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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6524–6540
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Simultaneous implementation of XOR and XNOR operations using a directed logic circuit based on two microring resonators

Lei Zhang, Ruiqiang Ji, Yonghui Tian, Lin Yang, Ping Zhou, Yangyang Lu, Weiwei Zhu, Yuliang Liu, Lianxi Jia, Qing Fang, and Mingbin Yu  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6524-6540 (2011)
http://dx.doi.org/10.1364/OE.19.006524


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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

Directed logic is a newly proposed logical paradigm which takes advantage of the propagation of light to carry out Boolean functions [1

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

3

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

]. In essence, the directed logic circuit is a network of switching elements which control the propagation direction of light going through them. The states of these switching elements are determined by the operands of the Boolean functions in question. Unlike traditional implementation of optical logic, the directed logic is specially adapted to the features and promises of optics since it depends on the propagation of light other than the nonlinear interactions between light and materials [4

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

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]

]. Compared to traditional digital logic circuit, directed logic circuit has markedly less state delay because the operands that determine the states of the switching elements do not pass through the preceding elements in the circuits [1

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

]. Therefore, all elements can perform their switching functions simultaneously and the results are given instantaneously [1

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

3

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

].

We have demonstrated an XOR/XNOR directed logic circuit where two microring resonators (MRRs) were employed as the switching elements [8

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]

]. The fabricated MRRs had low extinction ratio (ER< 5 dB) at the XNOR port due to the high loss factor in the ring waveguides, which made it fail to implement the XOR and XNOR operations simultaneously.

In this paper we report the optimized device based on the same architecture, which are capable of carrying out the XOR and XNOR operations at the same time. We propose two different operating modes for the circuit and prove that it is suitable for faster operation. The spectra of the fabricated device are analyzed by the scattering matrix method. A novel behavior of the spectra is well explained through numerical simulations.

2. Design and fabrication

The directed logic circuit is shown in Fig. 1 (a)
Fig. 1 (a) Schematic and (b) micrograph of the XOR/XNOR directed logic circuit based on two cascaded microring resonators (CW: continuous wave, EPT: electrical pulse train, OPT: optical pulse train, MRR: microring resonator).
. The two MRRs function as optical switching elements in this circuit. The four ports of each MRR are denoted as 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).

Two logic signals of 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 ‘1s’) and off-resonance at λ if X and Y are at low level (representing ‘0s’). According to these rules, we can obtain two logic outputs of ‘XY+X¯Y¯’ and ‘XY¯+X¯Y’ at the drop and through ports of MRR2, respectively. Those two signals are just the results of ‘XY’ and ‘XY’ operations, where the symbols ‘⊙’ and ‘⊕’ represent the XNOR and XOR operators, respectively. Details on the principle are presented in [8

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]

]. Those two output ports of the circuit are called drop port and through port hereinafter for simplicity.

In the prototype presented in [8

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]

], channel waveguides with the cross-section of 400 nm × 220 nm were adopted to fabricate the device. The gaps between the ring and straight waveguides were 400 nm. Only the XOR operation was achieved at the 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

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]

]. To reduce this effect, an elliptical crossing (long axis = 6.25 μm, and short axis = 1.5 μm) is adopted in the optimized device [9

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]

].

The device is fabricated on a silicon-on-insulator (SOI) wafer with 220-nm-thick top Si layer and 2-µm-thick buried dioxide layer. 248-nm deep ultraviolet (UV) photolithography is used to define the device pattern. Inductively coupled plasma etching process is used to etch the top Si layer. Spot size converters (SSCs) are integrated on the input and output terminals of the waveguides to enhance the coupling efficiency between the waveguides and the fibers. The SSC is a 200-µm-long linearly inversed taper with 180-nm-wide tip. After the waveguide is etched, a 1500-nm-thick silica layer is deposited on the Si core layer as the separate layer (SL) by plasma enhanced chemical vapor deposition. Then a 200-nm-thick titanium (Ti) is sputtered on the SL and Ti heaters are fabricated by ultraviolet photolithography and dry etching. Two Ω-shaped heaters are adopted to tune the MRRs [10

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]

]. Aluminum wires and pads are fabricated after the heaters are done. Finally, the end-face of the SSC is exposed by a 110-µm-deep etching process as the world-to-chip interface. Details on the fabrication process are given in [11

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]

]. The micrograph of the device is shown in Fig. 1(b).

