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XOR and XNOR operations at 12.5 Gb/s using cascaded carrier-depletion microring resonators Lin Yang,* Lei Zhang, Chunming Guo, and Jianfeng Ding State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China * [email protected]

Abstract: We report the implementation of the XOR and XNOR logical operations using an electro-optic circuit, which is fabricated by CMOScompatible process in the silicon-on-insulator (SOI) platform. The circuit consists of two cascaded add-drop microring resonators (MRRs), which are modulated through electric-field-induced carrier depletion in reverse biased pn junctions embedded in the ring waveguides. The resonance wavelength mismatch between the two nominally identical MRRs caused by fabrication errors is compensated by thermal tuning. Simultaneous bitwise XOR and XNOR operations of the two electrical modulating signals at the speed of 12.5 Gb/s are demonstrated. And 20 Gb/s XOR operation at one output port of the circuit is achieved. We explain the phenomena that one half of the resonance regions of the device are much more sensitive to the round-trip phase shift in the ring waveguides than the other half resonance regions. Characteristic graphs with logarithmic phase coordinate are proposed to analyze the sensitivity of the demonstrated circuit, as well as several typical integrated optical structures. It is found that our circuit with arbitrary chosen parameters has similar sensitivity to MRRs under the critical coupling. ©2014 Optical Society of America OCIS codes: (130.3750) Optical logic devices; (130.4815) Optical switching devices; (230.4555) Coupled resonators; (250.5300) Photonic integrated circuits.

References and links 1.

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Received 13 Dec 2013; revised 27 Jan 2014; accepted 28 Jan 2014; published 31 Jan 2014 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.002996 | OPTICS EXPRESS 2996

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1. Introduction Exclusive or (XOR) is a logical operation that outputs true whenever its two inputs differ with each other. The opposite of XOR is exclusive nor (XNOR), which outputs true whenever both inputs are the same. Such two logical operations are indispensable in digital communication and computing. The XOR operation can be employed in many occasions such as label processing [1, 2], parity checking [3, 4], data encryption [5, 6], and pseudorandom number

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generation [7, 8]. With the aim of eliminating the optical-electronic-optical (OEO) conversion in optical communication system, all-optical XOR and XNOR logical operations attract most attentions [9–21]. The semiconductor optical amplifier (SOA) is the most widely employed element to achieve these two operations [9–12]. The nonlinear behavior that is a drawback for the SOA as a linear amplifier makes it a good choice for an optically controlled optical gate [9]. Other components that can be used to implement all-optical XOR and XNOR logic gates are periodically poled lithium niobate (PPLN) waveguide [13–15], highly nonlinear fiber (HNLF) [16, 17], silicon waveguides [18, 19], and chalcogenide planar waveguide [20, 21]. From the development process of optical XOR and XNOR gates, it can be found that there is a general trend to achieve such functions in a more compact manner with low power consumption. Although there is a long history of pursuing all-optical information processing, there is a common view that electrical systems are adept in precise control and optical systems have an overwhelming advantage in massive information transfer. It is straightforward to come up with the idea that it may bring benefits to combine them together to do signal processing functions. In 2007, Hardy and Shamir proposed a logic paradigm called directed logic, which takes advantage of the propagation of light to carry out Boolean functions [22]. As an original proposal, they did not specify which scheme to be employed to control the propagation of light. By taking advantage of the high refractive index contrast between silicon and silicon dioxide—and using silicon-on-insulator (SOI) wafers similar to those employed for advanced transistors—engineers can now construct micrometer-scale integrated optical circuits with complex functions and ultra-low power consumptions [23]. The benefits that the SOI platform can offer us are silicon waveguides with small dimensions, low losses, high optical mode confinement, as well as the possibility of integrated with electrical controlling modules to form a highly integrated and complex system. Such a prospect will probably make traditional optical systems acquire a completely new outlook. Since it is convenient to integrated electrical controlling parts with optical waveguides in the SOI platform, several prototype directed-logic devices have been proposed and demonstrated using silicon photonic devices [24–28]. In all the demonstrations by far, microring resonators (MRRs) are employed to construct optical switches, which have tiny volume and high sensitivity to the phase vibration due to the multiple-beam interference mechanism in MRRs. Thermo-optic effect and electro-optic effect have been used to modulate the MRR-based optical switches. The thermo-optic effect has high tuning efficiency but low response time [24, 25]. Thermal tuning is also adopted in electro-optic modulating circuits to compensate the fabrication errors. Up to now, only electric-field-induced carrier injection is adopted to construct directed logic circuits, which can just achieve several hundred Mb/s operations [26–28]. Here in this paper, we report the demonstration of carrierdepletion directed logic circuit with the XOR and XNOR functions. Simultaneous bitwise XOR and XNOR operations at two output ports of the circuit with the speed of 12.5 Gb/s are demonstrated. And 20 Gb/s XOR operation at one output port of the circuit is achieved. We explain the phenomena that one half of the resonance regions of the device are much more sensitive to the round-trip phase shift in the ring waveguides than the other half resonance regions. Characteristic graphs with logarithmic phase coordinate are proposed to analyze the sensitivity of the demonstrated circuit, as well as several typical integrated optical structures. It should be pointed out that the SOI platform not only offer us the convenience in integrating electrical controlling modules and optical waveguides, but also provide the possibility of achieving all-optical directed logic circuits with high efficiency and small volume due to the high optical mode confinement in silicon waveguides. We think that both the electro-optic and all-optic schemes have their own niche applications [29, 30]. And the future work should focus on the implementation of directed logic circuits with more complex and more reconfigurable functions. 2. Design and fabrication The schematic of the XOR/XNOR directed logic circuit based on two cascaded carrierdepletion MRRs is shown in Fig. 1 (a). The four ports of each MRR are denoted as input,

