## Optical Snakes and Ladders: Dispersion and nonlinearity in microcoil resonators

Optics Express, Vol. 16, Issue 20, pp. 16247-16254 (2008)

http://dx.doi.org/10.1364/OE.16.016247

Acrobat PDF (245 KB)

### Abstract

Microcoil resonators are a radical new geometry for high Q resonators with unique linear features. In this paper I briefly summarise their linear properties before extending the analysis to nonlinear interactions in microcoil resonators. As expected such nonlinear resonators are bistable and exhibit hysteresis. Finally I discuss possible applications and extensions to such resonators.

© 2008 Optical Society of America

## 1. Introduction

1. M. Sumetsky, “Optical fiber microcoil resonator,” Optics Express **12**, 2303 (2004). [CrossRef] [PubMed]

2. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Optics Express **13**, 4331 (2005). [CrossRef] [PubMed]

## 2. Theoretical model

2. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Optics Express **13**, 4331 (2005). [CrossRef] [PubMed]

*n*turns in the linear regime is described by the following set of equations:

*A*

*(*

_{k}*s*) is the slowly varying amplitude of the electric field in the

*k*th coil at a distance

*s*round the coil and

*κ*is the usual coupling constant between two adjacent waveguides. Since at the end of each loop the output of the

*k*coil must equal the input of the

*k*+1 coil this implies that

*β*is the propagation constant of the mode and

*l*is the length of each coil. The input to the coil is given by

*A*

_{1}(0) while the output is given by

*A*

*(*

_{n}*l*)

*e*

*. The transmission coefficient is defined as:*

^{iβl}*T*|=1 while one can describe the effects of loss through an imaginary propagation constant

*β*=2

*π*/

*λ*+

*iα*where

*α*represents the loss. Fig. 1 shows a typical transmission spectrum for a lossy OMR with 8 coils and with

*κ*=5mm

^{-1}and

*l*=1mm. Introducing loss into the system not only makes the model more realistic but also provides a useful guide to understanding the resonant behaviour of the OMR. In the lossy case resonances lead to light having a longer effective path length and so experiences more loss leading to a drop in transmission.

*A*is a column vector of the amplitudes

*A*

*and*

_{k}*K*is the formal matrix exponent of the coupling matrix in Eq. (1). Eq. (4) and Eq. (5) can be solved simultaneously giving

2. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Optics Express **13**, 4331 (2005). [CrossRef] [PubMed]

## 2.1. Group velocity and dispersion of Microcoil resonators

**13**, 4331 (2005). [CrossRef] [PubMed]

*t*

*as the derivate of the phase of the transmission with frequency. Writing the transmission as*

_{d}*T*=

*t*exp[

*iϕ*(

*ω*)] the group delay is then given by

*l*divided by the time delay.

*c*/

*n*

*). The first thing to notice about the group velocity is that typically it is greater than unity. This shows that for most wavelengths light takes a short cut through the resonator by coupling from one coil to the next (i.e. the same as going up a ladder) and so the length it “sees” is less than the physical resonator length. This can also be seen in the transmission spectrum where off resonance the transmission is greater than exp(-2*

_{eff}*αl*) which is the expected loss of the OMR. In the lossy case one can define an effective length by

*t*

^{2}is the absolute value squared of the transmission.

*β*and

*κ*for which

*A*

_{1}(0)=

*A*

*(*

_{n}*l*)=0 and |

*A*| > 0) which store energy indefinitely in the OMR. The regions of negative group velocity are harder to interpret physically but correspond to regions where the transmitted phase decreases with frequency.

6. F. Xu, P. Horak, and G. Brambilla, “Optimised design of microcoil resonators,” IEEE J. Light. Tech. **25**, 1561–1567 (2007). [CrossRef]

## 3. Nonlinear Microcoil resonators

*n*nonlinear waveguide arrays with a Kerr nonlinearity[8, 9

9. A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays,” Phys. Rev. E **53**, 1172–1189 (1996). [CrossRef]

*γ*is the usual nonlinear coefficient and is proportional to the appropriate element of the

*χ*

^{3}suseptability tensor and

*A*

*(*

_{κ}*s*) represents the slowly varying amplitude of the light in the

*k*th coil as before. Here I have explicitly included the loss

*α*in the equations. Again the boundary conditions given by Eq. (2) hold. Note that renormalising the field amplitudes by

*A*

*→1/√*

_{k}*γA*

*| sets the effective nonlinear coefficient to unity which is done in the numerical simulations. Writing the formal solution to Eq. (13) as*

_{k}**A**(

*l*)=

*N*(

**A**)

**A**(0) the only possible self consistent solutions obey

*B*is given by Eq. (2). The numerical approach used to find these solutions is discussed in Appendix 1.

10. K. Ogusu, “Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator,” IEEE J. Quant. Elect. **32**, 1537–1543 (1996). [CrossRef]

11. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. **31**, 341–343 (2006). [CrossRef] [PubMed]

*λ*=1.530

*µ*m. While having zero transmission is not necessary it does maximise the contrast between the high and low transmission states.

*µ*m and 2.0

*µ*m which implies that for a tapered SMF fibre γ≈0.03W-1 and so the maximum power required to see the hysteresis behaviour in Fig. 3(c) would be 16W which is easily achievable with a pulsed microsecond source. This power could be reduced considerable by using highly nonlinear fibres or by optimising the micro-coil design (which is the subject of ongoing work).

