## Controllable mode multistability in microring lasers |

Optics Express, Vol. 21, Issue 8, pp. 9624-9635 (2013)

http://dx.doi.org/10.1364/OE.21.009624

Acrobat PDF (1380 KB)

### Abstract

We investigate mode multistability, i.e. coexistence of direction bistability and wavelength bi/multistability in microring lasers (MRLs) theoretically and numerically. We derive the expressions for conditions required for mode multistable operation in microring lasers based on a nonlinear multimode model with nonlinear effects stemming from carrier density pulsation, carrier heating and spectral hole burning included. We find theoretically that lasing mode can be selected from the multistable modes by external optical injection through gain saturation, and removal of the external optical injection will not affect the stability of the established lasing mode. Numerical results on all-optical multistate flip-flop function demonstrate that switching between multistable modes can be induced by trigger signals with each states self-sustained after the removal of the trigger signals in a 50µm-radius microring laser.

© 2013 OSA

## 1. Introduction

1. K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on
nanocavities,” Nat. Photonics **6**(4), 248–252
(2012). [CrossRef]

2. Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring
resonators,” Opt. Express **15**(3), 924–929
(2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-3-924. [CrossRef] [PubMed]

3. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a
photonic-crystal nanocavity,” Nat. Photonics **4**(7), 477–483
(2010). [CrossRef]

4. M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled
micro-ring lasers,” Nature **432**(7014), 206–209
(2004). [CrossRef] [PubMed]

6. Z. Wang, G. Yuan, X. Cai, G. Verschaffelt, J. Dancka, Y. Liu, and S. Yu, “Error-free 10Gb/s all-optical switching based on a
bidirectional SRL with miniaturized retro-reflector cavity,”
IEEE Photon. Technol. Lett. **22**(24), 1805–1807
(2010). [CrossRef]

7. L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. J. Geluk, T. D. Vries, P. Regreny, D. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory
on a silicon chip,” Nat. Photonics **4**(3), 182–187
(2010). [CrossRef]

8. D. Liang and J. E. Bowers, “Recent progress in lasers on
silicon,” Nat. Photonics **4**(8), 511–517
(2010). [CrossRef]

10. Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Storing 2 bits of information in a novel single
semiconductor micro-ring laser memory cell,” IEEE Photon.
Technol. Lett. **20**(14), 1228–1230
(2008). [CrossRef]

## 2. Theory

*I*is the injection current,

*I*is the threshold current,

_{th}*e*for the elementary charge,

*V*for the volume of active region containing carriers,

_{c}*τ*for differential carrier lifetime,

_{N}*ε*for permittivity of free space,

_{0}*n*for group index,

_{g}*n*for the effective mode index,

_{0}*ω*for frequency of the linear gain peak,

_{pk}*v*for group velocity,

_{g}*g*for material gain of mode

_{a}*a*,

*E*for the complex slowly varying electric field of mode

_{a}*a*. Equation (1) takes into account the static carrier density change due to the intensity of modes, i.e., beating of the same modes.

*E*in the laser is expressed as a sum of complex field of both lasing and nonlasing modes in terms of

*E*aswhere

_{a}*z*is the field propagation direction, with CCW as positive

*z*direction and CW as negative

*z*direction, andwhere

*ω*accounts for the optical frequency of mode

_{a}*a*.

*a*is described by:

*Г*accounts for confinement factor,

*dg/dN*for differential gain coefficient,

*α*for linewidth enhancement factor associated with carrier density pulsation,

_{N}*τ*for photon lifetime,

_{p}*b, c*and

*d*[14

14. C. Born, G. Yuan, Z. Wang, and S. Yu, “Nonlinear gain in semiconductor ring
lasers,” IEEE J. Quantum Electron. **44**(11), 1055–1064
(2008). [CrossRef]

*ζ*would be very small for integration over many wavelengths long unless resonant coupling condition i.e.

