## Generalized coupled nonlinear equations for the analysis of asymmetric two-core fiber coupler

Optics Express, Vol. 11, Issue 2, pp. 116-119 (2003)

http://dx.doi.org/10.1364/OE.11.000116

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### Abstract

Studies have shown that 1^{st} order coupling coefficient dispersion can cause significant effects on the propagation of short pulses in a twin-core coupler. In this paper, we have extended the study to the case of nonidentical cores and investigated the effect of 2^{nd} order dispersive coupling coefficient on switching dynamics. A pair of new coupled nonlinear equations have been presented and analyzed.

© 2002 Optical Society of America

## 1. Introduction

1. P. Shum, K. S. Chiang, and W. Alec Gambling, “Switching Dynamics of Short Optical Pulses in a Nonliner Directional Coupler,” IEEE J. Quantum Electron. **35**, 79–83 (1999). [CrossRef]

3. P. M. Ramos and C. R. Paiva, “All-Optical Pulse Switching in Twin-Core Fiber Couplers with Intermodal Dispersion,” IEEE J.Quantum Electron. **35**, 983–989 (1999). [CrossRef]

^{st}order dispersive coupling coefficient (i.e., intermodal disperson) can give rise to significant effects on pulse propagation in a twin-core directional coupler. The purpose of this paper is to analyze the influence of 2

^{nd}order dispersive coupling coefficient on two-nonidentical-core coupler. To include the effects arising from the dispersion properties of the coupling coefficient, new coupled-mode equations including 2

^{nd}order coupling coefficient dispersion have been presented.

## 2. Analysis

4. K. S. Chiang, “Intermodal Dispersion in Two-Core Optical Fibers,” Opt. Lett. **20**, 997–999 (1995). [CrossRef] [PubMed]

^{nd}order dispersive coupling coefficient and govern the pulse dynamics in two-nonidentical-core couplers with Kerr nonlinearity. The two parallel nonidentical cores embedded in a common cladding, are different in their radii denoted by ρ

_{1}and ρ

_{2}, but have the same refractive index.

_{1}(Z,T) and a

_{2}(Z,T) represent the normalized varying envelopes of the pulses carried by the fundamental modes of the two cores in isolation.

*k*″

_{1(2)}denotes the group velocity dispersion of the large (small) core individually and ν

_{g1(2)}describes the group velocity.

*C*

_{12}and

*C*

_{21}are the coupling coefficients.

*C*′

_{12}and

*C*′

_{21}account for the intermodal dispersion, and the terms that contain

*C*″

_{12}and

*C*″2l describe the 2

^{nd}order dispersive coupling coefficients. To simplify the two equations, a normalized coordinate system that moves with the waves at the group velocity of small core is adopted, i.e.,

_{0}is width of the input pulse. Then the coupled-mode nonlinear equations for two-nonidentical-core couplers which contain 2

^{nd}order dispersive coupling coefficient are obtained as follows:

5. H. Ghafouri-Shiraz, P. Shum, and M. Nagata, “A Novel Method for Analysis of Soliton Propagation in Optical Fibers,” IEEE J.Quantum Electron. **31**, 190–200 (1995). [CrossRef]

*a*

_{1}and

*a*

_{2}are expanded in a Fourier series and substituted into (3) and (4), then a set of first-order partial differential equations are obtained. The set of equations obtained can be solved by Runge-Kutta method in the frequency domain. Employing the inverse Fourier transform, we can get the time domain solution.

_{0}=10fs, ρ

_{1}=3.25µm and ρ

_{2}/ ρ

_{1}=0.9..

^{nd}order coupling coefficient dispersion, we launch a pulse into the large core. The initial conditions are given by:

*a*

_{1}(0,

*T*) =

*A*sec

*h*(

*T*) and

*a*

_{2}(0,

*T*) = 0, where A is the amplitude of input pulse. We vary the amplitude of input pulse and study the shift of threshold power point without/ with the presence of 2

^{nd}order coupling coefficient dispersion.

^{nd}order coupling coefficient dispersion, i.e., setting

*C*″

_{12}=

*C*″

_{21}=0. It is well known that a nonlinear fiber coupler can function as an all-optical switch, and the pulse can couple back and forth between the two cores under the threshold value. Above it the pulse will essentially trap in the input core and cannot couple to the other core.

