## Rotating beams in isotropic optical system

Optics Express, Vol. 18, Issue 4, pp. 3568-3573 (2010)

http://dx.doi.org/10.1364/OE.18.003568

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### Abstract

Based on the ray transformation matrix formalism, we propose a simple method for generation of paraxial beams performing anisotropic rotation in the phase space during their propagation through isotropic optical systems. The widely discussed spiral beams are the particular case of these beams. The propagation of these beams through the symmetric fractional Fourier transformer is demonstrated by numerical simulations.

© 2010 OSA

## 1. Introduction

1. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. **102**(3-4), 336–350 (1993). [CrossRef]

5. A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. **31**(14), 2199–2201 (2006). [CrossRef] [PubMed]

6. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. **60**(9), 1168–1177 (1970). [CrossRef]

**T**, which relates the position

## 2. Isotropic and anisotropic phase-space rotators

9. G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. **30**(10), 1207–1209 (2005). [CrossRef] [PubMed]

10. T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. **32**(10), 1226–1228 (2007). [CrossRef] [PubMed]

*φ*is limited:

*φ*defined by the propagation distance

*z*and the refractive index gradient

*g*:

*φ*

*= gz*[11]. Note that in this case

*φ*can cover the interval of several periods of 2

*π*. Other fractional Fourier transformers can be constructed using one or two spherical lenses.

**T**will be denoted by

## 3. Design of rotating beams

12. M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A **23**(8), 1875–1883 (2006). [CrossRef]

13. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. **30**(12), 1461–1463 (2005). [CrossRef] [PubMed]

14. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. **33**(17), 1603–1629 (2000). [CrossRef]

15. E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. **6**(5), S157–S161 (2004). [CrossRef]

*γ,−γ*) and therefore at any pair of angles

*φ*. It means that all the modes in this decomposition accumulate the same Gouy phase during the propagation through an IOS or the corresponding symmetric fractional Fourier transformer. Similarly, a linear superposition of the modes

*γ*.

*γ*as

*v*indicates the velocity of the phase-space rotation associated with

*φ*, we can rewrite

*ϕ*as

*m*and

*n*, which satisfy the relationis also an eigenfunction of the canonical integral transform

*v*is irrational we have a trivial case where only one mode

*n*is fixed. Similar expression for fixed

*m*and arbitrary

*n*can be obtained for

*v*indicates the direction of the rotation, while its absolute value corresponds to the number of complete loops which beam makes during the 2π-interval of

*φ*. It can be bigger or less than one if a beam has certain symmetry in the phase space. The velocity of rotation is defined by the indices of any pair of modes

*k*and

*l*. Note that a linear combination of any two modes

## 3. Conclusion

*v*. These beams can be generated using spatial light modulators. The application of these beams for light-matter interaction is under investigation. We only mention that spiral beams have been found useful for optical trapping.

## Acknowledgments

## References and links

1. | E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. |

2. | E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. |

3. | E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. |

4. | A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. |

5. | A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. |

6. | S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. |

7. | R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A |

8. | J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A |

9. | G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. |

10. | T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. |

11. | H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, |

12. | M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A |

13. | T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. |

14. | A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. |

15. | E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. |

16. | T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.2590) Fourier optics and signal processing : ABCD transforms

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

(070.3185) Fourier optics and signal processing : Invariant optical fields

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: October 11, 2009

Revised Manuscript: January 19, 2010

Manuscript Accepted: January 28, 2010

Published: February 4, 2010

**Citation**

Tatiana Alieva, Eugeny Abramochkin, Ana Asenjo-Garcia, and Evgeniya Razueva, "Rotating beams in isotropic optical system," Opt. Express **18**, 3568-3573 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3568

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### References

- E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102(3-4), 336–350 (1993). [CrossRef]
- E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125(4-6), 302–323 (1996). [CrossRef]
- E. Abramochkin and V. Volostnikov, “Spiral light beams,” Phys. Usp. 47(12), 1177–1203 (2004). [CrossRef]
- A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31(6), 694–696 (2006). [CrossRef] [PubMed]
- A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31(14), 2199–2201 (2006). [CrossRef] [PubMed]
- S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]
- R. Simon and K. B. Wolf, “Structure of the set of paraxial optical systems,” J. Opt. Soc. Am. A 17(2), 342–355 (2000). [CrossRef]
- J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Optical system design for orthosymplectic transformations in phase space,” J. Opt. Soc. Am. A 23(10), 2494–2500 (2006). [CrossRef]
- G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005). [CrossRef] [PubMed]
- T. Alieva and M. J. Bastiaans, “Orthonormal mode sets for the two-dimensional fractional Fourier transformation,” Opt. Lett. 32(10), 1226–1228 (2007). [CrossRef] [PubMed]
- H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
- M. J. Bastiaans and T. Alieva, “First-order optical systems with unimodular eigenvalues,” J. Opt. Soc. Am. A 23(8), 1875–1883 (2006). [CrossRef]
- T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” Opt. Lett. 30(12), 1461–1463 (2005). [CrossRef] [PubMed]
- A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. Math. Gen. 33(17), 1603–1629 (2000). [CrossRef]
- E. Abramochkin and V. Volostnikov, “Generalized Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(5), S157–S161 (2004). [CrossRef]
- T. Alieva and A. Barbé, “Self-fractional Fourier images,” J. Mod. Opt. 46, 83–99 (1999).

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