## Rapid generation of light beams carrying orbital angular momentum |

Optics Express, Vol. 21, Issue 25, pp. 30196-30203 (2013)

http://dx.doi.org/10.1364/OE.21.030196

Acrobat PDF (1054 KB)

### Abstract

We report a technique for encoding both amplitude and phase variations onto a laser beam using a single digital micro-mirror device (DMD). Using this technique, we generate Laguerre-Gaussian and vortex orbital-angular-momentum (OAM) modes, along with modes in a set that is mutually unbiased with respect to the OAM basis. Additionally, we have demonstrated rapid switching among the generated modes at a speed of 4 kHz, which is much faster than the speed regularly achieved by phase-only spatial light modulators (SLMs). The dynamic control of both phase and amplitude of a laser beam is an enabling technology for classical communication and quantum key distribution (QKD) systems that employ spatial mode encoding.

© 2013 Optical Society of America

## 1. Introduction

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

4. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. **4**, 2781 (2013). [CrossRef] [PubMed]

5. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. **88**, 127902 (2002). [CrossRef] [PubMed]

6. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A **35**, 10065–10076 (2002). [CrossRef]

*e*phase structure onto a laser beam. This task can be achieved by using computer generated holograms [7

^{iℓϕ}7. T. Ando, T. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. **34**, 34–36 (2009). [CrossRef]

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

9. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**, 163905 (2006). [CrossRef] [PubMed]

10. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express **12**, 3548–3553 (2004). [CrossRef] [PubMed]

12. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. **38**, 3546–35490 (2013). [CrossRef] [PubMed]

13. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express **20**, 24444–24449 (2012). [CrossRef]

14. M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express **20**, 13195–13200 (2012). [CrossRef] [PubMed]

15. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. **47**, A32–A42 (2008). [CrossRef] [PubMed]

16. B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science **340**, 844–847 (2013). [CrossRef] [PubMed]

17. P. Zhu, O. Fajardo, J. Shum, Y.-P. Zhang Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. **7**, 1410–1425 (2012). [CrossRef] [PubMed]

18. Y.-X. Ren, M. Li, K. Huang, J.-G. Wu, H.-F. Gao, Z.-Q. Wang, and Y.-M. Li, “Experimental generation of Laguerre-Gaussian beam using digital micromirror device,” Appl. Opt. **49**, 1838 (2010). [CrossRef] [PubMed]

19. P. J. Rodrigo, I. R. Perch-Nielsen, and J. Glückstad, “High-speed phase modulation using the RPC method with a digital micromirror-array device,” Opt. Express **14**, 5588–5593 (2006). [CrossRef] [PubMed]

20. V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre–Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett. **37**, 4826–4828 (2012). [CrossRef] [PubMed]

21. R. W. Cohn and M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. **35**, 2488–2498 (1996). [CrossRef] [PubMed]

## 2. Theory

*x*

_{0}. The parameters

*p*and

*w*are unitless quantities that set the position and the width of each pulse and are equal to constant values for a uniform grating. Here we show that it is possible to locally change the value of these parameters to achieve phase and amplitude modulation of the optical field.

*x*,

*y*) is the sign function. It is easy to check that in the limit where

*w*(

*x*,

*y*) and

*p*(

*x*,

*y*) are slowly varying, this formula reproduces the pulse train described above. We can find the corresponding

*w*(

*x*,

*y*) and

*p*(

*x*,

*y*) functions for a general complex scalar field

*𝒜*(

*x*,

*y*)

*e*

^{iφ}^{(}

^{x,y}^{)}according to the relations We have assumed that the field contains no singularity and thus its amplitude can be normalized to have a maximum of unity (Note that the factor

*π*in the argument of arcsin is dropped since the maximum amplitude is normalized to unity).

## 3. Experiment

*μ*m, and an array diffraction efficiency of 86 %. The holograms for generating modes are created by modulating a grating function with 20 micro-mirrors per each period. As shown in Fig. 2, a 4f imaging system along with an aperture separates the first order diffracted light. We use a charge-coupled-device (CCD) camera for measuring the intensity profile of the generated modes. In addition, we have used a Mach-Zehnder interferometer for verifying the phase patterns of the created beams. A collimated plane-wave reference from the same laser is interfered with the modes generated by the DMD to obtains interferograms.

*p*+ 1 rings in the radial direction where

*p*is the radial quantum number [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]

14. M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express **20**, 13195–13200 (2012). [CrossRef] [PubMed]

*u*(

_{ℓ}*r*,

*φ*) represents each OAM mode (either LG or vortex) and

*N*is the total number of them. The simultaneous use of both OAM and ANG modes is necessary for achieving security in an OAM-based QKD system [14

14. M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express **20**, 13195–13200 (2012). [CrossRef] [PubMed]

31. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

*θ*

_{j,Nℓ}(

*r*,

*φ*) ANG mode is roughly equal to

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

*ℓ*= 5, −5 and 0. The computer generated holograms for these modes were loaded onto the memory of the DMD and the switching was achieved by using a clock signal. We have used the mode sorter described in [32

32. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express **20**, 2110–2115 (2012). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgments

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

3. | R. W. Boyd, A. K. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE |

4. | M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. |

5. | N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. |

6. | M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A |

7. | T. Ando, T. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. |

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

9. | L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. |

10. | K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express |

11. | C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in |

12. | E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. |

13. | M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express |

14. | M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express |

15. | M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. |

16. | B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science |

17. | P. Zhu, O. Fajardo, J. Shum, Y.-P. Zhang Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc. |

