## Numerical and experimental studies of coupling-induced phase shift in resonator and interferometric integrated optics devices |

Optics Express, Vol. 20, Issue 5, pp. 5789-5801 (2012)

http://dx.doi.org/10.1364/OE.20.005789

Acrobat PDF (2272 KB)

### Abstract

Coupling induced effects are higher order effects inherent in waveguide evanescent coupling that are known to spectrally distort optical performances of integrated optics devices formed by coupled resonators. We present both numerical and experimental studies of coupling-induced phase shift in various basic integrated optics devices. Rigorous finite difference time domain simulations and systematic experimental characterizations of different basic structures were conducted for more accurate parameter extraction, where it can be observed that coupling induced wave vector may change sign at the increasing gap separation. The devices characterized in this work were fabricated by CMOS-process 193nm Deep UV (DUV) lithography in silicon-on-insulator (SOI) technology.

© 2012 OSA

## 1. Introduction

1. M. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express **14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

3. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **62**(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]

4. S. Darmawan, L. Y. Tobing, and T. Mei, “Coupling-induced phase shift in a microring-coupled Mach-Zehnder interferometer,” Opt. Lett. **35**(2), 238–240 (2010). [CrossRef] [PubMed]

4. S. Darmawan, L. Y. Tobing, and T. Mei, “Coupling-induced phase shift in a microring-coupled Mach-Zehnder interferometer,” Opt. Lett. **35**(2), 238–240 (2010). [CrossRef] [PubMed]

8. 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**(10), 2263–2265 (2004). [CrossRef]

1. M. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express **14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

2. Q. Li, M. Soltani, A. H. Atabaki, S. Yegnanarayanan, and A. Adibi, “Quantitative modeling of coupling-induced resonance frequency shift in microring resonators,” Opt. Express **17**(26), 23474–23487 (2009). [CrossRef] [PubMed]

3. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **62**(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]

1. M. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express **14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

2. Q. Li, M. Soltani, A. H. Atabaki, S. Yegnanarayanan, and A. Adibi, “Quantitative modeling of coupling-induced resonance frequency shift in microring resonators,” Opt. Express **17**(26), 23474–23487 (2009). [CrossRef] [PubMed]

## 2. Numerical study of CIPS in Mach-Zehnder interferometers and resonators

*k*

_{mm},

*ϕ*

_{r}is the intrinsic phase of the self-coupling coefficient. On the other hand, the perturbed phase in the waveguide bus (ϕ’) is

*ϕ*

_{r}.

_{res}= (δ’-δ), then translates to the shift of resonance wavelength, Δλ

_{res}= (

*ϕ*

_{res}/2π)Δλ

_{FSR}, where the free-spectral range Δλ

_{FSR}= λ

_{0}

^{2}/(

*n*

_{g}

*L*

_{cav}) is expressed in terms of resonance wavelength λ

_{0}, group index

*n*

_{g}for a given wavelength band Δλ and the cavity circumference

*L*

_{cav}. The coupling induced effects have been numerically demonstrated in isolated rings side-coupled with optical waveguides, or in a directional coupler, where the phase response is typically extracted from a suitable reference plane before and after coupling [1

**14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

*ϕ*

_{r}is assumed to be the same since the directional coupler is symmetrical with respect to both vertical and horizontal axis [1

**14**(3), 1208–1222 (2006). [CrossRef] [PubMed]

*T*

_{B}) and cross (

*T*

_{C}) transmissions can be written as

*ϕ*

_{r}can then be deduced from the ratio of bar and cross transmissions,where coupling coefficients (

*r*,

*t*) are independently obtained from FDTD simulations of directional couplers for a given coupler length

*L*

_{C}.

*L*

_{C}) and gap separation (

*g*) from 2μm to 14μm and 200nm to 400nm, respectively. In order to save computation time, the FDTD simulations are limited only to two dimensions, with the waveguide core index deduced by effective index method of the actual silicon-on-insulator material. The core index is calculated to be ~2.84 and the cladding is SiO

_{2}(

*n*~1.444). In general, disagreement between 3D and 2D simulation is to be expected, since the 2D simulations are limited to Semi-Vectorial wave propagation in contrast to the Full-Vectorial propagation in the 3D counterparts. As a result, the group index and the field confinement factor are expected to be lower, as is evident from a rather large coupling strength, compared with experimental values in the Section 3.

*L*-band (1500nm to 1600nm). These coupling coefficients are then inserted into Eq. (5) in order to get the induced phase shift. Figure 4 shows the plot of induced phase shift for different coupler lengths and gap separation. Each point in Fig. 4 represents the average phase values taken within a particular wavelength band of interest (within 1560nm-1580nm wavelength band), together with their standard deviation represented in the error bar.

