## Poynting vector and orbital angular momentum density of superpositions of Bessel beams |

Optics Express, Vol. 19, Issue 18, pp. 16760-16771 (2011)

http://dx.doi.org/10.1364/OE.19.016760

Acrobat PDF (2297 KB)

### Abstract

We study theoretically the orbital angular momentum (OAM) density in arbitrary scalar optical fields, and outline a simple approach using only a spatial light modulator to measure this density. We demonstrate the theory in the laboratory by creating superpositions of non-diffracting Bessel beams with digital holograms, and find that the OAM distribution in the superposition field matches the predicted values. Knowledge of the OAM distribution has relevance in optical trapping and tweezing, and quantum information processing.

© 2011 OSA

## 1. Introduction

*ħ*(–

*ħ*) per photon for left (right) circularly polarised light, and that the transfer of this momentum can be measured in the laboratory when the light passes through a birefringent plate [1

1. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. **50**(2), 115–125 (1936). [CrossRef]

2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. **96**(1-3), 123–132 (1993). [CrossRef]

3. 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]

*ilφ*), where

*l*is the azimuthal mode index. Such fields carry OAM of

*lħ*per photon, and may be found as beams expressed in several basis functions, including Laguerre-Gaussian beams [2

2. M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. **96**(1-3), 123–132 (1993). [CrossRef]

4. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. **177**(1-6), 297–301 (2000). [CrossRef]

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**(13), 9411–9416 (2008). [CrossRef] [PubMed]

6. 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]

7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

11. 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**(22), 5448–5456 (2004). [CrossRef] [PubMed]

12. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. **175**(4-6), 301–308 (2000). [CrossRef]

16. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. **28**(23), 2285–2287 (2003). [CrossRef] [PubMed]

17. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**(5), 4064–4075 (1997). [CrossRef]

18. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**(14), 8619–8625 (2007). [CrossRef] [PubMed]

20. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. **28**(8), 657–659 (2003). [CrossRef] [PubMed]

7. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef] [PubMed]

## 2. Theory

*u*(

*x,y,z*), which varies in time (in complex notation) as exp(i

*ωt*). This problem has been addressed by others previously [5

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**(13), 9411–9416 (2008). [CrossRef] [PubMed]

21. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. **4**(2), S82–S89 (2002). [CrossRef]

26. M. V. Berry, “Paraxial beams of spinning light,” SPIE **3487**, 6–11 (1998). [CrossRef]

*x*direction) laser mode as [27

27. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**(3), 1177–1179 (2003). [CrossRef]

*z*direction is small compared to the transverse directions to remove derivatives of the field in the

*z*direction. We are now in a position to calculate the time average of the Poynting vector:where for brevity we will drop the arguments of the functions. The subscript

*real*indicates that we are required to consider only the real part of the fields (cos(

*kz*–

*ωt*)), easily found to be:

*α*

_{0}representing the amplitude difference of the two fields (i.e., how the fields are weighted). The amplitude (

*u*

_{0}) and phase (

*ψ*) of the complex scalar field is determined from the characteristics of the mode under study, and will be defined for Bessel and Bessel-Gauss beams a little later in this section. Substitution of Eq. (7) into Eq. (6) and solving for each component of the Poynting vector, we find that:

*α*=

_{l}*α*

_{0}(–1)

*, whereas for Laguerre-Gaussian beams it takes the form*

^{l}*α*=

_{l}*α*

_{0}; here

*α*

_{0}is the constant amplitude weighting of the fields. From these equations we may readily calculate the total angular momentum density of the field, and since by definition our field has a zero spin component, we have the total OAM density (along the direction of propagation,

*z*) as:

*S*are the conventional W/m

^{2}, while the OAM density (

*L*) is now expressed as the angular momentum per unit volume, or Ns/m

_{z}^{2}. With these fundamentals in place, we are ready to calculate the OAM density for superposition fields.

### 2.1 Superposition of Bessel beams

*J*is the Bessel function of order

_{l}*l*. Similar expressions can be found in references [28

28. S. Orlov and A. Stabinis, “Propagation of superpositions of coaxial optical Bessel beams carrying vortices,” J. Opt. A, Pure Appl. Opt. **6**(5), S259–S262 (2004). [CrossRef]

29. S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. **209**(1-3), 155–165 (2002). [CrossRef]

*γ*

_{1}and

*γ*

_{2}, differ as a result of the generation process (see later), but can be set to be equal if so required. As a result of the slightly different cone angles, both the radial and longitudinal wave numbers also differ for the two beams: the radial wave numbers become