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through, add and drop according to their functions. Monochromatic light with the wavelength of λ coupled into the input and add ports will be directed to the through and drop ports (i.e. bypass the MRR), respectively, when the MRR is off-resonance at λ. And if the MRR is onresonance at λ, light coupled into the input and add ports will be guided to the drop and through ports, respectively. The resonance status of each MRR is controlled by an electrical signal X and Y, respectively. Although the architecture in Fig. 1 (a) is as same as that employed in our previous work [28], the operation mode is slightly different. This difference stems from the distinction between the carrier-injection and carrier-depletion modulation modes. In the previous work, the carrier-injection modulation is employed, which tunes the MRR to be off-resonance when a positive voltage is applied to the PN diode. In other words, a logic ‘1’ makes the MRR to be off-resonance. While in the current work, the carrier-depletion modulation is employed, which tunes the MRR to be off-resonance when a negative voltage is applied to the PN diode. In this case, a logic ‘0’ makes the MRR to be off-resonance.

Fig. 1. (a) Schematic and (b) micrograph of the XOR/XNOR directed logic circuit based on two cascaded carrier-depletion microring resonators (CW: continuous wave, MRR: microring resonator, EPT: electrical pulse train, OPT: optical pulse train).

The working principle of the device in the current work is summarized as follows. The two MRRs are initially on-resonance at the input wavelength of λ, when no electrical signals are applied. When the electrical signal is at low level, the applied voltage is negative, which tunes the MRR to be off-resonance through extracting carriers from the PN diode. And when the electrical signal is at high level, the applied voltage is 0 V, which does not change the resonance status of the MRR. So if the two applied electrical signals are same (both at high or low level), the input light will be directed to the through port of the circuit. If the two applied electrical signals are different (one is at high level and the other one is at low level), the input light will be directed to the drop port of the circuit. This means that we can achieve the XOR and XNOR operations of the two applied electrical signals at the drop and through ports, respectively. All the four situations are summarized in Table 1. It should be noted that the denotation of the output ports of the circuit is different from that in the previous work [28].

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Table 1. Relationships between Logical Operations, Applied Voltages, Resonance Statuses of MRRs, and Logical Operation Outputs Oper and X

