## Refractive elements for the measurement of the orbital angular momentum of a single photon |

Optics Express, Vol. 20, Issue 3, pp. 2110-2115 (2012)

http://dx.doi.org/10.1364/OE.20.002110

Acrobat PDF (1516 KB)

### Abstract

We have developed a mode transformer comprising two custom refractive optical elements which convert orbital angular momentum states into transverse momentum states. This transformation allows for an efficient measurement of the orbital angular momentum content of an input light beam. We characterise the channel capacity of the system for 50 input modes, giving a maximum value of 3.46 bits per photon. Using an electron multiplying CCD (EMCCD) camera with a laser source attenuated such that on average there is less than one photon present within the system per measurement period, we demonstrate that the elements are efficient for the use in single photon experiments.

© 2012 OSA

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

*et al.*in 1992 showed that beams with an transverse amplitude profile of

*A*(

*r*)exp(

*iℓϕ*) carry an orbital angular momentum of

*ℓh̄*per photon [1

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

2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**, 161–204 (2011). [CrossRef]

*r*and

*ϕ*as the radial and angular coordinates respectively. The integer

*ℓ*is unbounded, giving a large state space in which to encode information [3

3. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Phys. Rev. Lett. **88**, 013601 (2001). [CrossRef]

7. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science **329**, 662–655 (2010). [CrossRef] [PubMed]

*ℓ*-fold fork dislocation has become commonplace for the generation of beams carrying OAM [8

8. V. Bazhenov, M. Soskin, and M. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. **39**, 985–990 (1992). [CrossRef]

9. N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. **17**, 221–223 (1992). [CrossRef] [PubMed]

10. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316, (2001). [CrossRef] [PubMed]

*ℓ*values to be measured, but checking for

*N*states require at least

*N*photons [10

10. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316, (2001). [CrossRef] [PubMed]

11. S. S. Oemrawsingh, J. de Jong, X. Ma, and A. Aiello. “High- dimensional mode analyzers for spatial quantum entanglement, Phys. Rev. A **73**, 032339, (2006) [CrossRef]

12. L. Marrucci, E. Karimi, S. Slussarenko, B Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. **13**, 064001, (2011). [CrossRef]

*et al*. It required a

*N*– 1 Mach-Zehnder interferometers with a Dove prism in each arm [13

13. J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**(25), 257901 (2002). [CrossRef] [PubMed]

*N*states. In principle, this routing can be achieved with 100% efficiency and with no loss of the input beam’s mode structure. However, simultaneously maintaining the alignment of

*N*interferometers has proved technically challenging.

14. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. **105**(15), 153601 (2010). [CrossRef]

*x,y*) in the input plane to a position (

*u,v*) in the output plane, where

*v*=

*a*arctan(

*y/x*) [15

15. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. **64**(8), 1092–1099 (1974). [CrossRef]

17. Y. Saito, S. Komatsu, and H. Ohzu, “Scale and rotation invariant real-time optical correlator using computer generated hologram,” Opt. Commun. **47**(1), 8–11 (1983). [CrossRef]

*iℓϕ*), to give a complex amplitude at the output plane of the form exp(

*iℓv/a*). A lens can then separate the resulting transverse momentum states into specified lateral positions, allowing for the efficient measurement of multiple states simultaneously [18

18. M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. **13**, 064006 (2011). [CrossRef]

19. G. C. G. Berkhout, M. P. J. Lavery, M. W. Beijersbergen, and M. J. Padgett, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. **36**, 1863–1865 (2011). [CrossRef] [PubMed]

14. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. **105**(15), 153601 (2010). [CrossRef]

14. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. **105**(15), 153601 (2010). [CrossRef]

*λ*, passes through a material of height

*Z*, and with a refractive index

*n*, the effective optical path length changes with respect to the same distance of propagation in a vacuum. The change in path length can be expressed as a change in phase of ΔΦ = 2

*π*(

*n*– 1)

*Z/λ*, hence the first element requires a height profile of where

*f*is the focal length of the integrated lens. There are two free parameters,

*a*and

*b*, which determine the scaling and position of the transformed beam. The parameter