*q*=

_{i}*k*sin

*γ*, while the longitudinal wave numbers, given by

_{i}*k*=

_{i}*k*cos

*γ*differ from the central

_{i}*z*-dependent wave number (

*k*) by ±

*Δk*(the subscripts

*i*here refers to the first (second) beam in the superposition). We have maintained a normalization constant

*A*

_{0}in order to later compare the theoretical results to experiment. Thus the Poynting vector becomes:

*r*and

*z*directions, has an additional term

*Δkz*which is responsible for the slow rotation of the intensity distribution during its propagation. This behavior has been noted previously [30

30. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express **17**(26), 23389–23395 (2009). [CrossRef] [PubMed]

*S*, a single non-diffracting beam has no

_{r}*S*component. However, in the case of a superposition of two non-diffracting beams, the resulting beam has a non-zero

_{r}*S*component. This results in a small rotation in the

_{r}*z*direction, thus changing the intensity distribution of the superposition at a particular radial coordinate. We may easily then compute the OAM density to be:

### 2.2 Superposition of Bessel-Gauss beams

24. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express **15**(23), 15214–15227 (2007). [CrossRef] [PubMed]

21. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. **4**(2), S82–S89 (2002). [CrossRef]

### 2.3 Measuring the OAM density

12. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. **175**(4-6), 301–308 (2000). [CrossRef]

16. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. **28**(23), 2285–2287 (2003). [CrossRef] [PubMed]

*ilφ*), are orthogonal over the azimuthal plane, we may express our superposition field in terms of such harmonics:with

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**(13), 9411–9416 (2008). [CrossRef] [PubMed]

## 3. Experimental methodology

31. J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

29. S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. **209**(1-3), 155–165 (2002). [CrossRef]

32. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Opt. Lett. **30**(24), 3266–3268 (2005). [CrossRef] [PubMed]

*l|*, of the two azimuthal phases were of equal but of opposite handedness, a ‘petal’-structure was produced, where the number of ‘petals’ is denoted by 2|

*l*| (Fig. 2(b) and (c)), as expected from theory [30

30. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express **17**(26), 23389–23395 (2009). [CrossRef] [PubMed]

*R*/

*f*with

*R*the ring radius) of

*γ*

_{1}= 0.0069 rad and

*γ*

_{2}= 0.0075 rad, so that

*Δk*~42.8801 m

^{−1}. Another consequence of the slightly differing ring areas (see Fig. 2(a)) is that the energy contained in the two Bessel beams is not equal. Since the illuminating Gaussian beam was measured to be of radius

*w*= 13.29 mm on LCD 1, the energy contained in each ring could be computed, and the weighting coefficients found:

*ilφ*), for various

*l*values, and for particular radial (

*r*) positions on the field. There are two important aspects of this experiment: firstly, the detection is restricted to the origin of the resulting field after the Fourier transforming lens L4, and secondly, the

*r*dependence of the coefficient

*a*is found by implementing the match filter in the form of a narrow (20 pixels) annular ring of radius

_{l}*r*.

*l*index. Thus the decomposition of our field to find the weighting coefficients could be executed as a function of radial co-ordinate and azimuthal mode. This is the first time such a technique has been demonstrated experimentally, and allows for the OAM density to be measured directly: the OAM spectrum,

*a*(

_{l}*r,z*) can be found at any radial position across the beam (and of course any

*z*plane). The weighting parameter (

*α*

_{0}) of the superposition field was set by a suitable choice of the width of the two rings in the phase pattern on LCD 1:

*α*

_{0}= 0.96 (nearly equal weighting).

## 4. Results and discussion

*k*= 0) the OAM density can be engineered to be non-zero as long as the weighting of the two fields differ: the larger the difference, the greater the OAM density. By considering line D on the graph (varying weighting by varying the input Gaussian beam width), the OAM density changes from a large positive value to a large negative value as one considers a single ring in the field (single radial position), passing through a zero value in between. However for a given weighting of the two fields, as one moves radially across the field, so the OAM density remains either positive or negative (passing through nulls between the rings), but does not oscillate in sign as in the case of Fig. 4(a). The variations in the angular momentum, both in terms of sign and magnitude, results from the variations in the amplitude and cone angles of the superimposed Bessel beams (refer to Fig. 6(a) and (b) ). The manipulation of the angular momentum can be implemented efficiently by the method described above and is a useful tool for the controlling of optical forces [33].

