## Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states |

Optics Express, Vol. 20, Issue 1, pp. 452-461 (2012)

http://dx.doi.org/10.1364/OE.20.000452

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### Abstract

The effect of atmosphere turbulence on light’s spatial structure compromises the information capacity of photons carrying the Orbital Angular Momentum (OAM) in free-space optical (FSO) communications. In this paper, we study two aberration correction methods to mitigate this effect. The first one is the Shack-Hartmann wavefront correction method, which is based on the Zernike polynomials, and the second is a phase correction method specific to OAM states. Our numerical results show that the phase correction method for OAM states outperforms the Shark-Hartmann wavefront correction method, although both methods improve significantly purity of a single OAM state and the channel capacities of FSO communication link. At the same time, our experimental results show that the values of participation functions go down at the phase correction method for OAM states, i.e., the correction method ameliorates effectively the bad effect of atmosphere turbulence.

© 2011 OSA

## 1. Introduction

*et. al.*showed that light can carry orbital angular momentum (OAM), and that an azimuthal phase dependence

*exp*(

*ilθ*) of a light beam corresponds to

*ℓ*units of OAM [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]

*N*quantum digits per single photon for free space optical communications (FSO) and quantum communications [2

2. J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

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

6. J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A **78**, 062320 (2008). [CrossRef]

7. 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**, 5448–5456 (2004). [CrossRef] [PubMed]

8. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett **94**, 153901 (2005). [CrossRef] [PubMed]

12. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. **9**, 94 (2007). [CrossRef]

8. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett **94**, 153901 (2005). [CrossRef] [PubMed]

9. J. Anguita, M. Neifeld, and B. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. **47**, 2414–2429 (2008). [CrossRef] [PubMed]

10. G. Tyler and R. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. **34**, 142–144 (2009). [CrossRef] [PubMed]

11. Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. **26**, 074220 (2009). [CrossRef]

12. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. **9**, 94 (2007). [CrossRef]

13. S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT) , pp. 580–583 (2010). [CrossRef]

14. L. Andrews and R. Phillips, *Laser Beam Propagation through Random Media* (SPIE Press, 2005). [CrossRef]

15. R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. **88**, 541–562 (1978). [CrossRef]

## 2. Turbulent aberrations and phase correction methods

*N*×

*N*array of random complex numbers with statistics that matches the fluctuations of the index of refraction. Many numerical methods have been proposed to simulate atmosphere turbulence [14

14. L. Andrews and R. Phillips, *Laser Beam Propagation through Random Media* (SPIE Press, 2005). [CrossRef]

18. M. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. **281**, 3395–3402 (2008). [CrossRef]

15. R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. **88**, 541–562 (1978). [CrossRef]

14. L. Andrews and R. Phillips, *Laser Beam Propagation through Random Media* (SPIE Press, 2005). [CrossRef]

*(*

_{n}*k*,

_{x}*k*) is then given as

_{y}*l*

_{0}is equals to the inner scale of turbulence, and

*k*(

_{i}*i*=

*x*,

*y*) is the wavenumber in

*i*direction. Then, the phase spectrum Φ(

*k*,

_{x}*k*) is where Δ

_{y}*Z*is the spacing between the subsequent phase screens, and

*k*

_{0}is the wavenumber of the light. The phase screen

*φ*(

*x*,

*y*) can be evaluated as the Fourier transform of a complex random distribution with the variance

*σ*

^{2}as follows: and where

*C*is an

*N*×

*N*array of complex random numbers with zero mean and variance one, and Δ

*x*is the grid spacing. Here, the grid spacing in

*x*direction is assumed to be equal to that in

*y*direction.

20. V. Voitsekhovich, “Hartmann test in atmospheric research,” J. Opt. Soc. Am. A **13**, 1749–1757 (1996). [CrossRef]

*φ*(

*x*,

*y*) can completely be described by a linear combination of Zernike polynomials

*Z*

_{0},

*Z*

_{1},...,

*Z*[19] Now, assuming that

_{N}*φ*(

*x*,

*y*) is the deformation wavefront caused by turbulent aberrations. By model estimation and using the least-squares method, the coefficients

*a*can be obtained by solving the equantion and that is where,

_{i}*G*(

_{x}*m*) =

*φ′*(

_{x}*m*),

*G*(

_{y}*m*) =

*φ′*(

_{y}*m*),

*D*(

_{kx}*m*) =

*Z*(

_{kx}*m*), and

*D*(

_{ky}*m*) =

*Z*(

_{ky}*m*). This equation can be expressed in a simplified form and the coefficients of Zernike polynomial

*A*is can then be calculated by This way, we could get the phase of deformation wavefront caused by atmosphere turbulence using above coefficients and Eq. (5).

