1. Introduction

Recent progress on optical microcavity technologies [1, 2] has allowed us to study optical nonlinearities at a very low input power. Optical switching operation [3], bistable operation [4, 5], and all-optical diode operation [6] have been demonstrated by means of a photonic crystal nanocavity [7] and a micro-ring resonator [8], at ultralow input powers. However, the challenge still remains of devising all-optical logic gates [9–12] composed of a number of resonators. The recent development of fabrication and post-process technologies for microresonators [13–15] are opening the way to the experimental demonstration of systems with multiple resonators. Indeed all-pass filters [16], coupled resonator optical waveguides [17, 18], all-optical matrix switches [19] and preliminary random-access memory circuits [1] have been fabricated and demonstrated that are constituted from more than a resonator. On the other hand, a sophisticated all-optical logic gate for digital photonics, such as NAND gates and flip-flop operators, has been designed, but remains at the numerical calculation stage. There is still a large gap between theory and experiment. Before we can make these devices practical, we need to know the degree to which fabrication error affects system performance. We also have to reveal the tolerance of the system to input power fluctuations. No such analysis has yet been performed. If we achieve the above it will lead to a system with high tolerance.

In addition to the tolerance issue, there are at least two different factors that keep numerical and experimental studies far apart. The first is wavelength. Previously proposed systems often use different input and output laser wavelengths. This makes it extremely difficult use this system connected in tandem. An all-optical gate with two inputs and one output all at the same wavelength is needed to secure system scalability. However, logic operation based on a cavity system usually requires two different wavelengths to operate.

Figure 1 shows how the switching operates with an add-drop ring resonator system, where one wavelength is used as a signal and the other for control. We need to design the system very carefully to allow single wavelength operation. Although all the signals should have the same wavelength, we can use the driving laser as a second wavelength source for internal use, because this does not make the system complex even when we connect them in tandem.

 

Fig. 1 Schematic illustration of an all-optical switch made of an add-drop microring resonator. The dotted line represents the transmittance spectrum of a cold cavity. The solid line is the transmittance when inputs are applied. A resonant shift occurs due to the optical Kerr effect. Two different wavelengths λ₁, and λ₂ are used for the operation. (a) λ₁ will drop (high) when only λ₁ is inputted (high). (b) λ₂ will not drop (low) when only λ₂ is inputted (high). (c) λ₂ will drop (high) when both λ₁ and λ₂ are inputted (high). As a result, λ₂ can be switched off and on by turning λ₁ signal on and off.

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An optical logic gate should also have a simple design. Cavities used in a system are often designed differently to enable a sophisticated logic operation. However, this makes fabrication and design difficult. To increase the robustness, the cavities used in the system should have an identical design (i.e. resonance wavelength and size).

The purpose of this paper is to fill the gap between theoretical and experimental work by solving these two problems. Namely, we design a system composed of cavities with the same design, and operate it using a single wavelength laser for the signals. These characteristics should allow us to connect systems in tandem. We also study the robustness of the system when the structural dimensions fluctuate and show how the system will behave when the input power fluctuates. This is the first such analysis.

This paper is organized as follows. In section 2, we show the characteristics of the basic elements used to constitute the all-optical logic gates. Then in section 3, we discuss our structure and the operation calculated with CMT [20, 21], for a NAND gate. The operation of AND, OR and NOR gates are shown in Appendix B. In addition, the demonstration on cascaded operation is described in Appendix C. Section 4 is the main part of this paper. It discusses various fluctuations that can occur when we fabricate and use the system. We investigate how the input power fluctuations, resonant wavelength errors, and fabrication errors affects the operation of the system. Finally we conclude the paper.

