All-Optical Phase Memory Circuit Based on Two Coupled Lasers and External Optical Injection

We propose a volatile static all-optical memory capable of storing phase information of a slowly-varying electric field. The scheme and its realization (a memory circuit) are based on two mutually coupled lasers subject to external optical injection. The proposed circuit has a single optical input for write and hold operations and two opposite-sign outputs for reading the memory. The proposed circuit operates with a single wavelength of light, a single direction of propagation, and without a need to switch the state of polarization. We prove mathematically that the proposed arrangement has equilibrium points that may discreetly quantify and store the phase in a bistable manner. The circuit is studied numerically for solid-state and semiconductor lasers with zero and non-zero linewidth enhancement factors, respectively. Simulations based on a rate equation system confirm the essential findings. Using typical parameters of a semiconductor laser and optimizing for a possibly wide range of operation, the write-read operations were simulated using PRBS-9 at the rate of 1 Gb/s with negligible errors. The proposed circuit will enable integrated memory implementations for future all-optical signal processing and computing systems.


I. INTRODUCTION
T HE von Neumann bottleneck is a limitation of computers that arises from the fact that the central processing unit (CPU) can perform more instructions per second than data can be transferred from memory. A mitigation to the bottleneck is to use a fast cache memory near the processing cores.
A potential next step in the increase of execution speed is to use all-optical computing principles. That is, utilizing logic gates and memory circuits that process and store light, not charged particles. Using photons instead of electrons as information carriers bears a promise of higher data transmission rates at the bus between the CPU and memory. Here, we investigate a memory circuit that stores binary information and could be accessible with low latency.
The requirements for optical transistors as fundamental optical computing elements were formulated in [14]. The same rules can be applied to more complex computing elements, such as optical logic gates and optical memory circuits. These requirements include: 1) Cascadability, i.e., compatibility of the input and output wavelengths and shapes of beam and pulse. 2) Fan-out capacity, i.e., the output of one stage should be sufficient to drive the inputs of at least two subsequent stages. 3) Logic-level restoration, i.e., cleaning up the logic signal from distortions when propagating through the system. 4) Input/output isolation, i.e., separation of the input and output signals. 5) Absence of critical biasing, i.e., tolerance to the operating parameters. 6) Logic level independence of transmission losses that may differ due to varying signal paths. Moreover, the memory bandwidth, energy efficiency, integration density, and access time are of particular interest for optical memories [15].
Analogously to the electronic memory circuits, optical bitlevel memories can be divided into two categories: dynamic and static, depending on the presence or absence of the required memory state refreshing, respectively. Various schemes have been studied for optical dynamic memory capabilities, such as optical buffers based on recirculating loops [16], bistable systems based on absorption and transparency [17], [18], and electroluminescence [19] in a rare-earth-doped medium. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ The known static memory implementations are based on optical bistability with two discrete states and have been demonstrated using the following phenomena: i) cross-phase modulation (XPM) in the two coupled SOA-MZI switches [20], [21], [22], [23], or cross-gain modulation (XGM) in the two coupled SOAs [5], [24], or XPM due to polarization-dependent gain saturation in the two coupled strained SOAs [25], all with the data encoding in the wavelength domain; ii) light recirculation in a feedback loop with an SOA-MZI [26], [27], [28]; iii) gain quenching mechanism in two [29], [30], [31] or three [32] coupled lasers in master-slave configuration, or in the cavities sharing the same SOA [33]; iv) nonlinear absorption and the resulting output power bistability in a laser diode with saturable absorber (referred to as a bistable laser diode -BLD) [34], where optical set/reset functions are realized by a combination of a BLD with nonlinear directional coupler [35] or using cross-gain saturation between two lasing modes in a multimode interference BLD [36], [37]; v) hysteresis loop in the laser output power vs. the injected power, caused by nonlinear dynamics of an injection-locked Fabry-Perot laser diode [38], or a DFB laser [39], or a system of mutually coupled DFB laser and SOA with external injection [40]; vi) lasing direction bistability in a semiconductor ring [4], [41], or microdisk [2] laser, or two coupled ring lasers [42]; vii) carrier-induced refractive index blue-shift and corresponding hysteresis response in an injectionlocked photonic crystal (PhC) laser [43], [44], [45] or PhC nanocavity [3], [46], [47], [48]; viii) polarization bistability due to cross-gain saturation in an injection-locked Vertical-Cavity Surface-Emitting Laser (VCSEL) [49], [50], [51], [52].
