## Simultaneous implementation of XOR and XNOR operations using a directed logic circuit based on two microring resonators |

Optics Express, Vol. 19, Issue 7, pp. 6524-6540 (2011)

http://dx.doi.org/10.1364/OE.19.006524

Acrobat PDF (1969 KB)

### Abstract

We report the simultaneous implementation of the XOR and XNOR operations at two ports of a directed logic circuit based on two cascaded microring resonators (MRRs), which are both modulated through thermo-optic effect. Two electrical modulating signals applied to the MRRs represent the two operands of each logic operation. Simultaneous bitwise XOR and XNOR operations at 10 kbit/s are demonstrated in two different operating modes. We show that such a circuit can be readily realized using the plasma dispersion effect or the electric field effects, indicating its potential for high-speed operation. We further employ the scattering matrix method to analyze the spectral characteristics of the fabricated circuit, which can be regarded as a Mach-Zehnder interferometer (MZI) in whole. The two MRRs in the circuit act as wavelength-dependent splitting and combining units of the MZI. The degradation of the spectra observed in the experiment is found to be related to the length difference between the MZI’s two arms. The evolution of the spectra with this length difference is presented.

© 2011 OSA

## 1. Introduction

1. J. Hardy and J. Shamir, “Optics inspired logic architecture,” Opt. Express **15**(1), 150–165 (2007). [CrossRef] [PubMed]

3. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics **4**(5), 261–263 (2010). [CrossRef]

4. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. **8**(3), 705–713 (2002). [CrossRef]

7. M. P. Fok and P. R. Prucnal, “All-optical XOR gate with optical feedback using highly Ge-doped nonlinear fiber and a terahertz optical asymmetric demultiplexer,” Appl. Opt. **50**(2), 237–241 (2011). [CrossRef] [PubMed]

1. J. Hardy and J. Shamir, “Optics inspired logic architecture,” Opt. Express **15**(1), 150–165 (2007). [CrossRef] [PubMed]

1. J. Hardy and J. Shamir, “Optics inspired logic architecture,” Opt. Express **15**(1), 150–165 (2007). [CrossRef] [PubMed]

3. H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics **4**(5), 261–263 (2010). [CrossRef]

8. L. Zhang, R. Q. Ji, L. X. Jia, L. Yang, P. Zhou, Y. H. Tian, P. Chen, Y. Y. Lu, Z. Y. Jiang, Y. L. Liu, Q. Fang, and M. B. Yu, “Demonstration of directed XOR/XNOR logic gates using two cascaded microring resonators,” Opt. Lett. **35**(10), 1620–1622 (2010). [CrossRef] [PubMed]

## 2. Design and fabrication

*input*,

*through*,

*add*, and

*drop*according to their functions. A monochromatic light (

*λ*) coupled into the

*input*and

*add*ports will be directed to the

*drop*and

*through*ports, respectively, if the MRR is on-resonance at λ. And if the MRR is off-resonance at

*λ*, light coupled into the

*input*and

*add*ports will be guided to the

*through*and

*drop*ports, respectively (bypassing the MRR).

*X*and

*Y*are used to control the resonant states of the two MRRs, respectively. We assume that the MRRs are on-resonance at

*λ*if

*X*and

*Y*are at high level (representing ‘

*1*s’) and off-resonance at

*λ*if

*X*and

*Y*are at low level (representing ‘

*0*s’). According to these rules, we can obtain two logic outputs of ‘

*drop*and

*through*ports of MRR

_{2}, respectively. Those two signals are just the results of ‘

8. L. Zhang, R. Q. Ji, L. X. Jia, L. Yang, P. Zhou, Y. H. Tian, P. Chen, Y. Y. Lu, Z. Y. Jiang, Y. L. Liu, Q. Fang, and M. B. Yu, “Demonstration of directed XOR/XNOR logic gates using two cascaded microring resonators,” Opt. Lett. **35**(10), 1620–1622 (2010). [CrossRef] [PubMed]

*drop*port and

*through*port hereinafter for simplicity.

