## Autocorrelation properties of fully coherent beam with and without orbital angular momentum |

Optics Express, Vol. 22, Issue 3, pp. 2925-2932 (2014)

http://dx.doi.org/10.1364/OE.22.002925

Acrobat PDF (1081 KB)

### Abstract

The spatial correlation singularity of a partially coherent vortex beam was demonstrated recently [Phys. Rev. Lett. **92**, 143905 (2004)], and it was shown that the cross-correlation singularity disappears when the spatial coherence is high. In this paper, we demonstrate that the spatial autocorrelation function of a fully coherent beam in the far-field is equivalent to the Fourier transform of its intensity in the source plane. Our theoretical and experimental results show that, depending on both the radial and azimuthal mode indices (*p*, *λ*) of the incident light beam, the distribution of the far-field autocorrelation function displays a series of concentric, alternate bright and dark rings. This phenomenon may be used to determine the topological charge (the azimuthal index) of light beam with a nonzero radial index.

© 2014 Optical Society of America

## 1. Introduction

1. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science **331**(6014), 192–195 (2011). [CrossRef] [PubMed]

*iλφ*) may carry an OAM of

*λħ*per photon, where

*λ*is an integer number and denotes the topological charge (azimuthal index) of the field, and

*φ*is azimuthal coordinate [2

2. 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]

3. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**(22), 5448–5456 (2004). [CrossRef] [PubMed]

5. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

6. 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]

8. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. **3**(5), 305–310 (2007). [CrossRef]

9. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**(5), 826–829 (1995). [CrossRef] [PubMed]

10. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

11. 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]

14. 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). [PubMed]

13. 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. 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]

12. I. Maleev, D. Palacios, A. Marathay, and G. Swartzlander Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B **21**(11), 1895–1900 (2004). [CrossRef]

13. 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]

13. 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]

*et al*. showed that the ring dislocation of cross-correlation function will disappear as the spatial coherence increases. To our knowledge, the investigation on the autocorrelation properties of the fully coherent beam has not been reported yet.

*p*,

*λ*) of a vortex remains an intriguing problem in both the quantum and classical domain [16

16. M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. **100**(23), 231115 (2012). [CrossRef]

*λ*of individual Laguerre-Gaussian (LG) beams from the diffraction pattern [17

17. 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]

21. C.-S. Guo, L. Lu, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. **94**(23), 231104 (2009). [CrossRef]

*p*of the incident light field is equal to zero. More recently, Prabhakar

*et al*. [22

22. 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]

*p*is also equal to zero as well in [22

22. 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]

16. M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. **100**(23), 231115 (2012). [CrossRef]

*p*,

*λ*) affect the autocorrelation properties. Our experimental results show a good agreement with the theoretical results. We also show a way to measure the topological charge of light beams when the radial mode

*p*is taken into account.

## 2. Fourier transform of the intensity in the source plane and the far-field spatial autocorrelation function

*u*(

*x*,

*y*). Then we can get its field

*U*(

*X*,

*Y*) in the far zone through Fourier transform, i.e.,It is known that autocorrelation is the cross-correlation of a variable with itself and the spatial field autocorrelation means that a dependency exists between values of a field in the neighboring or proximal locations in values of a field across the observation plane. In optics, the autocorrelation function is given by [23]where * denotes the complex conjugate and ⊗ represents the convolution notation.

*λ*in the source plane

*z*= 0 is given aswhere

*w*is the waist width,

*L*(•) is the associated Laguerre polynomial,

_{p}^{l}*p*is the radial mode index,

*λ*is the azimuthal mode index,

*ρ*and

*φ*are radial and azimuthal coordinates, respectively, and

*θ*is an arbitrary phase.

*ξ*is spatial frequency. Substituting Eq. (4) into Eq. (5), one can obtain the following expression for the far-field autocorrelation function (or the FT of the intensity in the source plane),

*λ*and the radial index

*p*.

## 3. Numerical results and analyses

*N*) of dark rings of autocorrelation function and the radial and azimuthal mode indices (

*p*,

*λ*), namely,

*p*and

*p*= 0) [22

22. 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]

*λ*= 0), as two special cases of the general results. When

*p*= 0, Eq. (6) can be rewritten asFrom Eq. (9) we can see that when the radial mode index is equal to zero, the number of the dark rings of the far-field autocorrelation function, or the number of dark rings of the DRITFOI, is equal to the topological charge

*λ*definitely. It is noted that Eq. (9) is consistent with the results of Eq. (10) in [22

**36**(22), 4398–4400 (2011). [CrossRef] [PubMed]

*λ*= 0 and is expressed asEquation (10) shows that for the beam without OAM, namely,

*λ*= 0, the number of the dark rings in the far-field autocorrelation function is identical to the value of the radial index

*p*definitely. The distribution of the far-field autocorrelation function for non-vortex beams with

*p*= 2 and 5 are shown in Fig. 2, from which we see that there are 2 and 5 dark rings, respectively. Therefore, the dark rings of the far-field autocorrelation function are not the sole property of optical vortex, and for a priori unknown beam we cannot determine the topological charge by observing the pattern of FT of its intensity only.

