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

doi:10.1364/OE.19.006524

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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▸ Article Outline
– Abstract
1. Introduction
2. Design and fabrication
3. Experimental results
4. Simulation and discussion
5. Conclusion
– References and links

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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.

1. Introduction

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

]. 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

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]

]. 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

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

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]

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

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).

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 ‘

X • Y + \bar{X} • \bar{Y}

’ and ‘

X • \bar{Y} + \bar{X} • Y

’ at the drop and through ports of MRR₂, respectively. Those two signals are just the results of ‘

X ⊙ Y

’ and ‘

X \oplus Y

’ operations, where the symbols ‘⊙’ and ‘⊕’ represent the XNOR and XOR operators, respectively. Details on the principle are presented in [8

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

In the prototype presented in [8

], 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

]. 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

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

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

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.

We use the static response spectra obtained at the 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

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) ). 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.

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.

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 . It can be found that the XNOR and XOR operations are carried out correctly at the drop and through ports simultaneously.

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.

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

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]

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

], 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

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]

To conveniently utilize the plasma dispersion effect to achieve monolithic integrated devices on the SOI platform, we propose another operating mode in the next subsection. We show that the loss induced during modulation does not have any negative effects on the operation of the device in the second mode.

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) , 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.

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.

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 , 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.

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.

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

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

–21

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 shows the spectra obtained at the 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.

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 λ_w1 in (b) and (f), MRR₂ is tuned to λ_w1 in (c) and (g), and both MRRs are tuned to λ_w1 in (d) and (h).

4.1. The numerical model of the circuit

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

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

–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 ). 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

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

Fig. 7 Scattering matrix model of the circuit (Z₁~Z₄: coupling regions, S₁~S₂: straight waveguides, R₁~R₂: ring waveguides).

(\begin{matrix} E_{p 1} \\ E_{r 2} \end{matrix}) = (\begin{matrix} t & j k \\ j k & t \end{matrix}) (\begin{matrix} E_{in} \\ E_{r 1} \end{matrix}),

(1)

where the complex mode amplitudes are normalized such that their squared magnitudes correspond to the modal powers. The input wave E _in is chosen to be 1 so that all the field amplitudes are normalized to it [22

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

]. The coupling is assumed to be lossless so that the lumped self- and cross-coupling coefficients t and k satisfy t ² + k ² = 1 [22

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

–24

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

]. The four coupling regions shown in Fig. 7 are assumed to be identical.

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 , which agree well with the experimental results shown in Fig. 6. The parameters we adopt in the numerical simulation are summarized in Table 1 .

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 λ_w1 in (b) and (f), MRR₂ is tuned to λ_w1 in (c) and (g), and both MRRs are tuned to λ_w1 in (d) and (h).

Table 1 Parameters adopted in the numerical simulation

Variables	Explanation	Values
R ₁	radius of the ring waveguide R₁	10.0308 μm
R ₂	radius of the ring waveguide R₂	10.0318 μm
Δn ₁	index increase of R₁ when actuated	0.00306
Δn ₂	index increase of R₂ when actuated	0.00280
k	cross-coupling coefficient of the field amplitude	0.250
t	self-coupling coefficient of the field amplitude	0.968
lc _s	loss coefficient in the straight waveguides	3 dB/cm
lc _r	loss coefficient in the ring waveguides	10 dB/cm
L ₁	length of the straight waveguide S₁	552.870 μm
L ₂	length of the straight waveguide S₂	554.420 μm
α _c	transmission factor of the crossing of S₁ and S₂	0.890

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.

The degradation of half of the four resonant regions is well described by this model (see Figs. 8 (d) and (h)). To reveal the source of this degradation, the related electric fields shown in Fig. 7 are derived in the following. The two outputs of MRR₁ are expressed by

\begin{array}{l} E_{p 1} = \frac{t (1 - α exp(j θ))}{1- α t^{2} \exp (j θ)} \times E_{i n}, \\ E_{v 1} = \frac{- k^{2} α^{1 / 4} exp(j θ / 4)}{1- α t^{2} \exp (j θ)} \times E_{i n} . \end{array}

(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 = β • 2 π R = 4 π^{2} n_{e f f} \frac{R}{λ},

(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 n_eff , which is also a function of wavelength.