8. L. Zhang, R. Q. Ji, L. X. Jia, L. Yang, P. Zhou, Y. H. Tian, P. Chen, Y. Y. Lu, Z. Y. Jiang, Y. L. Liu, Q. Fang, and M. B. Yu, “Demonstration of directed XOR/XNOR logic gates using two cascaded microring resonators,” Opt. Lett. **35**(10), 1620–1622 (2010). [CrossRef] [PubMed]

*through*port with a moderate extinction ratio (~10 dB). And the ER at the

*drop*port is less than 5 dB. Such a low extinction ratio was caused by the high loss in the ring waveguides. It was for the same reason that the MRRs had rather low quality factors (Q ~2500). To reduce the loss in the ring waveguides, we adopt the rib waveguides to fabricate the device. The width and height of the waveguides are 400 nm and 220 nm, respectively, and the slab thickness is 70 nm. The gaps are chosen to be 330 nm to achieve a balance between the extinction ratios of the

*drop*and

*through*ports. The radii of the ring waveguides are both 10 μm. We did not optimize the crossing of the two straight waveguides in our former design, which caused striking ripples in the spectra [8

**35**(10), 1620–1622 (2010). [CrossRef] [PubMed]

9. T. Fukazawa, T. Hirano, F. Ohno, and T. Baba, “Low loss intersection of Si photonic wire waveguides,” Jpn. J. Appl. Phys. **43**(2), 646–647 (2004). [CrossRef]

10. N. Sherwood-Droz, H. Wang, L. Chen, B. G. Lee, A. Biberman, K. Bergman, and M. Lipson, “Optical 4x4 hitless slicon router for optical networks-on-chip (NoC),” Opt. Express **16**(20), 15915–15922 (2008). [CrossRef] [PubMed]

11. M. M. Geng, L. X. Jia, L. Zhang, L. Yang, P. Chen, T. Wang, and Y. L. Liu, “Four-channel reconfigurable optical add-drop multiplexer based on photonic wire waveguide,” Opt. Express **17**(7), 5502–5516 (2009). [CrossRef] [PubMed]

## 3. Experimental results

*n*

_{eff}) of the ring waveguide increases and the resonant wavelength of the MRR increases accordingly.

*drop*port to determine the working wavelength and the tuning voltages. As shown in the next two subsections, the working wavelength can be chosen in two different ways. The working wavelength is located in the off-resonance region in the first way, whereas it is located in the on-resonance region in the second way. Simultaneous operations of XOR and XNOR operations are achieved in both operating modes.

### 3.1. The first operating mode: working in the off-resonance region

*drop*port when neither of the two heaters is actuated (see Fig. 2(a) ). According to the aforementioned principle, a maximum (representing a ‘

*1*’) should be obtained at the

*drop*port when two applied electrical signals are both at low level (representing two ‘

*0*s’). We choose 1541.205 nm in the off-resonance region as the working wavelength and denote it as

*λ*

_{w1}hereinafter. It can be judged from Fig. 2(a) that the resonant wavelengths of the two MRRs are both located around 1540 nm.

*1*’ for each MRR should be determined. Firstly, only a voltage of 3.13 V is applied to MRR

_{1}to make it resonate at

*λ*

_{w1}(see Fig. 2(b)). Then only a voltage of 2.15 V is applied to MRR

_{2}to make it resonate at

*λ*

_{w1}(see Fig. 2(c)). A minimum appears at

*λ*

_{w1}in both situations, which means that a result of ‘

*0*’ is obtained when only one of the operands is ‘

*1*’. The voltages of 3.13 V and 2.15 V correspond to logic ‘

*1*s’ for the two MRRs, respectively. Finally, two voltages of 3.13 V and 2.15 V are applied to the two MRRs, respectively, at the same time. A maximum appears at

*λ*

_{w1}again (see Fig. 2(d)). That is because the light (at 1541.205 nm) is guided to the

*drop*port of MRR

_{1}and further dropped by MRR

_{2}to the circuit’s

*drop*port. It means that a result of ‘

*1*’ is obtained at the

*drop*port when both operands are ‘

*1*s’.