2. Basic elements constituting the logic gates

2.1 Silicon nitride mirroring resonator

We employ a silicon nitride (Si₃N₄) platform for our system. Microring resonators made of Si₃N₄ can use the optical Kerr effect due to the large bandgap of the material. The waveguide we used is 900 nm wide and 600 nm tall and we assumed the use of air cladding. A schematic illustration of the microcavity is shown in Fig. 2.The radius of the ring resonator R is 20 μm; hence the mode volume V of the cavity is 1.5 × 10² μm³. The linear and nonlinear refractive indexes are n₀ = 1.98, and n₂ = 2.5 × 10⁻¹⁵ cm²W⁻¹, respectively. We assume the cavity resonances at 1550 and 1580 nm. The unloaded quality factor (Q_unload) for both resonant modes is 1.0×10⁶. This value is given by Q = ω₀τ_loss, where ω₀ is the resonant angular frequency of the cavity and τ_loss is the loss rate towards the outside. The input wavelength is slightly detuned from the resonance and is set at λ₁ = 1550.01 nm and λ₂ = 1550.02 nm.

 

Fig. 2 Schematic dimension of add-drop system used for this calculation.

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Throughout this study, we use coupled mode theory (CMT) [20, 21] to analyze the logic gate operations. The calculation model is described in detail in Appendix A.

2.2 Bus / drop waveguides and tapered waveguides

The Si₃N₄ [22–24] waveguide is 900 nm wide and 600 nm tall. In addition to a regular waveguide, we use a tapered waveguide [25] for the termination. The taper is 14 μm long and the waveguide width decreases linearly. Figure 3 shows the reflection to radiation ratio of this waveguide, which is calculated by using 3D FDTD. The light adiabatically radiates outside and the reflectance is 4%, which is sufficiently small that we can neglect it in our CMT analysis.

 

Fig. 3 Reflection and radiation at tapered waveguide.

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3. Designs of all-optical logic circuits

The designs of our AND, OR, and NOR gates are described in Appendix B. Here we focus on the design of a NAND gate, since it is a basic element for logic gates. The photonic circuit is shown in Fig. 4(a).It is composed of five cavities: C1, C2, C3, C4, and C5. We set the τ_coup values for C1, C2, C3, C4 and C5 at 150, 100, 250, 250, and 100 ps, respectively. C1 and C3 switch the signals on and off, which contributes directly to the logic operation. C4 is used to switch λ₂ light with λ₁ light. C2 and C5 are used to filter out unnecessary drive light.

 

Fig. 4 (a) Design of NAND gate. (b) Input and output versus time. The black, red, and green lines represent input 1, input 2, and output, respectively.

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Input 1, input 2 and output signals are at the same wavelength λ₁. We use two different wavelengths to drive the system; but as discussed previously, this does not degrade the scalability of the system. Pulsed drive sources with an amplitude of 360 mW are applied at a cycle of 5 ns with a duty ratio of 60%.

Figure 4(b) shows the calculated output waveform when inputs 1 and 2 are applied as shown in the graph. First, we set high input 1 (ON) and low input 2 (OFF) from 0 to 3 ns. The outputs exhibit a high state. When we switch both inputs to a high state (from 5 to 8 ns), the output is in a low state. We obtained a low state only when inputs 1 and 2 were both high, which corresponds to a NAND gate operation.

Although thermo-optic effect is present in Si₃N₄ microrings, we do not take this effect into account. It is because of the input power; a different study showed that the thermo-optic effect becomes dominant only at an even higher input [26]. Since this is a proof-of-principle study we focus on the optical Kerr effect. However, to show the potential of this system, here we would like to comment on the operation power. Figure 4(b) shows that we need a 0.25-W input because we employed Si₃N₄ parameters, and tried to use the optical Kerr effect. However, this value can be significantly reduced by using the carrier-plasma dispersion effect in silicon microrings. A simple calculation suggests that we can decrease the input power to 4 mW, which is a reasonably low value for an experimental demonstration [27, 28]. Since we would like to focus on the principle, here we concentrate on the Kerr effect, and the detail on the operation based on the carrier-plasma dispersion effect in the presence of thermo-optic effect is described in Appendix D. Briefly, we would like to emphasize that the power can be reduced by about two orders of magnitude when we use a silicon microring cavity and the carrier-plasma effect.

