## Nonlinear polarization bistability in optical nanowires |

Optics Letters, Vol. 36, Issue 4, pp. 588-590 (2011)

http://dx.doi.org/10.1364/OL.36.000588

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### Abstract

Using the full vectorial nonlinear Schrödinger equations that describe nonlinear processes in isotropic optical nanowires, we show that there exist structural anisotropic nonlinearities that lead to unstable polarization states that exhibit periodic bistable behavior. We analyze and solve the nonlinear equations for continuous waves by means of a Lagrangian formulation and show that the system has bistable states and also kink solitons that are limiting forms of the bistable states.

© 2011 Optical Society of America

2. S. Afshar V. and T. M. Monro, Opt. Express **17**, 2298 (2009). [CrossRef]

3. S. Afshar V., W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, Opt. Lett. **34**, 3577 (2009). [CrossRef]

2. S. Afshar V. and T. M. Monro, Opt. Express **17**, 2298 (2009). [CrossRef]

4. F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. St. J. Russell, Phys. Rev. Lett. **105**, 093904 (2010). [CrossRef] [PubMed]

5. B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B **27**, 956 (2010). [CrossRef]

5. B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B **27**, 956 (2010). [CrossRef]

*j*,

*n*th order propagation constants,

*γ*coefficients calculated using Eq. (3), for elliptical waveguides surrounded by air, with chalcogenide glass (

*γ*values. Figure 1 also shows that

*γ*values than circular shapes, and that in elliptical waveguides the

*γ*values of the modes polarized along the major/minor axes are different, similar to the

*γ*values in waveguides with anisotropic materials. As a consequence, the nonlinear behavior in birefringent waveguides includes anisotropic properties, which we refer to as structurally induced anisotropic nonlinearity.

*γ*coefficients, where Eq. (2) are not necessarily satisfied, in particular for

*z*. Define the following dimensionless variables: where

*b*depends only on the nanowire parameters, whereas

*a*also depends on the total power

*τ*as a “time” variable, then it can be shown from Eq. (5) that

*v*always remains within the physical region. We solve Eqs. (5, 6), in terms of periodic elliptic functions by observing that

*Q*is the fourth degree polynomial

*v*, denoted

*Q*. Since

*Q*generally has at least two real zeroes in the interval

*v*increases, in order to find

*τ*as a function of

*v*, and also the period

*T*: where

*T*is expressible in terms of the complete elliptic integral

*K*, and so can be written as an explicit function of

*a*,

*b*,

*Q*.

*γ*values calculated using Eq. (3). We require

*v*and

*τ*. The period becomes arbitrarily large as the initial values

*a*,

*b*this behavior is very sensitive to the proximity of

*V*in Eq. (13). For other values, such as

*θ*with the boundary condition

*c*is any constant. We have

*a*,

*b*under consideration, namely,

*θ*from Eq. (14), the power

*v*is obtained from Eq. (6), and (5) is then also satisfied. The soliton (14) can propagate in time as a pulse over the length of the waveguide according to the evolution Eq. (1) and maintains its identity as a soliton provided that the boundary conditions remain intact.

*γ*values calculated from Eq. (3) giving

*T*as a function of

*T*is infinite. The insets in Fig. 2 show

*z*, where the periodicity of the power functions and the bistability of

*γ*coefficients that do not necessarily satisfy the relations (2). This results in structurally induced anisotropic nonlinearities for isotropic material-based linearly birefringent waveguides. The model allows continuous wave solutions of three types: steady state solutions, periodic (including bistable) solutions, and kink soliton solutions. The bistable states and the soliton solutions (14) exist only for the extended range of

*γ*coefficients. Properties of the bistable states can in principle be utilized to construct photonic devices such as optical logical gates.

1. | G. P. Agrawal, |

2. | S. Afshar V. and T. M. Monro, Opt. Express |

3. | S. Afshar V., W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, Opt. Lett. |

4. | F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. St. J. Russell, Phys. Rev. Lett. |

5. | B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B |

6. | I. S. Gradshteyn and I. M. Ryzhik, |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(130.4310) Integrated optics : Nonlinear

(190.1450) Nonlinear optics : Bistability

(190.3270) Nonlinear optics : Kerr effect

(190.4360) Nonlinear optics : Nonlinear optics, devices

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 6, 2010

Manuscript Accepted: January 7, 2011

Published: February 15, 2011

**Citation**

Wen Qi Zhang, M. A. Lohe, Tanya M. Monro, and Shahraam Afshar V., "Nonlinear polarization bistability in optical nanowires," Opt. Lett. **36**, 588-590 (2011)

http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-588

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### References

- G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
- S. Afshar V. and T. M. Monro, Opt. Express 17, 2298 (2009). [CrossRef]
- S. Afshar V., W. Q. Zhang, H. Ebendorff-Heidepriem, and T. M. Monro, Opt. Lett. 34, 3577 (2009). [CrossRef]
- F. Biancalana, T. X. Tran, S. Stark, M. A. Schmidt, and P. St. J. Russell, Phys. Rev. Lett. 105, 093904 (2010). [CrossRef] [PubMed]
- B. A. Daniel and G. P. Agrawal, J. Opt. Soc. Am. B 27, 956 (2010). [CrossRef]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

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