## Observation of parabolic nondiffracting optical fields

Optics Express, Vol. 13, Issue 7, pp. 2364-2369 (2005)

http://dx.doi.org/10.1364/OPEX.13.002364

Acrobat PDF (418 KB)

### Abstract

We report the first experimental observation of parabolic non-diffracting beams, the fourth fundamental family of propagation-invariant optical fields of the Helmholtz equation. We generate the even and odd stationary parabolic beam and with them we are able to produce traveling parabolic beams. It is observed that these fields exhibit a number of unitary in-line vortices that do not interact on propagation. The experimental transverse patterns show an inherent parabolic structure in good agreement with the theoretical predictions. Our results exhibit a transverse energy flow of traveling beams never observed before.

© 2005 Optical Society of America

1. J. Durnin, “Exact solutions for nondiffracting beams. I The scalar theory,”J. Opt. Soc. Am. A **4**, 651–654 (1987). [CrossRef]

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

4. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**, 1493–1495 (2000). [CrossRef]

5. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. **195**, 35–40 (2001). [CrossRef]

6. S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. **4**, S52–S57 (2002). [CrossRef]

7. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

8. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

9. D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, “Anomalous Modes Drive Vortex Dynamics in Confined Bose-Einstein Condensates,” Phys. Rev. Lett. **86**, 564–567 (2001). [CrossRef] [PubMed]

10. I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, “Ginzburg-Landau Theory of Spiral Surface Growth,” Phys. Rev. Lett. **80**, 1770–1773 (1998). [CrossRef]

11. C. O. Weiss, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, “Solitons and vortices in lasers,” Appl. Phys. B **68**, 151–168 (1999). [CrossRef]

7. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

*U*(

**r**)=exp(-

*ik*

_{z}

*z*)

*u*(

**r**

_{t}), where

**r**

_{t}denotes the transverse coordinates. The transverse field

*u*(

**r**

_{t}) can be expressed in terms of the Whittaker integral

*A*(

*φ*) is the angular spectrum of the PIOF and the transverse and longitudinalwave vector components satisfy the relation

*k*

^{2}=

**r**

_{t}=(ξ,

*η*) as

*x*=(

*η*

^{2}-ξ

^{2})/2,

*y*=ξ

*η*, where ξ∊[0,∞), and

*η*∊(-∞,∞), the transverse field distributions of the even and odd stationary PBs are found to be [7

7. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

*k*

_{t})

^{1/2}, Γ

_{1}≡Γ(1/4+

*ia*/2), and Γ

_{3}≡Γ(3/4+

*ia*/2). Having failed to find a better term, we will term the continuous parameter

*α*∊(-∞,∞) the order of the beam. Here, P

_{e}(

*v*;

*a*) and P

_{o}(

*v*;

*a*) are the even and odd real solutions of the parabolic cylinder differential equation (

*d*

^{2}/

*dx*

^{2}+

*x*

^{2}/4-

*a*)P(

*x*;

*a*)=0. Angular spectra for the PBs in Eqs. (2) and (3) are given by

**29**, 44–46 (2004). [CrossRef] [PubMed]

*α*.

6. S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. **4**, S52–S57 (2002). [CrossRef]

*x*≥0,

*z*) for

*α*>0. When observed at fixed transverse planes the phase seems to follow confocal parabolic trajectories. The sign in Eq. (6) defines the traveling direction. For

*α*>0, the transverse intensity pattern consists of well-defined nondiffracting parabolic fringes with a dark parabolic region around the positive

*x*axis [7

**29**, 44–46 (2004). [CrossRef] [PubMed]

*A*

^{±}(

*φ*;

*a*)=

*A*

_{e}(

*φ*;

*a*)±

*iA*

_{o}(

*φ*;

*a*).

*R*and focal distance

*f*resulted in the desired field distribution. Evidently, the aperture of the lens imposes a boundary for the spatial extent of the beam while ideally, its propagation is strictly invariant only if the beam would be of infinite transverse extension.