## 4. Discussion and conclusions

## 5. Appendix 1: Numerical solutions of the nonlinear coupled mode equations

*f*:

*ℂ*

*→*

^{n}*ℂ*

*given by*

^{n}**A**

^{n}-

**A**

^{n}^{-1}|→0 as

*n*→∞.

*N*=

*K*and when

**A**

^{0}=

**A**

*this method results in*

_{in}*BK*‖ < 1. This solution technique is attractive since when

**A**

^{0}=

**A**

*it mimics the physics of the situation since each iteration corresponds to another round trip of the light round the coil and so the stability of this procedure should be similar to that of the physical system. Numerical I have found this procedure to be robust but slow. However it fails completely in a number of cases such as when there is gain in the system or the solution is unstable. In these cases other methods need to be used.*

_{in}**X**satisfies

**X**-

*f*(

**X**)=0 which can of course be rewritten as 2

*n*real equations (the real and imaginary parts of each component of the vector) and which are easily reduced to 2

*n*-2 real equations since the input power is given. Furthermore there are 2

*n*-2 unknowns being the real and imaginary parts of

*A*

*(0). This then forms a set of 2*

_{κ}*n*-2 nonlinear equations in 2

*n*-2 unknowns and so any standard numerical method can be used to solve them. In order to obtain the solutions presented here I have used a modified Newton’s method[12

12. J. Nocedal and S. J. Wright, *Numerical Optimisation*, 1st ed. (Springer-Verlag, New York, 1999). [CrossRef]

**X**-

*f*(

**X**)|

^{2}is less than 10

^{-15}for all points which is usually achieved by between 1 and 20 iterations of Newton’s method.

**X**-

*f*(

**X**)=0 as 2

*n*-1 real equations (the additional equation is for the transmission amplitude since only the phase is known) and solving it using the same Newton’s method as described previously.

## Acknowledgments

## References and links

1. | M. Sumetsky, “Optical fiber microcoil resonator,” Optics Express |

2. | M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Optics Express |

3. | M. Sumetsky, “Basic elements for microfiber photonics: Micro/nanofibers and microfibre coil resonators,” IEEE J. Light. Tech. |

4. | F. Xu and G. Brambilla, “Manufacture of 3-D microfiber coil resonators,” IEEE Photonics Technology Letters |

5. | F. Xu and G. Brambilla, “Embedding optical microfiber coil resonators in Teflon,” Opt. Lett. |

6. | F. Xu, P. Horak, and G. Brambilla, “Optimised design of microcoil resonators,” IEEE J. Light. Tech. |

7. |
It should be noted that there are at least two misprints in Sumetsky’s Optics Express paper [2]. Firstly in Eq. 1 there is a factor |

8. | G. P. Agrawal, |

9. | A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays,” Phys. Rev. E |

10. | K. Ogusu, “Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator,” IEEE J. Quant. Elect. |

11. | Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. |

12. | J. Nocedal and S. J. Wright, |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 5, 2008

Revised Manuscript: August 27, 2008

Manuscript Accepted: September 1, 2008

Published: September 26, 2008

**Citation**

N. G. Broderick, "Optical Snakes and Ladders: Dispersion and nonlinearity in microcoil resonators," Opt. Express **16**, 16247-16254 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-16247

Sort: Year | Journal | Reset

### References

- M. Sumetsky, "Optical fiber microcoil resonator," Optics Express 12,2303 (2004). [CrossRef] [PubMed]
- M. Sumetsky, "Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation," Optics Express 13,4331 (2005). [CrossRef] [PubMed]
- M. Sumetsky, "Basic elements for microfiber photonics: Micro/nanofibers and microfibre coil resonators," IEEE J. Light. Tech. 26,21-27 (2008). [CrossRef]
- F. Xu and G. Brambilla, "Manufacture of 3-D microfiber coil resonators," IEEE Photonics Technology Letters 19,1481-1483 (2007). [CrossRef]
- F. Xu and G. Brambilla, "Embedding optical microfiber coil resonators in Teflon," Opt. Lett. 32,2164-2166 (2007). [CrossRef] [PubMed]
- F. Xu, P. Horak, and G. Brambilla, "Optimised design of microcoil resonators," IEEE J. Light. Tech. 25,1561- 1567 (2007). [CrossRef]
- It should be noted that there are at least two misprints in Sumetsky???s Optics Express paper [2]. Firstly in Eq. 1 there is a factor i missing on the L.H.S. Secondly in the equation for the transmission a factor of exp(i® l) is missing. Both of these misprints do not occur in his original paper.
- G. P. Agrawal, Nonlinear Fibre Optics, 3rd ed. (Academic Press, San Diego, 2001).
- A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping, soliton interactions and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53,1172-1189 (1996). [CrossRef]
- K. Ogusu, "Dynamic Behavior of Reflection Optical Bistability in a Nonlinear Fiber Ring Resonator," IEEE J. Quant. Elect. 32,1537-1543 (1996). [CrossRef]
- Q. Xu and M. Lipson, "Carrier-induced optical bistability in silicon ring resonators," Opt. Lett. 31,341-343 (2006). [CrossRef] [PubMed]
- J. Nocedal and S. J. Wright, Numerical Optimisation, 1st ed. (Springer-Verlag, New York, 1999). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.