*K*

_{r}can be calculated from the phase slope (Δ

*K*

_{r}= dϕ

_{r}/d

*L*

_{C}), which is shown in Fig. 4(b) as a function of gap separation. It can be seen that Fig. 4(b) roughly follows the exponential trend because of the exponentially decreasing coupling strength in the increasing of gap separation.

_{1}(λ

_{0}) is the resonance wavelength for the coupled (uncoupled) case . The resonance wavelength for coupled case [Fig. 5(a)], is deduced from normalization of the transmission spectrum (drop or through port) with the input spectrum, while the resonance wavelength for the uncoupled case [Fig. 5(b)] is deduced from normalization of intra-cavity field with respect to the input spectrum.

_{r}in the resonator as compared to that in the waveguide.

## 3. Experimental study of CIPS in resonator-enhanced Mach-Zehnder interferometers

10. URL, http://www.epixfab.eu.

11. S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. **16**(1), 316–324 (2010). [CrossRef]

*L*

_{C}) and gap separation (

*g*) are varied while the ring radius is fixed to 5μm. The waveguide dimensions are of 450nm width and 220nm thickness. The sample is clad by SiO

_{2}by default, unless specified otherwise. To facilitate input/output coupling, a curved second order gratings are fabricated at both ends of the devices [see Fig. 6(a) ]. The grating coupler is much more compact than those in our previous work due to the curvature introduced in the grating, which functions to focus the light onto the waveguide and reduce the possibility for higher order mode conversion. The length of the grating is less than 100μm, which gives the device density about three times than those in our previous fabrication batches.

*ϕ*

_{CIPS}. In the following subsections, waveguide loss, coupling coefficients, and MMI loss and phase imbalances are briefly characterized.

### 3.1. Characterization of waveguide loss

_{2},

*i*-Line resist, and the air cladding (bare silicon). The SiO

_{2}film is of 600nm nominal thickness and was deposited at 300°C temperature. The other two claddings are used to create less field confinement (for

*i*-Line resist) to increase the coupling, and more field confinement (for air cladding) to decrease the coupling.

_{2}cladded sample (α

_{wg}= 2.84dB/cm) is higher than that for the air cladded samples (α

_{wg}= 2.34dB/cm) because of the non-ideal PECVD condition, which is supposed to be done at much higher temperature (~700°C) for more conformal and stoichiometric film deposition. The non-stoichiometric SiO

_{2}film may generate silicon nanocrystals which then behave as absorption centers. However, higher loss in

*i*-Line resist cladding sample is to be expected because of the absorbing nature of

*i*-Line photoresist (α

_{wg}= 5.74dB/cm). It is also interesting to note that the fiber-waveguide coupling efficiency is higher for

*i*-Line resists clad samples (α

_{c}= 3.59dB/facet) compared with the other two (α

_{c}~5dB/facet). This is because of lower index contrast in

*i*-Line resist and much better phase matching condition with the gratings.

### 3.2. Characterization of coupling coefficients in ring resonator structures

12. L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express **14**(26), 12770–12781 (2006). [CrossRef] [PubMed]

*a*= exp(-α

*L*

_{cav}/2) is the roundtrip loss for resonator with cavity length

*L*

_{cav},

*r*is the self-coupling coefficients, and δ = 2π

*n*

_{g}

*L*

_{cav}/λ is the round trip phase for a given group index

*n*

_{g}. The drop amplitude is strongly dependent on loss, which may be influenced by other loss not originated from the resonator, such as grating loss. Therefore, loss extraction from 1R2B structure is less accurate compared to those from waveguides. Furthermore, since resonance linewidth depends on both roundtrip loss and coupling coefficients, it is rather difficult to isolate one from another. However the above problem can be avoided by linearizing Eq. (6) into the following form,so that the first term depends on the coupling coefficients and round trip loss, while the second term solely dependent on the coupling coefficients. Thus, from the slope of Eq. (7) (with

_{2}PECVD cladding sample. The coupler length is varied from 6μm to 14μm. Prior to coupling coefficient extraction, an accurate group index needs to be determined. The initial condition of the group index is obtained from free-spectral range by

*n*

_{g}= λ

_{0}

^{2}/(Δλ

_{FSR}

*L*

_{cav}), which is then readjusted to match δ = 2

*m*π [or δ = (2

*m*+ 1)π] for the resonance (or anti-resonance) wavelength condition. The group index is found to be ~4.25 for PECVD and

