCF of PCVB in the observation plane can be obtained after tedious integration
χ(ρ)=Γ(ρ,ρ,z)=E02πexp(Δζ)Δ1n=0mn!(Cmn)2Δ2n(Δ3ζ)mn,
(3)
where

{Δ=2k2w4(2w2+Lc2)k2w4Lc2+4z2(2w2+Lc2),Δ1=(4zkw2)2m[k2w4Lc2k2w4Lc2+4z2(2w2+Lc2)]m+1,Δ2=4z2k2w2Lc2,ζ=ρ2w2,Cmn=m!n!(mn)!,Δ3=k2w4Lc4+4z2(2w2+Lc2)2k2w4Lc4+4z2Lc2(2w2+Lc2).
(4)

3 Correlation between CCF and topological charge of PCVB

3.1 The case while m = 1

For PCVB while m = 1, the CCF takes a simple form as
χ(ζ)exp(Δζ)(Δ2Δ3ζ).
(5)
It is obvious in Eq. (5) that there exist two extrema with opposite sign in the distribution of CCF, which implies that a single zero point is located between the positions of the two extrema, regardless of the observation distance and coherence length, such that the OAM of the 1st order PCVB can be detected by CCF for all conditions. The radius of the ring dislocation is decided by
ρ0=Δ2/Δ3w.
(6)
Although the above theoretical analysis ensures the presence of the zero point (ring of dislocation), the ring is not easy to be detected because of influence of the coherence length and observation distance. Figure 1
Fig. 1 Distributions of normalized CCF of the 1st order PCVB in near and far field for different coherence length. (a) z = 0.01ZR; (b) z = 0.5 ZR; (c) z = ZR; (d) z = 20 ZR.
shows the simulation result of normalized CCF of the 1st order PCVB and the parameters are chosen as: λ = 800nm, w = 2.5mm (ZR is the Rayleigh distance).

In the near field [Fig. 1(a)], the CCF with low coherence (Lc = 0.2w) shows high resolution for detection of the zero point (ring of dislocation) because the magnitude of two extrema are both considerable. While the coherence length increases beyond w, the zero point can only be revealed from the magnified drawing in the top right corner of Fig. 1(a). This phenomenon indicates that the normalized magnitude of the first extremum of the CCF becomes smaller with increasing of coherence length, resulting in obvious decreasing of resolution for detection of the zero point. In the field near Rayleigh distance [Figs. 1(b) and 1(c)], the CCF with low coherence (Lc = 0.2w) shows low resolution for detection of the zero point because the normalized magnitude of the second extremum approaches to zero. The CCF with moderate coherence length (Lc = w) possesses high resolution. While the coherence length increases to 3w, the first extremum drops and the resolution decreases, obviously. In the field far from Rayleigh distance [Fig. 1(d)], high coherence (Lc = 3w) dominates the resolution.

As a whole from Figs. 1(a)-1(d), it can be seen that the CCF of the 1st order PCVB shows high resolution to detect the zero point for the case of low coherence in near field, moderate coherence in the field near Rayleigh distance, and high coherence in the field far from Rayleigh distance, respectively.

3.2 The case of high order

For simplification, we first consider the case while m = 2. The CCF takes the form as
χ(ζ)exp(Δζ)(2Δ224Δ2Δ3ζ+Δ32ζ2).
(7)
The CCF has two zero points at any observation distance for any coherence length. The radii of the ring dislocations of are decided by
ρ0=2±2Δ2/Δ3w.
(8)
Figure 2
Fig. 2 Distributions of normalized CCF of the 2nd order PCVB in near and far field for different coherence length. (a) z = 0.01 ZR; (b) z = 0.5 ZR; (c) z = ZR; (d) z = 20 ZR.
shows the simulation result of normalized CCF of the 2nd order PCVB and the basic parameters are the same as in Fig. 1.

In the near field [Fig. 2(a)], the CCF with low coherence (Lc = 0.2w) still shows high resolution and it contains three extrema and two zero points. With increasing of coherence length, similar phenomenon arises for the case (Lc = w, 3w) as shown in Fig. 1(a). The magnitudes of the first two extrema of normalized CCF are very close to zero in the neighborhood of x-axis so that the zero points cannot be easily identified. In the field near and far from Rayleigh distance [Figs. 2(b)-2(d)], the CCF with moderate coherence length (Lc = w) possesses relatively high resolution. By comparing the case of Lc = 3w in Figs. 1(b)-1(d) and Figs. 2(b)-2(d), the similar characteristic can be found that, for larger coherence length, the CCFs of both the 1st and 2nd order PCVB need propagate longer distance to provide high resolution for the detection of zero points.