*a*takes the value

*a*=

*d*/2

*π*, ensuring that the azimuthal angle range (0 ↦ 2

*π*) is mapped onto the full width of the second element,

*d*. The parameter

*b*is optimised for the particular physical dimensions of the sorter. The second of these elements has a height profile where

*u*and

*v*are the coordinates in the output plane. This element is placed a distance

*f*behind the first element. Each surface is wavelength independent, but dispersion effects in the material manifest themselves as a change in the focal length of the integrated lens for different wavelengths. Hence, the system can be tuned to a specific wavelength by changing the distance between the elements.

20. T.A. Dow, M.H. Miller, and P.J. Falter, “Application of a fast tool servo fordiamond turning of non-rotationally symmetric surfaces,” J. Precision Eng. **13**, 243–250 (1991). [CrossRef]

*ℓ*-forked hologram, realised using a SLM, by programming the SLM with both phase and intensity information. The beam generated in the first order of the hologram was selected with an aperture and the plane of the SLM is imaged onto the plane of the first element. The beam is then passed through the elements transforming it into the form exp(

*iℓv/a*), giving a transverse direction state which is then focussed into an elongated spot on a camera. The transverse position of the spot is dependent on

*ℓ*.

*N*adjacent regions, where each region is centred on one elongated spot, and the measured intensity of the pixels in the region was summed for each region. For a single input mode, one would expect the majority of the energy to be detected in the bin corresponding to the input mode and any energy readings in other regions represent cross talk between channels. Our transformation from orbital angular momentum states into transverse momentum states gives rise to inherent cross talk due to the diffraction limit. The inherent degree of cross talk can be deduced from Fourier theory, which predicts approximately 80% of the input light will be present in the bin corresponding to that input OAM mode value. A common method of evaluating the degree of cross talk in a communications system is the channel capacity, which is the maximum amount of information that can be reliably transmitted by an information carrier [21]. In a multi-channel system, a photon can be in one of

*N*input states and the maximum channel capacity value is log

_{2}

*N*bits per photon.

*ℓ*= −25 to

*ℓ*= 25. The choice of LG beams allows the beam waist to be controlled, and the experimental result to be very closely matched to numerical modelling of the system. The channel capacity was measured for

*N*modes, where

*N*= 2,4,6,...,50. For each measurement the range

*ℓ*= −

*N*/2 to

*ℓ*=

*N*/2 was used while leaving

*ℓ*= 0 free as an alignment channel. The values measured are shown in Fig. 3. The optical transformation we utilise is only perfect for rays which are normally incident on the transformation elements. Helically phased beams are inherently not of this type, and have a skew angle of the rays of

*θ*=

_{s}*ℓ/kr*, where

*k*is the wavenumber of the light and

*r*is the distance from the beam centre [22

22. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**, 36–40 (1995). [CrossRef]

23. J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express **14**, 11919–11924 (2006). [CrossRef] [PubMed]

**105**(15), 153601 (2010). [CrossRef]

*ℓ*causing errors in the transformation, hence increasing the channel cross talk at these

*ℓ*values. Simulations show that reducing the separation between the components or increasing the aperture size of the system can reduce these skew angle effects at higher

*ℓ*values, hence reducing the cross talk within the system.

^{−17}

*W*, corresponding to approximately 75 photons per second entering the first element. The camera was set to capture 100 frames per second, hence on average we record less than one photon per measurement.

