1. Introduction

Directed logic can construct a switching network that can control the propagation direction of light and carry the Boolean logical operation with the speed of light^[1]. In essence, the complex Boolean operation can be decomposed into simple logic operations, each of which can be achieved by a combination of simple logical elements. When compared with a traditional optical logical device, directed logic has better performance and has the advantages of high speed and lower latency, which arises because the input electrical signals that determine the states of elements do not pass through the preceding operating elements in the circuits. Consequently, all of the operating signals can modulate the switching elements simultaneously and the results are given instantaneously. A silicon-based microring resonator (MRR) is compatible with complementary metal-oxide-semiconductor (CMOS) process and has the advantages of low power consumption and high compactness^[2], which make it an ideal component to construct integrated optoelectronic circuits^[3-15].

We have demonstrated a thermo-optic decoder^[12] that is based on two cascaded MRRs whose speed is 10 kbps, which is due to the thermo-optic modulation scheme. Compared to the previous demonstration, the plasma dispersion effect^[16-18] is utilized here to modulate the microring resonators. In this paper, an optical decoder consisting of two cascaded MRRs that are modulated by electric-field-induced carrier injection in forward biased PIN diodes^[16] is fabricated and demonstrated, which can perform the decoding operation at a speed of 100Mbps.

2. Design and fabrication

The proposed architecture is schematically shown in Fig. 1, which consists of two cascaded electrical modulated add-drop MRRs and three waveguides. Monochromatic continuous light with a working wavelength of $\lambda _{\rm work}$ is coupled into the device through the input port, and is then modulated by two electrical pulse sequences that are applied to the MRRs through electric-field-induced carrier injection in the forward biased PIN diodes.

Every MRR can act as an optical switch and two electrical logic signals are used to control the resonance states of the two MRRs. It is assumed that the MRR is on-resonance at the working wavelength if an electrical logic signal applied on MRR represents logic 0 and off-resonance at the working wavelength if an electrical logic signal applied on MRR represents logic 1. The high and low levels of the output optical power at four output ports represent the operation result of logic 1 and 0, respectively. According to the definition above, optical logic 1 is achieved at output port Z$_{1}$ and optical logic 0 is achieved at the other three output ports when MRR1 and MRR2 are both on-resonance (electrical logic 0 s). Similarly, optical logic 1 is achieved at output port Z$_{2}$ and optical logic 0 is achieved at the other three output ports when MRR1 is on-resonance (electrical logic 0) and MRR2 is off-resonance (electrical logic 1), respectively. Optical logic 1 is achieved at output port Z$_{3}$ and optical logic 0 is achieved at the other three output ports when MRR1 is off-resonance (electrical logic 1) and MRR2 is on-resonance (electrical logic 0), respectively. Optical logic 1 is achieved at output port Z$_{4}$ and optical logic 1 is achieved at the other three output ports when MRR1 and MRR2 are both off-resonance (electrical logic 1 s). According to the above analysis, the applied voltages and the corresponding resonance states of the two microring resonators is summarized in Table 1. As we can see, the proposed architecture can perform the decoding function from a 2-bit electrical signal to a 4-bit optical signal.