_{1}(Z,T)|

_{2}and V(Z,T)=|a

_{2}(Z,T)|

^{2}. At A = 4.0, the pulse can still couple back and forth between the two cores with little distortion, while at A = 4.5, the pulse stays mainly in one core. These phenomena form the basis of an intensity-dependent optical switch. Our simulation results show that there is a great power transfer from large core to small core for A = 4.0 to 4.3. At A = 4.4, the pulse is almost trapped in the input core. Thus, we can conclude that the amplitude threshold is between 4.3 and 4.4.

^{nd}order coupling coefficient dispersion on pulse propagation. Similarly we vary the amplitude of the input pulse and the propagation characteristics are shown in Fig. 2 (a) and (b) for amplitude A = 4.5 and 4.9 respectively. At A =4.5, the pulse can still couple between the two waveguides. For A = 4.9, the pulse stays mainly in the input core.

*C*″

_{12}, and

*C*″

_{21}. The results show that a greater coupling of power from core 1 to core 2 for A between 4.5 and 4.7. For A=4.8, less power can be coupled. At A = 4.9, little power can be transferred from core 1 to core 2. Therefore the amplitude threshold level is between 4.8 and 4.9. Compared with Fig. 1, the 2

^{nd}order dispersive coupling coefficient has the effect of increasing the threshold amplitude above which the input pulse cannot couple to the other core and only stay in the input core.

## 3. Conclusion

^{nd}order dispersive coupling coefficient. In particular, we have studied the effects of the 2

^{nd}order coupling coefficient on soliton switching dynamics in a two-nonidentical-core coupler. We have found that 2

^{nd}coupling-coefficient dispersion can increase the threshold amplitude and they should be an important factor of consideration in the design of two-nonidentical-core nonlinear directional couplers.

## References

1. | P. Shum, K. S. Chiang, and W. Alec Gambling, “Switching Dynamics of Short Optical Pulses in a Nonliner Directional Coupler,” IEEE J. Quantum Electron. |

2. | K. S. Chiang, “Coupled-mode equation for pulse switching in parallel waveguieds,” IEEE J. Quantum Electron. |

3. | P. M. Ramos and C. R. Paiva, “All-Optical Pulse Switching in Twin-Core Fiber Couplers with Intermodal Dispersion,” IEEE J.Quantum Electron. |

4. | K. S. Chiang, “Intermodal Dispersion in Two-Core Optical Fibers,” Opt. Lett. |

5. | H. Ghafouri-Shiraz, P. Shum, and M. Nagata, “A Novel Method for Analysis of Soliton Propagation in Optical Fibers,” IEEE J.Quantum Electron. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 3, 2002

Revised Manuscript: January 11, 2003

Published: January 27, 2003

**Citation**

Min Liu and P. Shum, "Generalized coupled nonlinear equations for the analysis of asymmetric two-core fiber coupler," Opt. Express **11**, 116-119 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-2-116

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### References

- P. Shum, K. S. Chiang and W.Alec Gambling, �??Switching Dynamics of Short Optical Pulses in a Nonliner Directional Coupler,�?? IEEE J. Quantum Electron. 35, 79-83 (1999). [CrossRef]
- K. S. Chiang, �??Coupled-mode equation for pulse switching in parallel waveguieds,�?? IEEE J. Quantum Electron. 33, 950-954 (1997). [CrossRef]
- P. M. Ramos and C. R. Paiva, �??All-Optical Pulse Switching in Twin-Core Fiber Couplers with Intermodal Dispersion,�?? IEEE J.Quantum Electron. 35, 983-989 (1999). [CrossRef]
- K. S. Chiang, �??Intermodal Dispersion in Two-Core Optical Fibers,�?? Opt. Lett. 20, 997-999 (1995). [CrossRef] [PubMed]
- H.Ghafouri-Shiraz, P.Shum and M.Nagata, �??A Novel Method for Analysis of Soliton Propagation in Optical Fibers,�?? IEEE J.Quantum Electron. 31, 190-200 (1995). [CrossRef]

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