Oper and Y

0

0

0

1

1

0

1

1

Volta ge on MRR1 negati ve negati ve 0V 0V

Volta ge on MRR2 negati ve 0V negati ve 0V

Resonan ce status of MRR1 Off

Resonan ce status of MRR2 Off

XO R result 0

X NOR result 1

Off

On

1

0

On

Off

1

0

On

On

0

1

It can be noted in Fig. 1 that there are two arched segments in the waveguides connecting the four coupling areas of the two MRRs. Such two arched waveguides are designed on purpose to adjust the length difference between the two arms connecting the two MRRs, which has a remarkable impact on the response spectra of the device [25]. We will discuss its impact in detail later in this paper. The device is fabricated on an 8 in. (20.3 cm) silicon-on-insulator (SOI) wafer with 220nm-thick top silicon and 2-μm-thick buried oxide layer. Rib waveguides with a height of 220 nm, a width of 400 nm and a slab thickness of 70 nm are used to construct the circuit, which only supports quasi-TE fundamental mode. The gaps between ring and straight waveguides are chosen to be 400 nm to achieve a balance between the extinction ratios of the drop and through ports of each MRR. The radii of the ring waveguides are both 10 μm. An elliptical structure (long axis = 6.25 μm, and short axis = 1.5 μm) is adopted to reduce the scattering at the crossing of the waveguides. 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 (Fig. 2 (a) and (b)). Spot size converters (SSCs) are integrated on the input and output terminals of the waveguides to enhance the coupling 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, two PN diodes are formed in the two ring waveguides. The p-type doping concentration is 1 × 1018/cm3 and the n-type doping concentration is 8 × 1017/cm3 (Fig. 2 (c) and (d)). The PN junction is designed to be an abrupt junction and the peak doping concentration for both p- and n-doping regions locates at the center of the rib waveguide in the vertical direction. In the lateral direction, the PN junction is right to the center of the rib waveguide with the offset of 40 nm. In other words, 220nm of the total width of 400 nm is p-doped, and the other 180 nm is n-doped. Such a doping profile makes the ptype depletion region have the maximum overlap with the optical mode in the ridge waveguide since the p-type carrier has a modulation effect around three times larger than the n-type carrier at such a doping concentration level [31]. Next, the anode (boron, p+ ~5.5 × 1020/cm3) and cathode (phosphorus, n+ ~5.5 × 1020/cm3) implants are formed (Fig. 2 (e) and (f)). After the doping, a 1500-nm-thick silica layer is deposited on the Si layer as the separate layer (SL) by plasma enhanced chemical vapor deposition (PECVD). Then a 150-nm-thick titanium nitride (TiN) layer is sputtered on the SL and two microheaters are fabricated by deep UV photolithography and dry etching (Fig. 2(g)) [28]. Another silica layer of 300 nm is deposited by PECVD on the TiN heaters. Via holes to the PN diodes and microheaters are etched on the silica layer in two steps (Fig. 2 (h) and (i)). Then a 1000-nm-thick aluminum layer is sputtered and etched to be wires and pads connected to the microheaters and PN diodes (Fig. 2 (j)). Finally, the end-face of the SSC is exposed by a 110-µm-deep etching process as the world-to-chip interface (Fig. 2 (k)). The micrograph of the device is shown in Fig. 1(b). The 200-µm-long SSCs are not included in this micrograph. The two square pads located at the lower-left and lower-right of the micrograph have side lengths of 100 µm. The effective area of the device including the SSCs is about 1.4 × 0.4 mm2.

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Fig. 2. Process flow of the device: (a) and (b) etching of the top Si layer by 150 nm and 70 nm, respectively, (c) and (d) p- and n-doping and through boron and phosphorus implantation, (e) and (f) p+- and n+-doping and through boron and phosphorus implantation, (g) deposition and etching of the TiN layer to form the microheater, (h) and (i) etching of the SiO2 layer to form the via holes to the PN diodes and microheaters, (j) deposition and etching of the Al layer to form the wires and pads, (k) deep etching to form the end-face of the spot size converters.

3. Experimental results Broadband light from an amplified spontaneous emission (ASE) source is coupled into the device through a lensed fiber. The output light is collected by another lensed fiber and fed into the optical spectrum analyzer (OSA). A tunable voltage source is used to drive the microheater above the MRR with shorter resonance wavelength to make it resonate at the same wavelengths as the other MRR. The response spectra at the two output ports of the device are shown in Fig. 3, with MRR1 being tuned by a heating voltage of 2.04 V to align the resonance wavelengths of the two MRRs. As the two arms connecting the two MRRs have the same lengths, the first and the third resonance regions in Fig. 3 are degenerate, which has been shown and explained in [25]. The spectra of the through ports at these two degenerate resonant regions should be flat due to the constructive interference between two light beams from two different paths [25]. Shallow dips still appear at these two regions due to the difference of the two nominally identical connecting arms and MRRs caused by fabrication errors. The thermal tuning just aligns the resonance wavelengths of the two MRRs, their line shapes may still differ from each other.

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Fig. 3. Response spectra obtained at the through and drop ports of the fabricated device with MRR1 being tuned by a heating voltage of 2.04 V to make it resonate at the same wavelength as MRR2 at the third resonance region.

