## The nonlinear microring add-drop filter

Optics Express, Vol. 16, Issue 19, pp. 15130-15136 (2008)

http://dx.doi.org/10.1364/OE.16.015130

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### Abstract

A new model is developed for the effects of the Kerr optical nonlinearity in a microring (racetrack) resonator coupled to input and output waveguides, which takes into account the nonlinearity in the couplers as well as the microring sections. It is shown how the nonlinear microring add-drop filter can be used as a self-adjusting all-optical beam splitter to extend the dynamic range of photodetectors and protect optoelectronic circuitry from high input powers.

© 2008 Optical Society of America

## 1. Introduction: the nonlinear microring add-drop filter

1. J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. **24**, 847–849 (1999). [CrossRef]

5. J. E. Heebner and R. W. Boyd, “Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides,” in *Laser Resonators and Beam Control VI*, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE Vol.4969. pp. 185–194 (2003). [CrossRef]

6. V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. **19**, 1734–1739 (2001). [CrossRef]

## 2. Calculation of the transfer function

*C*

_{1}and

*C*

_{2}, and the other two sections are half-rings, labeled

*R*

_{1}and

*R*

_{2}.

### 2.1. Coupler sections:

7. S. M. Jensen, “The nonlinear directional coupler,” IEEE Trans. Microwave Theory Tech. **MTT-30**, 1568–1571 (1982). [CrossRef]

*M*

_{a,b}and

*κ*

_{ab,ba}are the usual self-coupling and cross-coupling coefficients of a linear directional coupler, e.g., see [8, Eq. (13.8.4–5)], and

*Q*

_{3}and

*Q*

_{4}are the corresponding nonlinear corrections [7

7. S. M. Jensen, “The nonlinear directional coupler,” IEEE Trans. Microwave Theory Tech. **MTT-30**, 1568–1571 (1982). [CrossRef]

7. S. M. Jensen, “The nonlinear directional coupler,” IEEE Trans. Microwave Theory Tech. **MTT-30**, 1568–1571 (1982). [CrossRef]

### 2.2. Half-ring sections:

*b*

_{2}/

*b*

_{1}is obtained as

*ϕ*

_{hr}=

*k*

_{0}

*n*

_{2}|

*b*

_{1}|

^{2}is the nonlinear phase shift accumulated over the length of propagation

*L*

_{hr}, which is the length of the half-ring portions of the racetrack resonator (from

*b*

_{1}to

*b*

_{2}in Fig. 2).

*R*

_{1}and

*R*

_{2}) is obtained from the following equation [9

9. S. Blair, J. E. Heebner, and R. W. Boyd, “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Opt. Lett. **27**, 357–359 (2002). [CrossRef]

### 2.3. Iterative algorithm:

*a*

_{0}, we assume a value of

*b*

_{0}(a first guess is 1/

*|κ|*times larger than

*a*

_{0}) and calculate

*a*

_{1}and

*b*

_{1}using the split-step method to solve Eqs. (1a)–(1b). The transfer function of the second coupler

*C*

_{2}is calculated like

*C*

_{1}, except with the assumption that there is no input at the add port indicated in Fig. 1. Finally, the transfer function of the second half-ring is calculated, similar to Eq. (2). The final result,

*b*

^{′}

_{0}, is different from the value of

*b*

_{0}assumed initially. Therefore, an iterative algorithm is used to converge to the correct guess for

*b*

_{0}, and thus obtain the correct nonlinear transfer functions. Under some circumstances, the system can become chaotic even if the coupler is assumed to be linear [10

10. H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of Bifurcation to Chaos in an All-Optical Bistable System,” Phys. Rev. Lett. **50**, 109–112 (1983). [CrossRef]

12. B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, “Optical multistability in a fiber-optic passive-loop resonator,” Opt. Commun. **59**, 309–312 (1986). [CrossRef]

## 3. Calculation results and discussion

*L*

_{cplr}= 50

*µ*m, the half-circumference of the microring

*L*

_{hr}= 22.5

*µ*m, the linear and nonlinear absorption coefficients are

*α*= 0.1/cm and

*β*= 10

^{-9}cm/W, the linear refractive index

*n*

_{0}= 3.47, the nonlinear refractive index coefficient

*n*

_{2}= 1.4 × 10

^{-13}cm

^{2}/W at a wavelength

*λ*= 1550 nm, and the waveguide cross-section = 0.3

*µ*m × 0.3

*µ*m. The center-to-center waveguide separation is 480 nm, so that in the linear regime, the coupling coefficient |

*κ*

*L*

_{cplr}| = 0.2.

13. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**, 321–322 (2000). [CrossRef]

*a*

_{1}is increased, the intensity in the microring

*b*

_{0}increases even more, which leads to dephasing of the directional coupler, and a smaller coupling coefficient between the input waveguide and the microring. (Such behavior was demonstrated first in a directional coupler [14

14. P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple quantum well waveguide switch,” Electron. Lett. **21**, 26–28 (1985). [CrossRef]

15. A. Villeneuve, C. C. Yang, P. G. J., G. I. Wigley, J. S. Stegeman, C. N. Aitchison, and Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. **61**, 147–149 (1992). [CrossRef]

16. A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, “Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 mu m,” Appl. Phys. Lett. **90**, 101112 (2007). [CrossRef]

^{2}in Fig. 3]. Qualitatively, whereas for low input intensities, the microring sends most of the light to the drop port, for high intensities, most of the light is transmitted along the waveguide itself, and only a small fraction is sent to the drop port. As shown in Fig. 1(b), the light that remains in the input waveguide can be detected by subsequent receiver sections also configured in the same way, i.e., “capped” by a microring front-end.