3. Experimental results

An amplified spontaneous emission (ASE) source, an optical spectrum analyzer (OSA) and two tunable voltage sources are used to characterize the fabricated device. The broadband light is coupled into the device through a lensed fiber. The output light is collected by another lensed fiber and fed into the OSA. The two tunable voltage sources are used to drive the two heaters above the MRRs. When an MRR is heated up, the effective refractive index (n eff) of the ring waveguide increases and the resonant wavelength of the MRR increases accordingly.

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

The working wavelength is determined from the spectra obtained at the drop port when neither of the two heaters is actuated (see Fig. 2(a)
Fig. 2 Response spectra obtained at (a-d) the drop port and (e-h) the through port of the device. Voltages applied to MRR1 and MRR2 are both 0 V in (a) and (e), 3.13 V and 0 V in (b) and (f), 0 V and 2.15 V in (c) and (g), and 3.13 V and 2.15 V in (d) and (h). The dashed arrow indicates the location of the working wavelength λ w1.
). 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 ‘0s’). 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.

After the working wavelength has been chosen, the analog voltage representing logic ‘1’ for each MRR should be determined. Firstly, only a voltage of 3.13 V is applied to MRR1 to make it resonate at λ w1 (see Fig. 2(b)). Then only a voltage of 2.15 V is applied to MRR2 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 ‘1s’ 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 MRR1 and further dropped by MRR2 to the circuit’s drop port. It means that a result of ‘1’ is obtained at the drop port when both operands are ‘1s’.

We can find in Figs. 2 (b) and (c) that the resonant wavelengths for the MRR2 and MRR1 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 MRR1 to λ w1. The responses at the through port are obtained in the same way, which are shown in Figs. 2 (e-h).

After the working wavelength and the analog voltages representing logic ‘1s’ 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
Fig. 3 Signals applied to two MRRs and detected at the drop and through ports in the first operating mode. Signals applied to (a) MRR1 and (b) MRR2. Results of (c) XNOR operation result at the drop port and (d) XOR operation result at the through port.
. It can be found that the XNOR and XOR operations are carried out correctly at the drop and through ports simultaneously.

As shown in Fig. 3 (c) and (d), there are positive spikes between two consecutive outputs of ‘0s’, as well as negative spikes between two consecutive ‘1s’, 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 MRR1 been actuated in the first time-slot, and only MRR2 been actuated in the second time-slot), a positive spike (between two consecutive ‘0s’) and a negative spike (between two consecutive ‘1s’) 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.

The duration times of those spikes, as well as the rising and falling times of the output signals, which limit the working speed of the device, are determined by the transition time of the waveguide’s index alteration. To achieve faster operations, we can employ other advanced modulation schemes such as the plasma dispersion effect or the electric field effects to modulate the MRRs [12

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

21

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]

]. However, due to the remarkable absorption induced by the plasma dispersion effect when it alters the index of the waveguide [12

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

17

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]

], a high Q factor is required when employing the plasma dispersion effect in this operating mode (see Fig. 2, a high Q factor means a small tuning is sufficient to achieve an acceptable ER, so the induced loss may be negligible). Nevertheless, we can also achieve high-speed operation adopting the electric field effects with the aid of CMOS-compatible optical polymers, which do not introduce significant absorption during modulation if proper organic material is used [18

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

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

To briefly explain the principle of the second operating mode, we assume that the resonant wavelengths of both MRRs equal to 1540 nm, which is chosen to be the working wavelength and denoted as 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.

The resonant wavelengths of the two MRRs are not identical in reality due to the fabrication errors (see Figs. 2 (b) and 2 (c)). To compensate for that, a non-zero bias voltage should be applied to the MRR with a shorter resonant wavelength to make it resonate at the same wavelength as the other MRR. We can judge from Fig. 2 that MRR1 has a shorter resonant wavelength (1539.920 nm). The spectrum obtained at the drop port with MRR1 being applied with an offset voltage of 0.8 V is shown in Fig. 4 (a)
Fig. 4 Response spectra obtained at (a-d) the drop port and (e-h) the through port of the device. Voltages applied to MRR1 and MRR2 are 0.8 V and 0 V in (a) and (e), 3.07 V and 0 V in (b) and (f), 0.8 V and 2.15 V in (c) and (g), and 3.07 V and 2.15 V in (d) and (h). The dashed arrow indicates the location of the working wavelength λ w2.
, 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.

When only MRR1 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 MRR1 away from λ w2. Precise alignment of the voltage amplitude is not required. When only MRR2 is actuated (with a voltage of 2.15 V applied to MRR2, and a voltage of 0.8 V applied to MRR1 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).