3.1 Static characterization

Fig. 4. Spectra obtained at (a, b) the through port and (c, d) the drop port of the device. The left two figures are the responses of MRR1, which is thermally tuned to be resonant around 1552.4 nm. The right two figures are the responses of MRR2, which is resonant around 1550.6 nm.

The characterization of the static response spectra of the devices consists of two parts. Firstly, we characterize each MRR’s electro-optical response under different reverse biases. Since the two MRRs have similar resonance wavelengths, we tune MRR1 to resonate far away from MRR2 through heating. And then the responses at the through and drop ports of the device are recorded when different reverse biases from 0 V to 8 V are applied to both MRRs. The results in Fig. 4 show that the two MRRs have similar electro-optical response. The loaded quality factors (Q factors) of a single MRR at the through and drop ports are about 15,000 and 12,000, respectively. The shifts of the resonance wavelength are about 50 pm and 60 pm, respectively, when the reverse biases applied to the MRR are 4 V and 8 V.

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Fig. 5. Response spectra obtained at (a-d) the through port and (e-h) the drop port of the device. Voltages applied to the two PN diodes are both 0 V in (a) and (e), 4 V and 0 V in (b) and (f), 0 V and 4 V in (c) and (g), and both 4 V in (d) and (h). The dashed arrow indicates the location of the working wavelength. In all cases, MRR1 has a heating voltage of 2.04 V.

Secondly, the static working statuses of the device are validated. The working wavelength is determined from the spectra when neither of the two MRRs is actuated (Fig. 3). The response spectra around the third resonance region in Fig. 3 are shown in detail in Fig. 5. According to the aforementioned principle, a maximum (representing a ‘1’) and a minimum (representing a ‘0’) should be obtained at the through port and the drop port, respectively, when the two applied electrical signals are both at high level (0 V, representing ‘1s’). We choose 1550.64 nm as the working wavelength (Fig. 5 (a) and (e)). As shown in Fig. 5 (a-d), a maximum is obtained at the through port when the two applied electrical signals are both at low levels or high levels, and a minimum is obtained otherwise. As shown in Fig. 5 (e-h), a minimum is obtained at the drop port when the two applied electrical signals are both at low levels or high levels, and a maximum is obtained otherwise. Therefore, the XNOR and XOR operations are performed correctly at the through and drop ports of the device, respectively.

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3.2 Dynamic operation A monochromatic light at 1550.64 nm from a tunable laser is coupled into the device. Two user-defined non-return-to-zero (NRZ) signals with a period of 8 bits at the speed from 1 Gb/s to 20 Gb/s are applied to the two MRRs simultaneously. The two output optical signals are fed into a wideband sampling oscilloscope (Agilent DCA-X 860100D), which has 65 GHz optical and 80 GHz electrical plug-in modules. Since the oscilloscope has only one electrical port and one optical port, the two electrical signals applied to the two MRRs are measured separately, as well as the two output optical signals. The patterns of the two high-speed electrical signals are ‘00100101’ and ‘01001111’, respectively. Their XOR and XNOR operation results should be ‘01101010’ and ‘10010101’, respectively. Typical experimental results when the operation speed is 5Gb/s are shown in Fig. 6, in which the patterns of the applied electrical signals and the output optical signals are marked. These waveforms are not aligned with each other at the time axis since they are measured one by one. It can be found from Fig. 6 (c) and (d) that the two logic operations are carried out correctly at the drop and through ports simultaneously. It should be pointed out that the optical power of the monochromatic light from the tunable laser is about 3 dBm. After the coupling between the lensed fiber and the spot size converter, light injected to the silicon waveguide is about 0 dBm. Such a power level is too low to excite the nonlinear processes in the silicon waveguide, even with the enhancement of optical power in the ring cavity.

Fig. 6. Typical experimental results when two 5 Gb/s electrical signals are applied to the two MRRs. (a) Signals applied to MRR1 with the repeated pattern of ‘00100101’. (b) Signals applied to MRR2 with the repeated pattern of ‘01001111’. (c) XOR operation result with the repeated pattern of ‘01101010’ at the drop port. (d) XNOR operation result with the repeated pattern of ‘10010101’at the through port.