18. | Y.-X. Ren, M. Li, K. Huang, J.-G. Wu, H.-F. Gao, Z.-Q. Wang, and Y.-M. Li, “Experimental generation of Laguerre-Gaussian beam using digital micromirror device,” Appl. Opt. |

19. | P. J. Rodrigo, I. R. Perch-Nielsen, and J. Glückstad, “High-speed phase modulation using the RPC method with a digital micromirror-array device,” Opt. Express |

20. | V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre–Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett. |

21. | R. W. Cohn and M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt. |

22. | B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. |

23. | W. H. Lee, “High efficiency multiple beam gratings,” Appl. Opt. |

24. | J. A. Davis, K. Olea Valadez, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt. |

25. | J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. |

26. | D. C. Chu and J. W. Goodman, “Spectrum shaping with parity sequences,” Appl. Opt. |

27. | J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. |

28. | L. J. Hornbeck, “From cathode rays to digital micromirrors: a history of electronic projection display technology,” Tex. Instrum. Tech. J. |

29. | D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” in |

30. | Joseph W. Goodman, |

31. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

32. | M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express |

**OCIS Codes**

(230.6120) Optical devices : Spatial light modulators

(050.4865) Diffraction and gratings : Optical vortices

(270.5568) Quantum optics : Quantum cryptography

**ToC Category:**

Physical Optics

**History**

Original Manuscript: September 12, 2013

Revised Manuscript: November 14, 2013

Manuscript Accepted: November 20, 2013

Published: December 2, 2013

**Citation**

Mohammad Mirhosseini, Omar S. Magaña-Loaiza, Changchen Chen, Brandon Rodenburg, Mehul Malik, and Robert W. Boyd, "Rapid generation of light beams carrying orbital angular momentum," Opt. Express **21**, 30196-30203 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-25-30196

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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]
- J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6, 488–496 (2012). [CrossRef]
- R. W. Boyd, A. K. Jha, M. Malik, C. O’Sullivan, B. Rodenburg, and D. J. Gauthier, “Quantum key distribution in a high-dimensional state space: exploiting the transverse degree of freedom of the photon,” Proc. SPIE7948, 79480L (2011).
- M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun.4, 2781 (2013). [CrossRef] [PubMed]
- N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett.88, 127902 (2002). [CrossRef] [PubMed]
- M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A35, 10065–10076 (2002). [CrossRef]
- T. Ando, T. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett.34, 34–36 (2009). [CrossRef]
- G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12, 5448–5456 (2004). [CrossRef] [PubMed]
- L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96, 163905 (2006). [CrossRef] [PubMed]
- K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express12, 3548–3553 (2004). [CrossRef] [PubMed]
- C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proc. IEEE Int. Conf. (Bangalore, 1984), pp. 175–179.
- E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett.38, 3546–35490 (2013). [CrossRef] [PubMed]
- M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express20, 24444–24449 (2012). [CrossRef]
- M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express20, 13195–13200 (2012). [CrossRef] [PubMed]
- M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt.47, A32–A42 (2008). [CrossRef] [PubMed]
- B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science340, 844–847 (2013). [CrossRef] [PubMed]
- P. Zhu, O. Fajardo, J. Shum, Y.-P. Zhang Schärer, and R. W. Friedrich, “High-resolution optical control of spatiotemporal neuronal activity patterns in zebrafish using a digital micromirror device,” Nat. Protoc.7, 1410–1425 (2012). [CrossRef] [PubMed]
- Y.-X. Ren, M. Li, K. Huang, J.-G. Wu, H.-F. Gao, Z.-Q. Wang, and Y.-M. Li, “Experimental generation of Laguerre-Gaussian beam using digital micromirror device,” Appl. Opt.49, 1838 (2010). [CrossRef] [PubMed]
- P. J. Rodrigo, I. R. Perch-Nielsen, and J. Glückstad, “High-speed phase modulation using the RPC method with a digital micromirror-array device,” Opt. Express14, 5588–5593 (2006). [CrossRef] [PubMed]
- V. Lerner, D. Shwa, Y. Drori, and N. Katz, “Shaping Laguerre–Gaussian laser modes with binary gratings using a digital micromirror device,” Opt. Lett.37, 4826–4828 (2012). [CrossRef] [PubMed]
- R. W. Cohn and M. Liang, “Pseudorandom phase-only encoding of real-time spatial light modulators,” Appl. Opt.35, 2488–2498 (1996). [CrossRef] [PubMed]
- B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev.13, 160–168 (1969). [CrossRef]
- W. H. Lee, “High efficiency multiple beam gratings,” Appl. Opt.18, 2152–2158 (1979). [CrossRef] [PubMed]
- J. A. Davis, K. Olea Valadez, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt.43, 2003–2008 (2003). [CrossRef]
- J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am.61, 1023–1028 (1971). [CrossRef]
- D. C. Chu and J. W. Goodman, “Spectrum shaping with parity sequences,” Appl. Opt.11, 1716–1724 (1972). [CrossRef] [PubMed]
- J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt.38, 5004–5013 (1999). [CrossRef]
- L. J. Hornbeck, “From cathode rays to digital micromirrors: a history of electronic projection display technology,” Tex. Instrum. Tech. J.15, 7–46 (1998).
- D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micromirror device (DMD) applications,” in Micromachining and Microfabrication, H. Urey, ed. (SPIE, 2003), pp. 14–25.
- Joseph W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys.74, 145–195 (2002). [CrossRef]
- M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express20, 2110–2115 (2012). [CrossRef] [PubMed]

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