## Nested ring Mach-Zehnder interferometer

Optics Express, Vol. 15, Issue 2, pp. 437-448 (2007)

http://dx.doi.org/10.1364/OE.15.000437

Acrobat PDF (595 KB)

### Abstract

We propose a new device configuration that incorporates a nested ring with a Mach-Zehnder interferometer. The nested ring is analogous to a dual-bus coupled ring resonator, with the ends of the two buses connected to form a semi-closed loop. With proper design of the length of the U-shaped loop, as well as the coupling coefficient between the ring and the waveguide, the device is capable of generating a box-shaped spectral response. This is shown to be mainly due to the double-Fano resonances that arise from constructive interference between the nested ring and the outer loop. The device is simple in that it requires only one ring, and unique in that it harnesses a pair of Fano resonances to generate a reasonably box-like filter response. The analysis is based on the transfer matrix formalism, and compared and verified with the FDTD simulations.

© 2007 Optical Society of America

## 1. Introduction

1. B. Little, S. Chu, J. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,“ Opt. Lett. **25**,344–346 (2000). [CrossRef]

2. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,“ IEEE. Photon. Technol. Lett. **7**,1447–1449 (1995). [CrossRef]

3. R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,“ Appl. Opt. **40**,5742–5747 (2001). [CrossRef]

4. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,“ Nature. **431**,1081–1084 (2004). [CrossRef] [PubMed]

5. W. Green, R. Lee, G. DeRose, A. Scherer, and A. Yariv, “Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermo optic switching and coupling control,“ Opt. Express **13**,1651–1659 (2005). [CrossRef] [PubMed]

*L*

_{c}) and the coupling coefficient (κ) between the ring and the bus. In this paper, we introduce a modified single-ring building block that has interesting phase and transmission properties and applications. We shall call it the nested ring resonator (NRR) as it looks like a ring nested by a U-shaped waveguide. It is equivalent to the dual-bus ring with the two buses connected by a loop with an arbitrary length

*L*, which is expressed as a multiple of the inner-ring circumference

_{v}*L*=

_{v}*v*(2

*πR*). A key result of this paper is that the NRMZI can give a box-like transmission when

*v*is an integer or half-integer. The loop feedback path provides an alternative route for the propagation of light to that through the ring. The interference between two pathways is required to generate Fano resonances as have been demonstrated in other ring-based structures [6

6. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,“ Opt. Lett. **30**,3069–3071 (2005). [CrossRef] [PubMed]

7. J. Heebner, N. Lepeshkin, A. Schweinsberg, G. Wicks, R. Boyd, R. Grover, and P. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,“ Opt. Lett. **29**,769–771 (2004). [CrossRef] [PubMed]

## 2. Theory of the nested ring resonator (NRR)

*iφ*) , which is defined as

*ρ*=

*b*

_{1}/

*a*

_{1}, where

*b*

_{1}is the output field and

*a*

_{1}is the input field [8

8. S. Darmawan and M. K. Chin, “Critical coupling, oscillation, reflection and transmission in optical waveguide-ring resonator systems,“ J. Opt. Soc. Am. B. **23**,834–841 (2006). [CrossRef]

9. Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,“ IEEE J. Quantum Electron. **41**,1410–1418 (2005). [CrossRef]

*R*and the transmission

*T*, which are given by

*t*= ∣

*t*∣ exp(

*iφ*),

_{t}*ρ*=∣

*ρ*∣ exp(

*iφ*). The corresponding phase terms are given by

_{ρ}*ρ*and

*t*given by Eqs. (2) and (3), respectively. The U-loop connecting the two buses has a length of

*vL*. Hence, by summation, the total transmission is given by,

_{c}*a*= 1 and the nested ring behaves as an all-pass filter with phase-only response. In this case it can be shown [9

9. Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,“ IEEE J. Quantum Electron. **41**,1410–1418 (2005). [CrossRef]

*ρ*∣

^{2}+∣

*t*∣

^{2}=1 and(

*ρ*

^{2}-

*t*

^{2})/

*ρ*= 1/

*ρ**, and with

*φ*-

_{ρ}*φ*=

_{t}*π*/2 from Eq. (4), the transmission of the NRR can be simplified to,

*φ*=

_{NRR}*φ*+

_{t}*φ*, where

_{load}*ϕ*(denoted as the loading term), which is similar to the phase response of a single-bus RR, Eq. (1b), with the modifications