*φ*. Based on the McCutchen theorem [12

12. G. Indebetouw, “Nondiffractiing optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A **6**, 150–152 (1989). [CrossRef]

*A*

_{e}(

*φ*;

*α*=0) in a variation of the setup originally used by Durnin

*et al.*[1

1. J. Durnin, “Exact solutions for nondiffracting beams. I The scalar theory,”J. Opt. Soc. Am. A **4**, 651–654 (1987). [CrossRef]

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

*et al.*[5

5. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. **195**, 35–40 (2001). [CrossRef]

*r*

_{0}=0.5 mm and thickness Δ

*r*

_{0}=25

*µ*m. The angular modulation was accounted for by a properly exposed photographic film with the angular function

*A*

_{e}(

*φ*;

*α*=0). In this case, additional spatial bounds are determined by the finite thickness of the annular slit. For the odd beam a tilted glass plate, introduced in the half-region

*φ*∊(-

*π*,0), makes up for the required relative phase-shift of

*π*radians. A larger ring (

*r*

_{0}=1.0 mm, Δ

*r*

_{0}=47

*µ*m) was used in this case to make more accurate the positioning placing of the glass plate. The resulting diffractive optical element is then illuminated by a plane wave from a He-Ne 15 mW laser source (λ=632.8 nm). In Fig. 1, the experimental transverse intensity profiles of the even and odd zero-order PBs are observed at different distances along the propagation axis. The resulting patterns clearly exhibit well defined parabolic nodal lines, and are symmetrical along the

*x*and

*y*axes. As expected, the odd mode vanishes along the

*x*axis. Note that within the sampled distance, the beam is practically invariant and changes in the intensity distribution are only minor.

6. S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. **4**, S52–S57 (2002). [CrossRef]

*α*=4. Notice that the value of

*α*is not restricted to take integer values[7

**29**, 44–46 (2004). [CrossRef] [PubMed]

*TU*-(

*η*,ξ ;

*α*=4) is clearly observed in the photographic sequence shown in Fig. 3(a). Note that the energy flows within the parabolic nodal lines and around the positive

*x*axis. The light intensity moves away from the region originally occupied by the beam, this is well observed at the upper section (

*y*>0) the corresponding parabolae have apparently vanished. In the far-field, the intensity pattern of the

*TU*-(

*η*,ξ ;

*α*=4) beam tends to acquire the shape of its angular spectrum, namely a semi-circular ring whose amplitude is proportional to |

*A*

_{e}(

*φ*;

*α*)|

^{2}for

*φ*

*∊*(-

*π*,0) and vanishes elsewhere.

*α*>0 the principal branch of vortices occurs along the positive

*x*axis (ξ=0) at points

*x*

_{j}=

*j*=1,2,…,∞, and

*η*

_{j}are the zeros of the even parabolic function P

_{e}(σ

*η*;-

*a*) in the interval

*η∊*[0,∞). The phase of an ideal PB then has an infinite number of in-line vortices lying along the positive

*x*axis, each with unitary topological charge. The interferogram between the initial traveling beam in Fig. 3 and a plane wave is shown in Fig. 4. As predicted by the zeros of the function P

_{e}(σ

*η*;-

*a*), after the first two vortices, the spacing between phase dislocations becomes nearly constant. Increasing the value of

*a*has the effect of displacing the locus of the first vortex towards larger values of

*x*, and increasing the spacing between adjacent vortices.

14. K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. **4**, S82–S89 (2002). [CrossRef]

15. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature **419**, 145–147 (2002). [CrossRef] [PubMed]

16. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particless,” Opt. Lett. **21**, 827–829(1996). [CrossRef] [PubMed]

17. M. Erdélyi, Z. L. Horváth, G. Szabó, S. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B **15**, 287–292 (1997). [CrossRef]

18. J. Y. Lu and S. He, “Optical X wave communications,” Opt. Commun. **161**, 187–192 (1999). [CrossRef]

## Acknowledgments

## References and links

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

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

3. | S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. |

4. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

5. | J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. |

6. | S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum Semiclass. Opt. |

7. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

8. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A |

9. | D. L. Feder, A. A. Svidzinsky, A. L. Fetter, and C. W. Clark, “Anomalous Modes Drive Vortex Dynamics in Confined Bose-Einstein Condensates,” Phys. Rev. Lett. |

10. | I. S. Aranson, A. R. Bishop, I. Daruka, and V. M. Vinokur, “Ginzburg-Landau Theory of Spiral Surface Growth,” Phys. Rev. Lett. |