The device is fabricated on an 8-inch silicon-on-insulator (SOI) wafer with a 220 nm top silicon layer and a 2 $\mu$m buried silicon dioxide layer. The width of the straight ridge waveguide is 400 nm, with the height of 220 nm and slab thickness of 70nm, which is chosen to only support the fundamental quasi-TE mode. The radii of the bending waveguides and the radii of the ring waveguides are both 10 $\mu$m. The resulting free spectral ranges (FSR) of the MRRs are about 10 nm. Additionally, the radii of the ring waveguides are big enough to ensure that the radiation loss of the ring resonator is not too high and that the $Q$ factor is acceptable. The gaps between the ring waveguides and the straight waveguides are 340 nm, which is chosen to have balanced extinction ratios between the drop and through ports. For such a structure, the coupling coefficient is calculated to be about 0.2. In addition, the gaps between the straight waveguides and the ring waveguides are chosen to be 340 nm so that the $Q$ factor is about 8000, which is high enough for the electric-field-induced carrier injection modulation, due to low tuning efficiency. After the waveguides are formed, the p-and n-doping regions with the doped concentrations of 5.5 $\times$ 10$^{20}$ cm$^{-3}$ are embedded around the ring waveguides to form PIN diodes, which are employed to modulate the MRRs by the plasma dispersion effect. All of the doped regions are located 400 nm away from the sidewalls of the ring waveguides, which is chosen to achieve the balance between electric-field-induced carrier injection tuning efficiency and the insertion loss of MRR. After the doping, the 200-nm-thick and 1-$\mu$m-wide titanium microheaters are fabricated to tune the working wavelengths of the MRRs through the thermo-optic effect. In Fig. 2, the waveguides are sheltered by aluminum trances and pads. The dashed lines show where the waveguides are, and the green arrows represent the input port and output ports of the device.

3. Experimental results

In order to determine the working wavelength and driving voltages of the two MRRs, the static response spectra of the device are measured with an amplified spontaneous emission (ASE) source, an optical spectrum analyzer (OSA), and three tunable voltage sources. The light from the ASE source is coupled into the input port of the device through a lensed fiber. The light from the output port of the device is coupled into another lensed fiber and then fed into the OSA. One of the three tunable voltage sources is used to tune the microheater of the MRR with a shorter initial resonance wavelength to shift to the resonance wavelength of the other MRR (i.e. the working wavelength). The other two tunable voltage sources are employed to inject carriers into the two PIN diodes. As the applied voltages of the PIN diode increase, the effective refractive index decreases and the resonance wavelength of the MRR moves to a shorter wavelength.

Although the two MRRs are designed to have the same structure parameters, they have slightly different initial resonance wavelengths due to the limited manufacturing accuracy. The spectra of the output ports Z$_{1}$-Z$_{4}$ without any tunable voltage sources being applied to the device are shown in Fig. 3. We can see two dips at 1550.40 nm and 1551.48 nm, with extinction ratios of 16.2 dB and 21.8 dB, respectively. MRR2 has three coupling regions while MRR1 has two coupling regions. Because MRR1 is closer to the critical coupling than MRR2, the extinction ratio of the dip from MRR1 is larger than that from MRR2, which can be seen from Fig. 3(d).

The working wavelength is chosen in order to obtain the largest extinction ratio. Because the bias of the PIN diode decreases the extinction ratio of the corresponding MRR, 1551.48nm is chosen as the working wavelength $\lambda _{\rm work}$ in order to have the larger extinction ratio. When an MRR is heated, the effective refractive indices of Si waveguides increases, and so the resonance wavelength of the MRR accordingly shifts to a larger wavelength. When a 2.4 V voltage is used to control the microheater of MRR2, the resonance wavelengths of two MRRs are both at working wavelength and the spectra of four output ports can be seen in Figs. 4(a), 5(a), 6(a), and 7(a).

A low extinction ratio makes it difficult to distinguish between a high level and low level of optical output powers. A high extinction ratio of MRR is important to obtain better performance of optical decoder and reduce decoding error. In order to obtain the largest extinction ratio, the MRR should be designed to be under critical coupling. It should be noted that many ripples are observed in the response spectra at the output ports Z$_{1}$ and Z$_{2}$, which is due to the interference effect caused by a closed cavity formed by two cascaded MRRs and port M, as the green line shows in Fig. 1.

The measured response spectra at the output port Z$_{1}$ show the though filtering characteristics of MRR1 and MRR2 (Fig. 4). According to the principle aforementioned, there is a peak when the MRR1 and MRR2 are both off-resonance at working wavelength (Fig. 4(a)), the optical power is at high level (representing 1). A maximum should be given at the output port Z$_{1}$ when MRR1 and MRR2 are both on-resonance (Fig. 4(a)). In other working statuses, the optical power at output port Z$_{1}$ is at low power.

