## Direct measurement of the skew angle of the Poynting vector in a helically phased beam

Optics Express, Vol. 14, Issue 25, pp. 11919-11924 (2006)

http://dx.doi.org/10.1364/OE.14.011919

Acrobat PDF (337 KB)

### Abstract

We measure the local skew angle of the Poynting vector within a helically-phased, exp(*ilφ*), beam using a Shack Hartmann wavefront sensor. It is the skew angle of the Poynting vector with respect to the beam axis that gives rise to the orbital angular momentum of a light beam. We confirm that this skew angle is *l*/*kr*, corresponding to an orbital angular momentum of *lħ* per photon. Measurement of orbital angular momentum in this way is an alternative to interferometric techniques giving a non-ambiguous result to both the magnitude and sign of *l* from a single measurement, without any restriction on the optical bandwidth.

© 2006 Optical Society of America

## 1. Introduction

*ilϕ*) possess an orbital angular momentum of

*lħ*per photon [1], where

*ϕ*is the azimuthal angle and

*l*is an integer. Unlike spin angular momentum that is linked to circular polarization, and ultimately to the photon spin, the orbital angular momentum is solely a function of the form of the optical phase front. This azimuthal phase structure can be studied with an interferometer, where the azimuthal phase term results in the characteristic spiral interference pattern with |

*l*| radial fringes [2

2. M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun.106, 161–166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996). [CrossRef]

*l*-fold rotational symmetry can also be utilized within a Mach Zhender interferometer to sort an input beam between two or more outputs depending on the value of |

*l*| [3

3. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon,” Phys. Rev. Lett. **92**, 013601 (2004). [CrossRef] [PubMed]

*l*|. Determining the sign of the orbital angular momentum, clockwise +ve

*l*or anticlockwise −ve

*l*requires further measurements.

*l*. Rapid, simple and non-ambiguous determination of the sign of

*l*and azimuthal direction of the momentum is particularly important in optical tweezing experiments [5

5. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef] [PubMed]

6. 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 (2002). [CrossRef] [PubMed]

## 2. The Poynting vector in helically phased beams

*u*(

*r,ϕ,z*)=

*u*(

*r,z*) exp(

*ilϕ*), of which Laguerre-Gaussian and high order Bessel beams are both examples, has

*r*-,

*ϕ*- and

*z*-components of linear momentum density,

**p**=

*ε*

_{0}

**E**×

**B**, given by [1] (where

**E**and

**B**are the electric and magnetic field strengths respectively, and

*ε*

_{0}is the dielectric permittivity),

*ω*and

*k*are the angular frequency and the wavenumber of the light and

*z*

_{r}is the Rayleigh range of the Gaussian beam. For a well-collimated beam,

*p*

_{r}≈0, and

*P*

_{ϕ}/

*P*

_{z}gives the skew angle,

*γ*, of the Poynting vector with respect to the beam axis to be

*γ*=

*l*/

*kr*[7

7. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun.121, 36–40 (1995), L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef]

*l*=1,

*λ*=632nm and

*r*=1mm,

*γ*is only 0.1 milliradians! Resolving the linear momentum of the photon

*ħk*, into its corresponding azimuthal and axial components and multiplying by the radius,

*r*, is compatible with the orbital angular momentum around the beam axis being

*lħ*per photon. The skew angle of the Poynting vector gives rise to an azimuthal shift of the beam behind an linear obstruction [8] and/or a shift in the inference pattern produced by Young’s double slits [9

9. H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. **31**, 999–1001 (2006). [CrossRef] [PubMed]

## 3. Generation of helically phased beams and the Shack Hartmann wavefront sensor

10. HoloEye Photonics AG, www.holoeye.com/spatial_light_modulators-technology.html

*l*-forked diffraction grating [11], the first-order diffracted beam has helical phase fronts described by exp(

*ilϕ*). The Laguerre-Gaussian (

*LG*) modes are a convenient basis set from which to describe beams with helical phase fronts, and are given by [12

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

*w*(

*z*) is the radius of the beam at position

*z*, where

*p*and

*l*are mode indices, where

*l*corresponds to the azimuthal phase terms and

*p*is the number of radial nodes.

*4f*imaging system (where

*f*is the focal length of one of the lenses) was used to image the plane of the SLM and control the size of the beam incident on the lenslet array. In order to generate pure

*LG*beams, it was necessary to control both the intensity and the phase of the incident light. This was achieved by adjusting the local contrast on the SLM to shape the intensity of the light beam accordingly [13

13. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Y. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. **28**, 1185–1187 (2003). [CrossRef] [PubMed]

14. J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Knotted threads of darkness,” Nature **432**, 165 (2004). [CrossRef] [PubMed]

## 4. Results

*LG*modes with

*l*values ranging between +5 and -5. The length and direction of each vector arrow corresponds to the movement of the spot produced by a single lenslet within the Shack Hartman array. Where the intensity was too low to make an accurate measurement of the spot position, the arrow has been omitted.