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Fig. 7. Dynamic operation results of the device at varying speeds. (a) and (b) 1 Gb/s signals applied to MRR1 and MRR2, respectively. (c) and (d) 1 Gb/s XOR and XNOR operation results, respectively. (e) and (f) 5 Gb/s XOR and XNOR operation results, respectively. (g) and (h) 12.5 Gb/s XOR and XNOR operation results, respectively. (i) 20 Gb/s XOR operation result.

As shown in Fig. 6 (c) and (d), there are positive spikes between two consecutive outputs of ‘0s’, and negative spikes between two consecutive ‘1s’, which also appear in our previous work and has been well explained [24, 25]. We also characterize the device’s performance at different speed from 1 Gb/s to 20 Gb/s. The results are shown in Fig. 7, in which three periods

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(i.e. 24 bits) are shown for the input electrical signals and output optical signals. We have measured those signals one by one and then aligned them with each other in the time axis. Since all the input signals have the same patterns and square-wave-like waveforms, we only show the waveforms of the input electrical signals at 1 Gb/s for simplicity. The operation results are rather good for 1 Gb/s and 5 Gb/s, which show fast rising and falling times and high extinction ratios. Signal qualities start to get poor for 12.5 Gb/s, especially for the XNOR result. This is because of two factors. Firstly, the static extinction ratio is less than 10 dB at the through port (see Fig. 5 (a)-(d)), which is rather small compared with the drop port (see Fig. 5 (e)-(h)). Secondly, the response time of the leading wires and the PN diodes are limited when the operation speed is as high as 12.5 Gb/s. High frequency components of the signals are attenuated remarkably. In Fig. 7(i), we can barely recognize the XOR operation result at 20 Gb/s. The XNOR operation result at 20 Gb/s is not achievable for this device. 4. Discussion It should be pointed out that the spectra shown in Fig. 3 do not have four uniform resonance regions, which has been discussed in detail in our previous work [25]. The first and third resonance regions in Fig. 3 are called degenerate regions. And the second and fourth resonance regions are called non-degenerate regions. It has been shown in [28] that the logical operations can be achieved in both the degenerate and non-degenerate regions using carrierinjection modulation. But this is not the case in the current work. The main reason is that the tuning efficiency of carrier-depletion mode is not as high as that of the carrier-injection mode. In the current work, only the degenerate regions can be used to achieve the logical operations. In this section, we will firstly analyze why the degenerate regions are more sensitive to the tuning of the MRR than the non-degenerate regions. And then we will put forward a general manner to describe the sensitivity of all-pass MRR, add-drop MRR and Mach-Zehnder interferometer (MZI) to the tuning of their phase shifting component. 4.1 Sensitivity analysis of the XOR/XNOR directed logic circuit

Fig. 8. Transfer matrix model of the circuit (Z1~Z4: coupling regions, S1~S2: connecting waveguides, R1~R2: ring waveguides). θ1 and θ2 are round-trip phase shifts in the MRR1 and MRR2. θs1 and θs2 are the phase shifts of the connecting waveguides of S1 and S2.

The modeling of the structure shown in Fig. 1(a) has been introduced in the previous work [25]. We redraw the model of the circuit here in Fig. 8 for reference. We denote the round-trip phase shifts in MRR1 and MRR2 to be θ1 and θ2, respectively. And we consider that the round-trip amplitude transmission factor of the two MRRs to be identical, which is represented by α in Fig. 8. The single-pass phase shifts in the connecting waveguides of S1 and S2 are denoted to be θs1 and θs2, respectively. The four coupling regions are supposed to be identical, which are describe by lumped self- and cross-coupling coefficients t and k. These parameters of θ1, θ2, θs1 and θs2 are all wavelength-dependent. And α is considered to be wavelength-independent. In the previous work, we just calculate the static response spectra of

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the circuit, where θ1 always equals to θ2 [25]. Here in this paper, we will analyze the sensitivity of the spectra to the perturbation of θ1 and θ2. We will only analyze the behavior of the spectrum at the drop port. The sensitivity of the spectrum at the through port can be calculated in the same way. As a linear system, the circuit obeys the superposition principle. The output at the drop port (Ep3) can be decomposed into two parts. The first part comes from Ep1, which contribute to Ep3 via the drop function of MRR2. The second part comes from Ev1, which contribute to Ep3 via the through function of MRR2. So the expression of the electric field Ep3 can be written as follows. Edrop = E p 3 =

t (1 − α exp( j θ1 )) −k 2 α 3 4 exp( j 3 4θ 2 ) × × exp( j θ s1 ) 1 − α t 2 exp( j θ1 ) 1 − α t 2 exp( j θ 2 )

