Introduction. Computational technology based on conventional binary logic faces a bottleneck in calculation speed, energy efficiency, device complexity, and the level of integration. A novel paradigm based on multi-valued logic may provide increased information density per unit device and surpass the limits of Moore’s law imposed by von Neumann architectures, since more than a double reduction of overall system complexity can be achieved just by shifting from a binary to a ternary logic concept [1]. Various multi-valued logic electronic unit devices [2] based on conventional and emerging materials [1], have been proposed. A recent study demonstrates the logic-in-memory ternary inverter based on a silicon feedback field-effect transistor, which can substantially reduce the power consumption and circuit density [3].

The so-called post-Moore technologies, residing on the shifting to the optical domain, are another proposed way to improve energy efficiency and information processing speed [4,5]. Photonic devices such as ternary content-addressable memory cells [6], injection-locked lasers based all-optical flip flops [7,8], random-access-memories [9], and majority gates [10] are proposed. Indeed, injection-locked lasers can provide optical bistability [11–13] which is recognized as a common working principle of light-based storage devices [9]. The concept of multi-valued logic shed a new light on the research of optical tristability [14–17] and three-state optical memories [18,19]. Moreover, a new approach for achieving optical bistability and tristability in plasmonic coated nanoparticles has been theoretically investigated [20].

In this Letter we theoretically predict and experimentally validate formation of double hysteresis loops, comprising of three steady states, by utilizing dual injection-locking in Fabry–Perot laser diode (FPLD). As in single injection-locked laser diodes [11,12], sweeping through the hysteresis loops is performed by variation of one of the external signal powers, while the adjustment of the second, low-power, constant injection signal controls the double hysteresis formation, which in our scheme adds a feature of reconfigurability of the optical transfer function by switching between optical bistability and tristability. In comparison with Ref. [20], apart from offering reconfigurability, the proposed approach offers low required input optical powers, with a tristability region reported between 1.03 and 1.25 mW.

Theoretical model. We consider FPLD with an active region consisting of the strain-compensated multi-quantum-well-structure, and calculate the material gain $g(n,\omega )$ with respect to the carrier concentration $n$ and photon angular frequency $\omega$, as in Ref. [11]. Dynamics of dual injection-locking is described with the multimode rate equation system extended with terms modeling the coupling of the externally injected light [16]:

On the basis of our model, applying the stationary analysis as in Refs. [11,12], we analyze the output of the slave laser under simultaneous injection in two longitudinal cavity modes, for which we take $m_1 = -6, m_2 = -2$, where minus sign stands for side-modes with lower energies (longer wavelengths) in comparison with the central mode. In both injection-locked side-modes we choose negative frequency detunings, $\Delta f_{-6} = -15\,\text {GHz}$ and $\Delta f_{-2} = -38\,\text {GHz}$, which can provide three stable slave laser steady-states [15,16], i.e., optical tristability. The ratio of the two negative frequency detunings, for the stated value of $\alpha$, which can provide tristability should be at least $2:1$. Slave laser instabilities occurring at smaller negative detunings [12] impose the limit for one of the two detunings, while the width of the intermodal space imposes the limit for the other detuning. For higher $\alpha$, the necessary ratio slightly decreases.

Our model predicts that sweeping through three stable steady-states of the slave laser can be achieved by variation of injection power corresponding to the side-mode with more negative frequency detuning (in our simulation $m_2 = -2$), but the hysteresis shape depends on the non-variational injection power value (control signal), corresponding to the side-mode with smaller detuning (in our simulation $m_1 = -6$). We show two distinctive scenarios, which we illustrate and explain by analyzing the slave laser output with respect to injection power $P_{\text {inj},-2}$ increase (from sufficiently low power outside of the locking region, to sufficiently high power in the locking region) and decrease, followed by another variation which, depending on the control signal value, outlines double optical bistability (Case 1) or tristability (Case 2). In this research, we focus on predicting the injection power levels at which hysteresis loop appear and discuss circumscriptions of commutation through loops based on the stationary analysis. Our previous works [16,21,22] presented the theoretical analysis of the switching routes in the dynamic regime and performance metrics such as switching times, and mainly considered the case of single injection locking, which can give a good estimate for this research. It is shown that slave laser switching times are estimated in the range from 200 ps up to few nanoseconds, for switching between stationary states (defined as the locking time), or in the range 30–40 ps, for considering rise/fall time of the slave laser output [16,22]. The detailed analysis of the operating speed of the presented bistable/tristable operation will be addressed in our future work.

