## Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength

Optics Express, Vol. 17, Issue 15, pp. 13236-13245 (2009)

http://dx.doi.org/10.1364/OE.17.013236

Acrobat PDF (531 KB)

### Abstract

We generate a broadband “white light” Airy beam and characterize the dependence of the beam properties on wavelength. Experimental results are presented showing that the beam’s deflection coefficient and its characteristic length are wavelength dependent. In contrast the aperture coefficient is not wavelength dependent. However, this coefficient depends on the spatial coherence of the beam. We model this behaviour theoretically by extending the Gaussian-Schell model to describe the effect of spatial coherence on the propagation of Airy beams. The experimental results are compared to the model and good agreement is observed.

© 2009 Optical Society of America

## 1. Introduction

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

2. J. Durnin, “Exact Solutions for Nondiffracting Beams.1. The Scalar Theory,” J. Opt. Soc. Am. **A 4**, 651–654 (1987).
[CrossRef]

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99 (2007). [CrossRef]

4. M. V. Berry and N. L. Balazs, “Non-Spreading Wave Packets,” Am. J. Phys. **47**, 264–267 (1979).
[CrossRef]

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99 (2007). [CrossRef]

5. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nature Photonics **2**, 675–678 (2008).
[CrossRef]

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

7. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**, 207–211 (1998).
[CrossRef]

8. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express **16**, 12880–12891 (2008).
[CrossRef] [PubMed]

9. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007).
[CrossRef] [PubMed]

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

2. J. Durnin, “Exact Solutions for Nondiffracting Beams.1. The Scalar Theory,” J. Opt. Soc. Am. **A 4**, 651–654 (1987).
[CrossRef]

*b*

_{0}, the characteristic length,

*x*

_{0}, and the aperture coefficient,

*a*

_{0}, for this intriguing optical field. We first present an extended Gaussian-Schell model where we use Huygen’s integral to propagate the Airy beam through general ABCD optical elements. After describing the experimental setup, we present, discuss and verify the results.

## 2. Theory

### 2.1. Propagation of the coherent Airy beam

9. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007).
[CrossRef] [PubMed]

*z*is the direction of propagation,

*k*=2

*πn/λ*the wavevector and exp(

*i*(

*ωt-kz*)) the carrier plane wave considered here. The electric,

**E**, and magnetic,

**H**, fields associated with the scalar field

*u*(

_{0}*x,y, z*) are determined via the magnetic vector potential

**A**and the Lorenz gauge conditions,

**x**̂ represents the x-axis unit vector and

*ε*

_{0}and

*µ*

_{0}are the permittivity and permeability of free space. For clarity, we have omitted the time dependence exp(

*iωt*).

*u*(

_{0}*x,y,z*)=

*u*(

_{x}*x,z*)

*u*(

_{y}*y,z*) can be decomposed into two independent parts,

*u*(

_{x}*x,z*) and

*u*(

_{y}*y,z*), each solutions of:

*x*. A symmetric 2D solution can be obtained by replacing the coordinate

*x*with

*y*and multiplying the two scalar fields together.

*z*=0 for the parabola defined by

*z*=

*b*

_{0}

*x*

^{2}), of an exponentially apertured Airy beam is

*Ai*is the Airy function,

*x*

_{0}its characteristic length and a0 the aperture coefficient which determines the beam propagation distance. The associated spatial Fourier transform,

*û*, of this beam is

_{x}*(z=zSLM*); the first exponential term represents the incident Gaussian beam and the second represents the cubic phase mask generated on the SLM. Here

*k*represents the transversal K-space wavevector in the

_{x}*x*direction and corresponds to the

*x*coordinate in the SLM plane. Note that changing the cubic phase term in the SLM plane changes directly the characteristic length

*x*

_{0}of the Airy beam, while the aperture coefficient is given by

*a*

_{0}=

*ω*

^{2}

_{0}/(4

*x*

^{2}

_{0}), where

*ω*

_{0}is the beam waist of the incident Gaussian beam. This equation links the incident beam parameters to the Airy beam parameters.

*x*(

_{m}*z*)=

*z*

^{2}/(4

*k*

^{2}

*x*

^{3}

_{0}) defines the lateral position of the Airy beam. This integral is fully equivalent with the paraxial equation, Eq. 5, and determines the scalar field

*u*at a distance

_{x}*z*as a function of the field at

*z*=0.