^{load}*δ*→

*vδ*+

*φ*and

_{t}*r*→∣

*t*∣. This implies that the outer feedback loop behaves like a single-bus RR with respect to the inner dual-bus RR (DBRR). The resonance in the outer loop depends on the transmission ∣

*t*∣ of the DBRR, and increases with the coupling factor κ

^{2}. The inner ring resonance, on the other hand, increases with the reflectivity (

*r*). Since

*r*and κ are complementary, it is impossible to have resonance in both loops simultaneously. As shown in Fig. 1, when

*r*is low, the phase response is dominated by the loading phase, which is highly nonlinear with a step-like feature, where each step represents a 2π phase shift caused by resonance in the outer loop. On the other hand, when

*r*is large, the phase response is dominated by the resonance in the inner ring.

*vδ*+

*φ*) in

_{t}*φ*contains primarily the factor (

_{load}*v*+ 1/2)

*δ*, which implies: (a) if

*v*is an integer (

*v*=

*m*), then the period of

*φ*is 4π, and the number of possible resonances in the outer loop (corresponding to the number of steps in the phase response) is 2

_{load}*m*in each period; (b) if

*v*is a half-integer (

*v*=

*m*-1/2), then the period of

*φ*is 2π, and the number of resonances

_{load}*per*period is

*m*. Only these two cases are of interest, and two examples are shown in Fig. 3. For other values of

*v*, the power spectrum would be irregular with relatively large periodicities.

*a*≠ 1, the transmission is no longer all-pass and depends critically on the value of

*a*. In particular, for

*a*< 1, specific combinations of

*r*and

*a*can result in the numerator of Eq. (5) becoming zero at specific values of

*δ*, leading to the

*critical coupling*condition similar to that in a single-bus ring [10

10. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,“ IEEE Photon. Technol. Lett. **14**,483–485 (2004). [CrossRef]

*a*> 1 (which is possible if an optical amplifier is incorporated in the ring cavity [11

11. D. G. Rabus, M. Hamacher, U Troppenz, and H. Heidrich, “Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP,“ IEEE J. Sel. Top. Quantum Electron. **8**,1405–1410 (2002). [CrossRef]

*r*and

*a*can result in the denominator of Eq. (5) becoming zero at specific values of

*δ*, leading to the oscillation condition [8

8. S. Darmawan and M. K. Chin, “Critical coupling, oscillation, reflection and transmission in optical waveguide-ring resonator systems,“ J. Opt. Soc. Am. B. **23**,834–841 (2006). [CrossRef]

*v*= 5.5 for (a)

*a*< 1 and (b)

*a*> 1. The zeros and poles corresponding to critical coupling and oscillation, respectively, occur in pairs and move closer to

*δ*= 0 as

*r*increases. The resonances at

*δ*= 2

*πm*correspond to the inner-loop, whereas other resonances correspond to the outer-loop. The inner-loop resonance requires a very high

*r*value as compared to the outer-loop resonances, which is consistent with the phase response shown in Fig. 2. The zeros and poles can be displayed on a pole-zero plot as a function of

*r*, as discussed in detail elsewhere [8

8. S. Darmawan and M. K. Chin, “Critical coupling, oscillation, reflection and transmission in optical waveguide-ring resonator systems,“ J. Opt. Soc. Am. B. **23**,834–841 (2006). [CrossRef]

*v*= 5.5. As

*r*increases the zeros (poles) move outward (inward) from the unit circle, except for the pair nearest to the zero angle which give rise to the inner-ring resonance at

*δ*= 0 when

*r*is sufficiently large. Critical coupling (oscillation) occurs when the zeros (poles) cross the unit circle. For stability, all poles must lie within the unit circle [12]. The resonances indicated as “unstable” correspond to poles which fall outside the unit circle.

## 3. Nested ring Mach-Zehnder interferometer (NRMZI): Theory and results

*lossless*NRMZI are given by,

*r*= 1,

*φ*=

_{NRR}*vδ*since no light is coupled into the ring as if the ring is not present, so both arms of the MZI are identical.

*r*. A smaller

*r*results in steeper phase nonlinearity, which gives rise to the shar p asymmetric Fano resonances that define the sharp edges of the passband. However, the sharp Fano resonances are inevitably accompanied by ripples in and outside the passband which are undesirable for a bandpass filter. Increasing the value of

*r*can flatten the ripples but at the expense of the band edge roll-off. This trade-off is an inherent limitation of this device, which fundamentally relies on the double Fano resonances to generate the box-like output.