_{2}and MRR

_{1}are 1540.025 nm and 1539.920 nm, respectively, which originates from the fabrication errors. That is why we have to use a larger voltage to tune MRR

_{1}to

*λ*

_{w1}. The responses at the

*through*port are obtained in the same way, which are shown in Figs. 2 (e-h).

*1*s’ are determined, a monochromatic light at

*λ*

_{w1}from a tunable laser is coupled into the fabricated device and the output light at the

*drop*and

*through*ports of the circuit is fed into a detector. Two pseudo-random binary sequence (PRBS) non-return-to-zero (NRZ) signals at 10 kbit/s are converted to two analog voltage signals bit-by-bit according to the rule presented above and then applied to the corresponding MRRs. The electrical signals converted by the detector and the two electrical signals applied to the two MMRs are fed into a four-channel oscilloscope for waveform observation. The dynamic operation results are shown in Fig. 3 . It can be found that the XNOR and XOR operations are carried out correctly at the

*drop*and

*through*ports simultaneously.

*0*s’, as well as negative spikes between two consecutive ‘

*1*s’, which are caused by the speed-limited transitions of two different tuning statuses of the MRRs. For example, if the two signals applied to the MRRs are ‘

*10*’ and ‘

*01*’ in two successive time-slots (with only MRR

_{1}been actuated in the first time-slot, and only MRR

_{2}been actuated in the second time-slot), a positive spike (between two consecutive ‘

*0*s’) and a negative spike (between two consecutive ‘

*1*s’) appear at the

*drop*and

*through*ports, respectively. All the four tuning-status transitions (from ‘

*10*’ to ‘

*01*’, from ‘

*01*’ to ‘

*10*’, from ‘

*00*’ to ‘

*11*’, and from ‘

*11*’ to ‘

*00*’) that cause spikes are covered in Fig. 3.

12. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**(1), 123–129 (1987). [CrossRef]

21. J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics **2**(7), 433–437 (2008). [CrossRef]

12. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**(1), 123–129 (1987). [CrossRef]

17. P. Dong, R. Shafiiha, S. Liao, H. Liang, N.-N. Feng, D. Feng, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Wavelength-tunable silicon microring modulator,” Opt. Express **18**(11), 10941–10946 (2010). [CrossRef] [PubMed]

18. M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. Jen, L. Dalton, and A. Scherer, “Terahertz all-optical modulation in a silicon-polymer hybrid system,” Nat. Mater. **5**(9), 703–709 (2006). [CrossRef] [PubMed]

20. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics **3**(4), 216–219 (2009). [CrossRef]

### 3.2. The second operating mode: working at the resonant wavelength

*λ*

_{w}. As shown in Fig. 2 (a), a maximum is obtained at

*λ*

_{w}when neither of the MRRs is actuated. And a minimum is obtained at

*λ*

_{w}when only one MRR is tuned away (see Figs. 2 (b) and 2 (c)). And a maximum is obtained at

*λ*

_{w}when both MRRs are tuned away (see Fig. 2 (d)). Thus far all the four XNOR operations have been verified.

_{1}has a shorter resonant wavelength (1539.920 nm). The spectrum obtained at the

*drop*port with MRR

_{1}being applied with an offset voltage of 0.8 V is shown in Fig. 4 (a) , in which both MRRs resonate at 1540.025 nm. We choose 1540.025 nm as the working wavelength in the second operating mode and denote it as

*λ*

_{w2}hereinafter.

_{1}is actuated (with a voltage of 3.07 V applied), its resonant wavelength is shifted away from

*λ*

_{w2}. A minimum appears there (see Fig. 4 (b)). The voltage of 3.07 V is chosen rather arbitrarily as its function is only to tune the resonant wavelength of MRR

_{1}away from

*λ*

_{w2}. Precise alignment of the voltage amplitude is not required. When only MRR

_{2}is actuated (with a voltage of 2.15 V applied to MRR

_{2}, and a voltage of 0.8 V applied to MRR

_{1}as the offset), its resonant wavelength is shifted away from

*λ*

_{w2}. A minimum appears there as well (see Fig. 4 (c)). The voltage of 2.15 V is also chosen arbitrarily. And when both MRRs are tuned away from

*λ*

_{w2}, a maximum appears at

*λ*

_{w2}(see Fig. 4 (d)). Thus far all the four XNOR operations have been verified at the

*drop*port. We can get the response spectra at the

*through*port of the circuit accordingly, which are shown in Figs. 4 (e-h).

