## Orbital angular momentum (OAM) spectrum correction in free space optical communication

Optics Express, Vol. 16, Issue 10, pp. 7091-7101 (2008)

http://dx.doi.org/10.1364/OE.16.007091

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### Abstract

Orbital angular momentum (OAM) of laser beams has potential application in free space optical communication, but it is sensitive against pointing instabilities of the beam, i.e. shift (lateral displacement) and tilt (deflection of the beam). This work proposes a method to correct the distorted OAM spectrum by using the mean square value of the orbital angular momentum as an indicator. Qualitative analysis is given, and the numerical simulation is carried out for demonstration. The results show that the mean square value can be used to determine the beam axis of the superimposed helical beams. The initial OAM spectrum can be recovered.

© 2008 Optical Society of America

## 1. Introduction

3. 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]

8. M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. **7**, 46/1–17 (2005). [CrossRef]

9. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. **6**, 103/1–7 (2004). [CrossRef]

8. M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. **7**, 46/1–17 (2005). [CrossRef]

3. 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]

10. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. **24**, 1027–1049 (1992). [CrossRef]

11. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. **96**, 113901/1–4 (2006). [CrossRef] [PubMed]

11. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. **96**, 113901/1–4 (2006). [CrossRef] [PubMed]

## 2. OAM spectrum and misalignment

5. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. **88**, 013601/1–4 (2002). [CrossRef] [PubMed]

6. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt Express **13**, 873–881 (2005). [CrossRef] [PubMed]

11. R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. **96**, 113901/1–4 (2006). [CrossRef] [PubMed]

*ψ*〉 denotes the normalized complex field of the beam, and

*l̂*=

_{z}*x̂p̂*-

_{y}*ŷp̂*the angular momentum operator with

_{x}*x̂*and

*ŷ*the position operators and

*p̂*and

_{x}*p̂*the momentum operators in the transverse plane. The OAM spectrum is the Fourier transform of

_{y}*M*(

*θ*) and reads

*p*,

*l*〉 with radial index

*p*≥0 and azimuthal index

*l*are complete and orthogonal

*T⃗*an operator representing the misalignment. Figure 1 shows two kinds of misalignment, shift and tilt. The arrow on the operator denotes the action on the function on the right hand. For example, the operator for shift reads

*r*

_{0}is the distance between the two axes,

*η*the azimuthal angle of the shifted axis and the superscript

*LD*means shift (lateral displacement). The operator for tilt is

*γ*the direction deflection and

*DD*means tilt (direction deflection).

*p*

_{0}

*l*

_{0}|Φ〉 is the expansion coefficient of the normalized laser beam on the Laguerre-Gaussian beams and 〈

*pl*|

*T⃗*|

*p*

_{0}

*l*

_{0}〉 can be interpreted as the probability amplitude for the transition |

*p*

_{0}

*l*

_{0}〉to|

*pl*〉 caused by the misalignment operation

*T⃗*. The matrix element

*F*(*) and

*F*

^{-1}(*)are the Fourier transform and its inverse. The symbol ⊗ denotes the convolution.

*w*

_{0}and 2/

*w*

_{0}are the waist radii of the Laguerre-Gaussian beams in the spatial domain and frequency domain, respectively. The spatial shift in the space domainwill cause a phase term in the frequency domain

*r*

_{0}.

8. M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. **7**, 46/1–17 (2005). [CrossRef]

## 3. Indicator for beam axis determination and interpretation

*p*,

*l*)=(0,0) and (0,1), respectively The dark cross and the white cross denote the beam axis defined by the intensity momentum and the real axis, respectively. It’s clear that the two axes are not collinear. Here the real axis denotes the axis of the pre-set helical beams which are superposed collinearly.

12. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. **88**, 053601/1–4 (2002). [CrossRef] [PubMed]

**96**, 113901/1–4 (2006). [CrossRef] [PubMed]

**96**, 113901/1–4 (2006). [CrossRef] [PubMed]

*V*value varies with the position of axis. We will use the variance to find the beam center. For the single Laguerre-Gaussian beam well aligned,

*V*is zero, but with misalignment

*V*reads [11

**96**, 113901/1–4 (2006). [CrossRef] [PubMed]

*k*is the wave number,

*z*the Rayleigh range,

_{R}*p*

_{0}the radial number of the Laguerre-Gaussian beam, and l0 the azimuthal number. Eq. (23) demonstrates that the variance

*V*is an indicator of the relevance for the choice of the rotation axis position within the spatial beam distribution, even when the average OAM is unchanged.