4. System robustness

4.1 Input power fluctuations

First we investigated the system tolerance against input power fluctuations by calculating the output amplitude with different input powers. We set the drive light power at 360 mW, and changed the amplitude of the input signal. Figure 5 shows the input and output characteristics for four different gates. The black square dots show when we set the power of inputs 1 and 2 at the same value and changed both powers simultaneously. The red circular dots indicate the output amplitude when we fixed the input amplitude at 2 at 250 mW and changed input 1. The blue triangles show the result we obtained when we fixed the input 1 at 250 mW and changed the power of input 2.

 

Fig. 5 Input and output amplitudes of different logic gate versus time. (a) AND gate. (b) OR gate. (c) NOR gate. (d) NAND gate. The black squares are the output when the power of inputs 1 and 2 were changed together, the red circles are the output when only input 1 was changed (input 2 is fixed at 250 mW). The blue triangles are the output when only input 2 was changed (input 1 is fixed at 250 mW).

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Figures 5(a) and 5(b) show the output amplitudes for AND and OR gates. The output of an AND gate is at a low state when one of the inputs (1 or 2) is less than 60 mW. The output signal rises when the input exceeds 100 mW. The output is unstable when the input is between 60 and 100 mW. When we set the output power threshold at 50 mW, we need an input larger than 170 mW. However, if we increase the input too much, the output amplitude decreases again. This is due to the supply of excess λ₂ light to the C4 cavity of the AND gate. This analysis reveals that the optimized input is in the 160 to 300 mW range for a high input and from 0 to 50 mW for a low input. This tolerable power range is sufficiently large for practical use.

Figure 5(c) shows the NOR gate output. The output is low only when inputs 1 and 2 are both at a low state. The graph shows that this gate is at a low state when both of the inputs is less than 80 mW. When the input increases to 300 mW, the output amplitude rises. As a result of this analysis, we found that the input power for this NOR gate must set within 60 to 300 mW for a low input.

Figure 5(d) shows the NAND gate output. The output is low only when inputs 1 and 2 are both at a high state. The graph shows that the system exhibits a low output when the input power exceeds 230 mW. When the input increases to 400 mW, the output amplitude rises. As a result of this analysis, we found that the input power for this NAND gate must set within 230 to 400 mW for a high input and at less than 50 mW for a low input.

The results for all the gates show that the upper limit of the OFF signal is 40 mW and is restricted by the OR gate, and the lower limit of the ON signal is 230 mW and is restricted by the NAND gate, when we set the output threshold value at 50 mW. Through this study, we found that the input power fluctuations are not a critical issue in our system.

4.2 Resonant wavelength fluctuations

Next, we investigated the effect of the resonant wavelength fluctuation of cavities. Resonant wavelength fluctuations occur as a result of imperfect fabrication. An all-optical logic gate usually contains multiple microcavities; and a cavity is extremely difficult to fabricate exactly at the designed wavelength. The typical error value is a few 100 pm [29]. Reducing the wavelength fluctuation is essential if we are to move numerical studies to the practical level. Different approaches have been tried such as the thermal tuning or post-tuning of the cavity resonance [30]. In contrast, our approach is to find a robust system against wavelength fluctuation. Therefore, the first step is to understand the error rate when resonant wavelength fluctuation is present. Our system uses 4 (or 5) cavities, and each cavity has two resonant modes. To investigate the effect of the presence of wavelength fluctuation, we added random fluctuations to all the 8 (or 10) resonant modes. Wavelength fluctuations are added based on a normal distribution, and their widths are characterized by standard deviation σ. We calculated the output with CMT, and counted whether or not the system had failed.