Case 1: double optical bistability (OB). For the control signal, we apply $P_{\text {inj},-6} = -9\,\text {dBm}$ which, in combination with negative frequency detuning $\Delta f_{-6}$, in the case of single injection locking corresponds to the region in which the slave laser has two stable states [12]. By setting the slave laser into a free-running state before applying injection, we keep the side-mode $m_1 = -6$ unlocked. We further simulate a cycle of gradual increase (solid lines) and decrease (dashed lines) of the injection power in side-mode $m_2 = -2$, and depict the output optical power in both modes $m_2$ [Fig. 1(a)] and $m_1$ [Fig. 1(c)]. For small injection power ($P_{\text {inj},-2} < -1.7\,\text {dBm}$) both injection-locked modes stay unlocked (solid line ①). Side-mode $m_2$ is low in optical power, while $m_1$ has somewhat higher optical power, since it is closer to the locking regime. Further increase in injection power to $P_{\text {inj},-2} = -1.7\,\text {dBm}$ provides sufficient depletion of the carrier concentration in the active region of the laser for locking of $m_1$. This results in suppression of the optical power of all other modes, except the locked $m_1$, which becomes dominant and abruptly increases in optical power, and $m_2$, which also experiences abrupt increase in the optical power, although staying still unlocked [transition from ① to ② in Figs. 1(a) and 1(c)]. Further increase in $P_{\text {inj},-2}$ will move the locking from $m_1$ to $m_2$, making another abrupt increase of the output power in $m_2$ and abrupt decrease of the optical power in $m_1$ [transition from ① to ② at $P_{\text {inj},-2} = -0.2\,\text {dBm}$ in Figs. 1(a) and 1(c)]. Upon decreasing of $P_{\text {inj},-2}$ (dashed line from ③ to ④), the shift in locking from $m_2$ back to $m_1$ [abrupt changes from ④ to ② in Figs. 1(a) and 1(c)] takes place at lower injection power ($P_{\text {inj},-2} = -1\,\text {dBm}$) in comparison with the vice versa shift, thus forming a hysteresis loop (shaded area). Hysteresis branch ② corresponds to the state in which $m_1 = -6$ is locked, while opposite branch ④ corresponds to the state in which $m_2 = -2$ is locked. For further decrease of the injection power, $P_{\text {inj},-2} < -1\,\text {dBm}$, laser does not return to the initial state with both modes unlocked, but stays in the state of locked $m_1$ and unlocked $m_2$ (dashed line ⑤). The following injection power variation cycles will continue to outline the hysteresis (shaded area) in the counterclockwise direction, transferring the locking from $m_1$ to $m_2$, and vice versa. Access to the third possible state (initial state ①) would require lowering the injection power in the control signal ($P_{\text {inj},-6}$), thus in the presented case, depending on the slave laser prehistory, switching between three stable states may require variation of injected power in both modes, $m_1$ and $m_2$. For the purpose of greater clarity, in the supplementary material we provide hysteresis formation animation (Visualization 1 and Visualization 2) from the perspective of the $\text {d}n/\text {d}t - n$ phase space, as in Ref. [11].

Case 2: optical tristability (OT). The second case occurs for low injection power in the control signal, $P_{\text {inj},-6} = -13\,\text {dBm}$, corresponding to the region in which single injection cannot provide locking conditions, for $\Delta f_{-6}$. In this case, regardless of the slave laser prehistory, the switching between three stable steady-states would require a variation of only one input parameter, $P_{\text {inj},-2}$. As in Case 1, for low values of $P_{\text {inj},-2}$ the slave laser is in the state with both modes unlocked. Increase of $P_{\text {inj},-2}$ [solid lines with ① in Figs. 1(b) and 1(d)], as opposed to Case 1, does not lead to locking of $m_1$, but the transition occurs to the state of locked $m_2$ (solid line with ②), at $P_{\text {inj},-2} = 1.6\,\text {dBm}$. Upon decreasing of $P_{\text {inj},-2}$, unlocking of $m_2$ and simultaneous locking of $m_1$ occur at $P_{\text {inj},-2} = -0.95\,\text {dBm}$ (dashed line, transition from ③ to ④). Further decrease of $P_{\text {inj},-2}$ below $-3.5\,\text {dBm}$ returns the slave laser in the state with both modes unlocked. Alternatively, an increase of $P_{\text {inj},-2}$ leads to unlocking of $m_1$ and re-locking of $m_2$ (at $P_{\text {inj},-2} = 0.9\,\text {dBm}$, transition from ⑤ to ②) and outlines the hysteresis loop which is now completely inside the region of three steady-states (shaded area).