While considering higher order, the case becomes complex. For example m = 3, the CCF takes the form as
χ(ζ)exp(Δζ)(6Δ2318Δ22Δ3ζ+9Δ2Δ32ζ2Δ33ζ3).
(9)
There is no direct solution to derive the zero points from Eq. (9), which is the same as for the case while m>3. By numerous calculations we find that the characteristic of normalized CCF of higher order PCVB becomes more dependent on the coherence length. Simulation results are presented for the cases of low coherence, moderate coherence and high coherence in Figs. 3
Fig. 3 Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with low coherence length (Lc = 0.2w). (a) z = 0.01 ZR; (b) z = 0.02 ZR; (c) z = 0.04 ZR; (d) z = 0.5 ZR.
-5, respectively.

Figure 3 shows the case of low coherence. While the propagation length is 0.01ZR [Fig. 3(a)], the CCF of the 3rd order PCVB exhibits three zero points, however, the CCFs of the 4th and 5th order PCVB show no clear zero points. While the propagation length increases to 0.02ZR [Fig. 3(b)], it is interesting to find that the CCFs of the 3rd, 4th, 5th order PCVB show clearly 3,4,5 zero points, respectively. With the continuous increasing of propagation length [Figs. 3(c) and 3(d)], the CCFs (m = 3, 4, 5) only obviously display the anterior zero points and other zero points are blurry and almost missing. It can be observed that the CCF of the higher order PCVB with low coherence has high resolution of zero points in the near field and there exists an optimum observation plane [Fig. 3(b)].

For the case of moderate coherence (Lc = w), the CCFs of higher order PCVB show no clear zero points in the near field [Fig. 4(a)
Fig. 4 Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with moderate coherence length (Lc = w). (a) z = 0.05 ZR; (b) z = 0.2 ZR; (c) z = 0.5 ZR; (d) z = ZR.
]. While the propagation length increases to 0.2ZR, the CCF of the3rd order PCVB takes the lead to show all zero points [Fig. 4(b)]. The CCFs of the 4th and 5th order PCVB follow to show the zero points with the increasing of propagation length, and while the propagation length increases to 0.5ZR, the CCFs show all the zero points obviously [Fig. 4(c)]). With the continuous increasing of propagation length, the CCFs can still show all the zero points but the resolution of the posterior zero points tends to decrease [Fig. 4(d)]. It can be concluded from Figs. 4(a)-4(d) that the best distance should be between the near field and Rayleigh distance to display all the zero points of CCF of higher order PCVB with moderate coherence.
Fig. 5 Distributions of normalized CCF of higher order PCVB (m = 3, 4, 5) with high coherence length (Lc = 3w). (a) z = 0.05 ZR; (b) z = 0.5 ZR; (c) z = 5 ZR; (d) z = 20 ZR.

The most troublesome is the case of higher order PCVB with high coherence (Lc = 3w), as described in [16

16. Y. J. 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). [PubMed]

]. At the propagation length shorter than 0.5ZR, the CCFs show no clear zero points in Figs. 5(a) and 5(b). While the propagation length increases to 5ZR, the CCF of the 3rd PCVB takes the lead to show all zero points [Fig. 5(c)], which is a little similar to that in Fig. 4(b). However, with the continuous increasing of propagation length (z = 20ZR), the CCFs of the 4th and 5th order PCVB have no inclination to show the zero points at all. This characteristic does not change a little even at the propagation length beyond 100ZR (calculated, but not plotted in Fig. 5), which accords with that in [16

16. Y. J. 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). [PubMed]

].

4. Conclusion

To summarize, the following phenomena can be found for the 1st and 2nd order PCVB that: (I) for the low coherence case, all the zero point(s) of the CCF can be detected in the near field, the field near Rayleigh distance and the far field. The best choice is the near field; (II) for the moderate and high coherence cases, the best choice changes to the field near Rayleigh distance and the far field, respectively. However, for higher order PCVB with low coherence, all the zero points of CCF can only be detected in the near field; for higher order PCVB with moderate coherence, the best choice is the field below Rayleigh distance; with the further increasing of coherence, the method introduced for the case of coherent vortex beam can be used as a substitution to detect the topological charge.

Acknowledgments

This research was supported by the National Natural Science Foundations of China (Grant Nos. 61307001 and 61178015) and the National Natural Science Foundation of Fujian Province (Grant No.2013J05094).

References and links

1.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]

2.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]

3.