Figure 6(a) shows the error rate versus 2σ for a NAND gate. The error rate gradually increases when larger fluctuations are applied, and the rate reaches 50% when 2σ is larger than 3.0 pm. Table 1 summarizes the result for NAND and other gates. σ_lim is the standard deviation that reaches an error rate of 50%. S_max is the full width at half maximum (FWHM) of the resonant spectrum of the cavity with the lowest loaded Q used in the system. S_min is the FWHM of the spectrum of the cavity with the highest loaded Q used in the system. From the result, we know that the NAND gate is the most sensitive of all types. And the allowed fluctuation is about 0.3 of the cavity width. Although fluctuation where σ < 3.0 pm is not easy to achieve, this value can be improved by designing a system where the cavities couple more strongly with the waveguides. Figure 6(b) shows the error rate when we increase the coupling and reduce τ_coup tenfold. With increased operation power (we need about 10 times larger power) σ_lim is improved to 55 pm. Nevertheless, this is first study to analyze the effect of the resonant wavelength fluctuations of the cavities with a view to achieving successful operation of the system.

 

Fig. 6 Error rate of NAND gate that have different resonant wavelength fluctuation. (a) Result for a NAND gate with optimized waveguide-cavity couplings. (b) Result for a NAND gate with strong waveguide-cavity couplings.

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Table 1. Maximum and minimum widths of resonant spectrum (FWHM) S_max, S_min, and the standard deviation σ_lim limit of each logic gate

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4.3 Fabrication tolerance: Gap distance fluctuations

Fabrication error not only causes resonant wavelength fluctuations but also causes fluctuations in the cavity-waveguide coupling, because the coupling Q is determined by the gap distance s between the cavity and the waveguide. To investigate the impact of the fabrication error on the coupling strength κ and hence on the operation of the system, we first formulated κ with respect to s [31, 32]. To simplify this formulation, we assumed that the waveguide width w, refractive index n, and propagation constant β are the same for the cavity and the waveguides. The transverse propagation constant k_x and the evanescent field decay constant α in the cladding are given as

Figure 7 is the error rate versus the standard deviation σ` of the gap fluctuations. The system is error free when σ` is less than 5 nm. The error rate exceeds 50% when σ` is 12 nm. Table 2 summarizes the results for different logic gates. Again the NAND gate is most sensitive to the error. However, it is possible to achieve fabrication precision of better than ± 12 nm with current state-of-the-art fabrication technology.

 

Fig. 7 Error rate of a NAND gate when the gap distances fluctuate.

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Table 2. The limit of the fabrication fluctuation (in standard deviation σ`_lim) for different logic gate

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5. Summary

We designed all-optical logic gates that operated at the same input and output wavelength, which makes these logic gates highly scalable and easier to implement in practical use. We demonstrated the operation by using coupled mode theory, and investigated various error tolerances. The upper limit of the OFF signal was 40 mW and the lower limit of the ON signal was 200 mW for our logic gate, whose power range is sufficiently robust for practical use. We formulated the coupling constant with the gap distance and analyzed the impact of the imperfect fabrication of the gap distance. We found that the system was error-free when the device was fabricated with ± 5 nm precision and the error rate increased to 50% when the fabrication precision was ± 12 nm. Although current state-of-the-art processing satisfies the required precision of the gap distance, we found that the sensitive dependence on the resonant wavelength fluctuation is critical. The error rate increased to 50% when the resonant wavelength fluctuated 3 pm. Although this value is not easy to achieve, we showed that we could increase the robustness by employing stronger coupling with the waveguide at the cost of operation power. Note that the power demonstrated in this paper is based on the optical Kerr effect in Si₃N₄ microrings, but the required power can be reduced by about three orders of magnitude by employing the carrier-plasma dispersion effect in silicon microrings.

We are attempting to increase the scalability of the system, and this is the first detailed theoretical study of the impact of fabrication error on the operation of such all-optical logic gates, which provides important information that will make it possible to build a bridge between numerical and practical studies.