In this operating mode, the voltages corresponding to logic ‘0s’ for the two MRRs are 0.8 V and 0 V, respectively. And the voltages corresponding to logic ‘1s’ 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
Fig. 5 Signals applied to two MRRs and detected at the drop and through ports in the second operating mode. Signals applied to (a) MRR1 and (b) MRR2. Result of (c) XNOR operation result at the drop port and (d) XOR operation result at the through port.
, 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.

It should be noted that if the absorption coefficients of the waveguides increase when the MRRs are actuated, the circuit can still work properly. Because the function of the actuations is to make the MRRs inactive at the working wavelength, the accompanying losses do no harm to the light we concern about. In other words, the light at the working wavelength will bypass the MRRs whose resonant wavelengths have been shifted away and the losses have probably increased. That is why this circuit can be implemented employing the plasma dispersion effect or the electric field effects, which always increase the absorption coefficients of the waveguides while altering their refractive indices [12

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

21

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

In this section, we show the spectra in a much larger region to get a comprehensive understanding of this photonic integrated circuit. Figure 6
Fig. 6 Response spectra obtained at (a-d) the drop port and (e-h) the through port when the device operates in the first mode. No voltages are applied in (a) and (e), MRR1 is tuned to λw1 in (b) and (f), MRR2 is tuned to λw1 in (c) and (g), and both MRRs are tuned to λw1 in (d) and (h).
shows the spectra obtained at the drop and through ports of the device when it operates in the first mode (λw1 = 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

The spectra of the circuit are obtained using the scattering matrix method [22

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

24

24. J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, London, 2008), Chap. 3.

]. The whole device can be classified into three subsidiary sets according to their functions (see Fig. 7
Fig. 7 Scattering matrix model of the circuit (Z1~Z4: coupling regions, S1~S2: straight waveguides, R1~R2: ring waveguides).
). The first set consists of four coupling areas which cause energy exchanges among the straight waveguides and the ring waveguides. These coupling areas are described by four scattering matrixes, whose elements are assumed to be independent of wavelength [24

24. J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, London, 2008), Chap. 3.

]. The second set consists of two ring waveguides and two straight waveguides, which cause amplitude attenuations and phase shifts to the light passing through them. The third set is the crossing of two straight waveguides, which possibly causes scattering when the light passes through it. The scattering matrix describing the energy exchange in the coupling region Z1 is in the form of

(Ep1Er2)=(tjkjkt)(EinEr1),
(1)

The simulation results of the spectra at the drop and through ports of the device based on the above model are shown in Fig. 8
Fig. 8 Response spectra obtained in simulations at (a-d) the drop port and (e-h) the through port when the device operates in the first mode. No voltages are applied in (a) and (e), MRR1 is tuned to λw1 in (b) and (f), MRR2 is tuned to λw1 in (c) and (g), and both MRRs are tuned to λw1 in (d) and (h).
, which agree well with the experimental results shown in Fig. 6. The parameters we adopt in the numerical simulation are summarized in Table 1

Table 1. Parameters adopted in the numerical simulation

table-icon
View This Table
.

The finite-element method (FEM) is used to calculate the 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.

Ep1=t(1αexp(jθ))1-αt2exp(jθ)×Ein,Ev1=k2α1/4exp(jθ/4)1-αt2exp(jθ)×Ein.
(2)

The single-pass amplitude transmission factors and the single-pass phase shifts in the ring waveguides R1 and R2 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

θ=β2πR=4π2neffRλ,
(3)

where 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 neff, which is also a function of wavelength.

The phase shifts acquired by Ep1 and Ev1 with respect to Ein are

ϕp1=arctan[αsin(θ)1αcos(θ)]arctan[αt2sin(θ)1αt2cos(θ)],ϕv1=π+θ4arctan[αt2sin(θ)1αt2cos(θ)],
(4)

which can be interpreted as the phase transfer functions of the through and drop ports of MRR1, respectively. The amplitude transfer functions of the through and drop ports of MRR1 derived from Eq. (2) are

|Ep1|=[t2(1+α22αcos(θ))1+α2t42αt2cos(θ)]1/2,|Ev1|=[k4α1/21+α2t42αt2cos(θ)]1/2.
(5)

4.2. Analysis of the degradation in the spectra

In the following, we analyze the response spectra at the drop port of the circuit based on Eqs. (4) and (5). We show that the spectra vary with the length difference (ΔL) between the two straight waveguides S1 and S2. 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 ΔL equals even multiples of half the perimeter of the ring waveguides.

Since the MRR is linear time-invariant (LTI) system in our case [25

25. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999), Chap. 3.