_{abcd}*F*is the spontaneous emission Langevin noise term. Because effects from backscattering only weakly couple the pair of counter-propagating modes with the same cavity resonance wavelength and do not have noticeable effect on the mode multi-stability, they are not included in this model.

_{a}*L*(

_{p}*p = 1,2*) and one of the competing multistable modes as mode

*M*(

_{q}*q = 1,2*), where 1 and 2 represent the CCW and CW directions for wavelengths

*L*and

*M*respectively. We are interested in the dynamics and stability of the established mode and the competing multistable modes so we concentrate our theoretical analyses on the coupling between these modes. Assuming the laser is biased high above threshold current with quasi-single mode operation with mode

*L*as the dominant mode, we consider the situation illustrated in Fig. 1, where both wavelength

_{p}*L*and wavelength

*M*can propagate in either CCW or CW direction.

*L*is assumed to be the one that is closest to the linear material gain peak with frequency

_{p}*M*is assumed to be farther away from the linear gain peak than mode

_{q}*L*with frequency

_{p}*L*and

*M*have different wavelengths,

*1*and

*2*are used to represent two modes with the same wavelength but different propagation directions, i.e.

*1*for CCW and

*2*for CW direction of

*L*or

*M*respectively.

*L*and

_{p}*M*are co-propagating with

_{q}*p = q*, which is the case discussed in [12

12. G. Yuan and Z. Wang, “Theoretical and numerical investigations of wavelength
bi/multistability in semiconductor ring lasers,” IEEE J.
Quantum Electron. **47**(11), 1375–1382
(2011). [CrossRef]

*L*and

_{p}*M*are given by:where noise is neglected because the device is biased high above the threshold current, and the operating point is chosen to be in the middle of a robust unidirectional region. Notations

_{q}*p*and

*q*are removed for simplicity.

*L*and

_{p}*M*are counter-propagating with

_{q}*p≠q*, nonlinear coupling between modes

*L*and

_{p}*M*through four wave mixing does not exist because phase matching condition is not satisfied. We find in this case:

_{q}*p*and

*q*are also removed for simplicity.

*L*and

_{p}*M*are counter-propagating, the time evolutions of the normalized intensities of

_{q}*L*and

_{p}*M*, which are derived based on Eqs. (11) and (12) and Eq. (14), possess the same form as Eq. (13).

_{q}*L*and

_{1}*L*or modes

_{2}*M*and

_{1}*M*) by evaluating Eq. (4) as:Where notations

_{2}*L*and

*M*are removed for simplicity.

## 3. Analytical analysis and numerical results

### 3.1 Mode multistability analysis

*L*and

_{p}*M*degenerate into

_{q}*L*and

*M*since the time evolutions of their normalized intensities can all be described by Eq. (13) with whatever combination of

*p*and

*q*. Consequently, the notation

*p*and

*q*are removed in the following analyses for simplicity.

*L*and

*M*described by Eq. (13) can be studied by phase plane analysis. As shown in Fig. 2, two straight lines representing

*L*and

*M*at (

*L*, 0) and (0,

_{l,m}*M*) respectively. Multistable operation of modes

_{l,m}*L*and

*M*can be achieved if

*L*>

_{l}*L*and

_{m}*M*<

_{l}*M*, i.e. the following conditions are satisfied:

_{m}*L*and

*M*has a mode degeneracy of two including CW and CCW directions as indicated by Eq. (13), which means that two multistable states are of the same wavelength. The stability of modes with the same wavelength but different propagation directions is analyzed in the same way. As shown in Fig. 3, two straight lines representing

*CCW*and

*CW*at (

*CCW*, 0) and (0,

_{1,2}*CW*) respectively. We find that direction bistability can be achieved if

_{1,2}*CCW*and

_{1}>CCW_{2}*CW*, i.e. the following conditions have to be satisfied:

_{1}<CW_{2}10. Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Storing 2 bits of information in a novel single
semiconductor micro-ring laser memory cell,” IEEE Photon.
Technol. Lett. **20**(14), 1228–1230
(2008). [CrossRef]

*L*and

*M*are given by:

*L*, flows are all attracted to the solution of

*S*, therefore mode

_{M}= 0*L*is selected to be the lasing mode. Whereas Fig. 4(b) shows that

*M*, flows are all attracted to the solution of

*S*, and therefore mode

_{L}= 0*M*is selected to be the lasing mode. Furthermore, because both modes

*L*and

*M*are multistable, the lasing mode will remain stable even after removal of the external optical injection signal as indicated by Fig. 2.

