All-optical majority gate based on an injection-locked laser

An all-optical computer has remained an elusive concept. To construct a practical computing primitive equivalent to an electronic Boolean logic, one should utilize nonlinearity that overcomes weaknesses that plague many optical processing schemes. An advantageous nonlinearity provides a complete set of logic operations and allows cascaded operations without changes in wavelength or in signal encoding format. Here we demonstrate an all-optical majority gate based on a vertical-cavity surface-emitting laser (VCSEL). Using emulated signal coupling, the arrangement provides Bit Error Ratio (BER) of 10−6 at the rate of 1 GHz without changes in the wavelength or in the signal encoding format. Cascaded operation of the injection-locked laser majority gate is simulated on a full adder and a 3-bit ripple-carry adder circuits. Finally, utilizing the spin-flip model semiconductor laser rate equations, we prove that injection-locked lasers may perform normalization operations in the steady-state with an arbitrary linear state of polarization.

nonlinear and provides useful phenomena, such as hysteresis that can be used for switching. The changes in the mode-competition process, and thus changes of gain for a specific wavelength of light, rely on mutual variations of the injected powers. That is, the injected light, the data and the control signals alike, are amplitude-shift keyed (ASK). When the laser emission is coupled with a suitable bandpass filter, the output will manifest ASK encoded power levels, yet, often with a converted wavelength. Given proper management of the wavelengths and injected signal powers, operations can be combined and even complex functionalities can be realized.
When aiming for an integrated large scale signal processing, a particular challenge of many optical processing schemes is to cascade operations economically. A practical optical cascade requires logic level restoration, unaltered wavelength of operation, and unmodified encoding of signal 42,43 .
In this Paper we propose an all-optical computing primitive based on normalization operations of injection-locked lasers. Based on universal approximation theorem the normalization operations can be used to construct an arbitrary continuous function   → f : n m , with any given accuracy and we demonstrate a Boolean logic with a VCSEL-based all-optical majority gate at a rate of 1 GHz with BER of 10 −6 without changes in the operating wavelength or in the signal encoding format, as required for practical cascadability. Using spin-flip model laser rate equations, we show that in steady-state a weakly injection-locked laser may perform a two-dimensional complex normalization operation of linearly polarized electric fields (Supplementary S1) and we simulate an optical full adder and a 3-bit ripple-carry adder (Supplementary S2).

operating principle
An injection-locked slave laser is known to adopt and retain the phase of the master, while it will quench variations in injected signal amplitude. Qualitatively, this can be shown with a pair of rate-equations 44 where E and u η are the slowly varying complex amplitudes of the emitted and injected fields, respectively; N is the normalized total carrier density, γ c is the cavity photon decay rate, γ is the carrier recombination rate, α is the linewidth enhancement factor, and μ is the normalized pumping rate. Here we assume a linearly polarized emission out of, and the same state of polarization injection into a laser with absence of polarization anisotropies. An increase in amplitude of the emitted field E of Eq. (1) leads to a decrease in carrier density N of Eq. (2), which in turn will couple negatively back to the field E. Thus, under constant pumping above the lasing threshold (μ > 1) the system will approach steady-state ( =  E 0 and =  N 0) where the perturbations of the optical injection u η are diminished at the emitted output. A more rigorous steady-state analysis of spin-flip model rate equations are presented in Supplementary equations S1.
In general, design of sophisticated circuits requires an operating principle that can be described with a simple and succinct formulation. For example, digital electronic circuitry is regularly expressed with Boolean algebra that allows elaborate manipulations and use of ready-made libraries in design of complex circuitry. To that end we purposefully express the above described approximation of injection-locked lasers with a normalization operation as where x and y are complex amplitudes of the master and slave electric fields, respectively; and p is a dimensionless amplification factor determined by the slave. Figure 1a schematically illustrates amplitudes of a master (blue line) and a weakly-coupled slave (red line). After a perturbation, the amplitude of the slave returns to a constant while the phase remains locked with the master. The normalization operation can be further rendered into the signum function (see Fig. 1b) if the phase of the master equals to an integer of π.
Our treatment focuses on optical systems with a single linear state of polarization, yet we readily acknowledge the possibility for two orthogonal states of polarizations in which case the signals would be expressed with two-dimensional complex amplitude vectors and normalizations of the same (see Supplementary Equation S1).
Normalization of a phase encoded signal is akin to phase preserving limited amplification that has been proposed for regeneration in telecommunications systems 22 . As the name implies, the purpose of regeneration is to remove phase and amplitude noise from a channel while not changing the actual information content. In contrast, in computations the input is coming from more than a single source. Figure 1c depicts two phasor diagrams with superpositions of multiple signals and their normalizations. When the input contains an odd number of binary signals, their normalized superposition will always result in a binary output of −1 or +1. An even number of phase encoded binary inputs could potentially be problematic, because the amplitude of their superposition could be zero, which would result in a non-lock and unstable operation of the laser.
To combine signals from various sources, a possible integrated optics realization may contain a number of directional couplers that connect into a laser (see Fig. 1d). A collection of such circuit primitives can be used to construct a desired operations as discussed below. In following it is assumed that the master and the slave lasers have the same angular frequencies, the signal phases stay controlled, and the oscillators in a cascaded system have nonreciprocal flow of information, i.e., a master controls slaves but not the opposite.
From a practical viewpoint, when the principles of operation of the normalization method and the gain-modulation method are compared, we may observe differences in (i) wavelength management, (ii) control of signal phases, and (iii) signal encoding formats.
(i) Single wavelength vs. multiwavelength. Using the normalization method, the coupled inputs injected into a laser require the same wavelength, while the gain-modulated logic gate requires a multiwavelength input (except for the NOT gate). Thus, a system using the normalization method is assumed to have a common reference with sufficiently high coherence, while the gain-modulation method requires multiple light sources. (ii) Controlled phases vs. relaxed phase control. The normalization method requires mutually controlled phases of input signals, while the gain-modulation method is indifferent to phases of lightwaves. (iii) Signal encoding in full complex plane vs. positive real values, usually Boolean. The normalization operation is agnostic to signal encoding format, given that the input amplitude variation is within the dynamic range of the laser. Thus, the full complex plane can be used for the signal encoding. The gain-modulation method is indifferent to signal phases and, therefore, the signal encoding is inherently limited to positive real values, usually Boolean.