Experimental results and discussion. To experimentally investigate the dual hysteresis predicted by the theoretical model, a dual injection scheme shown in Fig. 2 was designed. Two fiber-coupled tunable laser sources ($\text {TL}_1$—MX10B Thorlabs, $\text {TL}_2$—81940A Keysight) operating in the C-band were used as master lasers. The injected laser power was controlled through EVOA and monitored using photodiodes ($\text {PD}_1$, $\text {PD}_2$) connected to 10% taps. The $50:50$ fiberoptic coupler was used to combine the $\text {TL}_1$ and $\text {TL}_2$ signals. The total injected power was monitored using a PWM connected to one of the output ports of the coupler, while the other port provided an input to an OC. OC enabled the injection of $\text {TL}_1$ and $\text {TL}_2$ signals into a slave FPLD ( B-FP-55-08-FA-S9-A0 Chongqing Aoptotek Technology co., Ltd) through one port, while the other port was used to monitor the FPLD output spectrum. The FPLD bias current and temperature were set to $1.43I_{\text {th}}$ and $30.2^{\circ }\text {C}$, respectively, to achieve optimal conditions for dual hysteresis formation. $\text {TL}_1$ and $\text {TL}_2$ were injected into side-modes $m_2 = -2$ and $m_1 = -6$, with frequency detunings $\Delta f_{-2} = -38 \,\text {GHz}$ and $\Delta f_{-6} = -15 \,\text {GHz}$, respectively, to satisfy the requirements of the theoretical model. The injection power $P_{\text {inj},-6}$ ($\text {TL}_2$) was kept fixed, while the power $P_{\text {inj},-2}$ ($\text {TL}_1$) was varied. The output spectrum of the slave FPLD was recorded using an OSA (AQ6370C Yokogawa), and the peak powers ($P_{\text {out},-2}$ and $P_{\text {out},-6}$) corresponding to $\text {TL}_1$ and $\text {TL}_2$ wavelengths were monitored. Figure 3 shows the dependence of these output powers on $P_{\text {inj},-2}$ for two different values of $P_{\text {inj},-6} \in \{-3.68, -4.66\}\,\text {dBm}$, corresponding to simulations in Case 1 [Figs. 3(a) and 3(c)] and Case 2 [Figs. 3(b) and 3(d)], respectively. The injection power $P_{\text {inj},-2}$ was gradually increased and decreased two times, to complete the dual hysteresis loops (repeating paths in Fig. 3 are omitted for clarity). The loop formation follows the predicted trends but the model under-performs in the precise determination of transitional optical powers. For $P_{\text {inj},-6} = -3.68\,\text {dBm}$ which is inside of the mode $m_1$ bistability region, variation of $P_{\text {inj},-2}$ does not provide a region of three stable states [Figs. 3(a) and 3(c)]. In case of lower $P_{\text {inj},-6} = -4.66\,\text {dBm}$, which is outside of the mode $m_1$ bistability region, variation of $P_{\text {inj},-2}$ provides a region of three stable states, between $0.13$ to $0.95 \,\text {dBm}$ [Fig. 3(b) and 3(d)]. During the first decrease of the $P_{\text {inj},-2}$ power in this case, it was necessary to keep its value above the $-2.39 \,\text {dBm}$ threshold to prevent the unlocking of mode $m_1$, and the return of the FPLD to the state with both side-modes unlocked. In the second decrease, $P_{\text {inj},-2}$ was pushed below this boundary, restoring the initial state.