21. A. Jesacher, A. Schwaighofer, S. Frhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express **15**, 5801–5808 (2007). [CrossRef] [PubMed]

*φ*(

*x*,

*y*) of a light field by just determining the modulus

*A*(

*k*,

_{x}*k*) of its Fourier transform as In [21

_{y}21. A. Jesacher, A. Schwaighofer, S. Frhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express **15**, 5801–5808 (2007). [CrossRef] [PubMed]

*φ*(

*x*,

*y*) corresponds to the hologram function, and

*A*(

*k*,

_{x}*k*) to the amplitude of the observed doughnut mode. GS-algorithm is iterative, and its procedure is as follows. A perfect phase spiral is used as a starting point, and the illumination beam profile is selected as the magnitude in the Spatial Light Modulation plane. A complex Fast Fourier Transform (FFT) is used to generate the phase in the diffraction plane. The magnitude part of a perfect ring is discarded and replaced with the distorted image generated by the SLM, and then transformed back to the SLM plane, where the magnitude is replaced with the illumination beam profile. After a few iterations of the loop, the phase will generally converge to a value, and the phase of the aberration can be retrieved. Moreover, it is shown that this retrieval algorithm can be implemented by experiment [22].

_{y}## 3. Simulation results

### 3.1. A single OAM state propagating through turbulence and its purity

*L*

_{0}= 50m,

*l*

_{0}= 0.0002m,

*N*=140, Δ

*x*=0.0003m, and Δ

*Z*=50m. In the simulation, there are supposed five phase screens during the beam propagation. The results show that the purity of the input OAM state,

*l*= −3

*h̄*, is damaged by the turbulent atmosphere, and the OAM state can be recovered by an aberration correction method.

*l*=

_{z}*ℓh̄*, is obtained by summing all probabilities associated with that eigenvalue where the superposition coefficients

*a*

_{p,l}(

*z*) are given by the inner products

*ℓ*=0 state, the probability of the OAM state keeping on

*ℓ*=0 is only 25.3% in the atmospheric turbulence enviroment with

*L*

_{0}= 50m,

*l*

_{0}= 0.0002m,

*N*= 128, Δ

*x*=0.0003m, Δ

*Z*=100m. This probability can be improved to 63.8% by using SH-algorithm, where 12th order of Zernike polynomials have been used to estimate the wavefront. And this probability can be enhanced to 69% by using GS-algorithm. The results show that the two aberration correction methods have improved the beam quality significantly, then the phase correction method for OAM states show a better performance.

### 3.2. A communication system on OAM and its capacity

8. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett **94**, 153901 (2005). [CrossRef] [PubMed]

10. G. Tyler and R. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. **34**, 142–144 (2009). [CrossRef] [PubMed]

*H*= [

*H*], where the conditional probabilities

_{mn}*H*can be evaluated by Eqs. (12) and (13). Here,

_{mn}*m*is the number of transmitted OAM states and

*n*is the number of received states. For the simplicity, we select

*ℓ*= 0, 1 as the transmitted OAM states for

_{m}*L*= 1 and the received OAM states are selected from

*ℓ*= −15 to

_{n}*ℓ*= 15. Similarly, we will select

_{n}*ℓ*= 0, 1,2,3 for

_{m}*L*= 2 and decompose the received state from

*ℓ*= −15 to

_{n}*ℓ*= 15. We use the same scheme for

_{n}*L*= 3 and

*L*= 4. After we obtain the channel matrix, we can calculate the capacity of this discrete channel by the Blahut-Arimoto algorithm [23

23. P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory **54**, 1887–1918 (2008). [CrossRef]

*L*= 1) to decrease from 1 bits/symbol to 0.08 bits/symbol when the structure constant of the index refraction varies from 1 × 10

^{−16}m

^{−2/3}to 1 × 10

^{−11}m

^{−2/3}. The decreasing point A means atmosphere turbulence will cause noise to the communication channel at

*L*= 1 and

## 4. Experimental results

### 4.1. Experimental setup

*ℓ*= 1 LG mode to determine the ’hologram’ of the turbulent atmosphere aberration from the distorted shape of a focused doughnut mode. Unfortunely, this mode does not explain the change numerically. In order to illustrate the improvement of the phase correction method for OAM states to the turbulent aberration, we will design a referenced spot mode (

*ℓ*= 0) besides the doughnut, and we will use the participation function (also named as sharpness metric) to measure the spot quality. The participation function of the referenced spot is defined as following where

*I*

_{i, j}is the intensity of the (

*i*,

*j*)

*pixel of the referenced spot. It is shown that the smaller value corresponds to the more tightly focused spot [24*

^{th}24. J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A **20**, 609–620 (2003). [CrossRef]

*ℓ*(

*ℓ*= 1 in the Fig. 5) and a referenced spot (

*ℓ*= 0) under somewhat turbulent aberration. As usual, the vortex lens is superposed by a grating in order to spatially separate the optical vortex generated in the first diffraction order from other orders. In order to detect the referenced spot besides the LG mode, different grating is assigned to LG mode and the referenced spot. The Fourier plane corresponds to the far-field diffraction pattern. A len (

*L*

_{3}) is used to focus the image on CCD in the Fourier plane, and several mirrors are arranged so as to get the biggest image in CCD as possible.