Still, a non-volatile optical memory has been realized by absorption modulation in phase-change materials (PCM) [53], [54]. The phase-state transitions in PCM rely on Joule heating, which limits the lower bound of the write/erase pulse duration and energy by the range of several nanoseconds and several picojoules, respectively. Consequently, current PCM-based optical memory realizations resemble more all-optical counterparts of the traditional erasable programmable read-only memory rather than random access memories.
Bistable optical memory arrangements have been demonstrated with injection-locked lasers utilizing the following physical phenomena: 1) Wavelength [43], [44], [45], [55], [56], 2) State of polarization [49], [50], [51], [52], 3) Laser output power [38], [57], and 4) Propagation direction [2], [4], [41], [42]. The first approach provided operation speeds up to 10 Gb/s [45] with a comparably small footprint and energy consumption, and the technology was claimed to be ready for high-density integration [3]. However, this approach implies that the input and output signals are at different wavelengths [43], [44], [45], or the output signal is wavelength-encoded [43], [44], [45], [55], [56]. This can be desirable for all-optical wavelength routing systems but impairs cascadability in optical computing applications -a non-negotiable requirement of complex logic circuits. The second approach involves two orthogonal states of polarization, and therefore relies on optical medium of sufficiently low birefringence to achieve a polarization bistability. To fulfill this condition, usually VCSELs are used; yet, the vertical emission may complicate large-scale planar integration on existing photonic platforms. The third approach is advantageous that it uses intensity domain signal encoding; however, it also requires the input signal (i.e., the set and reset pulses) at wavelengths different from the laser output signal. The last approach, typically implemented using micro-ring or micro-disk lasers, has demonstrated a small footprint compared to other memory circuit schemes, low switching energy (down to 2 pJ [42]), and fast switching times (down to 20 ps [42]). The different memory states, in this case, can be distinguished by power levels at the two laser outputs, which is also equivalent to the data encoding in the intensity domain. However, the last two approaches use separate set and reset input ports for the writing procedure (i.e., working as an SR-latch), thus requiring an additional logic gate stage.
We propose utilizing the phase of an optical signal for encoding information, with 'zero' and 'one' represented by different phase values in reference to a global master oscillator signal. In this case, the input (write) and output (read) signals are at the same constant wavelength, polarization, and direction of propagation. The proposed scheme can be considered a D-type latch. Phase encoding of logic states is convenient in optical computing applications due to the requirements mentioned above [14], which we will analyze in the Discussion below. Consequently, the proposed scheme opens new opportunities for complex computing and signal processing architectures, similar to conventional electronics. Moreover, phase modulation with coherent detection is known to provide additional SNR gain compared to amplitude modulation (up to 3.2 dB at 1 b/symbol spectrum efficiency and bit-error rate (BER) of 10 −9 [58]).
To create a system with optical phase bistability, we employ two coupled optical oscillators (lasers) in the optical injection locking (OIL) regime, which implement two normalization operations [59]. The dynamical behavior of the coupled laser arrays with external injection locking has been extensively modeled and analyzed in the literature (see [60] and references therein). Usually, these systems are investigated for the conditions providing stable continuous wave (CW), oscillatory and chaotic operation states. In our approach, the task is more specific. The system of two coupled OIL lasers should be stable and provide optical phase memory functionality, i.e., keep the different phase states once written. This requirement, as simulation shows, narrows down the operation regions in the system's parameter space; however, they still exist for both ideal and semiconductor lasers (i.e., with zero and non-zero linewidth enhancement factors α, respectively).