 

Fig. 1. (a),(c) Theoretically predicted double OB and (b),(d) OT. Optical power hysteresis loops for both injection-locked modes: (a),(b) $m_2 = - 2$; (c),(d) $m_1 = -6$ subjected to the variation of $P_{\text {inj},-2}$ and constant $P_{\text {inj},-6}$. For more insight in hysteresis formation see Visualization 1 and Visualization 2 in the Supplementary material.

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Fig. 2. Experimental setup of dual injection-locking: TL, tunable laser source; EVOA, electronic variable optical attenuator; PD, photodetector; PWM, power meter; FPLD, Fabry–Perot laser diode; OC, optical circulator; OSA, optical spectrum analyzer.

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Fig. 3. Measured output optical power at the wavelengths of the injection signals in side-modes (a),(b) $m_2 = -2$ (b),(d) $m_1 = -6$ confirming (a),(c) optical bistability and (b),(d) optical tristability, with respect to $P_{\text {inj},-2}$.

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In experimentally confirmed tristability in respect to the measured output optical power at the wavelength of the injection signal in $m_2 = -2$, the three stable steady-states are the both side-modes unlocked state, resulting in the lowest output power $P_{\text {out},-2}$ [square marker in Fig. 3(b)], the side-mode $m_2$ locked state with the highest output power $P_{\text {out},-2}$ [triangle marker in Fig. 3(b)], and the side-mode $m_1$ locked state which yielded the $P_{\text {out},-2}$ power between the previous two values for the same value of $P_{\text {inj},-2}$ [circle marker in Fig. 3(b)]. Monitoring the output optical power at the wavelength of the injection signal in $m_1 = -6$ yields a higher output power ratio (up to 7 dB) and different order between the three states [Fig. 3(d)], providing design flexibility in future applications. The FPLD spectra for these three states are presented in Fig. 4 in the stated order, with markers denoting the measured optical power. The zoomed regions in Fig. 4 emphasize the FPLD free-running modes and injection signals, thus enabling the differentiation between the $m_1$ and $m_2$ unlocked/locked states. All three spectral distributions are obtained for the same injected powers $P_{\text {inj},-6} = -4.66\,\text {dBm}$ and $P_{\text {inj},-2} = 0.67\,\text {dBm}$, leading to a conclusion that the system has memory of its history which determines one out of three possible stable FPLD output states.

 

Fig. 4. Measured FPLD spectra in case of optical tristability for stable states denoted in Fig. 3 with (a) square, (b) triangle, and (c) circle markers. Markers correspond to measured output powers at the wavelengths of injection signals. Free-running cavity modes in (a) are denoted with solid arrows. Zoomed regions emphasize the separation of the free-running cavity mode and injection signal in the unlocked case.

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By drawing an analogy with previously reported binary logic applications based on dual injection locking in FPLD [7,8], optical signals $P_{\text {inj},-6}$, $P_{\text {inj},-2}$, and $P_{\text {out},-2}$ in bistability regime (Case 1) can be treated as Set, inverted Reset, and Output signals, respectively, which provides an all-optical SR latch functionality. Moreover, owing to the carefully chosen frequency detunings and control signal power, we go beyond Refs. [7,8] by addition of the second loop (transition from ① to ⑤, and vice versa) which also enables SR latch functionality, provided that the Set and Reset signals are substituted.

Conclusion. In this Letter we experimentally validate the existence of double optical bistability and optical tristability in dual injection-locked Fabry–Perot lasers. By theoretical modeling we provide guidelines for selecting the injection-locked parameters (optical power levels and frequency detunings), which can provide formation of two distinctive hysteresis loops, i.e., double optical bistability, or overlapped loops leading to optical tristability. Moreover, we show that by changing the optical power level of control signal we can dynamically reconfigure proposed device between regimes of bistability and tristability. The overlapped hysteresis loops with three distinctive output states could provide a fruitful basis for the development of a multitude of all-optical logic gates. The transition between the three states would be possible by pulsed increase and decrease in one of the injected optical signals (i.e., $P_{\text {inj},-2}$), while the other serves as an enable input ($P_{\text {inj},-6}$) ensuring the existence of two or three stable states.