## References and links

1. | L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

2. | A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. |

3. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Phys. Rev. Lett. |

4. | A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. |

5. | 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 |

6. | J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. |

7. | J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science |

8. | V. Bazhenov, M. Soskin, and M. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt. |

9. | N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. |

10. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

11. | S. S. Oemrawsingh, J. de Jong, X. Ma, and A. Aiello. “High- dimensional mode analyzers for spatial quantum entanglement, Phys. Rev. A |

12. | L. Marrucci, E. Karimi, S. Slussarenko, B Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. |

13. | J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. |

14. | G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. |

15. | O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. |

16. | W. Hossack, A. Darling, and A. Dahdour, “Coordinate transformations with multiple computer-generated optical-elements,” J. Mod. Opt. |

17. | Y. Saito, S. Komatsu, and H. Ohzu, “Scale and rotation invariant real-time optical correlator using computer generated hologram,” Opt. Commun. |

18. | M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. |

19. | G. C. G. Berkhout, M. P. J. Lavery, M. W. Beijersbergen, and M. J. Padgett, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. |

20. | T.A. Dow, M.H. Miller, and P.J. Falter, “Application of a fast tool servo fordiamond turning of non-rotationally symmetric surfaces,” J. Precision Eng. |

21. | C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. |

22. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. |

23. | J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(080.3630) Geometric optics : Lenses

(050.4865) Diffraction and gratings : Optical vortices

(060.5565) Fiber optics and optical communications : Quantum communications

**History**

Original Manuscript: September 1, 2011

Revised Manuscript: November 10, 2011

Manuscript Accepted: December 2, 2011

Published: January 17, 2012

**Citation**

Martin P. J. Lavery, David J. Robertson, Gregorius C. G. Berkhout, Gordon D. Love, Miles J. Padgett, and Johannes Courtial, "Refractive elements for the measurement of the orbital angular momentum of a single photon," Opt. Express **20**, 2110-2115 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2110

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### References

- L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45, 8185–8189 (1992). [CrossRef] [PubMed]
- A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3, 161–204 (2011). [CrossRef]
- G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Phys. Rev. Lett.88, 013601 (2001). [CrossRef]
- A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett.89, 240401 (2002). [CrossRef] [PubMed]
- 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. Express12, 5448–5456 (2004). [CrossRef] [PubMed]
- J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys.4, 282 (2008). [CrossRef]
- J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662–655 (2010). [CrossRef] [PubMed]
- V. Bazhenov, M. Soskin, and M. Vasnetsov, “Screw dislocations in light wave-fronts,” J. Mod. Opt.39, 985–990 (1992). [CrossRef]
- N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett.17, 221–223 (1992). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316, (2001). [CrossRef] [PubMed]
- S. S. Oemrawsingh, J. de Jong, X. Ma, and A. Aiello. “High- dimensional mode analyzers for spatial quantum entanglement, Phys. Rev. A73, 032339, (2006) [CrossRef]
- L. Marrucci, E. Karimi, S. Slussarenko, B Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001, (2011). [CrossRef]
- J. Leach, M. Padgett, S. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett.88(25), 257901 (2002). [CrossRef] [PubMed]
- G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett.105(15), 153601 (2010). [CrossRef]
- O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am.64(8), 1092–1099 (1974). [CrossRef]
- W. Hossack, A. Darling, and A. Dahdour, “Coordinate transformations with multiple computer-generated optical-elements,” J. Mod. Opt.34, 1235–1250 (1987). [CrossRef]
- Y. Saito, S. Komatsu, and H. Ohzu, “Scale and rotation invariant real-time optical correlator using computer generated hologram,” Opt. Commun.47(1), 8–11 (1983). [CrossRef]
- M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt.13, 064006 (2011). [CrossRef]
- G. C. G. Berkhout, M. P. J. Lavery, M. W. Beijersbergen, and M. J. Padgett, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett.36, 1863–1865 (2011). [CrossRef] [PubMed]
- T.A. Dow, M.H. Miller, and P.J. Falter, “Application of a fast tool servo fordiamond turning of non-rotationally symmetric surfaces,” J. Precision Eng.13, 243–250 (1991). [CrossRef]
- C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J.27, 379 (1948).
- M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun.121, 36–40 (1995). [CrossRef]
- J. Leach, S. Keen, M. J. Padgett, C. Saunter, and G. D. Love, “Direct measurement of the skew angle of the Poynting vector in a helically phased beam,” Opt. Express14, 11919–11924 (2006). [CrossRef] [PubMed]

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