The phase shifts acquired by E _p1 and E _v1 with respect to E_in are

\begin{array}{l} ϕ_{p 1} = \arctan [\frac{- α \sin (θ)}{1 - α \cos (θ)}] - \arctan [\frac{- α t^{2} \sin (θ)}{1 - α t^{2} \cos (θ)}], \\ ϕ_{v 1} = π + \frac{θ}{4} - \arctan [\frac{- α t^{2} \sin (θ)}{1 - α t^{2} \cos (θ)}], \end{array}

(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

\begin{array}{l} | E_{p 1} | = {[\frac{t^{2} (1 + α^{2} - 2 α \cos (θ))}{1 + α^{2} t^{4} - 2 α t^{2} \cos (θ)}]}^{1 / 2}, \\ | E_{v 1} | = {[\frac{k^{4} α^{1 / 2}}{1 + α^{2} t^{4} - 2 α t^{2} \cos (θ)}]}^{1 / 2} . \end{array}

(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

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 (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 MRR₂. The second part comes from E_p ₂, which contributes to E_v ₃ via the dropping function of MRR₂. 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:

\begin{array}{l} | E_{v 3 - v 2} | = \frac{t^{2} (1 + α^{2} - 2 α \cos (θ))}{1 + α^{2} t^{4} - 2 α t^{2} \cos (θ)} \times α_{1}, \\ | E_{v 3 - p 2} | = \frac{k^{4} α^{1 / 2}}{1 + α^{2} t^{4} - 2 α t^{2} \cos (θ)} \times α_{2}, \end{array}

(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 . It is intuitive that they resemble the spectra of a single MRR.

Fig. 9 Amplitudes of the two constituent parts of E _v3 (E_v _3- _v ₂ and E_v _3- _p ₂) as functions of the wavelength.

The 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_in , which are expressed by

\begin{array}{l} φ_{v 3 - v 2} = 2 \arctan [\frac{- α \sin (θ)}{1 - α \cos (θ)}] - 2 \arctan [\frac{- α t^{2} \sin (θ)}{1 - α t^{2} \cos (θ)}] + θ_{1}, \\ φ_{v 3 - p 2} = \frac{θ}{2} - 2 \arctan [\frac{- α t^{2} \sin (θ)}{1 - α t^{2} \cos (θ)}] + θ_{2}, \end{array}

(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

Δ φ_{v 3} = 2 \arctan [\frac{- α \sin (θ)}{1 - α \cos (θ)}] - \frac{θ}{2} + (θ_{1} - θ_{2}) .

(8)

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

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

–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

Δ φ_{v 3} = 2 \arctan [\frac{- 1}{\frac{ε}{2} + \frac{(1 / α) - 1}{ε}}] + (p - 1) m π \approx {\begin{matrix} - π + (p - 1) m π, \begin{matrix}  \end{matrix} w h e n \begin{matrix}  \end{matrix} ε = \sqrt{2 [(1 / α) - 1]}, \\ π + (p - 1) m π, \begin{matrix}  \end{matrix} w h e n \begin{matrix}  \end{matrix} ε = - \sqrt{2 [(1 / α) - 1]} . \end{matrix}

(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, Δφ_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 2π) can be expressed by

| E_{v 3} | = | | E_{v 3 - v 2} | - | E_{v 3 - p 2} | |

(10)

The amplitude of 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.

Fig. 10 Amplitude of the electric field E _v3 and the phase difference of its two constituent parts (E_v _3- _v ₂ and E_v _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).

If the variable p in Eq. (9) is an even integer, the phase difference Δφ_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).

However, if m (the resonance order) is an odd integer while p is still an even integer, the phase difference Δφ_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 2π) can be expressed by

| E_{v 3} | = | | E_{v 3 - v 2} | + | E_{v 3 - p 2} | |

(11)

The amplitude of 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.

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

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 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

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]

]. 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 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

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]

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.

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

J. Hardy and J. Shamir, “Optics inspired logic architecture,” Opt. Express 15(1), 150–165 (2007). [CrossRef] [PubMed]
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]
H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics 4(5), 261–263 (2010). [CrossRef]
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]
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]
Q. F. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007). [CrossRef] [PubMed]
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]
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]
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]
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]
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]
R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987). [CrossRef]
G. T. Reed and A. P. Knights, Silicon Photonics: an Introduction (Wiley, West Sussex, 2004), Chap. 4.
G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010). [CrossRef]
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]
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]
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]
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]
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]
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]
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]
A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]
A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002). [CrossRef]
J. Heebner, R. Grover, and T. A. Ibrahim, Optical Microresonators: Theory, Fabrication, and Applications (Springer, London, 2008), Chap. 3.
C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (Wiley, New York, 1999), Chap. 3.
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]
U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]
S. Fan, “Sharp asymmetric line shapes in side-coupled waveguide-cavity systems,” Appl. Phys. Lett. 80(6), 908–910 (2002). [CrossRef]
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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