*0*s’ for the two MRRs are 0.8 V and 0 V, respectively. And the voltages corresponding to logic ‘

*1*s’ for the two MRRs can be arbitrarily chosen, which are both set to be 3.0 V in the dynamic demonstration. Two PRBS NRZ signals at 10 kbit/s are converted to two analog voltage signals bit-by-bit according to the above rule and then applied to the corresponding MRRs. The dynamic operation results are shown in Fig. 5 , which shows that the XNOR and XOR operations are carried out correctly at the

*drop*and

*through*ports simultaneously. Spikes still turn up as in Fig. 3, which can be mitigated using high-speed modulation schemes.

12. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. **23**(1), 123–129 (1987). [CrossRef]

21. J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics **2**(7), 433–437 (2008). [CrossRef]

## 4. Simulation and discussion

*drop*and

*through*ports of the device when it operates in the first mode (λ

_{w1}= 1541.205 nm). The free spectral ranges (FSRs) of both MRRs are about 10 nm. We can notice from Figs. 6 (d) and (h) that two of the four resonant regions diminish. In other words, the feature of the spectra located around 1541.205 nm does not appear around 1531.2 nm and 1551.2 nm, but just appears around 1561.2 nm. This characteristic is also observed in Figs. 6 (a) and (e), where the degradation is not as obvious as in Figs. 6 (d) and (h) due to the small mismatch of the two MRRs’ resonant wavelengths.

### 4.1. The numerical model of the circuit

22. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321–322 (2000). [CrossRef]

_{1}is in the form of

*E*

_{in}is chosen to be

*1*so that all the field amplitudes are normalized to it [22

22. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321–322 (2000). [CrossRef]

*t*and

*k*satisfy

*t*

^{2}+

*k*

^{2}

*=*1 [22

22. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321–322 (2000). [CrossRef]

*drop*and

*through*ports of the device based on the above model are shown in Fig. 8 , which agree well with the experimental results shown in Fig. 6. The parameters we adopt in the numerical simulation are summarized in Table 1 .

*n*

_{eff}of the ring waveguides. Then the radii of the two ring waveguides are determined, as well as the index increases of the actuated ring waveguides, according to the location of the resonant wavelengths shown in Fig. 6. The difference between the two ring waveguides is 1 nm in our model, which is within the fabrication error. The cross-coupling coefficient is calculated by the finite-difference time-domain (FDTD) method. And the self-coupling coefficient is obtained based on the lossless coupling assumption (

*t*

^{2}+

*k*

^{2}

*=*1). The loss coefficients of the waveguides are chosen according to the experimental results. The lengths of the two straight waveguides are determined based on the layout design and a fine tuning is made to fit the experimental results. The transmission factor of the crossing of the two waveguides is obtained through the FDTD method as well. And the lateral scattering caused by the crossing is ignored in our model.

_{1}and R

_{2}are assumed to be identical and represented by

*α*and

*θ*, respectively. The single-pass amplitude transmission factor

*α*is assumed to be independent of wavelength. The single-pass phase shift

*θ*is a function of wavelength and is expressed by

*β*is the propagation constant,

*R*is the radius of the ring waveguide, and

*λ*is the wavelength in vacuum. The effective refractive index of the ring waveguide is represented by

*n*, which is also a function of wavelength.

_{eff}*and E*

_{p1}*with respect to*

_{v1}*E*are

_{in}*through*and

*drop*ports of MRR

_{1}, respectively. The amplitude transfer functions of the

*through*and

*drop*ports of MRR

_{1}derived from Eq. (2) are

### 4.2. Analysis of the degradation in the spectra

*drop*port of the circuit based on Eqs. (4) and (5). We show that the spectra vary with the length difference (Δ

*L*) between the two straight waveguides S

_{1}and S

_{2}. This is intuitive since the circuit shown in Fig. 7 can be regarded as an MZI with two MRRs acting as its splitting and combining units, and the two straight waveguides acting as its two arms. We show hereinafter that the degradation occurs when Δ

*L*equals even multiples of half the perimeter of the ring waveguides.