The measured response spectra at the output port Z$_{2}$ show the drop filtering characteristics of MRR1 and the through filtering characteristics of MRR2 (Fig. 5). A maximum should be given at the output port Z$_{2}$ when MRR1 is on-resonance and MRR2 is off-resonance (Fig. 5(c)). In other working statuses, the optical power at output port Z$_{2}$ is at low power.

The measured response spectra at the output port Z$_{3}$ show the through filtering characteristics of MRR1 and the drop filtering characteristics of MRR2 (Fig. 6). Due to the three coupling regions, the coupling loss of MRR2 is bigger than that of MRR1. Consequently, the $Q$ factor of MRR2 is smaller than that of MRR1 and then the full width at half maximum (FWHM) of MRR2 is bigger than FWHM of MRR1. Therefore, there is a dip in the drop filtering spectra, which can be seen in Fig. 6(a). A maximum should be given at the output port Z$_{3}$ when MRR1 is off-resonance and MRR2 is on-resonance (Fig. 6(b)). In other working statuses (Figs. 6(c) and 6(d)), the optical power at output port Z$_{3}$ is at low power.

The measured response spectra at the output port Z$_{4}$ show the through filtering characteristics of MRR1 and MRR2 (Fig. 7). According to the principle aforementioned, there is a dip when the MRR1 and MRR2 are both on-resonance at working wavelength (Fig. 7(a)), where the optical power is at low level (representing 0). With only a voltage of 1.5 V applied to PIN diode of MRR1, the resonance wavelength of MRR1 shifts to a shorter wavelength (Fig. 7(b)) and the optical power is still at a low level (representing 0). With only a voltage of 1.3V applied to PIN diode of MRR2, the resonance wavelength of MRR2 shifts to a shorter wavelength (Fig. 7(c)) and the optical power is still at a low level (representing 0). A maximum will only be obtained at the output port Z$_{4}$ when MRR1 and MRR2 are both off-resonance (Fig. 7(d)).

All of the four static response spectra for four output ports are analyzed above. The results indicate that the device can implement the decoding function from a 2-bit electrical signal to a 4-bit optical signal.

After the working wavelength and driving voltages of the two MRRs are determined, a tunable laser, two pulse pattern generators (PPGs), and a multi-channel oscilloscope are employed to characterize the dynamic response of the device (Fig. 8). A monochromatic light with a working wavelength 1551.48nm is coupled into a polarization rotator and light with TE polarization is coupled into the device. Two non-return-to-zero electrical signals at a speed of 100 Mbps, generated by two PPGs, are used to control the PIN diodes of MRR1 and MRR2. The light at the output ports is fed into a detector. The electrical signals generated by the two PPGs and the electrical signal converted by the detector are fed into a multi-channel oscilloscope for waveform observation. As we can see from Fig. 8, the device performs the decoding function from a 2-bit electrical signal to a 4-bit optical signal correctly. Due to the different response speed of the two MRRs, some small peaks can be seen at the dynamic response in Fig. 8(d). The insertion loss of Z$_{1}$-Z$_{4}$ at the working wavelength is not the same, which is due to the different extinction ratio of MRR1 and MRR2, this makes the optical power at low level different (Figs. 4-7). Consequently, the optical output of the "0"s at ports Z$_{1}$-Z$_{4}$ are slightly different (Fig. 8).

4. Conclusion

In conclusion, we have implemented a decoding function using an electro-optic directed logic circuit that is based on two cascaded microring resonators. PIN diodes embedded around the MRRs are employed to modulate the MRRs through a carrier-injection modulation scheme. Bitwise operations at 100 Mbps are demonstrated successfully. This shows that the plasma dispersion effect can be employed to modulate MRRs in directed logic circuits. This means that we can employ other advanced modulation schemes, such as carrier-depletion modulation, to achieve an even faster operation speed.