*V*is not zero because of the non-helical wave front. The OAM spectrum at the receiver is dispersed from the incident superimposed Laguerre-Gaussian beam through Eq. (10). When the axis of the receiver is not collinear with the incident beam, the dispersion of the OAM spectrum becomes more serious with increasing lateral displacement, and the

*V*value larger. Only when the two axes are collinear the variance

*V*will reach its minimum. Then we can define the position with the minimum variance

*V*value as the center of the beam. This method also can be interpreted using the information entropy. It is defined unit bits as

*V*can be used to determine the beam axis position, we also can expect it can be used to determine the beam axis direction.

## 4. Numerical simulation and discussions

*l*=0) or bright circle (for

*l*≠0), as shown in Fig. 4. It is obvious that the amplitude distribution is not symmetric and the phase structure not helical. There are more than one singularity and they are not centered. Then the beam axis will be determined. As discussed in Section 3, the beam center will not have the minimum

*V*value, but the OAM spectrum is the same as the initial one shown in Fig. 3. The following calculation will point out that the beam axis defined by the minimum

*V*value is indeed the real axis.

*V*value and the shift, Fig. 5 shows a contour plot (

*a*), a mesh plot (

*b*) and plots along the

*x*,

*y*-directions through the center (c). Here x and y are scaled by

*w*

_{0}, the beam radius of the fundamental Laguerre-Gaussian beam. From Fig. 4(a) and (b) we can find out that the

*V*value changes with the shift in the transverse plane. All contour lines enclose the beam center, which indicates no other extremes exist, except the central value. The mesh plot shows a parabolic -like surface without ripples along the radial direction, which indicates that the

*V*value increases with the distance between the beam center and the axis of the receiver system. These two point are consistent with the qualitatively analysis. The grads of

*V*value in special cases along the

*x*,

*y*-directionsare plotted in Fig. 5(c). The lines are conic and reach the minimum at the position (

*x*,

*y*)=(0, 0), just the beam axis used of the incident beam. This result means that we can find out the position with the minimum

*V*value and this position is just the axis position of the incident beam. The beam center defined by the intensity moment is about (

*x*,

*y*)=(-0.02, 0.07), which is not the real incident beam center in free space optical communication using OAM. And at this center the received OAM spectrum is dispersed.

*I*varying with the lateral displacement, (

*a*) contour plot, (b) mesh plot, and (c)

*x*,

*y*-directions. The variation od the information entropy

*I*is similar to that of the mean square value of the OAM, as we expect in Section 3. The information entropy will find its minimum at the position (

*x*,

*y*)=(0, 0), i.e. at the axis of the incident beam. But the information entropy is not so excellent for the indicator of the beam axis than the mean square value. Comparing Fig. 5(c) and Fig. 6(c) we find the mean square value changes more sharply with the x and y coordinates than the information entropy. This is because the information entropy does not care for the azimuthal number, which will make the mean square value even larger with increasing dispersion. In case of more azimuthal numbers used for optical communication, the effect of the mean square value is more obvious.

*V*value to indicate the direction of incident beam’s axis. And it is easy to understand that in this case the

*V*value will reach the minimum, when the axis of the receiver system is collinear with the incident beam. For the complicated case that both misalignments occur together, we can make the analysis in two steps, first shift and then tilt or vice versa. In any case the

*V*value can reach the minimum only when the axis of the receiver system is collinear with the incident beam. This conclusion can be used to correct the OAM spectrum in free space optical communication. In the next Section we will give an algorithm and design a schematic setup for OAM spectrum correction.

## 5. Schematic design of an OAM spectrum correction

*x*,

*y*) and two for the tilt (

*u*,

*v*). At first we adjust the system to a place with rough calibration, set the two bits binary number

*C*, which is used for controlling the dimension to optimize, set S which is used for deciding the direction to optimize, and set

*V*

_{1}and

*V*

_{2}as the initial

*V*value to make the system work automatically. Then the system performs the mechanical adjustment to a new state, measures the OAM spectrum and calculates the

*V*value. There are only two states for

*V*,

*V*≠

*V*

_{1}and

*V*=

*V*

_{1}. The first case denotes the optimization in one dimension and is a first approach. The second one means that the

*V*value reaches its minimum in this dimension but it does not indicate the minimum in all the four dimensions, as shown in Fig. 7(a). When

*V*≠

*V*

_{1}, the loop will go on for the same parameter till

*V*=

*V*

_{1}. Then the second decision will be made whether

*V*≠

*V*

_{1}. If not, it means that the optimization of the second dimension is not complete, and the loop will go on for till

*V*=

*V*

_{1}=

*V*

_{2}. Then the first misalignment is probably corrected and the procedure jumps to the other misalignment. In our consideration, the optimization is carried out for each parameters of the same misalignment repetitively, and then for the two misalignments in the same manner. Only when the minimum

*V*value is known, we can perform the correction of the misalignment till the minimum

*V*value are obtained. The geometry of these dimensions is shown in Fig. 7(b).