*κ*

^{2}

*a*

^{v+1/2}

*e*

^{i(v+1/2)δ}in the denominator indicates the effect of the outer-loop ‘coupling’ based resonance. For the inner-loop we have defined the build-up factor as

*B*

_{31}instead of the conventional circular intensity

*B*

_{21}since the latter is still a part of the outer-loop branch.

*v*; increasing

*v*reduces the bandwidth. The higher the

*v*value, the narrower and sharper would be the Fano resonances if the

*r*was fixed at a small value. Hence, to give a flatter box-like response, the optimal value of

*r*would have to be higher for larger values of

*v*. This is verified in Fig. 9(a) which shows some examples of optimized box-shaped response and the corresponding parameters. The higher the

*v*value, the narrower and sharper are the transmission bands. To minimize the sidelobes while preserving the box-shaped response, a cascaded configuration of identical NRMZI can be adopted. Two stages are sufficient to suppress the sidelobes reasonably, as illustrated in Fig. 9(b).

*Δϕ*on the other arm of the MZI [6

_{bias}6. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,“ Opt. Lett. **30**,3069–3071 (2005). [CrossRef] [PubMed]

2. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,“ IEEE. Photon. Technol. Lett. **7**,1447–1449 (1995). [CrossRef]

10. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,“ IEEE Photon. Technol. Lett. **14**,483–485 (2004). [CrossRef]

## 4. Comparison with FDTD Simulations

*r*= 0.6. The fields in the inner and outer rings are ‘measured’ at each wavelength and normalized by the input spectrum to give

*B*

_{31}and

*B*

_{61}, respectively. A Gaussian light pulse of 15fs pulse width with near-resonant input wavelength is launched from one of the waveguides. In our simulation, we chose a grid size less than λ/20

*n*and set

*v*= 1 to save computation time. As shown in Fig. 10(a), the build-up spectra are similar to those shown in Fig. 7(a) obtained by the transfer matrix formalism. The insets show the FDTD field distributions at the wavelengths corresponding to the two peaks and the one dip in

*B*

_{61}. The dashed curves show the results of the transfer matrix formalism, which are in good agreement with FDTD. To obtain these results, we have taken into account the wavelength dispersion using the expression

*n*

_{eff}(λ) = 3.1334 – 0.2874λ, for 1.7μm < λ < 1.8μm, which is obtained by numerically solving the

*n*

_{eff}for various λ. Furthermore, to achieve good agreement the ring radius is increased to 1.717μm, consistent with the fact that the mode in the ring is slightly skewed outward from the waveguide axis.

## 5. The effect of loss and fabrication variations

13. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,“ IEEE Photon. Technol. Lett. **16**,2263–2265 (2004). [CrossRef]

14. C. Y. Chao and L. J. Guo, “Reduction of surface scattering loss in polymer mirorings using thermal-reflow technique,“ IEEE Photon. Technol. Lett. **16**,1498–1500 (2004). [CrossRef]

15. J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,“ Opt. Lett. **29**,2861–2863 (2004). [CrossRef]

*a*> 0.99, and hence feasible to realize such devices.

*v*given by

*v*

^{±}= (

*L*± Δ

_{v}*L*)/

*L*, where Δ

_{c}*L*= 50 nm, and (b) maximum deviations in the lower MZI arm length. We see that such deviations shift the resonance frequency and distort the box-like transmission by giving a slant to the passband and an imbalance in the sidelobe. The sidelobe imbalance gives rise to a crosstalk penalty, which is 3.2 dB due to the imbalance between the MZI arms and 1.6 dB due to the imbalance in the NRR. This shows that the performance is more sensitive to the MZI arm imbalance and that the fabrication tolerance for this device is less than 50 nm. One remedy is to add external bias (i.e., thermal film heater or electrical bias) at the NRR and the lower MZI arm to tune the NRMZI into the desired performance.

