## The Poynting vector and angular momentum of Airy beams

Optics Express, Vol. 16, Issue 13, pp. 9411-9416 (2008)

http://dx.doi.org/10.1364/OE.16.009411

Acrobat PDF (644 KB)

### Abstract

We analyze and describe the evolution of the Poynting vector and angular momentum of the Airy beam as it propagates through space. A numerical approach is used to show the Poynting vector follows the tangent line of the direction of propagation. A similar approach is used to show that while the total angular momentum of the Airy beam is zero, the angular momentum of the main intensity peak and the Airy “tail” are non-zero.

© 2008 Optical Society of America

## 1. Introduction

1. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999). [CrossRef]

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

3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651 (1987). [CrossRef]

4. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**, 264 (1979). [CrossRef]

5. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**, 213901 (2007). [CrossRef]

15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**, 207–209 (2008). [CrossRef] [PubMed]

8. D. M. Greenberger, “Comment on ‘Non-Spreading Wave Packets’,” Am. J. Phys. **48**, 256 (1980). [CrossRef]

*j⃗*=

*⃗r*×

*p⃗*while the second is calculating the angular momentum spectrum of this beam of light [9

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

## 2. Airy solution to the paraxial wave equation

*ϕ*as,

*s*=

_{x}*x*/

*x*

_{0}and

*s*=

_{y}*y*/

*y*

_{0}are normalized transverse coordinates,

*ξ*=

*z*/

*k*(

*x*

_{0}+

*y*

_{0})

^{2}is a normalized propagation distance,

*x*

_{0}and

*y*

_{0}are normalization constants,

*k*=2

*πn*/λ

_{0}and ∇

^{2}

_{τ}is the second partial derivative in the transverse direction.

7. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. **62**, 519–521 (1994). [CrossRef]

*Ai*(

*x*) is the Airy function and is shown in the inset at the center of Fig. 1. This type of beam can be produced experimentally with a specially designed diffraction grating [5

5. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**, 213901 (2007). [CrossRef]

5. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**, 213901 (2007). [CrossRef]

6. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979 (2007). [CrossRef] [PubMed]

*u*(

*s*,

_{x}*s*,0)=

_{y}*Ai*(

*s*)

_{x}*Ai*(

*s*)exp[

_{y}*a*(

*s*+

_{x}*s*)], where

_{y}*a*<<1 and is a positive parameter that limits the infinite energy in the Airy tail, the electric field amplitude becomes

*ξ*-positions when

*a*=0.15 and

*x*=

_{o}*y*=0.1. All of the intensity peaks are changing position as

_{o}*ξ*increases. The main intensity peak in the

*s*-

_{x}*s*plane of the Airy beam travels at 45°, along the line

_{y}*s*=

_{x}*s*, following the trajectories,

_{y}## 3. The Poynting vector of Airy beams

1. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999). [CrossRef]

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

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

*c*is the speed of light. Given a vector potential

*A⃗*=

*(*ε ^
u

*s*,

_{x}*s*,

_{y}*ξ*)exp[

*ik*

^{2}

*ξ*(

*x*

_{0}+

*y*

_{0})], where

*is an arbitrary polarization and*ε ^

*u*(

*s*,

_{x}*s*,

_{y}*ξ*) is the Airy field amplitude given by Eq. (3), we can use the

*E⃗*and

*B⃗*-fields in the Lorenz gauge, as given by Ref. [1

1. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999). [CrossRef]

*S⃗*>. Assuming an

*x̂*-polarized field, <

*S⃗*> becomes [1

1. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999). [CrossRef]

*term in the above equation is the energy flow in the*ξ ^

*-direction which is just proportional to the linear momentum density in that direction. This is typically the main contributing component of*ξ ^

*S⃗*in Gaussian optics. The first term is what we are really interested in here as it contributes a non-zero

*ŝ*- and

_{x}*ŝ*-component and an additional

_{y}*-term to the Poynting vector.*ξ ^

*ŝ*- and

_{x}*ŝ*-components of the Poynting vector for

_{y}*a*=0.15 at

*ξ*=0.025, 0.050, 0.075, and 0.1, respectively. The direction and magnitude of the arrows (shown in red) correspond to the direction and magnitude of the energy flow in the transverse plane. The intensity of the Airy field is shown in the background of each frame to show the direction of the energy flow in relation to the peaks of the Airy beam. The flow of energy of the main peak at

*ξ*>0 is consistently pointed at 45° relative to the

*s*-

_{x}*s*plane at all

_{y}*ξ*-locations. In contrast, the direction of the energy flow for the Airy tails, or the peaks oriented along the horizontal or vertical axis approaches a direction perpendicular to that axis. The direction of the net energy flow is measured, however, to be constant and pointed in the direction that the main peak moves, i.e 45° or along the line

*s*=

_{x}*s*. This is in contrast to the locally varying direction of energy flow.