*i*-Line resist coated samples, in agreement with those in [13

13. E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express **14**(9), 3853–3863 (2006). [CrossRef] [PubMed]

14. J. D. Doménech, P. Muñoz, and J. Capmany, “Transmission and group-delay characterization of coupled resonator optical waveguide apodized through the longitudinal offset technique,” Opt. Lett. **36**(2), 136–138 (2011). [CrossRef] [PubMed]

*L*

_{C}, i.e., (2/π)cos

^{−1}(

*r*), for the purpose of finding the beating length (

*L*

_{π}) and phase offset (

*ϕ*

_{0}). The beating length

*L*

_{π}is the length at which all the light power is transferred to the other waveguide, while the phase offset is the lumped phase parameter which takes into account the coupling from non-straight parts of the directional coupler. By means of linear regression, it is then straightforward to obtain the beating length and phase offset from the slope and offset, respectively. Figure 8 presents the beating length and phase offset for different gap separations in the PECVD coated sample (here we choose only one sample to represent the whole curve fitting of 1R2B devices from the three different claddings). Note that the beating length increases exponentially with the gap separation, while the phase offset decrease exponentially. This is to be expected from field overlap integral which has exponential dependence on gap separation. The empirically deduced expression of coupling coefficient can then be written aswhere the beating length

*g*. This is in agreement with theoretical formulation of 2D directional coupler [15], in which the coupling constant

*κ*is an exponentially dependent on gap separation.

### 3.3. Characterization of MMI phase imbalance

16. URL, http://www.rsoftdesign.com/.

*L*

_{MMI}is 41μm. The phase imbalances (ϕ

_{MMI}) can be deduced by the ratio between the Bar [

*T*

_{B}= sin

^{2}(ϕ

_{MMI}/2)] and Cross [

*T*

_{C}= cos

^{2}(ϕ

_{MMI}/2)] transmission

### 3.4. Characterization of CIPS in REMZI structures

*ϕ*

_{CIPS}can be obtained by fitting the spectrum asymmetricity by using the MZI formulawhere the MMI phase imbalance

*ϕ*

_{MMI}has been taken into account, and

*r*=

*a*) [17

17. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**(4), 483–485 (2002). [CrossRef]

*r*) and cavity roundtrip loss (

*a*), which have been independently obtained. Therefore, since the MMI phase imbalance

*ϕ*

_{MMI}has been obtained the previous section, we only have the

*ϕ*

_{CIPS}to be curve-fitted in REMZI structure.

_{2}PECVD coated sample. Note that the transmission contrast decreases when the gap separation is increased. This is evidently caused by resonant enhancement of cavity loss originated from critical coupling condition, which occurs when the coupling strength is equal to the cavity roundtrip loss (

*r*=

*a*) [17

17. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**(4), 483–485 (2002). [CrossRef]

*g*= 400nm), where the coupling strength is small enough to match the roundtrip loss. This also explains the apparent discontinuity in the resonance lineshape for 400nm gap separation, which originates from the almost discontinuous phase response of the ring transmittance near critical coupling condition [17

17. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. **14**(4), 483–485 (2002). [CrossRef]

*ϕ*

_{CIPS}) for different coupler lengths and gap separations is shown in Fig. 11(a) . It can be observed that

*ϕ*

_{CIPS}is almost linear with coupler length (

*L*

_{C}), as expected from simulation results. However, it should be noted that only one MZI arm length is side-coupled with the ring, which makes the term

*ϕ*

_{r}originates from the self-coupling coefficient. It should be noted that, due to the double coupling nature in 1R2B in Fig. 5(c), the induced phase shift should be about twice the induced phase shift in REMZI (which is singly coupled). By comparing the induced phase shift in Fig. 5(c) with that in Fig. 11(a), we can see that the induced phases deduced from 1R2B and REMZI are of the same order.

_{r}is more sensitive to coupler length than to gap separation. Thus, at increasing gap separation, the ϕ

_{r}starts to dominate the whole induced phase contribution because the term

## 4. Conclusions

_{CIPS}the only parameter to be fitted in REMZI structures. The effect of fabrication errors manifested in optical proximity effects (OPE) and MMI phase imbalance have been taken into account in the fitting, and it is observed that the induced phase and the phase slope can be negative when the gap separation is increased. Based on 193nm DUV lithography process, the optical proximity effect only predominates when the gap is smaller than 150nm, suggesting that the induced phase shift could not have been caused by the OPE. This is also confirmed by the fact that the induced phase shift is positive, instead of negative (if OPE is assumed to exist).