(1) − k 2 α 1 4 exp( j 1 4θ1 ) t (1 − α exp( j θ 2 )) + × × exp( j θ s 2 ) 1 − α t 2 exp( j θ1 ) 1 − α t 2 exp( j θ 2 ) Hear we do not consider the loss caused by the two connecting waveguides. The phase difference between the two terms of Eq. (1) is −α sin(θ1 ) −α sin(θ 2 ) 3 1 ] − arc tan[ ] + [ θ 2 − θ1 ] + [θ s 2 − θ s1 ] (2) 1 − α cos(θ1 ) 1 − α cos(θ 2 ) 4 4 When the two nominally identical MRRs are tuned to be resonant at the same wavelengths, θ1 equals to θ2, which means that the first two terms in Eq. (2) disappear. At the resonance wavelengths, we let θ1 and θ2 equal to 2mπ, where the integer m is the resonance order. Since θ1, θ2, θs1 and θs2 are all proportional to their own waveguide length, θs1 and θs2 are proportional to θ1 and θ2. The proportionality factors are the ratio between the lengths of the connecting waveguides and the ring waveguides. We suppose that θs1 − θs2 are pmπ, where p is a real number. Then Eq. (2) can be simplified to Δϕ p 3 = arc tan[

Δϕ p 3 = m × ( p + 1) × π

(3)

Fig. 9. The response spectra at the drop port under different reverse biases. (a) The second resonance region (non-degenerate). (b) The third resonance region (degenerate). The legends in the figures show the reverse voltages applied to MRR1 and MRR2, respectively.

If p is an even number, the parity of the resonance order m will greatly affect the spectrum. If m is odd, Eq. (3) equals to π, the two constituent parts of Ep3 will interfere destructively with each other. Since the two terms in Eq. (1) have similar amplitudes, the destructive interference will produce trivial response, which is shown in the first and third resonance regions in Fig. 3. The small peak in the first resonance region is caused by small dissimilarity between the two MRRs, as well as the two connecting waveguides. If m is even, Eq. (3) equals to 0, the two constituent parts of Ep3 will interfere constructively with each other. The addition of the two terms in Eq. (1) will produce what we see in the second and fourth resonance regions in Fig. 3.

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If p is an odd number, the parity of the resonance order m will not affect the spectrum. The value of Eq. (3) always equals to 0, the two constituent parts of Ep3 will interfere constructively with each other at every resonance regions. Such characteristic has been analyzed in our previous work, where the other port is discussed [25].

Fig. 10. The sensitivities of the output at the drop port with different parity of the resonance order. (a) The resonance order m = 98, and p = 0. (b) The resonance order m = 99, and p = 0.

Since the two connecting waveguides in our device have the same length, the value of p is zero. So, half of the resonance regions are degenerate, and the other half are non-degenerate. We find from the static response spectra under different reverse biases (see Fig. 9) that only the degenerate resonance regions can be employed to achieve the XOR and XNOR operations. In other words, the degenerate resonance regions are much more sensitive to the tuning of the MRR than the non-degenerate regions. This is because at these regions, the two constituent parts interfere destructively with each other. When θ1 = θ2 = 2mπ, the drop port will nominally output nothing at the degenerate resonance region’s center wavelength. And when any of θ1 or θ2 has a little variation, the balance will be broken. The misalignment of the two MRRs’ resonance wavelengths will produce a drastic increase of the output optical power. From Eq. (1), we can readily obtain the relation between the output optical power and the round-trip phase shifts in the MRRs. The results are presented in Fig. 10, where the roundtrip amplitude transmission factor α and the cross-coupling coefficient k are taken to be 0.9856 (corresponding to an attenuation factor of 20 dB/cm in the ring waveguides) and 0.2 (t = 0.9798), respectively. The effective refractive index of the quasi-TE mode in typical submicron silicon waveguides is about 2.4 [32]. Since the radii of the MRRs are 10 μm, the resonance order is calculated as 2π·R·neff/λ, which is about 100 around 1550 nm. We choose the resonance order to be 98 and 99 in Fig. 10 to show the impact of its parity on the sensitivity of the drop port’s output to the MRRs’ round-trip phase shift. In the calculations, we let θs1 = θs2 (p = 0). Figure 10 show that the sensitivity is much higher at the resonance

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point when the resonance order m is odd. When p is an odd integer, all the resonance regions have the sensitivities like that shown in Fig. 10(a), regardless of the parity of m.