^{2}(corresponding to 2.5 mW power in the waveguide), the gap between the

*D*

_{1}drop (blue squares) and the black line (input) starts to widen. The power reaching

*D*

_{1}is now capped and most of the input power reaches the through port. Two-photon absorption is not a limiting factor for this range of input powers, but comes into play at higher powers, as shown in Fig. 3.

^{2}(90 mW power), the intensity at port

*D*

_{1}is less than 1 MW/cm

^{2}(1 mW power). An additional 89 mW of power reaches the through port for detection by

*D*

_{2},

*D*

_{3}, etc. which is indicated by the shaded area in Fig. 4. This represents a gain in dynamic range of 19.5 dB, increasing to nearly 40 dB at the right edge of Fig. 4. Note that the gain in dynamic range arises purely because of a self-adjusted splitting of the input power in the optical domain. There may be additional performance gains in the opto-electronic conversion at the detectors themselves, since no individual detector would “see” more than a milliwatt of optical power.

## 4. Conclusion

## Acknowledgment

## References and links

1. | J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. |

2. | V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Tech. Lett. |

3. | V. Van, T. Ibrahim, P. Absil, F. Johnson, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. |

4. | S. Pereira, P. Chak, and J. E. Sipe, “All-optical AND gate by use of a Kerr nonlinear microresonator structure,” Opt. Lett. |

5. | J. E. Heebner and R. W. Boyd, “Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides,” in |

6. | V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. |

7. | S. M. Jensen, “The nonlinear directional coupler,” IEEE Trans. Microwave Theory Tech. |

8. | A. Yariv, |

9. | S. Blair, J. E. Heebner, and R. W. Boyd, “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Opt. Lett. |

10. | H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of Bifurcation to Chaos in an All-Optical Bistable System,” Phys. Rev. Lett. |

11. | K. Ikeda and M. Mizuno, “Frustrated Instabilities in Nonlinear Optical Resonators,” Phys. Rev. Lett. |

12. | B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, “Optical multistability in a fiber-optic passive-loop resonator,” Opt. Commun. |

13. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. |

14. | P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple quantum well waveguide switch,” Electron. Lett. |

15. | A. Villeneuve, C. C. Yang, P. G. J., G. I. Wigley, J. S. Stegeman, C. N. Aitchison, and Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. |

16. | A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, “Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 mu m,” Appl. Phys. Lett. |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

(230.0040) Optical devices : Detectors

(230.1360) Optical devices : Beam splitters

(230.5750) Optical devices : Resonators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: June 13, 2008

Revised Manuscript: September 7, 2008

Manuscript Accepted: September 8, 2008

Published: September 11, 2008

**Citation**

Shayan Mookherjea and Mark A. Schneider, "The nonlinear microring add-drop filter," Opt. Express **16**, 15130-15136 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15130

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### References

- J. E. Heebner and R. W. Boyd, "Enhanced all-optical switching by use of a nonlinear fiber ring resonator," Opt. Lett. 24, 847-849 (1999). [CrossRef]
- V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, "All-optical nonlinear switching in GaAs-AlGaAs microring resonators," IEEE Photon. Technol. Lett. 14, 74-76 (2002). [CrossRef]
- V. Van, T. Ibrahim, P. Absil, F. Johnson, R. Grover, and P.-T. Ho, "Optical signal processing using nonlinear semiconductor microring resonators," IEEE J. Sel. Top. Quantum Electron. 8, 705-713 (2002). [CrossRef]
- S. Pereira, P. Chak, and J. E. Sipe, "All-optical AND gate by use of a Kerr nonlinear microresonator structure," Opt. Lett. 28, 444-446 (2003). [CrossRef] [PubMed]
- J. E. Heebner and R.W. Boyd, "Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides," Proc. SPIE 4969, 185-194 (2003). [CrossRef]
- V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, "Propagation loss in single mode GaAs-AlGaAs microring resonators: measurement and model," J. Lightwave Technol. 19, 1734-1739 (2001). [CrossRef]
- S. M. Jensen, "The nonlinear directional coupler," IEEE Trans. Microwave Theory Tech. MTT-30, 1568-1571 (1982). [CrossRef]
- A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford, New York, 1997).
- S. Blair, J. E. Heebner, and R. W. Boyd, "Beyond the absorption-limited nonlinear phase shift with microring resonators," Opt. Lett. 27, 357-359 (2002). [CrossRef]
- H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of Bifurcation to Chaos in an All-Optical Bistable System," Phys. Rev. Lett. 50, 109-112 (1983). [CrossRef]
- K. Ikeda and M. Mizuno, "Frustrated Instabilities in Nonlinear Optical Resonators," Phys. Rev. Lett. 53, 1340-1343 (1984). [CrossRef]
- B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, "Optical multistability in a fiber-optic passive-loop resonator," Opt. Commun. 59, 309-312 (1986). [CrossRef]
- A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000). [CrossRef]
- P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, "All optical multiple quantum well waveguide switch," Electron. Lett. 21, 26-28 (1985). [CrossRef]
- A. Villeneuve, C. C. Yang, P. G. J. Wigley, G. I. Stegeman, J. S. Aitchison, and C. N. Ironside, "Ultrafast alloptical switching in semiconductor nonlinear directional couplers at half the band gap," Appl. Phys. Lett. 61, 147-149 (1992). [CrossRef]
- A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, "Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 ?m," Appl. Phys. Lett. 90, 101112 (2007). [CrossRef]

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