*drop*port (

*E*

_{v}_{3}) can be decomposed into two separate parts. The first one comes from

*E*

_{v}_{2}, which contributes to

*E*

_{v}_{3}via the pass-through function of MRR

_{2}. The second part comes from

*E*

_{p}_{2}, which contributes to

*E*

_{v}_{3}via the dropping function of MRR

_{2}. We denote these two constituent parts as

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}, respectively, hereinafter. Their amplitudes derived from Eq. (5) are shown below:

_{1}and α

_{2}represent the amplitude transmission factors of the straight waveguides S

_{1}and S

_{2}, respectively. The two items in Eq. (6) as functions of the wavelength are shown in Fig. 9 . It is intuitive that they resemble the spectra of a single MRR.

*phases*of

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}are denoted as

*φ*

_{v}_{3-}

_{v}_{2}and

*φ*

_{v}_{3-}

_{p}_{2}, respectively. Here the

*phases*mean the relative phase shifts of the two electric field (

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}) with respect to

*E*, which are expressed by

_{in}*θ*

_{1}and

*θ*

_{2}are the phase shifts of the straight waveguides S

_{1}and S

_{2}, respectively. The difference between the two items in Eq. (7) is

*θ*equals to 2

*mπ*(the resonance order

*m*is an integer) [22

**36**(4), 321–322 (2000). [CrossRef]

*θ =*2

*mπ + ε*in the following derivation, where

*ε*means a perturbation around the resonant point. We also suppose that

*θ*

_{1}-

*θ*

_{2}=

*pmπ*(

*p*is a real number). Then Eq. (8) can be simplified as

*α*) is close to unity. If the variable

*p*in Eq. (9) is an odd integer, Δ

*φ*

_{v}_{3}approximately equals to

*π*regardless of the value of

*m*. It means that the two constituent parts of

*E*

_{v}_{3}(

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}) interfere destructively with each other. In other words, the amplitude of

*E*

_{v}_{3}in the four resonant regions (where

*ε*is much smaller than 2

*π*) can be expressed by

*E*

_{v}_{3}and the phase difference of its two constituent parts (

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}) are shown in Fig. 10 (a) when

*p*is an odd integer. In every resonant region, we can see a peak closely surrounded by two dips, which can be well understood by subtracting |

*E*

_{v}_{3-}

_{p}_{2}| from |

*E*

_{v}_{3-}

_{v}_{2}| in every resonant region (see Fig. 9). No degradation occurs in such occasions.

*p*in Eq. (9) is an even integer, the phase difference Δ

*φ*

_{v}_{3}equals to

*π*when the variable

*m*is an even integer. Then the two constituent parts of

*E*

_{v}_{3}(

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}) interfere destructively with each other, producing the same results in the resonant regions as in Fig. 10 (a).

*m*(the resonance order) is an odd integer while

*p*is still an even integer, the phase difference Δ

*φ*

_{v}_{3}equals to

*0*. It means that the two constituent parts of

*E*

_{v}_{3}(

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}) interfere constructively with each other. In other words, the amplitude of

*E*

_{v}_{3}in the four resonant regions (where

*ε*is much smaller than 2

*π*) can be expressed by

*E*

_{v}_{3}and the phase difference of its two constituent parts (

*E*

_{v}_{3-}

_{v}_{2}and

*E*

_{v}_{3-}

_{p}_{2}) are shown in Fig. 10 (b) when

*p*is an even integer. The degradation occurs in the resonant regions where

*m*is an odd integer. Since |

*E*

_{v}_{3-}

_{v}_{2}| and |

*E*

_{v}_{3-}

_{p}_{2}| are complementary to each other (see Fig. 9), it is intuitive that the summation of them produces trivial responses at the resonant regions where

*m*is an odd integer.