*u*

_{1}(

*x*

_{1}) and

*u*

_{2}(

*x*

_{2}) are respectively the fields in the initial and final transverse planes. The coefficients A, B, C and D are obtained through the matrix product of all associated matrices of the optical elements involved [10]. Considering again the exponentially apertured Airy beam,

*AD-BC*=1. We remark that the propagation through an optical system does conserve the intensity profile of the exponentially apertured Airy beam in the same way as the free space propagation does. This result is similar to [11

11. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express **15**, 16719–16728 (2007).
[CrossRef] [PubMed]

*z*after the ABCD element we can determine the path of the beam profile. This path is no longer parabolic and changes to a rational polynomial of the form

*C*=0, this is the case for the experiments presented here.

*A*+

*Cz*=0 where the lateral shift diverges and the beam seems to disappear at +∞ to reappear at -∞. This divergent behaviour is accompanied by the “wavelength” of the Airy oscillations becoming infinitely small rendering the paraxial approximation invalid. It is interesting to remark that this behaviour is also present in the case of Schrödinger’s equation where the equation does maintain its validity. In this case, it is the size of the initial wavefunction that determines the minimal wavelength probed.

### 2.2. Partially coherent Airy beams from Gaussian Schell-model sources

*r*

_{1}and

*r*

_{2}, in an optical field [12

12. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. **67**, 245–250 (1988).
[CrossRef]

*µ*(

**r**

_{1}–

**r**

_{2}) is the spectral degree of coherence which has a value of 1 for perfect coherence. Within the Gaussian Schell-model this spectral coherence is given by

*σ*is the coherence length. The average intensity of the partially coherent beam at a propagation distance

_{µ}*z*is given by

*I*(

*x,y,z*)=

*W*(

**r,r**,

*z*).

*ϕ*the maximal phase shift across the hologram.

*j*

^{3}=−1,

*Ai*(

*x*) is the Airy function and

*σ*. This shows that the deflection coefficient

_{µ}*b*

_{0}and the characteristic length

*x*

_{0}are not affected by the spatial coherence of the generating beam. This means that the general beam properties remain the same regardless of the degree of spatial coherence. In contrast, the aperture coefficient

*a*

_{0}, which determines the distance over which the Airy beam propagates, is affected by the degree of spatial coherence.

*ω*

_{0}and Rayleigh range

*z*and is given by

_{r}12. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. **67**, 245–250 (1988).
[CrossRef]

*σ*

^{2}

*=*

_{µ}*w*

^{2}

_{0}/((

*M*

^{2})

^{2}−1. Using this relation we can determine the aperture coefficient

*a*

_{0}

## 3. Experimental setup

*et al*. [13

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

*x*

_{0}, the deflection coefficient of the beam from a straight path is given by [3

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99 (2007). [CrossRef]

*b*

_{0}, was found in each case by determining the transverse displacement of the beam as it propagated. Images of the beam oriented at 90° and 270° were taken at 2cm intervals along the propagation axis using a CCD that was placed on rails parallel to that axis. This resulted in around 15 data points for each case. Correction for any mismatch between the axis of the camera rails and the beam propagation axis was achieved by taking data for Airy beams with opposite deflections (90° and 270° beams), the data points were then averaged. A parabola was fitted to this combined data and the deflection coefficient was determined to be the

*z*

^{2}coefficient of the fit.

*x*

_{0}and the aperture coefficient,

*a*

_{0}, were found for each beam. The experimental

*x*

_{0}values were determined by fitting the Airy function to cross sections of the two beams (oriented at 90° and 270°) at

*z*=0, the average value was then taken to be the experimentally measured

*x*

_{0}. The

*a*

_{0}values were determined by fitting the Airy function to a cross section of the beams.