M. Šiler, P. Jákl, O. Brzobohatý, and P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20(22), 24304–24319 (2012). [CrossRef] [PubMed]

4.

K. Y. Bliokh and F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012). [CrossRef]

5.

H. W. Yan, E. T. Zhang, B. Y. Zhao, and K. L. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express 20(16), 17904–17915 (2012). [CrossRef] [PubMed]

6.

Y. F. Jiang, K. K. Huang, and X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef] [PubMed]

7.

E. Mari, F. Tamburini, G. A. Swartzlander Jr, A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012). [CrossRef] [PubMed]

8.

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

9.

Y. X. Liu, J. X. Pu, and B. D. Lü, “Method for exploring the orbital angular momentum of an optical vortex beam with a triangular multipoint plate,” Appl. Opt. 50(24), 4844–4847 (2011). [CrossRef] [PubMed]

10.

S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36(22), 4398–4400 (2011). [CrossRef] [PubMed]

11.

L. Shi, L. H. Tian, and X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. 10(12), 120501 (2012). [CrossRef]

12.

H. Tao and J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012). [CrossRef]

13.

M. E. Anderson, H. Bigman, L. E. E. De Araujo, and J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B 29(8), 1968–1976 (2012).

14.

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]

15.

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]

16.

Y. J. 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). [PubMed]

17.

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]

18.

Z. Y. Chen and J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008). [CrossRef]

19.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

20.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(050.4865) Diffraction and gratings : Optical vortices
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: October 14, 2013
Revised Manuscript: December 21, 2013
Manuscript Accepted: December 21, 2013
Published: January 14, 2014

Citation
Pan Feng Ding and Jixiong Pu, "The cross correlation function of partially coherent vortex beam," Opt. Express 22, 1350-1358 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1350


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References

  1. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974). [CrossRef]
  2. E. Brasselet, N. Murazawa, H. Misawa, S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103(10), 103903 (2009). [CrossRef] [PubMed]
  3. M. Šiler, P. Jákl, O. Brzobohatý, P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20(22), 24304–24319 (2012). [CrossRef] [PubMed]
  4. K. Y. Bliokh, F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012). [CrossRef]
  5. H. W. Yan, E. T. Zhang, B. Y. Zhao, K. L. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express 20(16), 17904–17915 (2012). [CrossRef] [PubMed]
  6. Y. F. Jiang, K. K. Huang, X. H. Lu, “Propagation dynamics of abruptly autofocusing Airy beams with optical vortices,” Opt. Express 20(17), 18579–18584 (2012). [CrossRef] [PubMed]
  7. E. Mari, F. Tamburini, G. A. Swartzlander, A. Bianchini, C. Barbieri, F. Romanato, B. Thidé, “Sub-Rayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012). [CrossRef] [PubMed]
  8. Y. J. Han, G. H. Zhao, “Measuring the topological charge of optical vortices with an axicon,” Opt. Lett. 36(11), 2017–2019 (2011). [CrossRef] [PubMed]
  9. Y. X. Liu, J. X. Pu, B. D. Lü, “Method for exploring the orbital angular momentum of an optical vortex beam with a triangular multipoint plate,” Appl. Opt. 50(24), 4844–4847 (2011). [CrossRef] [PubMed]
  10. S. Prabhakar, A. Kumar, J. Banerji, R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36(22), 4398–4400 (2011). [CrossRef] [PubMed]
  11. L. Shi, L. H. Tian, X. F. Chen, “Characterizing topological charge of optical vortex using non-uniformly distributed multi-pinhole plate,” Chin. Opt. Lett. 10(12), 120501 (2012). [CrossRef]
  12. H. Tao, J. X. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106(4), 927–932 (2012). [CrossRef]
  13. M. E. Anderson, H. Bigman, L. E. E. De Araujo, J. L. Chaloupka, “Measuring the topological charge of ultra-broadband, optical-vortex beams with a triangular aperture,” J. Opt. Soc. Am. B 29(8), 1968–1976 (2012).
  14. D. M. Palacios, I. D. Maleev, A. S. Marathay, G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004). [CrossRef] [PubMed]
  15. C. Zhao, F. Wang, Y. Dong, Y. Han, Y. Cai, “Effect of spatial coherence on determining the topological charge of a vortex beam,” Appl. Phys. Lett. 101(26), 261104 (2012). [CrossRef]
  16. Y. J. Yang, M. Mazilu, 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). [PubMed]
  17. Y. Yang, M. Chen, M. Mazilu, A. Mourka, Y. Liu, 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]
  18. Z. Y. Chen, J. X. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372(15), 2734–2740 (2008). [CrossRef]
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

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