## 6. Conclusion

6. Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,“ Opt. Lett. **30**,3069–3071 (2005). [CrossRef] [PubMed]

7. J. Heebner, N. Lepeshkin, A. Schweinsberg, G. Wicks, R. Boyd, R. Grover, and P. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,“ Opt. Lett. **29**,769–771 (2004). [CrossRef] [PubMed]

2. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,“ IEEE. Photon. Technol. Lett. **7**,1447–1449 (1995). [CrossRef]

## Acknowledgment

## References and links

1. | B. Little, S. Chu, J. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,“ Opt. Lett. |

2. | R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,“ IEEE. Photon. Technol. Lett. |

3. | R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,“ Appl. Opt. |

4. | V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,“ Nature. |

5. | W. Green, R. Lee, G. DeRose, A. Scherer, and A. Yariv, “Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermo optic switching and coupling control,“ Opt. Express |

6. | Y. Lu, J. Yao, X. Li, and P. Wang, “Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer,“ Opt. Lett. |

7. | J. Heebner, N. Lepeshkin, A. Schweinsberg, G. Wicks, R. Boyd, R. Grover, and P. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,“ Opt. Lett. |

8. | S. Darmawan and M. K. Chin, “Critical coupling, oscillation, reflection and transmission in optical waveguide-ring resonator systems,“ J. Opt. Soc. Am. B. |

9. | Y. M. Landobasa, S. Darmawan, and M. K. Chin, “Matrix analysis of 2-D micro-resonator lattice optical filters,“ IEEE J. Quantum Electron. |

10. | A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,“ IEEE Photon. Technol. Lett. |

11. | D. G. Rabus, M. Hamacher, U Troppenz, and H. Heidrich, “Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP,“ IEEE J. Sel. Top. Quantum Electron. |

12. | C. K. Madsen and J. H. Zhao, |

13. | B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,“ IEEE Photon. Technol. Lett. |

14. | C. Y. Chao and L. J. Guo, “Reduction of surface scattering loss in polymer mirorings using thermal-reflow technique,“ IEEE Photon. Technol. Lett. |

15. | J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,“ Opt. Lett. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: October 12, 2006

Revised Manuscript: November 20, 2006

Manuscript Accepted: November 30, 2006

Published: January 22, 2007

**Citation**

S. Darmawan, Y. M. Landobasa, and M. K. Chin, "Nested ring Mach-Zehnder interferometer," Opt. Express **15**, 437-448 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-437

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### References

- B. Little, S. Chu, J. Hryniewicz, and P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000). [CrossRef]
- R. Orta, P. Savi, R. Tascone, and D. Trinchero, "Synthesis of multiple-ring-resonator filters for optical systems," IEEE. Photon. Technol. Lett. 7, 1447-1449 (1995). [CrossRef]
- R. W. Boyd and J. E. Heebner, "Sensitive disk resonator photonic biosensor," Appl. Opt. 40, 5742-5747 (2001). [CrossRef]
- V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature. 431, 1081-1084 (2004). [CrossRef] [PubMed]
- W. Green, R. Lee, G. DeRose, A. Scherer, and A. Yariv, "Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermo optic switching and coupling control," Opt. Express 13, 1651-1659 (2005). [CrossRef] [PubMed]
- Y. Lu, J. Yao, X. Li, and P. Wang, "Tunable asymmetrical Fano resonance and bistability in a microcavity-resonator-coupled Mach-Zehnder interferometer," Opt. Lett. 30, 3069-3071 (2005). [CrossRef] [PubMed]
- J. Heebner, N. Lepeshkin, A. Schweinsberg, G. Wicks, R. Boyd, R. Grover, and P. Ho, "Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators," Opt. Lett. 29, 769-771 (2004). [CrossRef] [PubMed]
- S. Darmawan and M. K. Chin, "Critical coupling, oscillation, reflection and transmission in optical waveguide-ring resonator systems," J. Opt. Soc. Am. B. 23, 834-841 (2006). [CrossRef]
- Y. M. Landobasa, S. Darmawan, M. K. Chin, "Matrix analysis of 2-D micro-resonator lattice optical filters," IEEE J. Quantum Electron. 41, 1410-1418 (2005). [CrossRef]
- A. Yariv, "Critical coupling and its control in optical waveguide-ring resonator systems," IEEE Photon. Technol. Lett. 14, 483-485 (2004). [CrossRef]
- D. G. Rabus, M. Hamacher, U Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1410 (2002). [CrossRef]
- C. K. Madsen and J. H. Zhao, Optical filter design and analysis: A signal processing approach (Wiley, New York, 1999).
- B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, "Very high-order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004). [CrossRef]
- C. Y. Chao and L. J. Guo, "Reduction of surface scattering loss in polymer mirorings using thermal-reflow technique," IEEE Photon. Technol. Lett. 16, 1498-1500 (2004). [CrossRef]
- J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, "Ultrahigh-quality-factor silicon-on-insulator microring resonator," Opt. Lett. 29, 2861-2863 (2004). [CrossRef]

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