All-Optical Majority Logic
We demonstrate the normalization operation with a majority logic [45][46][47][48] using an injection-locked laser and an emulated input of three optical signals. The input signals A, B, and C are phase encoded such that the binary numbers of 1 and 0 relate to phase-shifts of 0 and π, and complex electric field amplitudes of +1 and −1, respectively. As known in the art, when one of the inputs, now the signal C, is set to 0 (hence the phase shift is π and the related electric field amplitude is −1), the majority function will produce the Boolean AND(A, B); and C = 1 will produce the OR(A, B) operation as shown in Fig. 2.  www.nature.com/scientificreports www.nature.com/scientificreports/ We experimented the majority gate using a Vertical-Cavity Surface-Emitting Laser (VCSEL), which was injected with an emulated superposition of phase encoded signals of A, B, and C; where the signals A and B were uncorrelated and randomly generated, and the signal C was switched between 1 and 0 for each successive operation. The hybrid encoded signal A + B + C had relative intensity variation of 1:9; that is, having the extinction ratio (ER) of 9.54 dB, and phases of 0 and π. The VCSEL output was connected to a Bit Error Ratio Tester (BERT) that counted the errors. For detailed description of the measurement setup, see the Methods section.
The slave VCSEL's ability to quench the amplitude modulation was measured first by gradually increasing the ER of the hybrid amplitude-phase encoded signal. The BER was measured with and without the presence of the slave VCSEL (see Fig. 3). In absence of the VCSEL the BER is >10 −3 when ER is at 6.5 dB, and when ER is above 7.5 dB, the BER saturates. As evident, an uncompensated receiver performance is severely compromised with increase of the amplitude modulation. When the slave VCSEL is present, the BER is improved by 8 decades at ER = 7.0 dB; or at fixed BER of 10 −6 the improvement in ER sustainability is 4 dB. All BER measurements were performed under constant receiving power. The BER improvement, effectively a decrease of information entropy, is a proof of a successful injection-locking of the VCSEL, both in terms of the amplitude and the phase. The BER readings are similar when the hybrid amplitude-phase encoding was changed to logic gate sequence feed of Fig. 4. As noted above, the superposition of the inputs A, B, and C had the ER of 9.54 dB, which in the current setup is translated into BER of 10 −6 .  www.nature.com/scientificreports www.nature.com/scientificreports/ To build a practical optical computing circuit, one has to ensure cascadability of the scheme 42 . We investigate cascadability of the majority gates first with a simulation of a full adder, and further on, a simulation of a 3-bit ripple-carry adder based on aforementioned full adders. The schematic integrated optical circuits, simulation models, and results are presented in Supplementary Information.

Discussion
We have proposed an all-optical computing scheme based on normalization operations of injection-locked lasers and demonstrated the scheme with the optical majority logic, where the input and output signals share the same wavelength, the same optical power, and the same signal encoding format; all prerequisites for elaborate circuit designs. Semiconductor lasers are attractive candidates for all-optical processing because they offer scalable wafer level fabrication and integration, which enables inter-and intra-chip optical communications that bypasses expensive electro-optic conversions.