*S*to control the adding or subtracting of an increment. Take as an examplethe optimizing of x. When

*S*=‘+’ and the

*V*value at

*x*+Δ

*x*is larger than that at

*x*, it mean the direction with increasing

*x*is wrong, and we should change

*S*to ‘-’. Generally

*S*should be changed to

*S*×sign (

*V*

_{1}-

*V*) where sign(*) is the sign function. This mechanism proposes negative feedback to minimize the

*V*value minimum.

*C*is useful because of inversing the first bit for different misalignment and the second one for different dimension of the same misalignment.

*C*is also used for selecting the dimension to optimize, i.e. 00 for

*u*, 01 for

*v*, 10 for

*x*and 11 for

*y*.

13. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. **88**, 257901/1–4 (2002). [CrossRef] [PubMed]

14. G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. **132**, 8–14 (1996). [CrossRef]

15. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. **81**, 4828–4830 (1998). [CrossRef]

15. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. **81**, 4828–4830 (1998). [CrossRef]

## 6. Conclusions

*V*value on the shift

*r*

_{0}and tilt angle

*γ*is merely qualitatively and numerically validated without a rigid mathematical deduction.

## Acknowledgments

## References and links

1. | 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 |

2. | 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. |

3. | 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 |

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

5. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. |

6. | L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt Express |

7. | Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, “Realizing high density optical data storage by using orbital angular momentum of light beam,” Acta Phys. Sin. |

8. | M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, “Analysis of orbital angular momentum of a misaligned optical beam,” New J. Phys. |

9. | S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. |

10. | H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. |

11. | R. Zambrini and S. M. Barnett, “Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum,” Phys. Rev. Lett. |

12. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam,” Phys. Rev. Lett. |

13. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. |

14. | G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. |

15. | J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational Frequency Shift of a Light Beam,” Phys. Rev. Lett. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(350.5030) Other areas of optics : Phase

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 3, 2008

Revised Manuscript: April 17, 2008

Manuscript Accepted: April 17, 2008

Published: May 1, 2008

**Citation**

Yi-Dong Liu, Chunqing Gao, Xiaoqing Qi, and Horst Weber, "Orbital angular momentum (OAM) spectrum correction in free space optical communication," Opt. Express **16**, 7091-7101 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7091

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### References

- 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, 8185-8190 (1992). [CrossRef] [PubMed]
- 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, 826-829 (1995). [CrossRef] [PubMed]
- 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]
- 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]
- G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601/1-4 (2002). [CrossRef] [PubMed]
- L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt Express 13, 873-881 (2005). [CrossRef] [PubMed]
- Y.-D. Liu, C.-Q. Gao, M.-W. Gao, and F. Li, "Realizing high density optical data storage by using orbital angular momentum of light beam," Acta Phys. Sin. 56, 854-858 (2007) (in Chinese).
- M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, "Analysis of orbital angular momentum of a misaligned optical beam," New J. Phys. 7, 46/1-17 (2005). [CrossRef]
- S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, "Uncertainty principle for angular position and angular momentum," New J. Phys. 6, 103/1-7 (2004). [CrossRef]
- H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992). [CrossRef]
- R. Zambrini and S. M. Barnett, "Quasi-Intrinsic Angular Momentum and the Measurement of Its Spectrum," Phys. Rev. Lett. 96, 113901/1-4 (2006). [CrossRef] [PubMed]
- A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601/1-4 (2002). [CrossRef] [PubMed]
- J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the Orbital Angular Momentum of a Single Photon," Phys. Rev. Lett. 88, 257901/1-4 (2002). [CrossRef] [PubMed]
- G. Nienhuis, "Doppler effect induced by rotating lenses," Opt. Commun. 132, 8-14 (1996). [CrossRef]
- J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998). [CrossRef]

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