## 4. Results and discussion

*π*over the beam using a hologram of side length 0.0146m, similar to Siviloglou

*et al*. [13

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

*z*=0. The cross-section profile of the Airy beam shows the lobes indicating that dispersion is well compensated. It is interesting to note that changing the grating spacing of the SLM hologram allows direct control over the spreading of the wavelengths in the Airy beam. This may be useful for enhancing the difference in the beam trajectories between wavelengths for purposes of optical micromanipulation and even optical sorting using such beams [5

5. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nature Photonics **2**, 675–678 (2008).
[CrossRef]

*z*=0. The 515nm Airy beam had the largest deflection coefficient and as the wavelength was increased the deflection coefficient decreased, see Fig. 3. Error bars of ±10% are used for the deflection coefficient values. The parabolas have been arranged for their minima to overlap at the origin for clear comparison of the beam deflections. It is interesting to note that the x0 value also varies with wavelength (see Table 1), therefore the side lobes of the Airy beam are more closely spaced for shorter wavelength beams. If this was not the case then the deflection coefficient would vary only as λ

^{2}(see Eq. 21), resulting in the opposite trend of the deflection coefficient increasing with increasing wavelength. It is therefore extremely important to take into account the variation of x0 with wavelength. This wavelength dependence of the lobe spacing compared to that seen in a “white light” Bessel beam [14

14. P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express **13**, 6657–6666 (2005).
[CrossRef] [PubMed]

*x*

_{0}values and two calculated

*x*

_{0}values. The column labelled “Eq. 21” has

*x*

_{0}values determined by Eq. 21 using the deflection values shown in Fig. 3. The column labelled “Eq. 16” has

*x*

_{0}values determined using Eq. 16.

*a*

_{0}values based on the

*M*value, the beam spot size incident on the SLM,

^{2}*ω*and parameter

_{slm}*c*

_{0}(described in section 2.2). The value of

*c*

_{0}for our system was calculated to be 785.9m

^{−1}. The experimentally determined

*a*

_{0}values agree within 10% of the calculated values determined using Eq. 20. Table 2 shows the beam waists incident on the SLM

*ω*, the

_{slm}*M*values and the experimental (direct fit) and calculated

^{2}*a*

_{0}values.

*M*

^{2}values were determined for the horizontal axis of each beam after the SLM and Fourier lens. The

*M*

^{2}values were found using a beam profiler (Thorlabs BP104-vis) to determine how the beam waist varied with distance. The Rayleigh range was then used to calculate the

*M*values using Eq. 19. Where

^{2}*ω*

_{0}is the beam waist,

*λ*is the wavelength and

*z*is the Rayleigh range. The

_{r}*M*variability with wavelength probably originates from the different filters used to select the wavelengths.

^{2}## 5. Conclusion

*M*beam quality values of the generating Gaussian beam, we observe a variation of this coefficient with wavelength. Lastly, the Airy beam shape (determined by

^{2}*x*

_{0}and

*b*

_{0}) is not affected by the degree of spatial coherence but its propagation range (determined by

*a*

_{0}) increases with increasing coherence.

## Acknowledgements

## References and links

1. | J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. |

2. | J. Durnin, “Exact Solutions for Nondiffracting Beams.1. The Scalar Theory,” J. Opt. Soc. Am. |

3. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99 (2007). [CrossRef] |

4. | M. V. Berry and N. L. Balazs, “Non-Spreading Wave Packets,” Am. J. Phys. |

5. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nature Photonics |

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

7. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. |

8. | J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

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

10. | A. E. Siegman, |

11. | M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express |

12. | Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. |

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

14. | P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(050.1970) Diffraction and gratings : Diffractive optics

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 11, 2009

Revised Manuscript: July 14, 2009

Manuscript Accepted: July 15, 2009

Published: July 17, 2009

**Citation**

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cižmár, and K. Dholakia, "Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength," Opt. Express **17**, 13236-13245 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-13236

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### References

- J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- J. Durnin, "Exact Solutions for Nondiffracting Beams.1. The Scalar Theory," J. Opt. Soc. Am. A 4, 651-654 (1987). [CrossRef]
- Q1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99 (2007). [CrossRef]
- M. V. Berry and N. L. Balazs, "Non-Spreading Wave Packets," Am. J. Phys. 47, 264-267 (1979). [CrossRef]
- Q2. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008). [CrossRef]
- H. I. Sztul and R. R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express 16, 9411-9416 (2008). [CrossRef] [PubMed]
- Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998). [CrossRef]
- J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, "Self-healing properties of optical Airy beams," Opt. Express 16, 12880-12891 (2008). [CrossRef] [PubMed]
- G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986).
- M. A. Bandres and J. C. Guti’errez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007). [CrossRef] [PubMed]
- Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008). [CrossRef] [PubMed]
- P. Fischer, C. T. A. Brown, J. E. Morris, C. L’opez-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005). [CrossRef] [PubMed]

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