], the electric field at the drop port (Ev 3) can be decomposed into two separate parts. The first one comes from Ev 2, which contributes to Ev 3 via the pass-through function of MRR2. The second part comes from Ep 2, which contributes to Ev 3 via the dropping function of MRR2. We denote these two constituent parts as Ev 3- v 2 and Ev 3- p 2, respectively, hereinafter. Their amplitudes derived from Eq. (5) are shown below:

|Ev3v2|=t2(1+α22αcos(θ))1+α2t42αt2cos(θ)×α1,|Ev3p2|=k4α1/21+α2t42αt2cos(θ)×α2,
(6)

In Eq. (6), α1 and α2 represent the amplitude transmission factors of the straight waveguides S1 and S2, respectively. The two items in Eq. (6) as functions of the wavelength are shown in Fig. 9
Fig. 9 Amplitudes of the two constituent parts of Ev3 (Ev 3- v 2 and Ev 3- p 2) as functions of the wavelength.
. It is intuitive that they resemble the spectra of a single MRR.

The phases of Ev 3- v 2 and Ev 3- p 2 are denoted as φv 3- v 2 and φv 3- p 2, respectively. Here the phases mean the relative phase shifts of the two electric field (Ev 3- v 2 and Ev 3- p 2) with respect to Ein, which are expressed by

φv3v2=2arctan[αsin(θ)1αcos(θ)]2arctan[αt2sin(θ)1αt2cos(θ)]+θ1,φv3p2=θ22arctan[αt2sin(θ)1αt2cos(θ)]+θ2,
(7)

where θ 1 and θ 2 are the phase shifts of the straight waveguides S1 and S2, respectively. The difference between the two items in Eq. (7) is

Δφv3=2arctan[αsin(θ)1αcos(θ)]θ2+(θ1θ2).
(8)

For resonant wavelengths, θ equals to 2 (the resonance order m is an integer) [22

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

24

24. J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, London, 2008), Chap. 3.

]. To investigate the behavior of Eq. (8) around the resonant wavelengths, we suppose that θ = 2mπ + 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

Δφv3=2arctan[1ε2+(1/α)1ε]+(p1)mπ{π+(p1)mπ,whenε=2[(1/α)1],π+(p1)mπ,whenε=2[(1/α)1].
(9)

In the above derivation, we assume that the single-pass amplitude transmission in the ring waveguides (α) is close to unity. If the variable p in Eq. (9) is an odd integer, Δφv 3 approximately equals to π regardless of the value of m. It means that the two constituent parts of Ev 3 (Ev 3- v 2 and Ev 3- p 2) interfere destructively with each other. In other words, the amplitude of Ev 3 in the four resonant regions (where ε is much smaller than 2π) can be expressed by

|Ev3|=||Ev3v2||Ev3p2||
(10)

The amplitude of Ev 3 and the phase difference of its two constituent parts (Ev 3- v 2 and Ev 3- p 2) are shown in Fig. 10 (a)
Fig. 10 Amplitude of the electric field E v3 and the phase difference of its two constituent parts (Ev 3- v 2 and Ev 3- p 2) when the length difference of S1 and S2 is (a) odd multiples of half the perimeter of the ring waveguides (p = 1), (b) even multiples of half the perimeter of the ring waveguides (p = 0).
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 |Ev 3- p 2| from |Ev 3- v 2| in every resonant region (see Fig. 9). No degradation occurs in such occasions.

If the variable p in Eq. (9) is an even integer, the phase difference Δφv 3 equals to π when the variable m is an even integer. Then the two constituent parts of Ev 3 (Ev 3- v 2 and Ev 3- p 2) interfere destructively with each other, producing the same results in the resonant regions as in Fig. 10 (a).

However, if m (the resonance order) is an odd integer while p is still an even integer, the phase difference Δφv 3 equals to 0. It means that the two constituent parts of Ev 3 (Ev 3- v 2 and Ev 3- p 2) interfere constructively with each other. In other words, the amplitude of Ev 3 in the four resonant regions (where ε is much smaller than 2π) can be expressed by

|Ev3|=||Ev3v2|+|Ev3p2||
(11)

The amplitude of Ev 3 and the phase difference of its two constituent parts (Ev 3- v 2 and Ev 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 | Ev 3- v 2 | and |Ev 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.

To sum up, if the length difference between the two arms (S1 and S2) is odd multiple of half the perimeter of the ring waveguides (R1 and R2), light interferes destructively in all resonant regions, resulting no degradation. If the length difference between S1 and S2 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

In the last subsection, we have analyzed the degradation of the spectra using the scattering matrix method. To get a more comprehensive understanding of this circuit, we present more simulation results of the spectra in this part. Figure 11
Fig. 11 The spectra obtained at (a-c) the drop port and (d-f) the through port when the length differences between S1 and S2 are C/8, C/4, and 3C/8. C represents the ring waveguide’s perimeter.
shows the spectra of the circuit with varying length difference between the two arms (S1 and S2). 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

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]

]. While in our case, the asymmetry of the spectra is caused by the interference between two discrete level series (e. g. at the drop port, one discrete level series is Ev 3- v 2 and the other discrete level series is Ev 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

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]

].