Here, we intentionally do not link our memory circuit concept and numerical modeling parameters to any specific laser type because the circuit can be implemented with various possible lasers, including integrated InGaAsP/InP edge-emitting lasers, VCSELs, nanolasers, or photonic crystal nanocavity lasers. The latter has been demonstrated as a promising solution in terms of size and energy consumption [43], [44], [45]. The study of specific layout options and analysis of footprint and power consumption are beyond the scope of this Paper.

A. Concept of a Bistable Circuit
The operation of a bistable circuit can be mathematically described with a pair of coupled equations as where z k is the output amplitude of the nonlinear element k ∈ {1, 2}, b k is the external bias, and z k , b k ∈ C. The normalization operation is denoted as z 0 ≡ z/|z|. The circuit of (1) has solutions That is, in both equilibrium states, the nonlinear elements output a constant amplitude of one, and a phase that can be one of the two possible values. The equations have no time dependence, and the state will not change if the biases remain constant. Consequently, the circuit of (1) may retain its phase state indefinitely and serve as a memory, or a latch, assuming there would be a way to flip between the states purposefully.
We recently showed in [59], [61] that a weakly injectionlocked laser can perform the required normalization operation of (1). For this Paper, we study the bistable circuit with a system of rate equations (Appendix A) and provide a rigorous mathematical analysis of the circuit (Appendix B), albeit using a simplified laser model. We show the possibility for equilibrium points where the phase will be quantified and retained indefinitely, including the pair of states (e ±iπ/3 , e ∓iπ/3 ) when b = 1. Depending on the relative strength of the external bias b, the system may have five different sets of equilibria. Some of these steady-state solutions are trivial (e.g., when the injection is strong, or nil), but we also found solutions with meaningful phase differences for two-state phase information encoding.
An idealized model of the phase memory circuit is illustrated in Fig. 1(a) that shows the mutual coupling of the two nonlinear elements {L1, L2} (rectangular symbols) and the linear operations of phase shift and addition (round symbols). In other words, each nonlinear element receives a signal that combines the bias and the opposite amplitude of the other element's output.
When the nonlinear elements of the circuit are replaced with mutually coupled lasers, the circuit can be described purely with optical components. Two possible realizations are schematically illustrated in Figs. 1(b) and (c) for the ring and linear cavity lasers, respectively.
A notable difference between the physical and idealized models is time constants, or the lack thereof. The idealized model assumes instantaneous interaction between the nonlinear elements and equally instantaneous normalization operations. (c) An optical memory based on linear cavity lasers L1,2. The bias b 1,2 is generated by a reference laser whose output is split in two. Here, the reference signal is assumed to have sufficiently high degree of coherence. The cubical beam combiners perform the required addition operations (+).
In addition to the rigorous mathematical proof, we will explore the memory circuit properties under various operating conditions while increasing the level of realism and complexity of the system. The numerical modeling will assume noninstantaneous normalization operations using rate equations (see Appendix A and [59], [61], [62]) and the existence of a delay in signal propagation (i.e., distance) between the lasers. In other words, each laser is injected with a field that is the history of the other laser output combined with external injection. First, we will model the circuit with zero linewidth enhancement factor (α = 0), i.e., a solid-state laser, such as Nd:YAG or alike. Next, we will assume a semiconductor laser with non-zero linewidth enhancement factor α ≈ 3, which is known to potentially destabilize mutually connected semiconductor lasers [60].