*L*or

*M*,

*S*, therefore CCW direction of mode

_{2}= 0*L*or

*M*is selected to be the lasing direction. Whereas Fig. 5(b) shows that

*L*or

*M*, flows are all attracted to the solution of

*S*, and therefore CW direction of mode

_{1}= 0*L*or

*M*is selected to be the lasing direction. The lasing direction will remain stable even after the removal of the injection signal since the microring laser is directional bistable as shown in Fig. 3.

### 3.2 Numerical results

*m*~

_{0,CCW}*m*in CCW direction and modes

_{4,CCW}*m*~

_{0,CW}*m*in the CW direction where modes

_{4,CW}*m*and

_{0,CCW}*m*are the free running modes whereas

_{0,CW}*m*and

_{4,CCW}*m*are on the long wavelength side of the free running modes.

_{4,CW}*m*,

_{0,CCW}*m*,

_{3,CCW}, m_{3,CW}*m*,

_{2,CW}, m_{4,CCW}*m*,

_{2,CCW}, m_{1,CCW}*m*, and

_{4,CW}, m_{1,CW}*m*(Fig. 6(b)), which cover all the supported multistable modes in the device. Triggered by the pulse stream, the oscillating mode switches from

_{0,CW}*m*in the CCW direction to modes

_{0,CCW}*m*,

_{3,CCW}, m_{3,CW}*m*,

_{2,CW}, m_{4,CCW}*m*,

_{2,CCW}, m_{1,CCW}*m*and

_{4,CW},*m*consecutively and finally switches to mode

_{1,CW}*m*in the CW direction. Each state is self-sustained which means that the selected multistable mode is stable after the removal of the trigger signal as shown in Fig. 7(a).

_{0,CW}*m*,

_{0,CCW}*m*,

_{0,CW}, m_{0,CCW}*m*,

_{4,CCW}, m_{2,CW}*m*,

_{4,CCW}, m_{4,CW}*m*,

_{2,CCW}, m_{1,CW}*m*,

_{3,CCW}, m_{1,CCW}*m*,

_{3,CW}, m_{2,CW}*m*and

_{3,CW},*m*and shown in Fig. 6(c). The combination includes not only switching between all the supported multistable modes, but also several set-reset operations between a few pairs of modes such as

_{1,CCW}*m*,

_{0,CCW}*m*and

_{0,CW},*m*. As shown by Fig. 7(b), the lasing mode is successfully switched to the selected states by the trigger pulse signals and self-sustained flip-flop operations are achieved after the removal of the signals.

_{0,CCW}## 4. Conclusions

## Acknowledgments

## References and links

1. | K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on
nanocavities,” Nat. Photonics |

2. | Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring
resonators,” Opt. Express |

3. | K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a
photonic-crystal nanocavity,” Nat. Photonics |

4. | M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled
micro-ring lasers,” Nature |

5. | M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring
lasers: experiment and model,” IEEE J. Quantum
Electron. |

6. | Z. Wang, G. Yuan, X. Cai, G. Verschaffelt, J. Dancka, Y. Liu, and S. Yu, “Error-free 10Gb/s all-optical switching based on a
bidirectional SRL with miniaturized retro-reflector cavity,”
IEEE Photon. Technol. Lett. |

7. | L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. J. Geluk, T. D. Vries, P. Regreny, D. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory
on a silicon chip,” Nat. Photonics |

8. | D. Liang and J. E. Bowers, “Recent progress in lasers on
silicon,” Nat. Photonics |