Although this circuit has such a feature that the spectra at both ports vary with the arms’ length difference, it does not hinder the circuit’s function of carrying out the XOR and XNOR operations. Since the perimeter of the ring waveguide is about 60 μm (assuming that the ring waveguide has a moderate radius of 10 μm), the margin for the fabrication-caused length error of the two arms is quite large (a deviation of a couple of micrometers from the intended value is tolerable).

5. Conclusion

We implement simultaneous XOR and XNOR operations using an optimized directed logic circuit based on two cascaded microring resonators. Bitwise operations at 10 kbit/s are demonstrated in two different operating modes employing thermo-optic modulation. We prove that the circuit can be readily realized using advanced modulating schemes such as the plasma dispersion effect or the electric field effects. The attenuation associated with those advanced modulating schemes when they modify the index of the waveguide does not hinder the operation of the circuit in the second operating mode. In addition to the demonstration of logic operations, we analyze the behavior of the spectra of the device using the scattering matrix method. The degradation of the spectra observed in the experiment is well explained in the numerical model based on the fact that the MRR is an LTI system. The variation of the spectra is found to be associated with the length difference between the two straight waveguides in the circuit. This is an intuitive result since the proposed circuit can be regarded as an MZI with the two straight waveguides as its two arms. Such a circuit can produce Fano-like resonance line-shapes when the two arms’ length difference is not integer multiples of half of the ring waveguide’s perimeter.

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (NSFC) under grants 60877015 and 60977037, and by the National High Technology Research and Development Program of China under grant 2009AA03Z416.

References and links

1.

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

2.

H. J. Caulfield, R. A. Soref, and C. S. Vikram, “Universal reconfigurable optical logic with silicon-on-insulator resonant structures,” Photonics Nanostruct. Fund. Appl. 5(1), 14–20 (2007). [CrossRef]

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]

5.

X. Zhang, Y. Wang, J. Q. Sun, D. M. Liu, and D. X. Huang, “All-optical AND gate at 10 Gbit/s based on cascaded single-port-couple SOAs,” Opt. Express 12(3), 361–366 (2004). [CrossRef] [PubMed]

6.

Q. F. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007). [CrossRef] [PubMed]

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]

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]

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]

12.

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

13.

G. T. Reed and A. P. Knights, Silicon Photonics: an Introduction (Wiley, West Sussex, 2004), Chap. 4.

14.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010). [CrossRef]

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 15(2), 430–436 (2007). [CrossRef] [PubMed]

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 15(2), 660–668 (2007). [CrossRef] [PubMed]

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]

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. 27(4), 440–448 (2009). [CrossRef]

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]

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]

22.

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

23.

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

24.

J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, London, 2008), Chap. 3.

25.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999), Chap. 3.

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]

27.

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

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]

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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References

  1. J. Hardy and J. Shamir, “Optics inspired logic architecture,” Opt. Express 15(1), 150–165 (2007). [CrossRef] [PubMed]
  2. H. J. Caulfield, R. A. Soref, and C. S. Vikram, “Universal reconfigurable optical logic with silicon-on-insulator resonant structures,” Photonics Nanostruct. Fund. Appl. 5(1), 14–20 (2007). [CrossRef]
  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]
  5. X. Zhang, Y. Wang, J. Q. Sun, D. M. Liu, and D. X. Huang, “All-optical AND gate at 10 Gbit/s based on cascaded single-port-couple SOAs,” Opt. Express 12(3), 361–366 (2004). [CrossRef] [PubMed]
  6. Q. F. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007). [CrossRef] [PubMed]
  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]
  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]
  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]
  12. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
  13. G. T. Reed and A. P. Knights, Silicon Photonics: an Introduction (Wiley, West Sussex, 2004), Chap. 4.
  14. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010). [CrossRef]
  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 15(2), 430–436 (2007). [CrossRef] [PubMed]
  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 15(2), 660–668 (2007). [CrossRef] [PubMed]
  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]
  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. 27(4), 440–448 (2009). [CrossRef]
  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]
  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]
  22. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]
  23. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002). [CrossRef]
  24. J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, London, 2008), Chap. 3.
  25. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999), Chap. 3.
  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]
  27. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]
  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]

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