B. Operating Scheme
Simulations of the two-laser phase memory ran on arrangement schematically depicted in Fig. 2. Here, light from the external master laser is injected with the power coupling efficiency κ inj at a fixed linear state of polarization, aligned along a possible polarization axis of the coupled lasers (assumed now x-axis). The lasers are identically pumped with the normalized pumping rate μ. Optical injection from the external master laser may have the frequency detuning relative to the free-running frequency of the coupled slave lasers: Δν inj = ν inj − ν fr . The power coupling efficiency between the coupled lasers is κ 12 . The phase-shift φ 12 between the coupled lasers consists of a constant term φ 0 , which is set to π by default (unless mentioned otherwise), and a term proportional to the frequency detuning and the delay in signal propagation between lasers (see Appendix A): (2) The system is (bi)stable under suitable operating conditions and will ideally retain a state as long as the external injection is constant (static nature of the memory). A short phase perturbation serves as an event that erases and writes a new state into the memory (volatile nature of the memory). Theoretically, it is enough to subject only one laser to the write event. However, simulations showed that the system settles faster and more reliably if the write event is injected simultaneously on both lasers with opposing information states. For instance, assuming b = 1, if Laser 1 is subject to optical injection with the electric field phase of e +iπ/3 , then Laser 2 should be subjected to injection with the phase of e −iπ/3 . The numerical model assumes the carrier number decay rate γ of 2.0 ns −1 and the cavity electric field decay rate γ c of 500 ns −1 for both lasers.
The phase-encoded memory state is read with a standard coherent receiver arrangement [63] that consists of a beam combiner (BC) and a pair of balanced photodiodes (BPD) (see Fig. 2). The x-polarized output field E 1 of the Laser 1 is combined in the BC with the x-polarized reference signal E ref from the external master laser. The differential output provides a steady phase-to-power converted signal when the lasers are phase-locked.

A. Model Description
The differential equation system of (9) and (10) of Appendix A was integrated numerically to find stable regions in the system parameter space. For this, we used the delayed differential equation (DDE) solver in Julia [64], [65] with post-processing and analysis of the calculated output traces using MATLAB. In order to simulate possibly realistic operation conditions, we used pseudo-random binary sequence input (PRBS-9, consisting of 2 9 − 1 bits), each bit slot containing 'write' and 'hold' periods. The intensity of injection was kept constant to provide the required locking. The phase of the optical injection was set and changed in the spirit of encoding used in coherent communications. The phase was kept to "zero" if the bit value did not change relative to the previous bit. Otherwise, during the "write" part, the injected signal phase was temporarily changed to the opposing ±π/3 or ∓π/3 for the lasers 1 and 2, respectively. Here the upper and lower signs represent the input bit of "one" or "zero," respectively. In our simulations, the length of the 'write' period was 38% of the bit slot duration, and the 'hold' period (where the bit slot payload remains stored and can be read) occupied the rest 62% of the bit slot duration. We optimized this proportion for possibly large eye-opening within the 'read' period. Then, in order to smooth the input signal and make it implementable in real systems, we limited the bandwidth by a raised-cosine filter of 1/(2T w ), where T w is the duration of the 'write' pulse, and using a roll-off factor of 1.0 that provided optimal performance. Fig. 3(a) shows an example of the injected signal phase at the bit rate of 1 Gb/s.
At the system output, after the field E 1 of Laser 1 was combined with the reference signal E ref , the output from the balanced photodiode (BPD) obeyed the relation: (3) Fig. 3(b)-(e) show the output power and phase of the lasers, the corresponding voltage trace, and the eye diagram at the BPD, respectively.
To analyze the dynamic behavior and the memory function, we plot 2-D maps in the parameter spaces of 'Δν inj vs. κ inj ' and 'Δν inj vs. D 12 ', where D 12 is the inter-laser distance (i.e., the waveguide length between the two coupled slave lasers), which is proportional to the delay in signal propagation time τ 12 (here, we assume the group refractive index of the waveguide between lasers n g = 3.7, typical for InP-based integration platforms). While sweeping over the dimension of inter-laser distance, the phase shift between the lasers remains defined by (2). Theoretical BER is calculated as: where Q stands for the Q-factor: and V 1 BPD and V 0 BPD are the respective output voltage levels of 'one' and 'zero' bits. The mean values of output voltage levels and corresponding standard deviations σ 1 and σ 0 are calculated over the 'read' period for all the bits in the PRBS under consideration. Only those regions were analyzed for BER where min{V 1 BPD } − max{V 0 BPD } > 0. Non-analyzed regions are colorless on BER maps.
