## Switching with vortex beams in nonlinear concentric couplers

Optics Express, Vol. 15, Issue 20, pp. 12916-12921 (2007)

http://dx.doi.org/10.1364/OE.15.012916

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### Abstract

We demonstrate that a concentric ring coupler can be employed for nonlinear switching of the angular momentum of light carried by an optical vortex. We find different types of stationary vortex states in the nonlinear coupler and study coupling of both power and momentum of an optical vortex launched into one of the rings, demonstrating that the switching takes place well below the collapse threshold. The switching is more effective for the inner-ring excitation since it triggers more sharply and for the powers low enough to avoid the vortex instability and breakup.

© 2007 Optical Society of America

## 1. Introduction

*nonlinear switching*for the beam power and angular momentum operating with ring-like optical vortex beams. Such two-ring annular waveguide couplers can be created, in particular, by a proper modulation of nondiffracting ring Bessel-like optical lattices [4

4. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. **93**, 093904-4 (2004). [CrossRef] [PubMed]

5. X. Wang, Z. Chen, and P. G. Kevrekidis, “Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices,” Phys. Rev. Lett. **96**, 083,904 (2006). [PubMed]

## 2. Model and stationary states

*n*

_{1}and

*n*

_{0}, and both possessing a nonlinear Kerr response. For the sake of simplicity, we took

*n*

_{0}= 1 (base index) and

*n*

_{1}= 2 (so that the index difference is Δ

*n*= 1) in all our calculations. The study is, however, valid for any other values of the indices, since changing the index difference only supposes a change in the spatial scale of the system and a change in the base index only produces a shift in the propagation constants. The scalar optical field

*ψ*(

*r*,

*ϕ*,

*z*) propagating in the

*z*-direction may be described by the normalized equation,

^{2}

_{⊥}is the Laplace operator, and

*V*(

*r*) is the external potential of the double-ring coupler. In the linear regime, this kind of annular structures were studied by means of a conformal transformation [7

7. L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. **35**(5), 737–744 (1999). [CrossRef]

8. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. **11**, 75–83 (1975). [CrossRef]

*u*(

*r*) is the radial profile of the corresponding stationary state,

*β*is its propagation constant, and ℓ is the winding number. Substitution of this function into the model (1) yields to the following

*z*-independent equation,

*P*= 2

*π*∫

*u*

^{2}

*rdr*as a function of the propagation constant

*β*, as shown in the power diagram of Fig. 1(a). Examples of the nonlinear stationary states corresponding to different points in the power diagram are shown in Fig. 1(b). Due to the inherent asymmetry of the coupler, at low powers there exist two branches represented by asymmetric modes, one even with power concentrated on the external ring (

*e-asymmetric*), and one odd (regarded as

*anti-asymmetric*), whose limit at

*P*= 0 correspond to the linear modes of the concentric ring coupler. When power exceeds a particular threshold, there appear two new branches, one

*symmetric*and another asymmetric with power concentrated on the internal ring (

*i-asymmetric*) which join together at point B. The point B is out of the e-asymmetric branch due to asymmetry of the coupler which remove degeneracy of the bifurcation point of a symmetric nonlinear coupler [9].

*z*> 25000, while the mode G breaks at

*z*> 6000. Since the powers of those two states are similar, we conclude that e-asymmetric states are less favored to develop the azimuthal instability, remaining stable for much longer distances. Above the threshold where the symmetric and i-asymmetric states exist, the i-asymmetric states are less unstable: the mode F breaks at

*z*> 200 whereas the mode C breaks at

*z*> 50. For higher powers, all the modes become strongly unstable and they first break into a number of lobes, and then develop the collapse instability (dashed lines in Fig. 1(a)) as usual happens in self-focusing nonlinear Kerr media [3]. Due to the presence of the ring waveguides the beam spreading is stopped and so for low powers the beam remains stable.

## 3. Vortex switching

*u*(

*r*) =

*u*

_{0}(

*r*) to be that of the single-waveguide mode in order to assure a good coupling to the ring waveguide, although there would be no problem to take some other shape like Gaussian. The field is also initially scaled to establish the desired power and then propagates in a nonlinear regime. At each value of

*z*, we characterize the vortex beam by calculating the beam power,

*P*= ∫|ψ|

^{2}

*rdrdϕ*, and the angular momentum,

*L*= Im{∫

_{z}^{*}

*∂*}, which are monitored for each waveguide. To do so, we consider a circular boundary just half way between both the cores and use all the points located in the inner part of this boundary to calculate the power and momentum for the inner core, and those in the external part to calculate those values for the external core.

_{ϕ}ψrdrdϕ*ψ*

_{0}(

*r*,

*ϕ*) =

*u*

_{0}(

*r*)exp(

*iℓϕ*), they are initially proportional,

## 4. Conclusions

## Acknowledgments

## References and links

1. | M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in |

2. | A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in |

3. | Yu. S. Kivshar and G. P. Agrawal, |

4. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. |

5. | X. Wang, Z. Chen, and P. G. Kevrekidis, “Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices,” Phys. Rev. Lett. |

6. | Q. E. Hoq, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed, “Ring-shaped solitons in a dartboard photonic lattice,” Phys. Lett. A |

7. | L. Djaloshinski and M. Orenstein, “Disk and ring microcavity lasers and their concentric coupling,” IEEE J. Quantum Electron. |

8. | M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. |

9. | N. N. Akhmediev and A. Ankiewicz, |

10. | J. F. Nye, |

**OCIS Codes**

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(190.3270) Nonlinear optics : Kerr effect

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 20, 2007

Revised Manuscript: September 4, 2007

Manuscript Accepted: September 18, 2007

Published: September 24, 2007

**Citation**

José R. Salgueiro and Yuri S. Kivshar, "Switching with vortex beams in nonlinear concentric couplers," Opt. Express **15**, 12916-12921 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12916

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### References

- M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E. Wolf, ed., vol. 42, p. 219 (North-Holand, Amsterdam, 2001).
- A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," in Progress in Optics, E. Wolf, ed., vol. 47, pp. 291-391 (North-Holand, Amsterdam, 2005).
- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003), 520 pp.
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093904-4 (2004). [CrossRef] [PubMed]
- X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices," Phys. Rev. Lett. 96, 083,904 (2006). [PubMed]
- Q. E. Hoq, P. G. Kevrekidis, D. J. Frantzeskakis, and B. A. Malomed, "Ring-shaped solitons in a dartboard photonic lattice," Phys. Lett. A 341, 341-155 (2006).
- L. Djaloshinski and M. Orenstein, "Disk and ring microcavity lasers and their concentric coupling," IEEE J. Quantum Electron. 35, 737-744 (1999). [CrossRef]
- M. Heiblum and J. H. Harris, "Analysis of curved optical waveguides by conformal transformation," IEEE J. Quantum Electron. 11, 75-83 (1975). [CrossRef]
- N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, Cornwall, 1997).
- J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Taylor & Francis, 1999).

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