_{y}*s*or negative

_{x}*s*-direction on each Airy tail; in (b) of the figure the direction starts to turn partially towards the direction of the energy flow of the main Airy peak (45°); and in (c) and (d) as the beam propagates further, the direction swings around even more towards 45°. The change in the Poynting vector can be used to explain why in Ref. [15

_{y}15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**, 207–209 (2008). [CrossRef] [PubMed]

*S⃗*∝

*P⃗*which points more along the tangential directions than the

*ξ*-direction the further the beam propagates.

*a*=0, then the Poynting vector would point along the line

*s*=

_{x}*s*everywhere. This leads to infinite energy propagation as discussed in section 1 above and in Ref. [5

_{y}**99**, 213901 (2007). [CrossRef]

## 4. Angular momentum of the Airy beam

*P⃗*∝

*E⃗*×

*B⃗*from which follows that angular momentum density about the

*-axis is*ξ ^

*S⃗*> taken from Eq. (6) and shown in Fig. 2 is used to calculate the angular momentum in the

*-direction. Figure 3 (a-d) shows the*ξ ^

*-component of the angular momentum density with*ξ ^

*a*=0.15 at ξ=0.025, 0.050, 0.075, and 0.100, respectively. At ξ=0 the computed angular momentum is zero so it is not shown. In Fig. 3 reds are positive values (clockwise), blues are negative values (counter-clockwise), and green is zero. The non-discrete nature of these values will be discussed in the next section.

*ξ*-axis is always zero. The spatial distribution of the angular momentum is changing however, and locally has non-zero values of angular momentum. Not only is the angular momentum changing in the Airy tails, but there are also changes to the angular momentum in the main Airy peak. This change of angular momentum is a torque that corresponds to the force present due to the changing linear momentum. Note that changing the axis about which the angular momentum is calculated would merely alter the angular momentum by a constant.

## 5. Angular momentum spectrum of Airy beams

*spiral imaging*[9

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

14. 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**, 2285–2287 (2003). [CrossRef] [PubMed]

*a*(

_{m}*r*,

*z*)=1/(2

*π*)

^{1/2}∫

^{2π}

_{0}

*u*(

*r*,

*ϕ*,

*z*)exp(-

*imϕ*)

*dϕ*and the energy of each mode,

*m*, is described by

*C*=∫

_{m}^{∞}

_{0}|

*a*(

_{m}*r*,

*ϕ*,

*z*)|

^{2}

*rdr*. The power, or weight, of each angular momentum state for the arbitrary field

*u*is given by [9

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

*u*, is taken from Eq. (3), i.e. the weight of each spiral mode when the field is decomposed in these spiral harmonics, for

*ξ*ranging from 0 to 0.125. Note that the sum of all of the modes at each

*ξ*location is one and the net angular momentum is zero.

*J*, is shown to have non-discrete, non-integer values (positive and negative). Figure 4 shows that this Airy field,

_{ξ}*locally*, has an integer sum of discrete values of orbital angular momentum while the

*total*angular momentum is in fact zero.

## 6. Conclusions

7. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. **62**, 519–521 (1994). [CrossRef]

*ξ*-direction which should have implications when analyzing the velocity of this field. Some form of the velocity, be it phase, energy, or signal velocity, should be changing as the beam propagates and should be investigated in future studies. These beams have promise for applications in optical trapping, imaging, and spectroscopy where a sample might interact with a changing momentum and spatially varying angular momentum.

## Acknowledgments

## References and links

1. | L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999). [CrossRef] |

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

3. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

4. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

5. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. |

6. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

7. | I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. |

8. | D. M. Greenberger, “Comment on ‘Non-Spreading Wave Packets’,” Am. J. Phys. |

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

10. | J. D. Jackson, |

11. | M. Born and E. Wolf, |

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

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

14. | 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. |

15. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: April 17, 2008

Revised Manuscript: May 1, 2008

Manuscript Accepted: June 2, 2008

Published: June 11, 2008

**Citation**

H. I. Sztul and R. R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express **16**, 9411-9416 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-13-9411

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### References

- L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX (Elsevier Science & Technology, 1999) pp. 291-372. [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651 (1987). [CrossRef]
- M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264 (1979). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of Accelerating Airy Beams," Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
- G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979 (2007). [CrossRef] [PubMed]
- I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "Nondispersive accelerating wave packets," Am. J. Phys. 62, 519-521 (1994). [CrossRef]
- D. M. Greenberger, "Comment on ???Non-Spreading Wave Packets???, " Am. J. Phys. 48, 256 (1980). [CrossRef]
- L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt. Express 13, 873-881 (2005). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
- M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).
- L. Allen and M. J. Padgett, "The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum desnity," Opt. Commun. 184, 67-71 (2000). [CrossRef]
- H. I. Sztul and R. R. Alfano, "Double-slit interference with Laguerre-Gaussian beams," Opt. Lett. 31, 999-1001 (2006). [CrossRef] [PubMed]
- 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, 2285-2287 (2003). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008). [CrossRef] [PubMed]

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