Coherent optical systems are susceptible to changes in internal or external variables. The same applies to the proposed memory circuit that bears sensitivity in a detuning between the injected and the free-running laser frequencies. However, the circuit has some in-built robustness in the operational bias condition. Proposition 1 (Appendix B) suggests that the relative external bias b (16) determines the system behavior. The bias depends on the pumping rate μ, the effective injected field κ inj E inj , and the inter-laser coupling efficiency κ 12 . In practice, the coupling efficiency of the external field κ inj is usually limited by the laser cavity design. Thus, should we desire to keep the bias, we can compensate for this limitation by adjusting the value of κ 12 , or vice versa.
The simplified laser model of Appendix B assumes operation in the vicinity of the threshold pump current. The simulations indicate proper operation also at the higher end of the pumping rate. As numerical results show below, the pumping range μ = 1.3-1.7 provided sufficient operational freedom. Also, from a practical point of view, such a range is reasonable for maintaining low power consumption. A higher pumping rate is possible, yet this increase does not yield substantial performance enhancement. Finally, the constant part of the optical phase-shift between lasers φ 0 was optimized to provide a possibly wide operation tolerance to the external injection parameters.

B. Numerical Modeling Results
A collection of BER maps is shown in Fig. 4. The first and second columns (sub-figures (a), (c), (e), and (b), (d), (f)) illustrate the operating regions where the power coupling efficiency κ inj and the inter-laser distance D 12 , respectively, are varied over the injection frequency detuning Δν inj . The bit rate was 1 Gb/s at all instances. Fig. 4(a), (b) show the output maps of a non-semiconductor Laser 1 (α = 0) with inter-laser coupling κ 12 of −17 dB, phaseshift φ 0 of π, and normalized pumping μ of 1.5.
The sub-figure (a) shows the case with a fixed inter-laser distance D 12 of 3.0 mm, but varied injection coupling κ inj . The operational parameter region is centered around zero frequency detuning of the external injection, manifesting a simple Arnold tongue of phase-locking. The BER map indicates the tolerance of 4 dB × 5.5 GHz to the external injection parameters κ inj × Δν inj (at the BER level of 10 −7 ). Referring to the simplified mathematical model of the memory circuit (Appendix B), the results confirm Proposition 1. As the simplified model predicts, the increase of the injection coupling efficiency κ inj results in diminished phase difference (closing eye) between the two steady states. In addition, the numerical modeling shows that the increase also provides a wider tolerance to the injection frequency detuning. The finding is further corroborated in Supplementary Figures showing the system remains stable over a wider parameter range under strong coupling, yet, without memory function (see Figs. S1, S3, S5 sub-figures c,d).
The corresponding BER map of Fig. 4(b) is calculated with a fixed injection coupling κ inj of −13.2 dB, yet, with varying inter-laser distance D 12 . The system is stable and provides memory functionality with similar behavior within a reasonably wide range of distances between the coupled lasers. The injection detuning may vary over the range of 6.1 GHz, centered around 0 GHz.
As known, a semiconductor laser with linewidth enhancement factor α > 0 shifts the stable phase-locking region to the negative detuning values and makes the injection-locked laser otherwise prone to harmonic oscillations or chaotic behavior [66]. Fig. 4(c), (d) shows the modeling results of the memory circuit with α = 3, which is typical for multiple quantum well (MQW) lasers [67]. In these simulations, we used the following parameter values: inter-laser coupling κ 12 of −19 dB, phase-shift φ 0 of 1.24π, and normalized pumping μ of 1.3. As evident, the central stable detuning value shifts to the neighborhood of −31 GHz.
The sub-figure (c) shows the case with the same fixed interlaser distance D 12 of 3.0 mm as in sub-figure (a). Now, the memory functions within a relatively narrow stripe of approximately 0.9 dB × 5 GHz (at the BER level of 10 −7 ) centered at κ inj = −10.1 dB and Δν inj = −30.3 GHz.
The corresponding varying inter-laser distance BER map (d) is calculated with the fixed injection coupling κ inj of −10.1 dB. As in the previous case, the system exhibits agnosticism toward the changing distance until the BER space tapers out for distances greater than 3 mm.