Appendix A: Coupled mode analysis

The mode amplitude a in the cavity, the transmittance output wave s_tr and the drop output wave s_dr are given as

(7)$$\frac{da}{dt}=\left[j{\omega }_0-\frac{1}{2}\left(\frac{1}{{\tau }_{loss}}+\frac{2}{{\tau }_{coup}}\right)\right]a+\sqrt{\frac{1}{{\tau }_{coup}}}\exp \left(j\theta \right)s_{in}$$

(8)$$s_{tr}=\exp \left(-j\beta d\right)\times \left[s_{in}-\sqrt{\frac{1}{{\tau }_{coup}}}\exp \left(-j\theta \right)a\right]$$

(9)$$s_{dr}=\exp \left(-j\beta d\right)\times \sqrt{\frac{1}{{\tau }_{coup}}}a$$

where ω₀, τ_loss and τ_coup are the resonant frequency of the cavity, the photon lifetime of the cavity and the coupling to the waveguide, respectively. β is the propagation constant of the waveguide, d is the waveguide length and s_in is the input wave. θ is the relative phase between the mode amplitude in the cavity and the waveguide and is given as

(10)$$\theta =4{\pi }^2{n}_{0}R\left(\frac{1}{{\lambda }_0}-\frac{1}{\lambda }\right)$$

where n₀, λ₀ and λ are the refractive index, the resonant wavelength and the input wavelength, respectively.

We have to consider the XPM effect because our logic gates use two wavelengths. The XPM effect is twice the SPM effect. So the refractive index modulation Δn_kerr of the λ₁ mode is given as

(11)$$\Delta n_{\text{kerr}}=\frac{2n_{2}c}{n_0}\frac{\left(U_{{\lambda }_1}+2U_{{\lambda }_2}\right)}{V}$$

where, n₂ and c are the nonlinear refractive index and the speed of light, respectively. U is the energy of one mode in the cavity.

Therefore, the resonant wavelength modulation is given as

(12)$$\delta \lambda =\frac{\Delta n_{\text{kerr}}}{n_0}{\lambda }_0$$

Appendix B: Design of AND, OR, and NOR gates

The design of our AND gate is shown in Fig. 8(a).It is made of four cavities denoted C1, C2, C3, and C4. We set the τ_coup value of all four cavities at 200 ps. C1 and C3 are used to switch the signals on and off, and contribute directly to the logic operation, whereas C2 and C4 are used to filter out unnecessary signals. Input 1, input 2 and the output signals are at the same wavelength λ₁. We use two different wavelengths to drive the system; but as discussed previously, this does not degrade system scalability. Pulsed drive sources with an amplitude of 250 mW are applied at a cycle of 5 ns with a duty ratio of 60%.

 

Fig. 8 (a) Design of AND gate. (b) Input and output versus time. The black, red, and green lines represent input 1, input 2, and output, respectively.

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Figure 8(b) is the calculated output waveform when inputs 1 and 2 are applied as shown in the graph. First, we set a high input 1 (ON) and a low input 2(OFF) from 0 to 3 ns. The output exhibits a low state. When we switch both inputs to a high state (from 5 to 8 ns), the output is at a high state. We obtained a high state only when both input 1 and input 2 are high, which corresponds to an AND gate operation.

Next, we show the configuration and the output waveform of an OR gate. The design of our OR gate is shown in Fig. 9(a). An OR gate is composed of five cavities. We set the τ_coup values of C1, C2, C3, C4, and C5 at 20, 20, 30, 15, and 20 ps, respectively. C1, C2, and C4 are used to switch the signals on and off, whereas C3 and C5 are used for filtering. Figure 9(b) is the calculated output waveform, where the output is high unless both inputs are high, which corresponds to an OR gate operation.

 

Fig. 9 (a) Design of OR gate. (b) Input and output versus time. The black, red, and green lines represent input1, input2, and output. The input amplitude is 250 mW.

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The NOR gate is constructed of five cavities C1 to C5 (Fig. 10(a)). We use the nonlinearity in cavities C1, C2, and C3, whereas C4 and C5 are for filtering. We set the τ_coup values of C1, C2, C4, and C5 at 250 ps and 150 ps for C3. Figure 10(b) shows the input and output waveforms. As shown by the graph, the system realizes a NOR gate operation. We observed ringing in the output waveform when the output rose from low to high, which is different behavior from other types of gates. The ringing is caused because of the delayed interference of the bus waveguide and the light in cavity C5. Unlike other gates, NOR gates use transmitted light as their output; hence the output jumps to a high state until the cavity is charged with photons and interference takes place.