9. | C. Born, S. Yu, M. Sorel, and P. J. R. Laybourn,
“Controllable and stable mode selection in a semiconductor ring laser by injection
locking,” in CLEO Proceedings, paper |

10. | Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Storing 2 bits of information in a novel single
semiconductor micro-ring laser memory cell,” IEEE Photon.
Technol. Lett. |

11. | G. Yuan and S. Yu, “Bistability and switching properties of semiconductor
ring lasers with external optical Injection,” IEEE J. Quantum
Electron. |

12. | G. Yuan and Z. Wang, “Theoretical and numerical investigations of wavelength
bi/multistability in semiconductor ring lasers,” IEEE J.
Quantum Electron. |

13. | A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Longitudinal mode multistability in ring and
Fabry-Pérot lasers: the effect of spatial hole burning,”
Opt. Express |

14. | C. Born, G. Yuan, Z. Wang, and S. Yu, “Nonlinear gain in semiconductor ring
lasers,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.3560) Lasers and laser optics : Lasers, ring

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.1450) Nonlinear optics : Bistability

(140.3948) Lasers and laser optics : Microcavity devices

(130.4815) Integrated optics : Optical switching devices

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 6, 2012

Revised Manuscript: February 15, 2013

Manuscript Accepted: March 17, 2013

Published: April 10, 2013

**Citation**

Guohui Yuan and Zhuoran Wang, "Controllable mode multistability in microring lasers," Opt. Express **21**, 9624-9635 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9624

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### References

- K. Nozaki, A. Shinya, S. Matsuo, Y. Suzaki, T. Segawa, T. Sato, Y. Kawaguchi, R. Takahashi, and M. Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat. Photonics6(4), 248–252 (2012). [CrossRef]
- Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express15(3), 924–929 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-15-3-924 . [CrossRef] [PubMed]
- K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics4(7), 477–483 (2010). [CrossRef]
- M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature432(7014), 206–209 (2004). [CrossRef] [PubMed]
- M. Sorel, G. Giuliani, A. Scire, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron.39(10), 1187–1195 (2003). [CrossRef]
- Z. Wang, G. Yuan, X. Cai, G. Verschaffelt, J. Dancka, Y. Liu, and S. Yu, “Error-free 10Gb/s all-optical switching based on a bidirectional SRL with miniaturized retro-reflector cavity,” IEEE Photon. Technol. Lett.22(24), 1805–1807 (2010). [CrossRef]
- L. Liu, R. Kumar, K. Huybrechts, T. Spuesens, G. Roelkens, E. J. Geluk, T. D. Vries, P. Regreny, D. Thourhout, R. Baets, and G. Morthier, “An ultra-small, low-power, all-optical flip-flop memory on a silicon chip,” Nat. Photonics4(3), 182–187 (2010). [CrossRef]
- D. Liang and J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics4(8), 511–517 (2010). [CrossRef]
- C. Born, S. Yu, M. Sorel, and P. J. R. Laybourn, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” in CLEO Proceedings, paper CWK4, (2003).
- Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Storing 2 bits of information in a novel single semiconductor micro-ring laser memory cell,” IEEE Photon. Technol. Lett.20(14), 1228–1230 (2008). [CrossRef]
- G. Yuan and S. Yu, “Bistability and switching properties of semiconductor ring lasers with external optical Injection,” IEEE J. Quantum Electron.44(1), 41–48 (2008). [CrossRef]
- G. Yuan and Z. Wang, “Theoretical and numerical investigations of wavelength bi/multistability in semiconductor ring lasers,” IEEE J. Quantum Electron.47(11), 1375–1382 (2011). [CrossRef]
- A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Longitudinal mode multistability in ring and Fabry-Pérot lasers: the effect of spatial hole burning,” Opt. Express19(4), 3284–3289 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-19-4-3284 . [CrossRef] [PubMed]
- C. Born, G. Yuan, Z. Wang, and S. Yu, “Nonlinear gain in semiconductor ring lasers,” IEEE J. Quantum Electron.44(11), 1055–1064 (2008). [CrossRef]

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