For practical purposes, we naturally aim for a semiconductor laser memory circuit with a possibly wide operations range, yet, with a minimal footprint. To this end, we sought a new operations point with decreased inter-laser distance D 12 of 0.5 mm and conducted simulations using the following parameter values: inter-laser coupling κ 12 of −19 dB, phase-shift φ 0 of 1.33π, and normalized pumping μ of 1.5.
Sub-figure (e) shows an enlarged functional BER map with substantial improvement over the sub-figure (c). The functional region is centered around κ inj = −12.4 dB and Δν inj = −20 GHz.
The accompanying BER sub-figure (f) exhibits a widened tolerance to frequency detuning (up to 7.5 GHz) when the fixed injection coupling κ inj is −12.4 dB. The inter-laser distance working region is now limited to approximately D 12 < 1.5 mm, which is more than enough for most integrated optical implementations.
As mentioned in Section III-A above, the system tolerates the pumping rate μ within a relevant range. Supplementary Fig. S7 shows the BER maps, corresponding to the previously discussed cases and shown in Fig. 4(a), (e), yet, for the pumping rate μ values of 1.3 and 1.7.
The simulations show decent robustness of the proposed circuit to the asymmetry of the coupled lasers parameters. First, we introduced up to ±20% mismatch in pumping rates μ 1 and μ 2 for the case of ideal lasers (α = 0), and up to ±7% mismatch for semiconductor lasers (α = 3). The stable memory operating region shrank as if both lasers were pumped at the lower value of the μ 1 and μ 2 rates. Only a further increase in the pumping rate mismatch degraded the results rapidly. Second, we obtained similar results for the mismatch of cavity electric field decay rates γ c,1 and γ c,2 , which incorporate the laser cavity losses and geometric properties. The circuit was resilient up to a mismatch of about ×2 for α = 0, and up to ±5% for α = 3. Until then, the memory circuit would operate adequately, shrinking the operable parameter region as if both lasers were having the same lower γ c .
We tested cascadability as follows. First, we simulated the memory circuit and stored its output reading phase states. Then we ran a new simulation where we applied the output phase states of the previous run as inputs, thus having an effective feed-forward configuration. The numerical modeling confirmed the operation. Here, the phase shift between the circuits was controlled such that the "zero" and "one" states had symmetrical opposite phase values (see Supplementary Fig. S8). In this test, the noise accumulation was neglected. Assuming the results hold also for a physical circuit, the modulation format will be preserved even through a long cascade. The result agrees with our prediction that the mutually coupled oscillator system has energy minima and will always gravitate toward these states and indicates that the circuit bears a critically important property: the restoration of logic levels.
In summary, memory operation can be achieved using two interconnected lasers and external bias. Non-surprisingly, the parameter space is more generous when α = 0, but semiconductor lasers will function when the inter-laser phase is adjusted according to (2). Importantly for practical applications, and especially for integrated optical designs, the inter-laser distance D 12 can be chosen liberally.

IV. DISCUSSION AND CONCLUSIONS
We proposed an all-optical static memory circuit based on the phase bistability of two mutually coupled lasers and external bias, whose perturbations can realize write operations. Referring to cascadability and accompanying properties of an implementable optical computing system [14], the present scheme benefits from the same wavelength, the same state of polarization, the same direction of propagation, and an unaltered information encoding format at the input and the output. Also, the simulation results suggest restoration of the logic states in a cascade.
The circuit inherently provides two outputs, which may drive two subsequent logic elements -thus fulfilling the requirement of fan-out. Also, depending on the optical system design, the laser output may counter losses of the remaining system and restore the optical power levels.
Necessary for practical designs, the circuit is tolerant to many parameters, such as inter-laser distance, given that the phase can be adjusted. Inter-laser coupling efficiency and coupling of the external bias must be accommodated within a few decibels, and frequency detuning of the external bias should usually lie within a range of a few GHz. Noteworthy, the operation parameter region will widen by decreasing the inter-laser distance, naturally improving the system integration density. Finally, one of the main advantages of the proposed scheme is its relative independence of transmission losses due to phase encoding.