 

Fig. 10 (a) Design of NOR gate. (b) Input and output versus time. The black, red, and green lines represent input 1, input 2, and output, respectively. The input amplitude is 250 mW.

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Appendix C: Cascaded operation

We show the operation of a cascaded-logic system constituted of OR and AND gates. Figures 11(a) and 11(b) are the logic circuit chart and the truth table. The output is Output = (Input1+Input2)∙Input3. We use the same light wavelengths, light power, and τ_coup from that we used in our OR and AND gates. It should be noted that no back reflection occurs at the AND gate, which disturbs the operation of the OR gate. Figure 11(c) is the calculation result. Although we see some modulation in the output, the result shows a successful operation.

 

Fig. 11 (a) Logic circuit chart of the cascade system. (b) The truth table. (c) Input and output versus time. The black, red, green, and blue lines represent the input1, input2, input3, and output, respectively.

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Appendix D: Operation using silicon microring resonators

Here we show a preliminary study on the NAND operation with a system based on silicon microrings. Although we use carrier-plasma dispersion effect for the operation, we take thermo-optic (TO) effect into account, which is inherent in a real device. Carriers are generated by two-photon absorption (TPA). We use the same microcavity dimensions as that we used in Fig. 2 except the cross sectional size (400 nm wide and 200 nm tall). The refractive index is n₀ = 3.47, and Q_unload is 1.0×10⁶. We assume the cavity resonances at 1550 nm and 1580 nm. The input wavelength is slightly detuned from the resonance and are set as λ₁ = 1549.9 nm and λ₂ = 1579.95 nm.

Instead of using Δn_kerr given by Eq. (11), we use Δn_c(t), which is given by the following equations. N(t) is the TPA carrier density and U_T(t) is the energy density of the heat generated in the cavity.

(13)$$\frac{dN(t)}{dt}=\frac{2{\lambda }_{1}c{\beta }_{TPA}}{h{n}_0{}^2{V}^2}{\left(U_{{\lambda }_1}+U_{{\lambda }_2}\right)}^2-\frac{N(t)}{{\tau }_c}$$

(14)$$\frac{d{U}_T(t)}{dt}=E_g\frac{N(t)}{{\tau }_c}-\frac{U_T(t)}{{\tau }_T}$$

(15)$$\Delta n_c(t)=1.12\times {10}^{-16}{U}_T(t)-[8.8\times {10}^{-22}N(t)+8.8\times {10}^{-18}N{(t)}^{0.8}]$$

β_TPA, h, E_g, τ_c, and τ_t are TPA coefficient [33], Planck’s constant, bandgap energy of the Si, the effective carrier lifetime, and the thermal relaxation constant, respectively [27]. We use τ_{c =} 1 ns [34] and τ_{t =} 100 ns. N(t) and U_T(t) are given in cm⁻³ and pJ/cm⁻³, respectively.

The photonic circuit is shown in Fig. 4(a). We set the τ_coup for C1, C2, C3, C4 and C5 as 15, 20, 8, 35, and 30 ps, respectively. Pulsed drive sources with amplitude of 4 mW are applied at a cycle of 5 ns with duty ratio of 60%.

Figure 12 is the calculated output when input 1 and 2 are applied as shown in the graph. Although we observe some effect of the thermo-optic effect, we obtained successful NAND gate operation, where we have low output state only when we apply both high input 1 and input 2. Because the nonlinear TPA absorption coefficient and the carrier-plasma dispersion effect are larger than the Kerr effect, we can operate the device with much lower input power. Further optimization should be possible, and we expect further reduction on the operation power.

 

Fig. 12 Input and output waveforms. The black, red, green, and blue lines are the input1, input2, input3, and output, respectively.

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