## Appendix: Best fit parameters of SiO_{2}, i-Line resist, and air cladded samples

## Acknowledgment

## References and links

1. | M. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express |

2. | Q. Li, M. Soltani, A. H. Atabaki, S. Yegnanarayanan, and A. Adibi, “Quantitative modeling of coupling-induced resonance frequency shift in microring resonators,” Opt. Express |

3. | Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

4. | S. Darmawan, L. Y. Tobing, and T. Mei, “Coupling-induced phase shift in a microring-coupled Mach-Zehnder interferometer,” Opt. Lett. |

5. | L. Y. M. Tobing, P. Dumon, R. Baets, and M. K. Chin, “Boxlike filter response based on complementary photonic bandgaps in two-dimensional microresonator arrays,” Opt. Lett. |

6. | K. Oda, N. Takato, H. Toba, K. Nosu, and J. Lightwave, “A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for optical FDM transmission systems,” J. LightwaveTechnol. |

7. | M. Kohtoku, S. Oku, Y. Kadota, Y. Shibata, and Y. Yoshikuni, “200-GHz FSR periodic multi/demultiplexer with flattened transmission and rejection band by using a Mach-Zehnder interferometer with a ring resonator,” IEEE Photon. Technol. Lett. |

8. | 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. |

9. | D. Marcuse, |

10. | URL, http://www.epixfab.eu. |

11. | S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. |

12. | L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express |

13. | E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express |

14. | J. D. Doménech, P. Muñoz, and J. Capmany, “Transmission and group-delay characterization of coupled resonator optical waveguide apodized through the longitudinal offset technique,” Opt. Lett. |

15. | A. Yariv, |

16. | |

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

**OCIS Codes**

(230.3120) Optical devices : Integrated optics devices

(230.5750) Optical devices : Resonators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: October 6, 2011

Revised Manuscript: December 18, 2011

Manuscript Accepted: December 19, 2011

Published: February 24, 2012

**Citation**

L. Y. M. Tobing, L. Tjahjana, S. Darmawan, and D. H. Zhang, "Numerical and experimental studies of coupling-induced phase shift in resonator and interferometric integrated optics devices," Opt. Express **20**, 5789-5801 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5789

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### References

- M. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express14(3), 1208–1222 (2006). [CrossRef] [PubMed]
- Q. Li, M. Soltani, A. H. Atabaki, S. Yegnanarayanan, and A. Adibi, “Quantitative modeling of coupling-induced resonance frequency shift in microring resonators,” Opt. Express17(26), 23474–23487 (2009). [CrossRef] [PubMed]
- Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(55 Pt B), 7389–7404 (2000). [CrossRef] [PubMed]
- S. Darmawan, L. Y. Tobing, and T. Mei, “Coupling-induced phase shift in a microring-coupled Mach-Zehnder interferometer,” Opt. Lett.35(2), 238–240 (2010). [CrossRef] [PubMed]
- L. Y. M. Tobing, P. Dumon, R. Baets, and M. K. Chin, “Boxlike filter response based on complementary photonic bandgaps in two-dimensional microresonator arrays,” Opt. Lett.33(21), 2512–2514 (2008). [CrossRef] [PubMed]
- K. Oda, N. Takato, H. Toba, K. Nosu, and J. Lightwave, “A wide-band guided-wave periodic multi/demultiplexer with a ring resonator for optical FDM transmission systems,” J. LightwaveTechnol.6, 1016–1023 (1988).
- M. Kohtoku, S. Oku, Y. Kadota, Y. Shibata, and Y. Yoshikuni, “200-GHz FSR periodic multi/demultiplexer with flattened transmission and rejection band by using a Mach-Zehnder interferometer with a ring resonator,” IEEE Photon. Technol. Lett.12(9), 1174–1176 (2000). [CrossRef]
- 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(10), 2263–2265 (2004). [CrossRef]
- D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).
- URL, http://www.epixfab.eu .
- S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron.16(1), 316–324 (2010). [CrossRef]
- L. Y. Mario, S. Darmawan, and M. K. Chin, “Asymmetric Fano resonance and bistability for high extinction ratio, large modulation depth, and low power switching,” Opt. Express14(26), 12770–12781 (2006). [CrossRef] [PubMed]
- E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express14(9), 3853–3863 (2006). [CrossRef] [PubMed]
- J. D. Doménech, P. Muñoz, and J. Capmany, “Transmission and group-delay characterization of coupled resonator optical waveguide apodized through the longitudinal offset technique,” Opt. Lett.36(2), 136–138 (2011). [CrossRef] [PubMed]
- A. Yariv, Optical Electronics in Modern Communication, 5th ed. (Oxford, 1997).
- URL, http://www.rsoftdesign.com/ .
- A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett.14(4), 483–485 (2002). [CrossRef]

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