The proposed memory circuit is envisioned to provide highfrequency operation in the baseband range of tens of gigahertz. In our numerical model, we used the bit rate of 1 Gb/s; however, the normalization operation at the speed of 10 Gb/s was experimentally demonstrated even with a bulk Fabry-Pérot laser [68], and with a transition to micro-and nano-scale integrated lasers will likely increase the bandwidth further. We believe the proposed circuit is suitable for applications where high throughput is crucial, such as photonic counterparts of conventional electronic cache or random-access memories.
Regardless of the chosen laser type and properties of the optical medium, practical implementation of a functioning circuit will require specialized know-how. As an example, single-mode operation of a laser is rarely a coincidence, but a result of careful design and fabrication, albeit the external optical injection is assumed to assist in our case. For instance, a ring laser-based circuit will likely require unidirectional propagation of light. To quench the undesired direction of propagation, some mitigation is likely needed. An option would be to use a broadband multimode interference reflector (MIR) or a Bragg reflector. Even more, reflections elsewhere from the optical system must be controlled and possibly minimized to ensure reliable operation of the memory.
Regarding practical implementation and projected circuit characteristics, the proposed circuit's storage density and power consumption will strongly depend on the used type of lasers and the properties of the photonic integration platform. Therefore, we must leave the quantitative assessment for further elaboration. However, considering an electronically-pumped continuous-wave room-temperature semiconductor laser with sub-wavelength dimensions [69], the circuit storage density may achieve the magnitude of one wavelength per bit. That said, the memory circuit's size will also be determined by the optical interconnects between its elements and the power dissipation constraints.
Compared to the existing optical memory schemes, the proposed circuit uses phase encoding of logic states instead of intensity encoding. Thus, the involved lasers will work with constant energy feeding. The transient intensity peaks arising when switching the logic states (see Fig. 4(b)) are a short parasitic effect. The switching energy will be defined mainly by the external phase modulation procedure. The energy consumption of the memory circuit will be determined by the pumping energy of the involved lasers (the shared external master and two coupled slave lasers). Assuming, for instance, the properties of [69], normalized pumping rate μ = 1.5, and the information rate of 25 Gb/s, the constant energy consumed by the lasers will be about 1 pJ/bit for supporting "write," "hold," and "read" operations. Yet, the overall power consumption of the circuit will also depend on the realization of interconnects between its elements.
In this Paper we have increased the level of realism step by step, starting from a simple pair of coupled equations and ending up to use delay-differential equation system with nonzero linewidth enhancement factor. A possible next step on this path would be to use stochastic differential equations to model spontaneous emission and fluctuations in the number of carriers.
The proposed circuit appears feasible for integrated implementation. The passive and active building blocks used in the proposed memory circuit are currently available in the process development kits (PDK) of commercially available InP/InGaAsP-based platforms. Novel nanophotonic integration platforms are desired to improve energy efficiency and footprint further.

APPENDIX A LASER RATE EQUATIONS
We model a laser under external optical injection with rate equations [59], [61], [62]: Here E ω fr is the complex-valued slowly varying amplitude of a linearly polarized complex electric field E, where ω fr is the angular free-run frequency of the laser, z is the spatial variable, k is the wave number, u is the external injection, and N is the normalized difference of the sums of the upper and the lower state populations. The dot-operator denotes time derivativeẋ ≡ dx dt , α is the linewidth enhancement factor, γ is the decay rate of the total carrier number, γ c is the decay rate of the electric field in the cavity, and μ is the normalized injection current (1 at the threshold and about 3 at 1 mW output of a typical vertical-cavity surface-emitting laser).