*λ*and the radial index

*p*From a physical point of view, the dark rings in field autocorrelation function mean the disappearance of the correlation. It is also shown that the influence of radial index

*p*of an optical vortex on the number of dark rings of the far-field autocorrelation function is more significant than that of the non-vortex beam.

**36**(22), 4398–4400 (2011). [CrossRef] [PubMed]

*λ*of a vortex beam when the radial mode

*p*is not equal to zero. It is well known that the radial mode index

*p*of LG beams denotes the number of dark rings in the intensity profile. Therefore, one can directly determine the radial index

*p*of a LG beam from its intensity profile. Moreover, if there is a dark center in the intensity profile, which shows that the azimuthal index (topological charge) is not equal to zero. Then we can carry out the FT of the intensity by using MATLAB and measure the number of the dark rings, namely,

*N*, in the FT pattern. Therefore, the topological charge can be determined by recalling the relationship

## 4. Experiments

*p*and

*λ*. After passing through the thin lens (L), the generated coherent LG beam arrives at a beam profile analyzer (BPA), which is used to measure the focused intensity profile. The final intensity image of the LG beam recorded by the BPA can be stored in a computer (PC

_{2}) and its FT can be processed in MATLAB.

*p*= 1 and

*λ*= 2, and its FT pattern that is processed in MATLAB is shown in Fig. 4(b). The corresponding theoretical results of the intensity and its FT patterns are shown in Fig. 4(c) and 4(d), respectively. From Fig. 4, one sees that our experimental results agree well with the theoretical results. Therefore, for a given LG beam, we can determine its radial index from intensity profile, and determine its azimuthal mode index (topological charge) from the Fourier transform of the intensity profile, step by step. In other words, the FT method proposed in [22

**36**(22), 4398–4400 (2011). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science |

2. | 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 |

3. | G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

4. | I. B. Djordjevic, “Deep-space and near-Earth optical communications by coded orbital angular momentum (OAM) modulation,” Opt. Express |

5. | J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics |

6. | 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. |

7. | G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. |

8. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

9. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

10. | K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics |

11. | H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. |

12. | I. Maleev, D. Palacios, A. Marathay, and G. Swartzlander Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B |

13. | D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. |

14. | 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. |

15. | 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. |

16. | M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. |

17. | 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. |

18. | Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. |

19. | L. E. E. de Araujo and M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Opt. Lett. |

20. | A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express |

21. | C.-S. Guo, L. Lu, S.-J. Yue, and G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. |

22. | S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, “Revealing the order of a vortex through its intensity record,” Opt. Lett. |

23. | J. W. Goodman, |

24. | I. S. Gradshteyn and I. M. Ryzhik, |

**OCIS Codes**

(050.4865) Diffraction and gratings : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 28, 2013

Revised Manuscript: January 24, 2014

Manuscript Accepted: January 27, 2014

Published: January 31, 2014

**Citation**

Yuanjie Yang, Yuan Dong, Chengliang Zhao, Yi-dong Liu, and Yangjian Cai, "Autocorrelation properties of fully coherent beam with and without orbital angular momentum," Opt. Express **22**, 2925-2932 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2925

Sort: Year | Journal | Reset

### References

- B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011). [CrossRef] [PubMed]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, 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]
- G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
- 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]
- J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
- A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91(22), 227902 (2003). [CrossRef] [PubMed]
- G. Molina-Terriza, A. Vaziri, J. Rehácek, Z. Hradil, A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92(16), 167903 (2004). [CrossRef] [PubMed]
- G. Molina-Terriza, J. P. Torres, L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]
- H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]
- K. Dholakia, T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011). [CrossRef]
- H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003). [CrossRef] [PubMed]
- I. Maleev, D. Palacios, A. Marathay, G. Swartzlander., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21(11), 1895–1900 (2004). [CrossRef]
- 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]
- Y. 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]
- 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]
- M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100(23), 231115 (2012). [CrossRef]
- J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, 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]
- Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011). [CrossRef] [PubMed]
- L. E. E. de Araujo, M. E. Anderson, “Measuring vortex charge with a triangular aperture,” Opt. Lett. 36(6), 787–789 (2011). [CrossRef] [PubMed]
- A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19(7), 5760–5771 (2011). [CrossRef] [PubMed]
- C.-S. Guo, L. Lu, S.-J. Yue, G.-X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94(23), 231104 (2009). [CrossRef]
- 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]
- J. W. Goodman, Introduction to Fourier optics (McGraw-Hill, 1996).
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, 2007).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.