*r*

_{1}through

*r*

_{10}) are overlaid as vertical lines. The core result is Fig. 6 (d), where the analytical OAM density prediction (solid curve) as a function of the radial position on the superposition field is shown with the measured OAM density (red bars). It is very clear that there is excellent quantitative agreement between the two (with the exception of the first measurement radius which appears to have a large error). This agreement can also be noted by considering Fig. 6 (e) and (f), where the theoretical and experimental data are shown for the |

*a*

_{3}|

^{2}and |

*a*

_{–3}|

^{2}coefficients as a function of the radial position on the beam.

34. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express **13**(3), 873–881 (2005). [CrossRef] [PubMed]

35. J. A. Rodrigo, A. M. Caravaca-Aguirre, T. Alieva, G. Cristóbal, and M. L. Calvo, “Microparticle movements in optical funnels and pods,” Opt. Express **19**(6), 5232–5243 (2011). [CrossRef] [PubMed]

36. S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express **14**(2), 535–541 (2006). [CrossRef] [PubMed]

37. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express **14**(15), 6604–6612 (2006). [CrossRef] [PubMed]

16. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. **28**(23), 2285–2287 (2003). [CrossRef] [PubMed]

*p*+ |

*l*| + 1, where

*p*is the radial order. This means that modes of +

*l*and a –

*l*will have the same phase shift, so again it can be considered as a constant phase offset (but dependent on

*z*). However, while the OAM arises from the azimuthal phase variation, the rotation of the Poynting vector is equal to arctan(

*z*/

*z*) when considering the radius of maximum field of

_{r}*p*= 0,

*l*≠ 0 modes (since such modes are a single ring of light), i.e., the rotation looks like the Gouy phase shift of a Gaussian mode. The salient point is that the Poynting vector rotation is proportional to arctan(

*z*/

*z*), and that the maximum rotation is fixed, in the case of a

_{r}*p*= 0 mode, to π/2 either side of the beam waist. As the radial order is increased, so the maximum rotation of the field also increases [22

22. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**(1-3), 36–40 (1995). [CrossRef]

23. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. **184**(1-4), 67–71 (2000). [CrossRef]

*l*but also in

*p*, resulting in an OAM spectrum with weighting coefficients different to that of a purely azimuthal superposition, with a larger contribution from the

*a*

_{0}term in Eq. (16) due to the azimuthal symmetry of the radial orders.

## 5. Conclusion

## References and links

1. | R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. |

2. | M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. |

3. | 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 |

4. | J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. |

5. | H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express |

6. | 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. |

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

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

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

10. | 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 |

11. | 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. | S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. |

13. | 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. |

14. | 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. |

15. | C. Gao, X. Qi, Y. Liu, J. Xin, and L. Wang, “Sorting and detecting orbital angular momentum states by using a Dove prism embedded Mach–Zehnder interferometer and amplitude gratings,” Opt. Commun. |

16. | M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. |

17. | M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

18. | S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express |

19. | L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science |

20. | D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. |

21. | K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of high-order Bessel light beams,” J. Opt. B Quantum Semiclassical Opt. |

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

23. | L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. |

24. | R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express |

25. | 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. |

26. | M. V. Berry, “Paraxial beams of spinning light,” SPIE |

27. | L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A |

28. | S. Orlov and A. Stabinis, “Propagation of superpositions of coaxial optical Bessel beams carrying vortices,” J. Opt. A, Pure Appl. Opt. |

29. | S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. |

30. | R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express |

31. | J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

32. | J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Collinear superposition of multiple helical beams generated by a single azimuthally modulated phase-only element,” Opt. Lett. |

33. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. |

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

35. | J. A. Rodrigo, A. M. Caravaca-Aguirre, T. Alieva, G. Cristóbal, and M. L. Calvo, “Microparticle movements in optical funnels and pods,” Opt. Express |

36. | S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express |

37. | C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(270.0270) Quantum optics : Quantum optics

(050.4865) Diffraction and gratings : Optical vortices

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 21, 2011

Revised Manuscript: June 2, 2011

Manuscript Accepted: July 26, 2011

Published: August 15, 2011

**Citation**

Igor A. Litvin, Angela Dudley, and Andrew Forbes, "Poynting vector and orbital angular momentum density of superpositions of Bessel beams," Opt. Express **19**, 16760-16771 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16760

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### References

- R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]
- M. W. Beijersbergen, L. Allen, H. E. L. O. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and the transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]
- 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]
- J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]
- H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [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(5), 826–829 (1995). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]
- A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002). [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(4), 282–286 (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,” Science 329(5992), 662–665 (2010). [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(22), 5448–5456 (2004). [CrossRef] [PubMed]
- S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakkonen, and J. Turunen, “Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175(4-6), 301–308 (2000). [CrossRef]
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