11. | C. O. Weiss, M. Vaupel, K. Staliunas, G. Slekys, and V. B. Taranenko, “Solitons and vortices in lasers,” Appl. Phys. B |

12. | G. Indebetouw, “Nondiffractiing optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A |

13. | H. I. Bjelkhagen, |

14. | K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass. Opt. |

15. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

16. | K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particless,” Opt. Lett. |

17. | M. Erdélyi, Z. L. Horváth, G. Szabó, S. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B |

18. | J. Y. Lu and S. He, “Optical X wave communications,” Opt. Commun. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 17, 2005

Revised Manuscript: March 14, 2005

Published: April 4, 2005

**Citation**

Carlos López-Mariscal, Miguel Bandres, Julio Gutiérrez-Vega, and Sabino Chávez-Cerda, "Observation of parabolic nondiffracting optical fields," Opt. Express **13**, 2364-2369 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2364

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### References

- J. Durnin, �??Exact solutions for nondiffracting beams. I The scalar theory,�?? J. Opt. Soc. Am. A 4, 651�??654 (1987). [CrossRef]
- J. Durnin, J. J. Micely Jr., and J. H. Eberly, �??Diffraction-Free Beams,�?? Phys. Rev. Lett. 58, 1499�??1501 (1987). [CrossRef] [PubMed]
- S. Chávez-Cerda, �??A new approach to Bessel beams,�?? J. Mod. Opt. 46, 923�??942 (1999).
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, �??Alternative formulation for invariant optical fields: Mathieu beams,�?? Opt. Lett. 25, 1493�??1495 (2000). [CrossRef]
- J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, �??Experimental demonstration of optical Mathieu beams,�?? Opt. Commun. 195, 35�??40 (2001). [CrossRef]
- S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O�??Neil, I. MacVicar, and J. Courtial, �??Holographic generation and orbital angular momentum of high-order Mathieu beams,�?? J. Opt. B: Quantum Semiclass. Opt. 4, S52�??S57 (2002). [CrossRef]
- M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, �??Parabolic nondiffracting optical wave fields,�?? Opt. Lett. 29, 44�??46 (2004). [CrossRef] [PubMed]
- J. F. Nye and M. V. Berry, �??Dislocations in wave trains, �?? Proc. R. Soc. Lond. A 336, 165�??190 (1974). [CrossRef]
- D. L. Feder, A. A. Svidzinsky, A. L. Fetter and C. W. Clark, �??Anomalous Modes Drive Vortex Dynamics in Confined Bose-Einstein Condensates,�?? Phys. Rev. Lett. 86, 564�??567 (2001). [CrossRef] [PubMed]
- I. S. Aranson, A. R. Bishop, I. Daruka and V. M. Vinokur, �??Ginzburg-Landau Theory of Spiral Surface Growth,�?? Phys. Rev. Lett. 80, 1770�??1773 (1998). [CrossRef]
- C. O. Weiss, M. Vaupel, K. Staliunas, G. Slekys and V. B. Taranenko, �?? Solitons and vortices in lasers,�?? Appl Phys. B 68, 151�??168 (1999). [CrossRef]
- G. Indebetouw, �??Nondiffractiing optical fields: some remarks on their analysis and synthesis,�?? J. Opt. Soc. Am. A 6, 150�??152 (1989). [CrossRef]
- H. I. Bjelkhagen, Silver-halide recording materials (Springer, Berlin, 1993) Ch. 5.
- K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, �??Orbital angular momentum of a high-order Bessel light beam,�?? J. Opt. B: Quantum Semiclass. Opt. 4, S82�??S89 (2002). [CrossRef]
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, �??Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,�?? Nature 419, 145�??147 (2002). [CrossRef] [PubMed]
- K. T. Gahagan and G. A. Swartzlander, Jr., �??Optical vortex trapping of particless,�?? Opt. Lett. 21, 827�??829 (1996). [CrossRef] [PubMed]
- M. Erdélyi, Z. L. Horváth, G. Szabó, S. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, �??Generation of diffraction-free beams for applications in optical microlithography,�?? J. Vac. Sci. Technol. B 15, 287�??292 (1997). [CrossRef]
- J. Y. Lu and S. He, �??Optical X wave communications,�?? Opt. Commun. 161, 187�??192 (1999). [CrossRef]

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