Laser output signal (7) has the power defined as: where * denotes the complex conjugate. The memory circuit consists of two mutually coupled lasers that are injected externally. Both lasers are assumed to share common values for the parameters α, γ, γ c , and μ. The coupled rate equations for the lasers 1 and 2 are with u ω fr u ω fr where E 1,inj and E 2,inj are the slowly varying amplitudes of external sources that have the frequency ω inj , and Δω inj = ω inj − ω fr is the angular frequency deviation from the external source (Δω inj = 2πΔν inj , where Δν inj is the frequency detuning). The effect of electric fields E ω fr 1 and E ω fr 2 are delayed by constant τ 12 and multiplied with complex coupling coefficients κ 12 e iφ 12 = κ 21 e iφ 21 . The injections E 1,inj (t) and E 2,inj (t) from the external sources are instantaneous and are multiplied with respective coupling coefficients.
Equations (9)-(10) can be written in the reference frame of the angular frequency ω inj of the external source, which yields equations for the input phase-modulated signals. Equations (9b) and (9d) for N 1 and N 2 do not change, but equations for the slowly varying amplitudes E and u ω inj Note that in addition to the optical phase shifts φ 12 and φ 21 in (12), there is also an effective phase shift due to the terms exp(−iτ 12 Δω inj ).

APPENDIX B MATHEMATICAL ANALYSIS OF THE STEADY STATES OF THE MEMORY CIRCUIT
Let us consider the rate equations (6) with zero detuning, i.e., Δω inj = 0, and let us denote E(t) := E ω fr (t). If the injected field u ∈ C \ {0} is independent of time and small, then the system has a steady state with electric field [59], [61] Here (z) 0 := z/|z|, z ∈ C \ {0}, is the normalization operation, θ := − arg(1 + iα), and O(u) denotes a function of u with absolute value bounded from the above with some constant multiple of |u|.
We consider a simplified model of the memory circuit described by (9) and (10) in its hold-state, i.e., when the external injections E 1 ,inj and E 2,inj are independent of time and equal. We denote E inj = E 1,inj and we assume E inj ≥ 0. We choose κ 12 = κ 21 and κ 1,inj = κ 2,inj , we denote them by κ and κ inj , respectively. We assume that κ > 0 and κ inj > 0 are small enough so that we can approximate the steady state electric field of the injection locked lasers by In other words, we drop the small O(|u|)-term in (13). Furthermore, we choose φ 12 = φ 21 = π − θ. Under the approximation (14) and with the above choices for the parameters in (9) and (10), we see that if the memory circuit has stabilized into a steady state, then the electric fields E 1 and E 2 of laser 1 and laser 2, respectively, must satisfy System (15) can be solved analytically: 2 be the set of solutions of system (15), i.e., the set of steady states of the memory circuit.
Proof: We start by reducing (15) to a simpler system. Define and given E 1 , Then (E 1 , E 2 ) solves system (15), if and only if (z 1 , z 2 ) solves Note that the five order relations between κ inj E inj and κ √ μ − 1 in the statement of the proposition correspond to b ≥ 2, Let (E 1 , E 2 ) be a solution of system (15), and let (z 1 , z 2 ) and b be defined as above. Then We will show that (E 1 , E 2 ) ∈ S. If b = 0 this is clear, so we assume b = 0. Because and b is real, we have |b − z j | Im(z k ) = − Im(z j ), j = k, (20) and therefore |b − z 1 | |b − z 2 | Im(z 1 ) Im(z 2 ) = Im(z 1 ) Im(z 2 ).
From (20) it follows in particular that Im(z 1 ) = 0 if and only if Im(z 2 ) = 0. First suppose that z 1 and z 2 are real. By (18) there are only four possible values for (z 1 , z 2 ). Considering the cases b > 1, b = 1, and 0 < b < 1 separately shows that necessarily (E 1 , E 2 ) ∈ S.
This implies that also in this case (E 1 , E 2 ) ∈ S (note that Ω = π/3 if b = 1). On the other hand, a straightforward verification shows that in each of the five cases all elements of S solve system (15). He is the Director of the Centre of Excellence on Inverse Modelling and Imaging, funded by the Academy of Finland. His research interests include inverse problems, imaging applications, machine learning and mathematical methods on electromagnetism, in particular, differential geometrical methods on these subjects. He has also studied metamaterials and invisibility cloaking for electromagnetic and quantum fields and dynamical systems related to lasers.