Fig. 11. The sensitivities of the output at the drop port with different parity of the resonance order. (a) The resonance order m = 98, and p = 0. (b) The resonance order m = 99, and p = 0. In the calculation, the cross-coupling coefficient k and the attenuation factor α are 0.2 and 20 dB/cm, respectively.

As we have presented hereinbefore, the resonance wavelength change is about 50 pm when a reverse bias of 4 V is applied (see Fig. 4). And the free spectral range (FSR) of the MRR is about 10 nm. The round-trip phase shift is calculated to be about 0.01π. In consideration of that the device is much more sensitive around the resonance wavelength, we redraw Fig. 10 near the resonance point in Fig. 11 employing logarithmic coordinates in both the horizontal and vertical axes. In Fig. 11, we fix the value of θ2 to be 2mπ, only considering the change of the output with the variation of the value of θ1. We can find from Fig. 11 that the roll-up factor of the output when m is odd is about 20 dB/decade. In the above calculations, the cross-coupling coefficient and the attenuation factor equal to 0.2 and 20 dB/cm, respectively. We can get the same roll-up factor of 20 dB/decade with other combinations of these two parameters (e.g. k = 0.3, attenuation factor = 10 dB/cm). The lower the attenuation factor is, the higher the degree of linearity the sensitivity graph shows. As shown in Fig. 11, the sensitivity of the output at the drop port is quite dependent on the parity of the resonance order. The radii of the microring resonators and the dispersion characteristic of the waveguides codetermine the locations of the odd and even resonance orders. Besides the sensitivity, we can readily find from such a graph the extinction ratio and insertion loss of the device. In the next sub-section, we will present the same graphs for the all-pass MRR, the add-drop MRR and the MZI structures. 4.2 Sensitivity diagrams of typical photonic structures 4.2.1 All-pass microring resonator The output of an all-pass MRR with only one access waveguide is [33] t − α exp( j θ ) (4) 1 − α t exp( j θ ) All the parameters in Eq. (4) have the same meanings as those in Eq. (1). We can obtain the similar graph as Fig. 11, which is shown in Fig. 12. In the calculations, the attenuation factor in the ring waveguide varies from 3 dB/cm to 50 dB/cm. And the cross-coupling coefficient k has a constant value of 0.2 (t = 0.9798). Critical coupling happens when the Ethrough =

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attenuation factor equals to 28.22 dB/cm, which makes the amplitude transmission factor α equal to the self-coupling coefficient t. It can be found from Fig. 12 that only a small range around the critical coupling point has a high sensitivity to the variation of the round-trip phase shift. It means that optical modulator based on such structure has to be work around the critical coupling point to make it have high efficiency and high extinction ratio. Compared Fig. 12 with Fig. 11, it can be found that the drop port of the circuit in Fig. 1 has a similar sensitivity to a all-pass MRR under critical coupling condition. Since the attenuation factor in Fig. 11 is arbitrarily chosen, it means that such a high sensitivity of our circuit is not susceptive to the attenuation factor in the ring waveguides. This feature can be used to enhance the sensitivity of bio-sensors based on MRRs [34–36].

Fig. 12. Sensitivity diagram of all-pass MRR with the attenuation factor varying from 3 dB/cm to 50 dB/cm. For all the cases, the cross-coupling coefficient k equals to 0.2. Critical coupling condition is met when the attenuation factor equals to 28.22 dB/cm (α = t = 0.9798).

4.2.2 Add-drop microring resonator The two outputs of an add-drop MRR with two access waveguides are [33] Ethrough =

t1 − α t2 exp( j θ ) ; 1 − α  t1 t2 exp( j θ )

(5) −α 1 2 k1 k2 exp( j θ 2) . 1 − α  t1 t2 exp( j θ ) Since the add-drop structure has two coupling regions, the subscripts in Eq. (5) indicate which regions they describe. Firstly, we will calculate the sensitivity graph when the two coupling regions are identical (k1 = k2, t1 = t2), which is the most common case in the in the literature. The results are shown in Fig. 13. It can be found that the through port is more sensitive to the attenuation factor in the ring waveguide. There is a huge difference in the extinction ratio for different attenuation factors for the through port. While for the drop port, the difference is not so drastic. And the minimum powers obtained when θ = π are almost the same for different attenuation factors for the drop port. Another remarkable feature is that the insertion loss almost has nothing to do with the attenuation factor for the through port. While for the drop port, the insertion loss is closely related to the loss in the ring waveguide. Edrop =

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Fig. 13. Sensitivity diagrams of add-drop MRR at (a) the through port, and (b) the drop port, with the attenuation factor in the ring waveguide varying from 3 dB/cm to 50 dB/cm. For all the cases, the cross-coupling coefficients k1 and k2 equal to 0.2 (t1 = t2 = 0.9798).