*γ*=

*l*/

*kr*.

*l*value of a Laguerre-Gaussian mode. For each lenslet within the array, the expression for the skew angle of the Poynting vector can be rearranged to give

*l*=

*γkr*. The value of

*γkr*can then be averaged over all the lenslets to give a measure of

*l*and hence the orbital angular momentum per photon, see Fig. 4.

## 5. Discussion and conclusions

*ilϕ*), beam can be measured using a Shack Hartmann wavefront sensor. We have confirmed that this skew angle is in close agreement with that expected, corresponding to an orbital angular momentum of

*lħ*per photon. Beyond a simple confirmation of the structure of helically phased modes, the technique is particularly suited for making unambiguous measurements of the sense of the orbital angular momentum in experiments based on momentum transfer. This non-interferometric technique requires only a single, exposure and is hence ideally suited to use with pulsed or white light sources. It is also applicable to analysis of more complex modal superpositions with differing values of

*l*, such as elliptical [15

15. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. **144**, 210–213 (1997). [CrossRef]

16. C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express , **13**, 1749–1760 (2005). [CrossRef] [PubMed]

*l*.

## Acknowledgments

## References and links

1. | L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Progress in Optics 39, 291–372 (1999), M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Physics Today, 57, 35–40 (2004). |

2. | M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun.106, 161–166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996). [CrossRef] |

3. | J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon,” Phys. Rev. Lett. |

4. | e.g. “Adaptive Optics for Astronomical Telescopes.” John W. Hardy, Oxford University Press (1998). |

5. | D. G. Grier, “A revolution in optical manipulation,” Nature |

6. | 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. |

7. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun.121, 36–40 (1995), L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000). [CrossRef] |

8. | J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. |

9. | H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. |

10. | HoloEye Photonics AG, www.holoeye.com/spatial_light_modulators-technology.html |

11. | V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. |

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

13. | I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Y. Bekshaev, “Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam,” Opt. Lett. |

14. | J. Leach, M. Dennis, J. Courtial, and M. Padgett, “Knotted threads of darkness,” Nature |

15. | J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. |

16. | C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express , |

**OCIS Codes**

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(230.6120) Optical devices : Spatial light modulators

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Atmospheric and ocean optics

**History**

Original Manuscript: August 29, 2006

Manuscript Accepted: November 11, 2006

Published: December 11, 2006

**Citation**

Jonathan Leach, Stephen Keen, Miles J. Padgett, Christopher Saunter, and Gordon D. Love, "Direct measurement of the skew angle of the Poynting vector in a helically phased beam," Opt. Express **14**, 11919-11924 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11919

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### References

- L. Allen, M. J. Padgett and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999), M. Padgett, J. Courtial, and L. Allen, "Light's Orbital Angular Momentum," Physics Today, 57, 35-40 (2004).
- M. Harris, C. A. Hill and J. M. Vaughan, "Optical helices and spiral interference fringes," Opt. Commun. 106, 161-166 (1994), M. Padgett, J. Arlt, N. Simpson and L. Allen, "An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes," Am. J. Phys. 64,77-82 (1996). [CrossRef]
- J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold and M. J. Padgett, "Interferometric Methods to Measure Orbital and Spin, or the Total Angular Momentum of a Single Photon," Phys. Rev. Lett. 92, 013601 (2004). [CrossRef] [PubMed]
- e.g. "Adaptive Optics for Astronomical Telescopes." John W. Hardy, Oxford University Press (1998).
- D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003). [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 (2002). [CrossRef] [PubMed]
- M. J. Padgett and L. Allen, "The Poynting vector in Laguerre-Gaussian laser modes," Opt. Commun. 121, 36-40 (1995), L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum density," Opt. Commun. 184, 67-71 (2000). [CrossRef]
- J. Arlt, "Handedness and azimuthal energy flow of optical vortex beams," J. Mod. Opt. 50, 1573-1580 (2003).
- H. I. Sztul and R. R. Alfano, "Double-slit interference with Laguerre-Gaussian beams," Opt. Lett. 31, 999-1001 (2006). [CrossRef] [PubMed]
- HoloEye Photonics AG, www.holoeye.com/spatial_light_modulators-technology.html
- V. Yu. Bazhenov, M. V. Vasnetsov and M. S. Soskin, "Laser beams with screw dislocations in their wavefronts," JETP Lett. 52,429-431 (1990).
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
- I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Y. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003). [CrossRef] [PubMed]
- J. Leach, M. Dennis, J. Courtial and M. Padgett, "Knotted threads of darkness," Nature 432, 165 (2004). [CrossRef] [PubMed]
- J. Courtial, K. Dholakia, L. Allen and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997). [CrossRef]
- C. Alonzo, P. J. Rodrigo, and J. Glückstad, "Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express, 13, 1749-1760 (2005). [CrossRef] [PubMed]

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