_{1}and S

_{2}) is odd multiple of half the perimeter of the ring waveguides (R

_{1}and R

_{2}), light interferes destructively in all resonant regions, resulting no degradation. If the length difference between S

_{1}and S

_{2}is even multiple of half the perimeter of the ring waveguides, light interferes destructively only when

*m*is an even integer, and interfere constructively otherwise (when

*m*is an odd integer). When light interferes constructively, it produces trivial resonant regions (degradation of the spectra). The degradation at the

*through*port (see Fig. 6 (h)) can be explained in the same way. Similar mechanism has been utilized to double the FSR of the MRR filter [26

26. J. B. Feng, Q. Q. Li, and Z. P. Zhou, “Single ring interferometer configuration with doubled free-spectral range,” IEEE Photon. Technol. Lett. **23**(2), 79–81 (2011). [CrossRef]

### 4.3. The evolution of the spectra

_{1}and S

_{2}). These asymmetric spectral line-shapes resemble the Fano resonances involving interference between a continuum and a discrete level [27

27. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. **124**(6), 1866–1878 (1961). [CrossRef]

29. M. E. Solmaz, Y. F. Zhou, and C. K. Madsen, “Modeling Asymmetric Resonances Using an Optical Filter Approach,” J. Lightwave Technol. **28**(20), 2951–2955 (2010). [CrossRef]

*drop*port, one discrete level series is

*E*

_{v}_{3-}

_{v}_{2}and the other discrete level series is

*E*

_{v}_{3-}

_{p}_{2}). It has been suggested that the increased slope due to asymmetry can be utilized to create more sensitive sensors [28

28. S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. **80**(6), 908–910 (2002). [CrossRef]

29. M. E. Solmaz, Y. F. Zhou, and C. K. Madsen, “Modeling Asymmetric Resonances Using an Optical Filter Approach,” J. Lightwave Technol. **28**(20), 2951–2955 (2010). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

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3. | H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics |

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14. | G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics |

15. | Q. F. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/s carrier-injection-based silicon micro-ring silicon modulators,” Opt. Express |

16. | A. S. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, “High-speed optical modulation based on carrier depletion in a silicon waveguide,” Opt. Express |

17. | P. Dong, R. Shafiiha, S. Liao, H. Liang, N.-N. Feng, D. Feng, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Wavelength-tunable silicon microring modulator,” Opt. Express |

18. | M. Hochberg, T. Baehr-Jones, G. Wang, M. Shearn, K. Harvard, J. Luo, B. Chen, Z. Shi, R. Lawson, P. Sullivan, A. K. Jen, L. Dalton, and A. Scherer, “Terahertz all-optical modulation in a silicon-polymer hybrid system,” Nat. Mater. |

19. | J. Takayesu, M. Hochberg, T. Baehr-Jones, E. Chan, G. Wang, P. Sullivan, Y. Liao, J. Davies, L. Dalton, A. Scherer, and W. Krug, “A hybrid electrooptic microring resonator-based 1×4×1 ROADM for wafer scale optical interconnects,” J. Lightwave Technol. |

20. | C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics |

21. | J. F. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics |

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

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

24. | J. Heebner, R. Grover, and T. A. Ibrahim, |

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

26. | J. B. Feng, Q. Q. Li, and Z. P. Zhou, “Single ring interferometer configuration with doubled free-spectral range,” IEEE Photon. Technol. Lett. |

27. | U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. |

28. | S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. |

29. | M. E. Solmaz, Y. F. Zhou, and C. K. Madsen, “Modeling Asymmetric Resonances Using an Optical Filter Approach,” J. Lightwave Technol. |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(130.3750) Integrated optics : Optical logic devices

(230.5750) Optical devices : Resonators

(250.5300) Optoelectronics : Photonic integrated circuits

(130.4815) Integrated optics : Optical switching devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 27, 2011

Manuscript Accepted: March 14, 2011

Published: March 22, 2011

**Citation**

Lei Zhang, Ruiqiang Ji, Yonghui Tian, Lin Yang, Ping Zhou, Yangyang Lu, Weiwei Zhu, Yuliang Liu, Lianxi Jia, Qing Fang, and Mingbin Yu, "Simultaneous implementation of XOR and XNOR operations using a directed logic circuit based on two microring resonators," Opt. Express **19**, 6524-6540 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6524

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