For the add-drop MRR with two symmetric coupling regions, critical coupling is approached when the attenuation factor is close to 0. This is because the critical coupling condition for the add-drop MRR is α = t1/t2. So the critical coupling condition can only be met when asymmetric coupling is adopted. We can find from Fig. 13 (a) that the extinction ratio of the through port is increased dramatically when the critical coupling condition is approached.

Fig. 14. Sensitivity diagrams of add-drop MRR at (a) the through port, and (b) the drop port, when the critical coupling condition is satisfied. The cross-coupling coefficient k1 varies from 0.2 to 0.4, and the attenuation factor takes the values of 3 dB/cm, 10 dB/cm and 20 dB/cm (corresponding to α = 0.9978, 0.9928, and 0.9856 for ring waveguide with the radius of 10 μm). The cross-coupling coefficient and the attenuation factor are shown in the legends of the figures, respectively. When k1 changes, k2 is adjusted to meet the critical coupling condition.

Next we will analyze the sensitivities of the two ports of the add-drop MRR when the critical coupling condition is satisfied. The results are shown in Fig. 14. It should be pointed out that the three lines for the same k1 in Fig. 14 (a) coincide. It means that for a certain k1, the sensitivity does not change with the loss in the ring, as long as the critical coupling is satisfied. It can be found that, for different k1, the roll-up and roll-off factors have almost the same amplitudes, which are measured to be about 20 dB/decade at the through port and −20 dB/decade at the drop port, respectively. It has the same value as a simple first order network such as the RC circuit. This is because both the MRR around the resonate frequency and the RC circuit have lorentzian line shapes. 4.2.3 Mach-Zehnder interferometer The output of a Mach-Zehnder interferometer (MZI) is Eout = 1 4 × (α12 + α 22 + 2α1 α 2 cos( Δθ )). (6) In the above equation, α1 and α2 are the amplitude transmission factors in the two arms of the MZI. And ∆θ is the phase shift difference between the two arms. Since the MZI does not have an infinite impulse response (IIR) as the MRR, it is much less sensitive to the vibration

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of the phase shift. We draw its sensitivity graph in linear coordinate in Fig. 15 (a), with the same data plot in logarithmic coordinate in Fig. 15 (b) to be compared with the property of MRRs shown in Figs. 13 and 14. The insensitivity of MZI to the phase shift makes the MZIbased optical modulators have much larger volume than MRR-based ones.

Fig. 15. Sensitivity diagrams of MZI in (a) linear coordinate, and (b) logarithmic coordinate. The length of one arm of the MZI is fixed to be 1000 μm, with the length of the other arm varying from 1000 μm to 200 μm with an interval of 200 μm. The legends in the figures show the length differences between the two arms. The attenuation factors in the two arms are both 20 dB/cm. The beam splitting and combining are assumed to be lossless.

5. Conclusion We implement XOR and XNOR operations using an electro-optic directed logic circuit based on two cascaded microring resonators. Simultaneous bitwise XOR and XNOR operations at 12.5 Gbit/s are demonstrated employing carrier-depletion modulation. And 20 Gbit/s XOR operation is achieved. The phenomenon that only half of the resonance regions can be employed to implement the logical operations is well explained. The graph with logarithmic phase coordinate is proposed to analyze the sensitivity of the spectra of the circuit as well as the sensitivity of typical photonic structures such as the all-pass MRR, the add-drop MRR and the MZI structures. Further efforts should be made to construct large-scale and reconfigurable directed logic circuits, in the electro-optic or all-optical manner. Acknowledgment This work has been supported by the National Natural Science Foundation of China (NSFC) under grants 61204061, 61235001, and 61377067 and by the National High Technology Research and Development Program of China under grants 2012AA012202 and 2013AA014203.

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