Methods
We emulated the combined input of A + B + C with a tandem of an external cavity laser (ECL) with a linewidth of 150 kHz (OSICS ECL 1560/P6), an electro-absorption modulator (EAM), and a phase modulator (PM) that together produced the desired multilevel hybrid encoding (see Figs 4 and 5). Similar emulation of multiple signals is common in telecommunications, particularly in orthogonal frequency-division multiplexing (OFDM) systems.
To experiment the logic gate regime, the bit streams of A and B were each given 128 randomly generated bits (not using linear feedback shift registers, but a computer-generated sequence of Fig. 4) and the signal C was sequentially switched between 0 and 1 for each operation. The respective multilevel phase-modulated signal had relative power levels of 1 and 9, i.e., the extinction ratio (ER) of 9.54 dB, and the phase shifts of 0 and π. The EAM and PM were each controlled by separate bit pattern generators (BPG) Anritsu MP1755A and Advantest D3186, respectively, that provided independent bit patterns at a rate of 1 Gbit/s. The BPGs were synchronized in time such that the modulated optical amplitude and phase symbols coincided. The PM output was coupled to an erbium-doped fibre amplifier (EDFA), an optical tunable bandpass filter (OF), and a variable attenuator (ATT) in order to control and optimize the optical power of the seeding signal. During the preparation, the setup was experimented with shift-register-based PRBSs with dissimilar word lengths of 2 9 − 1 and 2 7 − 1 that were fed to EAM and PM, respectively.
The seeding signal was coupled into the slave laser (commercial single-mode VCSEL by VERTILAS GmbH) via an optical circulator (OC), which directed the returning signal into a second EDFA and a second OF. The phase of an optical signal cannot be measured directly from the optical power, but it has to interfere with some known reference, which in our case was a preceding symbol. Thus, we measured a differential phase of pulses in the spirit of differential phase-shift-keyed (DPSK) demodulated output using a delay-line interferometer (DLI, by Kylia) with free spectral range of 1 GHz. A single output of the DLI was measured with a p-i-n photodiode (HP 11982A) that was connected to a BERT (Advantest D3286). The expected bit sequence of the DPSK modulated output was preprogrammed into the BERT, which counted the errors.
In this setup the ratio between optical injection and optical output 21 P inj /P out ≈ 4 dB. We permitted the positive ratio, because our objective was to optimize the BER, not the power budget. In practice, BER was in minimum when the slave VCSEL was pumped only slightly above its lasing threshold that in turn enabled the maximal speed of operation. Indeed, the power budget, the speed of operation, and the achieved BER are interdependent; and furthermore, they are dependent on internal laser properties, such as the linewidth enhancement factor α, the decay rate of the cavity field, and the decay rate of the carrier number. In our experiment we optimized the BER in expense of the power. This should not be interpreted as a statement concerning the power budget of laser based normalization operations, merely just specifics of this particular experiment. By use of another objectives of the experiment, and possibly another type of a laser, one may tailor and optimize the performance. To optimize the power budget the linewidth enhancement factor α of a laser should be minimized-preferably to nil. As known in the art, the optimal regime of injection locking, and thus normalization operations, is obtained when the detuning between ECL master and VCSEL slave approaches zero. We optimized the injection locking by setting the wavelength of the ECL close to free-running wavelength of the VCSEL and then detuning was minimized by adjusting the VCSEL pump current (as the amount of pumping is known to determine the emission wavelength of a laser diode). While the semiconductor laser locking and thus the emission properties are highly dependent on the amount of detuning between the master and slave lasers, the fine-tuning of the pump power enables minimization of the BER. To avoid thermal drift of the VCSEL free-running wavelength due to environmental conditions, temperature of the VCSEL wafer was controlled with the TEC system. In free-running state the VCSEL emission has a vastly broader bandwidth than the ECL, yet, as known in the art, the slave adopts, or at least imitates, the properties of the master 21 . As a consequence, in the locked state the VCSEL emission bandwidth resembles the one of the ECL. Depending on both the pump current and the resulting detuning, an optimal optical injection power was set to minimize the BER of the received demodulated DPSK signal.
The measurement of the BER was performed by gradual degradation of the signal quality by increase of the ER of the hybrid encoded symbols. To compare the injection locked results with the non-injection locked reference, we measured the BER with and without the presence of the slave laser. In practice, the measurement setup had a 50/50 coupler whose other arm was used as the reference (dashed arrow in Fig. 5) that at times was connected to the input of the DPSK receiver (open dot in Fig. 3) instead of the optical output of the circulator.