### 4.2. improvement of the participation function

*ℓ*= 1 LG mode), and one for participation function calculations ((

*ℓ*= 0 LG mode).

^{−15}to 10

^{−12}. The experimental results show that the values of participation function go down with the correction method, and the correction method is an effective way to mitigate the turbulent aberration. We get 45% maximum improvement and 17% average improvement by the change of strength of aberration.

## 5. Conclusion

21. A. Jesacher, A. Schwaighofer, S. Frhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express **15**, 5801–5808 (2007). [CrossRef] [PubMed]

**94**, 153901 (2005). [CrossRef] [PubMed]

## 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. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011). |

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

4. | I. Djordjevic and M. Arabaci, “LDPC-coded orbital angular momentum (OAM) modulation for free-space optical communication,” Opt. Express |

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

6. | J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A |

7. | 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 |

8. | C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett |

9. | J. Anguita, M. Neifeld, and B. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. |

10. | G. Tyler and R. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. |

11. | Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. |

12. | C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. |

13. | S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT) , pp. 580–583 (2010). [CrossRef] |

14. | L. Andrews and R. Phillips, |

15. | R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. |

16. | J. Strasburg and W. Harper, “Impact of atmospheric turbulence on beam propagation,” Proc. SPIE |

17. | R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. |

18. | M. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. |

19. | B. Platt, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. |

20. | V. Voitsekhovich, “Hartmann test in atmospheric research,” J. Opt. Soc. Am. A |

21. | A. Jesacher, A. Schwaighofer, S. Frhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express |

22. | R. Bowman, “Aberration correction for spatial light modulators,” Master’s thesis (Churchill College, 2007). |

23. | P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory |

24. | J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A |

**OCIS Codes**

(010.1285) Atmospheric and oceanic optics : Atmospheric correction

(060.2605) Fiber optics and optical communications : Free-space optical communication

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: September 7, 2011

Revised Manuscript: November 19, 2011

Manuscript Accepted: December 1, 2011

Published: December 21, 2011

**Citation**

S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, "Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states," Opt. Express **20**, 452-461 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-452

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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. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).
- I. Djordjevic, “Deep-space and near-Earth optical communications by coded orbital angular momentum (OAM) modulation,” Opt. Express19, 14277–14289 (2011). [CrossRef] [PubMed]
- I. Djordjevic and M. Arabaci, “LDPC-coded orbital angular momentum (OAM) modulation for free-space optical communication,” Opt. Express18, 24722–24728 (2010). [CrossRef] [PubMed]
- M. Gruneisen, W. Miller, R. Dymale, and A. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt.47, A32–A42 (2008). [CrossRef] [PubMed]
- J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A78, 062320 (2008). [CrossRef]
- 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]
- C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett94, 153901 (2005). [CrossRef] [PubMed]
- J. Anguita, M. Neifeld, and B. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt.47, 2414–2429 (2008). [CrossRef] [PubMed]
- G. Tyler and R. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett.34, 142–144 (2009). [CrossRef] [PubMed]
- Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett.26, 074220 (2009). [CrossRef]
- C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys.9, 94 (2007). [CrossRef]
- S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010). [CrossRef]
- L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005). [CrossRef]
- R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech.88, 541–562 (1978). [CrossRef]
- J. Strasburg and W. Harper, “Impact of atmospheric turbulence on beam propagation,” Proc. SPIE5413, 93 (2004). [CrossRef]
- R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt.39, 393–397 (2000). [CrossRef]
- M. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun.281, 3395–3402 (2008). [CrossRef]
- B. Platt, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17, 573–577 (2001).
- V. Voitsekhovich, “Hartmann test in atmospheric research,” J. Opt. Soc. Am. A13, 1749–1757 (1996). [CrossRef]
- A. Jesacher, A. Schwaighofer, S. Frhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express15, 5801–5808 (2007). [CrossRef] [PubMed]
- R. Bowman, “Aberration correction for spatial light modulators,” Master’s thesis (Churchill College, 2007).
- P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory54, 1887–1918 (2008). [CrossRef]
- J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A20, 609–620 (2003). [CrossRef]

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